ON-LINE CONDITION MONITORING AND FAULT DIAGNOSIS IN HYDRAULIC SYSTEM COMPONENTS USING PARAMETER ESTIMATION AND PATTERN CLASSIFICATION by M A S O U D KHOSHZABAN-ZAVAREHI B.Sc. (Mechanical Engineering) Sharif University of Technology, Tehran, Iran, 1986 M A . S c . (Mechanical Engineering) The University of British Columbia, 1993 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Mechanical Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A October 1997 ©Masoud Khoshzaban- Zavarehi, 1997 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. 1 further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my. department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my. written permission. Department of (V\ f ' C q J " f ? ^ [ ^ ^ y e p The University of British Columbia , ' Vancouver, Canada Date ^ C ^ , \ M J DE-6 (2/88) ABSTRACT Safety and functionality of a fluid power control system can considerably be increased by implementing predictive maintenance routines. Modern predictive maintenance practices are based on automatic condition monitoring and fault diagnosis of the system components. In most cases, low-quality raw sensor data are directly monitored for constraint violations or threshold crossings. Subsequent fault diagnosis is often performed by a knowledge-based expert system based on "order-of-magnitude reasoning". This means that quantitative sensor data are first transformed into more understandable "linguistic terminologies" such as "low", high", etc., and are then assessed by production rules in order to diagnose system (or component) faults. A major problem with this technique is that it is not usually feasible to directly measure the desired quantity, e.g., the flow rate inside a valve. Another problem is the association of noise and variations with directly measured signals, which might be misinterpreted as faults, especially in highly dynamic systems. In practice, failure modes often involve a change in the model structure, which may be interpreted as change(s) in one or several system parameters. The theme of this thesis is on automatic generation of fault symptoms in the form of qualitative variation of system physical parameters by on-line processing of low-quality raw sensor data. To accomplish this, a novel model-based methodology has been proposed that has integrated four levels of information processing in a structured hierarchy: 1. State/parameter estimation of the hydraulic system components using state-space models, stochastic signal processing techniques such as Kalman filtering, and raw sensor data from the hydraulic system. 2. Monitoring and change detection in the identified parameters of the system components, using statistical tests, such as sequential probability ratio test. 3. Generation of fault symptoms in the form of qualitative changes in the physical parameter values, such as "increased", "decreased", etc. 4. Fault recognition by fault symptom classification using neural network pattern classifiers, 5. Fault diagnosis maintenance aiding using knowledge-based expert systems. ii By using a second-order linear system as an example, we have shown how each element of the proposed hierarchical methodology effectively processes the lower quality data received from the previous element and provides higher quality information for the next element in the hierarchy, so that an incipient fault or an abrupt failure can be successfully detected and diagnosed. The proposed fault detection and diagnosis (FDD) technique has also been applied on a real hydraulic test rig which has been built in the Robotics and Control Laboratory, at UBC. The hydraulic test rig has a two-stage proportional directional flow control valve, which has been thoroughly modelled for simulation of faults. A step-by-step methodology has been adopted to obtain the physical valve parameters from static measurements, as well as through numerical search techniques using dynamic measurements. In order to estimate the system parameters and states in real-time, nonlinear state-space models have been developed for various hydraulic components, including the two-stage servovalve, a hydraulic cylinder, and a manipulator. Extended Kalman Filtering (EKF) is applied on the state-space models to get the parameter estimates. Only low-cost robust sensors such as pressure transducers and position sensors have been used for this purpose. More expensive or hard-to-measure states such as flow rates and orifice areas are predicted using novel state-space models. One of the major achievements of this thesis has been incorporation of a novel state-space model for a valve orifice area that allows us not only to obtain accurate estimates of the flow rate through the valve, but also to detect several incipient faults and abrupt failures in the valve and its connecting ports. The valve orifice area is considered as a nonlinear unknown function of the valve spool position. No a priori knowledge about the orifice profile or the spool deadband size is assumed. The functional relationship, along with the deadband size are automatically revealed during the on-line estimation process, while the decision as to which port is open to the flow is made internally. Experimental results were promising and showed that the identified valve orifice area is an excellent measure in quick detection and diagnosis of incipient or gradual faults, as well as and abrupt failures, in servovalves and servo-actuator systems. iii Table of Contents ABSTRACT ii LIST OF TABLES x LIST OF FIGURES xi Acknowledgment xix Dedication xx Chapter 1 INTRODUCTION 1 1.1 Preliminary Remarks 1 1.2 Motivation and General Objective 2 1.3 Maintenance Strategies 3 1.4 Predictive Maintenance Elements 4 1.5 Scope of the Present Work 6 1.6 Thesis Overview 7 1.7 Notational Convention 9 Chapter 2 FAULT DETECTION AND DIAGNOSIS: METHODS AND CRITICISMS . 10 2.1 Introduction 10 2.2 Observer-Based State Estimation Approach 12 2.3 Parameter Estimation Approach 16 2.3.1 Data Processing 17 2.3.2 Fault Detection 19 2.3.3 Fault Diagnosis 20 2.4 Expert Systems Approach 21 2.4.1 Shallow Reasoning 23 2.4.2 Deep Reasoning 23 iv 2.5 Pattern Recognition and Neural Network Approach 27 2.5.1 Statistical Pattern Recognition 28 2.5.2 Neural Network Pattern Classification 28 2.5.3 Neural Network Function Approximation 31 2.6 Concluding Remarks 34 Chapter 3 A HIERARCHICAL FDD METHODOLOGY 35 3.1 Statement of the problem 35 3.1.1 FDD Related Characteristics 35 3.1.2 Criteria for an Appropriate FDD Scheme 37 3.2 An Integrated Hierarchical Solution 38 3.2.1 The Objective 39 3.2.2 The Algorithm 40 3.3 Condition Monitoring by State/Parameter Estimation 42 3.4 Fault Detection Using Statistical Decision Making 47 3.4.1 Residual Generation 48 3.4.2 Fault Detection 51 3.4.3 Fault Diagnosis Initialization Process 59 3.5 Fault Symptom Generation 60 3.5.1 Parameter Jump Level Classification 61 3.5.2 Residual Generation 62 3.5.3 Parameter Jump Level Assignment 62 3.6 Fault Recognition Using Neural Network Pattern Classification 73 3.7 Fault Diagnosis Using a Rule-Based Expert System 77 3.8 Concluding Remarks 80 v Chapter 4 AN EXPERIMENTAL HYDRAULIC TEST RIG 82 4.1 Introduction 82 4.2 System Description 84 4.2.1 Mechanical and Hydraulic Units 84 4.2.2 Analog and Digital Electronic Units 86 4.2.3 Operation 88 4.3 Hydraulic Circuit Components 89 4.3.1 Hydraulic Oil 89 4.3.2 Hydraulic Hoses 92 4.3.3 Filters 94 4.3.4 Gear Pump 95 4.3.5 Pressure Relief Valve 98 4.3.6 Check Valve 100 4.3.7 Line Relief/Anticavitation Valves 102 4.3.8 Electro-Hydraulic Servovalve 102 4.3.9 Cylinder Actuator 106 4.3.10 Manipulator I l l 4.4 Measuring Devices 115 Chapter 5 NONLINEAR MODELING OF A SERVOVALVE 116 5.1 Introduction 116 5.2 Valve Description 119 5.3 The Servovalve Model 120 5.3.1 Solenoid Model 121 5.3.2 Pilot Valve Model 125 5.3.3 Main Valve Model 130 vi 5.4 Model Validation and Performance Analysis 130 5.4.1 Determination of Model Parameters 130 5.4.2 Steady State Behavior 138 5.4.3 Frequency Response Behavior 139 5.4.4 Time Response Behavior 139 5.5 Sensitivity Analysis 142 5.6 Summary and Concluding Remarks 147 Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 149 6.1 Introduction 149 6.1.1 Problem Statement 149 6.1.2 Scope of the Present Chapter 151 6.2 Modeling 152 6.2.1 Solenoid Model 152 6.2.2 Pilot valve Model 154 6.2.3 Main Valve Model 156 6.3 State-Space Representations 159 6.3.1 The Solenoid Model 160 6.3.2 The Pilot Valve Model 161 6.3.3 The Main Valve Model 162 6.4 State and Parameter Estimation 163 6.5 Simulation Results and Discussion 167 6.5.1 The Data 167 6.5.2 Starting Values for the Kalman Filter 167 6.5.3 Normal System States and Parameters 169 vii 6.5.4 Condition Monitoring and Fault Detection Results 173 6.6 Experimental Results and Discussion 183 6.6.1 Experimental Conditions 183 6.6.2 Estimation of the New Valve Parameters and States 184 6.6.3 Detection and Diagnosis of Changes in the Servovalve Characteristics 188 6.6.4 Detection of a Sensor Fault followed by an Actuator Fault 199 6.7 Summary and Concluding Remarks 207 Chapter 7 CONDITION MONITORING IN A SERVO-ACTUATOR SYSTEM . . . . 209 7.1 Introduction 209 7.2 State-Space Representations • 211 7.2.1 The Manipulator 211 7.2.2 The Hydraulic Actuator 213 7.2.3 The Main Valve Connecting Ports 214 7.2.4 The High Pressure Filter 219 7.3 Model Partitioning and Estimation 220 7.3.1 Model Partitioning 220 7.3.2 Estimation Method 223 7.4 Experimental Results and Discussion 223 7.4.1 The Data 224 7.4.2 Starting Values for the Kalman Filters 224 7.4.3 Experiment 1: Estimation of Normal System Parameters 225 7.4.4 Experiment 2: Condition Monitoring and Fault Detection 237 7.5 Summary and Concluding Remarks 244 viii Chapter 8 CONCLUSION 245 8.1 Summary 245 8.2 Contributions 245 8.3 Practical Issues 248 8.4 Suggestions for Future Work 249 NOMENCLATURE 250 BIBLIOGRAPHY 253 Appendix A Extended Kalman Filtering Algorithm 259 Appendix B The Pocket Algorithm for Training Linear Machines 261 Appendix C Specifications for the Hydraulic Test Rig Components 263 Appendix D Specifications for the Measuring Devices 266 ix LIST OF TABLES Table 2.1 Comparison of major fault detection and diagnosis methods 34 Table 3.1 Numerical values for parameter jump levels and their confidence intervals . 72 Table 3.2 Fault symptom codes generated for five different faults in a second-order system 72 Table 5.1 Numeric values for parameters from the manufacturer or direct measurement 131 Table 5.2 Identified values for solenoid parameters 134 Table 5.3 Identified values for the pilot and main valve parameters 136 Table 5.4 Sensitivity of the valve outputs to changes in various variables 144 Table 6.1 Estimation values for the "normal" parameters of the "simulated" servovalve 175 Table 6.2 Estimated values for the parameters of the "new" servovalve from Experiment 1 188 Table 6.3 Estimated values for the parameters of the "used" servovalve from Experiment 2 193 Table 7.1 Component faults and affected component filters 222 Table 7.2 Sensor faults and affected component filters 222 Table 7.3 Some of the system constant parameters used in the experiments 224 Table 7.4 Mean values and standard deviations for the identified parameters of the manipulator 226 Table 7.5 Coefficients of the fitted polynomials to the main valve orifice area estimates 235 x LIST OF FIGURES Figure 1.1 Typical Elements of a predictive maintenance strategy 6 Figure 2.1 Major fault detection and diagnosis (FDD) schemes 13 Figure 2.2 Fault detection and diagnosis using a bank of filters or observers 15 Figure 2.3 Hybrid fault detection and diagnosis based on parameter estimation and expert systems 18 Figure 2.4 Input-output measurements from a hydraulically actuated arm while a failure occurs at / = 1 sec 27 Figure 2.5 A general artificial neural network topology 29 Figure 2.6 Neural network fault (.) and no-fault (+) classification when a chemical process is in steady state (left), and in transient dynamic state (right) . . . . 32 Figure 2.7 Using a bank of neural network function approximators for fault diagnosis in dynamic systems 33 Figure 3.1 Typical changes in system physical coefficients due to a failure 40 Figure 3.2 The proposed integrated hierarchic algorithm for on-line condition monitoring and fault diagnosis 42 Figure 3.3 Estimation results for a second-order system in normal operation 47 Figure 3.4 Correlation tests for E K F innovations. Horizontal dot-dash lines are 95% confidence limits 49 Figure 3.5 Autocorrelation functions to test viscous damping and spring constant residuals. Horizontal dot-dash lines are 95% confidence limits 52 Figure 3.6 Estimation error auto-covariances of the states and parameters 53 Figure 3.7 Typical behavior of the log-likelihood ratio corresponding to a change in the mean of a Gaussian sequence with constant variance. Vertical solid bar is the SPRT estimated fault time 57 xi Figure 3.8 Estimation results (—) and true values (- -) for the state vector due to "Fault 1" 65 Figure 3.9 Estimation results (—) and true values (- -) for the state vector xj due to "Fault 2" 65 Figure 3.10 Estimation results (—) and true values (- -) for the state vector xj due to "Fault 3" 66 Figure 3.11 Estimation results (—) and true values (- -) for the state vector x^ due to "Fault 4" 66 Figure 3.12 Fault mode 1. (a): SPRT fault detection, (b): parameter classification 68 Figure 3.13 Fault mode 2. (a): SPRT fault detection, (b): parameter classification 68 Figure 3.14 Fault mode 3. (a): SPRT fault detection, (b): parameter classification 69 Figure 3.15 Fault mode 4. (a): SPRT fault detection, (b): parameter classification 69 Figure 3.16 Detection of fault mode 1 by monitoring "long-term" drift in (a): viscous damping coefficient, and (b): spring constant, (c): Fault classification using "long-term" parameter drift threshold checking 70 Figure 3.17 Detection of fault mode 2 by monitoring "long-term" drift in (a): viscous damping coefficient, and (b): spring constant, (c): Fault classification using "long-term" parameter drift threshold checking 70 Figure 3.18 Detection of fault mode 3 by monitoring "long-term" drift in (a): viscous damping coefficient, and (b): spring constant, (c): Fault classification using "long-term" parameter drift threshold checking 71 Figure 3.19 Detection of fault mode 4 by monitoring "long-term" drift in (a): viscous damping coefficient, and (b): spring constant, (c): Fault classification using "long-term" parameter drift threshold checking 71 Figure 3.20 A winner-take-all group of cells for solving the "XOR" problem 75 Figure 4.1 Pictures of the experimental test rig and its two-stage proportional directional servovalve built in the Robotics and Control Laboratory, U B C 83 xii Figure 4.2 Schematic diagram of the experimental test rig 85 Figure 4.3 Schematic diagram of the hydraulic test rig's controlling units 87 Figure 4.4 Variation of the Imperial Oil's NUTO H32 hydraulic oil density with temperature at atmospheric pressure 90 Figure 4.5 Variation of the Imperial Oil's NUTO H32 hydraulic oil kinematic viscosity with temperature at atmospheric pressure 91 Figure 4.6 Variation of bulk modulus w.r.t. oil pressure and entrained air 93 Figure 4.7 Schematic diagram of a typical spur gear pump 96 Figure 4.8 Flow/pressure characteristics of the gear pump 96 Figure 4.9 Time history and power spectrum of a gear pump pressure ripples 97 Figure 4.10 Schematic diagram of a pilot operated relief valve (Courtesy of Parker Hannifin Corp., Cleaveland, Ohio) 99 Figure 4.11 Flow rate-pressure characteristic of the RA101 pressure relief valve at cracking pressure of Pset = 7.6 MPa 100 Figure 4.12 Flow rate-pressure drop relationship in the check valve 101 Figure 4.13 A proportional directional two-stage servovalve, Model 4 WRZ (Courtesy of Rexroth) 103 Figure 4.14 Orifice effective area vs. spool displacement for a wedge-shaped-porting valve 105 Figure 4.15 Schematic diagram of a single-rod double-acting hydraulic cylinder with acting forces 107 Figure 4.16 Measured friction force of the hydraulic cylinder in retraction and extension 110 Figure 4.17 The manipulator-like link attached to the cylinder rod of the hydraulic test rig I l l Figure 5.1 A two-stage electrohydraulic servovalve 119 xiii Figure 5.2 Electro-mechanical energy flow and conversion between various valve components 121 Figure 5.3 A simplified solenoid model 122 Figure 5.4 An improved solenoid model 122 Figure 5.5 Coil voltage, restoring, dissipation, and total coil currents for the test rig's servovalve solenoids 123 Figure 5.6 Pilot valve orifice area vs. pilot spool stroke 127 Figure 5.7 Solenoid current-flux (above) and force-flux (below) relationships 133 Figure 5.8 Schematic diagram of the solenoid identification algorithm 135 Figure 5.9 Iterated dissipation currents from initial condition to final optimal value, along with the current obtained from the measurement 135 Figure 5.10 Steady-state pilot pressure and main spool stroke as functions of input voltage 138 Figure 5.11 Frequency response of the pilot pressure differential due to a chirp input.. 140 Figure 5.12 Time response of the solenoid current, pilot pressure, and main spool displacement for four step input voltage levels 141 Figure 5.13 Time response of the solenoid current, pilot pressure, and main spool displacement due to arbitrary input voltage 143 - 0 Figure 5.14 Step response of the valve when: 1) / i = 0.027 N.s.m"z at 40 C, 2) fi = 0.350 N.s.m"2 at 0 °C 145 Figure 5.15 Time response of the pilot spool displacement and the pilot pressure for different supply pressure levels 146 Figure 5.16 Step response of the pilot valve pressure differential with and without the deadband 146 Figure 6.1 Comparison of typical forces acting upon the main valve spool 157 Figure 6.2 Input-output relationships for the pilot valve and the main valve in a dynamic condition 158 xiv Figure 6.3 Input-output signals obtained for the simulated valve (see Fig. 5.13). . . . 168 Figure 6.4 States and parameter estimates for the "simulated" solenoid in "normal" condition 170 Figure 6.5 State estimates for the "simulated" pilot valve in "normal" condition, (a): Spool position time history, (b): Flow rate time history, (c): Orifice effective area estimate vs. true spool position, (d): Orifice effective area estimate vs. spool position estimate 170 Figure 6.6 Parameter estimates for the "simulated" pilot valve in "normal" condition. 171 Figure 6.7 Parameter estimates for the "simulated" main valve in "normal" condition. 174 Figure 6.8 A two-layer feedforward neural network for function approximation. . . . 176 Figure 6.9 Neural network function approximation of the pilot orifice area (training set) 177 Figure 6.10 Neural network function approximation of the pilot orifice area (test set). . 177 Figure 6.11 Estimates of the pilot stage parameters and traceable state during an abrupt change in the viscous damping coefficient (based on simulation results). . 179 Figure 6.12 Fault detection and symptom generation during an abrupt change in the viscous damping coefficient (based on simulation results) 179 Figure 6.13 Estimates of the pilot stage parameters during a gradual increase in the pilot orifice deadband length (based on simulation results) 181 Figure 6.14 Pilot orifice area monitoring and fault detection during a gradual increase in its deadband length (based on simulation results) 182 Figure 6.15 Comparison of the pilot spool position and orifice area between a faulty and a normal valve (based on simulation results) 182 Figure 6.16 Inputs and outputs measured from the servo valve in Experiment 1 185 Figure 6.17 State and parameter estimates for the solenoid from Experiment 1 185 Figure 6.18 Parameter estimates of the pilot valve from Experiment 1 187 Figure 6.19 State estimates of the pilot valve from Experiment 1 187 xv Figure 6.20 Parameter estimates of the main valve from Experiment 1 189 Figure 6.21 Comparison between the input-output signals measured from the "new valve" and the "used valve" 191 Figure 6.22 Comparison between the traceable states of the "new valve" and the "used valve" 191 Figure 6.23 Comparison between the pilot stage parameters of the "new valve" and the "used valve" 192 Figure 6.24 Comparison between the total friction force in the pilot stage of the "new valve" and the "used valve" 192 Figure 6.25 Results of estimation, fault detection, and jump level classification for the spring constant of the main valve 195 Figure 6.26 Results of estimation, fault detection, and jump level classification for the spring preload of the main valve 196 Figure 6.27 Results of estimation, fault detection, and jump level classification for the Coulomb friction force of the main valve 197 Figure 6.28 Results of estimation, fault detection, and jump level classification for the viscous friction coefficient of the main valve 198 Figure 6.29 Comparison between the spool position-dynamic force relationships in the main stage of the "new valve" and the "used valve" 199 Figure 6.30 Schematic diagram of the setup for reduction of the solenoid circuit resistance 200 Figure 6.31 Input-output measurements from the servovalve in Experiment 3 201 Figure 6.32 Estimation results for the parameters of the pilot stage in Experiment 3. . 202 Figure 6.33 Comparison between the total friction force in a faulty valve and a normal valve in Experiment 3 204 Figure 6.34 Comparison between the traceable state estimates in a faulty valve and a normal valve in Experiment 3 204 xvi Figure 6.35 Results of estimation, fault detection, and jump level classification for the coil resistance in Experiment 3 205 Figure 6.36 Results of estimation, fault detection, and jump level classification for the pilot stage viscous damping coefficient in Experiment 3 205 Figure 6.37 Results of estimation, fault detection, and jump level classification for the pilot stage Coulomb friction force in Experiment 3 206 Figure 7.1 Schematic diagram of a servo-actuator system and its components 210 Figure 7.2 The data flow chart relates various component filters via measurement inputs and filter outputs 221 Figure 7.3 Measured signals used for estimation of the manipulator parameters. . . . 227 Figure 7.4 Estimation results for the state and parameters of the manipulator 228 Figure 7.5 Measured and estimated signals used for estimation of the actuator parameters 229 Figure 7.6 Estimation results for the parameters of the actuator 230 Figure 7.7 Actuator's total friction force estimate vs. the piston rod velocity 230 Figure 7.8 Measured and estimated signals used for prediction of the main valve orifice parameters 232 Figure 7.9 Estimation results for the main valve orifice profiles and profile slopes. . . 234 Figure 7.10 Comparisons between the main valve orifice area measurements, estimation, and function approximation 234 Figure 7.11 Measured and estimated signals used for identification of the high pressure filter resistance factor 236 Figure 7.12 Estimation results for the high pressure filter resistance coefficient 236 Figure 7.13 Measured and estimated signals used for tracking of the main valve orifice parameters during a faulty operation 240 Figure 7.14 Condition monitoring and fault detection results using the rod-side orifice area estimate residuals 241 xvii Figure 7.15 Condition monitoring and fault detection results using the head-side orifice area estimate residuals 242 Figure 7.16 Measured and estimated signals used to monitor the high pressure filter resistance factor during a faulty operation 243 Figure 7.17 SPRT and parameter jump level classification results using the high pressure filter resistance factor estimates 243 xviii Acknowledgment I would like to express my first and foremost gratitude to my supervisors, Professor F. Sassani and Professor P. D. Lawrence, for their guidance and encouragement in the course of this research. Their patience and support are sincerely appreciated. I express my special gratitude to my external examiner, Professor K. A. Edge, from the University of Bath, England, for his careful reading of my thesis and his very useful comments. My gratitude extends to the technical staff of the Robotics Group, in particular, Dan Chan, Simon Bachmann, and Icarus Chau. I also appreciate the assistance of our kind and hard-working secretary, Ms. Leslie Nichols. My deepest gratitude goes to my family members, especially my wife, Manzar, for her love, encouragement, and unconditional support during the whole course of my Ph.D. work at UBC. Finally, I wish to thank my friends, Dr. Shahram Tafazoli and Dr. Rudy seethauler, for their generous help of various kinds in the course of my Ph.D. research. xix Dedication To my beloved Manzar and Mahon xx Chapter 1: INTRODUCTION 1.1 Preliminary Remarks Industrial hydraulics is a relatively young field of energy transmission and control. The manufacturing of hydraulic components for various industrial applications has expanded very rapidly during the past five decades. Modern hydraulic equipment can economically convert mechanical energy into fluid energy, and with simple components this energy may be regulated to provide direction, speed, and force control. No other type of power transmission provides the range of control of force, speed, and direction that is possible with fluid power transmission. The ever-increasing use of industrial robots and automated machinery in plants is made possible in part by continuing development in the field of fluid power systems and controls. The objective of the work presented in this thesis is to develop an efficient and practical monitoring technique that will assist in the automated condition monitoring and diagnosis of faults in fluid power system components. System monitoring and diagnosis is a support technique that increases safety and extends the functionality of a system. In many applications, increased requirements on productivity and performance lead to plants operating near their design limits for much of the time. This may often result in system faults or failures, which are typically characterized by gradual or critical changes in the system parameters, or even by changes in the inherent dynamics of the system. Faults cause gradual degradation of the plant performance, and if not fixed in time, lead to failures that can not only result in loss of productivity and expensive equipment, but also seriously endanger human safety. In response to these concerns, tighter safety and reliability specifications have been imposed, which has resulted in increased activity on research dealing with maintenance procedures. The amount of work done or published in this area is overwhelming, and includes work on hydraulic systems. 1 Chapter 1 INTRODUCTION 2 1.2 Motivation and General Objective Today, fluid power equipment is used in nearly every branch of industry, which may be categorized into two major distinct areas. One area involves high risk, high capital cost applications such as nuclear power, aerospace, and marine. Although the number of catastrophic accidents are low when compared with the activity, the resulting loss of life is, however, unacceptable. In these situations, failure at first stage is not allowed to occur by tolerating the additional costs of premature replacement. Rare but catastrophic failures are accommodated only after they occur, mostly by relying on hardware redundancy through a fail/safe operation (which increases initial capital investment). No major effort has been made so far to accommodate incipient faults in order to avoid catastrophic failures through a comprehensive condition monitoring technology. One of the objectives of this work is to implement such a monitoring scheme to prevent further progression of faults in sensitive hydraulic equipment. The other area in which fluid power systems are often employed involves industrial applications including transportation, mining, construction, forestry, machine tools, and material handling. This is perhaps the area which is currently receiving the least attention from a monitoring point of view. Fluid losses alone due to unnecessary part replacement, leakage in damaged pipes and hydraulic components, etc., in such an area as mining result in hundreds of thousands of dollars per annum in replacement costs, or costs due to resulting failures or inefficiency of operation. Also, it has been estimated that when a hydraulic system fails, approximately 60% of the downtime is used to diagnose the problem while only 40% is used to repair or correct the problem [16 Fitch, 1986]. To reduce the diagnosis time and effort, a number of automated tools have been designed and implemented which mostly use knowledge-based expert systems based on qualitative models of system components and sensor data [24 Hogan et al., 1990][18 Freyermuth, 1991][75 Watton, 1993]. Considerable reduction in fault detection and diagnosis time and improvement in system performance have already been reported for a number of applications. One of the major drawbacks in automated fault diagnosis using expert systems is the difficulties in Section 1. 3 Maintenance Strategies 3 properly incorporating on-line sensor measurements into the rule-based system. The sensor data which carry valuable information about the system health condition are often time-varying, associated with noise, nonrobust (e.g., flow rates), and non-descriptive. Turning these poor quality numeric information into qualitative measures and interpreting them into understandable linguistic terms, though highly desired, has been a challenging task [4 Bull et al, 1996]. Another major objective of this research is to devise means to facilitate this data/information conversion process so that higher quality, more informative data can be generated and offered to the existing fault diagnosis expert systems. Of course, in the aforementioned two areas of hydraulics applications, by adopting an advanced maintenance strategy that monitors the health condition of a hydraulic machine (preferably on-line), as we have pursued in this research, the normal operating life can be extended and many of the following benefits may easily be achieved: • reduced downtime and unscheduled maintenance, • avoidance of redundant and premature part replacement, • continuous and comprehensive analysis of complex equipment, • residual machine and component life estimation, • extended machine life, • increased safety and reliability, • increased energy efficiencies, and • identification of improper operating conditions and part assembly. The degree of achievement, however, depends on the type of maintenance procedure adopted and the level of its sophistication. 1.3 Maintenance Strategies Currently, three types of system maintenance strategies are being employed: Chapter I INTRODUCTION 4 • Breakdown Maintenance, which is the simplest of strategies since the machinery is allowed to operate without any significant monitoring effort until it wears out or breaks down. The overall maintenance cost is ultimately the highest amongst all other maintenance practices. • Time-Based or Preventive Maintenance, which consists of predetermined "time-scheduled" tasks periodically performed on equipment to minimize the likelihood of on-line failure. Although it has a definite advantage over the "breakdown maintenance", a preventive maintenance program without knowledge of actual machine conditions can result in unnecessary and premature maintenance activities that are usually costly. • Condition-Based or Predictive Maintenance, a relatively new concept in the field of machinery maintenance, is a systematic program of regularly monitoring machine components to determine their health condition while the machine is operating. It incorporates condition monitoring with fault diagnosis in many cases. Condition monitoring involves the acquisition, processing, and analysis of sensor data related to machine parameters, such as vibration, electrical current, temperature, pressure, or other plant variables. By continuously comparing the results to previous or normal operation readings, developing problems can be detected and identified at early stages and appropriate decisions can be made accordingly to fix the problems far before a catastrophic failure causes a costly damage. Dramatic improvements in operation costs and safety has more than ever made the "predictive" condition-based maintenance strategy, among others, a viable and cost-effective choice for the optimum operation of modern plants. Besides, the advent of new sensor and computer technology has provided new opportunities for engineers and scientists to tackle more challenging real-time problems in the area of condition monitoring and fault diagnosis. This is the focus in this research. f 1.4 Predictive Maintenance Elements Condition-based (or predictive) maintenance technology is fast becoming an issue of primary significance Section 1. 4 Predictive Maintenance Elements 5 in the design of "intelligent" and "autonomous" control systems for a wide range of system engineering applications, including aircraft and propulsion systems, power plants, chemical processes, industrial machinery, and on- and off-highway vehicles. Consider the technical plant in Fig. 1.1, which is under control. The "measurable" input and output vectors of the plant are u(t) and y(f), respectively. It is assumed that faults affect the technical process and its control. Generally, a fault is to be understood as a non-permitted "deviation" of a characteristic property of the process itself, the actuators, the sensors, and the controllers [28 Isermann, 1984]. According to Fig. 1.1, the process of system failure characterization through a predictive maintenance strategy can be broken up into four steps: i. Condition monitoring consists of constantly processing measurable data or input-output signals from the plant until useful quantities that best describe the current health condition of the system are extracted. The processed information is then compared against some known or predetermined normal (or nominal) quantities, and finally, fault or failure indicating signals are generated. ii. Fault/failure detection deals with the fault/failure indicating signals, usually through calculation of some decision statistics, and determines if a fault in the form of some slight abnormal changes is pending, or if a major malfunctioning in the form of some abrupt changes has occurred in the system. iii. Fault diagnosis considers, through a decision making mechanism, the problem of isolating and/or identifying a fault/failure after it has been detected. iv. Fault/failure accommodation attempts to self-correct a particular failure through re-configuration of the control system, if possible, or to give proper alarms to the operators), or to shut down the whole process if a major failure is pending or has already occurred. In more advanced systems, automated "maintenance aiding", usually in the form of an expert system, follows the fault accommodation to help the plant technicians fix or replace the faulty component at an appropriate time. Depending on the application, a diagnostic system may include some or all of the above tasks. Chapter I INTRODUCTION 6 Fault/Failure Noise Noise Condition Monitoring Change indicating signals 1 (e.g., estimated states and parameters) Fault/Failure Detection Fault indicating signals (fault time and symptoms) Fault/Failure Diagnosis Identified fault 1 (type, location, causes, effects) Fault/Failure Accommodation Maintenance aiding, alarms, i control reconfiguration, etc. Fig. 1.1: Typical Elements of a predictive maintenance strategy. 1.5 Scope of the Present Work Pursuing all the elements of the predictive maintenance strategy shown in Fig. 1.1 requires a considerable amount of research time and effort in such broad areas as mathematics, statistics, probability theory, mechanical engineering, electrical engineering, computer science, and artificial intelligence. As a result, to keep the volume of the research manageable and also appropriate for a Ph.D. thesis, we have pursued only the first two tasks in details, i.e., applying condition monitoring and fault detection techniques to a class of dynamic plants, namely "hydraulic" systems, for the purpose of generating qualitative fault symptoms from raw sensor data. New innovative methods that use the results of the present work for on-line "fault diagnosis" and "failure accommodation" in fluid power systems are left for future investigations. Section 1. 6 Thesis Overview 1 This research is focused on a real-world application. The strong emphasis in this work has been on finding new applications for the existing theoretical results in such diverse areas as stochastic signal processing, estimation theory, nonlinear modeling and simulation, hydraulics, artificial intelligence, and decision theory. We believe that some of the existing theoretical tools for fault detection in dynamic systems should be applicable to hydraulic components, which are generally considered as nonlinear time-varying systems. We have mainly tailored appropriate theoretical methods from these fields to a particular real-work application, i.e., a fluid power system. More importantly, we have shown that theoretical methods work, with some modifications, according to our specifications and requirements. 1.6 Thesis Overview This thesis is organized as follows: Chapter 2. Condition Monitoring and Fault Diagnosis Methods and Criticisms: This chapter explains and compares different monitoring and fault diagnosis techniques that exist in the literature. These techniques are mostly based on quantitative models (e.g., observer-based, parameter-based), qualitative models (e.g., expert systems), or heuristic models (e.g., neural networks) of the monitored system. Their advantages and disadvantages when applied to fluid power systems are also discussed in detail. Chapter 3. A New Methodology for Condition Monitoring and Fault Diagnosis: In this chapter, first, we consider the unique characteristics of hydraulic systems relevant to the on-line monitoring and fault diagnosis problem. We also discuss the specific requirements that a comprehensive fault detection and diagnosis strategy should possess. Then, we introduce an integrated methodology which is a hierarchical combination of a number of techniques explained in Chapter 2. Here, the goal is to process low-quality raw sensor signals in order to generate fault symptoms in the form of qualitative changes in system physical parameters. A "model-based" technique that uses state-space representations for various hydraulic components and is based on estimation and tracking of system physical parameters is the key element in the proposed methodology. This is the heart of our research thesis which has been elaborated Chapter 1 INTRODUCTION 8 in further chapters. Statistical methods for robust detection of faults, and generation and classification of qualitative fault symptoms are also introduced in this chapter. Chapter 4. A Hydraulic Test Rig: Development and validation of any on-line condition monitoring strategy should be performed on a specific working system, both for simulation and experimental evaluation. In this chapter a computer controlled experimental hydraulic test rig is introduced, which has been setup in the UBC's Robotics and Control Laboratory. The setup has many major components that are found in a typical servo-actuator hydraulic system, including a motor-driven gear pump, filters, valves, hoses, a linear actuator, and a single degree of freedom robotic manipulator acting as a variable load. Mathematical models and specifications for each hydraulic component, along with some typical failure modes and causes of the failures, are explained in this chapter. Chapter 5. Nonlinear Modeling of a Two-Stage Servovalve: A novel simulation model for a typical two-stage proportional directional servovalve, which is one of the key components in our hydraulic test rig, is developed and validated. The model is required for investigation of the valve's behavior under various faulty situations. A step-by-step methodology to obtain the valve's physical parameters from direct and indirect measurements is introduced in order to validate the valve's simulation model. The results of the valve simulations, along with some experimental results, are used in further chapters to detect faults and evaluate the proposed condition monitoring strategy. Chapter 6. Condition Monitoring and Fault Detection in the Servovalve: Condition monitoring of the hydraulic test rig begins in Chapter 6 with physical parameter identification and state reconstruction of the two-state servovalve at normal (nominal) operating conditions. Nonlinear state-space representations for the valve's solenoids, pilot stage, and the main stage are derived in a form appropriate for real-time state/parameter estimation. Extended Kalman Filtering is used for this purpose, and the results are compared to the actual values, both from simulations and the experiments. Once the normal-operation parameters are obtained, we then look for changes in those parameters due to incipient faults or sudden failures. Statistical techniques based on Sequential Probability Ratio Test (SPRT) are used to detect Section 1. 7 Notational Convention 9 gradual and abrupt changes in component model parameters and valve orifice area signatures. Any fault or failure has its own unique signature, or symptom. A fault/failure pattern, based on changes in system parameters, is formed accordingly that should lead us to the true faulty situation. Chapter 7. Condition Monitoring and Fault Detection in a Servo-Actuator System: The same procedure as in Chapter 6 has been adopted to identify the normal parameters of some typical components in the hydraulic test rig, and to detect a purposely induced fault in the system. Once again, orifice area estimates are shown to be excellent measures to detect a fault in the form of including leakage and flow loss in the servo-actuator circuit. Chapter 8. Conclusion: A summary of the work is given first followed by a list of contributions of this thesis. Then we bring in some discussions about the proposed methodology and concludes the thesis with some recommendations for the future work. 1.7 Notational Convention Throughout this thesis, unless otherwise stated, the following notations have been pursued to represent different types of quantities: x: lower case italic — scalar variable, X: UPPER CASE ITALIC — scalar constant or parameter, x: lower case bold — vector, X: UPPER CASE BOLD — matrix. Also, please note that figures with multiple boxes should be counted in an alphabetical order from top to bottom, if not numbered so. Chapter 2: FAULT DETECTION AND DIAGNOSIS: METHODS AND CRITICISMS 2.1 Introduction Condition monitoring and fault diagnosis has become an issue of primary importance in modern computerized process automation as it provides the prerequisites for increased reliability, safety, minimization of maintenance activities and costs, and the enhancement of system performance in any complex engineering plant including hydraulic control systems. As mentioned in Chapter 1, the purpose of condition monitoring is to detect abrupt and gradual changes in a dynamic system by processing the raw sensor data. A number of approaches to fault detection and diagnosis (FDD) have been reported during the last three decades. In many practical applications, the traditional engineering approach of hardware redundancy is used to incorporate diagnostics into dynamical systems. Repeated hardware elements (actuators, sensors, mechanical or hydraulic elements, etc.) are usually distributed spatially around the system to provide protection against localized damage (see, e.g., aerospace applications in [58 SAE, 1991]). Such schemes operate typically in at least a triplex redundancy configuration and redundant measurements are compared for consistency. The major problems encountered with hardware redundancy are the extra cost and software, and usually the additional space required to accommodate the equipment. Besides, hardware redundancy cannot be used to detect failures that affect all instruments in the same way and it may have difficulties in detecting subtle degradations in instrument behavior [86 Zhang, 1989]. Most of the current research since the early 1970's is based on the use of functional redundancy. The rationale for this idea is that even though the sensors are dissimilar, they are all driven by the same static or dynamic state of the system and are therefore functionally related. The FDD methods 10 Section 2. 1 Introduction 11 based on functional redundancy can be grouped into methods which i) do not use plant models, and ii) model-based approaches. Representatives of the first category in the area of hydraulics are classical limit and trend value checks of direct measurements, such as fluid temperature, pressure, flow rate, oil contamination, actuator velocity, etc. [10 Collacott, 1977] [79 Weber, 1982] [15 Fekete, 1989] [75 Watton, 1992]. Signature analysis, which is based on autocorrelation and spectral analysis techniques, and uses dynamic measurements such as vibration or stress wave sensing, also belongs to the "non-parametric" fault diagnosis methods (see [75 Watton, 1992] for a comprehensive survey). The deficiencies of non-parametric techniques can be summarized as follows: • Normally, these methods need dedicated and specialized sensor systems, including some non-robust ones, such as flowmeters. This may become expensive and more sensitive to sensor-based faults or environmental changes [75 Watton, 1992]. • Closed-loop control systems generally compensate for changes in the measured outputs by changing the system inputs. Therefore deviations caused by faults cannot be recognized by range checking alone. • Although spectral analysis and autocorrelation techniques are powerful tools for condition monitoring of high-frequency systems such as rotating hydraulic pumps or motors, they do not provide adequate information regarding faults that affect low-frequency responses of such hydraulic components as cylinder actuators and servovalves. "Model-based" FDD approach is a representative of the second class of functional redundancy methods. According to this approach, sensor measurements are processed analytically to estimate the value of a desired variable. This method is "model-based" since normally a quantitative model or a qualitative model of the physical system is used for estimating the desired variable. The estimate is then compared with the observed value of the variable to generate a residual. A fault is declared if the residual exceeds a certain threshold value. This chapter provides a brief coverage of various model-based FDD schemes, their advantages and disadvantages, applicable (or already applied) to fluid power control systems and components. Al l of the Chapter 2 FAULT DETECTION AND DIAGNOSIS: METHODS AND CRITICISMS 12 work reported in this chapter and later on lies within the realm of model-based functional redundancy approach, although it is recognized that a combination of all functional and hardware redundancy methods can be employed together to advantage in many cases. Most results on functional redundancy are based on a few basic concepts, including state estimation, parity space, parameter identification, expert systems, and pattern recognition. The first three approaches require fairly accurate mathematical models of the system, which may be either state-space models or input-output models. Expert systems are A l techniques that use heuristic, qualitative, and sometimes quantitative models of the process. Pattern recognition methods, such as neural networks, mostly use heuristic (known as connectionist) models for the process and the faults. Fig. 2.1 shows major FDD schemes that currently exist in the literature. 2.2 Observer-Based State Estimation Approach State estimation methods were mostly developed in the 1970's [82 Willsky, 1976][9 Clark, 1980][17 Frank, 1980], and since then have been successfully implemented to detect faults in sensors and actuators of various dynamic systems such as nuclear reactors [14 Elmadbouly and Frank, 1983], aerospace systems [52 Patton et al., 1986], inertial navigation systems [34 Kerr, 1987], and automobiles [57 Ribbens, 1991]. This fault detection and diagnostic approach is mostly based upon a "linear" state-space model of the dynamic system being diagnosed; i.e., x = A(0)x + B(0)u + w (2.1) y = C(0)x + v where x, u, and y are the state, input, and output (measurement) vectors, respectively, and A and C are known state transition and measurement matrices that depend on the vector of known system parameters, 0. w and v are system and sensor noise, respectively, and are assumed to be white Gaussian. The time dependent state variables, x, are either well-defined "physical" variables like the position and speed of a hydraulic piston described in the "controllable canonical" form [70 Vossoughi and Donath, 1995], or more complicated functions of the input and output signals as, e.g., in the "observable canonical" form [32 Jelali and Shwartz, 1995]. Section 2. 2 Observer-Based State Estimation Approach 13 ( FDD Methods ) Hardware Redundancy Functional Redundancy Limit Checking Model-Based Methods Analytical Model Observer-Based State Estimation (Clark, Frank) Parameter Identification (Isermann) Non-Parametric Methods Autocorrelation Analysis e.g.: vibration stress wave Spectral Analysis e.g.: vibration (Martin) Limit Checking e.g.: temperature pressure flow Qualitative/ Simulation Model Heuristic Model Expert System Knowledge Base System (Burrows) Neural Network Neural Network Pattern Function Classification Approximation (Kramer) (Sorsa) Hybrid Method (Isermann) Neural Network Expert System (Gallant) Fig. 2.1: Major fault detection and diagnosis (FDD) schemes. The model in Eq. (2.1) can be run continuously in a computer that receives inputs possibly from the same sensors used for control of the system. Using observers (w = v = 0) or Kalman filters, the computer continuously estimates the system states and calculates the sensor output values in the form of: x = A(0)x + B(0)u + K e e = y - y y = C(0)x (2.2) Chapter 2 FAULT DETECTION AND DIAGNOSIS: METHODS AND CRITICISMS 14 where stands for estimate, and K is the observer or Kalman filter gain. As long as there is no failure, the difference between the measured and calculated sensor outputs (innovations) is negligible. System failures normally fall within one of the following categories: i. component failures, which are represented by changes in the system transition matrix (because of system structure and/or parameter changes), so that A becomes A + A A . ii. actuator failures, which are represented by changes in B matrix, so that B becomes B + A B . iii. sensor failures which affect the output vector y, so that C matrix becomes C + A C , As soon as a failure in any system component does occur, such that x = A(0)x + B(0)u + ii + w (2.3) where f, is the failure event vector, the measured and calculated sensor output values disagree and significant changes in the "innovation" sequences occur. These changes are then fault signatures which appear as changes in definite directions with certain patterns and/or with certain time dependencies, and may be detected through certain statistical analysis and decision making techniques [1 Basseville and Nikiforov, 1993]. Fault detection then achieves fault diagnosis by, for example, propagating the error residuals, or innovations, e, in a unique direction in output (or parity) space (see, e.g., [57 Ribbens, 1991]). Each hypothesized failure may be associated with a unique direction for the output error vector e. For example, the output error residual ej corresponding to the jth actuator failure A B , will always point in the direction £ = C ( X - X ) = Cfi (2.4) = C ( A B i ) d i where d, is a constant directional vector. There are certain practical problems associated with FDD using state estimation. One problem is that for many dynamic systems, there are failures which have the same innovation signatures. In order to achieve an unambiguous diagnosis, the individual failure must be isolated from all other failures; a task which is Section 2. 2 Observer-Based State Estimation Approach 15 Dynamic System Fault 0 Filter eO Fault 1 Filter el w Fault 2 Filter £2 • Fault n Filter en • • Voting Logic Fault Decision Fault Diagnosis Fig. 2.2: Fault detection and diagnosis using a bank of filters or observers. not always realized in practice due to sensor noise and model reductions. One way to accomplish the task is using a bank of observers [51 Patton etai, 1987] or Kalman filters [48 Montgomery, 1981], where each filter represents a unique failure (including the normal behavior). These filters must be run in parallel with the actual plant. As shown in Fig. 2.2, the basic idea is that, under a specific fault condition (including no-fault), the filter or observer which operates according to that specific fault model is the only one which produces a zero mean and independent innovation sequence. Statistical tests can be constructed so that the properties of the innovations generated by different filters are analyzed [1 Basseville and Nikiforov, 1993]. Decision can then be made through some form of voting logic using estimates from the filters. This method is very complicated and computationally demanding, since increasing number of parallel filters are needed for an efficient fault detection and diagnosis as the number of fault modes increases. Another major problem with the observer-based FDD schemes is that they require the prior quantitative knowledge of all possible system failures in a state-space form, a task which is also difficult to achieve in practice. The most important problem with this method, however, is that it is still limited to systems whose dominant parts are "linear" and "known". Unfortunately, many real dynamic systems, including hydraulic components, cannot be represented by a global linear model such as the one given by Eq. Chapter 2 FAULT DETECTION AND DIAGNOSIS: METHODS AND CRITICISMS 16 (2.1). In this case, sometimes a set of local linear models stored in the computer memory are used for different operating points, which adds even more to the complexity of the FDD scheme. 2.3 Parameter Estimation Approach Another popular approach for providing fault indicating signals (fault symptoms) is "parameter estimation", where residuals are generated by parameter identification techniques. The immediate advantage of considering system parameters over states for FDD purposes is the time independency of parameters by definition, so that it is possible to compare system behavior under different fault conditions only by looking at the parameter magnitudes. A very good survey of process fault detection techniques based on parameter estimation is given in [28 Isermann, 1984]. Parameter-based FDD has been applied by several authors to detect and identify faults in technical processes, including motor-pumps [20 Geiger, 1983], hydraulic pumps [62 Silva, 1986], machine tools [56 Rault et al, 1984], chemical reactors [86 Zhang, 1989], and robotic manipulators [31 Isermann and Freyermuth, 1991]. Different parameterized process models consisting of input-output A R M A X models [86 Zhang, 1989], steady-state algebraic (static) models [29 Isermann, 1989], and continuous-time dynamic models [31 Isermann and Freyermuth, 1991] have been used for these applications. To represent a dynamic system as a mathematical model, suppose that the system (e.g., a hydraulic component) can generally be described as: y = f(x,u,ff,e) (2.5) where u and y are vectors of measured input and output variables, x is the system state vector, Q is the vector of constant or slowly time-varying system physical parameters, and e is the noise and disturbance vector. Several versions of the plant description in Eq. (2.5) exist, including input-output A R M A X models, Vk = aiUk-i H h apyk-p + H h b r u k - r + ek + ciek-\ H V cqek_q, (2.6) Section 2. 3 Parameter Estimation Approach 17 with k as a time index, steady-state algebraic models, y(u) = b0 + biu + b2u2 + • • • + brur, (2.7) continuous-time dynamic models, y(t) = aiyW(t) + ••• + apy(p\t) + b0u(t) + + • • • + M ( r ) ( i ) , (2.8) and state-space models (Eq. (2.2)). The fundamental idea is that any fault in the plant represented by Eq. (2.5), which also contains actuators and sensors, affects one or more system physical coefficients g in the form of abrupt changes or gradual drifts. Examples are increase in the resistance of a solenoid coil due to plugging or temperature rise, increase in Coulomb friction of a hydraulic actuator due to component wear or poor lubrication, and sensor gain drift or bias because of environmental effects or miscalibration. These changes are then fault symptoms and appear as certain patterns with i) offsets in definite directions, ii) changes in variances, and/or iii) changes with certain time dependencies [28 Isermann, 1984]. A hierarchical on-line condition monitoring and fault diagnosis algorithm is given in [29 Isermann, 1989], which has three phases: data processing, fault detection, and fault diagnosis. This scheme is graphically illustrated in Fig. 2.3. A brief description of the method is given below. 2.3.1 Data Processing Relevant information in the form of estimated system parameters is extracted from input-output sensor data, using identification algorithms (see, e.g., [41 Ljung, 1987]). An analytical model of the plant is naturally required. In order to have high detection accuracy and differentiation power, the process model should structurally represent the physical system as precisely as possible. However, it has been shown that even with a heavily reduced model or a biased identification procedure, early fault detection and diagnosis is still possible using parameter estimation methods [85 Zhang et al., 1994]. Very often, the system in Eq. (2.5) can be expressed by a linear regression model of the form y(t) = *(*)*(*) + e(t) (2.9) Chapter 2 FAULT DETECTION AND DIAGNOSIS: METHODS AND CRITICISMS 18 Parameter Estimation s Theoretical System Model Normal System Calculation of System Coefficients Change Detection Faulty System Fault Signature Statistical Fault Decision Fault (event, time) 1 Expert System Fault Classification Fault (type, location, size, cause) Fig. 2.3: Hybrid fault detection and diagnosis based on parameter estimation and expert systems [29 Isermann, 1989]. where $ is the regression or data matrix consisting of known system inputs and states, such as position, velocity, acceleration, flow rate, pressure, etc., e is the equation error vector, and 0 is the system parameter vector which is a function of the system physical coefficients, or o = g(e). (2.10) Section 2. 3 Parameter Estimation Approach 19 It is worth noting that the system in Eq. (2.5) can be expressed in the form of Eq. (2.9) if it is (at least locally) linear in parameters, 0. Therefore, a certain class of nonlinear systems can also be included in this representation. Using the regression model in Eq. (2.9), one may determine the system parameters through a recursive algorithm (for on-line purposes), such as least squares, maximum likelihood, instrumental variable, etc. These algorithms have a general form of: 0t = 4 - i + Rt(yt - # t » t - i ) (2-11) where R is the estimation gain matrix and is updated recursively (see, e.g., [41 Ljung, 1987] for various update algorithms). The output of this phase, thus, is the estimated system parameters vector 0. 2.3.2 Fault Detection Utilizing only the reduced process information, which are estimated system parameters, features are extracted using 0 or the system physical coefficients vector g , which then allow the detection of faults in the process. If the parameter mapping function g in Eq. (2.10) is invertible, the physical coefficients of the system can then be obtained from e = S~1(o) (2.12) This, however, is not always possible since either the process parameters 0 are generally "nonlinear" algebraic functions of process physical coefficients g [30 Isermann, 1993], or the system is originally represented by a "black-box" input-output model, such as an A R M A X model, in which case, the elements of the parameter vector 0 generally have no associations with the system's physical coefficients. Changes in g (or 0) are then determined with reference to the normal system coefficients go (or 0o), such that A0(t) = 0(t) - 0o(t) (2.13) Chapter 2 FAULT DETECTION AND DIAGNOSIS: METHODS AND CRITICISMS 20 These changes are subsequently used to recognize a fault event and the time of its occurrence. For this task, statistical decision making methods, such as "Bayes decision" or "sequential probability ratio test" (SPRT), may be applied [1 Basseville and Nikiforov, 1993]. 2.3.3 Fault Diagnosis After a fault event has been detected, the parameter change patterns are submitted to a classification procedure with the aim to determine the fault type, fault location, fault size, and the cause of the fault. For this purpose, knowledge-based "expert systems" with "fault-tree" or "root-cause" analysis approaches have been suggested for a number of applications [18 Freyermuth, 1991] [31 Isermann and Freyermuth, 1991] [30 Isermann, 1993]. In these applications, "heuristic knowledge" of the process, along with the analytical results obtained from the fault detection phase, have been used to conclude about a specific fault in the system. Expert systems for FDD applications will be explained next in this section. However, it should be noted that sometimes fault decision and classification are so combined that their clear separation is not possible. One example is the application of artificial neural networks (ANNs) in fault detection and diagnosis, where ANNs monitor the plant, and detect and diagnose system failures simultaneously [37 Leonard and Kramer, 1993] [64 Sorsa and Koivo, 1993]. One great advantage of the parameter identification approach over the (mostly multi-model) observer-based scheme is that it is possible to use only "one" system model for the fault detection and diagnosis problem. This is because the faults are assumed to change the parameters only, and the change patterns are assumed to be already known. Hence, less model maintenance effort and less computational complexity is required. Another advantage is that the incipient or progressing faults can equally be detected and diagnosed, as well as sudden failures. This is because most parameter estimation schemes naturally provide means to track system parameter drifts over time, especially if they are interpretable as system physical coefficients such as flow resistance, viscous damping, etc. Section 2. 4 Expert Systems Approach 21 There are, however, some disadvantages associated with FDD using parameter estimation. One, for example, is the requirement for "persistent excitation" [41 Ljung, 1987], which can be partly achieved by adding low-amplitude auxiliary input signals [86 Zhang, 1989]. The other problem is an increased time delay for fault detection compared to observer-based schemes. This is because more time is normally required for the parameters to converge to steady-state values, and parameter tracking speed is usually much less than state tracking speed. 2.4 Expert Systems Approach Expert systems are alternative means to detect and diagnose faults in technological processes. A knowledge-based expert system (KBES) is a computer program which is designed using artificial intelligence (Al) techniques to emulate human-like reasoning. An expert system solves the problem of fault detection and diagnosis in the same way as a human expert makes or suggests decisions on the faults. The expert system then explains its fault conclusions and diagnostics to the user who may not be an expert and is only familiar with such a domain as hydraulic systems. On-line fault detection and diagnosis has been an active area of research in knowledge-based expert systems (see, e.g., [68 Tzafestas, 1989] for an overview). So far, almost all major efforts for automatic fault detection and diagnosis in fluid power systems have been focused on using knowledge-based expert systems [24 Hogan etal, 1990] [7 Burton and Sargent, 1990] [25 Hogan etal, 1992] [75 Watton, 1992] [26 Hogan et al., 1996]. This is partly because precise information about the components and failure modes is not always known, and partly because quite often the size and complexity of the hydraulic circuits make it difficult to precisely describe the fault generating mechanism [4 Bull et al., 1996]. The ability of expert systems to integrate both quantitative and qualitative models of the process in a "structured" manner and to assess both linguistic information and numeric data has made expert systems ideal for fault diagnosis purposes. On-line expert systems use direct process outputs and measurements and have the following components: Chapter 2 FAULT DETECTION AND DIAGNOSIS: METHODS AND CRITICISMS 22 • Knowledge base, which is the general knowledge about the problem, in the form of facts and rules. • Data base, which contains the current information about the problem, i.e., input data from sensors and possibly from the user. • Inference engine, which is an executable program of methods for applying the general knowledge for the problem. • Explanation component, which informs the user on why and how the conclusions are obtained. • Knowledge acquisition and user interface components, to acquire numeric and boolean data from the process and the user, and to present the diagnostic results in appropriate formats. "Rule-based" expert systems are probably the most popular form of expert systems, which use the knowledge representation in the form of production rules, such as: RULE: Rk (cfr) IF c, A N D c 2 A N D A N D cm T H E N hi A N D h2 A N D A N D hn where Rk is the rule number (or name), c, is the rule "condition" which may be a fault symptom or an event, and hj is the rule "consequence" which can be a deduced event or a fault. cfr is the rule certainty factor and is a measure of belief in the rule. This factor is usually between -1 (no belief) and +1 (full belief). Besides, each premise or condition c, has its own certainty factor which is often calculated on-the-fly or is obtained from the user. A rule is "triggered" when a combination of its condition certainty factors and cfr exceeds a threshold. Then, the certainty factors of the rule consequences, hj, will be computed from the conditions' certainty factors and from cfr using various reasoning methods such as Bayesian inference, fuzzy logic, Shafer-Dempster theory, etc. Rule-based expert systems have many advantages such as providing a homogeneous representation of knowledge and allowing incremental growth of knowledge through addition of new rules (see [68 Tzafestas, 1989] for more details). In order to accomplish the task of fault diagnosis, knowledge-based expert systems use two different approaches which are briefly explained below. Section 2. 4 Expert Systems Approach 23 2.4.1 Shallow Reasoning Shallow reasoning consists of a set of production rules describing a certain problem, and are based on heuristic relationships between fault symptoms and system malfunctions. Shallow reasoning is appropriate for evidentially oriented systems where a good understanding about the process being monitored is not available, and only direct heuristic relationships between system irregularities and faults exist. It normally consists of observing the evidences (fault symptoms) in the form of numeric and/or boolean data, finding the rules whose condition (or IF) parts are triggered by the data item, firing (or activating) the rule, and deducing new facts (e.g., faults) which are the consequences of the fired rules. This is called forward chaining. One example could be: IF inlet pressure IS "normal" A N D valve direction IS "retract" A N D cylinder motion IS "retract" T H E N operation IS "normal" in which measurements of the cylinder inlet pressure and valve and actuator direction of motion directly lead to the "normal" operation. The main disadvantages of the shallow reasoning are as follows: • Knowledge acquisition is difficult, • Knowledge-base cannot be generalized and is only specified for a certain plant, • Unstructured (heuristic) knowledge is needed, • There is no guarantee for diagnosis because of incomplete or incorrect a priori information, • Large systems require excessive number of rules and execution time. 2.4.2 Deep Reasoning Deep reasoning is usually used to diagnose precisely known man-made systems. It is based on a structural and functional (namely deep) model of the technological process, and can predict the system's behavior under various fault conditions for a given set of signals and parameters. Expert systems that use deep Chapter 2 FAULT DETECTION AND DIAGNOSIS: METHODS AND CRITICISMS 24 reasoning attempt to explicitly find the underlying principles of the process operation. Therefore, the need to predict every possible fault scenario or to use prescribed rules is eliminated. Two popular fault diagnosis techniques in the domain of hydraulic systems that use deep models are explained below. A . Hypothesis test technique: This method is only concerned with identifying the significant failure modes [4 Bull et al., 1996]. In this technique, a cause for a system malfunction is hypothesized, the symptoms of the postulated fault are determined, and the result is compared with the actual symptoms observed from the system. This technique requires qualitative simulation of the effects of the postulated malfunctions as a result of failures. Fault diagnosis then involves detecting all root causes using on-line sensor data appropriately discretized in classes such as "high", "normal", or "low". "Fault-Tree Analysis" and "Failure Modes and Effects Analysis" (FMEA) are two versions of this approach. In a rule-based expert system, for example, hypothesis test technique requires that a particular hypothesis is selected, and the rules are searched to see if the hypothesis is a consequent. If true, the conditions of the rule constitute the next set of hypothesis. The process is continued until some hypothesis is not true or all hypotheses are true based on the process data. This is called backward chaining as opposed to forward chaining in the shallow reasoning approach. A landmark expert system for fault diagnosis of hydraulic circuits based on the hypothesis test technique has been proposed in [24 Hogan et al., 1990] and is later explained in [25 Hogan et al, 1992]. This KBES utilizes a library of qualitative behavioral simulation models of hydraulic components, both during correct operation and under various failure conditions. The failure modes for each particular component are established by performing a F M E A on that particular component [24 Hogan et al., 1990]. A new version of the program which is suitable for automated Fault Tree Analysis of hydraulic systems and can handle multiple faults has been described in [26 Hogan et al, 1996]. For qualitative reasoning purposes, "pressure" and "flow" measurements are discretized into ranges with the labels "high", "normal", and "low", with the flow variable having an extra range labeled "zero". Section 2. 4 Expert Systems Approach 25 Despite the expert system's power in diagnosing single faults, it turns out that as the size of the circuit increases, and the number of concurrent faults grows, the computation time for fault diagnosis increases exponentially [26 Hogan et al, 1996]. Another drawback is the expert system's relying on direct "pressure" and "flow" measurements for reasoning about fault effects. These two quantities are associated with noise and can also substantially vary during transition periods, or due to dynamic effects. Thus, ambiguous results might be obtained when performing an F M E A . The third drawback is that the qualitative models used in the knowledge base cannot simulate and predict time varying phenomena such as leakage [26 Hogan etal., 1996]. The importance of incorporating mathematical models inside the knowledge base to deal with dynamic effects and can predict structural or parameter changes due to faults has been stated in [4 Bull et al, 1996]. B . Governing equation technique: This technique, as opposed to the failure modes and effect analysis and fault-tree analysis methods, utilizes quantitative models of the process in the form of (mostly simple algebraic) constraint equations. It is applicable to all cases where there is an association between each constraint equation of the monitored system and a set of faults that are sufficient to cause violation of the constraint [35 Kramer, 1986]. Consider, for example, a hydraulic motor whose input and output flow rates, qi and q0, are being monitored by two flowmeters. At steady-state operation, the constraint equation gt - ? o « 0 expresses the mass balance for flow into and out of the unit. When the left-hand side is found to be much less than zero, then we can infer that: (g,_sensor_fails_low) OR (^0_sensor_fails_high) On the other hand, if the left hand side of the equation is found to be much greater than zero, then we can conclude that: (#/_sensor_fails_high) OR (^0_sensor_fails_low) OR (motor_leakage) By appropriately combining the inferences drawn from the complete set of process constraints, along Chapter 2 FAULT DETECTION AND DIAGNOSIS: METHODS AND CRITICISMS 26 with direct measurements and some heuristic knowledge obtained from the user or a data base, one can deduce the correct fault(s). One example of such an approach is given in [18 Freyermuth, 1991], where the faults in a robot have been detected and diagnosed using the robot estimated and nominal parameters and a rule-based expert system. In [75 Watton, 1992], an on-line rule-based expert system based on "governing equation technique" has been introduced and explained. It has been shown that the expert system is capable of successfully detecting and diagnosing faults (e.g., leakages) in a number of fluid power components if on-line measurements (especially flow rates) and algebraic constraint equations for the components are available. Although the expert system relies heavily on "steady-state" behavioral and quantitative models, it has also been able to detect and diagnose some dynamic faults, e.g., leakage in a directional valve-controlled cylinder, using dynamic flow measurements. The price, however, has been an increased detection time delay, additional logic and rules, and excessive constraint equations because of the need for both rod extension and rod retraction data. More than 30 rules have been applied for this purpose, about half of which have been used only to represent constraint equations. Some other factors that limit the capabilities of expert systems in fault detection and diagnosis are as follows: • all failure possibilities have to be explicitly enumerated in the form of rules and facts, • fault detection mechanism is inherently slow (because of rule chaining process), • it may not be possible to identify all the failure modes for a complex system, • there is little capability for robust handling of noisy data, • KBES's are very weak in real-time mathematical computations and it is extremely difficult to represent system dynamics and dynamic faults in the form of rules. As an example of incapability of expert systems in capturing dynamic faults, consider the input-output measurements shown in Fig. 2.4 which are obtained from a robotic arm attached to a hydraulic cylinder (see Fig. 4.17 for a schematic diagram of the arm). At time t = 1 s, half of the load being carried away by Section 2. 5 Pattern Recognition and Neural Network Approach 27 Time History of the Movement of a Single Robotic Arm 40 £ 20 g» 0 < o -20 -40 0 x10 r j i i I • i i i I • 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time (sec) Time History of the Applied Force to the Robotic Arm Fig. 2.4: Input-output measurements from a hydraulically actuated arm while a failure occurs at t = 1 sec. the arm was suddenly detached from the arm. Apparently, there is no noticeable trend or any significant signs of abrupt changes in the input force or the output arm position signals. It is not clear how an expert system relying purely on such raw sensor data and/or qualitative models can detect or diagnose such dynamic system's failure. In Chapter 3, we introduce a model-based technique that extracts high-quality signals from the raw sensor measurements for automatic fault detection, and provides more informative data usually required to diagnose faults in an expert system. 2.5 Pattern Recognition and Neural Network Approach Pattern classification techniques can generally be classified into three categories: i) knowledge-based Chapter 2 FAULT DETECTION AND DIAGNOSIS: METHODS AND CRITICISMS 28 approach, ii) statistical pattern recognition, and iii) neural networks. In the previous section, it was explained that although knowledge-based expert systems can exhibit powerful capabilities, simulating the domain properly may be difficult and computationally intensive for real-time or on-line fault detection and diagnosis. Pattern classification using neural networks falls into the general categories ii) and iii) above and is a new and rapidly growing area of research. Both these methods have strong theoretical and mathematical basis, which makes pattern classification a popular tool for fault-related problem solving. 2.5.1 Statistical Pattern Recognition Pattern recognition techniques are widely used for classification discrimination, fault diagnosis, and decision making [80 Weiss and Kulikowsky, 1991]. The basic principle behind fault classification and diagnosis using pattern recognition is the selection of features, or fault symptoms, which are closely related to the classes, or faults. Pattern classifiers characterize classes by their probability density functions on the input features and use of Bayesian decision theory to form decision regions from these densities. Pattern recognition approaches based on statistical techniques are effective, but tend to be complex and cumbersome in situations involving large training sets, especially when there is little or no a priori information about the underlying fault generating process and system noise. Neural networks can overcome these limitations and are finding increasing applications in system fault diagnosis and incipient fault detection. 2.5.2 Neural Network Pattern Classification In all of the previously described fault detection and diagnosis methods, either a known qualitative or a known quantitative model for the process, or at least some a priori information about the statistical distribution of the process data was required. Sometimes, the difficulty in formulating a FDD problem is the lack of such a model that relates symptoms to the system faults, due to the lack of understanding of fault induction and propagation mechanisms in the system. In neural network approaches, data collected from the plant replaces the mathematical model or the expert's knowledge. This approach is Section 2. 5 Pattern Recognition and Neural Network Approach 29 OUTPUT OUTPUT LAYER HIDDEN LAYER INPUT LAYER VV44 Fig. 2.5: A general artificial neural network topology. particularly useful when expert diagnostic knowledge and prior models of fault-symptom relationships are not available, as often is the case for complex processes [37 Leonard and Kramer, 1993]. A neural network is a class of computational structure modelled on biological processes. "Trainability" and "adaptability" are the two important features of a neural network. Many different variations in neural network structures and topologies exist in the literature, such as single and multilayer perceptrons, feedforward networks, radial basis function networks, etc. Also, various learning techniques for training such networks can be found in the literature, such as backpropagation, perceptron learning, mean squared error, etc. Detailed explanation of those structures and these algorithms is not the purpose of this research and the interested reader is referred elsewhere (see, e.g., [19 Gallant, 1993]). Fig. 2.5 shows a neural network topology that consists of processing units, called "cells", which are joined by directed arcs, named "connections". Each connection has a numerical weight, wij, that specifies the influence of cell Cj on cell c, and determines the behavior of the network. A zero weight means there is no connection between the two cells. Each neural network has a set of input cells and a set Chapter 2 FAULT DETECTION AND DIAGNOSIS: METHODS AND CRITICISMS 30 of output cells. The "input cells" (boxes in the figure) pass the inputs that they receive from external sources to the cells in the next layers, without any alterations. The "output cells" provide the outputs for the whole network. Some intermediate or "hidden cells", which are neither inputs nor outputs, may also exist in the network to handle nonseparable problems. Each cell, c,, computes a numerical output called "activation". Cell inputs and outputs may be discrete, i.e., -1, 0, or +1, or continuous values usually normalized between -1 and +1. Every cell, c,-, except for input cells, computes its new activation c, as a function of the weighted sum of the inputs to cell c, from directly connected cells, such that / JL \ (2.14) The activation function, / , is usually a nonlinear, bounded, and piecewise differentiable function. Three typical examples are: threshold function, { -1 for x < 0 0 for x = 0 + 1 fora;>0 (2.15) sigmoidal function, and radial basis function, /(*) = 1 + e-f(x) = exp where xCj and 6j are the center and the width of the jth cell, respectively, and || Euclidean norm. (2.16) (2.17) || denotes the Regardless of the specific type of network and training algorithm considered, a neural network pattern classifier (NNPC) produces fault classes corresponding to the normal operation and different fault situations. Hence, the goal of pattern classification using neural networks is to assign input patterns to a finite number of classes. The input pattern, which is a vector of system features selected for distinguishing between fault classes, can be viewed as points in the multidimensional space defined by the input feature Section 2. 5 Pattern Recognition and Neural Network Approach 31 measurements. These inputs might include past as well as present values to capture temporal information. The classifier partitions the multidimensional space into decision regions that indicate to which class any of the input patterns belongs. NNPC's form nonlinear discriminant functions using single or multi-layer perceptrons and provide a greater degree of robustness or fault tolerance than other competitive fault detection methods because of their inherent parallel computational structures. The first ideas of applying neural networks in fault detection and diagnosis have considered the classification of continuous measurements of chemical processes according to the steady state operation of the process [27 Hoskins and Himmelblau, 1988][36 Kramer and Leonard, 1990][65 Sorsa and Koivo, 1993b]. Both backpropagation and radial basis function networks have been used, and the neural network classifiers are designed to produce classes corresponding to the normal operation and different fault situations. The problem with this classification method is that the dynamic behavior of the process is not taken into account, and in many cases, the effect of faults on the dynamics of the process and the changes in system dynamics are very difficult to detect with such classification methods. Besides, feedback controllers usually keep most of the steady-state measured outputs of the system fairly constant. To consider the process dynamics and detect faults during transient situations, an NNPC has been proposed [37 Leonard and Kramer, 1993] using radial basis functions, that classifies time-series type patterns (using past data) collected during process transitions. It has been shown that when a fault occurs, the operating point drifts away from the normal operating class to the fault situation class along some (usually) unknown trajectory [65 Sorsa and Koivo, 1993b]. Therefore, the classification during unpredictable transients is poor, but when a new steady state is reached, the classification performance is generally good (see Fig. 2.6). 2.5.3 Neural Network Function Approximation Another important application of artificial neural networks is in the area of nonlinear function approximation theory, where ANNs have been used to establish heuristic unstructured relationships between inputs and outputs of mathematical models of nonlinear dynamic systems [53 Pham and Liu, Chapter 2 FAULT DETECTION AND DIAGNOSIS: METHODS AND CRITICISMS 32 Temperature T ['Q Temperature T [*C] Fig. 2.6: Neural network fault (.) and no-fault (+) classification when a chemical process is in steady state (left), and in transient dynamic state (right) [64 Sorsa and Koivo, 1993a]. 1992][45 Masri et al, 1993][84 Xue and Watton, 1995][83 Xu et al, 1996]. In this case, both the inputs and outputs of the neural network are continuous values, much similar to a nonlinear multi-input multi-output (MIMO) function. Unlike an NNPC that classifies system measurements according to its operation state, a neural network function approximator (NNFA) tries to mimic the system behavior by learning the output(s) of the system under specific inputs. Learning is accomplished by estimating and adjusting the network weights during the training process (e.g., using backpropagation algorithm) and is similar to nonlinear black-box (NARMAX) modeling of a plant using statistical and parameter identification techniques [63 Sjoberg et al., 1995]. The most crucial requirement here is the "richness" and "availability" of the data that represent a broad range of system operation. In order to include the effects of dynamic faults during transition periods, a NNFA has been proposed in [64 Sorsa and Koivo, 1993a] that predicts the outputs of nonlinear dynamic systems. The technique can be seen as an extension of the state estimation methods (that uses banks of observers or filters) to more general nonlinear systems. Here, when appropriate process measurements are available, a bank of neural network models, such as the one shown in Fig. 2.7, can be generated. The bank contains both the model for nominal operation and the models for fault situations which must be precisely known or be trained Section 2. 5 Pattern Recognition and Neural Network Approach 33 ANNO Dynamic System y(t), y(t-i y(t-2),.., y(t-m) eO A N N 1 el A N N 2 e2 A N N n en • • Statistical Decision Fault Decision Fault Diagnosis Fig. 2.7: Using a bank of neural network function approximators for fault diagnosis in dynamic systems, beforehand in order to achieve accurate diagnosis results. A slowly varying chemical process has been chosen for this purpose, and in order to avoid parameter identification problems in large MIMO systems, the process is modelled as a multi-input single-output (MISO) nonlinear system of the form: yt - f(yt-i,---,yt-p,ut-i,---~ut-T) (2.18) The measured and estimated outputs from each network are subtracted and low-pass filtered to obtain residuals. For decision on the most probable fault, a Bayes's classifier has been applied to the residuals to decide upon the network that gives the smallest fault-model-to-measurement distance. Although a relatively simple simulation example has been used, a number of temporary false alarms and missed alarms have been observed during an 800-minute monitoring period. One problem in using such a NNFA scheme for fault diagnosis is that it is generally difficult to decide upon which measurement is the most appropriate one to be chosen as the network output. Besides, when the number of faults grow, training each NNFA for the corresponding fault and for the entire range of plant operation becomes quite a tedious task which requires a great deal of experience and patience. Neural network incapability in explaining faults should also be added to the above argument. Chapter 2 FAULT DETECTION AND DIAGNOSIS: METHODS AND CRITICISMS 34 Table 2.1 Comparison of major fault detection and diagnosis methods. Method —• Characteristics 1 State Estimation Parameter Estimation NNPC NNFA Expert System Unsteady-State Operation + + + + — + + — Noise in System/Sensor + + + + + - — Incomplete Measurement + + - + - -System Nonlinearities - + + + — Poor or No Math. Models — - + + + + + Parameter Uncertainty — + + + + + + + Incipient Fault Detection — + + + — + Fault Diagnosis — - + + — + + Fault Explanation — + — — + + + +: Very good in handling the characteristic. +: Good in handling the characteristic. — : Very poor in handling the characteristic. - : Poor in handling the characteristic. 2.6 Concluding Remarks In this chapter, we discussed major fault detection and diagnosis schemes that exist in the literature. We have summarized the advantages and disadvantages of these techniques in Table 2.1. Although evaluation of each scheme is a rather subjective argument, a general trend is obvious from the table: • When there is a good mathematical or physical model, estimation methods are more robust and outperform other methods in the fault detection phase. • Parameter estimation method is generally more efficient than other methods for incipient fault detection and diagnosis, since it combines both fault detection and feature extraction problems into one unique problem of parameter tracking. If the estimated parameters are physically interpretable, the fault diagnosis problem can also be decoupled most of the time. • In the fault diagnosis phase, where a certain amount of human expertise is still required for problem solving, pattern classification techniques based on heuristic knowledge and qualitative reasoning (NNPCs and expert systems) are more efficient and outperform analytical methods. In the next chapter, we introduce a new integrated hierarchical methodology for condition monitoring and fault diagnosis in a dynamic system that takes advantage of many strengths the existing F D D schemes. Chapter 3: A HIERARCHICAL FDD METHODOLOGY 3.1 Statement of the problem In Chapter 2, major advantages and disadvantages of a number of fault detection and diagnosis schemes were recognized. In this chapter, in order to decide on a suitable FDD technique for a typical hydraulic system, first we have summarized the characteristics of hydraulic systems and the criteria for the performance of an appropriate FDD scheme. Then, the elements of a novel hierarchical methodology for on-line condition monitoring and fault diagnosis are described that consider the hydraulic system characteristics and the FDD criteria. 3.1.1 FDD Related Characteristics Fluid power circuits and components have special features and characteristics that must be taken into account when choosing a monitoring and fault diagnosis scheme for on-line applications. Chapters 4 and 5 give some detailed information about the operation and analytical models of various components in a hydraulic system. The following FDD-related characteristics (CH) for a typical hydraulic system are noticeable by closely examining those mathematical models and experimental results. CHI—Nonstationarity: Hydraulic machines are mostly used in industries where normally heavy loads are manipulated or large forces are exerted on their surroundings (e.g., presses) in a very short period of time. Hydraulic systems contain many high bandwidth electrical, mechanical, and hydraulic components that are in direct interaction with one another most of the time. High bandwidth implies fast response time, which is critical in many applications. As a result, unlike such stationary processes as chemical process, and pulp and paper, that mostly operate at some specific set points, a hydraulic system is normally demanded to operate in an unsteady-state condition. 35 Chapter 3 A HIERARCHICAL FDD METHODOLOGY 36 CH2—Lack of proper measurements: Valuable information can be gained from the way a fluid power system behaves under dynamic conditions. Information of this nature is valuable but it does require more advanced instrumentation and sensors capable of capturing the transient data in real-time. However, such sensors are not always available, easy to install, or economic to use. For example, flow loss and leakage in various hydraulic components represent an important class of faults in practice [75 Watton, 1992]. However, high bandwidth flowmeters of sufficient accuracy and repeatability are expensive, and it is not safe to place them near actuators in harsh operating environments. As another example, consider the spool position of a pilot valve which cannot be directly measurable because of mechanical restrictions (see Chapter 5 for more details). Also, quite often, we need to know the velocity of such moving components as a cylinder rod (Chapter 4) or a valve spool (Chapter 5). However, due to the slow motion of such components and mechanical restrictions, usage of a velocity sensor is not recommended. CH3—Noisy measurements: Sensor noise and disturbance are inevitable in any kind of measurement performed on a dynamic system. However, due to extensive use of hydraulic systems in harsh environments, the sensors are more prone to damage or to pick up high frequency disturbances from the surrounding devices or environment. CH4—Nonlinearities: Hydraulic systems have severe nonlinearities which originate from both actuation and loading subsystems. As has been discussed in Chapters 4 and 5, accurate dynamic models for hydraulic components must include some or all of these differentiable and non-differentiable nonlinearities. The most significant differentiable nonlinearities come from flow through orifices (square-root law), spool position and orifice area relationship, asymmetric actuation (unequal piston areas), oil and hose compliance variations, flow forces, load kinematics, and load dynamics. The most important non-differentiable nonlinearities include Coulomb friction in moving components, preloaded springs in servovalves, and servovalve deadbands and porting overlaps. CH5—Time-variability: Aside from nonlinearities, many parameters of a hydraulic actuation system vary over time (see Chapters 4 and 5). Even at one operating point the system characteristics will often Section 3. 1 Statement of the problem 37 vary, for example, due to changes in load behavior or oil temperature. Over longer periods of time, Coulomb friction and flow resistance coefficients also increase due to excessive dirt and oil debris. CH6—Availability of good analytical models: Chapters 4 and 5 show that, despite all the above-mentioned drawbacks, good analytical models for various hydraulic components can be developed having different levels of complexity. This is due to the fact that even the most complex hydraulic system is still a man-made system, and that the physical laws governing the operations of such systems, though complicated, are well-known. 3.1.2 Criteria for an Appropriate FDD Scheme To design a fault detection and diagnosis scheme that is appropriate for a hydraulic system, some performance criteria must be considered in advance. A number of the leading criteria (CR) for assessing the performance of an FDD scheme are given below. CR1—Promptness of detection: The issue of promptness may be of vital importance assuming that a fault is detected successfully. This issue is more relevant to the detection of abrupt changes (or catastrophic failures) in the dynamic system. The time delay for detection is normally minimized for a fixed false alarm rate [1 Basseville, 1993]. More reliable detections and also fault identification and diagnosis require longer time delays, and therefore, a trade-off should be made. For example, in aerospace applications, shorter time delays for detection are required, while in industrial applications, more reliable detections along with fault identification are desired. In every case, prompt detection of a fault inhibits further fault propagation, and therefore, reduces the probability of having multiple faults which can usually cause more damage and substantially increase fault diagnosis time and expenses [26 Hogan et al., 1996]. CR2—Sensitivity to incipient faults: Sudden failures should be detected promptly due to safety reasons. In fluid power systems, however, it is equally desirable to detect small or slowly developing (incipient) faults in individual components. This is important from a predictive maintenance point of view in which a fault detection scheme is intended to enhance maintenance operations by early spotting, for example, Chapter 3 A HIERARCHICAL FDD METHODOLOGY 38 a worn part in a hydraulic circuit. Besides, early detection of a progressing fault prevents further faults in other system components. This usually means more focused diagnosis which saves time and money. CR3—False alarm rate: False alarms are generally indicative of poor performance in a fault detection scheme and may quickly lead to a lack of confidence in the detection system. Compromises must be made in detection system design among false alarm rate, sensitivity to incipient faults, and promptness of detection. A good FDD scheme should increase the sensitivity of the detector to actual faults and reduce the probability of false alarms by discriminating between true faults and disturbance effects due to noise and uncertain dynamics. CR4—Missed alarm rate: This is also directly related to the false alarm rate and promptness of detection of abrupt changes, and should be minimized to increase the reliability of the FDD scheme. Generally speaking, increasing the sensitivity of the detection scheme to reduce the missed alarm rate increases the fault alarm rate, and vice versa. CR5—Incorrect fault diagnosis: Another criterion for the FDD scheme, closely related to both false alarm and missed alarm rates, is the incorrect identification (or diagnosis) of a failed component. In this case, the detection system correctly registers a fault, but the diagnosis system incorrectly identifies the component which has failed. CR6—Robustness: The robustness of an FDD scheme is the degree to which its performance is unaffected by conditions in the dynamic system which rum out to be different from what they were assumed to be in the design of the FDD scheme [50 Patton et al., 1989]. Robustness problem in an FDD scheme are generally considered with respect to: i) physical parameter uncertainties, ii) unmodelled nonlinearities, iii) disturbances and noise, and iv) fault types [50 Patton et al., 1989]. 3.2 An Integrated Hierarchical Solution Considering the advantages and disadvantages of the FDD methods explained in Chapter 2, it turns out that Section 3. 2 An Integrated Hierarchical Solution 39 no single scheme is able to meet all the FDD criteria for a hydraulic machine with typical characteristics explained above. In this case, a multi-scheme hierarchical algorithm may be a better choice, as will be shown later. In the rest of the chapter, we have introduced and described an integrated condition monitoring and fault diagnosis methodology that has combined many advantages of the aforementioned fault detection and diagnosis schemes. The applicability of the methodology in detecting and identifying faults has been tested on a number of real-world hydraulic components in Chapters 6 and 7. 3.2.1 The Objective Our goal is to provide means to convert poor-quality sensor data into high-quality self-descriptive information such as system physical parameters, so that the qualitative reasoning process in a knowledge-based system regarding faults is facilitated. In practice, failure modes often involve a change in the model. The basic idea behind the proposed model-based methodology is that most often the diagnosis of a continuous dynamic system can be performed by tracking the behavior of its physical coefficients which describe the model during their evolution from one equilibrium state to another. That is, the model "synchronously" tracks the evolution of the system. In particular, this will allow faults that are developing, or are only observable, during the transient to be identified. A fault mode can then be modelled as a constant jump bias (abrupt change) or a ramp bias (drift) in the system coefficients. If these parameter changes are appropriately discretized into qualitative ranges such as "high", low", etc., fault diagnosis based on qualitative reasoning may be better accomplished [26 Hogan et al., 1996]. Fig. 3.1 shows one such example when a hydraulically actuated manipulator such as the one described in Chapter 4, Fig. 4.17, suddenly lost part of its load at tf = 1 s. The system coefficients are apparently changed after the fault time. Mathematically, this fault can be expressed as: (3.1) Chapter 3 A HIERARCHICAL FDD METHODOLOGY 40 (a) Load Mass (c) Coulomb Friction 0.5 1 1.5 Time (sec) (b) Center of Gravity 0.5 1 1.5 2 Time (sec) 0.5 1 1.5 Time (sec) (d) Viscous Friction 0.5 1 1.5 2 Time (sec) Fig. 3.1: Typical changes in system physical coefficients due to a failure. where g°'s are known or previously identified "normal" (or "nominal") system coefficients, and "I" is the unit step function at fault time tf. For incipient faults, tf normally takes much longer than the fault time for an abrupt failure. The variables tf, i, and A@i are all unknowns to be determined by the fault diagnosis procedure. In this way, the fault diagnosis problem leads to the joint problem of detection of tf, recognition of /', and estimation of AQ{. In Fig. 3.1, for example, / / = 1 s, AQI = — 7 Kg, AQ2 = 0 m, A # 3 = —0.35 N.m, and AQ4 = —3 N.m.s/rad. 3.2.2 The Algorithm The systematic hierarchical methodology proposed in this thesis is primarily appropriate for systems Section 3. 2 An Integrated Hierarchical Solution 41 with known dynamics which can be described explicitly by linear or nonlinear state-space models. This, fortunately, encompasses a large number of man-made dynamic systems. The diagnostic system, as shown in Fig. 3.2, is an integration of analytical, statistical, heuristic, and knowledge-based schemes and has four distinct phases: i) condition monitoring, ii) fault detection, iii) fault symptom generation, and iv) fault diagnosis. To solve the difficulties in recognizing the faulty component among many candidates, a joint parameter and state monitoring scheme is proposed at the first level. For fault detection at the second level, a statistical decision making based on Wald's sequential probability ratio test (SPRT) [71 Wald, 1947] is performed on the estimated parameter and state residuals. At this level, residuals are formed by comparing the current estimates with the normal parameters identified at different operating conditions. A fault is declared as soon as a residual exceeds a threshold. After a fault is detected, certain fault symptoms in the form of binary codes are generated at the third level by discretizing each parameter value into ranges, named "jump classes", using the "control charts" idea. The discretized fault symptoms will then be passed to a neural network pattern recognizer in the fourth level for fault identification. The candidate fault can then be diagnosed (confirmed or rejected) by a knowledge-based expert system using qualitative reasoning, production rules, and possibly other heuristic information. Fault explanation and maintenance aiding may also be provided by the expert system. This hierarchical methodology, in principle, is similar to the parameter-based fault detection and rule-based fault diagnosis approach proposed in [30 Isermann, 1993] and discussed in Section 2.3, Fig. 2.3. However, there are a number of differences that make our integrated algorithm automatic and more appropriate for dynamic systems; for example, inclusion of joint state/parameter estimation for unmeasured state reconstruction, automatic parameter value discretization and jump level classification after change detection, and fault pattern recognition using a neural network pattern classifier. Chapter 3 A HIERARCHICAL FDD METHODOLOGY 42 Fault/Failure Noise \ Noise Input /State-space/ - H Technical Plant Output / model / *j Monitoring |« Condition traceable states and / parameter estimates \ fault reset switch ,*X^ j diagnosis ^ _ _ _ -switch Fault/Failure Detection / Faulty States /_ i & Parameters / Fault Symptom Generation / Fault / / *• 1 / Patterns/ / Normal States / l & Parameters / fault code Fault Classification i Heuristic ' / L .^1 / Knowledge / / _J recognized fault event Fault/Failure Diagnosis identified fault (type, location, causes, effects, remedies) Fig. 3.2: The proposed integrated hierarchic algorithm for on-line condition monitoring and fault diagnosis. 3.3 Condition Monitoring by State/Parameter Estimation The first level in the proposed hierarchy is the condition monitoring phase, which consists of data collection and processing, state/parameter estimation, and generation of change indicating signals, using an appropriate model for each system component we are monitoring. Here, we assume that a system Section 3. 3 Condition Monitoring by State/Parameter Estimation 43 component can be represented by a nonlinear multi-input multi-output continuous-time state-space model: x(0 = f(x(t),u(O,0(O) + w(<) (3.2) y(O = h(x(O,u(t),0(t)) + v(O where x is the state vector, u is the input vector, y is the output or measurement vector, 0 is the system parameter vector, and w(« ) ~ Af(0, Q(f)), v(t) ~ AT(0, R(i)) (3.3) are system and measurement noise, which are assumed to be white Gaussian with covariances Q and R, respectively. The basic idea is to process and filter raw measurements (which are normally variable and noisy) so that more informative sets of data are generated. Here, the extracted (or estimated) information we are looking for can be either a constant or drifting system coefficient over time (e.g., flow resistances, damping factors, friction coefficients, masses, spring constants, etc.) or some variable states whose variations can be associated with variations of some known or measured state or input (e.g., a valve orifice area which is a function of the valve spool position). We have termed the dependent state "a traceable state". A nearly constant system parameter can usually be modelled as a random-walk process with a zero mean [41 Ljung, 1987]: 0i(t + At) = 0i(t) + Wi(t) (3.4) On the other hand, a traceable state can be obtained from: Xi(t) = g(xj(t)) + Wi(t) (3.5) where g represents a (nonlinear) functional relationship between the traceable state Xj and the "measured" state Xj. For each fault situation (including normal operation), a different set of parameters and traceable states can be obtained by conducting experiments. It is worth noting that in cases where accurate numerical models exist, such as the servovalve model in Chapter 5, less expensive simulation results can also be used in the initial stage to investigate the effects of different faults on the observed patterns of parameter changes. Chapter 3 A HIERARCHICAL FDD METHODOLOGY 44 A parameter estimation scheme would have served the purpose had we had access to all system states and variables in Eq. (3.2). Unfortunately, this is not the case for most hydraulic systems (see the Character CH2 in Section 3.1.1). Hence, we have to resort to nonlinear estimation techniques that not only estimate unknown system parameters and slowly varying faults, but also reconstruct unmeasured system states. For simultaneous state and parameter estimation, an enhanced state vector x , may be obtained by combining the state vector x, and the parameter vector 9t, so that x'(i) = [*(t),9(t)]T (3.6) Since enhanced state models are explicitly nonlinear due to the product of state variables, they require more complicated and computationally demanding techniques such as the celebrated Extended Kalman Filtering (EKF), which has been shown to be quite successful in many applications [21 Grewal and Andrews, 1993]. This technique has extensively been used in this thesis for real-time condition monitoring purposes. The E K F algorithm is given in Appendix A. In the training (learning) mode, the model parameters (and possibly physical coefficients) for normal and faulty situations are obtained by an appropriate state/parameter estimation technique, and their means and variances are stored in a data base for future fault detection. The recovered functional dependencies between the traceable variables and certain states, if any, will also be saved in the data base. Illustrative Example 3.1: Consider a linear, underdamped, second-order system with displacement x(i) (unit length), velocity x(t) (unit length/s), damping ratio C. undamped natural frequency un (rad/s), a driving input u(t) (unit length), and an additive white noise w(t) (unit length) which is normally distributed. This example has been used in [21 Grewal and Andrews, 1993] to show the ability of the EKF in estimating both velocity and damping ratio when the system position is measured and its natural frequency is known. We have adopted and further expanded this example throughout this chapter to illustrate how different modules of the proposed FDD algorithm work together in the case of a simplified model that well approximates a large number of complex structural and mechanical systems; e.g., a spool in a hydraulic valve (see Chapter 5). Section 3. 3 Condition Monitoring by State/Parameter Estimation 45 The equation of motion for this continuous-time dynamic system is: x(t) + 2C,unx(t) + u2nx{t) = Ku(t) + w(t) (3.7) where K is the input gain and w is some system noise. The parameters un and C in Eq. (3.7) can be replaced by a more convenient set of parameters suitable for identification, i.e., 0! = 2(un, 02 = u2n (3.8) It turns out that, in this special case, #i and 02. Since the mass is unity here, these parameters also have physical meanings, namely viscous damping coefficient (Cv) and spring constant (Ks), even if there might be no viscous fluid or mechanical spring present in the actual system. Suppose that the viscous damping coefficient and the spring constant are those parameters that we want to monitor for detecting any abnormality in the system. To obtain a state-space form for the model represented by Eq. (3.7), we use transformation in Eq. (3.6) to enhance the state vector as: [x(t) x(t) 0! 02]T^[xi(t) x2(t) x3(t) x4(t)]T (3.9) Assuming that the only measurements available are the driving input u(t) and the system displacement xi(t), we get: Xi(t) - X2(t) + Wi(t) £ 2 ( 0 = -X2(t)x3(t) - Xi(t)x4(t) + Ku(t) + W2(t) (3.10) Z 3 ( < ) = w3(t) x4(t) = w±(t) with the observation (measurement) equation y(t) = Xl(t) + v(t) (3.11) Note that the system parameters x3(t) and x4(t) are allowed to drift over time because of noise effects in the system. Chapter 3 A HIERARCHICAL FDD METHODOLOGY 4 6 As a numerical example, the dynamic system represented by Eq. (3.7) was simulated for 4 seconds and 400 data points for the system displacement were generated using the information shown in the following table. Method: The discrete nonlinear plant and linear observation x(0) i(0) C K «(*) w(t) R 0 0 0.2 5.0 1.0 0.1 sin (2nt) 0 3.6x 1(T9 equations for the model shown in Eqs. (3.10) and (3.11) were obtained as 22,fc+l = (1 - TsXz,k)^2,k ~ TsXitkX4tk + TsUk + W2,k X4,k+1 = X4}k + W4<k Vk - x1<k + vk using the "forward difference" approximation: (3.12) x(t) (3.13) where k is the time index and Ts is the sampling period (=0.01 s). Note that the plant states x1 and x2 vary due to plant dynamics while the system parameters x3 and x4 vary according to random-walk models. The extended Kalman filtering algorithm given in Appendix A was used to estimate all four system states and parameters, using the initial conditions given below: 100 x Results: The estimation results are plotted in Fig. 3.3, where we note that the parameter estimates have converged to steady state values while the velocity estimate closely tracks the actual (unmeasured) value. We also note that both parameter estimates are biased and the error in the viscous damping estimate x3 is larger than the error in the spring constant estimate x4, as "zi,o~ "0" Z2.0 0 £ 3 , 0 0 .x4,0. 0. "1 0 0 0" "0 0 0 0 " 0 1 0 0 , Qk = 0 0 0 0 0 0 1 0 0 0 0 .0001 0 0 0 0 1 0 0 0 0 .001 Section 3. 4 Fault Detection Using Statistical Decision Making 47 Time (s) Time (s) Fig. 3.3: Estimation results for a second-order system in normal operation, the former is a multiplier of the system velocity, which, by itself, is an unmeasured state. The bias in the parameter estimates is generally due to the discretization error, and can be easily shown to decrease as the sampling frequency is further increased [21 Grewal and Andrews, 1993]. • Once the system parameters (or possibly physical coefficients) and traceable states are estimated in the "training mode" and are stored for normal and faulty situations, they will be re-estimated recursively in the "operating mode" to watch for any abnormal condition. 3.4 Fault Detection Using Statistical Decision Making This computational module has three purposes: 1) generation of test signals by forming residuals from Chapter 3 A HIERARCHICAL FDD METHODOLOGY 48 estimated states/parameters, 2) fault detection using the residuals, and 3) fault diagnosis initiation. It is assumed that the dynamic system is initially running under a known fault (including no-fault) situation and the estimator/filter has reached a steady-state condition. Fault detection is performed first by generating residuals from the processed signals being received continuously from the previous computational module, and then by conducting an on-line statistical test such as "sequential probability ratio test" (SPRT), to see whether any residual is deviated substantially from its normal value. If not, nothing happens and the tests will restart until a deviation in one of the residuals is detected. An alarm will then be raised indicating that a possible fault is pending. The next step is to reset the particular component filter that the change (fault) is detected in, if required. All other component filters which are receiving input signals from the faulty component filter should be reset as well. The resetting is performed in order to obtain new sets of state/parameter estimates for the faulty component(s). Once a statistical test based on SPRT algorithm indicates that the reset filters are again in steady-state condition, the next phase which generates fault symptoms for fault diagnosis purposes will be activated. 3.4.1 Residual Generation First, the change indicating signals in the form of error residuals must be produced from the estimated and the stored quantities. Four sets of residuals may be used at this level, as explained below. The first set consists of the output errors (innovations) of the Kalman filters. From Eq. (3.2), we get e* = y* - Yk (3.15) where e,-^ is a zero-mean white Gaussian random sequence under no fault situation [41 Ljung, 1987]. After an abrupt failure, the innovations mean, variance, or both will change according to a dynamic profile which depends on the type of fault and system model [1 Basseville and Nikiforov, 1993]. Illustrative Example 3.2: For the second-order system in Example 5.1, the innovation Section 3. 4 Fault Detection Using Statistical Decision Making 49 EKF Innovations 0.5 c g } o o o "3 < -0.5 -1 — 1 1 1 1 — r i i i i n „ n _ n flr-i r i m „ _ .-I fl nn _ u ^ ip u u u Lju JJ u u uu y i i i i i i i i i 10 15 20 25 Lag 30 35 40 45 50 Fig. 3.4: Correlation tests for EKF innovations. Horizontal dot-dash lines are 95% confidence limits, sequence obtained from the EKF was tested for whiteness, using autocorrelation functions. The result shown in Fig. 3.4 indicates that the innovations are uncorrelated. • The second set of residuals can be obtained by comparing the current values of the estimated parameters with the previously estimated values, such that (3.16) (3.17) If the parameter is estimated using a Kalman filter, the parameter residual given by A§IJK = Kitk(yk - Vk) = Kiikik may be assumed a zero-mean white Gaussian sequence under no fault situation if the true parameter variation is also a zero-mean random walk process and the Kalman gain, K^, has already reached a steady-state value [41 Ljung, 1987]. After a fault or failure, the mean of the affected parameters) will change, so that: Oi,k = Oi,k-i + A'.fclty + Wi,k (3.18) Chapter 3 A HIERARCHICAL FDD METHODOLOGY 50 where W{ ~ A/"(0,Q,-,,-). For abrupt parameter changes due to sudden failures (e.g., when a contaminant securely lodges in a valve): lt = f 1 */ - 1 - lf + 6 t (3.19) ' 10 otherwise and for parameter drifts due to incipient (or progressing) faults (e.g., flow resistance coefficient in a filter), *' - {o 0 t<tf The estimation algorithm produces A ^ , f c = $itkltj (3.21) after the fault time tf. If the fault is incipient, i.e., fatk has small magnitude and long duration, the estimated drift parameter faik obtained from a Kalman filter can closely follow the actual drift parameter all the time, so that fa>k ~ fa,k- This happens provided that the parameter variation covariance, Q,,, in Eq. (3.3), is appropriately chosen in the filtering algorithm (see Appendix A) so that, while a reasonable signal/noise ratio level is maintained, the estimation error covariance matrix, P in Eq. (A.9), is still large enough to track the actual parameter(s) over time [3 Benveniste, 1987]. On the other hand, if a change in the parameter is abrupt, i.e., fa has considerable magnitude and short duration, and the E K F has already reached a steady-state level, it is possible that the estimation error covariance matrix is so small that it cannot thrust the estimate fa towards the true value fa within a short fault time St or even in any finite period of time [3 Benveniste, 1987]. Nevertheless, as a result of a permanent change in the system structure due to the fault, the current state-space model being used for estimation will no longer be valid, and therefore, some or all of the parameter estimates will start deviating from their current values, giving fa a certain significant change, which should be sufficient for fault detection purposes. Illustrative Example 3.3: For the second-order system in Example 5.1, the residual sequences for the estimated "viscous damping" and "spring constant" parameters from EKF were tested for whiteness, using autocorrelation functions. Two hundred samples for each parameter were used when the EKF became steady-state. The results shown in Fig. 3.5 Section 3. 4 Fault Detection Using Statistical Decision Making 51 indicate that the residuals are all white noise sequences. This was expected since, as we stated earlier, the true parameters were assumed to be constant, and the filter was in the steady-state condition, as is depicted from the estimation error auto-covariances shown in Fig. 3.6. • The third set of residuals suitable for fault detection may be computed by comparing the current estimates of the traceable states with their corresponding normal values, so that we get the state residuals as Aij.fc = Xi,k(xj,k) - x°,k(xj,k) (3-22) where the superscript "o" means normal situation. Because of its association with system dynamics, a state, unlike a parameter, can quickly vary in a short period of time. As a result, the state residual can be a good indication of any sudden or temporary changes in the system model. However, care must be taken in order to avoid false alarms during short-peak transition periods, where the state estimation error is normally higher than expected. The fourth set of residuals, which is particularly suitable for detecting long term gradual faults, may be obtained by, first, averaging the last n samples (n is large) of the estimated system parameters or coefficients, and then comparing the results with their normal values, such that A0;jnfc = 9itnk - ®i (3.23) for system parameters, or AQi,nk = Qi,nk ~ ft ( 3- 2 4) for system physical coefficients. Since the evaluation of this set of residual sequences occurs n times slower than the sampling and estimation process, or A ^ , n f c can well be used for detection of incipient faults or parameter drifts in the system over longer periods of time; for example, an oil filter flow resistance factor. 3.4.2 Fault Detection An alarm will be raised and an incipient or abrupt fault in the system will be declared as soon as a sufficient deviation from zero mean is observed in any of the residual sequences discussed above. Chapter 3 A HIERARCHICAL FDD METHODOLOGY 52 (a) Viscous damping estimation residuals -1 I I I I 1 1 1 I i n 1 I |~1 n f l i - i •-, n nn n n n u U L J L J — ULTyU U U L J U u -I 1 i i i i i i i 0 5 10 15 20 25 30 35 40 45 50 Lag (b) Spring constant estimation residuals n 1 r i r i r 0.5 c o a> 1 o o o -*—* < 0 , n n , i ~ L I I I* *i i — ^ — ' — • * *•—•• ii i* * « — ^ i i i — i i — i ^ — a — i ii ii ii n uuu^ y u [ T ^ L J 0 - ^ = y u u i J u -0.5 -1 j i i i _ j i _ 5 10 15 20 25 30 35 40 45 50 Lag Fig. 3.5: Autocorrelation functions to test viscous damping and spring constant residuals. Horizontal dot-dash lines are 95% confidence limits. Section 3. 4 Fault Detection Using Statistical Decision Making 53 x 10" (a) P(1,1) (c) P(3,3) 0 x 10 -5 2 Time (s) (b) P(2,2) 0 2 Time (s) (d) P(4,4) 1 2 3 4 0 1 Time (s) Fig. 3.6: Estimation error auto-covariances of the states and parameters. 2 3 Time (s) Detection of incipient faults in the form of gradual drifts can be performed simply by limit checking of the parameters, using 0,-injt or Aginle. For detection of abrupt failures, however, more sophisticated methods are required. A number of techniques are available to decide on a sufficient change in a residual sequence due to a fault. Heuristic approaches based on neural networks [72 Wang et al., 1994] or fuzzy logic [60 Schneider and Frank, 1996] have been used wherever statistical distribution of the residuals is not known, a good analytical model is not at hand, or human factors and heuristic knowledge play an important role in decision making on faults. On the other hand, statistical decision making based on Bayesian theory [29 Isermann, 1989], sequential Chapter 3 A HIERARCHICAL FDD METHODOLOGY 54 probability ratio test (SPRT) [86 Zhang, 1989], and generalized likelihood ratio (GLR) test [1 Basseville and Nikiforov, 1993] have widely been used in the literature to test between no-fault and fault hypotheses, whenever good statistical models for the residual processes have been available. Statistical techniques have the maximum efficiency if the residual sequences are independent and identically distributed (i.i.d.) [1 Basseville and Nikiforov, 1993]. Among decision making techniques, the SPRT algorithm has very interesting features that make it ideal for on-line detection of abrupt changes. For example, SPRT can be represented by a simple recursive formulation. It is also well known that the SPRT is optimal in deciding between two alternative hypotheses, in the sense that it yields the fastest average detection time of all tests with the same power [71 Wald, 1947]. In this section we give a brief description of the Wald's sequential probability ratio test for deciding between two hypotheses. Let us consider a sequence of random variables y^ (k = 1, .... n) with a probability density function Pfi(y) depending on only one scalar parameter, fi. Using SPRT, our purpose is to test between two simple hypotheses: The probability of observing y/ to y„ under the hypothesis i (/ = 0, 1) is given by the likelihood function: k=2 Then, at time t when n samples are collected, the log-likelihood ratio between the two probabilities of H Q and H\ can be computed from (3.25) n Hyi,y2,---,yn;iii) = Pni(yi)Y[Pni(yk I iri,•••>»*-!) (3.26) (3.27) The log-likelihood ratio can equivalently be written in a recursive form as Afc = Afc_i + In PnoiVk | 2/1, - - - , (3.28) Section 3. 4 Fault Detection Using Statistical Decision Making 55 provided that the initial condition for Ao is known (usually taken as zero). When the random samples yk are independent, we obtain A f c = A f c _ 1 + l n ^ ^ (3.29) It may intuitively be realized from Eq. (3.29) that, on the average, the log-likelihood ratio shows a negative drift before change and a positive drift after change. As an example, consider the test of the mean of a random independent sequence having a Gaussian distribution with a known constant variance, a2. Using Eq. (3.29), it can be easily shown [1 Basseville and Nikiforov, 1993] that the SPRT formula becomes: Xk = Afc_i + — — \yk 1 (3.31) From Eq. (3.31), we see that if fii > fio and E(yk) > (g(.) is expectation), on the average Xk will increase, and if E(yk) < £ i±£a. , on the average Xk will decrease. Given two constant thresholds Bt < 0 and Bu > 0, the Wald's SPRT algorithm may finally be written as the following procedure: i. Set k = 0 and Xk = 0. ii. Set k = k + 1. iii. Compute Aj. from Eq. (3.28). iv. If Xk > Bu, terminate the observation with the acceptance of H i . v. If Xk < Bu terminate the observation with the acceptance of Ho. vi. Otherwise go to Step ii without any decision. The constant thresholds Bi and Bu are determined by the prescribed test strength (a\, a2), such that [71 Wald, 1947]: 0 1 1 (3.32) B , > l n - ^ -1 — «*i Chapter 3 A HIERARCHICAL FDD METHODOLOGY 56 where « i is the probability of rejecting Ho when Ho is true (false alarm rate), 02 is the probability of accepting Ho when Hi is true (missed alarm rate). Illustrative Example 3.4: Suppose that the spring factor in the second-order system of Example 5.1 is a random-walk process and has undergone a permanent change due to a fault in the system, as shown in Fig. 3.7a. Method: Using Eq. (3.16), spring constant residuals are calculated and plotted in Fig. 3.7b. The log-likelihood ratio for these residuals are obtained from Eq. (3.31) and plotted in Fig. 3.7c, where the upper and lower thresholds are set to 6.9 and zero, respectively. Observations: From the figure, it can be realized that during transition period, the residuals deviate from zero mean, and as a result of positive deviation, \ t increases and passes the upper threshold after about 7 samples from the estimated fault time. It is also interesting to note that after the parameter (spring constant) has permanently changed and the residual mean has returned to a nearly zero value, the magnitude of the log-likelihood ratio has also decreased and hit the lower threshold, indicating that the parameter variation has become a stationary process again. • A few modifications to the original Wald's SPRT algorithm have been made [IBasseville and Nikiforov, 1993] to obtain a more appropriate and efficient algorithm for fault detection purposes. They are summarized as follows: • The key idea is to restart the above SPRT algorithm as long as the previously taken decision is Ho in Step v; i.e., to start a new SPRT as soon as the decision function A crosses the lower threshold. This is because we are primarily interested in detecting system changes from normal to faulty condition. Therefore, the first time at which the decision becomes H i , we stop observation at Step iv above and do not restart a new cycle of the SPRT. This time is then the alarm time, ta, at which the change (fault) has been detected. Section 3. 4 Fault Detection Using Statistical Decision Making 57 (a) Permanent change in parameter due to a fault 0.2 h 0.2 0.4 0.6 0.8 1 (b) Temporary change in parameter residuals due to a fault 0.2 0.4 0.6 0.8 1 (c) SPRT results based on parameter residuals I I I upper threshold 1 ' 1 '< detection time 0.2 0.4 0.6 Time (s) 0.8 1.2 Fig. 3.7: Typical behavior of the log-likelihood ratio corresponding to a change in the mean of a Gaussian sequence with constant variance. Vertical solid bar is the SPRT estimated fault time. It has been shown that the optimal value of the lower threshold for a repetitive SPRT is zero [43 Lorden, 1971]. As a result of taking the lower threshold B\ = 0, we restart the SPRT at Step v whenever A* < 0. Since a change in parameter po due to a fault can be in either positive or negative direction, it is relevant to use two simultaneous SPRT formulae, X+ and ; the first one for detecting Chapter 3 A HIERARCHICAL FDD METHODOLOGY 58 an increase in the parameter, and the second one for detecting a decrease in the parameter. For example, to detect a change in the mean of a random Gaussian sequence, //o, Eq (3.31) should be duplicated for X+ and AjT with p+ = po + 6 and p7 = po - respectively. • If only the change in the parameter is desired, no matter what the direction of change is, the following decision based on a two-sided SPRT can be used: Xk = max (A^, AjJ") (3.33) • Once Hi is accepted when either of the two positive or negative SPRT passes the positive threshold, the stopping rule and the alarm time, ta, will be obtained from: ta = T s m i n {k : (A+ > Bu) U (A" > Bu)} (3.34) where "U" means OR. Accordingly, the estimated fault time, if, is the last time that the alarming log-likelihood ratio, A^ or A^, has crossed the zero threshold. The estimated time delay for fault detection can then be calculated from: id = ta- if (3.35) • Although the magnitude of the parameter po before change is well known (e.g., using on-line estimation algorithms) in most practical cases, little is known about the magnitude of the parameter after change, or equivalently 8. However, three possible a priori choices can be made for using SPRT algorithm in this case [1 Basseville and Nikiforov, 1993]. The best thing to do is choosing 8 as the most likely magnitude jump. An a posteriori estimation of S leads to the optimal General Likelihood Ratio (GLR) algorithm, which is a computationally demanding off-line technique, and therefore, is not appropriate for real-time applications. Another choice for S is a kind of "worst-case" value with the cost of increasing undetected changes. The third option is choosing 8 as a minimum possible jump which is allowed to detect. This is good when we do not want to miss any alarm at the fault detection level. However, we should note that, in all the three cases, the resulting SPRT algorithm is optimal for only one possible jump magnitude exactly equal to 8 [1 Basseville and Nikiforov, 1993]. Section 3. 4 Fault Detection Using Statistical Decision Making 59 3.4.3 Fault Diagnosis Initialization Process Fault diagnosis follows fault detection provided that the fault symptoms can be generated using information on parameter change magnitudes. Detection of a change in process residuals, however, means that the system model under normal operation is no longer valid (at least for the current set of parameters) and that the current model should be replaced by a model with a new set of parameter values. If the parameter estimation process is to be performed on-line, a time delay is normally required before a new valid set of parameter values becomes available (see, e.g., the time lag between the estimated change time and the alarm time in Fig. 3.7a). The time delay is due to one of the two following reasons: 1. Parameter estimation algorithms, in general, have poor tracking capabilities during rapid transition periods from one parameter set to another [3 Benveniste, 1987] (see also Fig. 3.7a). This primarily stems from the fact that a "parameter", by definition, is supposed to be nearly constant over time. Sometimes, an exponential forgetting factor is employed to improve tracking capabilities of the identification algorithms. This has been shown to be equivalent to increasing the system noise covariance, Q, in the Kalman filter algorithm [42 Ljung and Gunnarsson, 1990]. This ensures that the estimation error covariance, P in Eq. (A.9), and steady-state Kalman gain, K in Eq. (A. 10), remain large enough to minimize estimation error during transitions. A large value for Q, however, means a poor signal/noise ratio for the estimated parameters. 2. The filter/estimator of the faulty component is reset after a change is detected but the filter gain and estimation error covariance require time to converge to their steady-state values. As was discussed in the previous section, resetting of P is required if the steady-state Kalman gain, K, is so small that it basically cannot generate new parameter estimates close to the actual values within a short period of time. In every case, an automatic mechanism should be devised in order to inform the symptom generation module as to when the new set of parameter values is valid for classification purposes. If the filter gain is high enough and the filter is not required to be reset (case 1 above), the same two-sided SPRT (that first detected a probable change in a residual sequence) is suggested to be continued until both positive Chapter 3 A HIERARCHICAL FDD METHODOLOGY 60 and negative log-likelihood ratios cross the lower bound (i.e., zero line) for the first time after change detection. Then, the SPRT will be terminated with the acceptance of the null hypothesis, Ho. This means that the parameter estimates are statistically at steady-state levels once again, and that the symptom generation process can now be initialized. The suggested method, which is an original contribution, is graphically illustrated in Fig. 3.7b and Fig. 3.7c, with the spring constant residual sequence being used as the test signal. If, after a fault is detected, the Kalman filter has to be reset due to a low magnitude of the Kalman gain (case 2 above), a two-sided version of the original Wald's SPRT algorithm (see Section 3.4.2) is suggested to be applied over the parameter estimate residuals A^ j t defined in Eq. (3.16). Using the log-likelihood ratio in Eq. (3.28), it can be easily shown that change detection in a parameter modelled as a random-walk process, i.e., Eq. (3.18), is equivalent to testing change in the mean of a random Gaussian sequence from 0 to f3 with known constant variance a2. In this case, y* in Eq. (3.31) should be replaced by the residual sequence A9itk given in Eq. (3.16). Noting that /?,• determines the drift in a parameter 0„ and that if E^AO^k) > P™/2 o r if E^AO^k) < -{3™/2 hypothesis Hi will be accepted, we continuously test whether or not the drift estimate A0 t | / t , on the average, exceeds prescribed levels, f3™ or — u s i n g a two-sided version of the original SPRT algorithm. We stop the test once the two-sided SPRT corresponding to the alarming parameter residual accepts the null hypothesis, Ho, and start generating fault symptoms corresponding to a specific fault with the new sets of estimated parameters and traceable states. 3.5 Fault Symptom Generation Fault diagnosis is based on observed symptoms which are usually expressed in terms of such quantities as pressure levels or flow rates. In knowledge-based expert systems, it is normally preferred to represent the deep knowledge about faults in qualitative terms such as "low", "high", etc., rather than with mathematical equations or quantities [26 Hogan et al, 1996]. This is mainly because qualitative reasoning is much Section 3. 5 Fault Symptom Generation 61 more understandable by human analysts. In [4 Bull et al., 1996], for example, qualitative reasoning has been achieved by discretizing quantitate fault symptoms (pressure levels, flow rates, etc.) into ranges between "too small", and "too high" using mathematical relations such as " « " , " » " , etc. Motivated by the foregoing qualitative reasoning approach, we have adopted a similar idea in this thesis in order to produce a fault symptom binary code according to a specific change pattern observed in the parameter/state residuals. For this reason, each parameter value is discretized into levels represented by a particular binary code. For example, the code {-1, -1, +1, -1, -1} might represent a qualitative class "no change" in a parameter value. The binary codes thus produced for each parameter are then combined to represent a specific fault signature. It should be noted that the "fault symptom generation" module in the hierarchy becomes active if only: i) a fault is detected by the "fault detection" module, and ii) it is found that the estimation filters are in steady-state condition. 3.5.1 Parameter Jump Level Classification All possible levels that a parameter, or a traceable state, XJ, might jump after a fault should be defined first. These levels may be' discretized into j distinct classes in a descending order. Suppose that we assign 5 jump levels (j = 5) for every parameter, or traceable state, and assign the following classes to these jump levels: Jump Level JL(/) -2 -1 0 1 2 Jump Class JC(0 Highly Decreased Decreased No Change Increased Highly Increased (HD) <P) (jvo (/) (HI) To improve diagnostic resolution or diagnostic power when parameter jumps for two or more faults are so close to each other that they produce the same fault codes, we may increase the number of jump levels for more sensitive parameters. Each class is then identified by a central or mean value: fi(HI), Chapter 3 A HIERARCHICAL FDD METHODOLOGY 62 fi(I), fi(NC), n(D), or fi(HD), and a confidence interval: C(HI), C(I), C(NC), C(D), or C(HD). If the average magnitudes of jumps for each parameter are known for some or all faults, they can be readily assigned as centers of the jump levels. 3.5.2 Residual Generation Each estimated parameter or traceable state is subtracted from every possible jump center, i.e., Qi,k(J) = Qi,k ~ A*?U) (3.36) where j refers to a jump class (e.g., HI, I, etc.). To reject noise effects, we might use an average of the last n estimated quantities in Eq. (3.36). This, however, means that the fault symptoms are generated and fault diagnosis is performed on-line but at a slower rate than the speed of the real-time control loop or the fault detection computational module. 3.5.3 Parameter Jump Level Assignment To determine which jump level (or class) a current parameter estimate should assume after a change is detected, we should check whether the magnitude of the parameter change has confidently fallen within the limits of the corresponding jump class. A "+1" will be assigned to that particular class if the answer is positive and a "-1" shall be used otherwise. In mathematical terms: Class.Value^) = { + 1 i f " C M * (<» *.•.*(*)) < W ) ( 3.37) (—1 otherwise The above classification has one major drawback: if a parameter value lies between two successive jump levels, it may be classified as "-1" for all jump classes, resulting an unknown state. This problem has been resolved using a combinatorial logic between the two successive levels that the parameter value is in between. The jump level assignment algorithm is given below: 1. Initialize: Class. Value,-(/) = -1 for every jump class j. 2. Start from the highest level of jump in positive direction; e.g., j = 2 (i.e., HI). Section 3. 5 Fault Symptom Generation 63 3. If Qi(j) > -C(j), assign: JL(7) = j, and Class. Value,(/) = +1, and exit the algorithm. 4. Go one class below; i.e., set: j = j - 1. 5. If -C(j) < Qi(j) < C(j), assign: JL(/) = ;', and Class.Value,(/) = +1, and exit the algorithm. 6. If Qi(j) > C(j), assign: JL(7) = j + 0.5, and Class. Value, (/') = +1, and C l a s s . V a l u e , = +1, and exit the algorithm. 7. If the current class is not in the lowest level, go to step 4. 8. If Qi(j) < C(j), assign: JL(/) = j, and Class. Value, (/") = +1, and exit the algorithm. Regarding the above classification algorithm, a few comments are in order: • It is now possible that a parameter value be classified in between two jump levels with both class values equal to +1. In this case, it indirectly refers to a new subclass assigned to the parameter value. For example, if the parameter / value is something between classes / (JL(/) = 1) and HI (JL(i) = 2), it will be classified in a new subclass named "medium high" or mh (JL(/) = 1.5). Thus, any class name with capital letters indicates a major class and any class name with small letters indicate a minor class. Accordingly, any integer jump level indicates a major jump level and any jump level with ".5" digits is a minor jump level. • The parameter value must be in one and only one class at a time. If the class is major, only the same class value will become +1. Once the class is minor, the two surrounding major class values will become +1. • There are no more than two major classes with values equal to +1. • There is no subclass higher or lower than the upper and lower major classes. NOTE: In case when the only change in a parameter value from normal condition is considered, the above algorithm may be replaced by a single binary code for each parameter, using Eq. (3.37), so that a means a change and a means no change. Illustrative Example 3.5 Let's consider fault detection and classification problem in the second-order system of Example 5.1. Five different fault modes have been listed in table below: Chapter 3 A HIERARCHICAL FDD METHODOLOGY 64 Fault Mode Fault Time Cause Effect 0 — normal operation no change in parameters 1 tf = 2.39 s spring strain hardening Ks - (1 + x2) 2 — increasing damping Cv Cc (1 + a t) 3 tf = 3.0 s increased input gain Ku - 1.1 Ku 4 tf = 3.0 s sensor drift y,^yt + 0.0002 t To monitor the system and detect the above fault modes, the procedures proposed in the hierarchical FDD methodology have been adopted consecutively and the results are as follows: 1. Estimation Results: For Fault 0, we have already given the results from the estimation process in Fig. 3.3. For the other fault types, the estimation results are plotted in Figs. 3.8 to 3.11. Fault 1 in Fig. 3.8 and Fault 2 in Fig. 3.9 are "plant" faults and have only affected the relevant parameters. Fault 3 in Fig. 3.10 is "input" (sometimes called "actuator") fault, and since the input gain is not estimated, it has affected the system parameter estimates in the form of a constant bias. Fault 4 in Fig. 3.9 is "sensor" fault, and has caused a correlation between the system parameter estimates with the input signals. 2. Fault Detection: Method — Detection of incipient or progressing faults has been achieved by limit checking of "long-term" parameter drift estimates, i.e., monitoring A8I<NK in Eq. (3.23) with n = 50 samples. To detect abrupt failures in the system, we have constantly monitored two types of residuals using two-sided SPRT's. These residuals are: 1) Kalman filter innovations, £ k in Eq. (3.15), and 2) "short-term" parameter drift estimates, A9I<K in Eq. (3.16). To implement the sequential tests, we have used the following information: Parameters I Residual Ao* Ml a Ot-2 Bj £ 0.0 ±a 0.00007 0.1% 0.1% 6.9 Cv: i = 1 AO 0.0 ±a 0.026 0.1% 0.1% 6.9 Ks: i = 2 0.0 ±a 0.085 0.1% 0.1% 6.9 * Estimated from sample data. t Calculated from Eq. (3.32). Section 3. 5 Fault Symptom Generation 65 x 10 (a)X1 (c) X3 2 4 6 Time (s) .g 3 Q. E CO ° 2 CO o CJ 31 4 Time (s) (d) X4 40 5 30 CO to c O 20 at _c c | l0 0 2 4 6 Time (s) Fig. 3.8: Estimation results (—) and true values (- -) for the state vector x, due to "Fault 1". Fig. 3.9: Estimation results (—) and true values (- -) for the state vector x, due to "Fault 2". Chapter 3 A HIERARCHICAL FDD METHODOLOGY 66 (c) X3 0 2 4 6 8 Time (s) (b)X2 .£ 3 Q. E CO Q 2 CO o o £ 1 2 4 6 Time (s) Fig. 3.10: Estimation results (—) and true values (- -) for the state vector x, due to "Fault 3". x 10 (a) X1 (c) X3 4 6 Time (s) 4 6 Time (s) (b)X2 (d) X4 Fig. 3.11: Estimation results (—) and true values (- -) for the state vector x, due to "Fault 4". Section 3. 5 Fault Symptom Generation 67 Please note that for Fault 4, in order to have a finite "waiting time" before re-sampling new parameter estimates, we have assigned a second threshold level about 3 to 4 times greater than the first level threshold in the above table. Once the innovations' log-likelihood ratio crosses the second threshold, a second alarm is generated indicating that the estimation is diverging. Fault Detection: Results — Figs. 3.12a to 3.15a show two-sided SPRT results for the four fault modes. It is clear from the figures that all abrupt failures (i.e., Fault 1, Fault 3, and Fault 4) were successfully detected within a short period of time, using the assigned thresholds for 0.1% false alarm rate and 0.1% missed-alarm rate. Regarding the sensor Fault 4 in Fig. 3.15a, the SPRT on the Kalman filter innovations (dashed lines) did cross the lower bound (the zero line) after the fault, indicating that the measured and the estimated output did not match after the sensor has became faulty. Another important point to note is that the SPRTs, in general, have been unable to detect the incipient Fault 2 due to a gradual drift of the viscous damping coefficient. Figs. 3.16a,b to 3.19a,b, on the other hand, show the trends of the "long-term" parameter drift estimates, A6itnk. In Fig. 3.17a, for example, the drifting magnitude of viscous damping coefficient was tracked for a long period of time before it reached a critical level (Fault 2), which then raised an alarm. 3. Parameter Value Discretization: Method — First, the right time for re-sampling the parameter estimates after change should be identified. Using the same SPRTs and the method mentioned in Section 3.4.3, these appropriate times have been determined and sketched as vertical dash-dot lines in Figs. 3.12 to 3.15. Next, parameter jump levels must be determined. We have arbitrarily assigned five jump levels corresponding to ±10% and ±20% of the normal parameter values, each with confidence intervals equal to ±3<3-;. These are design parameters that depend on designer's concern about the severity of fault, type of fault, and desired diagnostic resolution. The numerical values for parameter jump levels and their confidence intervals are listed in Table 3.1. Chapter 3 A HIERARCHICAL FDD METHODOLOGY 68 (a) SPRT results 1 1 1, 1 ih ll 1 1 \ 1 \ !'«I 1 I \ 1 i i i i k 20 h 15 CO •a § 10 6 2 3 4 5 Time (s) (b) Parameter jump level assignment: JL(1) = 0, JL(2) = 1 " i — r 2 1 0 -2 > CD _J a. e T ] p ~ T " -1 L I » "'if Tir?r 0 3 4 5 Time (s) Fig. 3.12: Fault mode 1. (a): SPRT fault detection, (b): parameter classification. (a) SPRT results 3 4 5 Time (s) (b) Parameter jump level assignment: JL(1) = 0, JL(2) = 0 > CD 1 o E I D C 3 4 5 Time (s) Fig. 3.13: Fault mode 2. (a): SPRT fault detection, (b): parameter classification. Section 3. 5 Fault Symptom Generation 69 (a) SPRT results 2 3 4 5 Time (s) (b) Parameter jump level assignment: JL(1) = -1, JL(2) = 0.5 -i 1 r-Q. 0 E 3 J 0 1 2 3 4 5 6 7 8 Time (s) Fig. 3.14: Fault mode 3. (a): SPRT fault detection, (b): parameter classification. (a) SPRT results 100 80 60 40 20 0 1 1 1 1 1 ! 1 ( 1 1 - ! i / /-' ' ' A 1 \ r\ 1 i ' \\' S\ t 1. I . - I A • \ h / i • ^ r\ r \ n >\ Ii 1 J _ i 1 ' / Y v \ 1 /' \ i \ 1 " \< I A I A P 1 \ ...v.* 4 ,U V v . / i \ /l . II W 7 i / ' V I . . \ I •• > - ;, Al-w V i v > it^W i l , , 2 3 4 5 6 Time (s) (b) Parameter jump level assignment: JL(1) = -2, JL(2) = 0 1 1 1 1—i 1 > CO •L o E 3 V'— I 3 4 5 Time (s) Fig. 3.15: Fault mode 4. (a): SPRT fault detection, (b): parameter classification. Chapter 3 A HIERARCHICAL FDD METHODOLOGY 70 (a) S o l i d c u r v e : X 3 o: X 3 b a r ( 5 0 ) ' i ' 1 ' ' ' i i n/\ v ^ f^n. /A i fJ^ ft f\ Ah.* An\ I I I 1 1 1 1 I I I I I I I 1 I I 0 1 2 3 4 5 6 7 8 (b) S o l i d c u r v e : X 4 o: X 4 b a r ( 5 0 ) T i m e (s) Fig. 3.16: Detection of fault mode 1 by monitoring "long-term" drift in (a): viscous damping coefficient, and (b): spring constant, (c): Fault classification using "long-term" parameter drift threshold checking. Fig. 3.17: Detection of fault mode 2 by monitoring "long-term" drift in (a): viscous damping coefficient, and (b): spring constant, (c): Fault classification using "long-term" parameter drift threshold checking. Section 3. 5 Fault Symptom Generation 71 (a) S o l i d c u r v e : X 3 o : X 3 b a r ( 5 0 ) | 2 . 5 8 > 2 — 3 0 O 2 5 2 0 2 9 1 CD ^ . o £ - 2 2 3 4 (b) S o l i d c u r v e : X 4 5 6 o : X 4 b a r ( 5 0 ) 2 3 4 (c) : V i s c o u s D a m p i n g 5 6 - : S p r i n g C o n s t a n t 4 T i m e (s) 1 1 1 1 1 1 1 _ . _ . _ — . — 1 —1 1 ' " 1 '-1 Fig. 3.18: Detection of fault mode 3 by monitoring "long-term" drift in (a): viscous damping coefficient, and (b): spring constant, (c): Fault classification using "long-term" parameter drift threshold checking. Fig. 3.19: Detection of fault mode 4 by monitoring "long-term" drift in (a): viscous damping coefficient, and (b): spring constant, (c): Fault classification using "long-term" parameter drift threshold checking. Chapter 3 A HIERARCHICAL FDD METHODOLOGY 72 Table 3.1 Numerical values for parameter jump levels and their confidence intervals Parameter p(HD) H(NC) CO) Cv: i=l 1.918 2.157 2.397 2.637 2.876 ±0.151 Ks: i=2 20.31 22.84 25.38 27.92 30.46 ±0.45 In Figs. 3.12b to 3.15b, the jump levels (classes) for each system parameter are then assigned using the classification algorithm described earlier in this section. Parameter Value Discretization: Results — Figs. 3.16c to 3.19c illustrate classification results for "long-term" parameter drifts using the same jump levels with confidence intervals equal to ±3<5-t obtained for sample number n = 50. For all abrupt failures, the SPRTs have apparently detected the faults sooner than the "long-term" threshold checking. The gradual drift in the viscous damping coefficient, however, is more obvious in Fig. 3.17c than in Fig. 3.13b, indicating the better capability of the "long-term" threshold checking algorithm compared to the SPRT algorithm in detecting progressing faults. 4. Fault Symptom Generation Results: Finally, fault symptoms in the form of binary codes are generated by combining discretized parameter values. The results are summarized in Table 3.2. For each fault, a different pattern according to each specific parameter jump level is obtained. Each one of these fault patterns, once observed in real-time from the process, may be passed to a pattern classifier for fault event recognition, or be directly used by a knowledge-based expert system for qualitative reasoning. • Table 3.2 Fault symptom codes generated for five different faults in a second-order system. fault mode jump class for Damping Coefficient (Cv) jump class for Spring Constant (Ks) Flags HD D NC / HI HD D NC / HI SA PD F0 -1 -1 +1 -1 -1 -1 -1 +1 -1 -1 -1 -1 F l -1 -1 +1 -1 -1 -1 -1 -1 +1 -1 -1 -1 F2 -1 -1 -1 +1 -1 -1 -1 +1 -1 -1 -1 +1 F3 -1 +1 -1 -1 -1 -1 -1 +1 +1 -1 -1 -1 F4 +1 -1 -1 -1 -1 -1 -1 +1 -1 -1 +1 -1 SA: second threshold is crossed and the secondary alarm is on. PD: detection by "long-term" parameter drift algorithm. Section 3. 6 Fault Recognition Using Neural Network Pattern Classification 73 From Table 3.2, we realize that for each fault type, a binary code consisting of twelve -1 or +1 digits is generated. It is interesting to note that although 2 1 2 = 4096 different patterns can theoretically exist using all possible combinations, at most only valid fault patterns is allowed to be generated in practice, assuming that both SA and PD flags cannot coexist. A general formula for the maximum number of patterns, Pmax, for a system with i monitored parameters (and traceable states) and j jump levels can be easily derived as: 3.6 Fault Recognition Using Neural Network Pattern Classification In the present approach, diagnostic problem solving has been conceptualized as the association or mapping of patterns of binary input data representing the "qualitative" behavior of the physical process to corresponding fault conditions. Binary inputs, as compared to real numbers, have the advantage that there is always a finite number of patterns involved with fault modes, and mapping is accomplished on a closed hyperplane. Association or connection of observed binary symptom patterns to related fault events can then be performed in several ways (see Chapter 2 for a brief review). Statistical pattern recognition is one approach that has been used for fault classification and diagnosis. They characterize fault classes by their probability density functions (PDFs) on the input features and use Bayes's decision theory to form decision regions from these densities. Application to diagnostic problem solving consists of representing the network components, which are to be identified in the case of a fault, as "a priori", "joint", and "class-conditional" probabilities, respectively [13 Duda and Hart, 1973]. These approaches are effective, but tend to be complex and cumbersome in situations involving large training sets, and where a priori information about the underlying process is not well-understood [19 Gallant, 1993]. Particularly, in technological system applications, the symptoms are not guaranteed to 3 x [5 + (5 - l)] 2 = 243 3(2j - iy if j > i 3(2)' if i = 1 (3.39) Chapter 3 A HIERARCHICAL FDD METHODOLOGY 74 be independent, and also a priori probabilities of faults and symptoms are not readily available. Hence, the Bayesian inference approach may not be a convenient method for fault diagnosis. Knowledge-based expert system approach, on the other hand, tries to match symptom patterns to certain fault classes using production rules. The number of these rules, however, tends to increase exponentially as the number of symptoms and fault modes increases, or there is noise and redundancy in the input data. For example, imagine that all 243 fault patterns in example 5.5 have been recognized and should be related, via some production rules, to several fault modes in the dynamic system! A major attraction of neural networks for fault identification is that they can serve as knowledge bases for classification expert systems. Most importantly, learning algorithms allow us to generate knowledge bases (rules) automatically from training examples. Neural network pattern classifiers also have potential features of statistical pattern recognition methods, and since NNPCs solely operate on (possibly) training examples, they can overcome limitations of statistical methods to some great extent. Neural network pattern classifiers have the ability to learn and store information about fault occurrence in the physical system via associative memory. This is very similar to what a human expert infers, almost instantly, of a particular fault as soon as a pattern of incidents related to the fault are realized. However, explanation, justification of the fault and its causes and remedies cannot be provided by an NNPC, and integration with an expert system for better explanation facility is usually desired. Among many different NNPCs proposed for fault recognition using binary input-output data, feedforward single-layer networks with perceptron cells are probably the simplest NNPCs. These networks are usually trained using one of the so-called "fast-learning" algorithms (as compared with, e.g., "slow-learning" backpropagation algorithm). However, it is well-known that single-layer networks with perceptron learning are unable to solve such "nonseparable" problems as "XOR" (exclusive OR). It has been shown that a "linear machine", also called winner-take-all groups (WTAGs), can overcome this problem while keeping the simple single-layer structure of the network with a "fast-learning" algorithm [19 Gallant, 1993]. A schematic diagram for such a neural network to solve the "XOR" problem is shown in Fig. Section 3. 6 Fault Recognition Using Neural Network Pattern Classification 75 ClassificationResult Inputs Fig. 3.20: A winner-take-all group of cells for solving the "XOR" problem [19 Gallant, 1993]. 3.20. WTAGs generalize single-cell models and are particularly useful for pattern recognition problems where binary inputs must be placed in one of several binary-coded classes [19 Gallant, 1993]. To automatically train WTAGs, a specialized "fast-learning" algorithm named pocket algorithm has been given in [19 Gallant, 1993], which is reproduced in Appendix B. In brief, the pocket algorithm for WTAGs first runs the perceptron learning algorithm and takes its correct classifications into account by keeping a separate set of weights, W"*^', for all output cells in a "pocket", along with the number of consecutive iterations for which Wocket correctly classified the chosen training examples, E . It also keeps a single run length for the entire group of cell weights in the "pocket". Now, whenever the current perceptron weights, W, have a longer run of correct responses than the run length for the pocket weights, we replace all of the weights in the pocket by the current weights for winner-take-all group cells and update the pocket run length [19 Gallant, 1993]. It has been proven that the pocket algorithm used for training WTAGs converges to a set of weights that correctly classifies a randomly chosen training example with maximum likelihood [19 Gallant, 1993]. Therefore, noisy training examples are handled in a natural way. As a matter of fact, WTAGs provide an estimate of the optimal Bayesian decision function as the number of (possible noisy) training examples Chapter 3 A HIERARCHICAL FDD METHODOLOGY 76 tend to infinity [19 Gallant, 1993]. The performance of a W T A G neural network, however, depends on the number and the quality of the training data presented to it. In general, WTAG's have a good capacity of the following merits [19 Gallant, 1993]: • good generalization to unseen data, • easy training scheme, • fast training speed, • ease of derivation of analytical bounds on performance, • robustness to noisy inputs and redundant data, • capability for handling relative importance and frequency of occurrence for each fault, i.e., fault statistics, • capability for handling large problem sets, The last three merits are particularly important when considering a rule-based expert system as an alternative way for identifying faults in technological plants. In particular, it is difficult, if not impossible, for an IF-THEN rule to take into account the relative importance of a fault and the frequency of its occurrence, i.e., fault statistics. For example, if the importance of diagnosing a lubrication problem in the second-order system of Example 5.1 is raised high enough, the FDD system should always conclude that such a problem exists. The corollary is that even if all signals are normal, there is a small chance that there is a lubrication problem and all readings are noisy. This situation can be automatically handled in WTAGs by proportionally increasing the number of training examples related to more important or more frequent faults. Capability for handling large problem sets is another important merit of WTAGs when there is not enough information regarding all possible fault modes in the physical system. In fact, it is possible that such a neural network is trained independent of the actual process, and purely based on (noisy) "simulation" examples, provided that a behavioral (as opposed to quantitative) model of the faulty process is available. For example, if from physical laws and experience we already know to what extent each system parameter (i.e., network input) can be affected due to a particular fault, we can generate the fault symptom code Section 3. 7 Fault Diagnosis Using a Rule-Based Expert System 11 accordingly, without resorting to the algorithms mentioned in the previous sections. We may then train the network for that particular fault. Noise effects in input data sets can also be considered by changing the sign of each individual binary input according to its noise probability. Illustrative Example 3.6: Five hundred input-output training example sets {(Cc and Ks), F/} for all fault modes were generated for the first 100 samples after each fault was detected in Example 5.5 (see Figs. 3.12b to 3.15b). A WTAG neural network with 12 input cells (corresponding to the 12-digit binary code generated in Table 3.2), 1 bias cell, and 5 output cells (corresponding to the five fault modes) was then trained using the training examples. For each iteration, one training example set was randomly picked and the weights were adjusted for 30000 iterations according to the pocket algorithm given in Appendix B. Then, the data from Table 3.2 were presented to the network, and all faults were correctly identified. In fact, the neural network correctly classified all the training examples presented to it. • 3.7 Fault Diagnosis Using a Rule-Based Expert System As was explained in Chapter 2, automated fault diagnosis in hydraulic circuits using knowledge-based expert systems has been the subject of an extensive research since a decade ago, and very promising results have already been reported [5 Burrows et al, 1989][25 Hogan et al., 1992][75 Watton, 1992][76 Watton, 1994][26 Hogan et al, 1996][4 Bull et al., 1996]. We also mentioned that one of the major problems encountered by such expert systems is that the fault symptoms being used are mostly based on direct measurement or observation of such system states as pressures and flow rates, which are noisy and highly variable during transients or due to dynamic effects. The symptom generation mechanism that we proposed in this chapter, in contrast, is much more robust and self-explanatory since it is based on system physical parameters obtained by processing raw sensor signals. The qualitative changes in such variables as pressures and flow rates may then be replaced in a fault diagnosis expert system by the qualitative changes in physical parameter values, as was shown Chapter 3 A HIERARCHICAL FDD METHODOLOGY 78 in Example 5.5. A better option for such expert systems is to utilize a neural network's fault classification results rather than directly use fault symptoms for fault diagnosis. Fault event recognition by a neural network generates a hypothesis about the most probable operating state of the monitored hydraulic component on the basis of classification of a set of fault symptoms. Fault diagnosis using a rule-based expert system then involves testing and explaining the hypothesized fault using (possibly) other components' fault observations, on-line heuristic information, consultation, and/or the component history (if available). Thus, for each testable hypothesis (or fault even), there is only a small rule base. This hierarchical fault diagnosis method allows a straightforward description of many faults and failures in a real hydraulic machine. By applying such a hybrid neural network expert system for fault diagnosis we can combine the usual advantages of neural processing (which is governed by learning algorithms and associative memory) with a certain degree of modularity and the simulation of stepwise inferential reasoning that an expert system provides. The fault diagnosis method that we have proposed in this section is similar to the "fault diagnosis based on symptom processing" which was explained in Section 2.3 [30 Isermann, 1993]. However, there is a major difference: here, the relationships between fault symptoms and the corresponding fault events are automatically established and tested by a compact set of neural connections rather than a (large) set of production rules. As a result, the need to confirm a hypothesized fault by forward chaining of rules using "fault symptoms" have been minimized. In fact, the neural network can (at least partially) take over the forward-chaining task from the expert system's inference engine. This way, a great deal of effort and time in making excessive rules and executing them may be saved. Illustrative Example 3.7: To compare the conventional expert system with the proposed neural network expert system, and to demonstrate the treatment of heuristic symptoms and the process history, consider the second-order system in Example 5.6 with Fault mode 2. Also, assume that the second-order system is actually a pilot valve in a fluid power circuit. Section 3. 7 Fault Diagnosis Using a Rule-Based Expert System 79 Fault diagnosis using a conventional expert system: Some typical rules to diagnose Fault mode 2 (F2) according to the binary-coded analytic symptoms in Table 3.2, heuristic symptoms, and process history, are listed below. • Analytic Symptoms: Rule 1: IF JL(CV) = 1 THEN V i s c o u s D a m p i n g IS I n c r e a s e d . Rule 2: IF JL(A:5) = 0 THEN S p r i n g C o n s t a n t IS NOT C h a n g e d . • Heuristic Symptoms: Rule 3: IF PD = 1 THEN P a r a m e t e r IS D r i f t e d . Rule 4: IF SA = 1 THEN S e c o n d a r y A l a r m IS O n . • Fault event recognition: Rule 5: IF V i s c o u s D a m p i n g IS I n c r e a s e d , AND S p r i n g C o n s t a n t IS NOT C h a n g e d , AND P a r a m e t e r IS D r i f t e d , AND S e c o n d a r y A l a r m IS NOT O n , THEN F a u l t E v e n t IS F2. Rule 6: IF F a u l t E v e n t IS F2 THEN S p o o l D a m p i n g IS G r a d u a l l y I n c r e a s e d . • Process History and fault diagnosis: Rule 7: IF N e e d l e O r i f i c e IS NOT R e c e n t l y C l e a n e d , AND S p o o l D a m p i n g IS G r a d u a l l y I n c r e a s e d THEN F a u l t M o d e IS N e e d l e O r i f i c e _ D i r t y . Chapter 3 A HIERARCHICAL FDD METHODOLOGY 80 Fault diagnosis using neural network expert system: Rules 1 to 4 can be bypassed using a WTAG, since it automatically generated F2 from the analytical and heuristic symptoms. The rules to diagnose fault F2 according to the hypothesized fault event and process history may take the form: Rule 1: IF F a u l t E v e n t IS F2 THEN S p o o l D a m p i n g IS G r a d u a l l y I n c r e a s e d . Rule 2: IF N e e d l e O r i f i c e IS NOT R e c e n t l y C l e a n e d , AND S p o o l D a m p i n g IS G r a d u a l l y I n c r e a s e d THEN F a u l t IS N e e d l e O r i f i c e _ D i r t y . Note that the number of rules in the second case is greatly reduced. However, the explanation capability of the latter is also reduced to some extent. In order to enhance the fault explanation capability, we may transfer rules 1 to 4 from the first rule base to the second one, and add a single "hyper-rule" for all existing cases: H.Rule 1: WHEN F a u l t E v e n t THEN F a u l t S y m p t o m IS V i s c o u s D a m p i n g (value) AND S p r i n g D a m p i n g (value) AND P a r a m e t e r D r i f t (value) AND S e c o n d A l a r m (value). • 3.8 Concluding Remarks This chapter introduced a hierarchical methodology for automated on-line condition monitoring and fault diagnosis of hydraulic system components based on parameter estimation and pattern classification. The objective was to provide means to convert poor-quality sensor data into high-quality self-descriptive information, such as system physical parameters, so that the qualitative reasoning process in a knowledge-based system regarding faults can be facilitated. The emphasis, however, was on statistical and Section 3. 8 Concluding Remarks 81 mathematical methods for processing quantitative data rather than on A l techniques which are used for processing qualitative information in a typical knowledge-based system. In current knowledge-based expert systems, a common problem usually encountered is dealing with highly variable and noisy quantitative data obtained directly from measurements or observations. Sometimes, the data required for knowledge processing is missing or is not reliable due to the lack of appropriate robust sensors. Those expert systems that directly incorporate such sensor data into rules through algebraic relationships or constraint equations [75 Watton, 1992] often suffer from having excessive amount of rules for data interpretation and false alarm avoidance. Qualitative reasoning based on discretization of observed or measured variables into ranges such as "high", "low", etc., is one appropriate solution to the problem of data uncertainty or inaccuracy in knowledge-based systems [26 Hogan et al, 1996]. However, this still involves using extensive rules based on mathematical and behavioral component models in order to relate the observed qualitative changes in system variables to possible faults. In contrast, we managed the problem of data inaccuracy, uncertainty, and time-variability by automatically processing the information at different levels of abstraction outside an expert system shell. This way, by appropriately transforming the directly measured variables into time-constant physical parameters, we not only obtained new data which were shown to be much more robust to the noise and dynamics of the system, but also facilitated the fault diagnosis procedure by reducing the number of rules. The latter was achieved by an automatic fault symptom generating mechanism that discretized the parameter values into proper jump levels based on statistical properties of parameter estimates and mathematical methods. The number of rules relating a fault mode to its effects (symptoms) in a knowledge-based expert system was also shown to be further reducible by using a neural network pattern classifier. Chapter 4: AN EXPERIMENTAL HYDRAULIC TEST RIG 4.1 Introduction Proper maintenance and diagnosis of hydraulic systems is dependent upon a thorough knowledge of the fluid and of the functions of the mechanical components. The aim of this chapter is a review of some important fluid power system components, including a description of their functions, mathematical models, significant parameters and variables, and a few typical fault/failure related problems and causes. The endless variety of fluid power circuits and components makes it impossible to cover all the variations available from manufacturers. However, regardless of any complexity, a hydraulic system contains seven basic components: 1) a reservoir to hold the fluid supply, 2) connecting lines to transmit the fluid power, 3) a pump to convert input power into fluid power, 4) a pressure control valve to regulate pressure, 5) a directional control valve to control the direction of fluid flow, 6) a flow control device to regulate speed or fluid flow, and 7) an actuator to convert hydraulic power into mechanical motion. This chapter considers an experimental hydraulic test rig, as shown in Fig. 4.1, which contains a variety of these essential components which are also common to many industrial, agricultural, and aerospace applications. There are many standard components used in this circuit, such as flow control valves, relief valves, check valves, filters, a pump, and a cylinder. A one-degree-of freedom (1-DOF) robotic manipulator is also added to the actuator to simulate variable load conditions that exist in many applications such as excavators and aircraft control surfaces. All investigations into the development of condition monitoring and fault detection strategies in this thesis have been performed utilizing this experimental test rig. A VME-bus based computer system has been used in the experiments to issue control commands, collect the sensory data, and perform monitoring and fault diagnosis tasks in real-time. 82 Fig. 4.1: Pictures of the experimental test rig and its two-stage proportional directional servovalve built in the Robotics and Control Laboratory, UBC. Chapter 4 AN EXPERIMENTAL HYDRAULIC TEST RIG 84 Schematic diagrams, brief descriptions, basic mathematical models, and some typical faults for each test rig component are all given in the following sections. Specifications pertaining to the hydraulic test rig components and the corresponding measuring devices are given in Appendix C and Appendix D, respectively. 4.2 System Description A schematic diagram of the mechanical/hydraulic elements of the test rig is shown in Fig. 4.2. The first (pilot) stage of the servovalve, however, is not shown in the figure and will be described in Chapter 5. Aside from the mechanical/hydraulic units, there exist a controlling joystick, amplifier cards, computer, and data acquisition/processing boards which are not shown in the figure but will be explained later in this section. 4.2.1 Mechanical and Hydraulic Units The mechanical/hydraulic elements in the test rig can be grouped into seven functional units: 1. Reservoir unit: consists of a tank, a low pressure filter, and a check valve. The tank is at atmospheric pressure. The low pressure filter maintains the tank clean by removing dirt and abrasive particles detached from various components of the hydraulic circuit. The pressure upstream of the check valve is kept slightly above the atmospheric pressure in order to avoid cavitation in the main circuit. 2. Power unit: consists of a 3-phase A C motor, a coupling with shafts, a gear pump, a pressure relief valve, and connecting lines. The A C motor drives the pump which draws the oil from the reservoir and delivers it to the pressure line. The relief valve after the pump is a safety device that relieves excessive line pressure by bypassing the oil in the pressure line to the reservoir unit. 3. Filtering unit: consists of a high pressure filter immediately after the pump. Its function is to keep the hydraulic circuit, and especially the servovalve, clean by removing (abrasive) particles from the hydraulic oil. Section 4. 2 System Description Fig. 4.2: Schematic diagram of the experimental test rig. Chapter 4 AN EXPERIMENTAL HYDRAULIC TEST RIG 86 4. Flow control unit: consists of a two-stage proportional directional flow control servovalve that converts amplified electrical signals (commands) into mechanical motion in its main spool, thereby controls the direction and the amount of fluid from the pump to the actuator, and from the actuator to the reservoir unit. 5. Actuation unit: consists of a linear actuator, which is an unequal-area hydraulic cylinder and rod, and connecting hoses to the servovalve and to the pressure control unit. Depending on the amount and direction of fluid received from the servovalve, the cylinder actuator moves the variable load (manipulator) attached to its rod-end through extension or retraction. 6. Pressure control unit: consists of two pressure relief and two check valves, and connecting hoses to the actuation unit. The relief valves maintain the actuator pressures within a set range by relieving the excessive oil into the reservoir unit. The check valves prevent cavitation in the cylinder by redirecting part of the flow from the exit port to that side of the cylinder which suffers from imminent negative pressure. 7. Loading unit: consists of a 1-DOF robotic manipulator attached to the end of the cylinder rod by a rotary joint, and a variable load attached to the manipulator's end effector by a single bolt and nut. The linear motion of the hydraulic actuator is transferred through a rotary joint to the robotic arm, which is loaded at the other end. The rotary joint is designed such that its frictional torque can be varied on the fly. 4.2.2 Analog and Digital Electronic Units The analog and digital electronics and computers shown in Fig. 4.3 are used to monitor, control, and obtain data from the hydraulic test rig. They are grouped into five functional units: 1. Manual command unit: is a two-directional joystick for moving the actuator (and load) in extension or retraction mode. It produces voltage (-10 V to +10 V) proportional to the position of the joystick, which reflects the desired velocity of the load. Section 4. 2 System Description 87 Hydraulic Testbed Sensory Data Input Command V M E Bus Card Cage (Stethoscope) (VxWorks OS) Joystick SUN SPARC 5 Server Computer (Unix) u 3 ^ § Z to en n Of U e o s i 5 s e n Q T3 U CS o Fig. 4.3: Schematic diagram of the hydraulic test rig's controlling units. 2. Analog to digital conversion unit: is a 32-channel, 12-bit board with the input voltage range of -10 V to +10 V. It is used to obtain sensory data from the hydraulic system and the manual command unit, and pass the digitized data to the host computer. 3. Sensing unit: it contains several sensors and transducers to measure system variables on-line and send the information in electronic format to the A/D unit. The sensing unit consists of: 1 tachometer to measure the motor-pump speed; 7 pressure transducers to measure pump, supply, exit, pilot A and B, and head-side and rod-side line pressures; 1 linear pot to measure the main spool position; 1 load pin to measure the actuator force; 1 rotary pot to measure the manipulator's joint angle; and 4 temperature sensors located in the reservoir and at supply, head-side, and rod-side lines. Chapter 4 AN EXPERIMENTAL HYDRAULIC TEST RIG 88 4. Digital to analog conversion unit: is an 8-channel 12-bit board with the output voltage range between -10 V to +10 V. It is used to send the computer command signals to the power amplifier of the solenoids. 5. Central processing unit: consists of three computers and a V M E bus cage. The server computer — a Sun Spark 5 workstation running under Unix operating system — is used to write, modify, and download programs and supervisory commands and information to the host computer, and receive, display, and store on-line information about the hydraulic test rig condition. The host computer — a Sun Spark 1E workstation running under VxWorks real-time operating system — is used to spawn the real-time tasks, perform I/O operations, communicate with the server and the array processor, receive sensory data from the A/D unit, and send control commands to the D/A unit. The third computer is a SkyBolt™ array processor for fast computation of equations involved in the monitoring and fault detection algorithms. Al l the foregoing units communicate through a V M E bus. 4.2.3 Operation The operator's desire to move the load in extension or retraction mode is converted from mechanical hand motions in forward or backward direction into positive or negative voltage via the hand joystick. The D C signals of the joystick pass through the host computer, and after amplification, are sent to the directional proportional electrohydraulic servovalve. The pilot stage of the servovalve converts the electrical power into hydraulic force that, depending on the sign of the voltage command signals, moves the main valve spool (shown in Fig. 4.2) from its null (closed) position to the right or left direction. Then, the high pressure flow runs from the pump across the main valve pressure port to the head-side or rod-side of the ram-type actuator while, due to motion of the cylinder, fluid from the other side of the cylinder discharges through the main valve exit port. When, due to any reason, the pressure on either side of the actuator exceeds a certain value, the corresponding relief valve opens and part of the flow is discharged into the exit port, resulting in pressure reduction in the line. When the joystick is in centre Section 4. 3 Hydraulic Circuit Components 89 position, the input voltage to the servovalve is zero, and the main valve spool is centered. Since the main valve is overlapped, no flow passes through it, and therefore, all the fluid coming out of the pump is bypassed to the tank through a relief valve. Clearly, the operation of a circuit such as shown in Fig. 4.2 can cause rapid fluctuations in such variables as flow rates, pressures, and the arm position. As was shown in Chapter 3, valuable information on the condition of components may be obtained by measuring and processing these dynamically changing variables using proper transducers and data acquisition/processing units. 4.3 Hydraulic Circuit Components 4.3.1 Hydraulic Oil Hydraulic fluids are regarded as the most important material in hydraulic systems, since they tie everything together in the system. It is estimated that more than 70% of all hydraulic system problems are directly related to improper choice or handling of hydraulic fluids [23 Hehn, 1984]. Heat and contamination degrade fluids while they are attacking other system components: the result is costly downtime. With proper selection and handling of the fluid, most of the potential problems with the hydraulic system can be prevented and the system can function near its design point. In selecting the proper fluid for a given application, a number of fluid properties and their variations with respect to (w.r.t.) such factors as temperature and pressure must be considered in advance. In what follows we have briefly explained some of those properties which quantitatively enter into our future analytical models of the hydraulic system components. The ranges of variation in all cases for oil temperature and pressure are assumed between - 3 5 ° C to 80°C and 0 MPa to 15 MPa (2200 psi), respectively. Mass Density: Mass density is defined as mass per unit of volume, p = M / V, with units of Kg/m 3 . Chapter 4 AN EXPERIMENTAL HYDRAULIC TEST RIG 90 -40 -20 20 40 Temperature (C) Fig. 4.4: Variation of the Imperial Oil's NUTO H32 hydraulic oil density with temperature at atmospheric pressure. Variation with temperature: For most hydraulic oils, the density variation is quite linear, as is seen from Fig. 4.4, which can be expressed as: p = 880 - 0.7* (4.1) where t is the oil temperature in °C. Variation with pressure is less than 1% up to 20 MPa, and considered negligible. Viscosity: The "dynamic viscosity" of a fluid is a measure of its resistance to shear force [46 McCloy, 1980]; i.e., shear stress shear rate and has N.s/m 2 units. In many hydraulic applications, however, the ratio p, I p is more common, which is called "kinematic viscosity", v, and is expressed as centistoke (cSt = 10 - 6 m2/s). Variation with temperature: is exponential for all hydraulic fluids. Fig. 4.5 shows a typical plot of kinematic viscosity vs. temperature for the fluid we have used in our experiments, along with the curve Section 4. 3 Hydraulic Circuit Components 91 -40 -20 0 20 40 Temperature (C) Fig. 4.5: Variation of the Imperial Oil's NUTO H32 hydraulic oil kinematic viscosity with temperature at atmospheric pressure. fitted to the data, which is of the form v = 434.4exp (0.0005^ - 0.0876*) cSt (4.3) Variation with pressure: Within the pressure range 0 MPa to 15 MPa (2200 psi), the variation is less than 10% and therefore can be considered negligible as compared with variations with temperature, which could be as high as 1000%. Bulk modulus: No liquid is perfectly incompressible. The amount of change in a fluid volume per unit increase of pressure, i.e., AV/AP, is called the "oil compliance". The compliance characteristic of the oil in a hydraulic control system is a vital parameter affecting the system's dynamic response. "Bulk modulus of elasticity" is a compliance measure that is expressed as the ratio of change of fluid pressure to the relative change of fluid volume, or: Ap (3 = -V AV (4.4) Chapter 4 AN EXPERIMENTAL HYDRAULIC TEST RIG 92 Variation with temperature: The bulk modulus decreases with temperature. For the temperature range we are concerned, the variation for most hydraulic oils can be assumed linear. For example, the average bulk modulus of the Imperial Oil's NUTO H32 hydraulic fluid between 0 to 10 MPa pressure can be approximated by: Pi = 1676.8 - 6.2* M P a (4.5) Variation with pressure: The oil is always held in some form of container (e.g., cylinder, pump, hoses, etc.). Hydraulic lines, particularly hoses, can expand radially under pressure, and therefore, have high compliance. Entrained (free) air in oil also reduces the oil bulk modulus, since gases are much more compressible than liquids. As a result, the "effective" bulk modulus of the oil is the resultant of all of the above effects and may be written [47 Merritt, 1967] as: — - 1 — 7? 1 f3e~ I3t + f 3 c + K 3 1 P where - = - + - + Rg— (4.6) j3e: effective (total) bulk modulus of the oil, Pa /?/: bulk modulus of air-free liquid, Pa f3c: bulk modulus of container, Pa Rg: the entrained air content by volume in oil, % 7: specific heat ratio of air (= 1.4) p: oil pressure, Pa Fig. 4.6 shows that as the percentage of entrained air in oil increases, the effective bulk modulus decreases dramatically. Also from the figure, at higher pressures, f3e increases with p at a slower rate. Determination of the bulk modulus f3c for fluid conducting elements (i.e., tube and hoses) is explained in the next section. 4.3.2 Hydraulic Hoses Hoses are flexible fluid carrying links between hydraulic components, which can adapt to machine members that move, such as actuators. Section 4. 3 Hydraulic Circuit Components 93 0 5 10 15 Oil Pressure (MPa) Fig. 4.6: Variation of bulk modulus w.r.t. oil pressure and entrained air. Determination of bulk modulus of fluid containers and connectors is a difficult task, because of mechanical compliance. The major source of mechanical compliance is long hydraulic hoses connecting pumps and valves to actuators. These hoses usually have teflon or hard rubber cores that give comparatively low bulk modulus, typically between 70 to 350 MPa [47 Merritt, 1967]. The value for a particular hose can be computed from cubic expansion coefficients of the hose under pressure, which are provided by the manufacturers. For the hoses that we have used in our test rig, from the manufacturer's catalogue and Eq. (4.4), we obtain j3c = 83 MPa, which is very low as compared to the oil bulk modulus of /?/ = 1400 MPa. As a result, the bulk modulus of the hose totally controls the overall bulk modulus in Eq. (4.6). Chapter 4 AN EXPERIMENTAL HYDRAULIC TEST RIG 94 4.3.3 Filters Two types of filters have been used in the hydraulic test rig. Pressure Filter: This filter is used between the outlet side of the pump and ahead of the proportional valve which is highly sensitive to dirt. The filter removes critical contaminants passing through or generated by the gear pump before they get into the remainder of the system. As a result, the rate of change of the filter "resistance factor" to the flow is a good indication of a wearing or failing pump. Return line filter: Any contamination entering the system downstream of the pressure filter, such as the dirt generated by the cylinder rod seals, will increase dirt levels in the reservoir unless controlled by a return line filter. Such a filter has been placed in the system return line to clean the hydraulic fluid coming from the proportional valve or the cylinder, before it enters the reservoir. As a result, the rate of change of the filter resistance factor to the flow can be a good indication of a wearing or failing proportional valve or hydraulic cylinder seal. Pressure loss: A filter in a hydraulic line causes resistance to flow which is reflected as a pressure drop across the filter. In most cases, the pressure drop-flow rate relationship is nonlinear second or third order. However, for a limited flow range between 0 to 0.0002 m3/s (the maximum allowable flow rate in our test rig), we may assume a linear relationship [12 Dransfield, 1981] such that Apf = Rfqf (4.7) where Ap/. pressure drop across the filter, Pa Rf. filter resistance, Pa.s/m3 qf. flow through the filter, m3/s Typical failure modes and probable causes: • Too high an upstream pressure or oil bypass: element dirty, oil viscosity too high. • Dirty oil: partial bypass, faulty (leaking) or ruptured filter. Section 4. 3 Hydraulic Circuit Components 95 • Broken housing: too much pressure, too much mechanical shock. 4.3.4 Gear Pump Gear pumps belong to the family of fixed positive displacement pumps, which provide almost constant supply pressure required to drive any load. They transform the mechanical power input to their shaft into fluid power (pressure and flow) output from their port. A schematic picture of a spur gear pump is shown in Fig. 4.7. In brief, a basic gear pump consists of two meshed gears enclosed in a closely fitted housing. The gears rotate in opposite directions and mesh at a point in the housing between the outlet and the inlet ports. As the gears rotate, a kind of vacuum is formed as the teeth unmesh, which causes liquid to be forced in through the inlet port from the reservoir. Fluid is then displaced as the teeth mesh at the outlet side and is forced out of the pump into the hydraulic system. A hydraulic pump normally runs continuously. In dynamic analyses where driven load response is the main objective of study, it can often be assumed that pump speed is constant. For a gear pump, inertia of the pump's moving parts and internal friction can then be neglected. This is equivalent to regarding the pump as experiencing a speed source. In practice, positive displacement pumps, including gear pumps, may be characterized by a single equation relating flow rate to load pressure and internal losses of the machine. For example, Fig. 4.8 illustrates the steady-state characteristics of the gear pump used in the test rig circuit. As is seen, pump output flow rate decreases as load pressure increases due to a flow loss characteristic caused by clearances between moving parts. The flow characteristic measured is sufficiently linear to allow defining the following equation: qp = Vpu> - CpPp (4.8) where qp: pump flow rate, m i3/s Vp: pump displacement, m3/rad Chapter 4 AN EXPERIMENTAL HYDRAULIC TEST RIG 96 x 10" 1000 2000 3000 Pump Speed (RPM) Fig. 4.8: Flow/pressure characteristics of the gear pump. 4000 Fig. 4.7: Schematic diagram of a typical spur gear pump. u> = pump speed, rad/s pp: pump outlet pressure, Pa Cp: pump leakage coefficient, m3/Pa.s Compliance effect: A pump invariably discharges into some kind of a volume. The volume will include the pump's own porting and outlet, and can include the volumes of connecting hydraulic lines and of a high-pressure filter. The compliance effect of the pump's discharge side volume immediately adjacent to the pump, cp, can be integrated into Eq. (4.8), to give the amount of qp lost to the compliance effect. This amount causes the pump pressure pp to change, so that PP = —(VPu - Cppp - qp) (4.9) Pump ripple: Gear pumps contain a finite number of pumping elements, and the summation of flow rates from each tooth forms the output average flow rate. A small-amplitude high-frequency ripple component is inherently superimposed on the mean flow rate due to meshing and unmeshing action of the teeth. For example, Fig. 4.9 shows the time history and power spectrum of the pressure ripple of the test rig's gear Section 4. 3 Hydraulic Circuit Components 97 Time History Power Spectrum 7.81 • • • •—i 1i • I i i i I Q I - • , • • U • 1 1 0 5 10 15 20 0 500 1000 Time (ms) Frequency (Hz) Fig. 4.9: Time history and power spectrum of a gear pump pressure ripples. pump. The periodic nature of the pressure ripple is evident from the figure, which may be approximated by a Fourier Series [44 Martin, 1996]. If a lumped parameter model is used in its simplest form, a sinusoidal function is obtained whose frequency can be estimated either from spectrum analysis of the pressure ripple data, or alternatively, by observing that pressure ripple fundamental frequency « number of teeth X pump speed Accordingly, Eq. (4.9) can be rewritten as pp « —{[Vpo + Vpl sin (Nturt) + Vp2 cos (Ntu>t)]u - Cppp - qp} (4.10) cp where N,: number of teeth in the gears Vpo: nominal (mean) pump displacement Vpl, Vp2'- ripple coefficients that have to be estimated from experimental data for a specific pump. The ripple characteristics, i.e., coefficients Vpo, Vp/, and VP2, can be a useful means for condition monitoring of a pump, since the shape of the ripple will change as particular pump problems occur Chapter 4 AN EXPERIMENTAL HYDRAULIC TEST RIG 98 [62 Silva, 1986][75 Watton, 1992]. Note that when considering the ripple components, the frequencies involved (even just the fundamental one) are usually suffuciently high to prevent lumped parameter modeling, such as the one given in Eq. (4.10), to be applied. Hence, Eq. (4.10) may not give a reliable prediction of the fluctuating component of pressure. Typical failure modes and probable causes: • Excessive pump noise: misalignment of pump-motor coupling, low level of oil, too high a speed, wrong type of oil, air leak in suction line or in case drain line or around the shaft packing, flow restriction in suction pipe, broken or worn parts, filter restriction. • Excessive heat: excessive pump pressure, pump discharging through relief valve, inadequate cooling, excessive friction, excessive system leakage. • Leakage at oil seal: abrasives on pump shaft, incorrect seal installation, damaged shaft seal, poor coupling alignment. • Insufficient system pressure: wrong system relief valve setting, oil drainage to reservoir, pressure relief through relief valve, slow running pump, defective pressure gauge. • No pump flow: suction line air leak, too slow speed, plugged line or suction filter, too high oil viscosity, damaged tooth or shaft, too loose pump cover. 4.3.5 Pressure Relief Valve Relief valves belong to the family of pressure control valves. A relief valve is installed in the circuit of the hydraulic test rig, right after the gear pump, to make sure that system pressure does not exceed safety limits. It is intended to relieve occasional excess pressures arising during the course of normal operation. Excess fluid is allowed to return to the reservoir through an outlet port in the valve while full adjusted pressure is maintained in the system. Fig. 4.10 shows the schematic diagram of a pilot operated relief valve used in the hydraulic test rig. Section 4. 3 Hydraulic Circuit Components 99 PRESSURE ADJUSTMENT * CONTROL VENT OR REMOTE CONTROL CONNECTION FLOW FLOW DISCHARGE TO TANK Fig. 4.10: Schematic diagram of a pilot operated relief valve (Courtesy of Parker Hannifin Corp., Cleaveland, Ohio). Valve model: The relief valve itself has internal dynamics, as it is composed essentially of a damped spring-mass excited by an external force, i.e., pressure x effective area. Flow forces also affect the dynamics of relief valve components. A detailed dynamic model of relief valves is given in [22 Handroos, 1990]. Relief valves normally have a dynamic response many times faster than the dynamic response of the system's driven load [12 Dransfield, 1981]. As a result, the dynamics of the relief valve can be neglected in most applications. Fig. 4.11 illustrates the form of the static flow rate, qr, to be expected from the test rig's relief valve installed right after the pump. It is reproduced from the manufacturer's data sheets. The relationship can be described by the discontinuous two-part equation: (4.11) where qr: flow rate through the relief valve, m i3/s Cr: relief valve flow coefficient, m3/Pa.s Pset: relief valve pressure setting, Pa Chapter 4 AN EXPERIMENTAL HYDRAULIC TEST RIG 100 x 10"3 0 1 2 3 4 5 6 7 8 9 Pump Pressure (MPa) Fig. 4.11: Flow rate-pressure characteristic of the RA101 pressure relief valve at cracking pressure of Pse, - 7.6 MPa. Typical failure modes and probable causes: • Low or erratic pressure: dirt, chip, or burr holding valve partially open, worn or damaged seat or poppet, sticking valve spool, damaged or weak spring, partially blocked orifice, air in system. • Excessive noise or chatter: too low oil viscosity, faulty or worn seat or poppet, fluctuating return line pressure. • Overheating of system: continuous operation at relief setting, oil viscosity too high, leakage at valve seat. 4.3.6 Check Valve The check valve, or non-return valve, in the test rig of Fig. 4.2 is intended to allow free flow in the direction from the main valve to the reservoir and to totally prevent flow in the opposite direction. It also maintains the return line pressure slightly above the reservoir (atmospheric) pressure (about 50 psi or 350 KPa) in order to compensate for momentary lack of positive pressure in the return line that otherwise could result in cavitation in the main valve and in the cylinder. From the flow-pressure characteristic Section 4. 3 Hydraulic Circuit Components 101 x 10 -4 8h 7 co 4 5 2 1 0 0 0.1 0.2 0.3 0.4 Pressure Drop (MPa) 0.5 0.6 0.7 Fig. 4.12: Flow rate-pressure drop relationship in the check valve. of the valve, which is shown in Fig. 4.12, it is evident that a small "cracking" pressure is required to open the valve, and a small positive pressure drop is present when flow is positive, and this pressure drop can increase somewhat with flow rate. Valve model: The model of the valve is very similar to that of a relief valve. Again, ignoring the valve's dynamics as compared with the loading system dynamics, we may obtain the flow rate through the valve using two non-continuous equations: where qc: flow rate through the check valve, m3/s Cc: check valve flow coefficient, m3/Pa.s Ap: pressure drop across the check valve, Pa Pcrack' check valve cracking pressure, Pa Typical failure modes and probable causes: • Flow stoppage: parts broken. (4.12) Chapter 4 AN EXPERIMENTAL HYDRAULIC TEST RIG 102 • Fails to hold pressure: damaged or eroded seat, excessive leakage. 4.3.7 Line Relief/Anticavitation Valves Two line relief/anticavitation valves have been used, as shown in Fig. 4.2, between the cylinder ports and the exit line. Each valve consists of a direct acting sequence valve and a check valve. While the former relieves excessive pressure in the cylinder chamber due to overloading conditions, the latter (the check valve) allows some fluid to return to the actuator from the exit line in case there is a sudden pressure drop in the cylinder. This way, the pressure lines and the cylinder do not experience any momentary negative pressures that otherwise might cause cavitation in the circuit. Valve Model: The flow model of the valve is the same as in Eqs. (4.11) and (4.12). Therefore, we have the following non-continuous equations: ' Cc(Ap + Pcrack) if Ap < Pcrack 0 if - Pcrack <AP< Pset (4.13) , Cr(Ap- Pset) UAp> la where qa: flow through the line relief/anticavitation valve, m3/s Ap: pressure drop across the valve, Pa Cc: check valve flow coefficient, m3/Pa.s Cr: relief valve flow coefficient, m3/Pa.s Pcrack'- check valve cracking pressure, Pa Pset: relief valve setting pressure, Pa Typical failure modes and probable causes: Similar to those of relief and check valves. 4.3.8 Electro-Hydraulic Servovalve The servovalve is an interface between low power electrical signals and high hydraulic power. It is electrically operated and designed to provide fluid flow proportional to the applied signal for smooth operation of the actuator. The two-stage valve that controls the motion of the hydraulic actuator in Section 4. 3 Hydraulic Circuit Components 103 Fig. 4.13: A proportional directional two-stage servovalve, Model 4 WRZ (Courtesy of Rexroth). the test rig of Fig. 4.2 is typically used in off-highway machinery and other heavy-duty applications, and permits high accuracy and relatively fast response in open and closed-loop control systems. In the two-stage valve, the pilot or first stage receives an electromechanical input, amplifies it, and controls the movement of the second (main) stage spool. The main or boost stage controls pressure on both sides of the actuator through its spool movement. A schematic diagram of the servovalve is shown in Fig. 4.13. The first (pilot) stage: The first stage is a four-port proportional directional pressure control servovalve which utilizes two force-generating proportional solenoids directly acting on the valve's control spool ends. Since the valve is directly actuated by proportional solenoids, it is dynamically independent of supply pressure. In the de-energized condition the spool is centered by springs and the force generated by the proportional solenoids is counterbalanced by pilot pressure feedback and spring resisting force. Since the governing equations of motion and flow-pressure relationships for the pilot valve are quite involved, further explanation of the servovalve is deferred until Chapter 5, where a novel nonlinear model for the servovalve in question is given and evaluated against the actual valve's behavior. Chapter 4 AN EXPERIMENTAL HYDRAULIC TEST RIG 104 The second (main) stage: The second stage is a closed-centre 4-way proportional directional flow control valve which utilizes pilot pressures directly acting on the valve's spool. In the de-energized condition the spool is centered by a preloaded push-pull spring and the net force generated by the pilot pressures A and B is counterbalanced by resisting spring force. With each spool stroke a certain wedge-shaped throttling cross-section is produced at the metering edges of the spool, resulting in different flow rates through the valve. Basic flow equations through an orifice: Flow through the main valve is controlled by the degree of opening of a set of orifices formed by the valve body-spool configuration. The rate of flow through an orifice depends on the area of the orifice and on the pressure drop across it. This functional relationship can be expressed as where q0: flow rate through the orifice a„: area of orifice opening normal to flow Ap(): pressure drop across the orifice n: 0.5 for turbulent flow and 1.0 for laminar flow across the orifice. It is a usual practice to assume a turbulent flow across the orifice of a flow control valve [47 Merritt, 1967], [46 McCloy, 1980]. Therefore, the flow rate in Eq. (4.14) can be obtained from the turbulent flow equation through an orifice [47 Merritt, 1967]; i.e., where p is the oil density and Q (< 1) is the orifice discharge coefficient, which depends on the pressure drop and flow Reynolds number across the orifice [47 Merritt, 1967]. In practice, however, Q can be assumed to be constant, especially when the flow through the orifice remains turbulent for most of the time [47 Merritt, 1967]. Note that the quantity Cda0, which we have called the "orifice effective area", is always less than the orifice area aQ in magnitude. q0 = f(a0,ApnQ) (4.14) (4.15) Section 4. 3 Hydraulic Circuit Components 105 Spool Position (mm) Fig. 4.14: Orifice effective area vs. spool displacement for a wedge-shaped-porting valve. The orifice area configuration is of prime importance to valve performance and depends fundamentally on the valve spool displacement, xm. The most common porting shapes are square, round, full annulus, or wedge-shaped. Fig. 4.14, for example, shows a geometrical relationship between the main spool position and the orifice effective area for the wedge-shaped-porting valve that we have used in the hydraulic test rig. To obtain the experimental data points shown in Fig. 4.14, we opened one of the valve's main ports to a container, applied a constant voltage to the valve solenoids, and measured the supply port pressure, the spool position, and the amount of fluid poured to the container during a certain period of time. We repeated the process for a number of valve openings and plotted the results. We then used the flow rate and pressure drop measurements to calculate the orifice effective area, Cd.a0, from Eq. (4.15). It should be noted that for closed-centre valves, ideally, the line-on-line configuration allows no flow into or from the control ports while the spool is centered. In practice, some overlap of spool lands over the port edges is necessary to "minimize" leakage in neutral position. In Fig. 4.14, for example, the overlap length, Lm, is about 0.4 mm from the spool's null position, and some leakage flow is evident Chapter 4 AN EXPERIMENTAL HYDRAULIC TEST RIG 106 within the overlap range. Typical failure modes and probable causes for the pilot valve: • No response to command signal: open coil or lead, jammed spool, plugged inlet orifices, filters silted with contamination. • High null bias: air gap contamination, filters silted with contamination, partially plugged main valve, partially plugged inlet orifice. • Poor response (delay in returning to neutral position): filters silted with contamination. • Non-repeatability: sticky spool. • Servovalve does not follow input command signal: open coil assembly or open coil leads. • Low flow gain: shorted coil assembly. • Jerky and/or oscillatory motion: sticky spool. Typical failure modes and probable causes for the main valve: • Spool not moving: solenoids inoperative, no pilot pressure, blocked pilot drain, dirty ports, distortion, siltted spool. • spool response sluggish: start-up oil viscosity too high, restricted drain, distortion in valve body, solenoid malfunctioning, dirt in system, low pilot pressure. • Faulty or incomplete spool movement: insufficient pilot pressure, faulty solenoid, defective centering spring, improper spool adjustment. • Actuator drifts: spool not centered, valve spool body worn, leakage past piston in cylinder, valve seats leaking (worn). • Oil heats up: valve seat leakage (pressure or return circuits). 4.3.9 Cylinder Actuator A single-rod double-acting cylinder is used in the actuation unit of the hydraulic test rig for transforming oil pressure into mechanical force. Cylinders produce a straight-line motion, but a lever-type manipulator Section 4. 3 Hydraulic Circuit Components 107 oil out h A oil in • 1 Cylinder Rod Fig. 4.15: Schematic diagram of a single-rod double-acting hydraulic cylinder with acting forces. is used to convert the linear motion into a limited rotary motion in order to simulate a nonlinear loading condition. The schematic diagram of a hydraulic cylinder is shown Fig. 4.15. The cylinder consists of a cylinder body, a piston, and a piston rod attached to the piston. As the cylinder rod extends and retracts, it is guided and supported by a bushing called a "rod gland". The side through which the rod protrudes is called the "rod-side". The opposite side without the rod is termed the "head-side". Inlet and outlet ports are located in the rod and head ends. For efficient operation, a leak-free seal exists at the rod gland. Hydraulic cylinders often have cast iron piston rings as a piston seal. Piston rings provide a durable seal but they have some leakage flow due to the clearances between the piston ring and cylinder body tube. The leakage coefficient depends on the pressure difference, and velocity and direction of motion of the piston [47 Merritt, 1967]. Flow equations: When the main valve opens in one direction, flow passes across its open orifice to one of the cylinder sides, say the rod-side. The accumulation of oil in the chamber raises the rod-side pressure and pushes the cylinder towards its head end (see Fig. 4.15). Oil from the head-side then Chapter 4 AN EXPERIMENTAL HYDRAULIC TEST RIG 108 escapes across the other orifice formed in the main valve to the reservoir. We assume there is leakage across the piston seal between the rod-side chamber and the head-side chamber. We also assume that the fluid and hoses have compliance, and that the piston is moving in the positive direction from rod-end towards the head-end. The net amount of fluid entering the rod-side circuit (chamber + hoses) from the main valve is then equal to the amount of fluid displaced by the piston, leakage from the rod-side chamber to the head-side chamber, and the amount of fluid compressed in the circuit [47 Merritt, 1967]: qr = ArXr + Cltrh(Pr - Ph) + CrPr + qa,r (4.16) where qr\ net inflow to the rod-side circuit, m3/s Ar: net pressure area on the piston's rod-end, m 2 xT: actuator's rod velocity, m/s pr: rod-side pressure, Pa Ph'. head-side pressure, Pa Q r / , : internal leakage coefficient from rod-side to head-side, m3/Pa.s cr: rod-side circuit compliance and is given by [47 Merritt, 1967]: ' 0 7* "T~ J\.T*&T Cr = ' „ (4.17) Pe where xr: rod displacement from its reference position, Voy. volume of the rod-side circuit when the piston is at its reference position, m 3 Similarly, the net amount of fluid escaping from head-side circuit (chamber + hoses) is equal to the amount of fluid displaced by the piston, leakage from rod-side to head-side, and the amount of fluid expanded in the circuit: qh = Ahxr + Ci>rfl(pr - ph) - chph - qa,h (4.18) Section 4. 3 Hydraulic Circuit Components 109 where qh'. net outflow from the head-side circuit, m3/s Ah', net pressure area on the piston's head-end, m 2 Ch'- the head-side circuit compliance which is given by: Ch = (4.19) where V0,h'. volume of the head-side circuit when the piston is at its reference position, m 3 . Equation of motion: From Fig. 4.15, the forces acting upon the moving piston-rod assembly consist of the pressure forces at both sides of the piston, the piston Coulomb and viscous forces, and the external force at the load end of the rod. From Newton's second law of motion, the net force equalizes the inertial force of the assembly; therefore, where Mr: mass of the piston-rod assembly, Kg, xr: rod (piston) acceleration, m/s2, F / J P : Coulomb friction force for the piston, N, CViP: viscous friction coefficient for the piston, N.s/m, //: load force acting at the rod end, N. Friction measurement: Friction coefficients are actuator-dependent; i.e., they vary substantially when the load, piston seal condition, oil temperature, or even the piston's direction of motion is changed. Fig. 4.16 shows the total friction force, i.e., Coulomb + viscous, acting on the actuator. We obtained the data points from Eq. (4.20) by measuring cylinder pressures when the load-free actuator was moved at various constant velocities from nearly zero to 0.08 m/s. The Stribeck effect due to boundary lubrication is evident from the figure, where the friction force first drops and then gradually increases as the piston velocity increases. Note that in retraction mode, i.e., when the piston moves from the rod-end towards Mrxr = A r P r - Ahph - FftPsgn(xr) - CVtPxr - ft (4.20) Chapter 4 AN EXPERIMENTAL HYDRAULIC TEST RIG 110 1001 1 ' 1 ' ' 1 ' ' 1 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Piston-Rod Velocity (m/s) Fig. 4.16: Measured friction force of the hydraulic cylinder in retraction and extension. the head-end, the total friction force is smaller than in extension mode, i.e., when the piston moves from the head-end towards the rod-end. Typical failure modes and probable causes: • drift: leakage in the piston seals, rod seals, connecting lines, or the control valve. • no movement: excessive friction load in the linkage, too low pressure, piston seal leak, scored cylinder bore due to contamination in the hydraulic circuit, piston rod broken. • Erratic cylinder action: valve sticking or binding, cylinder sticking or binding, too high viscosity during warm-up period, too low pilot pressure, internal leakage in the piston, air in system. • Leak in the cylinder body: loose tie rods, excessive pressure, pinched or extruded seal, seal deterioration due to wear or loss of elasticity. • Rod gland seal leak: torn or worn seal, seal deterioration. • Excessive or rapid piston seal wear: excessive back pressure, wear due to contaminated fluid, bent rod. Section 4. 3 Hydraulic Circuit Components 111 h Fig. 4.17: The manipulator-like link attached to the cylinder rod of the hydraulic test rig. 4.3.10 Manipulator A manipulator-like link, as illustrated in Fig. 4.17, is attached to the free end of the actuator rod and is used to simulate a variable load on the hydraulic cylinder, for example, in an excavator, in a hydraulic robot with a rotary joint, or in an aircraft's control surface. The link is pulled or pushed by the actuator at joint A and can freely rotate about joint B which has a friction torque Tf. A load having a mass M/ can be bolted to the free end of the link. The distance between the load's center of gravity and the anchor point B is L/, and the load applies some resisting force / / to the cylinder rod at joint A. This force is measured using a load pin connected to the cylinder rod at joint A. The angle of rotation of the load's center of gravity, 6, with respect to a vertical line is measured using an angular pot attached to the link at joint B. Manipulator kinematics: To obtain the linear position and velocity of the cylinder rod at any time, we Chapter 4 AN EXPERIMENTAL HYDRAULIC TEST RIG 112 realize that for the triangle A B D in Fig. 4.17, we can calculate the instantaneous length of the rod as: a = y/b2 + d2 -2bdcos{3 (4.21) where P = Po~0 (4.22) and flo is the angle fd when the link is in vertical position. Also, when 0 = 0, from Eq. (4.21), the rod has a length do = ^b2 + d2 -2bd cos [30 (4.23) Therefore, the elongation of the rod, when the link is not at vertical position, can be determined using xr = ao — a t (4.24) = oo - yjb2 + d2- 2bdcos(p0 - 0) where ao, b, d, and Po are all known constants, and the joint angle 6 is measured. Differentiating xr in Eq. (4.24) with respect to time t gives the rod velocity, as xr = h9 (4.25) where 6 is the angular velocity of the link and h is the shortest distance between joint B and the line of action of force / / . Thus a Manipulator dynamics: The forces acting upon the link are due to the hydraulic actuator, gravity, and friction at joint B. The resultant moment generated about this point causes both the manipulator and the attached load to accelerate. Assuming that the angular velocity of the link is not too high, the mass of the aluminum link is negligible compared to the mass of the load, and the load mass is concentrated at a point, we may write the balance of moments about joint B, to obtain: M,L]0 = hfi - MigLi sin 0-Tf (4.27) Section 4. 3 Hydraulic Circuit Components 113 where Mf. load mass, Kg L/: load center of gravity, m 8: the angular velocity of the load (or link), rad/s //: actuator force at joint A, N g: acceleration of gravity = 9.81 m/s2 Tf Coulomb + viscous friction torque at joint B, N.m Joint friction: The friction torque at joint B is due to the viscous and Coulomb effects. It is not only a function of the link's angular velocity but also a function of the normal resultant force applied to the surface of the bearing at joint B. The normal force, by itself, is a function of the link position as well as actuator force, since the loading condition on the bearing of joint B is time-varying. In mathematical terms, 7} can be rewritten as: Tf = Rbpcfnsgn9 + Cj (4.28) where /?/,: radius of the bearing at joint B, m fic: Coulomb friction coefficient /„ : resultant force normal to the surface of the bearing B, N C v : viscous friction coefficient, N.s/rad The Coulomb friction coefficient pc is generally a function of the type of contact between the journal and the bearing, the contacting materials, the lubrication condition, and the size of the bearing, and can vary between 0.08 to 0.8 for metal contacts [2 Baumeister et al., 1978]. Now, from the geometry of the link and the actuator in Fig. 4.17, we realize that the normal resultant force /„ is equal to the resultant of all external forces applied to the link. /„ has two components: 1) fnn in the direction of line BC, and 2) fnt perpendicular to line BC. Chapter 4 AN EXPERIMENTAL HYDRAULIC TEST RIG 114 For /„„, taking the centrifugal force of the load mass into account, we obtain: fnn = fi sin <p + Mtg cos 9 + M[Li (02 + a B , (4.29) where L/: distance between the load's center of gravity and joint B &B,n- component of joint B acceleration normal to the joint bearing = 0 (joint is stationary here) The first term in Eq. (4.29) is the component of the rod force in the direction BA, while the second term is the component of the force at C (gravity force) in direction BC. The third term is the centrifugal force which is always in direction BC. Notice that in case of a multiple link manipulator, Eq. (4.29) is still applicable provided that A//g is replaced by a known force at joint C and as.n is known. Similarly, the tangential component of the normal bearing force, /„ , , which is perpendicular to /„„, is given by: where the last term in Eq. (4.30) is due to the load acceleration, and ag,, = 0 if joint B is stationary. Finally, the magnitude of the normal bearing force, / „ , at joint B is given by After substituting for 9 in Eq. 4.30 from Eqs. (4.27) and (4.28), and some mathematical manipulations, we finally obtain (4.30) (4.31) fn — (4.32) « « * - £ ) ' ' + i H ' + ' - ( i f * ) ' * 1 Section 4. 4 Measuring Devices 115 4.4 Measuring Devices A variety of measuring instruments has been used in the hydraulic test rig to accurately measure the process variables in real-time. The specifications of the sensors are given in Appendix D. In choosing appropriate sensors for the purpose of condition monitoring and fault diagnosis of the hydraulic test rig components, the following guidelines were considered: • The sensors should be robust; i.e., their measurements should be reliable and consistent. Typically, a pressure sensor is considered a robust sensor while a flowmeter is a nonrobust sensor. • The sensors should be commercially available and low in cost. • Position sensors are usually required wherever a moving element such as a spool or a piston exist in the system. • Flowmeters and velocity sensors can be avoided by using appropriate state estimation techniques. By examining the state equations for each hydraulic component, the appropriate sensors for the required measurements were chosen using the above guidelines, as listed in Appendix D. Chapter 5: NONLINEAR MODELING OF A SERVOVALVE Static and dynamic models for some typical hydraulic components that have been used in our hydraulic test rig were investigated in Chapter 4. Derivation of such models for the test rig's servovalve was postponed until this chapter, due to the complexity of the valve model and the novelty of the modeling and validation procedure. This chapter presents a step-by-step methodology for nonlinear modelling, parameter determination, and performance evaluation of a typical two-stage proportional directional flow control servovalve used in the hydraulic test rig. This model has been utilized in Chapter 6 to simulate a number of valve faults and to validate the proposed on-line estimation and fault detection algorithms. The first stage of the valve is called "the pilot stage", while the second stage is called "the main stage". For the pressure control valve in the pilot stage, a nonlinear formulation is proposed that takes into account pilot spool deadband, spool friction, flow coefficient variability, and leakage. This new model is complete in the sense that we have incorporated all nonlinear interactions between various electro-mechanical elements in a compact yet comprehensive physical model, which accepts a command voltage to the solenoids as input and gives the second stage spool displacement as output. Then, a systematic approach based on state measurement and curve-fitting techniques is adopted to obtain those parameters that are valve dependent and are not directly measurable. Next, experimental and simulation results are compared and a good match is reported. At the end of the chapter, we have investigated the valve's performance due to changes in several valve parameters, using the valve model. 5.1 Introduction Electrohydraulic servovalves are electro-mechanical devices which can be driven by computers for precise motion and force control of powerful hydraulic actuators used in modern aircraft, spacecraft, 116 Section 5. 1 Introduction 117 off-highway machinery, machine tools, and other heavy-duty stationary and mobile equipment. For low power applications, a single-stage servovalve is usually appropriate, while for high pressure - high flow demands, a two-stage valve is often used. In this case, the first (or pilot) stage is often a stabilized flapper-nozzle or a pressure control servovalve powered by an electromagnetic device such as a proportional solenoid. The second (or main) stage is then boosted by the pilot pressure, and is almost always a proportional flow control valve. Performance of an entire hydraulic system mostly relies on the characteristics of its servovalve component. A computer model that accurately represents the "steady-state" and "dynamic" performance of such a valve can be used, for example, to deduce the effect of the changes in the valve's behavior due to incipient faults. This allows the computer model to be used to predict the effect of parameter changes which may be caused in practice by damage and/or wear without actually having to damage the real component. A list of cause-and-effect characteristics may then be established, which later can be incorporated into a fault detection and diagnosis (FDD) algorithm for the real servovalve, as is suggested in [5 Burrows et al, 1989]. Both "steady-state" and "dynamic" behavior of electrohydraulic servovalves are still the subject of a significant ongoing research and development (see, e.g., a literature review in [73 Watton, 1989a]). Complexity, nonlinearity, and sensitivity of electro-mechanical servovalves have so far imposed major drawbacks in achieving especially accurate dynamic models1. One reason is that the linearized input/output model for a single- or two-stage servovalve has generally been unable to "explain" and faithfully "represent" the actual valve's behavior under different working conditions and over the valve's entire range of operation [47 Merritt, 1967] [46 McCloy and Martin, 1980] [39 Lin and Akers, 1989] [70 Vossoughi and Donath, 1995]. So far, a number of nonlinear models for single- and two-stage flapper-nozzle servovalves have been proposed by researchers. [40 Lin and Akers, 1991] performed a nonlinear dynamic analysis on the 1) Even in today's modern aircraft and spacecraft, mechanical servovalve models are still preferred to digital valve models for performance monitoring of their operational counterparts [58 SAE, 1991]. Chapter 5 NONLINEAR MODELING OF A SERVOVALVE 118 flapper-nozzle stage of a two-stage servovalve. Improved results were reported when valve nonlinearities were considered. [67 Tsai et al, 1991] evaluated the nonlinear model of a two-stage two-spool pressure control servovalve. The torque motor that actuated the flapper-nozzle stage was simply modelled as a constant-gain electrical current amplifier. Moderately good agreement for steady-state and frequency response of the model with experimental data was obtained. No comparison, however, between the time response of the model and the actual valve was reported, and spool deadband (overlap/underlap) effects and orifice flow regimes (laminar/turbulent) were not considered. Our main focus here is on a two-stage proportional directional flow control servovalve, which is especially popular in heavy-duty applications such as off-highway machinery. Very few nonlinear models for these valves are available in the literature. [74 Watton, 1989b], [77 Watton and Salters, 1990], and [22 Handroos and Vilenius, 1990] gave some detailed models for a variety of pressure control relief valves, which are to some extent similar in operation to the first stage component of the servovalve at hand. For the solenoid component of a pressure control servovalve (the pilot stage), [69 Vaughan and Gamble, 1996] suggested a nonlinear semi-empirical model and evaluated a solenoid actuated spool valve using the armature displacement measurements. No oil flow or pressure effects were considered, and it was assumed that the armature and the valve spool always remained in contact. One of our purposes here is to generalize this solenoid model when two of them are connected to a four-way pressure control servovalve that uses "oil pressure" as a natural feedback. In this case, the valve spool or solenoid armature displacement may not be directly measurable due to mechanical inaccessibility, and the solenoid armature may not always remain in contact with the spool. The main contribution of this chapter is that it presents a complete model for a general spool-operated servovalve that accepts voltage commands from the solenoid amplifier card as input and gives the second (boost) stage spool a displacement as output. By comparing the simulation to experimental results, we have shown that the proposed model closely follows the actual valve dynamics over a wide operating range of the valve. Such an accurate model has enabled us for the first time to easily predict a valve performance due to such parameter variations as oil viscosity, bulk modulus, and supply pressure, as Section 5. 2 Valve Description 119 Fig. 5.1: A two-stage electrohydraulic servovalve. well as pilot spool deadband, that would have been difficult and expensive using a real valve. Another contribution is that a procedural methodology has been introduced to obtain reliable valve parameters and physical coefficients from direct and indirect measurements combined with simulations. 5.2 Valve Description The schematic diagram of a typical two-stage proportional directional flow control servovalve is shown in Fig. 5.1. It has a pilot operated four-way spool valve as the second (main) stage, which controls the start, stop, direction, and amount of fluid flow for smooth acceleration and deceleration of a hydraulic actuator. The main spool is centered by a push-pull preloaded spring, as shown in the figure, and the main ports to and from the actuator are closed with a small overlap at the spool null (center) position. Chapter 5 NONLINEAR MODELING OF A SERVOVALVE 120 The first (pilot) stage of the valve is a proportional pressure control spool valve, which is used to control the pressure and direction of fluid flow in the first stage. The pilot valve basically consists of two proportional solenoids with associated pistons, capillary tubes, and compression springs, a control spool, a pressure supply port, a tank port, and two pilot ports which are connected to the end caps of the main spool, and are open to the tank port at the pilot spool's null position. An input voltage signal to an electronic amplifier card provides current to the solenoids which in turn is converted into a proportional force. If, for instance, Solenoid B is activated, its corresponding force pushes directly against the pilot spool through a push-pin and shifts the spool to open pilot Port A to the pressure supply. To stabilize the servovalve operation, the pilot pressure building up in the end cap of the main spool is also directly fed back to the A end of the pilot spool, as shown in Fig. 5.1. When the pilot pressure at Port A is high enough to equal or exceed the net force of Solenoid B, the pilot spool pulls back towards the center to no-flow condition within the deadband region, and closes the connection from the supply pressure to Port A, while modulating to hold the pilot pressure nearly constant. 5.3 The Servovalve Model Fig. 5.2 illustrates the electro-mechanical and hydraulic energy exchange/conversion between the four components of the servovalve. The proportional solenoids convert electrical power input into a weak mechanical power via coils and armatures. The pilot stage converts this mechanical power into a low- to medium-level hydraulic power by a control spool. The main stage takes the hydraulic power from the pilot stage and converts it again into a high-level hydraulic power through another spool. The hydraulic power thus obtained can then move a hydraulic actuator. The governing equations of motion for each individual component, when either of the proportional solenoids is activated by an input voltage, are derived next. Section 5. 3 The Servovalve Model 121 voltage force current velocity Pilot Spool ^ o o 1/1 1/1 o OH Pilot Valve i <D 3 o CO CO \ Main Spool force velocity Solenoid •I A voltage input current o CO CO OH output r Fig. 5.2: Electro-mechanical energy flow and conversion between various valve components. 5.3.1 Solenoid Model A. Solenoid Force: In its simplest form, the solenoid coil may be considered as a resistor in series with an inductor. If the magnetic core has a very high permeability and there is very little saturation in the magnetic circuit, the solenoid voltage/current relationship can be derived using Kirchhoffs voltage law [81 White and Woodson, 1959] as vs = Ris + Lis + i. dL dx« (5.1) In Eq. (5.1), the effects of changing the air gap between the coil and the armature is modelled by including the rate of change in the coil inductance L with respect to the armature displacement xs. A schematic diagram for this simplified solenoid model is given in Fig. 5.3. Chapter 5 NONLINEAR MODELING OF A SERVOVALVE 122 input voltage t< Nonlinear Inductor armature displacement and velocity . dL • R Xs, Xs 0.5 i .2 dL armature force fs solenoid current Resistor Fig. 5.3: A simplified solenoid model. Nonlinear Inductor input voltage 1 s h(X,Xs) —X— armature force f(K xs) restoring current g(vt) dissipation current 1+TS lr ld + R armature position fs Xs solenoid current Resistor Fig. 5.4: An improved solenoid model [69 Vaughan and Gamble, 1996]. In reality, however, due to the losses in the iron core, magnetic saturation, and hysteresis effects, the solenoid inductance is also a strong function of the coil current. An improved nonlinear model for a proportional solenoid is given in [69 Vaughan and Gamble, 1996] and is illustrated in Fig. 5.4, where good agreements between the measured and the simulated current and armature displacement are reported. Following their methodology and considering a nonlinear model for the solenoid inductor, we may obtain Section 5. 3 The Servovalve Model 123 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 Time, t (s) Fig. 5.5: Coil voltage, restoring, dissipation, and total coil currents for the test rig's servovalve solenoids. the voltage drop across the nonlinear inductor, as: vi - vs - Ris (5.2) Assuming a lumped parameter hysteresis model for the inductor, [8 Chua and Stromsmoe, 1970] gave the solenoid current is as the sum of a restoring current and a dissipation current: is = ir + id (5-3) where, in general, ir and id are strictly monotonic functions of the flux linkage in the inductor, A, and voltage drop across the inductor, v/, respectively. Typical plots of coil voltage, restoring, dissipation, and total coil currents for one solenoid of our test rig's servovalve are shown in Fig. 5.5, when the solenoid has been subjected to a step input voltage. Chapter 5 NONLINEAR MODELING OF A SERVOVALVE 124 [69 Vaughan and Gamble, 1996] showed that the restoring component of the solenoid current is also a strong function of the air gap between the solenoid coil and armature; i.e., t r = /(A,ar.) (5.4) id = g(vi) - Tid (5.5) where the flux linkage can be easily derived from Faraday's Law: X = vi (5.6) It has been shown that the force generated by an ideal solenoid is proportional to the square of flux linkage and the solenoid armature position [81 White and Woodson, 1959]. For a non-ideal solenoid, we may use a similar nonlinear force/flux relationship [69 Vaughan and Gamble, 1996]: /, = h(X2,xs) (5.7) The nonlinear functions / , g, and h, along with the dissipation time constant r, are solenoid dependent, and are empirically determined in Section 5.4.1. B. Armature Equation of Motion: Either of the solenoid armatures can apply force fsa or fsb to the pilot spool through an on-off contact of its own push-pin. The contact characteristic between the spool end and the push-pin head has been modelled as a very stiff contact-noncontact spring with stiffness Ksp. In reality, the magnitude of Ksp is very high. In simulation, however, a compromise should be made in choosing an appropriate value for this coefficient such that it is neither so large that it gives rise to numerical instability and excessive computation time, nor so small that it significantly alters the natural frequency, the static response, and the dynamic response of the moving components. Other forces that act upon the armature are the pilot pressure force at Port A or Port B, viscous damping force, Coulomb friction force, and the solenoid push-spring force. The resultant force cause the solenoid armature to accelerate. Therefore, for Solenoids A and B we can write: Msxsa = Appa - fsa - Csisa. - Ffssgn(£sa) - Ksxsa + A' s pmax(0, xp - xsa) (5.8) Section 5. 3 The Servovalve Model 125 Msxsb = fsb - Appb - Csxsb - Fjssgn(xsb) - Ksxsb - A' s pmax(0, xsb - xp) (5.9) In Eqs. (5.8) and (5.9), a positive solenoid force has been assumed when the voltage input to the solenoid, v„ is positive and the armature pushes against the pilot spool. It is worth noting that, in the above equations and equations that follow, the use of mathematical switches "max", "min", and "sgn" allows us to nicely include discontinuous nonlinearities of the system in a compact yet comprehensive manner. The viscous damping term Csxs in (5.8) and (5.9) is mostly due to the connection of solenoid chambers to the tank port through the narrow pipes in Fig. 5.1. Solenoids driving hydraulic servovalves are usually designed to have high damping to increase stability and reduce unwanted oscillations. The capillary tubes in Fig. 5.1, for instance, build up pressures at the end caps of the pilot valve as the armatures move and push/pull the oil through the pipes. With laminar flow inside the capillary tubes, the resisting pressure buildup, pcp, at the end caps of the pilot valve can be computed from the well-known Hagen-Poiseuille law [47 Merritt, 1967]: P c p = nDL 9 c p = vDi yup ua) cp " cp 2 n 2 i (5.10) .4 This pressure causes a damping force, fcp = (-K/4)(DP2 - Da2)pcp, that opposes the armature motion. Substituting for pcp from Eq. (5.10) and comparing fcp to the viscous damping terms in Eqs. (5.8) and (5.9), we obtain {Dl-Dl\2 C, = 8*nLj PD^ a \ (5.11) Note that the armature viscous friction effect, though much smaller than the capillary damping effect in Eq. (5.11), may be incorporated in this equation as an additional constant parameter. 5.3.2 Pilot Valve Model A. Flow Equations: Depending on the sign of the input voltage, when one of the two solenoids is activated, a net force is exerted on the left or right end of the pilot spool through the corresponding push-pin. If for instance, a Chapter 5 NONLINEAR MODELING OF A SERVOVALVE 126 positive voltage is applied, Solenoid B will be activated and the pilot spool starts moving from the null position to the right. The spool first closes the connection between the tank port and Port A at xp = Lt, and then after passing the deadband region (Lt < xp < Ls), opens Port A to the pressure supply port at xp = Ls. Thus, three distinct net inflow states to Port A exit, i.e., qa\, qa2, qa3, which correspond to the above three spool positions (see Fig. 5.6). The net inflow to Port A from Ports P and T, is therefore given by: qa =max[0, sgn(Xi - xp)]qa\ + max[0, sgn(xp - Ls)]qa3+ (5.12) {max[0,sgn(xp - Lt)\ - max[0,sgn(xp - Ls)]}qa2 where Qai = Cls(ps - Pa) ~ Cdtdt^Pa, (xp < Lt) qa2 - Cls(ps ~ Pa) ~ CltPa, (Lt < Xp < Ls) (5-13) qa3 = CdsaS)J^(Ps ~ Pa) ~ CltPa, (Ls < Xp) Meanwhile, the net inflow to Port B from Ports P and T can similarly be written as: qb =max[0 , sgn(Z« + xp)]qbl + max[0, sgn(-z p - Ls)]qb3+ {max[0,sgn(-xp - Lt)] - max[0, sgn(-a:p - Ls)]}qb2 where qi,i, qbi, and qb3 are similar to qai, qa2, and qa3, with pa replaced by pi,-(5.14) The dependency of the orifice area to the spool displacement is an important issue that has usually been simplified as a linear relation, assuming rectangular ports. In practice, the majority of variable area orifices of commercial spool servovalves, including ours, take the form of a segment of a circle due to the ease of manufacturing of circular ports. In this case, the orifice area can easily be obtained (see the insert in Fig. 5.6) from a0 = R20 cos"1 (1 - y0/R0) - (1 - y0fR0W(y0lX)(2 - y0/R0) (5.15) When the orifice is open to the supply port, a„ = as, R0 = Rs, and y 0 = ys. Similarly, when the orifice is open to the tank port, a„ = a,, Ra = Rt, and y0 = yt, where for Port A it gives: ys = min[2 JR s,max(0,x p - Ls)], yt = mm\2Rt,max(0, Lt - xp)} (5.16) Similarly, for Port B we have: ys = mm[2Rs,m!Lx(0,-xp-L3)], yt = mm[2Rt,ma.x(0,Lt + xp)] (5.17) Section 5. 3 The Servovalve Model 127 Pilot Spool Stroke, xp (mm) Fig. 5.6: Pilot valve orifice area vs. pilot spool stroke. A typical plot of the pilot valve orifice area for Port A vs. the pilot spool stroke is given in Fig. 5.6, where a significant difference between the tank-side and the pressure-side orifice areas is seen. Also, there is a deadband length, Ls - Lh in which the valve is open neither to the tank port nor to the pressure supply port. Another important effect that is often ignored is the variability of flow regimes between laminar and turbulent through the servovalve's variable orifice. In fact, the flow coefficients Cds and Cd, in Eq. (5.13) are strong functions of the orifice Reynolds Number, especially when the flow is laminar. The orifice Chapter 5 NONLINEAR MODELING OF A SERVOVALVE 128 Reynolds number is denned as Re = ESSBL (5.18) p.a0 where D/, is the orifice hydraulic diameter and denned as 4 X flow section area Aa0 Dh = 7, : : = (5.19) flow section perimeter s0 and q„ is the flow across the orifice area aa. Depending on which orifice we are considering, we may write q0 = Cdoa0yJ^APo (5.20) where Cd,, is discharge coefficient and Apa is the pressure drop across that particular orifice. Substituting (5.19) and (5.20) in (5.18), we get E e = ^ * (5.21) For a segment of a circle, sa can be calculated from s0 = 2RC (5.22) From the experiments in [38 Lewis and Stern, 1962] for flow through the segment of a circle, we may approximate the discharge coefficient as n - r (nrC ^ / 0 . 1 6 + 0.281og10(i2e), (Re < 100) Cd0 - Cds(or Cdt) « | Q { R e > 1 0 Q ) (5.23) B. Pilot Pressure Equations: Let's first consider the control volume in chamber A between the pilot spool and the main valve spool. Aside from the net inflow to Port A, the motion of the pilot spool adds oil to the chamber, while the motion of the main spool takes oil from the chamber into the main valve. There is also a slight leakage from the chamber to the tank through the main valve spool clearance. The net difference between the flows in and out of the chamber causes the oil to compress. We can derive the pilot pressure building up at Port A as: Pa - y , A ~A~( Z T ^ a ~ A M X M + AP(XP ~ Xsa) ~ C'm(Pa ~ Pt)] (5.24) Section 5. 3 The Servovalve Model 129 where pa[Vao + A.mxm - Ap(xp - xsa)]//3e is the portion of the inflow to the control volume that has been compressed due to the chamber-oil compliance, and Ap(xp — xsa) is an excessive flow to Port A when the armature and the pilot spool are detached from each other. The pilot pressure at Port B can similarly be written as: Pb = T7 ~A T~T1 \ + AmXm - Ap(xp - xsb) - Cim(pb - Pt)] (5.25) » 6 0 — A-mXm +• Ap^Xj, Xsb) C. Equation of Motion for the Pilot Spool: Either of the solenoids can apply a driving force on the pilot spool. The pilot pressure, on the other hand, has the role of a natural feedback force, which opposes the solenoid force. The total force acting on the pilot stage spool is due to the solenoid push-pins, pilot pressure differential, Coulomb friction, steady-state and transient flow reaction, and viscous damping [47 Merritt, 1967]. The flow reaction force exists due to fluid acceleration across the flow area and contains a static component due to momentum change plus a dynamic component due to fluid acceleration, although the latter tends to be neglected. The net external force causes the spool to accelerate. Applying Newton's second law, we obtain: MpXp =A ' s p[max(0, xsb - xp) - max(0 ,Zp - xsa)] - Ap(pa - pb)-r ^ r • H*DpLp . (5-26) F / p sgn(x p ) - fjslow - pLspqD -f-+xp Of where the flow rate through a segment of a circular orifice, neglecting the discharge coefficient rate and pressure rate terms, may be approximated by 2 C d s R s ^ ( ^ ) (2 - 2 ^ ) yjfy. - ps)ip, (xp > Ls) q ° * j 2CdsRs]J(=2£±) (2 - =S£±) ^ J ( p . - pb)xp, (xp < -L.) ( 0, (otherwise) The steady-state flow force in Eq. (5.26) can be calculated from f S S f 2 c o s 0[Cdsas(ps - pa) + Cdtat(pb - pt)], (xp > 0) \ 2 cos 6[-Cdsas(Ps-Pb)-Cdtat(pa-pt)], (xp < 0) P " Z 8 ; where 9 is the jet angle across the orifice. When the flow is two-dimensional across a rectangular orifice, the recommended value for 6 is 69° [47 Merritt, 1967]. Although in a circular segment orifice, where the flow is no longer two-dimensional, we might still use this jet angle value as an approximation if there are no other data available. (5.27) Chapter 5 NONLINEAR MODELING OF A SERVOVALVE 130 5.3.3 Main Valve Model In Chapter 4, we derived flow equations for rod-side and head-side ports of the main valve. We also showed that the oil pressure and flow through the main valve depend not only on the main valve opening (spool displacement), but also on the motion of the hydraulic actuator and other components present in the hydraulic circuit. Since in this chapter we are interested in the dynamic behavior of the servovalve, only the equation of motion of the main spool is of importance, as long as flow forces in the main valve are negligible (e.g., by locking the actuator). D. Equation of Motion for the Main Spool: The spool of the second stage valve is initially centered by a precompressed spring. When the oil pressure in Chamber A is built up enough to overcome this preload, the spool starts moving to the left (positive direction), and opens Port Y to the pressure supply port and Port Z to the exit port. The total force acting upon the valve is therefore due to the pilot pressure differential, spring compression, spring preload, Coulomb friction, viscous damping, and flow force through the main valve. Newton's second law then gives the inertial force as: MmXm =Am(Pa - Pb) - Kmxm - Fpmsgn(xm) - Ffmsgn(xm)-(5.29) Cmxm - 0ASaom[ps - (py - Pz)sgn(xm) - Pe]sgn(xm) As was mentioned above, the flow force term in the above equation can be neglected if, for example, the hydraulic cylinder is locked. 5.4 Model Validation and Performance Analysis o Note: all the experiments were performed when the oil temperature was 40 ± 5 C, and the sampling frequency for data acquisition was 1 KHz. 5.4.1 Determination of Model Parameters The structure of the valve model was given in the previous section. This model, however, requires a Section 5. 4 Model Validation and Performance Analysis 131 Table 5.1 Numeric values for parameters from the manufacturer or direct measurement. Ap = 23 mm 2 Mp = 0.025 kg Lp = 40 mm Clm, Qs, C] t = 20 mm 3/s/MPa M , = 0.085 kg Ls = 0.67 mm Cr = 0.012 mm R = 19.6 ft L, = 0.47 mm Ffi = 0.01 N V«o, V M = 16990 mm 3 /?, = 1.4 mm Fpm = 46 N Da = 1.5 mm R, = 3.0 mm K,„ = 132 N/mm D c p = 1.7 mm B = 1200 MPa K, = 7.5 N/mm Dp = 12 mm H = 0.0267 Ns/m 2 M,„ = 0.220 kg LCp = 17 mm p = 852 kg/m 3 number of parameters and physical quantities to be determined in order to produce reliable simulation results. Determination of all the important coefficients in flow and force equations is the first task when considering dynamic performance, and experience is required in addition to a study of the technical literature. Basically, two sets of parameters have been recognized for the current situation. The parameters in the first set are those quantities that are fixed for a series of identical valves and are either available from the product manufacturer, or directly measurable from a typical valve. The values for these parameters are given in Table 5.1. The parameters in the second set are those that are not directly measurable nor given by the manufacturer, but can be determined indirectly through measurements of a number of system states and by using a variety of curve-fitting techniques such as linear and nonlinear least-squares [11 Dennis Jr., 1977]. A. Solenoid Parameters: First, consider the solenoid empirical functions / and h in Eqs. (5.4) and (5.7), respectively. We took apart one of the two solenoids from a typical servovalve and applied incremental step voltages from 1 to 15 V to its coil at a number of different but fixed armature positions, between -0.4 and 1.5 mm (w.r.t. the armature null position), and recorded the time histories of the solenoid voltage/current. The steady-state values for the armature force were also measured using a load cell. To obtain functions that properly Chapter 5 NONLINEAR MODELING OF A SERVOVALVE 132 describe the current/flux and force/flux relationships, the following identification procedure was adopted: i. Obtain coil voltage time histories from Eq. (5.2) using solenoid voltage/current step response measurements (R is given in Table 5.1). ii. Integrate Eq. (5.6) to obtain the flux linkage A. iii. Set ir = is at steady-state (ij = 0). iv. Fit a two-dimensional function to the steady-state experimental magnetization curves of ir vs. A at different xs positions (Fig. 5.7a). A polynomial of the form: ir = f(X,xs) (5.30) = ( / 4 i a ? . + / 4 o)A 4 + (/31a;, + / 3 0 ) A 3 + (f2lx, + / 2 0 ) A 2 + (fnxs + /io)A was a natural choice and gave the best fit with the coefficients obtained using the "linear least squares" technique. v. Fit a two-dimensional function to the steady-state experimental force/flux curves at different armature positions (Fig. 5.7b). A polynomial of the form: /s = h(X2,xs) -(h41xs + h40)(X2)4 + (h31xs + h30)(X2)3+ (5.31) (h2ixs + h20)(X2f + (hnx. + h10)(X2) was fitted to the data, again, using the linear least squares technique. To obtain the dissipation function g(vi) in Eq. (5.5), [69 Vaughan and Gamble, 1996] proposed piecewise power functions gn |uj|a'2sgn(u;) at various voltage levels to represent "S-shape" transient curves of ij vs. v/, and used simulations to adjust the dissipation time constant, r , to give a good fit with the experimental results. In contrast, we have used a single continuous sigmoidal function with fewer parameters, and a nonlinear iterative least-squares plus integration method to optimally estimate the g(vi) parameters and T simultaneously. The following procedure was adopted: i. Follow Steps i. and ii. above. ii. Compute the transient restoring current time histories, ir, using the identified restoring function f(X, xs) from Eq. (5.30). iii. Calculate the transient dissipation current ij from Eq. (5.3). Section 5. 4 Model Validation and Performance Analysis 133 o: Experiment : Fitted Curves 0.81 1 1 1 1 1 r gl i I i i i i 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Flux Linkage (Wb) o: Experiment : Fitted Curves 701 1 1 1 1 1 r Flux Linkage (Wb) Fig. 5.7: Solenoid current-flux (above) and force-flux (below) relationships. Chapter 5 NONLINEAR MODELING OF A SERVOVALVE 134 Table 5.2 Identified values for solenoid parameters. ft, =-187.2 fio = 0.272 A/y = -3.90X104 fto = 0.336 h41 = -1.37X106 A/o = 5.49X101 fsi = 1869 h40 = 1.24X103 gi = 0.061 fso = 0.296 Aj/ = 1.26X106 g2 = 3.521 hi = -1703 h30 = -1.48x10s r = 1.4 ms fio= 1-030 A 2 / = -2.44X 105 / / / = 162.2 A2o = 6.27 x l O 2 iv. Select a function that visually, best describes the plots of ij vs. v/. In this case, a sigmoidal function was chosen as: ^ ' ) = ^ ( i T ^ - 1 ) ( 5 3 2 ) v. Assume initial values for the parameters g\, g2, and r. vi. Using vi time histories for each step input, calculate g(vi) from (5.32), and solve for ij by integrating Eq. (5.5) (e.g., the 4th order Runge-Kutta algorithm ode45 in Matlab). vii. Compare the measured dissipation current from iii. with the estimated one from vi. and generate an error vector. viii. Use the error vector and the nonlinear least-squares algorithm (e.g., the function l e a s t s q in Matlab's Optimization Toolbox) to update the parameter estimates. ix. Iterate steps vi. to viii. above until convergence is achieved. x. If convergence not achieved after certain iterations, go to step v. All the identified values for the solenoid parameters are tabulated in Table 5.2. The identification algorithm mentioned above is schematically illustrated in Fig. 5.8. An example of the estimated dissipation current obtained from iterated coefficients gi, g2, and r in Steps vi to ix above, along with its final optimum value and the one obtained from measurement, are plotted in Fig. 5.9. Section 5. 4 Model Validation and Performance Analysis 135 input voltage v, + armature position (fixed) f(Kxs) restoring current T " nonlinear least-squares 1 1+xs dissipation current , solenoid current (measured) Ae„=A9 0 +Ke y + i, Error Vector "e" Resistor Fig. 5.8: Schematic diagram of the solenoid identification algorithm. Coil Voltage, VI (V) 0.1 r Time, t (s) Fig. 5.9: Iterated dissipation currents from initial condition to final optimal value, along with the current obtained from the measurement. Chapter 5 NONLINEAR MODELING OF A SERVOVALVE 136 B. Pilot Valve Parameters: For 15 different step input voltages vs, ranging from +1 to +15 V (maximum allowable voltage was 15.7 V), the current Is of Solenoid B, the supply pressure ps, the pilot pressures pa and pb, and the main spool displacement x,„ were recorded. We were not able, however, to measure the pilot spool displacement xp due to the presence of the solenoids at both ends of the pilot spool (see Fig. 5.1). Also, the solenoid force was computed from voltage/current measurements using Eqs. (5.2) to (5.7). Next, by setting all the differential terms equal to zero, we wrote the steady-state versions of Eqs. (5.9) and (5.26) when the valve was closed, and added l.h.s. and r.h.s. terms to obtain A p P a + Ksxps = acfj + (Ffs + Ffp) (5.33) The linear least squares technique was applied to estimate the solenoid force correction factor ac (due to the use of a solenoid from a different valve of the same model) and the combined solenoid and pilot valve Coulomb friction force (FfS + FfP). The solenoid force /* 6 S was calculated from simulations using Eqs. (5.2) to (5.7) with solenoid voltage and current measured. The steady state position of the pilot spool xps (or the solenoid push-pin) was considered on the average to be 0.67 mm off the centre position, and in all cases the spool was moving towards the centre when it stopped. Thus, the l.h.s. of Eq. (5.33) were all known measured quantities. The identified parameters are given in Table 5.3. Table 5.3 Identified values for the pilot and main valve parameters. Pilot: | Ffp = 0.55 N ac = 1.07 Main: | A m = 308 mm 2 Cm = 30 Ns/m Ffm = 2.5 N Finally, although the solenoid viscous damping coefficient Cs could be directly determined from Eq. (5.11), we used simulated and experimental pa(t) (or pb(t)) curves as an alternative way to obtain a value for Cs, assuming all other parameters are already known. There are two reasons for doing so. First, some solenoids may not have capillary tubes as a high damping means. Second, the viscous friction effects may be comparable to other viscous damping effects. In every case, we have noticed that C, Section 5. 4 Model Validation and Performance Analysis 137 increases the pressure-buildup time delay by slowing down the movement of the combined armatures and pilot spool from the null position. Using this fact, we applied different step input voltages and simulated the valve model with several trial values of Cs until the pressure rise start times matched the measurements. The final value for Cs was about 7% more (due to viscous friction and other effects) than what we could obtain from Eq. (5.11). C. Main Spool Parameters: The last set of parameters to determine are the pressure area of the main spool A,n (which, as opposed to the pilot spool area, was not easy to measure due to a complex design), the Coulomb friction force Ffm, and the viscous damping coefficient Cm. These identified numbers are given in Table 5.3. A,„ and F/m were obtained from a number of steady-state spool position and pilot pressure measurements by applying the linear least squares method to the steady-state version of Eq. (5.29), i.e., Kmx% + Fpm = Am{psas - psbs) - Ffm (5.34) The flow force term in Eq. (5.29) has been dropped from Eq. (5.34) because the flow force was kept zero during the experiments by blocking the rod-side and the head-side ports of the main valve. Finally, in order to obtain the viscous friction coefficient Cm from Eq. (5.29), we applied a number of step input voltages and measured the time histories of the pilot pressure differentials and the main spool displacements. Assuming that all other coefficients in Eq. (5.29) are either known or previously identified, we then applied the same "iterative nonlinear least squares plus integral" algorithm that was previously used for estimation of the dissipation function coefficients. First, an initial value for Cm was postulated, and Eq. (5.29) was integrated by MATLAB's ode23 function to obtain the main spool displacements using pilot pressure differential measurements. The errors between the measured and the computed main spool positions were iteratively used by MATLAB's l e a s t s q function to update Cm in a least squares sense. This nonlinear least squares plus integral algorithm, unlike the linear least squares method used in [69 Vaughan and Gamble, 1996], does not require differentiation of noisy spool position measurements to get the velocity and acceleration of the main spool in Eq. (5.29). Chapter 5 NONLINEAR MODELING OF A SERVOVALVE 138 ffl2.5 CL g. 2 CD CD C~ 1 O K 0.5 Simulation o Experiment 0©-0 10 20 30 40 50 60 Input Voltage (%) 70 80 90 100 0.5 1 1.5 2 Pilot Pressure Differential (MPa) 100 Fig. 5.10: Steady-state pilot pressure and main spool stroke as functions of input voltage. 5.4.2 Steady State Behavior To evaluate the steady state performance of the model, Equations (5.2) through (5.29) were solved simultaneously using SIMULINK® simulation software and the fourth-order Runge-Kutta integration algorithm. The parameter values given in Tables 5.1 to 5.3 were used for this purpose. To get stable results in a reasonable time, the contact stiffness Ksp was set to 10000 N/mm in all simulations. Sixteen different step input voltages were applied to Solenoid B, and the steady state pilot pressure for Port A and the main spool stroke for both the model and the actual valve were obtained and plotted in Fig. 5.10. The root mean square (RMS) error for the output pressure is 0.04 MPa, or less than 1.5% of the maximum pilot pressure of 3.00 MPa. The RMS error for the main spool stroke is 0.12 mm, or less than 2.0% of the maximum spool stroke of 6.35 mm. In Fig. 5.10, apparently, the relationship between the Section 5. 4 Model Validation and Performance Analysis 139 solenoid input voltage and the pilot valve output pressure is slightly nonlinear, especially at limit values. On the other hand, a more linear relationship between the pilot pressure and the main valve spool stroke can be observed, as is expected from Eq. (5.34). 5.4.3 Frequency Response Behavior To compare the frequency response of the valve model with the actual valve, a sinusoidal voltage input with a time varying frequency [40 Lin and Akers, 1991] between 0.1 to 20 Hz and a constant amplitude of 1.5 V (about 10% of the maximum allowable input voltage) was applied to one of the solenoids of the model and the actual valve. A DC voltage of 8 V was also added to the periodic signals to avoid centering the pilot spool. The frequency response plots of the pilot pressure differential for simulation and experiment are shown in Fig. 5.11. This spool-operated servovalve is obviously about two orders of magnitude slower than a typical flapper-nozzle servovalve (see, e.g., [67 Tsai et al, 1991]). The amplitude error between the model and the actual valve is within 2 dB, and the phase lag error is less than 20 degrees. 5.4.4 Time Response Behavior A. Response to Step Inputs: Time response characteristics of the valve and the model were investigated by applying four step voltage inputs to one solenoid. The time response plots for the solenoid current, pilot (1st stage) pressure, and the main (2nd stage) spool stroke are given in Fig. 5.12, where the simulated solenoid current matches very well with the measured current, indicating that the model suggested in [69 Vaughan and Gamble, 1996] has an excellent predictive accuracy. From Fig. 5.12, there is also a good agreement between the steady-state and dynamic responses of the simulated valve and the actual valve over a wide range of voltage inputs. From the step response curves, both the pressure buildup time delay and its rise time become smaller Chapter 5 NONLINEAR MODELING OF A SERVOVALVE 140 Section 5. 4 Model Validation and Performance Analysis 141 3 E 100i 1 1 1 1 1 1 1 oT so- r 1 > S 6 0 -cg | 4 0 - — | 2 Q -•g o o 1 0) I i i i i 1 1 1 1 o 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time (sec) T 1 1 1 1 1 r Time (sec) 2.51 1 1 1 1 1 1 r Time (sec.) Fig. 5.12: Time response of the solenoid current, pilot pressure, and main spool displacement for four step input voltage levels. Chapter 5 NONLINEAR MODELING OF A SERVOVALVE 142 as the input voltage increases. An average rise time of 100 ms also indicates that this spool-operated servovalve is much slower than a typical flapper-nozzle two-stage servovalve. However, as opposed to the latter, the spool valve lacks the undesirable overshoots and undershoots when a step input is applied. One major reason is the existence of a small deadband, in which the pilot Ports A and B are both closed to the tank and the pressure supply ports (see Fig. 5.6). In Section 5, it has been shown that removing the deadband in the model adds undesirable oscillations to the pressure outputs. B. Response to an Arbitrary Input: To validate the overall performance of the valve model, an arbitrary positive voltage was applied to the valve for 10 seconds. Simulation and experimental results are compared in Fig. 5.13. The RMS error for the pilot pressure and the main spool stroke was less than 63 KPa and 0.17 mm, respectively. Again, the simulated solenoid current closely matches the measured current, while the simulated pilot pressure and the main spool displacement agree fairly well with the experimental values. The discrepancy between experiment and theory shown in Fig. 5.13 is representative of fluid power circuit analysis [12 Dransfield, 1981] [75 Watton, 1992]. Here, practical variations such as system noise, uncertainty and variability of such parameters as orifice jet angle and viscous damping coefficients, simplification of friction effects at very low speeds, variable leakage and also severe nonlinearities in flow gain, pressure sensitivity, and orifice discharge coefficients, especially around deadband area (xp « Ls and xp « Lt), cannot usually be modelled very well. 5.5 Sensitivity Analysis One of the great advantages of having an accurate numerical model for a valve is that it enables us to easily evaluate the valve behavior under severe operating conditions and measure the valve sensitivity to changes in design parameters. This kind of analysis might be very costly and time-consuming, if not impossible, having only one actual valve at hand. Several independent cases are considered below with the steady-state results tabulated in Table 5.4. Section 5. 5 Sensitivity Analysis 143 > 101 1 1 1 1 1 1 1 1 r Time (sec) 0.51 1 1 1 1 1 1 1 r Q1 i i i i i i i i i I 0 1 2 3 4 5 6 7 8 9 10 Time (sec) Fig. 5.13: Time response of the solenoid current, pilot pressure, and main spool displacement due to arbitrary input voltage. Chapter 5 NONLINEAR MODELING OF A SERVOVALVE 144 Table 5.4 Sensitivity of the valve outputs to changes in various variables. Variable Change in variable (%) Change in pilot pressure differential (%) Change in main spool displacement (%) Oil Viscosity +1200 -1.5 -2.0 Oil Supply Pressure -33 +0.10 +0.25 +100 -0.05 -0.11 Pilot Deadband -100 -0.05 -0.17 Oil Bulk Modulus +900 -0.2 -0.10 Pilot Spool Friction +100 +1.7 -2.4 A. Effect of Cold Start: Lowering the oil temperature causes a dramatic increase in hydraulic oil viscosity (see, e.g., Eq. (4.3)). As a result, orifice Reynolds number proportionally decreases and the flow state might become laminar across the orifice, [47 Merritt, 1967]. This, in turn, reduces the flow coefficient according to Eq. (5.23), and causes sluggish pressure buildup in the pilot ports. On the other hand, the viscous damping coefficients increase proportionally with the oil viscosity according to Eqs. (5.11) and (5.26), thus slowing down the solenoid armature and pilot spool motion. This has an adverse effect on the pilot spool stability, since it now takes longer for the pilot spool to move towards the deadband region and close the pilot ports to the supply and the tank ports. The slight overshoot and time delay in the pilot pressure and the main spool displacement shown in Fig. 5.14 is the result of such sluggish motion of the spool (also shown in the same figure) and lack of strong pilot pressure feedback, as described above. Reducing the oil temperature below -20 °C (the minimum recommended oil temperature by the valve's manufacturer) caused high amplitude oscillations and made the valve model unstable. B. Effect of supply pressure: The effect of supply pressure on the pilot stage response was investigated using the model. Fig. 5.15 shows that higher pressure supply results in a smaller spool displacement and produces a higher overshoot of the output pressure. This is also true, though at a larger scale, for a typical flapper-nozzle servovalve, [40 Lin and Akers, 1991]. Amongst different supply pressures tested, the one recommended by the valve Section 5. 5 Sensitivity Analysis 145 Legend: Pilot Spool Solenoid "A" -.-Solenoid "B" 21 1 1 1 1 1 1 1 1 r 0 0.1 0.2 0 .3 0.4 0.5 0.6 0.7 0.8 0.9 1 •—-1.51 1 1 1 1 1 1 i i i i £j 3 i 1 1 1 1 1 i i i r Time (sec) Fig. 5.14: Step response of the valve when: 1) / J = 0.027 N.s.nr2 at 40 °C, 2) p = 0.350 N.s.nr2 at 0 °C. manufacturer, i.e., ps = 7.6 MPa, gave the least overshoot, and therefore was an optimum supply pressure. C. Effect of the Porting Deadband: As stated earlier, the deadband in the pilot spool plays an important role in the stability and repeatability of the servovalve. Also, its reduction in time can be a good indication of the valve wear and aging. Removing the deadband, i.e., making Ls = L,, caused the pilot pressure to irregularly oscillate, as is shown in Fig. 5.16. Chapter 5 NONLINEAR MODELING OF A SERVOVALVE 146 Ps = 2.5 MPa Ps = 3.8 MPa Ps = 7.6 MPa Ps = 15.2 MPa 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Time (sec) T 1 1 1 1 r 1 1 r 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 Time (sec) Fig. 5.15: Time response of the pilot spool displacement and the pilot pressure for different supply pressure levels. With Deadband Without Deadband 0.2 0.4 0.6 0.8 1 1.2 Time (sec) Fig. 5.16: Step response of the pilot valve pressure differential with and without the deadband. Section 5. 6 Summary and Concluding Remarks 147 D. Effect of Bulk Modulus: We artificially increased the value of the bulk modulus of the oil by a factor of 10. However, very little change in the static and dynamic response of the servovalve model was observed. This is mostly due to the fact that the pilot and main valve pressure chambers are very small and there is no external connecting line between those two valves. As a result, we could turn the differential equations (5.24) and (5.25) into algebraic equations by setting the pressure buildup rates on the l.h.s. equal to zero, and then solving for the unknown pressures Pa and Pb-E. Effect of Friction: Finally, the Coulomb friction coefficient of the pilot spool was doubled to simulate the oil contamination effects and wear of the pilot valve. As a result, the time delay for the initial valve response to a step input was increased about 10%. The steady-state values for the pilot pressure output and the main valve spool displacement also showed some increase, the consequence of which is reduced accuracy. 5.6 Summary and Concluding Remarks In this chapter, a nonlinear dynamic model for a two-stage spool-operated flow control servovalve has been developed which accurately predicts the spool displacement of the second stage valve to a voltage input. All the interactive valve components, including two proportional solenoids, a pilot stage pressure control valve, and a boost stage flow control valve, have been separately modelled and incorporated into a set of nonlinear simultaneous differential and algebraic equations that represent the complete system. Many nonlinear effects such as magnetic saturation, hysteresis properties, flow force, friction force, viscous damping force, orifice flow coefficient, bulk modulus of oil, and oil leakage have been incorporated into the analysis. Steady-state, frequency response, step response, and arbitrary input response analyses were performed on the complete model, and all the simulation results were compared with the corresponding experimental Chapter 5 NONLINEAR MODELING OF A SERVOVALVE 148 values. Agreement between simulations and measurements to within 10% was observed. In the process of analysis, the semi-empirical solenoid model suggested in [69 Vaughan and Gamble, 1996] was found to be an accurate model for solenoid actuated servovalves. The effects of various parameter variations on the valve performance were investigated separately, using the simulation model. We observed from these analyses that the valve stability is sensitive to the deadband size of the pilot valve, and, in general, is not affected by considerable variations (between 50% to 200%) in oil supply pressure, oil bulk modulus, oil viscosity, and pilot spool friction. Only at o very low temperatures (typically, less than 0 C) did the valve start oscillating. Among various factors, valve wear and very low oil temperatures, as reflected by increased spool friction coefficient and oil viscosity, were found to have more significant effects on the valve's steady-state accuracy. Variation of the supply pressure and oil bulk modulus has very small effect even on the valve's dynamic response. This indicates that the compressibility of the oil inside the valve chambers may be neglected in the analysis to get more simplified models. The model developed here can serve to establish functional relationships between various measurable and non-measurable valve characteristics in a systematic manner. This aspect is particularly important for on-line monitoring, supervisory control, and failure detection purposes, which are described in the next chapter. Chapter 6: CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 6.1 Introduction 6.1.1 Problem Statement Modern hydraulic systems are usually controlled by computers, where electronic signals or commands are transformed into powerful mechanical force or large motion via electrohydraulic servovalves. These electromechanical valves are probably the most vital elements in any hydraulic machine, since the satisfactory performance and cost of operation of the complete system depends mainly on how the servovalves respond to control commands and external or internal disturbance. For example, the electrical resistance of a solenoid-activated valve might be increased due to heating, which reduces the solenoid magnetic power for the same input voltage command. Wear, oil debris, and overheating also affect the orifice shape and friction coefficients of the internal moving components of a servovalve. These might change the response time, flow rate, and the line pressure generated by the device. In extreme cases, a sticking valve might result in a catastrophic accidents1. On-line monitoring of the valve's behavior through tracking certain system parameters and state relationships, and comparing them against known nominal values is an effective way to predict and prevent any further performance degradation over time (see Chapter 3 for detailed methodology). It not only helps to apply more efficient motion and force control strategies by adjusting control parameters, but also reduces substantially the maintenance cost and repair time. Usually, the root-cause of a failure in a knowledge-based expert system that, for example, uses Fault Tree Analysis (FTA) is assumed to 1) For example, in September 9th, 1994, a US Air B-737 aircraft crashed into the ground near Pittsburgh Airport due to the jammed spool of its rudder actuation system. 131 lives were claimed as a result of the accident, according to the media. Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 150 be a change in a system parameter, such as an increase in friction coefficient. Direct access to the valve's physical system states and coefficients can further reduce the fault diagnosis time by eliminating some fault tree branches from the top level event to the root-cause failure. In Failure Mode and Effects Analysis (FMEA) programs [26 Hogan etal., 1996], early warning of slight changes in system parameters may also decrease the number of multiple F M E A runs by reducing the chance of running into secondary or concurrent faults. This monitoring strategy based on parameter and state tracking, however, requires a reliable state/parameter estimator and sufficiently accurate models of the valve and other components over their entire range of operation. Input-output models for hydraulic components have already been used by a number of researchers [54 Plummer and Vaughan, 1995][55 Plummer and Vaughan, 1996]. The parameters of the so-called "black-box" model can then be determined using one of the many parameter identification algorithms [41 Ljung, 1987]. One problem with input-output models is the need to measure such rarely available states as actuator velocity and acceleration, and flow rates [78 Watton and Xue, 1995]. Another problem, from a fault diagnosis point of view, is that the mathematical parameters of the model, which usually have no "physical" meanings, reveal very little, if any, information about the condition of a component in the hydraulic system. State-space representation is another approach, in which linear or nonlinear models for various components of a hydraulic system have been used for the design of state observers [32 Jelali and Schwarz, 1995], filters [22 Handroos and Vilenius, 1990], and controllers [70 Vossoughi and Donath, 1995]. In these works, all the model parameters, along with some crucial states such as flow rates, orifice areas, and flow forces in the valve, have been assumed to be either "known" or "measurable". One of the most important quantities required in almost any fluid power diagnosis routine is oil flow rate through different components [4 Bull etal., 1996][26 Hogan etal., 1996][78 Watton and Xue, 1995]. However, accurate experimental data for flow rate, especially transient flow inside a servovalve, is rarely obtainable, principally due to mechanical restrictions and the lack of commercially available low-cost Section 6. 1 Introduction 151 high-speed robust flowmeters. Computation of flow rate and flow force in a valve is possible using measurements of pressure drop across the valve, valve spool displacement, and a "known" functional relationship between the spool position and the valve opening area (see, e.g., Eqs. (4.15) and (5.28)). This functional relationship, which we have termed "orifice effective area", even in its simplest linear form, might be altered over time due to sludge, wear debris, oil viscosity change, erosion, etc., and if assumed constant, introduces error in the estimate of flow rate and flow force in the valve. Besides, monitoring the orifice effective area, by itself, can provide us with valuable information about the current condition of the valve, as well as, about the working oil contamination level. The few attempts that have been made to estimate the orifice area characteristics are off-line and still require flow rate measurement across the valve [22 Handroos and Vilenius, 1990][83 Xu et al., 1996]. 6.1.2 Scope of the Present Chapter The applicability and effectiveness of the proposed integrated FDD methodology was shown in Chapter 3 using a simple damped oscillator example. In this chapter (and the one that follows), the same methodology has been applied to monitor and detect electrical and mechanical faults in a number of real hydraulic components described in Chapters 4 and 5. The focus has been on condition monitoring, fault detection, and qualitative fault symptom generation aspects of the proposed hierarchical methodology. Fault pattern classification, fault recognition, and fault diagnosis techniques can then be applied to the processed signals in a straightforward manner, according to the methods explained in Chapter 3. First, the on-line condition monitoring issue is addressed, which utilizes low-level input-output measurements and produces higher level system state and parameter estimates, using stochastic signal processing techniques. For this purpose, we have introduced novel nonlinear state-space representations for the solenoid, pilot stage, and the main stage components of the two-stage electrohydraulic valve described in Chapter 5. The state-space models have states and coefficients which closely correspond to the continuous physical variables. Only a few low-cost measurements, such as solenoid input voltage, Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 152 coil current, pilot and line pressures, and main spool displacement, have been assumed to be available. Extended Kalman Filtering (EKF) has then been applied "on-line" to estimate expensive-to-measure and/or unavailable states, such as solenoid magnetic force, pilot valve flow rate, and orifice area, as well as a number of crucial system parameters, such as coil resistance, friction coefficients, spring constants, and valve orifice deadbands. The capability and accuracy of the estimation procedure to recover unmeasured states and unknown system coefficients from the proposed state-space models have been tested using numerical data obtained from the valve simulation results of the previous chapter. To demonstrate the applicability of the real-time method in practice, experimental estimation results for a normal healthy valve have also been reported using the computer setup and the hydraulic test rig explained in Chapter 4. Next, on-line fault detection and fault symptom generation problems have been addressed. A number of fault cases have been investigated using both simulation and experimental data. System parameters and traceable states are estimated in real-time and are compared to their corresponding normal values. SPRT is applied to detect abrupt jumps in estimated parameters, and the parameter jump classification method introduced in Chapter 3 is used to generate different fault symptom patterns. 6.2 Modeling The two-stage proportional directional flow control servovalve is a very common valve which is typically used in heavy-duty mobile machinery. A detailed description and model for the valve is given in Chapter 5. The valve consists of two proportional solenoids, a proportional directional pressure control valve, used as a pilot stage, and a proportional directional flow control valve, which is used as a main (boost) stage. A schematic diagram of the valve is shown in Fig. 5.1. 6.2.1 Solenoid Model A semi-empirical nonlinear model for a proportional solenoid is proposed in [69 Vaughan and Gamble, Section 6. 2 Modeling 153 1996]. This model has accurately predicted the solenoid magnetic force and current in an actual two-stage servovalve (see Chapter 5). The model's block diagram is shown in Fig. 5.4, where the solenoid coil is represented by a nonlinear inductor in series with a linear resistor. The solenoid current is therefore the summation of a restoring current, ir, and a dissipation current, The induced force, fs, at the tip of the solenoid armature pin is a nonlinear function of the solenoid flux linkage and the armature position, and should be empirically determined for each set of identical solenoids (see Section 5.4.1 for more details). For an activated solenoid shown in Fig. 5.4, the input is the command voltage vs > 0, which causes current, is, in the coil. Both and is are easily measurable. On the other hand, the armature position is an external disturbance that affects both solenoid magnetic force and restoring current [69 Vaughan and Gamble, 1996]. To obtain the dynamic equations for the solenoid coil, we use Figure 5.4 and Eqs. (5.2) to (5.7). The coil flux linkage, A, is directly given by Faraday's Law, as A = vi (6.1) where vi-va- Ris - vs - R(ir + id) (6.2) and R is the coil resistance. The solenoid restoring current, ir, is assumed to be a "known" function of the flux linkage and the armature position, xs, i.e., ir = f(X,xs) (6.3) This functional relationship in the form of a polynomial for a particular solenoid is given in Eq. (5.30) with coefficients listed in Table 5.2. Accordingly, the solenoid dissipation current, ij in Fig. 5.4, can be computed from id = [g(vi)-id]/T (6.4) where the solenoid dissipation function, g, and dissipation time constant, r, are already known from Eq. (5.32) and Table 5.2. Finally, when h is known, the solenoid magnetic force may be determined from fs = h(X2,xs) (6.5) Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 154 Unlike and is, direct measurement of the armature position, xs, is not always possible due to mechanical restrictions or the lack of proper sensors. In this case, according to the simulation results of Chapter 5, the armature pin can be assumed to remain in contact with the pilot spool as long as the solenoid is active (see, e.g., Fig. 5.14a). In the next section, we have shown that the position of the active armature/pilot spool assembly can be obtained from the dynamic equation of motion for the combined armature and the pilot spool. In this case, xs may be replaced by the pilot spool position, xp, as xs = — xp (for solenoid A) (6.6) xs = +xp (for solenoid B) 6.2.2 Pilot valve Model Equations of motion for the armatures and the pilot spool, including all forces acting upon each individual component, are previously given in Chapter 5. The magnetic force, fs , from an activated solenoid causes the corresponding solenoid armature and the pilot spool to attach and move together. The resisting force on the assembly is mainly due to pilot pressure feedback, spring compression, viscous damping, Coulomb friction, and steady-state fluid flow. Assuming that the tank pressure p, is zero, and at any given time, one and only one of the solenoid armatures is attached to the pilot spool, we can derive the equation of motion for the combined pilot spool and the armature. Then, if solenoid B is attached, by removing the common terms in Eqs. (5.9) and (5.26), we obtain McXp =/ s6 - A p P a - KsXp - CfaXp - Ffasgn(xp)-(6.7) [max(0, aoa)(Ps - pa) - min (0, aoa)(Pa)](2 cos 0a) and when solenoid A is attached, from Eqs. (5.8) and (5.26), we obtain McXp = - fsb + ApPb - Ksxp - CfbXp - FfbSgn(xp)+ (6.8) [max(0, aob)(Ps - Pb) + min (0, aob)pb](2 cos Ob) where Mc = combined mass of armature-pilot spool, Ap = pilot valve pressure area, Pa, b - pilot pressures at ports A and B, Section 6. 2 Modeling 155 Ks = the spring constant, Cf = the viscous damping coefficient, Ff = the Coulomb friction force, ps = the supply pressure (nearly constant), and 8 = flow jet angle across the open orifice. aoa (or aab) is the pilot valve's orifice effective area for port A (or port B), which also takes the direction of flow in and out of the pilot chamber into account. aQ, representing aoa or aQb\s defined as where Cds and Cdt are the orifice discharge coefficients to the supply and tank ports, respectively. As was discussed in Chapter 5, the actual orifice areas of the pilot port to the supply port, as, and to the tank port, a,, are generally nonlinear geometric functions of the spool position, i.e., For the particular valve we are dealing with, the orifice profile and the size of its deadband as a function of the pilot spool position is plotted in Fig. 5.6. From this figure, we notice that although the magnitude of an orifice area, in general, may substantially change for a certain amount of spool displacement, the orifice profile slope varies only a few percent over a wide range of spool displacement, either in the tank side or in the supply side. Also, note that the orifice discharge coefficients Cds and Cdt in Eq. (6.9) are either constant (for turbulent flow) or vary slowly (for laminar flow) w.r.t. the orifice Reynolds number, which, by itself, is a strong function of spool position (see Eqs. (5.21) and (5.23)). A direct outcome of the foregoing discussion is that for the orifice effective area, we can now establish the following relationships using Eq. (6.10), as (\xp\ > Ls pilot port open to supply) (\xp\ < Lp pilot port open to tank) (otherwise) (6.9) as = as(xp), at = at(xp) (6.10) CL0 — tt0(#p) (6.11) and, (6.12) Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 156 where the orifice profile slope is given by (6.13) As mentioned above, s0 is a "weak" function of the spool position, and as such may be assumed as a "random-walk" parameter in a parameter/state estimation scheme. Eqs. (6.12) and (6.13) are novel from the standpoint that they are able to automatically reveal the orifice profile in Eq. (6.11) if the spool velocity xp is known and an appropriate state/parameter estimation algorithm is applied. One of the contributions of this chapter has been to derive the proper set of state-space equations for such an estimation algorithm, and to show through simulation examples that the orifice effective area can be accurately estimated without flow rate measurements. Another issue is that since the pilot spool is surrounded by two solenoids (Fig. 5.1), direct measurement of its position without affecting the valve dynamics is a very difficult task. Instead, position measurement of the main spool in the second stage can easily be made by removing one of its end caps and inserting a position sensor, such as a linear pot. The pressure building up in the opened pilot chamber to the supply port pushes both the pilot spool and the main spool simultaneously. We take the pressurized chamber as a control volume and ignore the oil compressibility effects and leakage into and out of the chamber, from the results of Chapter 5. When solenoid B is active and attached to the pilot spool, port A is pressurized, and Eq. (5.24) gives A m x m = [max(0, aoa)y/2(ps -pa)/p+ (6.14) min(0,a o a)v /2p a//9] -j- ApXp Similarly, when solenoid A is attached to the pilot spool, port B is pressurized, and Eq. (5.25) gives Amxm = - [max(0,a o i ) ) v / 2(p 3 - pb)/p+ (6.15) min(0,cu)\/2p6//>] where A,„ is the main valve pressure area, xm is the main spool position, and p is the oil density. 6.2.3 Main Valve Model From the results of Chapter 5, the total force acting upon the main valve spool is due to the pilot pressure differential, spring compression, spring preload, Coulomb friction, viscous damping (friction), Section 6. 2 Modeling 157 0 1 2 3 4 5 6 7 8 Time (s) Fig. 6.1: Comparison of typical forces acting upon the main valve spool, flow force through the main valve, and spool inertial force (Eq. (5.29)). Among the above, the pilot pressure differential force and the compression spring force are the only major forces acting upon the main spool (see, e.g., Fig. 5.10b). These forces are typically between 200 to 400 N. The spring preload is about an order of magnitude lower than the above forces (Table 5.1). The remaining forces are about two orders of magnitude lower than the pressure and the spring reaction forces, as is revealed in Fig. 6.1, using simulation and experimental results of Chapter 5. Notice that the main spool "inertial force" is almost an order of magnitude smaller than the viscous damping force. As a result, the inertial force term in Eq. (5.29) can totally be ignored, which simplifies the dynamic force equation (5.29) into the following equation: (Pa ~ Pb) =K + sgn(xm)Fpm + sgn(xm)Ffm + C (6.16) ( a o m ) s g n ( x m ) [ ( p s - pe) - sgn(xm)(pr - Ph)](?cos Oj) where the coefficient and variable definitions are given in Chapter 5, and aom and 8j are main valve orifice effective area and flow jet angle, respectively. The velocity of the main spool, xm may be approximated by differentiating and low-pass filtering the spool position measurements, xm, in the ^-domain, e.g., Xm(s) ft : ; - xm(s) (6.17) 1 + lvs where Tv is the time constant of the filter. Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 158 The question of why we considered the inertial and other dynamic effects in the pilot/armature spool equations of motion (6.7) and (6.8), as opposed to the static equation (6.16) for the main spool, can be answered by comparing the static and dynamic relationships between the inputs and outputs of the pilot valve and the main valve, as are illustrated in Figs. 5.10 and 6.2, respectively. Fig. 6.2 has been obtained by plotting time histories of the measured signals previously shown in Fig. 5.13. From Figs. 5.10 and 6.2, we realize that the dynamic forces in the pilot stage due to the motion of the pilot spool are quite noticeable as they have altered the static linearity and introduced substantial hystereses and delay in the input-output relationship between the solenoid voltage (or current) and the pilot pressure. For the main spool, on the other hand, both pressure-spool relationships in static and dynamic conditions have Section 6. 3 State-Space Representations 159 remained linear and almost invariant, which means that inertial forces developed due to the motion of the main spool are negligible as compared to the pressure and spring forces. 6.3 State-Space Representations As we saw in Chapter 3, a multi-input multi-output (MIMO) dynamic system can be described by a continuous-time nonlinear state-space model comprising a state equation and a measurement equation as x(*) = f(x(0,u(«),*) + G(«)w(*) (6.18) y(<) = h(x(0.u(0.<) + v(<) with w(t) ~ AT(0, Q(t)), v(t) ~ Af(0, K(t)) (6.19) where f is the state function vector, h is the observation function vector, u is the input vector, y is the output vector, and the state vector x may be enhanced by including all the unknown system coefficients. It should be noted that the system noise vector, w(r), as well as the measurement noise vector, v(f), are normally assumed independent sequences if there is no other a priori information available. As such, the system and measurement noise covariance matrices, Q(t) and R(7), become diagonal [41 Ljung, 1987]. A discrete version of Eq. (6.18), which is more convenient for real-time digital computations, may be obtained using the finite-difference approximation in Eq. (5.13): Xfc+i = Xfc + T5f(xjfc, Ufc) + Wk (6.20) y* = h(xfc,u fc) + v f c where k is sample number, and Ts is sampling period which should be as small as possible. The discrete version of the velocity estimator for the main spool in Eq. (6.17), which is expressed in continuous-time domain, can be obtained by applying the bilinear transformation * = ~— y zr (6-2i) 2 1 - z'1 T3l + z-where z - 1 is the delay operator. This gives a recursive form for on-line computation of the main spool velocity, as xm,k = 2T" -|- T Xm'^ 2T + T *-Xm'^ ~ Xm'k\ (6.22) Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 160 6.3.1 The Solenoid Model Using Eqs. (6.1) to (6.4), the solenoid model can be cast into the nonlinear state-space representation (6.18), as: ii = {ui - [f(xi,x4) + x2]x3} + Wi X2 = {g(ui - [f(xi,x4) + X2]X3)/T - X2/T} + w2 (6.23) X3 = W3 Vl = [/(X1,X4) + X2] + Vi where u\ = v., (solenoid voltage input measurement), yi = is (solenoid coil current output measurement), x\ = X (solenoid flux linkage), * 2 = id (solenoid coil dissipation current), x$ = R (solenoid coil resistance), x4 = xs (solenoid armature position). The time index t is dropped from Eq. (6.23) for brevity. Note that x\ and x2 are true state variables, while JC3 is a system constant which is modelled as a random-walk parameter. Inclusion of the solenoid's coil resistance as a random-walk parameter not only allows us to track its drift over time due to coil heating and other effects, but also informs us, through an abrupt change in its magnitude, of any malfunctioning of the solenoid. The armature position in Eq. (6.23) is taken as a state variable, x4, since its direct measurement is not available for the present valve. Note that while states x\ to X3 can be directly computed from Eq. (6.23), state x4 needs auxiliary pilot valve state equations, as well as the conditional Eq. (6.6), to be determined. Here indeed, x4 acts like a bridge between the solenoid and the pilot valve state equations. As is shown later in this chapter, this fact has allowed us to obtain an estimate for the solenoid/pilot position despite the lack of an appropriate position sensor for the pilot spool. Section 6. 3 State-Space Representations 161 6.3.2 The Pilot Valve Model The state-space representation for the pilot valve, when, e.g., solenoid B is activated and attached to the pilot spool, can be derived from Eqs. (6.5), (6.7), (6.12), (6.13), and (6.14) as: 1 4 =x 5 + w4 1 5 = {h(x\,x4) /Mc - (Ap/Mc)u2 - (Ks/Mc)x4 - x5x9/Mc-[sgn(x5)/Mc]x10-in[max(0,a:7)(ps - u2) - min (0, x7)u2]/Mc} + w5 1 6 = {[ma,x(0,x7)y/2(ps - u2)/p + min(0, x7)y/2u2/p]/Am + (Ap/Am)x5} + w6 x7 - x 5 x 8 + w7 (6.24) x8 =w8 X9 =WQ i n =w\i y2 =x6 + v2 where ui = Pa (pilot port A pressure input measurement), yi = xm (main spool position output measurement), x4 = xp (pilot spool position), *5 = ip (pilot spool velocity), Xf, = xm (main spool position), x7 = aoa (pilot port A orifice effective area), *8 = s<>a (pilot port A orifice area profile slope), x9 = Cfa (viscous damping coefficient for solenoid-pilot combination), *io = Ffa (Coulomb friction force for solenoid-pilot combination), x\\ = 2cos6a (flow force coefficient for the pilot valve). Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 162 Here again, the state vector in Eq. (6.24) is enhanced by noise driven (random walk) system parameters to x\ i , which represent orifice profile slope, viscous damping coefficient, Coulomb friction force, and flow force coefficient, respectively. Similarly, a state-space model for the pilot valve, when solenoid A is activated and attached to the pilot spool can be derived from Eqs. (6.5), (6.8), (6.12), (6.13), and (6.15). In this case, all equations given in (6.23) and (6.24) remain exactly the same except that the inputs, the outputs, and the states should belong to Solenoid A and Port B of the pilot valve, and u2 = Pb- Also, x5 and x& terms in Eq. (6.24) should be replaced by the following equations: x5 ={(AP/Mc)u2 - h(xl,x4)/Mc - (Ks/Mc)x4 - x 5x 9/M c-[sgn(x s)/M c]x 1 0+ x n[max(0, x7)(ps - u2) + min (0, x7)u2]/Mc} + w5 (6.25) x6 ={-[max(0,x 7)v /2(p s - u2)/p + m.m{Q,x7)y/2u2lp]/Am + (Ap/Am)x5} + w6 6.3.3 The Main Valve Model Eq. (6.16) which expresses static balance of force on the main spool is not a dynamic equation. However, representation of its constant coefficients as random walk parameters allows us to derive a special state-space form for Eq. (6.16), as: Xl2 =W12 i l 3 =Wl3 Xl4 -Wl4 X15 =wis (6.26) Xl6 =V>16 V3 ={(xm/Am)xi2 + [sgn(a:m)/Am]2:13 + [sgn(iT O)/Am]a;i4 + (xm/Am)x15+ [(aom/Am)sgn(xm){(ps - pe) - sgn(a:m)(pr - Ph)}]xw} + v3 where )>3 = Pa - Pb (pilot pressure differential measurement), x\2 = Km (main spool spring constant), Section 6. 4 State and Parameter Estimation 163 JC13 = Fpm (main spool spring preload), *14 = Ffm (main spool Coulomb friction force), *15 = Cm (main spool viscous friction coefficient), *16 = 2cos0y (main spool flow force coefficient). Note that in Eq. (6.26), the main spool pressure area Am is assumed to be known and constant, since it is very unlikely that Am changes over time compared to other main valve parameters. 6.4 State and Parameter Estimation Extended Kalman Filtering (EKF) has widely been used for real-time concurrent estimation of states and parameters in nonlinear state-space models of dynamic systems. It is based on linearization around the current state estimates (see Chapter 3 and Appendix A for more detailed explanation and algorithm). The real-time computational burden and cost of using discrete E K F has been substantially reduced with the help of high-speed low-cost microcomputers. The continuous time state-space representations for the solenoid, pilot valve, and the main valve, i.e., Eqs. (6.23) to (6.16), can be appropriately discretized, using Eq. (6.20). Choosing a very small sampling period (e.g., Ts = 1 ms) reduces approximation error and rough state/parameter jumps across the discontinuity of f and h functions in Eq. (6.20) due to "max", "min", and "sgn" discontinuous effects. Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 164 In order to apply discrete E K F , we need to compute the Jacobian matrices from Eqs. (A.5) and (A.8) for the state function f and the observation function h, respectively. The Jacobian matrix elements for the solenoid model, using Eq. (6.23), are given below: , , rp df(xi,x4) 4>i,i = 1 - Ts—K- '-x3 ox i 4>\,2 - -Tsx3 ^1,3 = -T3f(xi,x4) , rrdf(x1,x4) 01,4 = -Ts ^ - x3 Tsdg(vi)df(x1,x4) <!>2,i = ~ ^ ~x3 T dvi dx\ <?>2,2 = 1 JT^X3 T T OVl 4>2A = - ^ d f l ) d f ( X ^ X 4 ) X 3 T OVl OX4 <f>3,3 = 1 8f(xuX4) "1,1 = K OXi HX,2 = 1 df(xi,x4) 1^,4 = a OX4 where, from Eqs. (5.30) and (5.32), df(xi,x4) A •_ 1 Q = 2^ *(/«•!*4 + /to)A 1=1 4 (6.27) df(xi,x4) ^ • —8xT~ = ^f'lX (6-28) _ j i dg(vt) 2gx e « dvi g2 All coefficients in Eq. (6.28) are assumed to be known and constant (see Table 5.2). Section 6. 4 State and Parameter Estimation 165 The Jacobian elements for the pilot valve state-space model are as follows: 04.4 04.5 05,1 05.4 05.5 05,7 05,9 05.10 05.11 06.5 06.6 06.7 07,5 07.7 07.8 4>i,i #2,6 1 Ts Ts dh(xl,x4) Mc dx2 2xr dh(x\, x4) dx4 K, = 1 xg = -jj-[sgn(ma,x(0,x7))(ps - u2) - sgn(min (0, x7))u2]xu [max (0, x7)(ps - u2) - min (0, x7)u2] A p 1 Tsxs 2 2 sgn(max(0,a:7))^/-(p s - u2) - sgn(min (0, x7))^j-u2 - 1 Tsx5 1, (* = 8 , - - - , l l ) 1 (6.29) where, from Eq. (5.31), dh(x\, x4) dx\ dh(x\,x4) 3XA ^2i(hnx4 + hi0)(X2y 1 i=i X > ( A 2 y (6.30) t=i Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 166 Finally, the Jacobian elements of the main spool model are computed as: <f>i,% 1 (t = 12, - - -, 16) -#3,12 X m #3,13 m (6.31) #3,15 X m A. m #3,16 a. om sgn(a:m)[(ps - pe) - sgn(xm)(Pr - ph)] A; -m where the main port pressures, pr and ph aire measured, and the main valve orifice effective area, aom, can be computed using the main spool position measurements and the empirical equation (7.19) derived from experimental tests. Note that in Eqs. (6.27), (6.29), and (6.31), fcj and Hjj are scalar elements of the Jacobian matrices f and h, respectively, and are identically equal to zero if not stated in those equations. Al l terms in Eqs. (6.27) to (6.31) should be evaluated at sample number k, which is dropped from these equations for brevity. In order to estimate the eleven states of the solenoid and the pilot valve models, Eqs. (6.23) and (6.24) must be solved simultaneously due to observability problem. This is because the active solenoid and the pilot valve are in direct interaction through the armature-spool assembly, whose position, x4, is "not measured". Knowledge of x4 is especially crucial when estimation of the pilot valve states and parameters is concerned; otherwise, Eq. (6.24) will not be observable. As a result of simultaneous solution of the solenoid-pilot valve models, the size of the Jacobian matrices becomes $ i i x i i and H 2 x n- If pilot spool position measurements were available, Eqs. (6.23) and (6.24) could be decoupled, and we would have $ 3 x 3 and H 1 X 3 for the solenoid, and $ 8 x 8 and H I X 8 for the pilot valve. Solution of the state-space equation (6.16) for the main spool, on the other hand, can be done independently, since all the measurements are available and the parameters are expressed linearly in the observation equation. As a result, for the main valve spool we have $ 5 x 5 and H ] X 5 . It is interesting Section 6. 5 Simulation Results and Discussion 167 to note that in this case, the Extended Kalman Filtering problem is greatly simplified and is reduced to the original Kalman filtering problem for linear systems [21 Grewal and Andrews, 1993]. 6.5 Simulation Results and Discussion E K F has been applied to the servovalve model described by Eqs. (6.23), (6.24), and (6.16), using input-output signals obtained from the simulation model explained in Chapter 5. The primary purpose is to benchmark the accuracy and performance of the proposed state-space model and the real-time capability of the E K F in estimation of the servovalve's parameters and states at a high sampling/computation rate of 1 KHz. The other reason is to investigate the effects of artificially produced faults (which are difficult, expensive, or dangerous to actually produce) on the behavior of the valve and the magnitude of its estimated parameters. 6.5.1 The Data Fig. 6.3 shows all the input-output signals which are assumed measurable from the actual valve, using the valve's data listed in Tables 5.1 to 5.3. In order to effectively excite the valve, a low-pass filtered random voltage input, vs was applied to the valve solenoid, and the solenoid current, the pilot pressure, and the main valve displacement were obtained accordingly. The stored input-output data was sampled at 1 KHz in the VxWorks™ real-time operating system environment, using a SPARC IE workstation, and the states and parameters were successfully updated within each sample time using a SkyBolt™ math co-processor. Please see Chapter 4 and Fig. 4.3 for the description and a schematic diagram of the experimental-computer setup. 6.5.2 Starting Values for the Kalman Filter The starting values for the state covariance matrix, P, in Eq. (A.9) did not affect the final steady-state parameter estimates if not set very high or very low. For the solenoid-valve model and the main spool Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 168 Solenoid Voltage Solenoid Current Pilot Pressure Main Spool Position 4 6 Time (s) 4 6 Time (s) Fig. 6.3: Input-output signals obtained for the simulated valve (see Fig. 5.13). model, they were set to P n x n = 104 x I n x i i ' a r , d 1*5X5 = 104 x Isx5> respectively. Al l the initial states/parameters in the enhanced state vectors, x, were set to small random numbers. In setting appropriate initial values for the system noise covariance matrix, Q, and the measurement noise covariance matrix, R, we realized that proper choice of Q and R played an important role in the speed of convergence and final estimation results. For these reasons, the values for Q and R were tuned by trial and error to give a good convergence rate and an acceptable level of accuracy. As a result, the system noise covariance matrix, Q, was chosen to be diagonal and constant. For the solenoid-valve model, we used: Q\,\ = 10-4, 02,2 = 10-6, 03,3 = 10-2, 08,8 = K T 1 , 09,9 = 2.0,10 = Gll.ll = 10~3. and for the main spool model, we used: Section 6. 5 Simulation Results and Discussion 169 fil.l = 22,2 = 23,3 = fi4,4 = Q 5,5 = 10"6. The measurement noise covariance matrix, R, was chosen to be constant and diagonal as well. For the solenoid-valve model, we set: = 10~3 and R.2,2 = 10~5. For the main spool model, we used R = 1. 6.5.3 Normal System States and Parameters A. The Solenoid Results: Fig. 6.4 shows the estimation results for the simulated solenoid in a normal condition. There is an excellent match between the true values and the estimation results for the solenoid current. Also, the solenoid estimated flux linkage and dissipation current closely follow the corresponding true values. The small discrepancy between the estimate and the true value of the solenoid flux linkage is due to the error in the armature (pilot spool) position estimate. From the figure, the coil resistance, which is a key parameter for condition monitoring of the solenoid, has rapidly converged to its true value. B. The Pilot Valve Results: Figs. 6.5 and 6.6 show the estimation results for the pilot stage. There are two unmeasured states which are estimated and shown in Figs. 6.5(a) and 6.5(b): pilot spool position, and flow rate through the orifice area. The main reason for relatively poor estimation of the armature-pilot spool position is the lack of measurements for £ 4 , so that it cannot be directly adjusted during the course of estimation/prediction. Therefore, its estimate, x4, relies solely on information coming from the solenoid current and the pilot pressure measurements, and on the accuracy of the interpolating functions f{x\,x4) and h(x\,x4). In contrast, the pilot orifice flow rate in Fig. 6.5(b) has been accurately predicted from Eq. (4.15) using pilot pressure measurements and the estimated orifice effective area, xj. The maximum error was always less than 10% full scale during transitions, and the RMS error was slightly above 1% full scale for the period of nine seconds. This accuracy is compatible with commercially available flowmeters, provided that they could measure transient flow rates inside a pilot stage within less than a Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 170 Flux Linkage (Lambda) Dissipation Current (Id) 20 19 118 O co 17 X 16 15 2 4 6 8 Coil Resistance (Rs) true value estimation 4 6 Time (s) 4 6 Coil Current (Is) 4 6 Time (s) Fig. 6.4: States and parameter estimates for the "simulated" solenoid in "normal" condition. 2 4 6 Time (s) E. •0.05 0.4 0.5 0.6 0.7 0.8 (d) £ - 0 . 0 5 o < -0.1 to tank to supply — true\aLue — estiirate 0.4 0.5 0.6 0.7 0.8 Pilot Spool Position (mm) Fig. 6.5: State estimates for the "simulated" pilot valve in "normal" condition, (a): Spool position time history, (b): Flow rate time history, (c): Orifice effective area estimate vs. true spool position, (d): Orifice effective area estimate vs. spool position estimate. Section 6. 5 Simulation Results and Discussion 171 z 1.5 c o o 1 E o o O 0.5 0 I I I A \ \ i i true \alue — estimatim i i i i v " i i i i i i i i i 8 0 1 2 3 4 5 6 7 8 Time (s) Fig. 6.6: Parameter estimates for the "simulated" pilot valve in "normal" condition. few milliseconds. One reason for the accuracy of the flow rate is that the flow rate estimate originates from the flow continuity equation (6.14), which mostly depends on the pilot pressure and main spool position sensor measurements, which are robust and inexpensive. This is an important result and, by itself, an original contribution, since we have virtually devised a "soft flowmeter" that uses a model, an estimation algorithm, and a few low-cost robust sensors. It was mentioned that the accuracy of the flow rate estimate to a great extent stems from the precise estimation of the orifice effective area, which is a "traceable state". This fact is confirmed again in Fig. 6.5(c), which shows that the E K F has successfully approximated the nonlinear geometric relationship (6.11) between the actual valve opening and the true spool position, by taking advantage of the geometric Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 172 dependency (though initially unknown) between the orifice area and the spool position, i.e., x7 = A0(x4). This is another original contribution, since, as can be realized from Fig. 6.5(c), not only have the opening profiles to both supply and tank ports been revealed, but the size of the ports' deadband, i.e., the tank-port underlap and the supply-port overlap, has also been automatically identified within a short period of time. The importance of this result will become more clear later in this chapter, when slight shifts in the valve orifice profile and the deadband have been detected, indicating wear effects and/or increased oil contamination level. In practice, since pilot spool position measurement is not available, we have to rely on the relationship between the orifice area estimate and the pilot spool position estimate, as is plotted in Fig. 6.5(d). This plot indicates that this functional relationship can be substantially impaired by not using the pilot spool position measurements, although the estimation of the effective valve area (and therefore, the valve flow rate) is still valid and accurate, as was shown before. Finally, considering parameter estimate results in Fig. 6.6, we realize that the viscous damping coefficient estimate, x9, the Coulomb friction force estimate, i io , and the flow force estimate, i n , converge slowly towards their final values, and £9 and i io are biased. We tested a number of data sets with different starting values for the filter. However, the parameters consistently converged to the same values for a rather long period of estimation, indicating a global minimum. The lack of full observability of the Coulomb friction force and the viscous damping coefficient, as evidenced by slower decreasing rates in their estimation error covariances, is responsible for their slow convergence rate and bias. There are several physical and numerical reasons for this happening, including severe nonlinearities in the valve and the valve model (such as Coulomb friction force), very low spool velocity (about a few mm/s), weak effects of viscous damping and Coulomb friction forces compared to magnetic and pressure forces, state-space model reduction and simplification compared to a much more complex simulation model of the valve given in Chapter 5, and error due to discretization and sampling rate. Section 6. 5 Simulation Results and Discussion 173 C . The Main Valve Results: Eq. (6.16) was solved recursively by applying Kalman filtering and using pilot pressure differential and main spool position measurements given in Fig. 6.3. The velocity of the main spool was estimated from Eq. (6.22) with Tv - 5 ms. In order to avoid discontinuity effects in Eq. (6.16) due to spring preload when xm « 0, and due to stick-slip friction force when xm(k) sa 0, we switched off the estimation process whenever \xm\ < 0.025 mm or \xm\ < 2.5 mm/s. The estimation results are plotted in Fig. 6.7. The convergence rate is fast and estimation error is small for the spring constant £12, spring preload i i 3 , and flow force coefficient xie, which have dominant effects in the force balance equation (6.26). Again, the Coulomb and viscous friction estimates, £ 1 4 and £ 1 5 have slower convergence rates and larger steady-state estimation errors, because of the reasons explained earlier. 6.5.4 Condition Monitoring and Fault Detection Results In this section, we have again used the data obtained from simulation of the valve model described in Chapter 5 and identified in the previous section. This time, some artificial faults were introduced in the model while the simulation program in the SIMULINK environment was running. The results of fault detection for the simulated valve have been used to show the capability and effectiveness of the proposed model-based methodology in fault detection and diagnosis, even with the presence of bias and uncertainty in parameter and state estimates. A. The Monitored Parameters: The identified parameters from the previous section have been used to monitor and detect any incipient fault or abrupt failure in the servovalve. The mean and standard deviation for each parameter in the normal condition have been calculated from the last 3 seconds of the estimation results (3000 data points) and are listed in Table 6.1. Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 174 Fig. 6.7: Parameter estimates for the "simulated" main valve in "normal" condition. Section 6. 5 Simulation Results and Discussion 175 Table 6.1 Estimation values for the "normal" parameters of the "simulated" servovalve. Parameter Mean Value STD Relative STD. solenoid resistance, £3 (Ohm) 19.6 0.1 0.51% viscous damping coefficient*, £9 (N.s/m) 142 2 1.4% Coulomb friction force*, £10 (N) 0.236 0.069 29% orifice flow force coefficient*, i n 0.437 0.055 13% spring constant, £12 (N/m) 132000 300 0.23% spring preload*, £13 (N) 47.4 0.3 0.63% Coulomb friction force*, £14 (N) 2.21 0.24 11% viscous friction coefficient*, £15 (N.s/m) 42.3 3.4 8.0% orifice flow force coefficient*:?: 16 0.447 0.049 11% *: pilot valve *: main valve STD: standard deviation B. The Monitored Traceable States: The orifice effective area estimate, £7 is a traceable state that is a function of the pilot spool position at any time. The shape and the size of the valve orifice, once estimated from the proposed equations, can give us valuable information about the health condition of the pilot valve and its inlet-outlet ports. To detect an on-line change in £7 due to a fault in the valve, the current orifice area estimates should be compared with the corresponding normal values at the same spool position. Therefore, we need: 1) to know the pilot spool position measurements xp, and 2) to have an interpolating function fg describing the geometrical relationship between the normal (nominal) orifice area and the pilot spool position, i.e., £7(0 = fg(xp(t)). In the absence of direct spool measurements, we have to use its predicted value, £4, instead of xp. This, however, was shown in Fig. 6.5(a) to have a limited accuracy and can give scattered results (see Fig. 6.5(d)). In fact, we did try a number of polynomials, up to a sixth order one, to approximate the functional relationship £7(2) = fg(x4(t)) shown graphically in Fig. 6.5(d), but we could not obtain a satisfactory fit. As another option, we considered using a higher order interpolating function such as the neural network function approximator (NNFA) shown in Fig. 6.8. By trial and error, a feedforward two-layer network Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 176 with one input cell for x4, one output cell for x7, and one hidden layer with five cells was found appropriate in terms of the minimum RMS error and training time. Its RMS error, for example, was about three times as low as that of obtained from a sixth order polynomial. The backpropagation t r a i n b p algorithm in Matlab's Neural Network Toolbox was run to train the network using the data points plotted in Fig. 6.5(d). The data points between t =2 s to t = 4.5 s (2500 input-output data points) were used as the training set, while the data points between t =4.5 s to t = 9 s were used to test the network. The input-output training set (a 2 x 2500 matrix) was normalized within the range of [-1, +1], using a linear mapping function. The test set (a 2 x 4500 matrix) was normalized accordingly. The network was trained for 3000 iterations before we observed that no significant reduction in the RMS output error could be achieved. The final RMS error for the 2500 outputs was 0.17. In search of a global minimum, the network was trained for several times with different initial random values for its weights, but the RMS errors were always slightly above 5.26. The outputs of the NNFA are compared to the E K F training data set and the actual valve profile in Fig. 6.9. The final weight values are listed below: W2,i — —3.207, W3 ,i = —3.352, w4ti = 3.337, w$t\ — —3.511, W6 ,i = —3.365 w7t2 = -0.385, w7,3 = -0.124, w7A = -0.409, w7,5 = -0.310, w7fi = -0.670 b2 = 3.362, 6 3 = -0.908, b4 = -2.860, b5 = 2.633, b6 = -3.031, b7 = 0.159 Section 6. 5 Simulation Results and Discussion 177 -0.5 0 0.5 1 Normalized Spool Position 3 8 m I • cc co 9 2 0 positive direction negative direction i / 1 n o 2 3 Time (s) Fig. 6.10: Neural network function approximation of the pilot orifice area (test set). Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 178 The performance of the NNFA was then examined using the test data set with 4500 data points representing 4.5 seconds of the valve simulation time. The results are shown in Fig. 6.10. It is clear from the figure that, despite a few spikes in the error signals, the average error is generally low (standard deviation = 0.25). To tune the threshold for detection of changes in the orifice area error signals, the SPRT was performed over the (non-independent, non-identically distributed) error signals of Fig. 6.10(c), and the threshold and minimum identifiable jump level after a fault were conveniently set to Bu = 8 and fi\ = 0.06, in order to avoid unnecessary false alarms in normal condition (see Fig. 6.10(d)). C. Detection of an Abrupt Failure in the Pilot Stage: The fault: The same input voltage shown in Fig. 6.3(a) was applied to the normal valve explained in Chapter 5. While the simulation program was running, an abrupt increase of 20 N.s/m in the viscous damping coefficient of the solenoid armature was introduced to the simulation program. This increase corresponded to about 24% decrease in the area of the solenoid capillary tube, according to Eq. (5.11), and intended to simulate the partial clogging of the tube due to an external particle. Condition monitoring: The on-line estimation process was resumed for the new set of data. The starting values for the Kalman filter were set equal to the last identified values for the "normal" parameters and the estimation error covariance. Fault detection: Fig. 6.11 shows the estimation results for those parameters and the traceable state (pilot orifice area) that were not altered during the simulation period. No significant change is observed before and after the fault time, (tf = 1 s), as was expected. Fig. 6.12, on the other hand, shows an abrupt change of about 5% in the magnitude of the viscous damping coefficient for the pilot stage despite its bias. The SPRT results in the figure are based on the parameter estimate residuals, Eq. (3.16). The failure has been successfully detected 32 ms (32 samples) after the actual fault time, and the filter settling time for parameter jump-level assignment and fault symptom generation has been 0.588 s (588 samples) after the detection time. Section 6. 5 Simulation Results and Discussion 179 (a) Solenoid coil resistance (c) Coulomb friction <—fault time , n i"\ .... „ f"-. D 1 2 3 A Time (s) (b) Flow force coefficient ! ,-\ .'"\ • . .. . 0.5 0.4 0.3 0.2 0.1 0 1 2 3 Time (s) (d) Normalized orifice area -nominal estimate 1 2 3 Time (s) -1 -0.5 0 0.5 1 Normalized spool position Fig. 6.11: Estimates of the pilot stage parameters and traceable state during an abrupt change in the viscous damping coefficient (based on simulation results). a) Viscous damping coefficient 1 1 I I i i i \A ! .positive direction V j —negative direction A A I \ A A JWj\ A ., 0.5 1 1.5 2 2.5 3 3.5 4 Time (s) Fig. 6.12: Fault detection and symptom generation during an abrupt change in the viscous damping coefficient (based on simulation results). Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 180 Fault symptom generation: For the pilot stage viscous damping coefficient, the jump class "Increased" (/) next to the jump class "No Change" (NC) was conveniently set 5 times as high as the relative standard deviation of x9 in Table 6.1 (see Section 3.5.1 for more details). As a result, the parameter jump level for the viscous damping coefficient was classified as "Increased" for this particular fault, as is indicated by the upper horizontal " - . - " line in Fig. 6.12(a). Jump level classification results (qualitative fault symptoms) for the pilot stage parameters are listed below: parameter solenoid resistance viscous damping Coulomb friction flow force coefficient supply-side orifice area tank-side orifice area jump class NC / NC NC NC NC For this particular fault, the parameters of the main valve were also monitored. However, no significant deviation was observed during the course of monitoring, as the main valve filter is basically independent of the pilot stage filter. D. Detection of an Incipient Fault in the Pilot Stage: The fault: Once again, the same input voltage shown in Fig. 6.3(a) was applied to the normal valve explained in Chapter 5. This time, while the simulation program was running, the size of the pilot spool-supply port overlap, Ls, was linearly increased from the normal size of 0.67 mm to 0.77 mm (about 15% increase) within 4 seconds of the simulation time. The purpose was to simulate a progressing dirt/oil debris accumulation around the pilot supply port opening in an artificially accelerated manner. Condition monitoring: The on-line estimation process was resumed for the new set of data. The starting values for the Kalman filter were set equal to the last identified values for the "normal" parameters and the estimation error covariance. Fault detection: Fig. 6.13 shows the estimation results for all parameters of the pilot stage. No significant change is observed before and after the fault time (tf = 0.5 s), as was expected. The on-line Section 6. 5 Simulation Results and Discussion 181 (a) Solenoid coil resistance (b) Flow force coefficient 1 2 3 Time (s) 180 (c) Viscous damping coefficient (d) Coulomb friction force 1 2 3 Time (s) Fig. 6.13: Estimates of the pilot stage parameters during a gradual increase in the pilot orifice deadband length (based on simulation results). estimation results for the pilot orifice area from the E K F , along with its nominal estimation results from the previously identified NNFA are plotted in Figs. 6.14(a) and 6.14(b). A noticeable difference between E K F and NNFA predictions of the orifice profile shape on the supply side (i.e., the right tails of the graphs in Fig. 6.14(b)), can be observed from the figure, and the error increases over time, as Fig. 6.14(c) indicates. The reason is that in the faulty valve with an increasing deadband length, the pilot spool must move further rightward before the pilot port opens to the supply port and develop pressure in the pilot chamber. This is evident from Fig. 6.15, where the pilot spool position and orifice area in the faulty value are compared to a normal valve under the same input signals. Meanwhile, the nominal (or normal) NNFA model assumes that the pilot deadband is still of the same size as before, and therefore, Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 182 0 1 Normalized Spool Position 2 3 Time (s) Fig. 6.14: Pilot orifice area monitoring and fault detection during a gradual increase in its deadband length (based on simulation results). (a) Pilot spool position time histories with and without fault 0.9 0.8 | 0 . 7 ^0.6 x 0.5 0.4 0.3 1— r • i '<— fault start time M M ' —i— u T 1 l\A/ -v | v • i i i i v V -0.5 1 1.5 3.5 2 2.5 3 Time (s) (b) Comparison between orifice profiles of a normal and a faulty valve 4.5 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 x7 (mm) Fig. 6.15: Comparison of the pilot spool position and orifice area between a faulty and a normal valve (based on simulation results). Section 6. 6 Experimental Results and Discussion 183 with the current estimate of the pilot spool position obtained from the E K F (which is obviously more than what it used to be), gives a larger size for the orifice area on the supply side. To demonstrate how this discrepancy can be automatically detected on-line, SPRT has been applied on the error signals of Fig. 6.14(c), and the results are shown in Fig. 6.14(d). The estimated fault time was if — 0.854 s, compared to the actual fault time of tf — 0.5 s. The reason is that when the incipient fault was first injected into the simulation model at tf = 0.5 s, the spool was moving towards the tank port, and therefore, the filter was unaware of the fault at that time, until the spool returned towards the faulty supply port for the first time after the fault started to grow. Apparently, the SPRT results in positive direction has been constantly increasing since the incipient fault was first realized at tf = 0.854 s. The fault, however, has not been detected until sufficient evidence for a graduating fault has been accumulated at tj = 4.161 s. Regarding the main valve, no significant change or deviation in any normal parameter value was observed during the monitoring and fault detection process, as was expected. Fault symptom generation: Jump classification results (qualitative fault symptoms) for the pilot stage parameters are listed below: parameter solenoid resistance viscous damping Coulomb friction flow force coefficient supply-side orifice area tank-side orifice area jump class NC NC NC NC / NC 6.6 Experimental Results and Discussion 6.6.1 Experimental Conditions We used the same two-stage servovalve that was previously modelled in Chapter 5. This servovalve, before being installed in our experimental hydraulic test rig, was part of a hydraulic machine, a "faller Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 184 buncher", and had been in use for about six months. We have named this servovalve the "new valve". Three separate experiments with the same input voltage were conducted on the same servovalve at different times. The purpose of the first and the second experiments, which were conducted a year apart, was to compare the signals obtained from a "new valve" with the signals obtained from the same valve after one year of operation (a "used valve"). The purpose of the third experiment, which was conducted on the "used valve", was to detect an abrupt fault in the servovalve. In between Experiment 1 and Experiments 2 and 3, the hydraulic test rig (with the same servovalve) had been constantly in use for some other purposes, often under harsh loading conditions and elevated temperatures. Also, we intentionally did not filter the hydraulic oil during the one-year period. As a result of the poor maintenance practice, the "new valve" eventually wore out after one year. All three experiments were conducted when the oil temperature at the pump outlet port was 40° ± 5° C. All other operating conditions were the same. The sampling frequency was at 1 KHz. 6.6.2 Estimation of the New Valve Parameters and States A. The Data: The new valve was monitored in real-time using the input-output signals plotted in Fig. 6.16. The starting values for the E K F were the same as the ones used for the simulated valve, which were given in Section 6.5.2. B. The Solenoid Parameters and States: The estimation results from the E K F for the "new" solenoid are plotted in Fig. 6.17. From the figure, all the solenoid states, which are flux linkage, restoring current, and dissipation current, and the only solenoid parameter, i.e., coil resistance, have been successfully estimated. The coil resistance estimate, xs = R, quickly converged to its steady-state value, whose average and standard deviation over the last 4 seconds of the experiment are given in Table 6.2. Section 6. 6 Experimental Results and Discussion 185 (a) Solenoid input voltage (c) Solenoid Current 2 4 6 8 (b) Pilot pressure 0 2 4 6 8 (d) Main spool position 4 6 Time (s) 2 4 6 Time (s) Fig. 6.16: Inputs and outputs measured from the servovalve in Experiment 1. (a) Solenoid flux linkage (c) Dissipation current 2 4 6 8 (b) Coil resistance -10 2 4 6 8 (d) Restoring current 2 4 6 Time (s) Fig. 6.17: State and parameter estimates for the solenoid from Experiment 1. Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 186 C. The Pilot Valve Parameters and States: The estimation results for the parameters of the pilot stage of the "new" servovalve are plotted in Fig. 6.18. The parameters have converged to their final steady-state values after about 5 seconds. However, as we expected from the simulation results, the steady-state estimates for the viscous damping coefficient and the Coulomb friction force are biased and far from their actual values obtained in Chapter 5. The reasons explained in Section 6.5.3 plus noisy measurements and inaccuracy of the friction model at very low velocities [33 Karnopp, 1985] are responsible for the bias and the slow convergence rate. Again, we estimated the same parameters with different input signals and initial conditions, and we obtained the same final values. The average and standard deviation of the parameter estimates for the last 4 seconds of the experiment are listed in Table 6.2. Fig. 6.19 shows the estimation results for the unmeasured states of the pilot valve. From the simulation results, we expect that the predicted net flow rate from the supply and the tank ports to the pilot chamber, shown in Fig. 6.19(b) should be much more accurate than the predicted pilot spool displacement in Fig. 6.19(a). The relationship between the orifice effective area and the pilot spool position in Fig. 6.19(c) is not as defined as what we observed from simulation results (compare with Fig. 6.5(d)), and therefore our attempt for fitting a curve to the data points using a NNFA or any other function approximators was not successful. This was due to error in the pilot spool position estimation, noise effects in the measurements, unmodelled valve leakage, a variable discharge coefficient, and other unmodelled nonlinearities. Nevertheless, a general hysteresis loop is noticeable from Fig. 6.19(c), which has a sharp highly twisted tail in the lower left corner of the plot (indicating the tank port) and a sharp slightly inclined head in the upper right corner of the plot (indicating the supply port). The plot of area profile slope estimate, xg, vs. the pilot spool position estimate, x4, which is shown in Fig. 6.19(d), re-confirms the claim, as it reveals slight and sharp positive trends in the orifice opening at the high and low ends of the spool traverse. Section 6. 6 Experimental Results and Discussion 187 300 (a) Viscous damping coefficient 3 4 5 6 (b) Coulomb friction force - i 1 1 r" -i r~ _l L_ 3 4 5 6 (c) Flow force coefficient Fig. 6.18: Parameter estimates of the pilot valve from Experiment 1. (a) Pilot spool position (c) Pilot orifice area 0 2 4 6 8 x -| o'W F ' o w to pilot chamber 0 0.2 0.4 0.6 0.8 (d) Area profile slope 4 6 Time (s) 0 0.2 0.4 0.6 0.8 Pilot spool position (mm) Fig. 6.19: State estimates of the pilot valve from Experiment 1. Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 188 Table 6.2 Estimated values for the parameters of the "new" servovalve from Experiment 1. Parameter Mean Value STD Relative STD. solenoid resistance, £ 3 (Ohm) 19.6 0.2 1.0% viscous damping coefficient*, xg (N.s/m) 191 3.7 1.9% Coulomb friction force*, xw (N) 0.267 0.072 27% orifice flow force coefficient*, x\\ 0.964 0.155 16% spring constant*, x\2 (N/m) 132000 400 0.3% spring preload*, x 1 3 (N) 49.0 0.4 0.8% Coulomb friction force*, £ 1 4 (N) 0.703 0.154 22% viscous friction coefficient*, £ 1 5 (N.s/m) 85.1 19.4 23% orifice flow force coefficient*^ 0.650 0.115 18% *: pilot valve *: main valve STD: standard deviation D. The Main Valve Parameters and States: Eq. (6.26) was used to estimate the "new" main valve parameters. The estimation results are plotted in Fig. 6.20. The parameters have converged to their final steady-state values after about 5 seconds. The estimations for the compression spring constant, the spring preload, and the flow force coefficient are close enough to the corresponding values obtained in Chapter 5 (the dashed lines in Fig. 6.20). However, as we expected from the simulation results and the discussion in Section 6.5.3, the steady-state estimates for the viscous damping coefficient and the Coulomb friction force are biased and slightly far from the values listed in Table 5.3. The average and the standard deviation of the main valve parameter estimates for the last 4 seconds of the experiment are also listed in Table 6.2. 6.6.3 Detection and Diagnosis of Changes in the Servovalve Characteristics In order to show how the proposed model-based monitoring and fault diagnosis method distinguishes between a "new valve" and a "used valve", a second experiment under the same conditions and with the same input signals was conducted on the same valve about a year later (see Section 6.6.1). The input-output measurements from both experiments are compared in Fig. 6.21. Section 6. 6 Experimental Results and Discussion 189 Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 190 A. Pilot Stage Test Results: The traceable states for the pilot stage of the used valve are estimated and compared with the same states for the new valve in Fig. 6.22. The orifice effective area curve in Fig. 6.22(a) for the used valve has apparently been more stretched towards the tank side, while its supply port portion has basically remained the same as for the new valve. The orifice profile slope curve in Fig. 6.22(b) also reveals the same effect. Recalling the simulated progressing fault in the supply opening of the pilot stage, Section 6.5.4, we may infer that the valve deadband in the tank side has slightly increased. In other words, the tank port underlap length, L,, has been slightly reduced, probably due to oil debris or dirt accumulation. One reason is that the oil pressure in the tank port is usually much less than the oil pressure in the supply port. As a result, the debris in the latter can be better washed off by the flow, while the oil debris and particles in the tank-side have better chance to settle down because of lower oil pressure and velocity. This reduction, however, does not have any significant effect in the valve performance at this level, since the tank-side flow gain is several times greater than the supply-side flow gain. The estimation results for the parameters of the used valve pilot stage are compared with the previously estimated parameters for the new valve in Fig. 6.23. Apparently, the solenoid coil resistance had increased from 19.6 Q, to 19.9 (I, or about 1.5%. This was actually confirmed using a hand-held volt-ohmeter. In Figs. 6.23(c) and 6.23(d), the viscous damping coefficient was slightly decreased while the Coulomb friction force was increased, indicating that in the pilot stage, either the solenoid armature passage, or the pilot spool passage, or both became somewhat deteriorated compared to the previous year. The slight increase in the total friction force estimate, / / = xexg + sgn(x6)£io, which is plotted against the spool velocity estimate, x5, in Fig. 6.24 is a good indication of the valve degradation. The statistics of the parameter estimates for the pilot stage of the "used valve" obtained from the last 4 seconds of the experimental data are listed and compared to those of the "new valve" in Table 6.3. Section 6. 6 Experimental Results and Discussion 191 (a) T I I I I I I I T _Q ^ I I I I I I I I I I I -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 x4 (mm) Fig. 6.22: Comparison between the traceable states of the "new valve" and the "used valve". Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 192 (a) Solenoid resistance (c) Viscous damping coefficient (b) Flow force coefficient 4 6 Time (s) 2 1.5 (d) Coulomb friction force new valve — used valve 2 4 6 Time (s) Fig. 6.23: Comparison between the pilot stage parameters of the "new valve" and the "used valve". CO f 0 CO + a> x * CO X new valve used valve -5 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 x6 (m/s) Fig. 6.24: Comparison between the total friction force in the pilot stage of the "new valve" and the "used valve" Section 6. 6 Experimental Results and Discussion 193 Table 6.3 Estimated values for the parameters of the "used" servovalve from Experiment 2. Parameter Mean Value STD Relative STD solenoid resistance*, £ 3 (Ohm) 19.9 (+1.5%)* 0.3 (+50%)* 1.5% viscous damping coefficient*, £ 9 (N.s/m) 181 (-5.2%) 4.4 (+19%) 2.4% Coulomb friction force*, x 1 0 (N) 0.386 (+45%) 0.112 (+56%) 29% flow force coefficient*, £ n 0.886 (-8.9%) 0.197 (+27%) 22% spring constant*, £ 1 2 (N/m) 131000 (-.8%) 1800 (+350%) 1.4% spring preload1', £ 1 3 (N) 53.6 (+9.4) 2.2 (+450%) 4.1% Coulomb friction force*, £14 (N) 1.9 (+171%) 0.309 (+101%) 16% viscous friction coefficient1', £15 (N.s/m) 306 (+260%) 46.3 (+139%) 15% flow force coefficient*, £16 0.711 (+9.2%) 0.277 (+132%) 39% *: pilot valve parameter. *: main valve parameter. *: The numbers in parentheses are the percentage of change w.r.t. the "new valve". B. Main Stage Test Results: In this section, we not only have estimated the parameters of the main valve, but also have shown that the on-line SPRT method for fault detection and subsequent fault symptom generation is still possible using the previously estimated normal system parameters (i.e., the "new valve") and the new estimates of the same parameters belonging to the "used valve". For this purpose, we utilized the last record of parameters and estimation error covariance matrix as the starting values for estimation of the new set of parameters for the main valve. The statistics of the new parameter estimates for the main stage of the "used valve" obtained from the last 4 seconds of the experimental data are fisted and compared to those of the "new valve" in Table 6.3. From the results, we see that the estimation variance generally increased. Also, the viscous friction coefficient, the Coulomb friction force, and the spring preload demonstrated sharp increases over the previous values for the "new valve". In order to detect any significant changes, the residuals of the parameter estimates, i.e., Eq. (3.16), were formed on-line, and the SPRT was applied over the residuals. After change detection, the new values for the parameters were recorded when the filter was resettled, and the parameter jump levels were assigned Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 194 accordingly. The results of the parameter estimation, change detection, and parameter jump classification are shown in Figs. 6.25, 6.26, 6.27, and 6.28. From Fig. 6.25(c), we see that the spring became slightly softer over time. There were a number of false alarms for the used valve primarily because we incorporated the estimation variance of the previously identified spring constant in the SPRT algorithm, i.e., Eq. (3.31), which is less than that of the "used valve". As a result, the spring constant jump level in Fig. 6.25(c) was classified as "Slightly Decreased". From Fig. 6.26(c), on the other hand, the increase in the spring preload estimate is noticeable. This sharp change was truly detected by the SPRT, and about two seconds after the detection time was classified as "Moderately Increased". The magnitudes of the Coulomb friction force and the viscous damping coefficient both increased over time, as depicted in Figs. 6.27(c) and 6.28(c). As a result, the Coulomb friction force magnitude was classified as "Moderately Increased", while the viscous friction coefficient magnitude was classified as "Highly Increased". There are several reasons for the increase in the friction force of the "used valve", including wear effects, oil dirtiness, and poor maintenance. Jump level classification results for the parameters of the main valve are listed below: parameter spring constant spring preload Coulomb friction viscous friction jump class SD MI MI HI The significant changes observed in the parameter estimates of the "used main valve" are independently confirmed in Fig. 6.29, which compares the spool position-dynamic force relationships in the main stage of the "new valve" and the "used valve". The two curves in the figure were obtained by plotting the time histories of the measured main spool positions vs. the measured pilot pressure force previously depicted in Figs. 6.21(d) and 6.21(c), respectively. In Fig. 6.29, both the spring preload and the Coulomb friction force increased for the "used valve", as a higher force was required to move the spool forward at the Section 6. 6 Experimental Results and Discussion 195 x 10" (a) Parameter estimate residuals new valve used valve -1 -2 0 2 3 4 5 6 7 8 9 (b) SPRT results on parameter estimate residuals 10 10 8h co 6 -a .Q E CO I i i ml i i i i LI o x 10 2 3 4 5 6 7 8 9 10 (c) Parameter jump level classification 1.45 1.4 If 1.35h 1.3 x 1.25 1.2 I I I I I I I I I I I -1 1 1 1 1 • u ! _ 1 X " 1 1 -V ^ 1 1 1 1 r 1 1 i I I I 1 1 1 I I 2 3 4 5 6 7 Time (s) Fig. 6.25: Results of estimation, fault detection, and jump level classification for the spring constant of the main valve. 8 10 Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 196 Fig. 6.26: Results of estimation, fault detection, and jump level classification for the spring preload of the main valve. Section 6. 6 Experimental Results and Discussion 197 (a) Parameter estimate residuals CO X Q 0.01 0.005 0 -0.005 -0.01 I 50 new valve L used valve 40 h ^ 30 TJ E CO 20 10 \ ft *V 1 n A \ L _ l l\ > h I k 1 2 3 4 5 6 7 8 9 (c) Parameter jump level classification 4 5 Time (s) Fig. 6.27: Results of estimation, fault detection, and jump level classification for the Coulomb friction force of the main valve. 2 3 4 5 6 7 8 9 10 (b) SPRT results on parameter estimate residuals 10 Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 198 Fig. 6.28: Results of estimation, fault detection, and jump level classification for the viscous friction coefficient of the main valve. Section 6. 6 Experimental Results and Discussion 199 31 1 1 1 1 ! 1 1 1 r -0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01 Velocity (m/s) Fig. 6.29: Comparison between the spool position-dynamic force relationships in the main stage of the "new valve" and the "used valve". beginning of the stroke, and to move it backward again during its unloading. The increase in the size of the hysteresis loop was due to the increase in the magnitude of the friction force. 6.6.4 Detection of a Sensor Fault followed by an Actuator Fault A. Statement of the Problem: In the third experiment, which was conducted on the "used valve", we investigated the effects of a sensor fault on the behavior of the parameter and traceable state estimates. The faulty sensor was the ammeter that measured the current of the solenoid. The correct measurement of the solenoid current is crucial since any error in its measurement can easily be interpreted as an actuator fault. This is because the solenoid magnetic force estimate, fs in Eq. (5.7), depends on the flux linkage, which by itself changes as a function of the coil current. Any change in the solenoid force, will then affect the force balance on the pilot spool in Eq. (6.7) or (6.8). If the change in the computed solenoid force is due to a true fault in the solenoid actuator, e.g., a change Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 200 Switch 700 Q Nonlinear Solenoid mm L(t) R © + Input Voltage Command Fig. 6.30: Schematic diagram of the setup for reduction of the solenoid circuit resistance. in the coil resistance, the resultant pilot pressure and the main spool displacement will then be affected, as we saw earlier in Fig. 6.21. If the change in the computed solenoid force, on the other hand, is because of an error in coil voltage or current measurement, the pilot pressure and the main spool position will not be affected at all, but some pilot (or possibly main) stage parameter and/or state estimates in Eqs. (6.23) to (6.26) should then be changed. The purpose of this experiment is therefore to investigate which parameters/states change and how we can distinguish between this type of sensor fault and the aging of the valve observed in the previous experiment. B. Fault Generation: To artificially simulate the ammeter fault while the "used valve" was in operation, we suddenly introduced a parallel resistor (Rr = 700 Q.) to the solenoid circuit at time // = 4.6 s. Because of reduction in the circuit's total resistance from 19.9 ft to 19.35 ft, the total current of the solenoid circuit measured by the ammeter after the fault time should be higher for the same input voltage command that was applied during the previous experiments. This is while the actual current through the solenoid coil should virtually remain unchanged, compared to Experiment 2. The measurement setup is schematically shown in Fig. 6.30. The measured input-output signals from Section 6. 6 Experimental Results and Discussion 201 (a) Solenoid input voltage (c) Pilot pressure Time (s) Time (s) Fig. 6.31: Input-output measurements from the servovalve in Experiment 3. the servovalve are shown in Fig. 6.31. Note that the input voltage command is the same as the one used in Experiment 2. The single spike in the voltage/current measurements at tp. =4.815 s was due to some short circuit or loose connection that occurred accidentally and caused the input voltage to the solenoid to become zero for about 1 ms (one sample time). This secondary fault, although unintentional, was treated in this experiment as a short duration "actuator fault". C. Estimation Procedure: The parameter and state error covariance estimates for the "used valve", which were obtained from the last set of data points, were utilized to initialize the E K F . The new sets of parameters and states were estimated on-line using the input-output measurements given in Fig. 6.31. Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 202 (a) Solenoid coil resistance (c) Viscous damping coefficient 1.5 0.5 (b) Flow force coefficient (d) Coulomb friction force 2 3 4 5 6 2 Time (s) Fig. 6.32: Estimation results for the parameters of the pilot stage in Experiment 3. 3 4 5 Time (s) D. Fault Effects on the Valve Parameters: Fig. 6.32 shows the estimation results for the monitored parameters of the pilot stage. The solenoid coil resistance estimate, from the figure, experienced a noticeable reduction in magnitude due to the sensor fault, and again an abrupt short duration increase due to the actuator failure. This is while the observed jumps in the friction parameters of the pilot stage right after the faults were well within their variation limits. The fluctuations seen in the friction parameter estimates in Figs. 6.32(c) and 6.32(d) are normally due to stick-slip and other temporary disturbances present in the valve. Fig. 6.33 confirms that the total friction force estimate (Coulomb + viscous damping) virtually remained unchanged before and after each fault. Section 6. 6 Experimental Results and Discussion 203 The parameter estimates of the main valve did not change at all, due to the independency of the main valve model from the pilot stage model. E. Fault Effects on the Pilot Orifice Area Profile: Fig. 6.34 compares the estimation results for the unmeasured states of the pilot stage obtained from the faulty "used valve" with the results previously obtained from the normal "used valve". In Fig. 6.34(a), we notice that the magnitudes of the pilot spool position estimates increased after the sensor fault, due to the excessive magnetic force that was computed from the faulty ammeter measurements. This was while the pilot orifice area estimate in 6.34(b) virtually remained unchanged compared to a normal valve. The reason is that the orifice area was fundamentally computed from the flow balance equation (6.14), or (6.15). Either of these equations was strongly dependent on the main spool displacement measurement and the pilot pressure measurement, both of which remained unchanged during the course of the experiment. The result of the above discussion is that, as we can see from Figs. 6.34(c) and 6.34(d), the shapes of the orifice area profile and its slope basically remained unaffected by the sensor fault, but rather shifted towards the supply port (rightward). This is in contrast with the results shown in Fig. 6.22, where the corresponding shapes were altered and rather stretched towards one of the ports (i.e., the tank port). F. Fault Detection and Symptom generation: In order to detect any significant changes due to the abrupt sensor and actuator faults, the residuals of the parameter estimates from Eq. (3.16) were computed on-line, and the SPRT was applied on the residuals. After change detection, the new values for the parameters were recorded when the filter was resettled, i.e., in steady-state condition. The parameter jump levels were assigned accordingly. Fault detection results and parameter jump level assignments for the pilot stage are shown in Figs. 6.35, 6.36, and 6.37. Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 204 1 1 1 1 1 1 1 1 1 1 1 1 1 — Used valve — Used valve with fault _ g I I I I I I I — I — I -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 x5 (mm/s) Fig. 6.33: Comparison between the total friction force in a faulty valve and a normal valve in Experiment 3. (a) Spool position estimate (c) Orifce area slope estimate • • 1 0.81 • • — Time (s) Spool position estimate (mm) Fig. 6.34: Comparison between the traceable state estimates in a faulty valve and a normal valve in Experiment 3. Section 6. 6 Experimental Results and Discussion 205 (a) Parameter estimate residuals 2.5 20 15 9 | 1 0 at _ l 5 0 3 3.5 4 4.5 5 (b) SPRT results on parameter estimate residuals , — , 1 ! 1— 1 1 -i<—resampling time V 2.5 3 3.5 4 4.5 5 5.5 (c) Parameter jump level classification before and after fault S. 20 9 19 18 17 1 1— 1 1 1 —I 1 L "7 .... _ 1 1 1 1 1 2.5 3.5 4 Time (s) 4.5 5.5 Fig. 6.35: Results of estimation, fault detection, and jump level classification for the coil resistance in Experiment 3. (a) Parameter estimate residuals 1 0.5 -0.5 -1, 40 30 I !20 i j 10 0 2.5 3 3.5 4 4.5 5 (b) SPRT results on parameter estimate residuals 5.5 • 1 1 1 1 • 1 i 1 A A 2.5 3 3.5 4 4.5 5 5.5 (c) Parameter jump level classification before and after fault Fig. 6.36: Results of estimation, fault detection, and jump level classification for the pilot stage viscous damping coefficient in Experiment 3. Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 206 (a) Parameter estimate residuals 0.021 1 1 1 1 r -0.01 --o.oi r -0.02rL 15 B 1 0 E 2.5 3 3.5 4 4.5 5 5.5 (b) SPRT results on parameter estimate residuals L . A , L 1 \ M \ R IU Ah. 2.5 3 3.5 4 4.5 5 5.5 (c) Parameter jump level classification before and after fault Fig. 6.37: Results of estimation, fault detection, and jump level classification for the pilot stage Coulomb friction force in Experiment 3. In Fig. 6.35(b), the sensor fault was quickly detected by the SPRT, and the coil resistance jump class was subsequently assigned as "Slightly Decreased" by the symptom generation algorithm (Fig. 6.35(c)). The SPRT was also able to successfully detect the second short duration actuator fault. This secondary fault detection, however, did not have any subsequent effects on fault classification and diagnosis results. This is because after resampling time, the parameter value remained unchanged since the last fault classification and diagnosis activity. Here, we assume that there is enough time between two subsequent faults so that the fault diagnosis for the first fault has been completed before the secondary fault can be detected. The SPRT of the viscous damping coefficient residuals in Fig. 6.36(b) only detected the second fault, which was classified as "No Change" according to Fig. 6.36(c). In general, the value of the viscous damping coefficient for this particular experiment was constantly jumping between "No Change" and "Slightly Increased" classes. Section 6. 7 Summary and Concluding Remarks 207 Finally, only the sensor fault was successfully detected by the SPRT of the Coulomb friction force. However, the jump level for this parameter after the detection and resampling time was classified as "No Change". A number of false alarms were also raised, all of which were classified as "No Change". The qualitative symptoms of the sensor fault for the pilot stage parameters generated from the results of Figs. 6.35(c), 6.36(c), and 6.37(c) are listed in the following table. parameter solenoid resistance viscous damping Coulomb friction flow force coefficient supply-side orifice area tank-side orifice area jump class SD NC NC NC / D 6.7 Summary and Concluding Remarks A novel nonlinear state-space model for a two-stage proportional directional valve was introduced that has physical states and parameters. This model is especially suitable for on-line condition monitoring and subsequent fault diagnosis of the valve based on the parameter tracking technique proposed in Chapter 3. We demonstrated that how some crucial but expensive-to-measure states, such as spool position, valve flow rate, and the system physical coefficients can simultaneously be estimated using some alternative low-cost state measurements and extended Kalman filtering. The pilot spool position, for example, was fairly well reconstructed by taking advantage of a known functional dependency between the solenoid magnetic force and the pilot spool position, although we recommend its direct measurement wherever possible. The valve flow rate, on the other hand, was accurately predicted by letting the valve's orifice effective area be an unknown general function of the valve spool displacement. The switching decision between one or another state (i.e., tank open or supply open) was automatically made by a sign change of one of the system states (i.e., the orifice effective area). No specific assumptions about the size or the shape of the orifice or the flow regime was made. Simulation results showed that even for the fast transient spikes near the valve's closing and opening regions, the maximum error was always less than 10% full Chapter 6 CONDITION MONITORING AND FAULT DETECTION IN A SERVOVALVE 208 scale, and the RMS error for the predicted flow rate was just above 1% full scale. This indicates that, whenever the use of a conventional flowmeter is not possible or economic, but information regarding the flow rate is still required, its estimation results from the proposed method can be used instead to an acceptable degree of accuracy. Regarding the system coefficient estimates, we found that while a number of parameters, such as coil resistance, main spring constant, and spring preload could accurately be estimated, the friction coefficient estimates were generally poor and biased with higher estimation error variance, indicating less observability. The complexity of the valve structure, unmodelled dynamics, noise effects, and numerous differentiable and non-differentiable nonlinearities in the servovalve were postulated to be responsible for the bias and high variance. By trying different starting values for the parameters and system noise covariances within a reasonable range, we found that the parameters consistently converged to certain values, which were not necessarily the true estimates. For the proposed FDD methodology described in Chapter 3, it is the change in a parameter value which is of importance, and therefore, as long as the estimated parameter values remain consistent and within a certain level, the parameter-based approach can be a good candidate for such a monitoring strategy. By using a number of simulation and real examples, we showed that fault detection and symptom generation, even by using noisy and uncertain physical parameter and state estimates is not only possible but also promising. In particular, we found that the valve orifice profile estimate when plotted against the spool position can give valuable on-line information about the health condition of the component. This particular process can also be performed off-line. One good example may be regular checking of a servovalve at the end of the day or the week, by giving the same input signals to the valve and measuring a number of system inputs and outputs. The predictive maintenance routine may then contain functions for comparing the orifice area plots vs. the spool position, looking for discrepancies. A similar process was performed in this research and was shown to be successful in finding flaws and incipient faults in a valve being in operation for about one year. Chapter 7: CONDITION MONITORING IN A SERVO-ACTUATOR SYSTEM 7.1 Introduction The problem of state-space modeling for condition monitoring and fault diagnosis of a typical electro-hydraulic servovalve was investigated in Chapter 6. A component-wise modeling and monitoring strategy was adopted, and it was shown how the estimated orifice effective area of a pilot valve, as well as other valve parameters, is able to reveal even minor and progressive faults in a hydraulic component. In this chapter, a number of other hydraulic components which are very common in industry and are present in the hydraulic test rig have been considered. Continuous-time models suitable for simulation of hydraulic circuits have already been proposed by a number of researchers (see, e.g., [12 Dransfield, 1981][5 Burrows et al., 1989][61 Sepehri, 1990]). The modeling requirements for "real-time" simulation of such systems has been addressed in [6 Burton et al, 1994]. In this chapter, we have developed a simplified state-space model which is appropriate for "on-line" condition monitoring and fault detection in the servo-actuator subsystem of the hydraulic test rig, which is shown in Fig. 4.2. The servo-actuator subsystem, as illustrated in Fig. 7.1, consists of: 1. the rigid manipulator and load (the link), 2. the actuator (the cylinder and piston), 3. the main valve head-side and rod-side circuits, and 4. the high pressure filter. 209 Fig. 7.1: Schematic diagram of a servo-actuator system and its components. Section 7. 2 State-Space Representations 211 The other components of the hydraulic test rig, such as the electric motor, the gear pump, and the safety valves were not considered in this investigation. State-space formulae representing the dynamic behavior of the four hydraulic components, i.e., the manipulator, the actuator, the main valve circuit, and the high pressure filter, are developed first, starting from the manipulator. Again, a component-wise modeling and monitoring strategy has been adopted, as we believe this, wherever possible, greatly reduces the time and the capital spent on a comprehensive fault diagnosis of a complex system. By assigning a Kalman filter to each component's model, on-line estimation of normal parameters and traceable states of the test rig components are performed experimentally. The applicability and effectiveness of our model-based FDD technique then shown using a real example from the machine. 7.2 State-Space Representations Note: all the notations used in this chapter are previously defined in Chapters 4, 5, and 6, and in Appendix A. 7.2.1 The Manipulator The 1 -degree-of-freedom rotary manipulator which is connected to the hydraulic cylinder and acts as a nonlinear load on the actuator was explained in Chapter 4 and Fig. 4.17. The manipulator and the attached load is a good representation of a general nonlinear load applied to a hydraulic actuator. Its condition monitoring is important especially in mobile hydraulics, robotics, and aerospace applications, where the load mass (or aerodynamic force), the load center of gravity (or aerodynamic center), and the linkage friction may change quite often. The equation of motion for the link is given in Eqs. (4.27) and (4.28), and may conveniently be rewritten Chapter 7 CONDITION MONITORING IN A SERVO-ACTUATOR SYSTEM 212 (7.1) in a continuous nonlinear state-space form as: X\ = X2 + Wi &2 = hn(ji + f°ff^jx3 -gsm(x1)x4 - x2x5 - Rbfnsgn(x2)x6 + w2 x3 = w3, x4 = w4, x5 = w5, xe = we yi = xi + v-i where xi = 6 is link joint angle from vertical (a state), x2 = 6 is link angular velocity (a state), x3 - (a parameter), x4 = (a parameter), x5 = j§dt ( a parameter), xe = J^JJI (a parameter), y1 = 9meas is the current measurement of link joint angle (the output), / ; is the current measurement of load force applied by the actuator (the input), f°** is the current load measurement error or offset (see Eq. (7.3)). In Eq. (7.1), the time index t is dropped for brevity and the load force arm hn is given by Eqs. (4.21), (4.22), (4.26), where all the kinematic parameters of the link are fixed and known, and the link joint angle 9meas is measured at time t. The normal force on the joint bearing, fn in Eq. (7.1), can be computed from Eq. (4.29). Note that the current estimates of link angular velocity, load mass, and load center of gravity are to be used in Eq. (4.29) to obtain / „ . These estimates, along with the current estimates of joint viscous damping and joint Coulomb friction coefficients, at time t are computed from xi - 1 - XK Xe. 8 = x2, Mt = ^-,Ll= - Cv = ^, Ac = (7.2) x3 x4 x3 x3 Section 7. 2 State-Space Representations 213 7.2.2 The Hydraulic Actuator Monitoring the internal friction in the hydraulic cylinder is important, since it gives valuable information about the condition of the piston seals, which might directly affect the working efficiency of the actuator. The equation of motion for the linear hydraulic cylinder is given in Eq. (4.20). Following the same argument as in Section 6.2.3, we notice that the inertial force of the piston and rod assembly (Mr = 0.7 Kg) is negligible compared to the inertial force of the load and internal frictional forces in the actuator. Therefore, there is virtually no state equation present (except for the random-walk parameter models), and the dynamic equation of motion of the actuator piston is simplified to a static force equilibrium relationship, as Xj = Wj Xg = Wg (7.3) X9-WQ y2 = xrx7 + sgn^x^zg + x9 where x7 is the viscous friction coefficient between the piston and the cylinder wall, £ 8 is the Coulomb friction between the piston and the cylinder wall, £9 is the force measurement residual due to possible offset, drift, and/or misalignment of the directional load pin w.r.t. the piston rod (assuming that the pressure transducers are error-free), 2/2 = Arpr — AhPh — fi is the net external force applied to the piston (measured), xr is the current estimate of the piston rod velocity which is computed from Eq. (4.25) using the current link angular velocity estimate, x2 and the current value for hn. Note that the current force measurement error, = x9 in Eq. (7.3), is added to the current actual force measurement / ; in Eq. (7.1) in order to compensate for the force measurement error. Due to the static nature of Eq. (7.1), and a relatively high piston velocity, as compared to the main spool velocity, here we may use a velocity threshold in order to avoid nonlinear and complicated friction effects Chapter 7 CONDITION MONITORING IN A SERVO-ACTUATOR SYSTEM 214 due to boundary lubrication at very low piston velocities [33 Karnopp, 1985]. This velocity threshold allows us to solve Eq. (7.1) only at the occasions when \xr\ > v t h r e s h _ 7.2.3 The Main Valve Connecting Ports On-line monitoring of the orifice profile of a valve can provide extremely useful information about the health condition of the connecting ports, orifice area shape, level of contamination in the oil, and any excessive leakage or fluid loss in the valve or the connecting port circuits. In this section, we have derived the state-space equations appropriate for on-line estimation of the main valve orifice profile during valve operation. A. The Valve Structure: The main valve shown in Fig. 7.1 has four ports: 1. the supply port with measured line pressure ps, which is connected to the high pressure filter outlet, 2. the exit port with measured line pressure pe, which is connected to the check valve (not shown), 3. the rod-side port with measured line pressure pr, which is connected to the rod-side (retraction) chamber of the hydraulic actuator, and 4. the head-side port with measured line pressure ph, which is connected to the head-side (extension) chamber of the hydraulic actuator. The dynamic equation of the main valve spool has already been given in Chapter 5, and has been used to on-line monitor the spool parameters in Chapter 6. Here, the flow equations through the valve are written in a nonlinear state space form suitable to monitor the valve parameters and the traceable states. As we have shown in Chapter 6, the orifice effective area of the pilot stage can be estimated accurately enough from the pressure and actuator position measurements, to give valuable information about the health condition of the valve. The same concept has been used here to predict the orifice effective area of the main valve using low-cost robust measuring devices other than flowmeters. Section 7. 2 State-Space Representations 215 B. The Assumptions: A number of assumptions have been made to obtain appropriate flow equations: 1. Since the lengths of the hydraulic hoses connecting the main valve to the actuator are short (less than 1 meter) and the total oil volume in each line is small (less than 0.35 liters), pressure transmission delay in the circuit is ignored [6 Burton et al., 1994] and a simple lumped parameter model has been used, as suggested in [47 Merritt, 1967]. 2. All internal and external leakages in the main valve, the hydraulic cylinder, and the safety valves have been ignored compared to the amount of flow passing through the main valve orifice. 3. The oil density and effective bulk modulus vary with temperature and pressure according to Eqs. (4.3), (4.5), and (4.6). 4. The main valve is closed center, and therefore has overlaps. The main spool displacement in positive direction opens the rod-side port to the supply port and the head-side port to the exit port, and vice versa. The first assumption is common, and has widely been used in the literature (see, e.g., [47 Merritt, 1967][46 McCloy and Martin, 1980][12 Dransfield, 1981]). Using this assumption and Eq. (4.4), we may lump the compliance of the connecting hose between the main valve and the actuator with the compliance of the actuator chamber to get the lumped compliance, as where Vh0 and Vao are the static volumes of the hose and the actuator chamber, respectively. Usually, AVh, the change in the volume of a short hose is negligible compared to AVa, the change in the volume of an actuator chamber due to the motion of its piston. Regarding the second assumption, as we shall show later, most of the leakage in the valve and the hydraulic lines is implicit in the "orifice effective area" estimate. The third assumption is based on the lump Vh Va = Pe.Vh + PekVa Peh PeJea (Vho + aVao) + (AVh + aAVa) (7.4) a Chapter 7 CONDITION MONITORING IN A SERVO-ACTUATOR SYSTEM 216 empirical relations and mathematical formulae derived in Chapter 4. The fourth assumption is dictated by the type of the valve installed in the actual circuit. C. The Rod-Side Port Flow Equation: The flow continuity equation in the rod-side circuit may be written linguistically as: flow through the valve = compliance flow in the circuit + flow due to piston motion + leakage The instantaneous amount of fluid compressed in the volume between the rod-side port of the valve and the rod chamber of the hydraulic cylinder gives rise to the rod-side pressure. Depending on which port, supply or exit, the valve opens to, two different equations for the pressure rise can be derived from Eqs. (4.15) to (4.17). Combining the two cases, we obtain Pr =77 T ^ - J {aor[max(0,sgn(a:m))sgn(p s - pr)^(2/p)\ps - pr\-V0}T + aArxr (7 5) min(0,sgn(a;m))sgn(pr - pe)y/(2/p)\pr - pe\) - Arxr - qw{pr)} where Vb,r = [Vh0 + aVao)r, and qir(pr) is a strong function of the line pressure pr, which accounts for the flow loss due to cylinder internal leakage, main valve leakage, relief valve opening, and hose leakage in the rod-side circuit. aor is the rod-side orifice effective area. Similar to the pilot valve orifice effective area definition in Eq. (6.9), aor is conveniently defined as: dor — \ [Cd ,sr^o,sr (if xm ^ Lsr) ~Cd,reao,re {if xm < ~LTe) (7.6) 0 (otherwise) where Cd,Sr is the discharge coefficient form the supply to the rod-side port, Cd,re is the discharge coefficient form the rod-side port to the exit port, a0tSr is the actual orifice area when the rod-side port is open to supply port, a0tre is the actual orifice area when the rod-side port is open to the exit port, Lsr is the supply-to-rod port spool overlap length (always positive), Lre is the rod-to-exit port spool overlap length (always positive). Section 7. 2 State-Space Representations 217 Note that a negative aor refers to an orifice area when the rod-side port opens to the exit port, while a positive aor refers to an orifice area when the rod-side port opens to the supply port. In every case, the orifice effective area is a strong function of the spool position (see Fig. 4.14), and as such, can be written as: dor — / r C ^ m ) (7*7) Taking the time derivative of the above, we obtain QIQT — Sor Xm _ dfr(xm) (7-8) OXm where sor is the rod-side orifice area profile slope, which usually varies at a much slower rate than does the orifice area w.r.t. the main spool displacement. D. The Rod-Side Port State-Space Model: A state-space representation for the rod-side model of the hydraulic circuit, neglecting the leakage flow rate qir, can now be written using Eqs. (7.5) to (7.8), as £ 1 0 = 7 7 — ^ { x n [ m a x ( 0 , s g n ( x m ) ) s g n ( p 5 - p r ) v / ( 2 / p ) b s - P r | -VQ,r + OiArXT where min (0, sgn(a;m))sgn(p7. - pe)V(2/p)\Pr ~ Pe\] - Arxr} + wxo i n = x m x 1 2 + tun xu =w12 V3 =Xio + v3 xio is the rod-side port pressure, a: 11 is the rod-side orifice effective area, #12 is the rod-side orifice area profile slope, i/3 is the rod-side port pressure measurement, £ r is the current estimate of the piston rod position obtained from Eqs. (4.21) to (4.24), (7.9) Chapter 7 CONDITION MONITORING IN A SERVO-ACTUATOR SYSTEM 218 xm is the current estimate of the main spool velocity obtained from Eq. (6.17), xm is the current main spool position (measured), E. The Head-Side Port Flow Equation: Similar to the rod-side port, the instantaneous amount of fluid compressed in the volume between the head-side port of the valve and the head chamber of the hydraulic cylinder gives rise to the head-side pressure. Depending on which port, supply or exit, the valve opens to, two different equations for the pressure rise can be derived from Eqs. (4.15), (4.18), and (4.19). Combining the two cases, we obtain where V0th = (Vh0 + aVao)h, and qihijph.) is a strong function of the line pressure ph and accounts for the flow loss due to cylinder internal leakage, main valve leakage, relief valve opening, and hose leakage in the head-side circuit. a„h is the head-side orifice effective area. Similar to the rod-side orifice effective area definition in Eq. (7.6), aQh is conveniently defined as: where Cd,sh is the discharge coefficient form the supply to the head-side port, Cd,he is the discharge coefficient form the head-side port to the exit port, a0ySh is the actual orifice area when the head-side port is open to the supply port, a0ihe is the actual orifice area when the head-side port is open to the exit port, Lsh is the supply-to-head port spool overlap length (always positive), Lhe is the head-to-exit port spool overlap length (always positive). Note that a negative aQh refers to an orifice area when the head-side port opens to the exit port, while a positive aah refers to an orifice area when the head-side port opens to the supply port. Again, the orifice effective area is a strong function of the spool position, and as such can be written as: Ph = (7.10) max(0,sgn(xTO))sgn(pA - pe)y/(2/p)\ph - pe\] + Ahxr - qlh(Ph)} (Cd ,sh^o,sh (if xm ^ Lsh) doh = \ -Cd,heao,he (if xm > Lhe) 0 (otherwise) (7.11) aah = fh(Xm) (7.12) Section 7. 2 State-Space Representations 219 Taking the time derivative of the above, we finally obtain "oft = Soh%m _ dfh(xm) (7-13) OXM where sor is the rod-side orifice area profile slope, which usually varies at a much slower rate than does the orifice area w.r.t. the main spool displacement. F. The Head-Side Port State-Space Model: Similar to the rod-side port, a state-space representation for the head-side model of the hydraulic circuit, neglecting the leakage flow rate q^, can be written using Eqs. (7.10) to (7.13), as £ i 3 =77 r-{a;14[mm(0,sgn(a;m))sgn(ps - Ph)V(2/p)\Ps ~ Ph\~ Vb.ft - OLAKXT (7.14) where max (0, sgn(xm))sgn(p f c - Pe)^/(2/p)\ph - Pe\] + Ahxr} + w13 X\4 =Xm.Xl5 + W\4 Xl5 =^15 2/4 = Z l 3 + V4 x13 is the head-side port pressure, x14 is the head-side orifice effective area, xi5 is the head-side orifice area profile slope, ?/4 is the head-side port pressure measurement. 7.2.4 The High Pressure Filter The high pressure filter is located between the pump outlet port and the servovalve supply port and has the duty of cleaning the oil before it enters into the very sensitive servovalve. The filter has its own compliance and resistance. However, its compliance is usually neglected [12 Dransfield, 1981]. The monitoring task here is to track the filter resistance factor to the flow over a long period of time and give appropriate alarms before the filter gets too dirty to pass the needed flow when demanded. Chapter 7 CONDITION MONITORING IN A SERVO-ACTUATOR SYSTEM 220 Using Eq. (4.7), the pressure drop across the filter can be related to the flow rate passing through the filter: Pp-ps = Rfqf (7.15) where /?/ is the filter resistance that increases over time as more particles are filtered. The flow rate passes through the filter is equal to the net flow rate passes through the supply port of the main valve (ignoring the pilot stage flow). Using Eqs. (7.5) and (7.10), we obtain qj =a o rmax(0,sgn(xm))sgn(p s - pr)V(2/P)\Ps ~ Pr\-(7.16) aoh min (0, sgn(xm))sgn(p s - Ph)V(2/P)\Ps ~ Ph\ where the estimates of the orifice areas are utilized. Then, the state-space representation of the filter, which estimates only one parameter, i.e., the filter resistance, becomes a>i6 =w16 2/5 = [ £ n m a x ( 0 , s g n ( a ; m ) ) s g n ( p s - pr)V(2/P)\P* ~ Pr\~ (7.17) x14 min (0, sgn(zm))sgn(p s - Ph)V(2/P)\P* ~ PhW^ie + v5 where xie is the filter resistance to the flow, y5 = pp — ps is the measured pressure drop across the filter. 7.3 Model Partitioning and Estimation 7.3.1 Model Partitioning Similar to Chapter 6, discrete Extended Kalman Filtering can be used to estimate both the system parameters and states simultaneously. Although it is possible to solve all the 16 state equations using the 5 measurement equations in (7.1), (7.3), (7.9), (7.14), and (7.17) simultaneously, this is not recommended due to unnecessary heavy computational burden and memory consumption for handling large matrix manipulations and inversions. The way the state-space equations are derived in the previous section has enabled us to partition between the different models and to design dedicated Kalman filters that estimate the states and parameters for Section 7. 3 Model Partitioning and Estimation 221 v / Fig. 7.2: The data flow chart relates various component niters via measurement inputs and filter outputs, each individual component. Some component models might require inputs which are not available, due to lack of sensing devices. This problem has been solved using the estimates of those inputs provided by the other component niters. A relational chart is then needed to track the data flow between various component filters. For example, in Fig. 7.2, although the servovalve is the source of power and motion, it turns out that the link filter/estimator is the most independent computational module in the system, since all the necessary pieces of information for filtering and estimation directly come from sensors, provided that the load sensor is correctly installed and calibrated, i.e., £9 « 0. On the other hand, the actuator, the rod-side, and the Chapter 7 CONDITION MONITORING IN A SERVO-ACTUATOR SYSTEM 222 Table 7.1 Component faults and affected component filters. Any fault occurs in: Parameter estimates may be affected in: Manipulator Manipulator, Actuator, Main valve, Hi.-Filter Actuator Manipulator, Actuator Rod-Side Hydraulic Circuit Rod-Side Hydraulic Circuit, Hi.-Press. Filter Head-Side Hydraulic Circuit Head-Side Hydraulic Circuit, Hi.-Press. Filter High Pressure Filter High Pressure Filter Table 7.2 Sensor faults and affected component filters. Any fault occurs in: Parameter estimates may be affected for: Joint angle and/or the load force sensor(s) All comonents Rod-side port pressure sensor Actuator AND R-S* Circuit AND H-P* Filter Head-side port pressure sensor Actuator AND H-S* Circuit AND H-P Filter Supply port pressure transducer R-S AND H-S Circuits AND H-P Filter Pump pressure transducer High Pressure (H-P) Filter Main spool position sensor R-S AND H-S Circuits AND H-P Filter AND Sevovalve Filters (See Chapter 6) *R-S: Rod-Side *H-S: Head-Side *H-P: High Pressure head-side filters, although independent from one another, all require the position and velocity estimates of the piston rod, which is provided by the manipulator filter. Therefore, any major fault in the link might affect the performance of any of the other three component filters, but not vice versa. The estimator for the high pressure filter is the most vulnerable computational module, since its output might be affected by a fault in any one of the rod-side or the head-side filters, or even worse, by a fault in the link. This hierarchical dependency is more advantageous than being a drawback. Aside from reduction of computational burden and real-time programming hurdles, a component focused fault diagnosis process should greatly enjoy this kind of partitioned/relational data flow, as indicated in Section 7.1. Table 7.1, which has been derived from the data flow chart in Fig. 7.2, shows how a fault in one component might affect the estimation results of the other components. Table 7.2 shows similar results as a sensor becomes faulty. This way, one can easily trace the fault path back to the faulty component if one observes discrepancies in the parameter or traceable states in one or more component filters. Using Table 7.2, for instance, one may readily conclude that the "rod-side pressure transducer" is faulty as soon Section 7. 4 Experimental Results and Discussion 223 as deviations in the parameter/traceable state estimates are observed for both "actuator" and "rod-side circuit" component filters. 7.3.2 Estimation Method Eqs. (7.1), (7.3), (7.9), (7.14), and (7.17) have been discretized using Eq. (6.20) with the sampling period of Ts. Since the state-space models for the actuator, the rod and head side circuits, and the high pressure filter are all linear in states (or parameters), the E K F algorithm for those models converts to classical Kalman filtering algorithm (see, e.g., [21 Grewal and Andrews, 1993]. Therefore, there is no need to compute the Jacobians on-the-fly, which greatly reduces the computational effort. The Jacobian elements of the state function f(x, u) and the measurement function h(x, u) for the manipulator, using the discretized version of Eq. (7.1), are given as: 01,1 = 1 01,2 = r s 02,1 = -Tsgcos(xi)xA 02,2 = 1 - 2 > 5 02,3 = Tshnfi 02,4 = - T s g sin (zi) 02,5 = ~Tsx2 02,6 = -TsRbfnsgn(x2) 0«',« = 1, (» = 3,---,6) #1,1 = 1 (7.18) with all other elements of the Jacobian matrices $ 6 x 6 and H i X 6 equal to zero. 7.4 Experimental Results and Discussion The results of the two experiments are reported in this section. The first experiment was intended to Chapter 7 CONDITION MONITORING IN A SERVO-ACTUATOR SYSTEM 224 Table 7.3 Some of the system constant parameters used in the experiments. Parameter Description Magnitude Source Mi Overall load mass 11.1 kg meaurement L, Overall load center of gravity 0.553 m measurement Vo.r Initial volume of oil in rod-side circuit 1.963 x 1CT4 m 3 measurement V0.h Initial volume of oil in head-side circuit 2.038 x 10"4 m 3 measurement Ar Rod-side pressure area in the actuator 1.140 x 10~3 m 2 catalogue Ah Head-side pressure area in the actuator 0.942 x 10"3 m 2 catalogue p Hydraulic oil density at 40° C 852 kg/m3 catalogue Bulk modulus of oil in actuator chamber 1400 MPa catalogue Effective bulk modulus of oil in hoses 65 MPa catalogue identify the normal parameters and traceable states of the hydraulic test rig components that described in Section 7.1. In the second experiment, a faulty operation was detected and diagnosed, using the identified normal system parameters and traceable states from the results of the first experiment. 7.4.1 The Data In both experiments, the identification and fault detection processes were performed on-line using the computer setup explained in Chapter4. The sampling and monitoring frequency was Ts = 1 ms, and all computations in the computer were performed in "double precision" arithmetic. The oil temperature at the pump outlet was 40° ± 5° C. The pump and actuator relief valve settings were 6.8 MPa and 6.4 MPa, respectively. All system constants and other measurable or known parameters of the hydraulic components that have been used in computations or comparisons are either given in Appendix C or listed in Table 7.3. 7.4.2 Starting Values for the Kalman Filters In the first experiment, the initial values of the state vectors, x, for all component filters were set equal to small random numbers. The starting values for the estimation error covariance matrix, P, did not affect the final steady-state parameter estimates if not set very high or very low. The initial values for Section 7. 4 Experimental Results and Discussion 225 P matrices are listed below: For the manipulator EKF: Pgx6 = 103 x '6X6> For the actuator estimator: P 3 X 3 = 104 x l4X4> For the rod-side and head-side circuit estimators: P 3 X 3 = 10 1 2 x 13x3* For the high-pressure filter estimator: P i x i = 10 3 0 x IJ.XI> where I,X, is the unit square matrix of size i. In setting appropriate initial values for the system noise covariance matrices, Qs, we realized that proper choice of Qs played an important role in the speed of convergence and accuracy of the final estimation results. The values for Qs, therefore, were carefully tuned by trial and error to give a good convergence rate and an acceptable level of accuracy, as well as a reasonable signal-to-noise ratio. The system noise covariance matrices were chosen to be diagonal and constant. Their values for different filters are listed below: For the manipulator estimator: = 0, $2,2 = 0, Q,-,,- = 10 - 6 , (i — 3, • • • ,6) For the actuator estimator: Qi . i = 1, $2,2 = 1, $3,3 = 1 For the rod-side and head-side and head-side circuit estimators: <3i,i = 0, Q2,2 = 0, Q3t3 = 1 0 - 1 0 For the high pressure filter estimator: Qi , i = 106 Finally, the measurement noise covariance matrices, Rs, were estimated on-line and applied in the filtering/estimation algorithm using Eq. (A. 16). In Experiment 2, the identification results of the first experiment were used. The final values for different covariance matrices and constant parameters were utilized for filter initialization. 7.4.3 Experiment 1: Estimation of Normal System Parameters Sinusoidal test signals with zero DC magnitude at 0.5 Hz for a period of eight seconds were applied to the solenoids of the servovalve while the hydraulic pump was running. As a result of pressure buildup in the actuator, the link started to move back and forth. During the eight-second operation, the SPARC IE Chapter 7 CONDITION MONITORING IN A SERVO-ACTUATOR SYSTEM 226 Table 7.4 Mean values and standard deviations for the identified parameters of the manipulator. Parameter Mean Value STD Relative STD load mass, M ; (kg) 12.86 0.31 2.4% load center of gravity, L\ (m) 0.514 0.003 0.58% joint bearing viscous frcition coefficient, Cv (N.m.s/rad) 19.38 0.66 3.4% joint bearing Coulomb friction coefficient, JJLC 0.327 0.011 3.4% server computer constantly received on-line data from the system sensors every 1 ms, passed the data to the S K Y B O L T math coprocessor for filtering and monitoring tasks, received the processed information from the coprocessor at the same sampling rate, and stored the processed data in a file. In order to keep the numerous input-output data/information tractable, we followed the data flow chart in Fig. 7.2 as a guide. Thus, the input sensory data passed to each component's filter, along with the output estimation results are shown and described separately, in the order of filtering precedency. A. The Manipulator Results: The applied force by the piston rod and the joint angle measurements for the manipulator are shown in Fig. 7.3. Although the input signals had zero DC magnitude, the link moved towards the negative (extension) direction on the average. The main cause was the asymmetric area actuator that received more flow from the main valve in the head-side chamber than in the rod-side chamber. Estimation results for the states and parameters of the manipulator, obtained from Eqs. (7.1) and (7.2), are plotted in Fig. 7.4. After eight seconds, all the parameters of the manipulator converged to their final steady-state values. The average values and standard deviations of the estimated parameters for the last 1000 data points are listed in Table 7.4. The overall load mass estimate, M ; , due to the combined mass of the load and the link, is somewhat more than the actual value of 11.1 Kg, while its center of gravity estimate, Li, is slightly less than the actual value of 0.553 m. This is mainly because of the effect of link moment of inertia which was neglected in the model equations. Using a heavier mass of 31.5 Kg in a separate experiment, we found that the relative estimation errors for both Mi and X; decreased. Section 7. 4 Experimental Results and Discussion 227 (a) Measured load force T 1 1 1 1 1 r I i i i i i i i 1 0 1 2 3 4 5 6 7 8 (b) Measured joint angle T 1 1 1 r Time (s) Fig. 7.3: Measured signals used for estimation of the manipulator parameters. Although we did not measure the actual values for the friction coefficients independently, it seems that the Coulomb friction coefficient estimate, nc, is biased and its magnitude is somewhat more than the value of a typical metal-to-metal contact friction coefficient. The main reason for such a discrepancy is that at low velocities, the simplified friction model in Eq. (7.1) and similar equations that we have used for parameter identification purposes are not accurate enough, and the friction phenomenon is much more complicated1 [33 Karnopp, 1985]. However, we tried a variety of starting values for the E K F on a number of different experimental data, and we always ended up with the same final results with insignificant variations, indicating that the estimates are stable and consistent, and therefore can be used for condition monitoring purposes. 1) For estimation of the friction force at low velocities in hydraulic actuators with more advanced models, see, e.g., [66 Tafazoli et al., 1996] Chapter 7 CONDITION MONITORING IN A SERVO-ACTUATOR SYSTEM 228 Section 7. 4 Experimental Results and Discussion 229 (a) Measured rod-side port pressure (c) Measured load force 0 2 4 6 8 (b) Measured head-side port pressure 0 2 4 6 8 (d) Estimated piston rod velocity 2 4 6 8 0 Time (s) Fig. 7.5: Measured and estimated signals used for estimation of the actuator parameters. 2 4 6 Time (s) B. The Actuator Results: Fig. 7.5 shows the measured hydraulic pressures and load force applied to the actuator's piston and piston rod, along with the piston rod velocity that has been estimated from Eq. (4.25) using the current estimate of the link angular velocity, 9 — x2. Utilizing the four signals shown in Fig. 7.5 as the outputs and the input, Eq. (7.3) has been solved recursively by applying Kalman filtering. A piston velocity threshold of vthresh — 0.01 m/s was chosen as a switch to turn on and off the estimation process. The viscous friction coefficient, the Coulomb friction force, and the load cell offset have been estimated and plotted in Fig. 7.6. From this figure, it can be realized that the estimated friction parameters have Chapter 7 CONDITION MONITORING IN A SERVO-ACTUATOR SYSTEM 230 300 200 CO X 100 0 -100 r-200 X -300 -400 -500 (a) Viscous friction coefficient 3 4 5 (b) Coulomb friction force 3 4 5 (c) Force sensor offset \*P—V 1 2 3 4 5 6 Time (s) Fig. 7.6: Estimation results for the parameters of the actuator. - 1 1 1 1 1 i i « - i i i i i 1 1 200 100 CD p C o o ir -ioo CD O -200 -300 -0.1 -0.08 -0.06 -0.04 -0.02 0 Piston rod velocity, DXr (m/s) 0.02 0.04 Fig. 7.7: Actuator's total friction force estimate vs. the piston rod velocity. Section 7. 4 Experimental Results and Discussion 231 lower absolute values in the retraction mode (xr > 0) than in the extension mode (xr < 0), as reported earlier in Fig. 4.16. The total friction estimates in both extension and retraction modes are plotted vs. the piston rod velocity in Fig. 7.7. Apparently, the friction estimates are in good agreement with the independent measurement results shown in Fig. 4.16. This indicates that, whenever dynamic effects can be ignored and the velocities are rather high, the friction estimates obtained from static equations using a velocity threshold are less biased than those values computed from dynamic equations and/or without velocity thresholding. The plots in Fig. 7.7 may be used in practice to monitor changes in the total friction force due to insufficient lubrication or excessive wear over a long period of time. One interesting point from Fig. 7.6 is that the measured load force had a considerable offset due to a misalignment of the load cell w.r.t. the piston rod end pin. Apparently, the more the link angle increased in the negative direction (due of the rod extension), the more the load-pin offset increased in the negative direction. This fact, if taken as a fault in a sensor, excellently shows the capability and the effectiveness of the proposed model-based condition monitoring technique for detecting system faults even at the normal parameter identification stage. C . The Main Valve Results: The amount and direction of oil flow through various main valve ports and the pressures building up in the corresponding port lines are all controlled by the position of the main spool through the voltage commands given to the servovalve solenoids (see Chapter 6 for more details). Al l the signals, either measured or estimated, which were required to solve for the states in Eqs. (7.9) and (7.14) are plotted in Fig. 7.8. The velocity of the main spool was computed from Eq. (6.22) by differentiating and low-pass filtering the main spool position measurements. The velocity of the piston rod was obtained from Eq. (4.25) using the Kalman filter estimation of the link angular velocity. The piston rod position has been computed from Eq. (4.24) using the joint angle measurements. Chapter 7 CONDITION MONITORING IN A SERVO-ACTUATOR SYSTEM 232 Measured supply port pressure Measured main spool position 0 2 4 6 8 Measured rod-side port pressure 0 2 4 6 8 Measured head-side port pressure 2 4 6 Measured exit port pressure 2 4 6 Time (s) 2 4 6 8 Estimated main spool velocity 0.05 F 0 Xr (i -0.05 Q -0.1 2 4 6 8 Estimated piston rod velocity 0 2 4 6 8 Estimated piston rod position 4 6 Time (s) Fig. 7.8: Measured and estimated signals used for prediction of the main valve orifice parameters. Section 7. 4 Experimental Results and Discussion 233 Using the signals shown in Fig. 7.8 and Kalman filtering, we were able to on-line estimate the main valve orifice effective areas and their approximate profile slopes when the rod-side port or the head-side port was open to the supply port or to the exit port (four different profiles). The estimation results are plotted in Fig. 7.9, which shows that a fast convergence for all the state estimates has been achieved. We also estimated the approximate valve deadband and overlaps between the spool lands and the valve ports. The estimation results are plotted versus the main spool displacement and are compared with independent measurements of the main valve orifice areas, all shown in Fig. 7.10. The independent measurements were obtained using the method described in Section 4.3.8. As can be seen in Fig. 7.10, there is a close match between the measured and the estimated valve orifice area profiles. Small discrepancies are due to sensor noise, sensor accuracy and resolution, actuator internal leakage, uncertainty of constant coefficients (e.g., pressure area, bulk modulus, etc.) used to solve the state equations, model approximations (e.g., turbulence vs. laminar flow), and discretization error. An encouraging point from Fig. 7.10 is that the valve orifice effective area estimates enabled us to detect the eccentricity of the main spool lands w.r.t the rod-side and the head side ports. Apparently, the spool null (zero) position was slightly shifted leftward. As a result, with the same input voltage magnitude in either direction, the head-side port passed more flow than the rod-side port did. If this is taken as a fault in a servovalve, which is the case in certain control applications, once again our proposed model-based condition monitoring method has successfully detected this flaw at the early stage of the normal system identification. D. Mathematical Representation of the Normal Orifice Area Estimates: In order to be able to detect the orifice profile change due to a faulty situation, the current estimations should be compared to the nominal values of the orifice effective areas in real-time. A functional relationship between the orifice area estimates and the main spool displacements should therefore be established for a normal case. In Section 6.5.4, a feedforward neural network was trained to Chapter 7 CONDITION MONITORING IN A SERVO-ACTUATOR SYSTEM 234 (a) Rod-side orifice area (c) Head-side orifice area 1.5 1 | 0.5 E r o X -0.5 -1 4 Time (s) (b) Rod-side orifice profile slope ) 2 4 6 8 Time (s) (d) Head-side orifice profile slope 4 Time (s) Fig. 7.9: Estimation results for the main valve orifice profiles and profile slopes. (a) Rod-side port orifice area vs. spool postion 1.5 0 ... , 1 . o open to exit 1 i i port closed 1 o -1 o 1 o 1 o 1 o O static experiment - on-line estimation — curve fitting 1 o open to supply _ _ i i 1 i i 1 i _ l L_ ! I I ! I I -1 -0.5 0 0.5 1 1.5 (b) Head-side port orifice area vs. spool postion mm Fig. 7.10: Comparisons between the main valve orifice area measurements, estimation, and function approximation. Section 7. 4 Experimental Results and Discussion 235 Table 7.5 Coefficients of the fitted polynomials to the main valve orifice area estimates. CASE CO Cl C2 ^overlap Supply to Rod-Side: -3.892 x 10"7 -1.077 x 10"3 2.117 0.75 mm Head-Side to Exit: -4.242 x lO"7 -2.501 X 10"4 1.073 0.76 mm Supply to Head-side: -7.737 x 10"7 -1.425 x 10"3 7.120 X 10"1 0.44 mm Rod-Side to Exit: -2.830 x 10"7 -2.537 x 10"4 9.418 x 10"1 0.43 mm approximate the relationship for the simulated pilot valve, due to unavailability of the pilot spool position measurements. In this chapter, however, because of the direct access to the main spool position measurements, the functional relationship is more clear and defined (compare Fig. 7.10 with Fig. 6.19(c)). As such, simple polynomials in the form of: anorm _ 1 C2xm + C\%m + c 0 (if \xm\ > Loverlap) (J ]0/) ° 10 otherwise were fit in a least-squares sense to the estimated normal orifice area data points for separate cases. M A T L A B ' s p o l y f i t function was used for this purpose. The coefficients of the polynomials are tabulated in Table 7.5, and the polynomial approximations are plotted as solid curves in Fig. 7.10 for comparison. E. High Pressure Filter Results: The pressure measurements across the high pressure filter and the instantaneous amount of flow passing through the filter, computed from Eq. (7.16), are shown in Fig. 7.11. These signals were used by a Kalman filter to estimate the high pressure filter resistance coefficient. The estimation results, which are plotted in Fig. 7.12, show that the flow resistance factor converged to a final value (approximately 1200 MPa.s/m3) that was an order of magnitude higher than an average resistance factor for a new filter provided by the manufacturer (which is about 100 MPa.s/m3 from Appendix C). The main reason for the discrepancy was the fact that the pressure transducers had been placed right after the pump outlet port and right before the servovalve supply port entrance (see Fig. 7.1). As a result, the Chapter 7 CONDITION MONITORING IN A SERVO-ACTUATOR SYSTEM 236 (a) Measured Pump pressure x 10 2 3 4 5 6 (c) Estimated flow rate through the supply port 3 4 Time (s) 8 Fig. 7.11: Measured and estimated signals used for identification of the high pressure filter resistance factor. 3000 r Fig. 7.12: Estimation results for the high pressure filter resistance coefficient. Section 7. 4 Experimental Results and Discussion 237 total pressure drop between the two sensors was considered in the computations, which was due to the resistance of the high pressure filter, as well as the resistance of pump outlet port, servovalve inlet port, pipe and hose connections and bents, and a safety flow restrictor. In a separate experiment, we calculated the circuit resistance without the high pressure filter in the line. We conducted the experiment by allowing a fixed amount of fluid to pass through the servovalve within a certain period of time. We then divided the measured pressure difference between the pump outlet and the valve inlet ports by the amount of fluid passed through the valve per second. The line resistance thus obtained was about 970 MPa.s/m 3. By comparing this resistance with the estimation results shown in Fig. 7.12, we concluded that the filter resistance was around 230 MPa.s/m 3. This number is not too far from the manufacturer's figure (about 100 MPa.s/m3), if we consider the fact that the filter had been in use for more than a year prior to the experiment. 7.4.4 Experiment 2: Condition Monitoring and Fault Detection After identification of the normal system in the previous section, all the system parameters, traceable states, and the last values for covariance matrices were stored in a file to be used for a condition monitoring experiment. A. Description of the Experiment: In Experiment 2, in order to assess the ability of the proposed FDD technique in practice, we introduced a fluid loss in one of the actuator's connecting hoses, i.e., the rod-side hose. In a plant, the fluid loss might, be due to hose rupture, substantial leakage in hose/pipe connections or a valve, or abrupt opening of a safety valve due to overloading. As such, fluid loss in the rod-side circuit was introduced by intentionally reducing the pressure setting of the rod-side sequence (pressure reducing) valve at a certain time during a retraction process. The line pressure was reduced manually by turning the adjustment screw on the anticavitation valve. Chapter 7 CONDITION MONITORING IN A SERVO-ACTUATOR SYSTEM 238 High frequency arbitrary voltage signals superimposed on a D C voltage were chosen as input to one of the servovalve solenoids in order to intentionally hide the effects of pressure reductions during the transition period (i.e., during turning the pressure setting screw). The whole experiment took 4 seconds and the approximate starting time of pressure reduction was recorded manually in a computer memory, which was at tf « 2.7 seconds from the starting time of the experiment. B. Fault Detection and Diagnosis Results of the Main Valve: On-line estimation results from the manipulator and the actuator filters did not show any significant changes in the parameter estimates, as we expected from the data flow chart in Fig. 7.2 and Table 7.1. Therefore, the fault detection and diagnosis effort was focussed on the main valve and high pressure filter components and their connecting circuits. All the measured signals from the main valve sensors, along with the actuator position and velocity estimates obtained from the manipulator filter are shown in Fig. 7.13. As can be seen from the rod-side pressure data, despite a very fast response pressure reducing valve (with 2 ms response time), there was a transition period (about 0.4 seconds) from the approximate starting time of the fault imposition until the pressure settles down at its set point value. This time lag was due to the fact that the pressure resetting was performed manually with some unavoidable glitches. Using the data shown in Fig. 7.13 and Kalman filtering, we re-estimated the valve orifice areas in real-time. We compared the estimation results in Figs. 7.14(b) and 7.15(b) with the corresponding nominal values obtained from the fitted polynomials to the normal orifice data, using the main spool position measurements. The estimation residuals, i.e., the error between the current orifice area estimates and the nominal values, are plotted in Figs. 7.14(c) and 7.15(c). Obviously, there was a close match between the current estimates and the corresponding nominal values up to the fault time. The mismatch between the current rod-side orifice area estimate and its nominal value began about the same time that the pressure setting started to change, indicating how sensitive the orifice area estimates were to the changes in the flow condition. This was while the head-side orifice area estimates did not show any significant deviations from the nominal values. Note that it was very difficult Section 7. 4 Experimental Results and Discussion 239 to detect the fault at early stages by only using the pressure data shown in Figs. 7.14(a) and 7.15(a). The reason for negative deviation of the orifice area estimates from the nominal values in Fig. 7.14(b) should be sought in state equation (7.9), which was obtained from Eq. (7.5) by neglecting the excess flow rate qir{Pr)- 1° this experiment, by intentionally discharging to the exit port some portion of the flow that was passing from the rod-side orifice to the actuator chamber, we actually produced a negative flow rate qir{Pr) in Eq. (7.5) which had not been considered in the state equation (7.9). As a result, the amount of the valve opening was automatically reduced by the rod-side estimation algorithm to compensate for the flow "lost" in the circuit, and to keep the continuity equation valid. In order to detect the flow loss in the circuit, SPRT algorithm was applied to the orifice area estimate residuals. The results are shown in Figs. 7.14(d) and 7.15(d). In Fig. 7.14(d), the test function A - for the rod-side SPRT passed the threshold at ta = 3.012 s, and the estimated fault time was if = 2.681 s. The test functions for the head-side SPRT never crossed the threshold, indicating a "no-fault" situation in the circuit. C. Fault Detection and Diagnosis Results of the High Pressure Filter: The measured and estimated signals required to on-line estimate the filter resistance coefficient are shown in Fig. 7.16. As was mentioned earlier, we expected that the filter coefficient estimate would change due to an artificial change in the estimated flow rate through the supply port. For this reason, the parameter estimation residuals were generated using Eq. (3.16). SPRT was then applied to the estimation residuals. The estimation and fault detection results are shown in Fig. 7.17. From the figure, apparently the high pressure filter resistance was affected only after the discharged flow became significant. Its estimated value increased for a short period of time to reflect the artificial flow deficiency across the supply port. Even after the resettling time (represented by the vertical dash-dot line), the magnitude of the estimated resistance coefficient did not significantly jump. Chapter 7 CONDITION MONITORING IN A SERVO-ACTUATOR SYSTEM 240 Measured supply port pressure 6.91 • • — i i i i 1 0 1 2 3 4 Measured head-side port pressure 6 i i i : 1— : 1 0 1 2 3 4 Measured exit port pressure 2 3 4 Time (s) Measured main spool position E 0.5 X I I 1 1 0 1 2 3 4 Estimated main spool velocity Estimated piston rod velocity 0.061 • • — l i i : i 1 0 1 2 3 4 Estimated piston rod position 0.081 • • — 0.06 Eo.04 *0.02 0 0 1 2 3 4 Time (s) Fig. 7.13: Measured and estimated signals used for tracking of the main valve orifice parameters during a faulty operation. Section 7. 4 Experimental Results and Discussion 241 (a) Rod-side port pressure measured estimated 0.5 1 1.5 2 2.5 3 (b) Rod-side port orifice effective area 3.5 0.5 1 1.5 2 2.5 3 (c) Orifice area estimation error (residuals) 0.41 x -0.2 -0.8 MM '0 0.5 1 1.5 2 2.5 3 3.5 (d) SPRT results on the orifice area estimation residuals 20 _ 1 5 TJ 110 CO negative direction positive direction fault time V 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (s) Fig. 7.14: Condition monitoring and fault detection results using the rod-side orifice area estimate residuals. Chapter 7 CONDITION MONITORING IN A SERVO-ACTUATOR SYSTEM 242 (a) Head-side port pressure measured estimated 1 1.5 2 2.5 3 (b) Head-side port orifice effective area 0.5 0.4 E 0 2 E x 0 E £ - 0 . 2 -0.4 1 1.5 2 2.5 3 (c) Orifice area estimation error (residuals) 3.5 i r 1 .rru ' 1" Ji i — i — i 11 1 i . Li W H1' 1 0.5 1 1.5 2 2.5 3 3.5 (d) SPRT results on the orifice area estimation residuals 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (s) Fig. 7.15: Condition monitoring and fault detection results using the head-side orifice area estimate residuals. Section 7. 4 Experimental Results and Discussion 243 (a) Measured pump pressure 1 1.5 2 2.5 3 (c) Estimated flow rate through the supply port Fig. 7.16: Measured and estimated signals used to monitor the high pressure filter resistance factor during a faulty operation. (a) Filter resistance estimation residuals S 10 8 g 6 4 2 0 1 1.5 2 2.5 3 (b) SPRT results on the parameter residuals , ~_ n , , , ! 1 i\ i i k ki 0.5 1 1.5 2 2.5 3 (c) Estimated high pressure filter coefficient 3.5 Fig. 7.17: SPRT and parameter jump level classification results using the high pressure filter resistance factor estimates. Chapter 7 CONDITION MONITORING IN A SERVO-ACTUATOR SYSTEM 244 As was explained in the previous section, and according to our experience, the filter resistance estimates, in general, did not tend to vary quickly over time. This, however, is in agreement with the actual observations in which a filter resistance to the flow increases slowly over longer periods of time. 7.5 Summary and Concluding Remarks In this chapter, the model-based condition monitoring and fault diagnosis methodology, proposed in Chapter 3, was applied to a number of typical fluid power components present in a hydraulic test rig. Nonlinear state-space equations, suitable for component-wise monitoring and fault diagnosis, were derived from mathematical relations and physical laws, and then were solved using Kalman filtering. The valve orifice areas were accurately estimated on-line using a novel modeling technique that allowed the valve variations to be a nonlinear function of the spool displacements. The orifice profile and its overlap lengths were also revealed that matched well the experimental results. A novel data flow chart was introduced in this chapter that can greatly simplify the fault diagnosis efforts by focussing attention on those components that may potentially be affected by the fault. The experimental fault detection and diagnosis results once again demonstrated the capability of the orifice area estimates in detecting not only the valve faults, but also the other faults that are related to the connected hydraulic circuits. This was achieved, however, by using only low cost robust sensors. Meanwhile, the valve estimation results showed that the main spool position played a critical role in accurately determining the valve orifice areas, and therefore, all the efforts must be made to gain direct access to its measurements. Chapter 8: CONCLUSION 8.1 Summary The aim of this thesis was to provide on-line self-descriptive robust information about the health condition of a hydraulic component from raw sensor data using different levels of data processing and abstraction. Condition-based maintenance issues in hydraulic systems were addressed in this thesis. First, fault detection and diagnosis techniques based on prediction of faults and failures in technological plants were explained, and their advantages and disadvantages when applied to fluid power control systems were discussed in detail. Then, a new hierarchical model-based methodology for on-line condition monitoring and fault diagnosis of hydraulic systems was introduced and its capabilities in handling various faults in a dynamic system were investigated. Next, a hydraulic test rig, which was built in the Robotics and Control Laboratory at UBC to test the concepts of the proposed FDD methodology, was described. A nonlinear model for accurate simulation of faults in one of the test rig components, a two-stage proportional directional servovalve, was developed and validated. State-space nonlinear models of hydraulic components for monitoring purposes based on parameter and state tracking were proposed, and normal parameters for various hydraulic components were identified. The on-line capabilities of the proposed model-based methodology to detect and diagnose faults in those identified components were investigated using both simulations and experiments. 8.2 Contributions A number of major contributions have been made to the fields of both hydraulics and fault detection and diagnosis, as follows: 1. A novel hierarchical methodology for early detection and diagnosis of both incipient faults and abrupt failures in fluid power components was proposed that accepts raw sensor data as input 245 Chapter 8 CONCLUSION 246 and automatically generates qualitative fault symptoms and fault patterns in the form of physical parameter changes as output. It was shown by a number of examples that the method was able to detect both incipient and abrupt faults, was generally robust to false alarms, and requires few rules to diagnose a fault. Early detection of gradual faults as well as sudden failures was achieved by tracking parameter magnitudes and applying sequential probability ratio tests on parameter/state residuals. The method's robustness was the consequence of using a hierarchical fault detection technique that rejects false alarms by applying higher level decision making strategies such as N N P C s and expert systems. On the other hand, tracking physical system parameters (rather than diagnosing faults by directly using time-varying, noisy, nonrobust sensor data) was shown to provide much more insight into the root causes of faults in system components. As a result, fewer rules and less execution time were required to set up and run a knowledge-based system for fault diagnosis purposes. 2. The same SPRT that detected a change in a parameter value due to a probable fault was used for the first time to automatically detect the end of transition period from one magnitude to another for the same parameter. This way, there was no need to manually determine the appropriate time for resampling a parameter magnitude after a probable fault was detected. We showed through several examples that the SPRT is quite able to detect the time, in a statistical sense, when no further significant changes in parameter estimates could be observed after a major jump had been detected. This was achieved by testing parameter estimate residuals rather than direct parameter estimates in the SPRT algorithm. The corollary was that a fault changes a steady-state parameter value from one level to another within a finite period of time, causing a temporary change in its residual estimates. 3. As another contribution, an effective method was proposed for quantization of numerical parameter estimates before and after each change by assigning a jump class (or level) to each parameter value using the Control Charts concept. The method replaces numerical quantities with simple linguistic terms in the form of "Decreased", "Increased", "Not Changed", etc., which can be used more conveniently in a fault diagnosis expert system. Section 8. 2 Contributions 247 4. A fast-learning neural network pattern classifier was suggested to be used inside an expert system as part of its knowledge-base. It was shown through an example that the NNPC can automatically relate fault symptoms to fault modes (or fault events) within an expert system without requiring explicit rules or understanding of the underlying fault process. In one comparison, it was shown that the number of rules could be reduced by half. 5. A new comprehensive model for accurate simulation of a two-stage proportional directional flow control valve was developed and validated using experimental data. The model predicted the pilot pressures and main spool positions well with an RMS error within 10%. The model was especially suitable for simulation of faults and evaluation of FDD strategies in a typical servovalve. A novel step-by-step approach was adopted to determine the unknown physical parameters of the valve at different levels. The parameter identification levels ranged from simple static measurement techniques to sophisticated numerical search algorithms that used dynamic measurements. 6. A novel modular state-space formulation was derived that allows on-line estimation and monitoring of time-varying physical coefficients for each individual component of a fluid power system. It also reconstructs many expensive-to-measure states such as solenoid force, valve flow rates, and spool positions and velocities in the servovalve. Very high accuracy (RMS error within 1 %) was reported for flow rate estimates using simulation examples. 7. For the first time, the effective orifice area of a valve, including the deadbands (overlaps/underlaps), was accurately estimated on-line without any a priori knowledge about the size and the profile of the orifice. This was achieved by using only low-cost sensor data and a novel state-space model that related the variations of the orifice area to the spool velocity and the orifice profile slope modelled as a random-walk parameter. Simulation and experiment showed a close match between the estimates and measurements. 8. It was shown through simulations and experiments that the orifice area estimates obtained from the proposed state-space models were excellent indicators (in terms of explanation power, detection speed and accuracy) for many incipient and progressing faults in valves, as well as for finding early leakages and fluid losses in hydraulic lines and components. For example, by tracking the Chapter 8 CONCLUSION 248 functional dependency between the spool positions and the orifice area estimates and comparing these signatures over time, we easily distinguished a used valve from a new valve — which would otherwise be a very difficult task to achieve. 8.3 Practical Issues The goal of this research was to provide more descriptive and less volatile information on a machine condition for existing fault diagnosis expert systems, so that the rule-making effort and computing time could be reduced. However, it turned out that about the same effort was then required to obtain good state-space models and to fine-tune the estimator gains (many different noise covariances) in order to produce satisfactory results. Our experience showed that almost 40% of our total time was spent on modeling and simulation, while another 40% was consumed in order to get working Kalman estimators. Only 20% was needed to monitor the actual machine and to detect and diagnose faults during experiments. Regarding implementation issues, we have recognized that the current technique that we have developed for real-time estimation purposes is computationally intensive, and requires that all the computations be performed in "double-precision" arithmetic and at high frequencies. The sampling and computation frequency, according to our experience, cannot be less that 500 Hz, or the accuracy and convergence can be in real jeopardy. Although for the time being, parallel computing is an option that we took advantage of, we are not sure at this moment if this can be quickly realized in commercial applications. Even if the structured FDD technique that we have proposed in this thesis cannot be fully implemented for on-line applications, we believe it still has many attractive features for off-line purposes. For instance, it is not always necessary to continuously monitor a component, such as a servovalve, during its operation. The valve can be tested every now and then at a test site and during an off-line procedure, using the same input signals. This is exactly what we did in Chapter 6, when we compared a used valve with a new one, using orifice area signatures and other valve parameters. This option seems to be more realizable for the time being. Section 8. 4 Suggestions for Future Work 249 8.4 Suggestions for Future Work The work that we have accomplished was not the first and will not be the last attempt in the field of condition monitoring and fault diagnosis for hydraulic systems. There are many ways to improve the proposed technique or to introduce new methods based on previous research, as well as the current one. Regarding the proposed technique, which is quite general in its application, a number of improvements and additions may be made, including: • The model-based method for condition monitoring can be extended to include other hydraulic, and even pneumatic, components. All that is required is to obtain good representative state-space models and appropriate estimators for those models. • Connection between the condition monitoring and fault detection modules and a neural network pattern classifier and a knowledge-based expert system is now manual. Integration of those modules with an on-line expert system shell that has a number of rules regarding fault diagnosis is desirable. • Currently, there is no user friendly interface between the real-time program and the operator. Designing and implementing a graphical user interface is also desirable. • Implementation of the condition monitoring and fault detection algorithm on a less expensive PC platform, rather than a workstation platform, as it is now, can pave the way for the method to receive industrial acceptance. NOMENCLATURE Am = main spool pressure area Ap = pilot spool pressure area Cds, Cd, = orifice discharge coefficients for pressure supply and tank ports Q,„ = main spool leakage coefficient Cis, C{, = coefficient for leakage between pilot port and supply/tank port Cm = main spool viscous damping coefficient Cr = radial clearance between the pilot spool and its sleeve Cs = solenoid viscous damping coefficient Da = armature's piston net diameter Dcp = capillary pipe diameter Dp = pilot spool diameter L = solenoid inductance Lcp = capillary pipe length Lm = main spool porting overlap Lp = pilot spool length Ls = pilot spool/supply porting overlap L, = pilot spool/tank porting underlap 251 Ffm, Ffp = main/pilot spool Coulomb friction force FfS = solenoid Coulomb friction force Fpm = main valve spring preload Km = main valve spring constant Ks = solenoid spring constant M,„ = main spool mass Mp = pilot spool mass Ms = solenoid armature mass R = solenoid resistance RE = orifice Reynolds number RS, R, = radii of the pilot and the tank ports VaO, = initial volumes of pilot valve chambers a f f l B = main spool orifice area as, at = pilot orifice areas to the pressure supply and tank ports fs, fsa, fb ~ solenoid force id, ir = dissipative and restoring currents Pa, Pb = pilot port pressures pe = main valve exit pressure ps = supply pressure 252 pr, ph = main valve rod-side port and head-side port pressures qcp = flow through capillary pipes j v/ = inductive voltage vv = solenoid voltage input x,n = main spool displacement xp = pilot spool displacement xs, xsa, xsb = solenoid armature displacement 8e = oil effective bulk modulus A = solenoid flux linkage \x = oil viscosity p = oil density T = solenoid dissipation time constant 251 BIBLIOGRAPHY [I] M . 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"Application of artificial neural networks in process fault diagnosis". Automatica, 29(4):843-849, 1993. [65] T. Sorsa, J. Suontausta, and H. N. Koivo. "Dynamic fault diagnosis using radial basis function networks". In Proceedings of the International Conference on Fault Diagnosis (TOOLDIAG'93), pages 160-169, Toulouse, France, 1993. [66] S. Tafazoli. Identification of Frictional Effects and Structural Dynamics for Improved Control of Hydraulic Manipulators. Ph.D. Thesis, The University of British Columbia, Vancouver, B C , 1997. [67] S. T. Tsai, A. Akers, and S. J. Lin. "Modeling and dynamic evaluation of a two-stage two-spool servovalve used for pressure control". ASME Journal of Dynamic Systems, Measurement, and Control, 113(4): 709-713, 1991. [68] S. G. Tzafestas. "System fault diagnosis using the knowledge-based methodology". In Ron Patton, Paul Frank, and Robert Clark, editors, Fault Diagnosis in Dynamic Systems, chapter 15, pages 509-572. Prentice Hall, London, 1989. [69] N. D. Vaughan and J. B. Gamble. "The modeling and simulation of a proportional solenoid valve". ASME Journal of Dynamic Systems, Measurement, and Control, 118(1): 120—125, 1996. [70] G. Vossoughi and M . Donath. "Dynamic feedback linearization for electrohydraulically actuated control systems". ASME Journal of Dynamic Systems, Measurement, and Control, 117(4):468-477, 1995. [71] A. Wald. Sequential Analysis. Wiley, New York, 1947. [72] H. Wang, M . Brown, and C. J. Harris. "Fault detection for a class of unknown non-linear systems via associative memory networks". Proceedings of the Institute of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 208:101-107, 1994. [73] J. Watton. Fluid Power Systems — Modelling, Simulation, Analog and Microcomputer Control. Prentice-Hall, 1989. [74] J Watton. "Transient analysis of a back-pressure controlled relief valve for controlling the supply pressure rate of rise". In Proceedings ofJHPS 1st International Symposium on Fluid Power, pages 373-378, Tokyo, Japan, 1989. [75] J. Watton. Condition Monitoring and Fault Diagnosis in Fluid Power Systems. Ellis Horwood, Chichester, England, 1992. [76] J. Watton, O. Lucca-Negro, and J. C. Stewart. "An on-line approach to fault diagnosis of fluid power cylinder drive systems". Proceedings of the Institute of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 208:249-262, 1994. [77] J. Watton and D. G. Salters. "Transient response of a pressure relief valve via spindle modification using a cad package". In Proceedings of 1990 American Control Conference, pages 2622-2625, 25S U.S.A., 1990. [78] J. Watton and Y. Xue. "Identification of fluid power component behaviour using dynamic flowrate measurement". Proceedings of the Institute of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 209:179-191, 1995. [79] B. D. Weber and R. Wilke. "Intelligent mobile hydraulics — the logical progression". SAE Transactions, 1982. [80] S. M . Weiss and C. A. Kulikowsky. Computer Systems that Learn: Classification and Prediction Methods from Statistics. Morgan Kaufmann Publishers, 1991. [81] D. C. White and H. H. Woodson. Electromechanical Energy Conversion. John Wiley & Sons, Inc., New York, NY, 1959. [82] A. S. Willsky. "A survey of design methods for failure detection in dynamic systems". Automatica, 12:601-611, 1976. [83] X. P. Xu, R. T. Burton, and C. M . Sargent. "Experimental identification of a flow orifice using a neural network and the conjugate gradient method". ASME Journal of Dynamic Systems, Measurement, and Control, 118(2):272-277, 1996. [84] Y. Xue and J. Watton. "A self-organizing neural network approach to data-based modelling of fluid power systems dynamics using the G M D H algorithm". Proceedings of the Institute of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 209:229-240, 1995. [85] Q. Zhang, M . Basseville, and A. Benveniste. "Early warning of slight changes in systems". Automatica, 30(1):95-113, 1994. [86] X. J. Zhang. Auxiliary Signal Design in Fault Detection and Diagnosis. Springer-Verlag, Berlin, Germany, 1989. Appendix A: Extended Kalman Filtering Algorithm The essential idea of the Extended Kalman Filter (EKF) was proposed by S. F. Schmidt, and since then it has been called "Kalman-Schmidt" filter [59 Schmidt, 1970]. The goal is to minimize the estimation or tracking error for the states of a nonlinear system along a trajectory by applying linearization techniques, i.e., to evaluate the Taylor series expansion about the estimated trajectory on-the-fly. Then, if the problem is sufficiently observable (as evidenced by the covariance of estimation uncertainty, P), then the deviations between the estimated trajectory (along which the expansion is made) and the actual trajectory will remain sufficiently small that the linearization assumption is valid [21 Grewal and Andrews, 1993]. The models and implementation equations of the discrete E K F , which have been used throughout this thesis, are summarized below: • Nonlinear dynamic model: Xfc =f(x f c _i , fc- l) + w f c w f c ~ A/"(0,Qfc) • Nonlinear measurement model: yk = h{xk,k) + vk v f c ~Af (0 ,R f c ) • Nonlinear implementation equations: Computing the predicted state estimate: xP = t(x[%k-i) (A.3) Computing the predicted measurement: y* = h(4_),fc) (A.4) • Linear approximation equations: « — i - g " .(-) (A.5) ax x - x * - i Computing the measurement error: £fc = Yk ~ 9k (A.6) Conditioning the predicted estimate on the measurement: (A.l) (A.2) <+) = 4 _ ) + K f c £ f c (A.7) H 4 - ^ U , - , Computing the a priori covariance matrix: = * L i + Q * - i (A.9) Computing the Kalman gain: Sk = H f c P J - ^ + Kk Kk = P^ETlSfc 1 Computing the a posteriori covariance matrix: (A. 10) P i + ) = P i _ ) - KfcHfcPJf* (A.11) 259 An unbiased estimator for R is given in [49 Myers and Tapley, 1976], as k »* = r h : £ { ( e i ~ * ' ) ( e ^ ^ ( A - 1 2 ) where 1 k 3=1 Recursive estimation of R* from Eqs. (A. 12) and (A. 13) is possible if the sample size k is large enough (e.g., > 100). Then, we can easily show that R f c « R f c _ i + -j^j{(£k - ek)(ek - ek)T - HjtPJ^HJf - R f c _i} (A.14) where ik = i k - i + ^(£k ~ 4 - i ) (A. 15) A maximum likelihood estimator for R* is also available [1 Basseville and Nikiforov, 1993]. For single output systems, i.e., when is scalar, the maximum likelihood estimation of R* can thus be obtained [41 Ljung, 1987] from Rk = Rk-i + ^(elS];l1-Rk-1) (A.16) 260 Appendix B: The Pocket Algorithm for Training Linear Machines An algorithm, named "pocket", to train "linear machines" or "winner-take-all-groups" has been given in [19 Gallant, 1993]. The algorithm is listed below: Data for the pocket algorithm for training linear machines: INPUT: Training examples [e(k), c(k)]. e(k) is a vector with eo(A:) = 1 and other components, e\(k), ... ep(k), taking values in {-1, 0, +1}. c(k) is an output vector with c entries, cp+i(k), ... cp+c(k), one of which is +1 and the other c - 1 entries taking value -1. OUTPUT: c cells, [wp+i, wp+2, w^c], where w,- = {vv,)0, vv,,i, witP) is a vector of integral "pocket" weights and w(>o is the "bias". TEMPORARY DATA: n: C vectors of integral perceptron weights, irp+\, irp+2, TTP+C, where each vector 7r, = {7r,,0, 7T,',1, 7 T , „ / ? } . r u n ^ : number of consecutive correct classifications using perceptron weights n. rurin,: number of consecutive correct classifications using pocket weights W. num_ok7r: total number of training examples that IT correctly classifies. num_okw: total number of training examples that W correctly classifies. The Pocket Algorithm: 1. Set TT, = {0, 0, 0} for i = p + 1,• • • ,p + c; ruri7r = r u n w = run_ok7r = run_ok v v = 0. 2. Randomly pick a training example e(k) with corresponding classification c(k). 3. If 7r correctly classifies e(k), i.e., if the correct class is i and 7r;.e(A;) > irj.e(k) for / not equal to j, Then: a. ruri7r = ruri7r + 1. b. If ruri7r > run^, Then: • Compute num_ok7r by checking every training example. • If n u m _ o k 7 r > num_okH,, Then: • Set W = TT . • Set r u n w = ruri7r. • Set num_okn, = num_ok7r. Otherwise: a. C H A N G E STEP: Form a new vector of perceptron weights. Select one cell so that j is not equal to i and satisfies 7r,.e(fc) < •Kj.e(k) and modify the weights of cells / and j as follows: 7T; = TTi + e(k) *j = *j - e(fc) 261 b. Set runw =0. If the specified number of iterations has not been taken, then go to Step 2. 262 Appendix C: Specifications for the Hydraulic Test Rig Components Hydraulic Hoses: Make: Parker Type: Parflex 540N-6 Material: Polymeric core tube, fiber reinforcement, abrasion-resistance urethane cover Internal Diameter: 9.5 mm Outer Diameter: 16.5 mm Maximum Working Pressure: 15 MPa Typical Volume Expansion at Maximum Working Pressure: 1.345 x 10~5 m 3/m = 1.9% High Pressure Filter: Make: Parker Model: 15P Maximum flow: 1.172 x 10~3 m3/s Maximum pressure: 20 MPa Average resistance factor (Rf): 100 MPa.s/m 3 Return Line Filter: Make: Parker Model: 12AT Maximum flow: 1.113 x 10~3 m3/s Maximum pressure: 1 MPa Average resistance factor (Rf): 12 MPa.s/m3 Gear Pump: Make: Parker Model: H31, fixed displacement gear pump Flow @6.9 MPa (1000 psi) and @1800 RPM: 3.462 x 10"4 m3/s Nominal Displacement, Vpo: 1.967 x 10 - 6 m3/rad Internal leakage coefficient (Cp): 1.840 x 10 - 6 m3/MPa.s Pressure Relief Valve: Make: Parker Model: RA101 Series pilot operated spool-type Rated flow: 1.875 m3/s Maximum setting pressure: 20 MPa Reseat pressure: 80% of set pressure Flow coefficient (Cr): 8.9 x 10"4 m3/MPa.s Leakage coefficient: 2.6 x 10~7 m3/MPa.s 263 Check Valve: Make: Sun Model: C X D A - X D N non-adjustable nose-to-nose Rated flow: 1.3 m3/s Cracking pressure: 350 KPa Flow coefficient (C c): 2.1 x 10~3 m3/MPa.s Leakage: negligible. Direct Acting Sequence Valve with Reverse Free Flow Check Valve: Make: Sun Model: S C C A - L A N Rated flow: 1.0 nr 3 /s Check valve cracking pressure: 275 KPa Relief valve pressure setting: 6.9 MPa Check valve flow coefficient (C c): 2.3 x 10~3 m3/MPa.s Relief valve flow coefficient (Cr): 2.8 x 10~3 m3/MPa.s Response time for sequence valve: 2 ms Leakage: negligible. Pilot Valve: Make: Rexroth Model: 3 DREP 6 C Max. operating pressure: 10.0 MPa (1450 psi) Max. flow @ A p 5.0 MPa: 0.25 x 10~3 m3/s Pilot fluid volume: 1.7 x 10"6 m 3 Fluid temperature range: -20 to 70 °C Hysteresis: < 3% Repeatability: < 1% Sensitivity (resolution and reversal): < 1% Main Valve: Make: Rexroth Model: 4 WRZ 10 Max. operating pressure: 31.7 MPa (4600 psi) Max. flow: 4.5 x 10~3 m3/s Fluid temperature range: -20 to 70 °C Hysteresis: < 6% Repeatability: < 3% Overlap, Lm = 1.53 mm Cylinder Actuator: Make: Parker Hannifin 264 Model: SB 3L Stroke: 0.254 m (10 in) Bore diameter: 0.0381 m (1-1/2 in) Rod diameter: 0.016 m (5/8 in) Internal leakage: Negligible up to 3000 psi Primary Seal: New TS-2000 Seal Manipulator: Link material: Aluminum Link weight: 4.3 Kg Link center of gravity (C.G.): 0.325 m (from the joint center at point B) b: 0.133 m d: 0.540 m x0: 0.488 m j30: 1.053 rad 265 Appendix D: Specifications for the Measuring Devices Pump Outlet port and Supply Port Pressure Transducers: Make: GP:50 New York Ltd. Model: 211-D-RV Output: 0 to 5 Vdc Supply voltage: 10.5-32 Vdc Pressure range: 0-34 MPa (0-5000) psi Accuracy: 0.1% FSO Dynamic response: 2 kHz Head-Side Port and Rod-Side Port Pressure Transducers: Make: GP:50 New York Ltd. Model: 211-D-RV Output: 0 to 5 Vdc Supply voltage: 10.5-32 Vdc Pressure range: 0-34 MPa (0-5000) psi Accuracy: 0.1% SFO Dynamic response: 2 kHz Pilot Ports and Exit Port Pressure Transducers: Make: GP:50 New York Ltd. Model: 211-D-RH Output: 0 to 5 Vdc Supply voltage: 10.5-32 Vdc Pressure range: 0-3.4 MPa (0-500) psi Accuracy: 0.1% FSO Dynamic response: 2 kHz Pump shaft tachometer: Make: Kearfott Model: CR09610011 Linearity: 0.05% Ripple: 3% Resolution: 1.3 RPM Temperature sensors: Make: National Semiconductor Model: LM235LH Range: - W C to +125°C Accuracy: ±2°C 266 Cylinder rod load pin: Make: Sensor Developments Model: 20006 Force capacity: -13500 N to +13500 N Nominal output: 2 mV/V Bridge resistance: 700 Bridge excitation voltage: 10 V Hysteresis and nonlinearity: 0.25% F.S. Compensated temperature range: +21 to +77 Usable temperature range: -54 to +121 °C Dynamic response: 2 KHz
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On-line condition monitoring and fault diagnosis in hydraulic system components using parameter estimation… Khoshzaban-Zavarehi, Masoud 1997
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Title | On-line condition monitoring and fault diagnosis in hydraulic system components using parameter estimation and pattern classification |
Creator |
Khoshzaban-Zavarehi, Masoud |
Date Issued | 1997 |
Description | Safety and functionality of a fluid power control system can considerably be increased by implementing predictive maintenance routines. Modern predictive maintenance practices are based on automatic condition monitoring and fault diagnosis of the system components. In most cases, low-quality raw sensor data are directly monitored for constraint violations or threshold crossings. Subsequent fault diagnosis is often performed by a knowledge-based expert system based on "order-of-magnitude reasoning". This means that quantitative sensor data are first transformed into more understandable "linguistic terminologies" such as "low", high", etc., and are then assessed by production rules in order to diagnose system (or component) faults. A major problem with this technique is that it is not usually feasible to directly measure the desired quantity, e.g., the flow rate inside a valve. Another problem is the association of noise and variations with directly measured signals, which might be misinterpreted as faults, especially in highly dynamic systems. In practice, failure modes often involve a change in the model structure, which may be interpreted as change(s) in one or several system parameters. The theme of this thesis is on automatic generation of fault symptoms in the form of qualitative variation of system physical parameters by on-line processing of low-quality raw sensor data. To accomplish this, a novel model-based methodology has been proposed that has integrated four levels of information processing in a structured hierarchy: 1. State/parameter estimation of the hydraulic system components using state-space models, stochastic signal processing techniques such as Kalman filtering, and raw sensor data from the hydraulic system. 2. Monitoring and change detection in the identified parameters of the system components, using statistical tests, such as sequential probability ratio test. 3. Generation of fault symptoms in the form of qualitative changes in the physical parameter values, such as "increased", "decreased", etc. 4. Fault recognition by fault symptom classification using neural network pattern classifiers, 5. Fault diagnosis maintenance aiding using knowledge-based expert systems. By using a second-order linear system as an example, we have shown how each element of the proposed hierarchical methodology effectively processes the lower quality data received from the previous element and provides higher quality information for the next element in the hierarchy, so that an incipient fault or an abrupt failure can be successfully detected and diagnosed. The proposed fault detection and diagnosis (FDD) technique has also been applied on a real hydraulic test rig which has been built in the Robotics and Control Laboratory, at UBC. The hydraulic test rig has a two-stage proportional directional flow control valve, which has been thoroughly modelled for simulation of faults. A step-by-step methodology has been adopted to obtain the physical valve parameters from static measurements, as well as through numerical search techniques using dynamic measurements. In order to estimate the system parameters and states in real-time, nonlinear state-space models have been developed for various hydraulic components, including the two-stage servovalve, a hydraulic cylinder, and a manipulator. Extended Kalman Filtering (EKF) is applied on the state-space models to get the parameter estimates. Only low-cost robust sensors such as pressure transducers and position sensors have been used for this purpose. More expensive or hard-to-measure states such as flow rates and orifice areas are predicted using novel state-space models. One of the major achievements of this thesis has been incorporation of a novel state-space model for a valve orifice area that allows us not only to obtain accurate estimates of the flow rate through the valve, but also to detect several incipient faults and abrupt failures in the valve and its connecting ports. The valve orifice area is considered as a nonlinear unknown function of the valve spool position. No a priori knowledge about the orifice profile or the spool deadband size is assumed. The functional relationship, along with the deadband size are automatically revealed during the on-line estimation process, while the decision as to which port is open to the flow is made internally. Experimental results were promising and showed that the identified valve orifice area is an excellent measure in quick detection and diagnosis of incipient or gradual faults, as well as and abrupt failures, in servovalves and servo-actuator systems. |
Extent | 17196839 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-04-01 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0080942 |
URI | http://hdl.handle.net/2429/6700 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1997-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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