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Investigating the use of compliance for fault detection in hydraulic machines Pirouz, Zarin 1997

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Investigating the Use of Compliance for Fault Detection in Hydraulic Machines  Zarin Pirouz B.A.Sc, University of British Columbia, 1988  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA August 1997 ©Zarin Pirouz, 1997  In  presenting- this  degree freely  at the  University  available for  copying  of  department publication  this or of  thesis  this  fulfilment  of  Columbia, I agree that the  British  reference  thesis by  in . partial  his  for  and study. scholarly  or  thesis for  her  of  I further  purposes  requirements  agree that  may  representatives.  financial  the  be  It  is  gain shall not  of  £ le.crV*-i'CflJL  Out\ck £>wv.puA-gr  The University of British Columbia Vancouver, Canada  Date  DE-6  (2/88)  ftrwj^A  2-1,  an  Library shall make  by  understood  the that  be allowed without  £yj>^\t\e£*\*e^  advanced  permission for  granted  permission.  Department  for  it  extensive  head  of  my  copying  or  my  written  Abstract  The problem of measuring the compliance of a hydraulic system using flow and pressure measurements was studied to determine if this method could be used to detect faults in mobile heavy-duty hydraulic machines. As a start, previous research that used Genetic Algorithms (GA) for compliance identification in a hydraulic machine and GA principles were studied in detail to determine if using the GA is necessary for compliance identification. Some discrepancies and misconceptions in the previous research were discovered. To eliminate the misconceptions, many elements contributing to the compliance of the machine and obstacles to using pressure and flow measurements in compliance calculation were studied. It was shown that many inter-related elements have to be modeled to be able to interpret the compliance value obtained from pressure and flow measurements in hydraulic machines. Some models believed to be most suitable for fault detection were developed. It was shown that despite the complexity of the elements, as far as fault detection is concerned, the system can be modeled in such a way that in most cases it is linear in the parameters which allows the use of simple identification methods. Experiments were performed on a hydraulic test-bed to examine the suitability of the proposed models. A few unexpected obstacles in using flow and pressure measurement to measure compliance for fault detection were encountered. It was discovered that even small amounts of leak that can be present during normal machine operation may cause enough distortion in flow measurement to overshadow the compliance value. It was also discovered that using flow and pressure measurements in compliance measurement is only possible if very accurate measurements are available. A few practical and simple methods were proposed to cope with these problems and extract useful information from these measurements for fault detection.  Table of Contents Abstract  ii  List of Tables  vi  List of Figures  vii  Relevant Hydraulic Circuit Symbols  ix  Acknowledgments  x  1 Introduction 1.1 Background 1.2 The Scope of This Study  1 1 5  2 Mobile Heavy-Duty Hydraulic Machines 2.1 General Description of Heavy-Duty Mobile Hydraulic Machines 2.2 Hydraulic Fluids 2.2.1 Density 2.2.2 Bulk Modulus 2.2.3 Air Solubility and Aeration Threshold 2.2.4 Viscosity 2.2.5 Other Properties of Hydraulic Fluid 2.3 Hydraulic Pumps 2.4 Hydraulic Actuators 2.5 Hydraulic Valves 2.6 Hydraulic Pipes and Hoses 2.7 Fluid Reservoirs 2.8 Accumulators 2.8 Some Operational Problems in Mobile Heavy-Duty Hydraulic Machines 2.8.1 Problems That Change Machine Compliance 2.8.1.1 Air in the System 2.8.1. 2 Over-heating 2.8.1.3 Worn-out Hoses 2.8.2 Problems That Only Appear to Change Machine Compliance 2.8.2.1 Punctured Hose and Leakage Out of the System 2.8.2.2 Leakage In the Cylinder 2.8.2.3 Leakage in the Valve 2.8.2.4 Dirt Built-up in the Valve 2.9 Subsystems of Interest to This Study  7 7 8 8 9 11 12 12 13 13 14 16 16 17 19 20 20 27 27 28 28 29 31 31 32  3 Parameter Estimation Methods 3.1 Simple Overview of Parameter Estimation Methods 3.2 Recursive Least Squares 3.3 Genetic Algorithms 3.3.1 Background 3.3.2 Basic Mechanism 3.3.3 Theoretical Properties of Genetic Algorithms  34 35 37 38 38 39 40  iii  3.4 Discussion  46  4 Compliance and Fault Detection in Hydraulic Systems 4.1 The Effective Bulk Modulus and "Effective" Compliance 4.1.1 Understanding The Effective Bulk Modulus 4.1.2 Existing Models for Effective Bulk Modulus 4.1.3 Proposed New Model for Effective Bulk Modulus 4.2 Calculating Compliance and J3 From Flow and Pressure Measurement 4.3 Interpreting The "Lumped" Compliance Value 4.4 Developing Compliance Models Suitable for Fault Detection 4.4.1 Leakage Across the Piston 4.4.2 Leakage to Outside 4.4.3 The Proposed Parametric Compliance Model e  5 Experimental Estimation of Compliance 5.1 Continuing Previous Work in Estimating Compliance in an Excavator 5.1.1 Methods and Results of the Previous Work 5.1.2 Using Recursive Least Squares in Estimating Compliance in the Excavator 5.1.3 Discussion 5.2 Estimation of Compliance in an Experimental Hydraulic Test-bed 5.2.1 Fundamental Questions to Be Answered 5.2.3 The Hydraulic Test-Bed 5.2.3 The Experiments on the Test-bed 5.2.3.1 Objective 5.2.3.2 Methods 5.2.3.3 Results 5.2.3.4 Discussion 5.2.3.5 Using Genetic Algorithms 5.2.4 Answers to Some of the Questions Posed 5.3 Proposed New Experiments and Methods 5.3.1 A Few Simple Observations for Detecting Leak 5.3.2 Proposed Simple Method for Compliance Identification 5.3.3 Compliance Identification Using the Fluid Sonic Bulk Modulus  47 49 49 54 60 62 64 66 67 68 69 71 71 71 . . 75 78 80 80 82 85 85 86 90 96 114 117 121 121 122 124  6 Conclusions and Suggestions for Future Research 6.1 Conclusions  125 125  6.2 Suggestions for Future Research  129  7 References  132  Appendix A A.l Recursive Least Squares A.2 The RLS Algorithm Used in This Study A.3 The Effect of Genetic Operators on Schemata A.4 Description of the GA Selected for Implementation Appendix B Additional Details on the Bulk Modulus Models Studied  134 134 139 140 142  iv  145  Appendix C The Geometrical Properties of the Hydraulic Test-bed  148  Appendix D D.l D.2 D.3 D.4  150 150 151 156 157  Data Collection Flow Calculation Selecting a Suitable Filter to Remove Noise Pressure Change Due to Flow Restrictions, Inertia Forces  v  List of Tables Table 1 Major components effective bulk modulus  51  Table 2 Measurements for fluid volume calculations in the test-bed  vi  149  List of Figures Figure 1 A Simplified Hydraulic System (redrawn from [29])  7  Figure 2 Tangent Bulk Modulus vs Secant Bulk Modulus  10  Figure 3 Double Acting Differential Area Linear Actuator  14  Figure 4 A Simplified Four-Land-Four-Way Spool Valve  15  Figure 5 Bladder-type Accumulator  17  Figure 6 Volume and Pressure Changes in an Accumulator  18  Figure 7 Air solution properties for MIL-H-5606 fluid  23  Figure 8 Cylinder Piston and Leakage  29  Figure 9 Leakage between two moving (cylindrical) surfaces in contact  29  Figure 10 Effects of speed, viscosity and pressure on leakage (from [29] )  31  Figure 11 Main hydraulic actuators of the CAT 215B excavator  32  Figure 12 Block diagram for pump, valve and actuator connection  (re-drawn from [30])  32  Figure 13 Volume change of container, liquid and gas as pressure increases  50  Figure 14 Volume change due to piston travel  52  Figure 15 Results from Jinghong et al [11] (reprinted with legend modification from [11])  59  Figure 16 Net flow into the linear actuator  63  Figure 17 Subsystem to be modeled  66  Figure 18 Interconnection circuit diagram for the valves of the CAT 215B excavator  72  Figure 19 Simplified Boom circuit diagram  72  Figure 20 Results from study by Wan [30]  77  Figure 21 Simplified Diagram of Arm and Actuator in the Hydraulic Test-bed  82  Figure 22 Hydraulic test-bed circuit diagram  83  Figure 23 Test-bed circuit representation for both fluid flow directions  84  Figure 24 Simplified Circuit for Experiment 2  89 vii  Figure 25 Basic Results for Experiment 1  92  Figure 26 Basic Results for Experiment 2  93  Figure 27 Basic Results for Experiment 3  94  Figure 28 Basic Results for Experiment 4  95  Figure 29 Results for Model 1 in Experiment 2  103  Figure 30 Results for Model 2 in Experiment 2  104  Figure 31 Results for Model 1 in Experiment 3  108  Figure 32 Results for Model 2 in Experiment 3  110  Figure 33 Compliance estimation for Experiment 4  112  Figure 34 Compliance calculation with GA  115  Figure 35 Estimating two compliances with GA  116  Figure 36 Using "Bleed Valves" to Depressurize the Actuator  122  Figure 37 The Slotted Roulette Wheel Method  142  Figure 38 The hydraulic test-bed actuator and arm geometry  148  Figure 39 Rod side pushing  152  Figure 40 Determining valve characteristics manually  154  Figure 41 Main valve characteristics obtained experimentally  155  viii  Relevant Hydraulic Circuit Symbols Hydraulic line (pipe or hose)  Pilot line  Lines crossing  Lines connecting  Reservoir (Tank)  Fixed displacement pump (unidirectional)  Double acting differential area linear actuator  Manual shut-off valve  Filter or Strainer  Three-position, four-way control valve  Spring-centered, pilot-controlled (valve)  Check valve  Pressure relief valve  Variable flow control orifice  Gas charged accumulator  ix  Acknowledgments  The author is greatly indebted to Dr. P. D. Lawrence, whose guidance, support, and patience throughout this research, made this work possible. This work was also made possible with the financial support of Forintek Canada Corp. and considerable patience and support of many great people at Forintek. The author would also like to express gratitude to Mr. Masoud Khoshzaban-Zavarehi for guidance and assistance in performing the experiments.  x  1 Introduction  1.1 Background Mobile heavy-duty hydraulic machines such as excavators, loaders and other common forestry, construction, agriculture, and mining equipment are often used in rugged outdoor environments where they are exposed to extreme conditions and high wear and tear. These harsh operating conditions can lead to frequent machine breakdown. These breakdowns lead to lost productivity that can be especially costly if the machine breaks down while operating in hard-to-access locations. Although scheduled preventative maintenance can reduce the breakdown frequency, it cannot predict or prevent many of the problems that can occur while the machine is operating.  When operators control the machine directly, they are "in-tune" with the machine and can often notice changes in machine response which are manifested as performance degradation or unusual sounds. This can alert the operators to the presence of a problem and using the manufacturer's diagnostic procedures, maintenance personnel can diagnose the problem to isolate and repair damaged or faulty components. However, the success of this type of fault detection and isolation depends on the level of experience and knowledge of the operators and how well they are able to "sense" and diagnose any problems.  Meanwhile, to increase precision, safety, and ease of operation, modern industrial machines are increasingly automated which means that their control systems are automated (often computerized). Along with their many advantages, the automated or computerized control systems have the disadvantage of isolating the operators from the equipment that they are operating or even from the environments that the machines are operating in. Efforts to improve operator safety and comfort often leads to operators working in a noise and vibration free environment away from the machine and its  1  surroundings. Thus, the operator is left with little or no physical appreciation and sensory feedback such as the full-sight, sound, smell, heat, or vibration of the equipment. Furthermore, the operator's control commands are translated to set points of closed loop control systems which then take over the task of ensuring that the correct level of input signal is applied to achieve the desired machine output. This will in turn isolate the operator from sensing and gauging the system response. Thus, even the experienced operators are unable to anticipate the problems of modern machines.  One way of helping the operator in predicting and avoiding some of the common breakdown problems of these machines is to have an on-board monitoring system to monitor some of the "vital signs" of the machine and display them to the operator. Some examples of these vital signs are pressure, flow, and temperature of the fluid at strategic locations. However, it is unlikely that the operator can devote enough attention to monitoring these values while performing tasks. Furthermore, as far as the operator's ability to associate certain feedback to a specific system problem is concerned, it is not clear if the abstract sensor information displays and alarms are as effective as the physical sensory feedback such as the direct sight, sound, vibration, smell and heat of the operation.  Another option to help the operator in diagnosis is to use a computer to try to interpret the raw sensor data. For example, in a manually controlled machine, the change in response time is a property that is normally noticed and intuitively tracked as a performance measure by operator. Therefore, it would be desirable for an automated diagnosis system to track a similar property of the machine. The response time of a system is dependant on its natural frequency which is affected by its stiffness. Sometimes rather than considering the stiffness, it is more convenient to consider the machine compliance which is inversely proportional to the stiffness.  Compliance of the machine is a measure of how much flexibility or "give" the machine has. Generally, 2  the compliance of the mechanical structure can be ignored in comparison with the compliance of the hydraulic fluid. However, depending on the system configuration, compliance of the flexible hoses may also have to be taken into account. The compliance or capacitance of the hydraulic fluid (which is analogous to electrical capacitance) is defined as the ratio of the fluid volume to the Bulk Modulus of the V hydraulic fluid [17], or C . The fluid bulk modulus, which will be discussed further in flmd  fluid  $ fluid  Section (2.2.2), is a measure of the stiffness of a fluid and has the same units as pressure (Pascals or psi). In general, for hydraulic fluids free of any entrapped air bubbles, the bulk modulus is a known and rather large (compared to oil containing air) constant. The SI units of hydraulic compliance are m s /Kg. In 4  2  theory, the compliance of subsystem of fluid and container can also be experimentally measured by measuring the amount of fluid that has to flow into the system to produce a unit increase in pressure. This can be written as C=Q/P, where C is compliance, Q is flow, and P is the pressure derivative. However, this relation can only be used if the inertia effects of the hydraulic fluid can be ignored, [17].  Some problems such as trapped air bubbles in the fluid and worn-out hydraulic hoses can increase the overall compliance of a hydraulic subsystem. If the pressure and flow measurements are used to calculate the fluid compliance in a hydraulic subsystem, the value measured reflects the overall compliance, including the portion due to the above problems. Furthermore, broken rod and piston seals in a hydraulic actuator, fluid leakage out of the system, and dirt build-up in the valve can also interfere with flow measurement and appear as a change in compliance. Thus, in theory, if the inertia effects can be ignored and flow and pressure measurements are used to calculate the compliance of a subsystem, the calculated value may be used as an indicator of system problems in an on-board diagnosis system. However, it was found that in practice, all the problems mentioned above also exist under normal operating conditions, albeit to a lesser extent. This means that while compliance is constant under ideal conditions, under normal operating conditions its value is pressure dependant. Thus, to use compliance for fault detection, its normal and abnormal variations should be distinguished which requires a full  3  understanding of the extent of the effect of the above problems in the measured compliance.  Some of the physical components such as the properties of the fluid, entrained air, or the flexible hoses that contribute to compliance have been studied in detail by various researchers. However, often one or just a few components are studied in isolation, only with specialized equipment, and under very carefully controlled conditions. There is no indication that any of these components have ever been identified on a real hydraulic machine. Adaptive control systems on hydraulic machines [24] may implicitly estimate some value for the overall machine compliance or stiffness for control purposes, but it is very difficult to determine if the lumped parameters of an adaptive controller truly reflect the system compliance. The only study found that attempted to estimate compliance for fault detection in a hydraulic system, is the work by Wan [30], which has some inadequacies due to a superficial treatment of compliance.  Thus, it appears that the following questions have never been dealt with: 1- How should hydraulic compliance be measured on-line, and can it be measured from flow and pressure measurements i.e., can the inertia effects can be ignored? 2- Can hydraulic compliance be considered "reasonably constant" under any circumstances and what is "reasonably constant"? 3- How can "real" compliance be distinguished from "apparent" compliance (changes in compliance measurements caused by errors from leakage)? 4- What is the range of compliance during normal operation? 5- How can changes in compliance be traced to specific faults?  It seems that only way to begin to answer these questions is to use models that take the leakage in the system and the pressure dependance of compliance into account. Depending on the complexity of the 4  model selected, a suitable parameter estimation method should be chosen to identify the parameters of the model.  1.2 The Scope of This Study This work is concerned with the determination and use of hydraulic compliance for on-line faultdetection in hydraulic machines. The objective is to find models, and methods of identifying model parameters, that can help in using compliance in on-line fault detection or diagnosis of heavy duty mobile hydraulic machines.  Previous work by Wan [30] concluded that the main challenge in determining compliance was the shortage of computational power needed for the on-line estimation of compliance. That study suggested that compliance can only be estimated from non-linear models which required using estimation methods such as Genetic Algorithms (GA) to globally search the parameter space. This study began as a continuation of Wan's work and as a natural first step it was decided that computationally intensive G A should be compared to Recursive Least Squares (RLS), which is one of the most simple, computationally inexpensive, and commonly used estimation methods.  However before the scope of the problem could be understood, hydraulic machines, their principles of operation, and the role of compliance had to be studied. It was discovered that the simple model used by Wan, which assumes constant compliance and no leakage, cannot explain the full dynamic relationship between the flow and pressure measurements. Thus, the phenomena contributing to machine compliance and affecting its measurement had to be studied in more detail.  Chapter 2: In this chapter, mobile hydraulic machines, hydraulic components relevant to this study, operational problems in the machine that may effect compliance, and the problems that may affect 5  compliance measurement are studied.  Chapter  3: Genetic Algorithms and Recursive Least Squares are studied in this chapter to understand  their relative strengths and weaknesses before attempting to decide their applicability to the compliance estimation problem.  In C h a p t e r 4: The role of compliance in fault detection and existing models for determining compliance  are studied and based on the study of existing literature and models, new models expected to be more useful to fault detection are proposed in this chapter.  Chapter  5: In this chapter, experimental work for compliance estimation is presented. Initially, the  work done by Wan [30] on estimating the compliance of a hydraulic excavator is reviewed. The results of that work are studied in detail to determine if using GA is necessary for compliance estimation. Some problems in the approach to compliance identification taken in that work are discussed which shows that a more comprehensive study of the basic concepts is required. Then, an original study of compliance on a hydraulic test-bed and the experiments performed are presented. The problems that were encountered and the unexpected findings are discussed and some practical methods for determining compliance are proposed.  Chapter  6: The conclusions of this study and recommendations for future research are presented in this  chapter.  6  2 Mobile Heavy-Duty Hydraulic Machines  2.1 General Description of Heavy-Duty Mobile Hydraulic Machines Mobile heavy-duty hydraulic machines carry an independent hydraulic power system on board that can deliver substantial force with a great deal of flexibility and control. As explained by Sullivan [29], fluid power systems, unlike a mechanical drive train can deliver considerable force to actuators away from the power source. At the same time, fluid power systems can be made to accomplish precise movements. A hydraulic power system uses pressurized hydraulic fluid that is confined to a system or subsystem to accomplish the desired work. The hydraulic pump, which is often powered by a gasoline or diesel engine, pressurizes the fluid. The hydraulic fluid used in these machines are often petroleum based oils. These fluids have low compliance and are very effective in transmitting power from the pump to the actuators. The fluid is transmitted through pipes, tubing, fittings, and flexible hoses. One or more pumps force the fluid from the reservoir through control valves which are used to direct it to the actuators to perform work. The excess fluid is then returned to the tank. Figure (1) shows an example of a simple hydraulic system illustrating the function of the main components. A hydraulic machine Operator Lever  or system is generally composed of a number of hydraulic circuits. Hydraulic circuits can contain a variety of components such as Control Valve  pumps, valves and actuators. The details of these components are cyu^er  beyond the scope of this work. In  F i g u r e  j  A  77777777777/77/  simplified Hydraulic System (redrawn from [29])  1  the study of compliance, flow in and out of a subsystem as well as the pressure should be measured. These measurements can be obtained most conveniently if the subsystem from the control valve up-to and including the actuator is studied.  The following discussion provides an overview of the system components that are relevant to this work. Particular details are provided on some components such as the hydraulic fluid due to their relevance to the discussion that follows. In practice, hydraulic systems will have many more components to ensure safety and to regulate output which are not discussed here.  2.2 Hydraulic Fluids The performance of a hydraulic machine is fully dependent on its hydraulic fluid. In this study, only petroleum based fluids which are commonly used in heavy hydraulic machines are considered and the terms "hydraulic fluid", "fluid" or "oil" are used interchangeably. Hydraulic fluids have to meet many performance criteria and their properties should match the desired operating conditions. Thus, understanding the properties of the fluid is very important in understanding many sources of problems in hydraulic machines. Some of the most relevant properties of hydraulic fluids are presented below.  2.2.1 Density The mass density is defined as mass per unit volume and here it will be represented with the symbol p and simply referred to as density. The density of a liquid is a function of pressure and temperature and is determined by its equation of state. According to Merritt [19], the equations of state for liquids cannot be derived directly from physical principles. Instead, it is assumed that the change in liquid density as a function of pressure and temperature is small enough that the relationship can be approximated by the first three term of a Taylor's series expansion of density with respect to pressure and temperature. Then for density p, temperature T, and pressure P, and near the initial values p , T , and P , the linearized 0  8  0  0  equation of state for a liquid is  p^ii^Hit^).  (.)  The flow rate of the fluid is dependent on its density, but variations in density in the operating range of the machine is often assumed to be negligible when calculating flow rates.  2.2.2  B u l k Modulus  One of the most important properties of the hydraulic fluid is its high stiffness that allows it to be an effective means for fast and accurate power transmission. As it will be shown later, entrained air in the fluid greatly reduces this desirable property. A fluid's stiffness or its resistance to volume reduction with increase in pressure is described by the quantity known as bulk modulus, which has the same units as pressure (Pa or psi). Bulk modulus is defined as the volumetric modulus of elasticity and is the reciprocal of compressibility k, which has the units Pa" or psi" . 1  1  The isothermal bulk modulus is used if the changes in volume and pressure are considered under constant temperature. According to Merritt [19], it is generally believed that during operation, the fluid will reach an equilibrium temperature and the machine will be able to dissipate enough heat to maintain that temperature. Thus, the isothermal bulk modulus is the fluid property of interest in this study. The problems caused by fluid temperature should be handled by the fault detection system independently, otherwise a more complex model of the bulk modulus that incorporates the temperature effects should be used. Thus, the results of this work which are based on this assumption may not be applicable to a machine before it warms up. To derive the isothermal bulk modulus Equation (1), (the equation of state for the fluid) may be written as follows:  P  = [i I(p-p )-a(r-r )] Po  +  0  0  9  for  and  p=pA ap  «=--J-A, p  •  a r  0  The initial density p is the ratio of mass Mover initial volume V , that is p = M/V , and the 0  0  0  0  relationship for /?, the isothermal bulk modulus of the liquid can be written as  V\ QM) V  T  y{  J V  a  T  V \dVh Q  where V is the volume at pressure P and the negative sign denotes a reduction in volume for an increase in pressure and P is positive. Thus for volume V , and a temperature that is constant at T, the isothermal 0  bulk modulus of the liquid is  The Isothermal secant bulk modulus refers to the change in volume for pressure changes from the atmospheric pressure and using the symbol A to indicate large changes, (see Figure (2)) can be defined as  r secant 0\ *y) ' LVi T  According to Wright [32], this value is frequently used in some engineering calculations in the United States, but it is not the "thermodynamically correct isothermal bulk modulus". The Isothermal tangent bulk modulus represents the change of volume with incremental pressure changes from the specific pressure value of interest and can be described as  Volume  Wright [32], states that the isothermal tangent bulk 10  *  Figure 2 Tangent Bulk Modulus vs Secant Bulk Modulus  modulus is the "thermodynamically correct isothermal bulk modulus and represents the true rate of change at the pressure of interest". The difference between the tangent bulk modulus and secant bulk modulus is demonstrated graphically in Figure (2). In Equation (3), the volume V from Equation (2) has 0  been replaced by V to represent the fluid volume at the pressure of interest. All the literature encountered in this study used the isothermal tangent bulk modulus, which is also the quantity of interest to this study, because a fault detection system should track changes in the system during the operation with incremental changes of pressure over time. This quantity will be simply referred to as bulk modulus or /?when there is no ambiguity. According to Merritt [19], the value of bulk modulus is about 220,000 psi (or 1.517 x 10 Pa) for petroleum fluids . 9  The Isentropic or adiabatic bulk modulus is used if the pressure change is too rapid for the heat to be dissipated and the temperature to return to equilibrium. It describes the volumetric modulus of elasticity under constant entropy conditions and is larger than the isothermal tangent bulk modulus by the ratio of specific heats CJC of the fluid [32], where C is the fluid specific heat under constant pressure and C is p  V  v  the specific heat under constant volume. The literature reviewed in this study indicate that researchers have a conflicting view of whether the conditions in hydraulic machines calls for using this definition of the bulk modulus. In this study, it is assumed that the temperature changes during the operating range can be neglected and this definition of bulk modulus is not used. The Sonic Bulk Modulus is defined as pc , where p is the fluid density at the temperature and pressure 2  of interest and c is the velocity of sound at the same temperature and pressure. According to Wright [32], the sonic bulk modulus is identical to the isentropic tangent bulk modulus.  2.2.3 Air Solubility and Aeration Threshold The solubility of air in hydraulic fluids is a major source of difficulty in troubleshooting many problems with hydraulic machines. Therefore, low air solubility is a desired characteristic of hydraulic fluids. Often fluid temperature is assumed to be constant during machine operation, but the pressure changes 11  considerably. The effect of pressure change on solubility of a gas in a liquid is described by Henry's law [5], which states that the concentration of the gaseous solute in the solution C , is directly proportional to g  the partial pressure of the of the gas above the solution. This can be written as C = k p where k is a g  g  g  g  constant and p is the partial pressure of the gas. However, the formation of air bubbles in the fluid g  complicates the interaction between gas and the hydraulic fluid. The problems with air in the fluid will be discussed in Section (2.8.1.1) in more detail.  2.2.4 Viscosity Viscosity of the hydraulic fluid is a measure of its resistance to flow and affects its suitability for the desired operation. Viscosity has to be high enough to prevent excessive leakage between moving parts inside the hydraulic circuits, and yet low enough to avoid excessive energy dissipation for overcoming the fluid friction. Viscosity decreases with increasing pressure and temperature. The relevance of viscosity to this study is its role in leakage and its interaction with air in the fluid. As shown by Hayward [8], air solubility in the fluid increases as viscosity decreases and air bubbles can slightly decrease the viscosity of the fluid. Furthermore, according to some experts the more viscose the fluid the longer it takes for air bubbles to travel to the surface and escape, but this assertion could not be corroborated with a published reference. Viscosity is measured in two ways: Dynamic viscosity, u. is defined as the ratio of shear stress to shear rate in the fluid [17] and has units of N.s/m (Pa.s) in the SI unit system, but the commonly used units are centistokes (mPa.s) [22]. 2  Kinematic viscosity, v is dynamic viscosity divided by density. The SI units for v is m /s, but the 2  commonly used units are centipoises (mm /s) [22]. 2  2.2.5 Other Properties of Hydraulic Fluid There are several other properties of the hydraulic fluid that can affect its performance. These properties describe the fluid's performance as a lubricant, its resistance to oxidation, foaming, and fire, its 12  compatibility with the other material in the system, its stability at high temperatures or when exposed to oxygen or water, its environmental safety, and many other specialized performance characteristics.  2.3 Hydraulic Pumps Hydraulic pumps raise the energy level of the fluid by increasing its pressure and transfer mechanical power to fluid power. The pumping action transfers fluid from one location (the reservoir) to another (valve or actuator) where it performs work and dissipates energy before returning to the reservoir. Pumps used in hydraulic systems are generally of the positive displacement category. Positive displacement pumps have separate chambers for suction and delivery so that the fluid is forced to circulate in the system in discrete quantities. The other category of pumps are hydrodynamic pumps that provide continuous fluid flow but are only suitable for low pressure and high volume operation.  Different pumps and their designs are not discussed here, but to study compliance, the pressure has to be measured and it is worth noting that the type of pump may affect the pressure profde. For example, a gear pump such as the one shown in Figure (1), the number of gears and the rotation speed of the pump will affect the ripple frequency of the pressure profde. Accumulators are often used in hydraulic machines to smooth the ripple in the pump pressure.  2.4 Hydraulic Actuators Hydraulic actuators convert the energy stored in pressurized fluid to movement of the "implements" of hydraulic machines. Actuators may be rotary or linear. Rotary actuators, or hydraulic motors, convert the stored energy to turning motion of a shaft. Linear actuators, or cylinders, convert the stored energy to linear motion of the piston. In this work only linear actuators are considered to simplify the problem. Figure (3) shows a double- acting differential area hydraulic cylinder. Double-acting indicates that piston motion in both directions is controlled by the fluid. Differential area indicates that the rod is only 13  on one side and the area of the piston exposed to the fluid is different on each side. Other common types of cylinder construction may be single- acting single-ended where the cylinder presses against a spring and returned via spring force and double-acting equal area where the rod extends to both sides of the piston.  Pi : Po: Qi:  Pressure at intake side Pressure at exit side Flow into intake side  Qo: Ai : Ao: v :  Flow out of exit side Piston area of intake side Piston area of exit side Piston velocity  fi»  pj  Qo  •  Po  1  L^B^M™a«M5Hi»»Bi™J  T  °'  h e  L o a d  ./ '  Ao  Figure 3 Double Acting Differential Area Linear Actuator  2.5 Hydraulic Valves Valves control the pressure, flow rate and direction of the fluid flow in a hydraulic system. They do this by directing the flow from a fluid source through different openings or by changing the area of these openings. Pressure control valves are used to control the pressure in a hydraulic circuit. Relief valves are examples of pressure control valves and are used to ensure that the system components are not damaged or the output force from an actuator does not exceed specifications. They achieve this by letting some flow out of the system if the maximum allowable pressure is exceeded. Flow control valves are used to accurately control the flow rate of the fluid that reaches one or more actuators from a single fixed displacement pump. They may be used to control the speed of actuators by controlling the flow from the pump to the actuator and from the actuator to the reservoir as well as divert the excess flow to the reservoir. Controlling the amount of flow is achieved by controlling the size of the opening the fluid goes through. This type of opening is called orifice which provides a sudden restriction with a very short length in the flow path. Flow through an orifice is generally described by  Q=C^P P ) r  (4)  2  14  where Q is the flow, A is the orifice area, p is the fluid mass density, P, and P are the fluid pressure into 0  2  and out of the orifice, and C is a constant known as d  "discharge coefficient".  Return  Supply Land  Direction control valves are used to manage the fluid path inside a hydraulic circuit. For example, a check valve is  Spool Movement  used to allow fluid to flow only in one direction which is useful in maintaining the pressure in a hydraulic subsystem when the flow from the pump stops. Direction control valves are often used to divide or control the flow in a number of paths and have also the function offlowand  To Load/Actuator  Spool  Figure 4 A Simplified Four-Land-Four-Way Spool Valve  pressure control built into them. Thus, they are often the main control valve in hydraulic systems. Pilot operated direction control valves may be operated remotely and actuated hydraulically by pilot pressure, pneumatically by air pressure, electrically by a solenoid or even manually. Often a sliding spool as shown in Figure (4) is used to control the flow in different ports which are the exit and entry points of the individual fluid pathways that connect the valve to the hydraulic circuit. The term way is used to describe the number of different directions in which the fluid is guided by the valve. For example a check valve is a one-way valve whereas a four-way valve allows the fluid to go back and forth in two pathways and is often used to operate a cylinder. Proportional control valves are designed to allow continuous control throughout their operating range. Proportional electro-hydraulic control valves are operated electrically and control thefluidflow proportionally to the electrical signal.  The valves that are of interest in this study are four-way pilot operated proportional (direction and flow) control spool valves. In pilot operated valves of interest here, a pilot valve which itself is controlled via a solenoid is used to produce the required pressure andflowto move the spool of the main valve. The  15  signal to the solenoid can be provided remotely, directly from the operators console, or via an automated control system. The movement of the spool can be measured to determine the spool position and thus the orifice area and flow out of the valve at any point in time. This flow measurement can be used for feedback control of the spool position as well as for compliance estimation.  2.6 Hydraulic Pipes and Hoses Pipes and hoses are pathways of fluid flow in a hydraulic system. Steel pipes rather than hoses are generally used when possible to transfer the fluid over longer distances. This minimizes loss of power, fluid, and dynamic performance, caused by hose expansion. The compliance of steel pipes is generally small enough to be neglected in analyzing machine compliance. Since flexible hoses must be used near the moving parts, they are designed to minimize expansion as much as possible. They are generally made of an elastic core tube with fiber braided reinforcement outside sheaths, further protected by an abrasion resistant cover. Nevertheless, the compliance of the flexible hose is substantially higher than that of the hydraulic fluid and it cannot be neglected in calculations. According to Merritt [19], the hose bulk modulus is generally in the range 10,000 to 50,000 psi.  2.7 Fluid Reservoirs The fluid reservoir or tank is where.the fluid is stored before and after the pump circulates it through the system. As described in detail by Yeaple [33], the reservoir performs several important functions in the system and should meet specific performance criteria. For example, the reservoir should be designed to cool and clean the oil before it is returned to the pump. The functions that are most relevant to this study are the reservoir's ability to expel entrained air from the oil and its ability to prevent air from being sucked back into the line by the pump. This is important since, as it will be seen later, air is a major source of performance problem in hydraulic systems. The tank should be large enough to supply the pump for at least several minutes without recirculating the 16  returning fluid [33]. This time is necessary to allow any dirt particles to settle, the heat to dissipate, and the entrained air bubbles to be escape. He adds that this "rule" is generally ignored to allow compactness in mobile hydraulic machines, which can degrade the performance of the mobile machines. There are some important design features that help reduce the amount of air in the hydraulic system. • Baffles are used to divert the return fluid from the pump inlet to prolong circulation time. • The fluid inlet is placed at the bottom of the reservoir to ensure that there is no possibility for air to get sucked into the pump even if the fluid is at its lowest level in the reservoir.  2.8 Accumulators Accumulators are used to supply a known source of compliance to a hydraulic circuit. They can be used to store energy to provide a standby or an emergency source of power to the system as well as provide additional volume in case of thermal expansion. They can also be used together with an orifice or a "resistive element" to achieve the hydraulic equivalent of an electrical low pass or Resistive-Capacitive (RC) fdter to smooth ripples produced by the pump. Since the compliance of the accumulator can be calculated, it is a convenient tool to test a compliance estimation method. Sullivan [29] categorizes accumulators as weighted, spring loaded, and gas-charged. As the name suggests weighted and Gas charge valve  spring loaded accumulators use the weight and the spring force to store energy. Gas-charged accumulators use the force required to compress inert compressible gases such as nitrogen  Rubber bag  to store energy. In this type of accumulators, pistons, diaphragms, bladders, or even no medium may be used to  Outlet valve  separate the gas from the hydraulic fluid. Figure (5) shows a Figure 5 Bladder-type Accumulator simplified diagram of a bladder-type accumulator which is the type that is of interest to this study. 17  Accumulators are generally pre-charged to a specific pressure depending on the application. It is assumed that when the gas is compressed, heat can dissipate, temperature remains constant, and the compression is isothermal. Boyle's law can then be used to calculate the volume change due to change in pressure. Thus, if the pre-charge pressure of gas is P and the volume of the accumulator is V at time 0  g  t the volume of fluid that will flow into the accumulator can be calculated for a change in pressure to P, 0>  at time tj by the using Boyle's law : V P„ = V, Pi 0  Compliance of the accumulator can be calculated by determining the volume change per unit time for a given pressure change (fi= VdP/dt and C=F/p=» C=dV/dP). Volume change per unit time is essentially the flow into the accumulator denoted here by Q. From Figure (6) it can be seen that for any change from time t, to t  2  V -V ' 1  Q=  t  t  2  1  2 }  P V  PV  1 -[- 0 0 t2 f, P j  0  r  r  0  •  7~ \  P  P  and since compliance is C=QIP, making the approximation P ~ t -t 2  x  for short time intervals, the  accumulator compliance is 1  C accumulator  t -t  [  2  x  P V P V °] / p 0  0  0  {  2  P -P  1  1  1  t -t 2  x  P -P  P V j- 0 * 0 _ J  P  PV 0r0j  Simplifying the above equation leads to Time = t„  C accumulator  Time = t,  Time = t,  P V pp r  l  r  (5)  P V 2  P.V,  2  AV  2  V////A  Y/////X  n  Figure 6 Volume and Pressure Changes in an Accumulator 18  2.8 Some Operational Problems in Mobile Heavy-Duty Hydraulic Machines Mobile heavy-duty hydraulic machines often work in very rough environments and are exposed to varying loads and environmental conditions. Under these conditions, hoses may wear out or become punctured, and other components such as valves and cylinders may accumulate rust and dirt. Such problems may lead to poor performance or failure of the machine, that can be costly and dangerous. If the operators do not estimate the load correctly while they perform tasks, the machines may also be exposed to loads that they were not designed to handle. According to Yeaple [33], in some cases, the end-users add or remove various implements from the machine without replacing the relevant cylinders to match the new load requirements. Yeaple [33] also writes that sometimes the operators may "shockload" or "otherwise abuse" the machine to try to break the load free. This type of misuse leads to premature failure of components such as seals. Although skilled operators, good maintenance, and careful operation can reduce problems in the system, in practice, it is impossible to eliminate unexpected failures. Furthermore, as previously mentioned, automated control systems render operators less effective in predicting many of the errors. Thus, an onboard diagnosis system would be a useful tool to assist operators in looking for early warning signs of and preventing impending machine failure.  Problems of interest to this study fall into two categories: 1- Problems such as trapped air or worn-out hoses that actually change the compliance. 2- Problems such as dirt build-up in the valve or leakage that introduce a systematic error in the measurements and only "appear" as changed compliance. The distinction between these two categories is important because the problems in the first category lead to changes in the compliance value that are easy to make sense of using a very simple compliance model. However as it will be shown later, problems in the second category make modeling and interpreting the apparent change in compliance very difficult. Thus, to use compliance as a meaningful indicator of the 19  machine performance, one has to first understand all the problems in both categories and their effect on the apparent compliance value. As it will be seen from the descriptions of the problems below, many of these problems are inter-related and are likely to occur simultaneously.  2.8.1 Problems That Change Machine Compliance Some of the most important problems that actually change the machine compliance are air in the system, fluid over-heating, and worn-out hoses. Here these problems are discussed with special emphasis on air in the system due to its complexity and importance.  2.8.1.1 Air in the System Since air is compressible, the presence of air in the system has a profound effect on the compliance of the machine. One of the potential benefits of estimating the compliance is to determine the presence of excessive air in the system. To fully understand the problems caused by air and how air in the system affects its compliance, the interaction between air and hydraulic fluid has to be studied in detail. Air in the hydraulic system is responsible for a variety of problems such as poor dynamic performance, loss of power, increase in temperature, foaming or loss of the fluid, physical damage to system components, and oxidation of the fluid. The problems caused by the interaction between air and hydraulic fluid are known to those who maintain and operate the hydraulic equipment. The only solution known seems to be simply to reduce ways for air to get into the system and increase ways for air to get out of the system. Unfortunately the research in this area which was mostly done in the 1950's and 1960's has not offered any better solution.  What is particularly notable about the problems caused by air is that they can occur in perfectly maintained, perfectly functioning systems without warning, and in ways that appear to be almost random. The reason for this only becomes apparent as all the phenomena involved are understood. 20  Verbal and electronic communication with some practitioners, oil manufacturers and researchers revealed that the interaction between air and hydraulic fluid is not fully understood by many of those who have to deal with these problems. According to Korn [12], "the presence of air in the system is sometimes not recognized as such but attributed to oil vapor". He dismisses this "theory" and goes on to argue that even at low pressures of a suction system, the hydraulic oils are unlikely to vaporize.  Citing rather obscure references from the 1930's to 1940's, most of the literature in hydraulics attributes the "mystery" associated with fluid aeration to the solution and evolution of air in the fluid. While not all the original literature could be located, the importance of air to this study motivated much effort to locate at least some of the original studies on this topic which are a basis for the following discussion.  One fact that all researchers agree on, is that air can be present in hydraulic fluid in dissolvedform, entrainedform as air bubbles, or free form as air pockets not in full contact with the fluid. In the extreme case where the bubbly oil has more than 30% air by volume at atmospheric pressure, the oil becomes foam [9]. It is believed that, depending on the operating conditions, air can move from one of these forms to another. Thus, if the fluid contains any air in free or bubble form and the pressure rises, some of this air dissolves in the fluid up to the point that the fluid is saturated with air. Conversely, if fluid contains a certain percentage of dissolved air, and the pressure is reduced to the point that the fluid is "super-saturated" with air, some of the air is released from the solution in the form of air bubbles. However, according to Rendel et al [21], if the fluid is not violently disturbed both unsaturated and super-saturated fluid can exist in contact with air.  To better understand the mechanism by which air gets into the system and the problems it causes, the following four topics are worth separate attention: •  Solubility of air in the hydraulic fluid 21  •  Evolution of air from the hydraulic fluid  •  Air bubbles and free air in the hydraulic fluid  •  Cavitation in the hydraulic fluid  Solubility of Air in Hydraulic Fluid Although it is an accepted fact that the of solubility of a gas in a liquid (the saturation point) changes with changing pressure according to Henry's law [5], the mechanism by which air moves from entrained to solution form and vice-versa is to some extent subject to speculation. No models were found that explain the exact process. Instead, researchers have used carefully designed apparatus to experimentally verify some observable properties of air in the system. While according to Hayward [8] oxygen dissolves more readily than nitrogen in almost all liquids, this difference is generally ignored in the study of solubility of air in hydraulic fluids. According to Hayward [9], air bubbles do not dissolve considerably in the fluid at atmospheric pressure as the air bubbles rise rapidly and escape the fluid. As the pressure increases above 100 psi, about 25% of the air bubbles dissolve rapidly, but the rest take longer to dissolve. He also points out that agitation increases the rate of solution and in low viscosity oils air can dissolve so rapidly that "the absorption seems instantaneous".  Rendel et al [21] state that 8-12% air can dissolve in oil by volume, without further explanation. Hayward [9] reports that at atmospheric pressure and ambient temperature, a typical petroleum fluid with average viscosity dissolves about 8.5% air by volume. Hayward's experimental method for arriving at dissolved air content is described in detail in [8] and [9]. Magorien [16] also performed studies to determine dissolved air content with the experiments that are reported in [15] and [16]. Both sets of experiments involve removing all air bubbles from oil, reducing pressure to vacuum, and measuring the volume of evolved air. Magorien [16] presents the graphs shown in Figure (7) without specifying the source. Similar graphs can be found in [9] and [33], also without specifying the source. 22  Many researchers report that the i  s  solubility of air also decreases with  o  increasing temperature which is  ,180 ' F  tt.  •o u >  evident from the formation of air  BO"F  0  B u  bubbles in the fluid as temperature  01  rises. The effect of two different  _3  u E  15  30  45  60  75  90  >  0.5  0.6  0.7  0.8  0.9  1.0  temperatures on air release from Saturation Pressure -PSIA  solution is shown graphically in Figure (7b). Hayward [8] reports tests that  Pressure -Atmospheres  (a)  (b)  Figure 7 Air solution properties for MIL-H-5606 fluid (a) Pressure and air solution in MIL-H-5606 fluid. (b) Vacuum and air release from MIL-H-5606 fluid, (redrawn from [16])  establish that dissolved air does not appreciably change the compressibility and viscosity of hydraulic oil. Margorien [15] also reports that his data does not indicate any change in compressibility or volume of fluid due to dissolved air.  In summary, the above discussion indicates that the relevance of the air solution to this study is: • It is likely that the volume of free air in a pressurized hydraulic system is very small. • Dissolved air cannot be detected by compliance estimation at high pressure. • If air gets into a pressurized system, it may dissolve before being detected by a fault detection system. • There are no models to describe the air solution in the fluid. • There is at least 8%-12% air by volume (normalized to atmospheric volume) in all hydraulic systems.  Evolution of Air From Hydraulic Fluid Unfortunately knowing the saturation point or the maximum solubility of air in oil, based on temperature and pressure under equilibrium conditions, is not enough to predict the air content or form at a given time. This is partially due to the non-uniformity of the fluid pressure and temperature within the 23  hydraulic system during short time intervals. For example, even if no air can get into the system from outside, large pressure gradients and even vacuum may exist in parts of the system which will may cause air to come out of the solution as air bubbles.  Sub-atmospheric pressures may be created, if fluid is accelerated over a point such as an orifice or a pump inlet, or into a double-acting actuator driven by an external load in such a way that more fluid leaves than enters the vicinity. According to Magorien [15], air bubbles generated as such may be redissolved into the fluid if the fluid slows down. However, if the fluid is further accelerated to above 100 ft./sec.(30.5 m/s), the air bubbles expand and are less likely to be re-dissolved in the surrounding fluid (which remains saturated). In [16], Magorien states that such partial vacuum (gas filled pockets) inside the system (also known as cavitation) cause erosion and severe surface damage to the system components. He adds that the reason behind this is not clear and has been attributed to "vapor bubble implosion" and "accelerated oxidation" by some. Cavitation is discussed further below. Yeaple [33] states that if there is not enough time for the air bubbles to dissolve back to the fluid, the sudden adiabatic compression of air bubbles in high-pressure pumps can cause high enough temperatures for oxidation and nitration of the oil. The degraded oil then causes system damage and failure. This type of failure is very illusive since it may happen in systems that are very well maintained and start out with clean fluid. Air also causes oxidation of the rubber in the hose which is speeded up by heat [33].  Other phenomena besides pressure gradients, also complicate the task of predicting air content and form. The rates of air solution for under-saturated oil and air evolution for supersaturated oil depends on the level of agitation, the type of liquid, the liquid-air interface area as well as the temperature and pressure [8]. Thus, the rate of air evolution is higher than solution, since the formation of air bubbles increases the air-oil contact surface. It may be possible to exploit this inherent asymmetry for distinguishing air from other sources of compliance in the system which may change more uniformly with pressure. 24  The above discussion indicates that the evolution of air out of solution is a major source of problem in hydraulic machines, but even the experts do not agree on the mechanism. Therefore, given the current state of knowledge, modeling air evolution is not possible without further research.  Air Bubbles and Free Air in Hydraulic Fluid Air bubble evolution from solution and some of the problems it causes were discussed above. However, there are other sources of air bubbles in mobile machines. Air mostly gets into the system from outside, through leaks or the pump inlet in the tank while mobile hydraulic machines working under rough conditions are especially prone to damage and leakage. Furthermore, many mobile machines work on uneven surfaces, tilting the tank and exposing the pump inlet to air. As discussed in Section (2.7), the tank is also likely to be too small in these machines. This means lower fluid levels that allow more air into the system, and insufficient rest time in the tank that prevents already entrapped air from escaping.  Some of the confusion caused by air bubbles and their effect on compliance is due to the fact that there are two methods by which the air bubbles "vanish" from the system. The first and most desired method is to let them escape the system. Besides escaping in the tank, bleeding points or valves may be used at high points in the system since the air bubbles tend to rise to the top and collect as air pockets. Unless they are bled out of the system, air pockets may dissolve slowly or eventually break-up and return into bubble form or be flushed out by the fluid. However, according to Merritt [19], it is possible that air may be trapped in some location for long periods of time and degrade system performance.  The other method is absorption of air bubbles back into the oil due to pressure increase. According to Hayward [8], experiments in the late 1940's with a volume of oil and a fixed volume of air on the top, determined that the rate of solution and evolution of air was proportional to the degree of under25  saturation and super-saturation of air in the oil. However, Hayward's own tests with bubbly air (rather than separated air and oil) did not confirm this result. This shows the additional complication introduced by the air bubbles. However, Hayward maintains that for pressures of higher than 100 psi a solution rate of 6% per second or higher should be expected. Magorien [16] calls the process of air bubbles going back into solution adsorption rather than absorption since adsorption means adhesion of an extremely thin gas film to the surfaces it is in contact with. He states that the rate of adsorption is proportional to the pressure and inversely proportional to the size of the bubbles.  In short, entrained and free air are the major source of compliance reduction in the hydraulic fluid, but it is very difficult to predict their volume at a given time. As far as compliance is concerned, the entrained and free air behave exactly as though the same volume of air were in one container separate from the fluid and subjected to the same pressure [21]. According to Merritt [19], even a 1% air in the fluid volume can increase its compliance by a few orders of magnitude. Estimates for the percentage of entrapped air in the fluid at atmospheric pressure have been as high as 20%. The effects of air on compliance are discussed in Chapter (4).  Cavitation Cavitation is caused by sudden collapse of gas bubbles in the fluid. It is known [23] to be a source of erosion and damage to the system components in the hydraulic system. Cavitation is worth mentioning here, since it is often blamed on the collapse of air bubbles. However, some sources [23] indicate that the air bubble growth and collapse caused by gas diffusion or slow pressure change is not rapid enough to be a problem in the system. The air in the system is not directly responsible for cavitation, but the entrained air creates the cavity in the fluid that can under certain conditions start the nucleation process. Nucleation is the process by which vapor filled bubbles are formed. With rapid pressure increase which 26  could even be caused by vibration, the bubbles may collapse and cavitation can occur. Since air bubbles may indirectly contribute to cavitation, any on-line method for their detection can help in identifying and reducing the sources of air in the system, and thus reducing the occurrences of cavitation.  2.8.1. 2 Over-heating An increase in oil temperature can cause an expansion of air bubble volume in the fluid compared to the fluid which may affect the overall compliance of the fluid. However, according to Yeaple [33], the main problem caused by overheating may be the degradation of oil due to high-temperature cracking, oxidation and nitration. This is because the air is compressed rapidly under pressure without enough time to either dissolve or dissipate heat to the surroundings. Thus, it is more likely that increased compliance and overheating may both be symptoms of the same problem, namely entrapped air.  2.8.1.3 Worn-out Hoses While hoses are designed to achieve the maximum possible stiffness (average bulk modulus of 10,000 to 50,000 psi [19]), they can wear out over time. Hoses are exposed to worse operating conditions than almost any other component in the hydraulic system [33]. If the hoses wear out, their elasticity can change. For example, if the outside sheath is partially damaged, the hose may become considerably more elastic or compliant. If the control system of a hydraulic machine continues to increase the flow to compensate for the additional compliance, it is quite possible that the hose may rupture without much warning. The result may be physical harm to the people working near the machine, damage to the equipment or surroundings, or even fire. However, if the gradual increase in compliance is detected by a diagnosis system and the operator is warned, the dangers may be avoided. It would be even more desirable if the diagnosis system could "guess" when damaged hoses are a likely source of the increased compliance.  27  2.8.2  Problems  T h a t O n l y A p p e a r to C h a n g e M a c h i n e  Compliance  In this study flow, pressure, and actuator position are used for estimating compliance and errors in measuring them could "appear" as a change in compliance. Sensor problems are not dealt with in this study, but other sources of error in flow measurement are often results of fault in the system. Examples of these faults are leakage in the actuator, in the valve, and to the outside and dirt build-up in the valve. It should be noted that although these problems do not affect the stiffness of the system, they affect the response of the system because they interfere with the delivery of power to the actuator. Therefore, a human operator may not be able to distinguish between these problems and the "true" sources of compliance. If an automated diagnosis system can distinguish these two types of problems, it can outperform humans in fault detection.  2.8.2.1  Punctured  Hose and Leakage Out  of the  System  If the fluid escapes the system through punctured hoses or other sources, the calculated flow into the actuator will be higher than the actual flow. This leads to incorrect compliance calculations. The actuator will also not move as much as expected for the given control signal. In such situations, the operator or the control system will increase the input signal gain to compensate for the sluggish response which may even accentuate the problem. If the operator controls the system directly, he might notice the problem. However, if the system is controlled by an automated feedback control system, the operator may not notice the difference in response time. Leakage can cause several problems such as increased operating cost, potential environmental damage, and fire risk (for flammable fluid). It may also be a result of a weakening hose and a prelude to a rupture. Furthermore, leakage in parts of the circuit that may be subject to pressure drops to values even slightly below the atmospheric pressure, can draw large volumes of air into the system [8]. In this case a problem that starts as only "apparent" increased compliance may quicky turn into a "true" source of increased compliance in the system. 28  2.8.2.2 Leakage In the C y l i n d e r The moving parts inside a hydraulic circuit must be separated by an oil fdm to provide lubrication. As thin as this fdm is, some leakage will be unavoidable. Thus, there will always be some leakage between the piston or  Rod Seals Piston Seals  rod and the walls of the cylinder as shown in Figure (8).  Figure 8 Cylinder Piston and Leakage Leakage between the rod and the walls lets fluid out of the system and can potentially be detected by careful observation. The piston-cylinder wall leakage is more troublesome since fluid escapes from one side of the piston to the other, which may not be easily detectable, yet cause erratic system performance. Seals such as O rings and U rings are used in special configurations to prevent this leakage as much as possible. Over time, seals tend to wear out and fail and leakage increases. As will be seen later, besides causing performance problems, the leakage inside the cylinder will also have a very dramatic effect on compliance estimation. The piston side that is losing flow will appear to be very compliant and the side gaining the flow can appear to be very stiff. This type of leakage is caused by laminar flow between the two surfaces in contact (piston-cylinder wall or the rod-cylinder end ). Martin [17] categorizes this type of leakage in the gap between two surfaces into two groups: 1- drag leakage due to relative movement of two surfaces  —  i-—... .  ... .\ :  2- slip leakage due to a pressure drop across the gap He derives the following relationships: bhv Q,drag  _bh dp  (6) Figure 9 Leakage between two moving (cylindrical) surfaces in contact (a) Side view; cross section (b) Front view: eccentricity  3  n  (7)  29  where Q  drag  is the leakage flow in the direction of movement, Q is the flow due to the pressure slip  difference across the gap, p is fluid viscosity, h is the gap width, b is the width of the contact surface, v  max  is the maximum velocity of oil fdm in the gap (which is essentially the relative velocity of the two  surfaces), and dp/dx is the rate of change of pressure in the direction of interest. Figure (9a) shows the side view of a narrow cross-section of a solid cylindrical object such as a rod or piston moving inside a hollow cylinder. For the entire cylinder b = nd where d is the cylinder diameter. According to Martin [17], the diametrical clearance in a cylinder is generally about 0.1% of the cylinder bore and the gap is Vi of this clearance. He gives the value of 50-250um as a general range for this gap. When the cylinder is operating there will be some leakage from both categories. Martin [17] also shows that for a piston inside a cylinder with diameter d, diametrical clearance 2h, contact length / and a pressure drop of P - P , from Equation (7), the slip leakage is u  d  TTV//, P 3  Q  »»  12u "/  ~P  A  ( 1 + 1  '  S c  >'  ( 8 )  where eccentricity e, which is shown in Figure (9b), indicates how far is the rod center off the center of the cylinder. It can be seen that the leakage flow can grow, if the rod is not centered. The practical hydraulics literature warns that in practice the rod is unlikely to be fully centered. However, it is not clear if the effect of eccentricity on the piston-cylinder leakage is also great in practice.  Equation (6) shows that the drag leakage increases with the piston velocity. Equation (8) shows that the slip leakage increases as the pressure difference between the two sides of the piston increases. Consequently, seals are designed and configured in such a way as to increase their sealing force as the pressure increases to compensate for the additional leakage between the surfaces of piston and cylinder. However, even in the presence of working seals there will always be some leakage. According to Sullivan [29], as pressure rises, at first leakage increases considerably, but as the seals also react to the 30  increase in pressure by increasing sealing force the leakage increases u A M ed at a lower rate. He adds that -2  Viscosity  •  Speed  *-  Pressure •  •  leakage increases proportionally to oil viscosity and roughly as a  Figure 10 Effects of speed, viscosity and pressure on leakage (from [29])  square of cylinder speed. Figure (10) which was drawn based on a similar figure in [29] shows these relationships graphically. One important point to note is that the magnitude and the even sign of the leakage in a linear actuator is highly dependant on the direction of motion of the piston. This is not only due to the drag leakage, but also due to the fact that the slip leakage depends on the pressure difference between the two sides of the piston. Depending on the surface areas of the two sides and the operation loads, this pressure difference and thus the leakage may change sign during the operation.  2.8.2.3 Leakage in the Valve Since the valves have moving parts, they also have a certain amount of leakage. The types of leakage between the surfaces is similar to that seen in the previous section. In particular if the spool in the valve is not centered, the valve appears to have large leakage. However, as long as the spool is centered, the valve leakage is not reported as a major source of problem in the literature reviewed. The valves are not as much subject to damage as the cylinders. Valves have several chamber, ports, and surfaces in contact and calculating the leakage flow for valves requires a detailed study of all the leakage sources for each type of valve. To focus the scope of this work, the valve leakage is not taken into account in this study.  2.8.2.4 Dirt Built-up in the Valve Dirt build-up in the valve may block the fluid passage and reduce the effective orifice area in the valve. This can result in erroneous flow measurement since the spool movement is used to calculate the orifice area. In this case the flow is over-estimated and this will lead to an erroneous estimate of compliance. 31  2.9 Subsystems of Interest to This Study Due to the availability of a Caterpillar (CAT 215B) excavator for research at the University of British Columbia and the previous research work done on modeling and tele-operation of this machine, the CAT 215B is used here as an example of a heavy-duty mobile hydraulic machine. The ultimate goal is to have an automatic diagnosis and tele-operation Figure 11 Main hydraulic actuators of the system on such machines. Figure (11) illustrates the  CAT 215B excavator  main hydraulic actuators of the excavator conceptually. The Boom, Stick and Bucket are actuated by linear actuators, specifically, double-acting differential area cylinders. The Cab movement or the Swing is actuated by a hydraulic motor. The inputs and outputs of the actuators are connected to the main valves with flexible hoses which are not shown in Figure (11). The main valves are operated by pilot  Pump 1  valves that are connected to the operator Pilot Valve  controls. The connections between the valves, pumps and actuators is shown in Figure (12). In the particular machine  PUot Valve  Operator Controls  Pilot Valve  Pump 2  Spool  Bucket  Spool  Maln  Spool  Vahree  Stick  Hydraulic 1 I  1  Boom  It—^  Swing  —|ptlot Valve j - — ' ^  studied, the pilot valves are replaced Flexible Hotel  with servo-valves to ensure that the pilot pressure corresponds to the voltage  Figure 12 Block diagram for pump, valve and actuator connection (re-drawn from [30])  signal received from the operator controls. The system is supplied by two variable displacement (constant flow) pumps in two branches. The first branch contains one pump and the Bucket and Boom valves and actuators. The second branch contains the second pump and the Stick and Swing valves and actuators. This machine utilizes a cross-over network between the two branches in which pressure 32  control valves redirect surplus power from the pump in one branch to the actuators in the other branch to maximize the power available to the individual actuators. This machine has been instrumented for the purpose of modeling and tele-operation. The sensors of interest to this study are pressure sensors at the pumps, tank, and actuator inputs and outputs, position sensors on the spools to be used for flow calculations, and resolvers on the joints to calculate the joint angles. These sensors are reasonably inexpensive and robust and it is conceivable that they may be installed in working machines.  The hydraulic subsystem of interest to this study is the fluid path from the control valve to the actuator. This subsystem contains some of the major sources of problems in the hydraulic systems such as a considerable volume of fluid which may be aerated, flexible hoses, the main valve, and the actuator. The linear actuators such as the Boom, Stick and Bucket are preferred for their simplicity. As will be shown later, the pressure and flow values in these subsystems can be used to obtain a measure of compliance. The goal is to obtain this compliance measure by using the simplest possible sensors. However, the cross-over network poses a challenge to calculating the pressure and flow values in the circuit since depending on internal pressure at the pressure control valves (check valves), various portions of the crossover network may not be active. This problem was solved by Sepehri [25] who devised a method of determining the portion of circuit that is active by comparing the relevant pressures in the circuit. This method requires an iterative search method where all the possible configurations are considered simultaneously and the corresponding pressure values are calculated to find the particular pressure distribution that matches the measurements.  33  3  Parameter Estimation  Methods  Depending on the compliance model chosen, a suitable parameter estimation method should be selected. Thus, some knowledge of parameter estimation methods is necessary. When parameters of a dynamic system are to be identified, this method is also known as system identification. Abundance of parameter estimation methods makes selection of the best method for many complex problems difficult, since the variations of each method can be "tweaked" almost infinitely to suit different problems. The literature on this topic can be confusing due the large number of methods, variations, and terminology used in different fields. According to Ljung et al [13], the field of recursive identification has been somewhat justifiably called a "fiddler's paradise". With increasing computational power, the scope of "fiddling" is even wider as using computationally intensive methods such as Genetic Algorithms becomes possible.  In this chapter, Recursive Least Squares and Genetic Algorithms are studied to determine and compare their suitability for the on-line identification of the parameters of models for hydraulic compliance. Since each method has a large number of variations, to focus the study only their "classic" forms are considered. Comparing such different methods only makes sense in the context of the requirements of a specific problem. Thus, the effort is focused on the value of these methods to the on-line identification of the parameters of models for hydraulic compliance. Furthermore, comparison of two estimation methods in an application will not be meaningful unless it is done in the context of the available options. While a detailed analysis of all the available methods falls outside of the scope of this work, a general overview of the basic philosophies behind the major classes of relevant methods is appropriate.  34  3.1 Simple Overview of Parameter Estimation Methods Parameter estimation revolves around finding the values of the parameters of a parameterized model of a system or process that best fit the measured data from that system or process. How well the selected values for parameters fit the data is determined by the performance measure. In some cases, if statistical knowledge of the system is available, specific statistical performance measures are used. A detailed discussion can be found in [13]. Here the discussion is restricted to the commonly used performance measure, or the magnitude of the estimation error (for a block of data), called the error function. The error function is often defined as the sum of the magnitudes or squares of the difference between the measured output and "estimated" output (the output predicted by the model using selected parameter values and the corresponding measured inputs) in a "block" of data. Despite the simplicity of the basic concepts, some of the specialized estimation methods become more complicated to handle special restrictions of specific types of problems or to improve performance by using specific knowledge about the problem.  For applications requiring on-line or real-time estimation of the parameters of a process or system, one major variation in the estimation method is to perform the calculations recursively. Since the data are generally measured repeatedly (sampled) over time, the amount of data collected increases with time and eventually becomes too large to be processed in real-time. Recursive methods perform the estimation for each step using the results of the calculations of the previous step and the measurements for the current step, thus eliminating the need to process or store the measurements of all the previous time steps.  Another basic difference between estimation methods is in the strategy used to search for the parameters that minimize the error function. One group of methods known as "gradient search methods " use a directional search based on the assumption that they can move in the parameter space in the direction of 35  the steepest decent of the error function and reach the global minimum. If the derivative of the error function is computable, the local derivative at each point in the parameter space can be used to direct the search for minimum error. This leads to a class of optimization methods that are often known as "hillclimbing" methods. The weakness of this type of local search is that upon arrival at the local minimum of the error function, there is no guarantee that the solution is optimum, or that the local minimum is also a global minimum. Therefore, the starting point for the search has a large impact on the success of these methods in converging to the global minimum of an error function with multiple local minima.  If the error function is the sum of the squares of the errors for each set of input-output measurements, and the error function is differentiated in the parameter space to find the minimum, the estimation method is known as Least Squares (LS). The recursive version of this method is known as Recursive Least Squares (RLS). The LS method is often used when the model is linear in the parameters and the minimum can be determined analytically. In such cases RLS provides a particularly simple and inexpensive method for real-time estimation of system or process parameters. A "weighted" version of this method is often used when the measurements are time based and the system or process changes with time. In such cases, the historical data can be weighted so that they can be "forgotten" in time and the model parameters can "adapt" to changes in the process or system.  If the error function is not differentiable, methods such as the "Downhill Simplex" [20] may be used. These methods use a variety of algorithms to utilize local information to search towards the direction of the minimum of the error function. On the other hand, if the model is not linear in the parameters and the error function has many local minima, a globalized search for the global minimum will be more desirable than a local search. Simulated Annealing [6] and Genetic Algorithms are two examples of such methods. Genetic Algorithms impose a structure on an essentially randomized searching technique to solve a given optimization problem effectively, but with a high computation cost. 36  3.2 Recursive Least Squares The recursive least squares (RLS) method is an inexpensive parameter estimation method which is suitable for cases where the model is described as a linear combination of the parameters that are to be estimated online. The RLS method is simply a recursive version of the Least Squares Method which is widely used in most scientific fields for fitting experimental data to models often known as regression models and according to Astrom [2] was first introduced by Gauss in the eighteenth century. Since RLS is a very common and widely known algorithm, the details of the algorithm and its interesting properties are only provided in Appendix (A.l).  RLS is a very well studied method which is based on well founded mathematical principles. While there are some restrictions on the conditions for which RLS performs acceptably, many methods have been devised to manage the restrictions or at least test the suitability of RLS to the problem at hand. It appears that in cases were the model is linear in the parameters, the parameters do not change extremely rapidly, the measurements are independent of the noise, and when the noise is randomly distributed with zero mean, there is no reason to use an estimation method which is more complex than the simple RLS. It should also be noted that if, for example, the excitation conditions or the restrictions on measurement noise are not met, the parameter estimation problem is inherently difficult to solve regardless of the method used.  37  3.3 Genetic Algorithms  3.3.1 Background Genetic Algorithms (GAs) are search and optimization algorithms that were inspired by nature's basic adaptive algorithm of natural selection through survival of the fittest and genetic modification. Genetic Algorithms were first proposed by John Holland in "Adaptation in Natural and Artificial Systems" [10], published in 1975. He set up a mathematical framework to demonstrate how complicated structures could be encoded by simple representations such as bit-strings and how by using a population of these bit strings and the proper transformations under a controlled structure, the "genetic algorithm", the bitstrings would evolve to adapt to the problem being solved. Holland showed how the global and "parallel" nature of the genetic algorithm in search may lead to a nearly optimal solution with a relatively rapid convergence rate. Due to this strength and the fact that GAs do not impose much requirements or model restrictions on the user, genetic algorithms provide a viable alternative for many nonlinear search and optimization problems in science and engineering.  One disadvantage of GAs is their high computational requirement, since they are based on a population rather than a single point. The distributed or population-based computation of the GAs is often used to implement Parallel Genetic Algorithms (PGAs) to speed up the GA computation. Recently, faster computers and parallel machines, has prompted an increased interest and research in this area. Another disadvantage of GAs, which is shared by many optimization algorithms, is that their convergence can not be guaranteed or predicted for many types of problems. Furthermore, many parameters that affect their performance can not be determined analytically. Thus, users resort to guessing and experimentation.  38  3.3.2  Basic Mechanism  In Genetic Algorithms, encoded representation of the parameters are treated as chromosomes in natural genetics. A population of individuals, each represented by a randomly generated chromosome, is subjected to genetic operators such as reproduction, cross-over and mutation for a number of generations. Every population member is measured based on a given fitness function and depending on the level of fitness of the individual, the individual is reproduced for the next generation.  The parameters may be encoded by any alphabet, but the binary representation as bit-strings of O's and l's is the simplest most commonly used encoding. Here, the discussion will be limited to the binary parameter representation. For example, an individual / may be represented with a bit string length of /: 1= hhh  h-i  The i's are sometimes called genes and the values of the /'s are sometimes called alleles. For binary encoding the alleles are either 0 or 1 and the parameters are bit-strings such as 101001010 or 001010.  If the upper bound for the parameter space being searched is max/ and the lower bound is min/, then the resolution of / represented by a binary coding is: , ,. maxl-minl resolution= 2'-l In simple terms the genetic operations can be described as follows:  Reproduction: The survival of an individual into the next generation and the number of copies it may have is decided based on its fitness value /, relative to the total fitness Yft of the population. Thus, the individual will be reproduced with the probabilityp =f lYJi • The population size is generally kept r  constant for all generations. Therefore, the fittest individuals may have more than one copy in the next generation while the unfit individuals will have no copies. 39  Cross-over: Cross-over is the combination of genes of two randomly chosen fit individuals with the probability p to produce potentially fitter individuals. After the reproduction step, population members c  are paired at random. A point p is selected randomly such that 0 zp zl-l. The parents' chromosomes are split into two sub-strings at point p and exchanged to obtain the offspring chromosomes: parent 1: parent 2:  I = J =  i i,i ....i  offspring 1: offspring 2:  K = Q =  i i, i .... i j .... J,., jojtjt-jpip+i  0  2  i  p  ui  joJiJ2--J --Ji-i P  0  2  p  p+l  Mutation: Mutation is a random change of the allele of each gene in all chromosomes from one generation to another with a probability p . For a binary representation mutation simply complements m  the bit value in the bit-string. Thus, if the chromosome is to be mutated at position p, then the string I~ hhh---i  p  will be modified to  /'= i i,i ...i 0  2  p  i .,. t  Mutation ensures that specific alleles or genetic material that might be lost after domination of certain types of individuals may be introduced to the population.  3.3.3 Theoretical Properties of Genetic Algorithms Both Holland [10] and Goldberg [7] have an extensive discussion on the theoretical analysis of GA performance and the work on this subject is still continuing by many researchers. Here some of the most important theoretical properties that are fundamental to the search mechanism of GA and are relevant to the discussions in this thesis are worth reviewing.  The Schema Theorem Holland [10] introduced the concept of schemata. He used it to establish a framework for rigorous analysis of the performance and the search power of Genetic Algorithms. A schema is a template that describes a subset of strings which are similar in specific positions. A schema can be thought of as a genetic feature within the chromosomes. In a binary parameter encoding, the parameters are encoded as 40  strings using the alphabet {0,1}. For the schema representation, a third element * or don't care is added to the alphabet, so that a string's schemata are defined over the ternary alphabet {0,1, *}. For a schema to match a specific string, the schema O's and 7's must be at the same position as the string O's and 7's respectively but the schema *'s can be at positions corresponding to either O's or l's in the string. For example, the schema * 10*11 matches the strings 010011, 010111, 110011, and 110111 but does not match 110110, 100011, or any other string. Conceptually, the symbol * is used to mask bits that are irrelevant to a specific schema or template. A specific string can match or contain a large number of schemata. For example, 100011 matches or contains ****]]  T  * * $ * * * JOOO**  and many other schemata.  The total number of possible schemata of length / for a binary encoding is 3', since each position can be 0, 1, or *. Thus, for length / and an alphabet of length k, there are (£+1)' possible schemata. At any given time, the entire population will contain a subset of these schemata. This subset changes at each time step as a genetic algorithm searches for optimum individuals. For a specific binary string of length /, the number of possible schemata in the string can be calculated by considering that each bit in the schemata should either match the string's corresponding allele or be a * (don't care). Thus there are two possible values for each bit in the schemata which gives 2' possible schemata. In a population of n there are up to n.2 possible schemata. 1  For a schema H, the order of the schema, 0(H) is defined as the number of fixed (specified as 0 or 1) bits and the defining length of H, 6(H) is defined as the difference between the positions of the first and last fixed bits. Goldberg [7] defines the position starting from 1 and increasing left to right. For example, O(*l**10**l**) =4, 0(1***) = 1, O(**l*0*) =2, S(*l**10**l**) = 9-2 = 7, 6(1***) = 1-1 = 0, and 6(**1*0*) = 5-3 = 2. The concepts of defining length and order of the schema can be used together with probability theory to determine how the number of instances of a specific schema is affected from one generation to another through the application genetic operators of reproduction, crossover and mutation. Assuming that at time 41  step t a population A (denoted by A(t) ) of size n contains m instances of the schema H, the effects of reproduction, cross-over, and mutation on the schemata can be determined as demonstrated in Appendix (A.3). Combining the effects of all operators, leads to a theoretical description of GA's search mechanism. The overall effect of the genetic operators on the number of instances of a given schemata //from time t to time t+1 can be described as m(H,t \) > +  (H,t).&-[\-p .^-0(H)p ] f ~  m  c  l  where n denote the population size,  m  l  / = ^2f/n,  is the average population fitness, m(H,t) denote the  number instances of schema //at time t, f(H) denote the average fitness of all strings that contain the schema H and YJt denote the sum of fitness values of all population members. Goldberg calls this the most important result derived by Holland known as the "Schema Theorem". It shows the three genetic operators of reproduction, cross-over and mutation cause an exponential increase in the number of instances of short, low-order, above-average schemata in the population. According to Goldberg [7], the general hypothesis for GA performance is that instead of looking at all possible strings for the optimum in the population, these short low-order and fit schemata within the strings are building blocks that are selected, combined and re-combined like to build fitter strings.  Implicit Parallelism in GA Search The phrase "implicit parallelism" wasfirstused by Holland [10] to convey the idea that although GA processes only n = population size strings at each time step or generation, it is actually processing a much larger number of schemata at the same time. He estimated this number to be of the order of n . 3  However the term "implicit parallelism" is perhaps a misnomer. It is not to be confused with the computational aspect of the algorithm which may be parallel or serial.  From the previous discussion it can be seen that for a schema to survive through the generations it has to 42  be of a relatively short length. Thus, although there could be between 2' and n.2' schemata in a randomly generated population, the genetic operators will reduce the number of schemata that can survive. Assuming that for a survival probability p„ only the schemata of length l or less are likely to survive, an s  estimate for the minimum number of schemata processed by GA can be obtained. Goldberg estimates this number to be n <> n(l - l +l).2' ' . For a population size of 2 , which is a population that on s  s  2  hn  s  average will have no more than one instance of schemata of length ls/2, he concludes n = (I - l +l).n /4 3  s  s  which is 0(n ). The details for this derivation can be found in [7]. 3  GA Deceptive Problems Although the discussion above indicates that GA is capable of identifying and recombining fit, short, and low-order schemata over the generations, it is possible that due to unfortunate combination of parameter coding method and fitness function, the combination of the selected schemata does not lead to the optimum solution. An extreme example would be a case where the optimum solution depends on values of two "far" bits in the bit string which are surrounded by or close to low fitness schemata. In such a case no schemata close to the desired bits is "encouraged" to survive by GA to provide the correct building blocks to create the optimum schemata. Furthermore, since short schemata are favored, no schemata is likely to be long enough to contain both bits. This type of problem is known as GA Deceptive. The GA deceptive problems correspond to functions whose optima are very isolated and no optimization is generally very successful for these types of problems. However, Golberg [7] suggests that many GA deceptive problems are not GA-hard (GA does not converge to the global optimum for GA-hard problems).  3.3.4  Observations  The above discussion indicates that GA can be a promising tool for estimating parameters of models that are highly nonlinear in the parameters or models whose parameters change rapidly. This is due to the 43  ability of GA to explore "almost" the entire parameter space without actually examining all values. Besides being able to handle non-linear models, black-box models, and local minima, another advantage of GA seems to be that as long as the measurements contain sufficient information, GA can be made to converge almost as rapidly as required. This rapid convergence is not in terms of computation time, rather it is in terms of the number of measurements used to arrive close to the optimum. Also, as Wan [30] points out, the strength of this form of search is that by increasing the length of the chromosome (number of bits in the binary coding) linearly, the computation effort is increased linearly, while the search space itself grows exponentially. To use GA in parameter estimation, some decision has to be made regarding the error function definition. The error function (which is independent of the search mechanism) can be defined by summing the squares of the errors from the predicted and measured outputs (as in the Least Squares method). GA can then be used to search a population of individuals (encoded parameters) to find the individual that minimizes the error function. Multiple parameters can be handled by concatenating the encoded parameters in a long string. On the other hand GA also has many disadvantages. • GA is very computationally intensive. If a given length of historical data needs to be taken into account, GA not only needs to "handle" the entire historical data at each time step, but it also needs to do this "handling" for every single population member. Population size depends on the chromosome length, but even a 20 bit chromosome requires a population of at least about 100. • It is very hard to determine beforehand what conditions should be used for terminating a GA search, since there is no guarantee that the algorithm converges within a certain number of generations or even if what it appears to converge to is correct. • It can only find near optimum solution and it is hard to define how near the optimum solution it can get. This problem could potentially be handled by using GA for global searching and switching to another search method to get the optimum solution. 44  • Many parameters such as the chromosome length, population size, mutation probability, crossover probability, and number of generations affect GA's performance, but are very hard to determine beforehand. • It is difficult to search for multi-parameter strings because they have long defining length and they can be destroyed easily or lead to GA-deceptive problems. The bits can be potentially re-arranged, but does not offer an immediate solution. • The range and resolution of the parameters has to be fixed beforehand due to the requirement for encoding the parameters. Newer methods that use floating point numbers rather than binary encoding are becoming more popular to deal with such problems. • When multiple parameters are required, the chromosome length grows rapidly which means that the population size and computational time grows accordingly. It is possible to attempt to reduce the number of bits required by designating special bits as "scale factors" to scale the range of the other bits so higher bit resolution can be achieved as the parameter search comes closer to the correct value. However, more study is required before a reliable method can be designed.  Thus, GA is not a very easy method to use, and is not a good choice, if there are other alternatives available for solving the problem. However, since GA is a very rapidly developing field, more robust and efficient variations are being developed every day. While the basics of the genetic algorithm is still similar to Holland's schemata representation and genetic operations, various methods for both GAs and PGAs (Parallel GAs) have been proposed by different researchers, each "tuned" to solve a particular problem. Davis et al [6] classify the major areas of research in genetic algorithms into theoretical research, applications, and classifier systems. Through their research many variations on representation schemata, fitness evaluation function, mutation process, cross-over process and determining the number of offsprings of fit individuals have been proposed.  45  3.4 Discussion From the above descriptions of the two methods of RLS and GA it can be seen that they are very different. It is almost impossible to compare the two algorithms since they are inherently suited to different types of problems. For example to use RLS with a nonlinear model, the model will have to first be linearized and in doing so the nature of the problem changes. On the other hand if the problem can be formulated as a linear combination of the parameters, there is very little justification to use GA. However, if a nonlinear estimation problem were to be solved using a more general form of least squares hill-climbing method, a more fair comparison could be made with GA since the fundamental search principles used in RLS would be compared to GA. In this case using a mechanism similar to RLS, at each step the sum of the squares of the errors would be numerically differentiated to find the "down-hill" direction in the parameter space.  In anticipation of continuing Wan's [30] work on using GA in estimation of compliance in an excavator, and to gain a better understanding of GA details, a GA program was developed. The particular implementation used for the GA program is described in Appendix (A.4). To further understand the problems involved in implementing a Parallel GA (PGA) algorithm, a PGA was developed and tested on a variety of configurations on a network of Transputers (microprocessor manufactured by INMOS for parallel processing) of up to 32 nodes. The standard Recursive Least Squares (RLS) algorithm discussed in Appendix (A.l) was used (additional detains in Appendix (A.2)) to estimate parameters of the proposed linear models in various experiments, and in studying the work by Wan [30].  46  4 Compliance and Fault Detection in Hydraulic Systems  Compliance of a hydraulic machine or a mechanical system is a measure of its overall flexibility. The compliance of the system is composed of the compliance of the structure and the compliance of the hydraulic components. According to Sepehri et al [24], the compliance in a hydraulic manipulator is mostly due to its hydraulic system as opposed to the physical structure of the linkages. Thus, in this study, the flexibility of the physical structure is considered to be negligible and the word "compliance" is generally used to describe "hydraulic compliance".  However, the hydraulic compliance is itself composed of the compliance of hydraulic oil, entrained air, flexible hoses and other oil containers. Furthermore, depending on the method of measurement, leakage, or other problems that can cause flow measurement errors, will also appear as a change in compliance. Thus, before compliance can be used in fault detection, some questions should be answered: 1- How can the compliance of a hydraulic system be measured on-line? 2- What should the compliance be in a correctly performing system? 3- What can be determined from the changes in compliance? Answering these questions is not easy since most of the factors affecting compliance calculation can exist simultaneously and change while the machine is operating acceptably. This can be seen from the discussions in Section (2.8) which indicated that there will always be some: 1- leakage in the actuator which changes as the machine operates since leakage depends on: • direction of piston motion • the pressure difference between the two sides of the piston • rod eccentricity of rod which can be affected by the type of operation • possibly the piston position 2- air bubbles in the fluid whose compressibility changes with pressure 47  3- dissolved air in the fluid, evolving and dissolving as pressure changes.  To use compliance for fault detection all these factor and their relative impact on compliance during normal and faulty operation should be fully understood. This requires separating real compliance from the elements such as leakage that only appear as compliance. Considering the definition of compliance V Cfluid' fifluid  ' [17] where V  fluid  is the fluid volume, (3  j]uid  is the fluid bulk modulus and C  fluid  is the  compliance, it can be seen that compliance also depends on the fluid volume. Fluid volume not only changes from system to system, but also changes in a subsystem as the piston moves. Therefore, to have a basis for comparison of stiffness for different systems regardless of their fluid volume, researchers often use the bulk modulus fi. The effective bulk modulus, P is used to describe the stiffness of the fluid e  subsystem, which includes the fluid, entrained air, and the container.  Therefore, to measure and use compliance for fault detection, not only the fluid bulk modulus has to be understood and modeled, but also the "compliance model" has to be expanded to include the actuator geometry as well as all the components such as leakage that affect the measurement of compliance. This chapter attempts to achieve this goal in the following steps: 1- The effective bulk modulus, its relation to compliance and some existing models for it are studied and a model that is believed to be useful for fault detection is developed. 2- The proposed method for determining compliance from flow and pressure measurements is derived which also shows what types of other problems can affect compliance measurements. 3- The inadequacies of using a lumped compliance model, that is, using a single parameter to describe the relationship between flow and pressure measurements is discussed. 4- Based on the findings of this study, a model expected to be suitable for fault detection in mobile hydraulic machines is derived and some possible variations are discussed.  48  4.1 The Effective Bulk Modulus and "Effective" Compliance As previously mentioned, the isothermal tangent bulk modulus as described in Equation (3) is the most suitable definition for describing the incremental changes in the fluid bulk modulus for fault detection in hydraulic machines. According to Merritt [19], the effective bulk modulus of the fluid, air, hoses, and the container is defined in a similar way to the isothermal tangent bulk modulus as  V.bP bV > t  (9) T  where V, is the total fluid volume and SP is the change in pressure that compresses the volume by SV, at v  md  constant temperature T. Using the compliance definition Cjj = uid  , for the effective bulk modulus fi , e  and the total fluid volume V , the effective compliance of the fluid together with air and hoses can be t  _V  then written as  l  C ffective~~r7~ '  0 0)  e  4.1.1 Understanding The Effective Bulk Modulus A good way to understand the effective bulk modulus is to derive a general model for it and study its individual components. The general model is derived here using Equation (9) and based on a method described by Merritt [19]. For a constant temperature and total volume reduction AV,, Equation (9) can VkP R = —— AV, '  be written as  e  Let the effective bulk modulus f3 be defined for the combination of a volume of fluid V and a volume of e  t  gas V in a container of volume V . If the total volume is V = V, + V , then for a pressure increase AP, g  c  t  g  V increases, but V and V, decrease. Thus, as shown in Figure (13) the total change is c  g  K =-|AKJ-|A^ |A^| (  +  49  T y p i c a l l y , the c o n t a i n e r is a c o m b i n a t i o n o f steel p i p e s , actuator a n d f l e x i b l e h o s e s a n d the t o t a l v o l u m e is V, = V = V + V„ , w h e r e V is the hose v o l u m e , a n d V is the v o l u m e o f the m e t a l l i c c o m p o n e n t s . c  h  h  m  A s m e n t i o n e d i n S e c t i o n (2.8.1.1), a l t h o u g h the e n t r a i n e d a i r is d i s t r i b u t e d t h r o u g h o u t the f l u i d as a i r b u b b l e s , it m a y be c o n s i d e r e d as o n e i n d e p e n d e n t v o l u m e o f a i r , V b e i n g s u b j e c t e d to the s a m e  AV,  g  AV  pressure as the f l u i d . T h u s , c o m p r e s s i b i l i t y f o r the t o t a l v o l u m e is  -bV -bV.+bV  1  g  VAP  l  V.bP  c • AV„  Figure 13 V o l u m e  T h e a b o v e r e l a t i o n c a n be a l s o w r i t t e n as  change o f container, l i q u i d and  gas as pressure increases  bV_ v V/ VbP  6V, v  V,i  T}  V.i  bV  V}VbP  vjbP'  t  C  o r u s i n g the d e f i n i t i o n o f b u l k m o d u l u s f o r e a c h c o m p o n e n t , 1/J3 m a y be w r i t t e n as e  P.  VP,  y}f>J  w h e r e f3 , J3 a n d fi are the b u l k m o d u l u s v a l u e s f o r the gas, the l i q u i d , a n d the c o n t a i n e r . F o r accurate g  h  c  r e p r e s e n t a t i o n , the pressure related e x p a n s i o n o f the m e t a l l i c c o m p o n e n t s c a n be c o n s i d e r e d , but i n m o b i l e m a c h i n e s , the p u m p is o n - b o a r d a n d t r a n s m i s s i o n l i n e s are short. T h i s m e a n s that the m e t a l v o l u m e v a r i a t i o n is s m a l l c o m p a r e d t o hose a n d a i r v o l u m e v a r i a t i o n a n d t r a c k i n g it d o e s n o t h e l p fault d e t e c t i o n . T h e r e f o r e , the o n l y p o r t i o n o f c o n t a i n e r b u l k m o d u l u s o f interest to fault d e t e c t i o n is the hose b u l k m o d u l u s , fi . C o n s e q u e n t l y , a s u i t a b l e g e n e r a l e f f e c t i v e b u l k m o d u l u s m o d e l f o r fault d e t e c t i o n is h  p.  — =  hJ  v  w  } + —(—) + —(—) .  50  (11)  The challenge in modeling the effective bulk modulus is that every component in Equation (11) may change depending on the operating conditions of the machine. Therefore, the main difference between various models is how these components are modeled. The following table summarizes the model or values that can be used for various components. Further discussion and explanation of the more complex components follow the table.  Component  Bulk Modulus  Volume  Total  P.=PV  V,:  p. Gas  P: g  w  w  8  m  V =V ±A(x- )  w  Isothermal case: P  V, = V„ + V m  V:  P,:  Xa  Isothermal case:  g  =P  V (t)=P(t-l)V (t-l)/P(t) g  g  Isentropic (adiabatic) case:  Adiabatic case:  p = Py.  V/t) = (P(t-l)V/(t-l) /P(t) ) >*.  8  Fluid  m0  - Pi value is generally provided by oil  V,: - The change in V, can often be ignored  manufacturer  compared to air and container volume change.  - P, depends more on pressure and temperature  - A convenient way to model thefluidvolume  than oil type [32]  is V, = V , - V  g  - Pi increases with pressure and decreases with temperature [32]. - The effect of Pi on compliance is small enough to ignore its pressure and temperature dependence. Hose  P: h  - Manufacturers provide expansion coefficient  V: h  - V depends on each hydraulic circuit. h  - If variable P is assumed V change based on  at working pressure.  h  - Martin[18], derives a practical pressure  h  pressure change can be calculated from  dependent model -2P  P,=(614D-2.18)P (l.ll- '"") l5  max  /  e  v (t) - v «-w+ ~!! - ~ h (ii) m  h  Table 1 Major components effective bulk modulus  51  h  ( t  i* 3  l  Total Volume, V , The total volume of the fluid body being considered at each time step is the sum of the volumes of the hoses V , and the volume of the metallic portion V which is the volume in the piston, metallic pipes and h  m  connectors.  v =v +v t  m  h  Since depending on the piston position the volume of  Valve. fluid being considered changes, the total volume V  t  should be represented as V = V ±A(x-x ) m  m0  ,  0  where A is the piston area, x is the piston position at a given time, and V and x represent total volume of m0  0  metallic parts and piston position at some initial point.  F i  g  u r e  1  4  Volume change due to piston travel  This volume change is shown in Figure (14). As mentioned before, the pressure and temperature dependancy of these volumes are neglected in this study. In most hydraulic machines the components are protected by additional circuitry such as relief valves and check valves. Care must be taken to include the additional volume due to the auxiliary fluid paths for the protective circuitry which is often ignored from a functional point of view.  Entrained A i r (Gas) Volume, V  g  The volume of the entrained air in the fluid can be treated as an unknown parameter that should be estimated on-line. As explained in Section (2.8.1.1), there are many factors that can cause large and rapid changes in the amount of air in the fluid and the portion of the air that is in the form of air bubbles. However, in addition to these factors, the volume of a given amount of entrained air changes due to changes in the temperature and pressure as described by the equation of state for gases, or since air is close to an ideal gas, by the ideal gas law [3], PV= kT. Here P is the gas pressure, Vis the volume, Tis  52  the temperature in Kelvins and A: is a constant. If the compression is isothermal and the temperature remains constant, then PV= constant. If the compression is adiabatic (isentropic) and heat can not escape the system, then it can be shown [31] that for gases behaving close to a perfect gas such as air PV = constant, and y=C/C r  v  is the ratio of specific heats where C is the specific heat at constant p  pressure and C is the specific heat at constant volume for air. The value of }^for air is 1.4 according to v  Merritt [19]. Researchers have conflicting opinions about the adiabatic versus isothermal nature of compression of air in the fluid. Hayward [8] reports of work that suggests the "normal-sized" air bubbles compress isothermally, whereas the work by Jinghong et al [11] suggested that this process is adiabatic. Merritt [19] and Yeaple [33] also consider this process to be adiabatic, but Blackburn [3] and Martin [17] suggest that this is situation dependent. Therefore from the gas law in the isothermal case the volume of air at time t, V (t), can be calculated as g  V (t)=P(t-l)V (t-l)/P(t), g  g  from the pressure at time t and pressure and volume values at time t-1. Similarly, in the isentropic case, V (t) can be calculated from g  v (t) = (P(t-i)v (t-iy/p(t)) g  g  "y.  The Gas (Air) Bulk Modulus P  g  The bulk modulus of the gas can be obtained from the equation of state for gases, which was shown in this section to be PV= kT'm the isothermal case and P V = k in the adiabatic (isentropic) case. The y  constant y was explained above. Then, the gas bulk modulus can be obtained using the definition for bulk modulus B = -  v d p  dV  and differentiating the above relations.  In the isothermal case, B can be found as g  PV = k -> V——--P - B = P dV Hg  53  and in the isentropic case  rlP  tiP  + PyV - =0 - V— = -Py - P = ^Y-  pyy=k - V — y  y  The Hose Bulk Modulus p  (14)  x  h  Hose manufacturers often provide a typical value for hose expansion coefficient at working pressure of the hose which does not reflect the pressure dependancy of the hose bulk modulus. The only research work on this subject found during this study is the paper by Martin [18] in which he experimentally determines an empirical formula for the relationship between hose bulk modulus and pressure. According to this formula  -ip  P,=(614£-2.18)P  ,5 max  (l.ll-e ™ \ P  ( 1 5 )  where D is the internal diameter of the hose in meters, P is the fluid pressure in MPa, and P  max  is the  maximum rated pressure for the hose in MPa. In [17], the same formula appears without the "2.18" which may make more sense since using the specifications for some of the smaller hoses the equation with the "2.18" results in a negative bulk modulus.  4.1.2  Existing Models for Effective B u l k  Modulus  Many researchers have modeled the effective bulk modulus of hydraulic fluids to take into account the air in the fluid and the flexibility of the container. However, this is often done in the context of providing guidelines for the dynamic performance and control requirements for the engineers who design the hydraulic equipment. In these situations, a general estimate of the overall effective bulk modulus of the subsystem, or at best a simple model as a function of pressure is derived to give designers an idea of the stiffness of the system. Examples of this can be seen in work by Wright [32], Rendel et al [21], 54  Hayward [8], Merritt [19], and Martin [17]. Some researchers such as Jinghong et al [11] or Songnian et al [26] have proposed methods of on-line estimation of effective bulk modulus. Jinghong et al [11] in particular present a very complex model. However, even this type of estimation as well as the implicit estimation methods used by Sepehri et al [24] and others who are interested in developing controllers for hydraulic systems are also presented mostly in the context of simulation and modeling. In particular, it is worth noting that in all of the previous work found on experimental determination of effective bulk modulus or air content measurement, the work relied on very accurate measurements and specially constructed equipment.  To use compliance or bulk modulus in fault detection or system diagnosis, these models may not be directly applicable or relevant. This is because when the bulk modulus is estimated for design, modeling and simulation purposes, the objective is to arrive at a good representation of the composite value of all the contributing bulk modulus values under normal operating conditions. Whereas for fault detection, the objective is to decompose the observed composite value to the most likely components under abnormal operating conditions. In the former approach, the model generally relies on using reasonable guesses for parameters such as percentage of air content, or hose elasticity to describe the system during normal operation and as long as the overall estimate meets the accuracy required, some sources of compliance may be ignored. In the latter approach, the model parameters are not chosen as much for overall accuracy as they are chosen for their role in diagnosis and identifiability . Furthermore for fault detection, the measurements have to be taken on industrial equipment operating in the field. This implies that one cannot rely on accurate measurements so models that depend on accuracy may not be applicable to fault detection. Also if the model is developed for fault detection, normal operating conditions may not be assumed. Thus, the inclusion of the sources of compliance in the model depends on their expected role in fault detection. For example in modeling a hydraulic system, many 55  researchers ignore the hose flexibility. This may be reasonable since flexible hoses are designed to be as stiff as possible and are generally used in a very small portion of the fluid transmission system and under normal operating conditions their effect may be negligible. However, as mentioned in Section (2.8.1.3) hoses are a likely source of problem and it is not unreasonable to speculate that as they wear out they may become a more significant source of compliance.  In the previous section a generic model for the effective bulk modulus was derived and some of the options for dealing with the components of this model were discussed. Incorporating all the components discussed leads to a complex and nonlinear model which does not even take gas solution and evolution due to pressure change into account. Here, some of the existing models for effective bulk modulus are studied to learn how other researchers have handled these complexities.  The Modular Approach to Modeling fi  e  Merritt [19] considers one material for the container so that V= V, = V, + V or V, = V,- V and using g  g  Equation (11) arrives at  p,"  y^J  y, V  +  where using P, » P„ to allow  +  P " P C  C  +  P,  +  >-, P " P/ L  G  _!_--]_<=_]_ gives  h P, P  8  1  Pe  1  1  + — + Pc P/  \ ± y y, P«  (.6)  The methods suggested by Merritt [19] for obtaining various components of the above relation are similar to those already discussed. The advantage of this approach is that the modular representation of bulk modulus components makes  56  the contribution of each component clear. Thus, it is easy to choose relevant components to suit different applications. This model also lends itself to practical use since it takes advantage of the fact that in practice, in a hydraulic circuit, the total volume can often be derived from the container volume.  The Analytical Approach to Modeling fi  e  Many researchers use an analytical method to derive the effective bulk modulus. In this approach, they first derive an analytical expression for the total volume and then differentiate it with respect to pressure to obtain a relationship forp =-  , where V is the total volume. This approach leads to very  v d p  e  complicated formulae, despite the fact that these researchers only consider the combined effect of fluid and air. Using this method, in 1951 Rendel et al [21] derived the relation  P. = p, £  • i] / g  £ • £w g  r  where P is the atmospheric pressure. This result was later followed up and corrected by Hayward [8], 0  who in 1961 derived the expression  P = p, (-5!  +  e  Ve  Vl  v  v  11) I  P  (_$!  +  V  g  1°^) , P  2  g  r  which also takes the correct volume expansion for the fluid over time into account by integrating the bulk modulus expression. Further details of these methods is provided in Appendix (B). Despite their complexity, the above relations do not add any useful information for fault detection compared to Merritt's [19]derivation. In hydraulic machines, the total volume of the container may be used to obtain the fluid volume as shown in equation (16) and the complex relation derived by Hayward [8] is not required. However, the method used by Hayward in integrating the fluid volume over time may be applicable in deriving exact hose volume at a given pressure. Hayward [8] asserts that the 57  practical value of this relation is doubtful since the solution and evolution of air bubbles which takes place very rapidly is not taken into account. Thus, it is worth studying a model that attempts to take the air solution problem into account.  An Attempt to Model Air Solution The most comprehensive effective bulk modulus model for the air-fluid interaction found was by Jinghong et al [11]. The model includes a coefficient to estimate the portion of air going in and out of solution. They argue that in the dynamic analysis of hydraulic systems there is not enough time for heat dissipation and the isentropic tangent bulk modulus should be used. Assuming an inflexible container, isentropic gas compression, and the relation -c, V P for gas solution, where c, is "coefficient of air g  bubble volume variation due to change in the ratio of entrained to dissolved air" they arrive at B,(l+l(r / ) 5  p = 8  >  ,+,/Y  (17)  L£  (1+10- P) 5  1+,/Y  +10' R( 1 -c,/>)(P/Y -10 -P) 5  5  where, R (entrained air content by volume at atmospheric pressure %), y(ratio of specific heats for air), and P, (bulk modulus of air free oil) are parameters to be estimated. The parameters are estimated by minimizing the error of P estimates compared to the measurements of the bulk modulus using sonic e  bulk modulus of the fluid. Instead of sound waves Jinghong et al [11] use a pressure wave and using two pressure transducers at two meters apart they calculate the wave propagation speed in the fluid. More detail on deriving Equation (17) is provided in Appendix (B). Since this model is highly nonlinear, Jinghong et al [11] report that the estimation method was carefully selected to avoid differentiating the objective function. By assuming that the change in the volume of the entrained air due to pressure change can be described as -c  2  P, where c is an air bubble volume 2  variation coefficient, they derived a simpler linear model. They then compare the performance of the original model with this simplified model. The model is further simplified for further performance 58  comparison. In the extreme case they use the relation P = c  Jinghong et al [11] show how simplifications in their model makes estimating the parameters easier, but lowers the performance of the model progressively. The most noticeable performance degradation is for the case where P = p, is assumed. It is worth noting e  that in the graphical representation of their results  1700 -  shown in Figure (15), none of the models matches the  1650 -  if  1600 -  measurements too well at the lower pressures of up to about lOMPa. The average error of their models is low due to the wide pressure range (0 - 30MPa ) used  15501500 •  portion of the air bubbles is likely to dissolve in the  — — „  ~s a. 2  <  * i  5  in these experiments. At such high pressures a large  *  1  0  Results: Equation (17) — — 1st Simplification  X  "i 10  Measured  • 20  • 30 Pressure (MPa)  0, models and experimental results  Figure 15 Results from Jinghong et al [11] (reprinted with legend modification from [11])  fluid, which reduces the effect of air. An exception  to this is the first approximation to their full model where they represented the change in the gas volume by -c V P. In that case, not only is there not a good match at low pressures, but as the pressure 2  g0  increases their estimated bulk modulus also diverges from their measured values. In fact, it is shown in Appendix (B) that the simplification of letting the change in the gas volume due to pressure change be -c  2  VgQ P is not a very good approximation.  While this is a very interesting model because it attempts to take into account the air solution in the fluid, the results are not convincing enough to be used in this study. A complex model with four parameters is fitted to the data to minimize the overall error, but no attempt is made to independently verify the values of any of the parameters. As a minimum, the value obtained for p,, could have been compared with the value provided by the manufacturer. From Figure (15), it seems that the deviations of all proposed models (except the model assuming constant P) from the experimental value is more than their e  59  deviations from each other. In general, the complexity of the bulk modulus models studied appears to be due to attempts to accurately model the fluid volume or the air solution. Two important points should be noted here. 1- Since the volume of fluid can be determined from the difference between container and gas volumes, it is possible to avoid modeling the fluid volume. 2- When compliance is measured dynamically in a circuit as will be described in Section (4.2), the volume of the section being studied is essentially fixed, but the volume of gas is reduced due to compression and solution as the pressure is increased. This is possible since additional fluid can enter the system. Thus as the pressure increases, the relative volume of gas V /V is reduced and as it can be g  t  seen from Equation (11), the contribution of the gas bulk modulus to the effective bulk modulus is also reduced. Thus if compliance is to be estimated in a pressurized system, the entrained air volume may be so small that as far as fault detection is concerned, the effect of the portion that goes in and out of solution may be negligible compared to the compression.  4.1.3 Proposed New Model for Effective Bulk Modulus Having studied the major elements contributing to effective bulk modulus and some of the existing models for it, the goal here is to obtain an expression for f3 that is practical and useful for fault detection. e  Using the simplification used by Merritt [19] discussed in Section (4.1.2 ) which assumes that — - — " — the relation  P P/ P g  g  or  may be derived in which  V,  =  V  0  ± A ( X - X Q )  +  V  H  as shown in Table (1). In Equation (18) the only 60  unknowns to be estimated are V and, if desired, fi . If the hose is not expected to be a source of problem, g  h  the manufacturers' value for J3 can be used. On the other hand, if it is expected that an aging hose will h  lose its stiffness and lead to problems over time, it may be desirable to treat P„ as a parameter to be monitored for fault detection. Alternatively, the relation derived by Martin [18] (Equation (15)) discussed in Section (4.1.1) for the hose bulk modulus, which was -2P  P K614£-2.18)P J ( U 1 ~ ^ 5  A  m  max  ),  may be used to derive a parametric pressure dependent model for hose bulk modulus as P^Lll-e^), where one or both of H, = (614D-2.18)P J  5  m  (19)  and H = 2/P 2  max  can be treated as parameters to be estimated  to track changes in hose elasticity. Here it is proposed that to simplify the model, the theoretical P or h  the estimated P from the previous step can be used to iteratively calculate hose volume V at a given n  h  time using Equation (11) or  In Equation (18), the isothermal or isentrophic relationship for air compression can be used by selecting either P or Py for the air bulk modulus P , as shown in Equations (13) and (14), which leads to g  p,  P, p  ]  (  '  or 1  1  _L = _L  ( y V, V _* _i _8| h  +  +  ( 2 2  )  for the isothermal (used here) and adiabatic (isentropic) conditions respectively. 61  Note that if there is enough evidence that air solution due to pressure can be represented as a coefficient proportional to air volume and pressure, the effective bulk modulus in Equation (17) derived by Jinghong et al [11] can be used to represent the combined effect of air and fluid. The effect of hose bulk modulus can then be added to get the effective bulk modulus. However, that would lead to a highly complex and nonlinear model. Since it is not clear how effective the Jinghong et al [11] model is in describing the amount of gas dissolving, it may be just as effective to use a parameter similar to bulk modulus, fi , to capture the effect as dV = V dP/p' leading to g  g  g  g  However, without further study to determine the correct method for modeling air solution and evolution, it is hard to justify the additional cost of adding another parameter to an already complex model.  4.2 Calculating Compliance and j8 From Flow and Pressure Measurements e  To select a method of measuring the compliance or bulk modulus in a working hydraulic machine, one needs to consider the available sensors, their effectiveness, their price and maintenance requirements. Pressure and position transduces are widely available and inexpensive, and it would be highly desirable to measure the compliance of the system using such simple sensors. Using the valve characteristics provided by manufacturer, it is generally possible to get the orifice size of the valve based on its spool position. The flow through the valve can then be measured using Equation (4) mention in Section (2.5). Thus, if compliance can be measured from flow and pressure measurements only, then position and pressure transduces will be sufficient for this task.  Compliance or bulk modulus in a hydraulic subsystem can be determined from flow and pressure measurements as follows. Consider a container of fixed volume of fluid V. If fluid is allowed to flow 62  into the container at the rate Q and under the pressure P increasing at the rate of AP per unit time unit, then the volume of additional fluid that "fitted" into the fixed volume Fin the unit time is Q. This means the volume of the fluid originally contained in the volume Kmust have been reduced by AV = Q in the VdP  unit of time. Then from the definition B =  in Equation (9), for each time unit dt we have dV  3 -  ydPldt _ P_ dVldt Q V  and from Equation (10) we get Ceffective ^  '  (23)  Since the expansion of the container and compression of air can be included in the effective bulk modulus, here only the fluid volume change is considered to arrive at C  effective  .  If the volume being considered is a hydraulic subsystem such as the fluid between the control to t = t + dt ! dx ! 0  valve and the piston head as shown in Figure (16), then the flow into the "container" is the net flow  AQ = Q - A dx/dt  1  into this subsystem at a given time. If the piston head with area^4 is moving at speed x, the amount dx/dt  of volume by which this movement enlarges the  Figure 16 Net flow into the linear actuator  subsystem volume per unit time is Ax. This  volume of fluid, leaves the original volume and represents the flow out of the system. Thus, the net flow into the subsystem is AQ = Q-Ax. From C  =^f-,  effeclive  where Q„ r^Q e  l s  the net flow into the system,  the relation for measuring compliance from flow and pressure can be derived as  63  •  C J=Q-Ax effect  (24)  However, if the leaks in the system are not estimated or measured, the value calculated for C ff e  be inaccurate. Furthermore, according to Martin [17], measuring C  effeclivi:  eclive  will  from Q and P is only valid if  the fluid is introduced slowly into the container. If the fluid flows into the system rapidly, Martin states that the inertia effects or fluid inertance cannot be ignored. Fluid inertance is the equivalent of electrical inductance, just as fluid compliance is the equivalent of electrical capacitance. The mathematical relationship for inertance is AP = L dQ/dt where L = Ip/a is inertance, p is fluid density, / is pipe h  h  length, a is the pipe area, Q is the flow, and AP is the pressure drop across the pipe. According to Merritt [19], when the fluid flow in a pipe is turbulent rather than laminar (mostly governed by viscosity), fluid inertia which leads to pressure variations in the fluid path has to be taken into account.  4.3 Interpreting The "Lumped" Compliance Value If Equation (24) from Section (4.2) is used to calculate or estimate C  ,, the value obtained will be a  effecuvl  "lumped" value for compliance in which effects of circuit compliance and leakage are combined. The discussion in Section (4.1.1) demonstrated how the value obtained for the effective bulk modulus VAP  P =—— incorporates fluid compressibility, entrained air and hose flexibility (Equation(l 1)). It was seen e  that even when measured correctly, f3 is hard to interpret since its components may all change with e  pressure. The discussion in Section (4.2) demonstrated that the hydraulic compliance may be determined from flow and pressure measurements. However, as discussed in Section (2.8.2) if flow measurements are used for compliance calculations, problems such as leaks and valve malfunction are also "lumped" as compliance. Thus, Equation (24) should be considered to be a "lumped compliance model"  (25)  C J-Q-Ax lump  Wan [30] argued that the error caused by lumping the leakage in compliance is at least proportional to 64  the amount of leakage and thus monitoring the compliance is a good way of detecting leakage. However, as it was seen in Section (2.8.2.2) the amount of leak in a linear actuator given by the equations Q =—— dra  and Q u =— s  P  —(1+L5e ) 2  is dependent on the pressure difference between the two sides of the piston, the eccentricity of the piston, the clearance between the piston and cylinder wall, the fluid viscosity, and the velocity of the piston. Thus the magnitude and sign of the actuator leak is highly dependent on the machine operation and can change rapidly during the operation. If the leakage value is to be lumped in flow measurement, and even if the leakage is only between the cylinder and piston, the resulting limped compliance value will be very difficult to interpret.  Furthermore, since the effective bulk modulus is already a complex and hard to interpret value, lumping a few sources of leakage in compliance measurement makes the measured value even harder to interpret and perhaps practically meaningless. This is because the combined values which sometimes have the same sign and sometimes not, will be highly pressure and operation dependent and without a clear trend.  Another component that should be considered when trying to understand a "lumped" measurement of compliance is the fluid volume. If the relative size of the cylinder is large compared to the volume of the rest of the fluid between the main valve and the actuator, then the position of the piston will have a significant effect on the fluid volume which has an effect on the magnitude of the compliance.  Volatile as it is, the range the lumped compliance value may be reasonably constant for a given machine. Thus, if one of the elements contributing to this value is substantially higher than normal, it is possible that measuring compliance may give an indication that "something is wrong" without necessarily helping in determining the source of the problem. Though if this lumped value is to be estimated it is important to remember that it is very likely to be a volatile parameter which may even change as rapidly as the 65  dynamic variables that one would like to use in its estimation.  4.4 Developing Compliance Models Suitable for Fault Detection The discussion in Section (4.3) showed that a lumped model for compliance estimation may be far too simplistic to be useful to fault detection. In the following discussion a new model is developed that instead of trying to fit a single parameter to the flow and pressure measurements of Equation (24), tries to take into account all the significant elements that change flow and pressure. The advantage of Ax  developing such a model is that it can be designed to have  P, = Q, - Ax ± Q, - Q',  meaningful parameters which change only if specific fault  Figure 17 Subsystem to be modeled related properties of the system changes. The subsystem of interest between the valve and the piston of the actuator, is shown in Figure (17). It was seen in Section (4.2) that the effective compliance of a body of fluid in a hydraulic system, including the contribution of the entrained air and flexible hoses, can be found from the net flow into, and pressure change of that system. The relation for compliance was described by Equation (24) or  C  , P = Q - Ax. effective ff  However, if the system has leakage, the net flow into the system is also affected by the leakage into and out of the system being studied. Figure (17) shows the types of leaks that are being considered in this study. Using the above equation and adding the leakage the general relationship  Ceffective^i  Qi  ±  Qslip^  Qdrag  Ql  66  (26)  can be obtained. In equation (26) only the pressure P, net flow Q, piston position x, and the area A are known from the measurements and all other components have to be found from approximate models with one or more unknown parameters. Q and Q slip  drag  can be found from Equations (8) and (6) given in  Section (2.8.2.2), and Q' a leak to the outside can be modeled as turbulent or laminar flow through a h  hole (here assumed to be similar to a valve orifice) [19].  If compliance is estimated from Equation (26), the effects of leakage is taken out of the estimate and Ceffeaive  w  d l truly represent the compliance of the system being studied at a given time. However, as  previously discussed, dependence of compliance on the fluid volume which changes due to piston movement means that comparing C ff e  eclive  values at different times is difficult. Thus, a better value to  track may be P„ which can be found from  C ^e  =  ecljve  V  t  /P  e  (defined in Equation (10)), where  V  t  is the  volume of the system at a given time. Therefore Equation (26) can be written as  ^  P, = Q, -  Ax  ±Q±  Q -  dl  drag  While P is a more useful parameter than C jf e  e  eclive  Q, .  (27)  for tracking the changes in the system stiffness, it is still  hard to interpret since it lumps the effects of fluid, air, and hose together and changes with pressure. Thus, to develop a model with meaningful parameters a parametric model for P such as Equation (21) e  or (22) should be used. This leads to a complex model with several unknown parameters, which are unlikely to be uniquely estimated from the available information. Therefore, the model components are analyzed to find a reasonable simplification of each component that can be used in fault detection.  4.4.1 Leakage Across the Piston In the event that the seals between the piston and the cylinder are broken, the values Q and Q slip  drag  to  describe the leakage between the piston and the cylinder as shown in Figure (17) should be determined. bhv  mm  The drag leakage Q =—^—^ , (from Equation (6)) may be written as drag  67  (28)  Q = i* K  drag  where K, is a constant. Based on the values shown in Figure (17), the slip leakage 0  shp  = ^ ^ l 1 . 5 e 12u / (  2  +  )  (from Equation (8)) can be written as Q ,  I  P  = ^ ( P  (29)  - P )  0  where K is a parameter that reflects changes in the viscosity u and the eccentricity e. The assumption of 2  slowly changing eccentricity may not be valid, since the load and position of the actuator are likely to change the rod eccentricity. However, trying to model this is very difficult and out of the scope of this study so it is lumped in K for convenience. If the rod side of the cylinder is being studied, the rod leak 2  can also be modeled similarly using P and P 0  as the relevant pressures for the  a l m o s p h e r i c  Q  s l j p  .  4.4.2 Leakage to Outside Assuming that the leakage to outside is due to a puncture in the hose, or a hole, the leakage can be approximated by a turbulent flow through an orifice as presented in [19]. This leads to a similar relation to that used for modeling the flow through a valve orifice. This leads to Ql  where a  kak  C a \J-(Pi  ~  d  ~  !eak  P  atmospheric)  e  r  is the area of the hole in the hose, or Q,  -  ^ ( P - P  a  l  m  o  s  p  h  i  c  (30)  )  for a parameter K to be identified experimentally. If the hole is very small, or if the flow goes through 3  two very close surfaces such as Qi_  Rod  in Figure(17), the contact between the rod and the cylinder, this  type of leakage may have the laminar flow characteristic which may be better approximated by  Q i( =K  P  i~  P  atmospheric) •  68  ( ) 31  4.4.3 The Proposed Parametric Compliance Model Based on the general equation for effective compliance (or Equation (26)) C „ , P = Q  z^i  effective i  - Ax ± Q, ± Q,  and equation (10) which relates C ff e  '  -  Q. *Z /  and /3 , one can arrive at  ecllve  <  -  *-<shp *^drag  e  •  V  ^ effective? i ~ ~^ i  ~ Qi  P  ^  ±  Q lip  Qdrag  ±  S  QI •  Based on the previous discussion on leakage, including all the leakage components from Equations (28), (29), (30), and (31) leads to  C „ ,. P. = —P  effective I  = Q  p l  - Ax + K.x+ KJP  -P)-  *w  2 o l'  I  Kj(P-P  , )+K'(P-P  3y l  V  v  , ) . (32)  aim' J /  atm'  v  \^^J  Combining the model for fi described by Equation (21) and the definition of effective compliance C e  effec  from Equation (10) leads to the relation  effective  K  v.  ^  ^  Vt,  v, v  v. v, v  ^  B„  p  B,  />  ^  and from Equation (32)  v v  V  C  P = effective i  (—+—+—^)P V„  „  >  ?f  = Q, - Ax  P  +  _ I  '  Kx t  ,  (34)  r  +  K (P -P)2  K ^PYPZ) Ki(P-P J  o  3  +  a  ,  where V, = V ±A(x- ) 0  Xo +  and  V  h(l)  V^V^l  +  ^-Vll)  can be calculated at each step. Since the gas volume V is pressure dependent, Boyle's law and the g  69  relation V = V P JP;, where P g  g0  a  is the atmospheric pressure can be used to find the equivalent gas  mm  volume at atmospheric pressure, V . After rearranging, Equation (34) can be written as g0  (35)  V - Ax -P.— =  Q  'P,  -K x  - K (P -P)  x  2  o  K  +  3  J ( P ^ J  Ki(P P J  +  r  + -±P V  a  t  -Z^LP  V  +  Ph  in which K,, K , K , 2  p  P  , V and p are independent parameters to be estimated from a linear model.  3  g0  If the relation P =//j(l.l 1-e A  h  2  ) from Equation (19) is used, the pressure dependance of hose bulk  modulus can be expressed explicitly and the model parameters K K , K , H,, H , h  2  3  and V will all be  2  g0  pressure independent. However, the model becomes nonlinear in the parameters.  Since the hose volume V , is dependent on the parameter fi , it is possible that theoretical or intermediate h  h  estimates offi may not produce correct results in the hose volume V . Especially, if due to aging the h  h  correct p deviates from its theoretical value, this may result in an incorrect estimate for fl . Considering h  h  that the range forfl '\sabout 10,000 to 50,000 psi and during the machine operation pressure changes of h  about 1000 psi are common, the expected volume change due to pressure could be as high as 10%. Therefore, it may be desirable to rewrite Equation (35) in terms of the original hose volume V , and the h0  parameter fi , instead of an iteratively calculated V . Then using the relations h  h  V = V + V and t  m  V =  h  V (\ ^^),  h  h0  +  Equation (35) becomes  Q, -  (36)  V +V  Ax -P  V  i  m  V h 0  V  P /  -K  LX  -  K (P -P) 2  o  +  K J(P7PZ) 3  * Ki(P-P j a  +  P  Ph  70  P  fa+lj^P, Pi  V +  v  ^  P p  P  r P a J  p A  V P  j  +  lZsap  i  P,  which is nonlinear in the parameters and requires using more complex parameter estimation methods.  5 Experimental Estimation of Compliance  5.1 Continuing Previous Work in Estimating Compliance in an Excavator This study essentially continues the work done by Wan [30], who used Genetic Algorithms (GA) for identification of compliance in a hydraulic excavator. It is important to note that while Wan attempted to demonstrate the potential of GA for a practical application in estimating the compliance of the excavator, the focus of his work was to experiment in the application of GA to nonlinear system identification. In this study, the focus of the work is the online estimation of compliance. Therefore in studying the previous work done by Wan, his results are mainly scrutinized from a compliance evaluation point of view and not as a parameter identification technique.  5.1.1 Methods and Results of the Previous Work The machine used in Wan's tests was a CAT 215B excavator as described in Section (2.9). Figure (18) shows the inter-connection circuit diagram of the main hydraulic valves in the excavator. As mentioned in Section (2.9) the cross-over network complicates the task of determining the internal flows and pressures in the circuit. To simplify the problem, only one of the special cases where the cross-over network in the excavator is not active was used. Branch 1, which contains the main valves for the Boom and Bucket actuators, was considered in Wan's study. The active portion of the circuit which was considered in Wan's study is shown as the shaded portion in Figure (18) and corresponds to the condition that P > P and P > P . Under these conditions the two cases of Boom only activation and l2  23  22  l3  simultaneous Boom and Bucket activation were studied, for the case where head side of the actuator is connected to the pump, and the Boom is being moved upwards.  71  In Wan's study, flow and pressure measurements were used with the "lumped" compliance model described by Equation (25) in Section (4.3). Pumpl (Branch 1)  Symbols:  Pump2 (Branch 2)  Q  Q =Flow P = Pressure a = Orifice Area  Q  B O =Boom B U = Bucket ST = Stick SW - Swing  Subscripts: i = input e = output m - main  Figure 18 Interconnection circuit diagram for the valves of the CAT 215B excavator  STICK  For the case where only the Boom is active without crossover flow, a simplified diagram for the Boom is shown in Figure (19). The equations describing the flow and pressure relationships were written as  Q ,  =  A , X  Q  +  P  I  C  i  = Ka  =  Ka  I  Y  JP  Z  P  I  JP~T  (37)  (38) Tank  Q  = A X-P C = Ka fP~-P  Pump  Figure 19 Simplified Boom circuit diagram  (39) 72  Q  =  (40)  Qi Qe +  where, P„ P and P are pressure into and out of the actuator and exit pressure at the tank (measured), Q m  e  is the total pump flow (a known constant), X is the piston speed (calculated from the Boom angle measurements and Boom geometry), a„ a , and a , are the orifice areas for the flow entering and leaving eo  0  the actuator and exit flow to the tank (calculated from the spool position measurements), K is the discharge coefficient for the vales, Q,, Q , and Q are the flow out of the above actuators (unknown), C, 0  e  and C are the compliance values of the subsystems between the main valve and the two sides of the 0  actuator (to be estimated), and P is the pump pressure (measured). However, Wan reported that the pump pressure was "too noisy" to be used assumed that the pump pressure is also to be estimated. This assumption drastically changes the nature of the problem since if the pump pressure measurements are used the compliance estimation problem becomes a simple linear estimation problem and the use of GA for parameter estimation is very difficult to justify. In this case the flow Q, can be simply calculated from the second part of equation (37) and P can be numerically calculated, and the compliance C, can t  be determined simply from the first part of Equation (37) which gives C. = (Q-AfflP,  .  On the other hand, if both the compliance and pump pressure are unknown, the problem becomes more complex and nonlinear and using methods such as GA where the error function does not need to be differentiated will simplify the estimation task. Using a population of randomly generated compliance values C,', Wan arrives at the estimate (41)  Q. = A^PC\  for each population member. Based on this estimate, he then calculates a value for the pump pressure as Qt  P - (  p— ) + P. and exit flow as Q = KaJP-P which are then used to determine the error 2  Ka,  K  function as  '  e  eV  E = \ Q-\Q,\~\Q \ I • The error function is then used to determine the fitness of each e  window  population member using the relation fitness(C) = 73  {B-E ) where B is a bias value and window is 2  the number of previous data points values used in the estimation. Wan used data collected from the excavator to estimate a constant value for compliance for the case where only the Boom is active and the Boom pressure increases from about 1000 to 2000 psi. However, no information that might indicate the actual or expected value of the compliance such as the volume of the fluid, the length or bulk modulus of the flexible hoses was provided as a basis for comparison.  Since compliance is the ratio of volume over the bulk modulus, the knowledge of the fluid volume would at least allow calculating the effective bulk modulus which is a common quantity for comparison with other circuits or the same circuit under different conditions. To verify the compliance estimation, the estimated value was fed to a simulator which produced a corresponding Boom pressure which was then compared with the measured pressure. No information regarding the simulator is provided in [30], but from some of the graphs it appears that the simulation assumes constant compliance and no leakage in each of the actuator circuits. Furthermore, it is not clear if the data used to estimate the compliance and the data used for the simulator were significantly different, but comparison of the only two data files recovered from Wan's experiments shows very close pressure profdes. Thus, it appears that the test to verify the compliance does not actually verify the value obtained for compliance, but at best evaluates the coherence between the simulation and estimation programs that are both designed based on the assumption of constant compliance. One truly independent test that could have been used in verification of his estimate would have been a simple comparison between his estimated pump pressure 2  Q P = (—'-) + P and the actual pump pressure measurements. Even if the pump pressure is believed to Ka.  be too "noisy" to use, it could have been used for comparison.  To estimate two compliance values in the case where both the Boom and the Bucket are active, data generated from a simulator was used. As in the single compliance estimation, this exercise is also not a way of estimating the actual machine compliance but a method for verifying the particular algorithm 74  used in the GA program with a simulator.  Based on the above tests, [30] concluded that GA was a promising tool for estimation of compliance in a hydraulic excavator and the main obstacle to overcome was the slowness of GA as a real time estimation tool. The fact that Genetic Algorithms are naturally easy to parallelize was used to implement a parallel genetic algorithm on a multi-Transputer system that achieved an impressive 12 fold speedup over the serial GA. The behavior and searching properties of a binary encoded GA were discussed using a hypercube for geometrical representation.  5.1.2 Using Recursive Least Squares in Estimating Compliance in the Excavator Since the focus of this study is the online estimation of compliance for fault diagnosis, the first question to be asked in following up Wan's work is if the use of GA and a parallel processing environment is necessary, practical or economical for an on-line diagnostic system. As shown in the previous section, the inclusion of pump pressure measurements makes the problem significantly simpler from a parameter estimation point of view. Analyzing the frequency spectrum of the pump data indicated that the noise had a strong 440Hz component which is probably due to the pump's discrete fluid displacement rather than random noise. Furthermore, for a pressure increase from 0 to 2000 psi the amplitude of the ripple was only about 30 psi which is not very high. It seems unreasonable that rather than filtering the noise or unwanted ripple such important data should be discarded, especially when no consideration was given to the effect of noise on numerical differentiation of the data for Boom angle and the pressure into the cylinder, P„ which are much more sensitive to noise. The effects of noise on P. is particularly important since regardless of the estimation method used the compliance value is multiplied with (scaled by) this value. Thus, as a first step towards determining if using GA is necessary it was decided to use the pump data and to attempt to use Recursive Least Squares (RLS) to estimate compliance and compare the results to 75  those obtained by Wan. To ensure that similar problems were being compared, data from Wan's programs and files were used. These data corresponded to the case where only the Boom is active. The various equations and thresholds for the Boom valve were not independently verified. The algorithm and thresholds used by Wan to determine the orifice are of valve ports was duplicated for use in this study.  Since in this situation only one parameter was to be estimated and it was possible to solve for it directly using known relationships, it was decided to use direct calculation first as a basis for comparison. Using the Equations (37) to (40) and the relation C\ = {Q-AX)IP , the compliance value was directly j  calculated for every time step. Then this value was estimated using the RLS algorithm described in Appendices (A.l and A.2) using a variety of previous data points and forgetting factors. All results indicated that the compliance value was highly variable. While the direct calculation and the estimated values agreed, no agreement was observed with the results reported in [30].  This discrepancy prompted a closer scrutiny of the data collected by Wan. As an independent test the  -  Q,  2  pump pressure estimated by Wan using the relation P = (—-) + P was compared with the measured Ka. pump pressure. These two values were also not in agreement. In fact, the value Wan estimated for pump pressure diverges from the actual pump pressure while the compliance from which this pressure was derived is supposed to converge to the "true" value. This can be seen in Figures (20 a and b), where the shadowed portions indicate regions where the algorithm is not active. As can be seen in Figure (20b), the estimated pump pressure is originally close to the actual pump pressure, but over time, it approaches the Boom pressure. Since the difference between the pump and Boom pressures determines the flow into the Boom actuator, this would suggest that the algorithm is essentially "cutting off the estimated flow into the actuator to compensate for another source of error which is increasing over time. Considering the small scale of the transient flow due to the estimated compliance (maximum of about 15 in /s using 3  values from Wan's study) relative to the flow into the Boom calculated from the measured pressure, this 76  error cannot be due to any errors in Wan's estimate of compliance. To further explore the discrepancy using (a) Compliance  the same constants, threshold values, and assumptions as [30] (with the exception of using the pump pressure), the flow into the actuator (represented by QBOi in (b) Boom and Pump Pressures  Figure (18) and Q in Figure (19)) was  2500  t  2000  calculated in two different ways based on  31500 |  the equations  1000  t V i ~ M ec P ' K I ' J I j MeaF..-pdP11!Pu.-np)  500  fin =  Vea-  toiJFT,  0.2  and  0.4  0.6  0.8 1 1.2 Time (seconds)  red PuX> U un)  1.4  1.6  1.8  (c) Flow into the Boom Actuator  Q = Q-Q = Q-Ka JFT a  e  e  .  e  These two values were also not in agreement as seen in Figure (20c), but it is interesting to note that Wan's estimate for Figure 20 Results from study by Wan [30] this flow value {Q"\r\ Equation (41)), also (a) Estimated compliance (b) Estimated pump pressure compared to measured shown in Figure (20c) starts as being close actuator and pump pressures (c) Estimated actuator flow compared to Q and Q to Qu and then approaches Q . It should u  i2  i2  also be noted that both of the above equations for flow are implicitly used by Wan in Equations (37), (38), and (40) to derive all the estimations. The valve characteristic equations were not independently verified in this study, but no other errors were found in flow equations used in [30] or their duplication in this study.  The above discrepancies, indicated that there was an inherent inconsistency between the Equations (37) to (40) and the data. One explanation could be that the valve directing flow to the bucket may not have 77  been fully closed as assumed in the study. This would imply that not only Equation (40) was not correct, but the pump pressure was also not equal to P, as seen in Figure (18). Another possible explanation 2  could be that either the cross over valve from the stick to branch 1 or that from the boom to branch 2 may not be fully closed. Sepehri [25] points out the search for solution between the alternative circuits is done by examining all alternatives to identify the case where the pressure distribution satisfies a particular configuration. However, there is no indication that this is done in [30]. Furthermore Sepehri points out that if the Boom-Up motion (which is studied in [30]) is speeded up, the configuration with cross-over flow from Boom to branch 2 is obtained. If either of these cross-over circuits are active the flow equations used in [30] are incorrect and the discrepancy in the results is explained.  5.1.3  Discussion  Before closely studying Wan's results, his conclusions regarding the requirements for compliance estimation were taken as facts to build on. Therefore, in anticipation of extending Wan's [30] achievements in developing a Parallel Genetic Algorithm (PGA) for compliance identification, a GA and PGA program were developed and tested on a Transputer network. The Transputer system used by Wan was also studied. However, the discrepancies observed in the results reported in [30], indicated that the portion of the results regarding compliance estimation does not provide a firm foundation to build on.  It was already observed that the overly complex nonlinear model used in [30] was not required. Furthermore, the literature review and development of various models to express compliance discussed already, showed that the compliance was not a simple constant value. This indicated that many basic questions about the validity of the models being used needed to be answered before focusing on developing complex parameter estimation methods. To investigate the discrepancies observed in results from [30] further, the large number of constants, threshold values and assumptions regarding the excavator had to be verified. However, given the complexity of the excavator circuitry, and the fact that 78  at the time the machine was not actively used and no other researchers were working on this machine, some of the required information could not be obtained in a timely manner. While previous researchers had already used and developed simulators for this system, using a simulator for this problem was deemed unwise since a basic study of a real system was necessary to answer several questions. This study was required since it was not even established that flow and pressure measurements could successfully lead to compliance estimation. The type of model suitable to compliance estimation for fault detection was also not known. Furthermore, it appeared that in [30], the use of a simulator could have been the reason that the discrepancies the results were not obvious.  Therefore, it was decided to continue the research on an available hydraulic test-bed. This test-bed was fully instrumented and considerably simpler than the excavator with fewer "unknowns". Furthermore, the test-bed was being actively used so that all the relevant information was available. While using the test-bed would not resolve the discrepancies regarding the excavator data observed here, it would allow for a much needed basic study of compliance estimation.  79  5.2 Estimation of Compliance in an Experimental Hydraulic Test-bed Since no work that actually estimates and interprets compliance on a hydraulic machine has been found, it appears that many of the questions that should be answered to do this task successfully, have not been answered or even posed yet. Based on the knowledge and experience gathered up to this point in the study, some fundamental questions were posed. Experiments were then proposed to attempt to answer these questions. Considering that the experience with the excavator showed that the hydraulic systems can be very complex with many unknown parameters, a hydraulic test-bed was used for these experiments to limit the number of unknown parameters. This test-bed was designed to perform experiments for another project. It was expected that this system would serve as an example of a hydraulic machine in which the operating conditions could be controlled for experimentation.  5.2.1 Fundamental Questions to Be Answered The following discussion describes the questions that were posed regarding on-line estimation of compliance with some explanation of why they were a source of concern.  Question 1: Can pressure and flow measurements be used to estimate compliance? One of the most fundamental questions was if, in a dynamic system, pressure and flow measurements could lead to the correct compliance calculation. As mentioned in Section (4.2), according to Martin [17], measuring compliance from flow and pressure measurements is only valid if the fluid is introduced "slowly" into the container and the inertia of the fluid can be ignored. Therefore, before using flow and pressure to calculate compliance of a system, the system should be studied to determine if the fluid inertia has negligible effect on the measurements.  Question 2: How to ensure meaningful results and a robust on-line algorithm? Before building a robust fault detection system, the following questions should be answered so that the 80  :em can reliably process the data on-line. Question 2.1: How can usable data portions be found systematically? While a slow fluid introduction is desired to avoid inertia, it is likely to result in slow pressure change, which makes the calculations difficult. This is because using the relation C ~ . = — P  derived in Section (4.2) requires using the pressure derivative in the denominator and when the derivative is too low, the compliance cannot be determined. Wan [30] used trial and error to find a reasonable lower bound for dP/dt, but for building a reliable fault detection system this should be determined systematically. Question 2.2: How should noise be removed? To obtain a meaningful pressure derivative, the noise and ripples in the pressure data should be removed without losing relevant information. A lowpass digital filter is the common tool for removing noise, but some investigation is required to select a suitable cut-off frequency for the filter. The phase distortion caused by a filter was also a source of concern since compliance is dependent on the relationship between the various measurements at each point in time. Question 2.3: Can the use of pressure derivative be avoided? Since low pressure derivatives lead to numerical instability an alternative that avoids using the derivative was suggested. Considering the relation (Equation(24))  C  derived in  P=Q-Ax  effecljve  Section (4.2), by summing both sides the relation over the time period Vt I  'o or  C  (P P,)  eJfeclive  r  = t  U  I  'o  QM - A(x,-x )  (43)  t  can be obtained. This relationship can be used to obtain an average value for C  effective  for a given  number of time steps if compliance is assumed to be time independent over this period. The only other research work found that used this method is the work by Songnian et al [26].  81  Question 3: What is a suitable model for compliance? Assuming that the flow and pressure data can be used to calculate the compliance, there were three questions to be posed and answered regarding the model to be used for compliance calculation: Question 3.1: Can a lumped parameter model for compliance provide useful information? Question 3.2: Can all or some of the parameters of the multi-parameter model be identified? Question 3.3: Can the results be sufficiently improved by using nonlinear models to justify the effort required for non-linear parameter estimation?  Question 4: How does air solution and evolution affect compliance estimation? As seen in Section (4.1.2), Hayward [8], who studied the effects of air in hydraulic fluids, was very skeptical about using any relation that did not take air solution and evolution into consideration. However, except for Jinghong et al's [11] approach (Section (4.1.2)) which only had moderate success, there are no known methods of modeling the effect of air solution and evolution. Thus, the questions to be answered here, is if air solution and evolution affected the system to the point that compliance could not be estimated correctly, and if the entrained air volume could be estimated.  5.2.3 The Hydraulic Test-Bed The hydraulic test-bed used in this study was designed to be an Rod Side  approximate representation of the Stick actuator in an excavator.  Head Side  Due to its simplicity, it provided a convenient platform for studying some of the basic questions regarding compliance. A simplified diagram of the actuator and arm is shown in Figure (21). As shown in the diagram, the actuator is a double-acting differential area cylinder with a rod that only extends in one side of the piston. To make the discussion clear, the side of the 82  Figure 21 Simplified Diagram of Arm and Actuator in the Hydraulic Test-bed  actuator with the rod is called the 'rod side" and the other side is called the "head side". A simplified circuit diagram is shown in Figure (22). The pilot valve and some of the pump's  Valve Cylinder  Figure 22 Hydraulic test-bed circuit diagram support circuitry are not shown in the diagram. As the relief valves only affect the circuit during an emergency, their effect can be ignored unless extreme operating conditions occur. The test-bed consisted of a 3 GPM (11.36 Liters/Minute), 1000 psi (6.89 MPa) gear pump (Parker D27), a cylinder (Parker 1 Vi" SB3LLT27Axl0") with a 25.4 cm stroke, a four-way pilot operated proportional (direction and control) spool valve (Rexroth 4WRZ 10E25-3X/6A24ETZ4), an actuator arm with a revolute joint, a 10 Gallon (37.85 Liters) reservoir, and the supporting circuitry such as check valves, relief valves and filters all connected with flexible hoses (PARFLEX 540N-3 and PARFLEX 540N-6). Since the overall length of the fluid path in the system is small, the volume of metal connectors is not negligible compared to total fluid volume. A more detailed description of the physical properties of the test bed are provided in Appendix (C). The subsystem of interest to this study is the portion from the valve to the actuator including the actuator. The circuit representation for the test-bed system for the portion of the system that is of interest to this 83  study is shown in Figure (23). Figure (23a) shows the circuit representation for when the fluid flows in such way that the head side of the piston is "pushing" and Figure (23b) shows the representation for when the rod side is pushing.  Pump P„ Qo  Tank  A x  A x  H  Head Side  R  ^"  Rod Side  Head Side  Rod Side  Pump  Tank  CHeadPi = Q , - A H X ± Q ,  Q = C a,v/2/p(P-P )  CRodP^Qi-A^iQ,  C „dP„ = A x - Q ± Q ,  Q = C ay2/p(P -P )  C„eadP„ = A „ X - Q ± Q ,  R  R  i  0  0  d  i  d  0  (A) Head Side "Pushing"  0  e  (B) Rod Side "Pushing"  Figure 23 Test-bed circuit representation for both fluid flow directions  In this diagram the values P and Q indicate pressure and flow, the subscripts i and o indicate in and out of the actuator, and the subscripts p and e indicate pump or exit side of the valve. The valve orifice area is denoted by a, and a for the valve ports going into and out of the actuator, the piston area on the head u  side and rod side is denoted by A &n&A , and the leakage flow in the actuator is denoted by Q,. The H  R  actuator length is denoted by x, which is calculated from the measurements from the angle of the arm and the geometrical relationships shown in Appendix (C), Figure (38).  84  5.2.3 The Experiments on the Test-bed  5.2.3.1 Objective The objective of performing experiments was to answer as many of the questions posed in Section (5.2.1) as possible. To achieve this, four experiments were proposed that would allow the test-bed system to be studied under normal (fault-free) conditions as well as under specific fault conditions. The experiments were designed based on the assumption that flow and pressure measurements could be used to determine compliance using Equation (24) derived in Section (4.2).  Experiment 1: Fault-free System The objective of this experiment was to study compliance in a fault-free system and answer the first two fundamental questions posed in Section (5.2.1). The results of this experiment were also to be used as a basis for comparison with later experiments where specific faults were introduced in the system. Experiment 2: System With Large Compliance The objective of this experiment was to introduce a large and known source of compliance such as an accumulator into the system to test the effectiveness of various models and algorithms in measuring the additional compliance. Experiment 3: Air in the System The objective of this experiment was to study the effects of free air in the system by introducing a known volume of air into the system. Unlike the gas in an accumulator, the air was free to form bubbles, dissolve, or move to other parts of the system which was expected to have a unique effect on the system compliance. Experiment 4: Leakage in the System The objective of this experiment was to introduce a known amount of leakage into the system to study the effectiveness of various models and methods in detecting and measuring this leakage. 85  5.2.3.2 Methods The methods used for the experiments, data collection, and basic data analysis were similar in all experiments. A 0.5 Hz sinusoidal input signal was applied to the main pilot valve for all experiments. This was believed to be a good representation of an input signal used by an operator working on a hydraulic machine. The temperature of the fluid in the tank was maintained in the range 40±5°C to ensure similar operating conditions for all experiments. The pump pressure, the tank pressure, the pressures on the head side and rod side of the actuator, the spool position, and the arm angle were measured at a 1000 Hz sampling rate. Details regarding the data collection can be found in Appendix (D.l). The actuator length was determined from the geometry of the test-bed system described in Appendix (C).  Due to a size mismatch between the valve and the actuator in the test-bed, the experiments were limited to small valve input signals to avoid damaging the equipment. This meant that the valve was operating close to its dead-band zone and according to the manufacturer's specifications the flow through the valve should have been near zero. Consequently, the operation curves provided by the manufacturer could not be used and the flow through the valve had to be calibrated experimentally. Details regarding flow calculation are provided in Appendix (D.2).  The data analysis was performed off-line for convenience, but the methods studied were all selected for their suitability to on-line calculation. To remove noise from the data an FIR (finite impulse response) low pass digital filter with 10Hz cut off was used for all experiments. An FIR differentiator with the same cut-off frequency was used to find the head and rod side pressure derivatives P (  h  and P ) and the r  piston speed d(actuator length)ldt=x. This cut-off frequency was found to be satisfactory based on preliminary data collected on the system before performing experiments. Further details on the filter type and cut-off frequency selection is provided in Appendix (D.3). Due to the linear phase response 86  characteristic of the FIR filter, introducing the appropriate time delay preserved the temporal relationship between different measurements.  The pressure change and the net flow into the subsystem of fluid from the main valve to the actuator were used to calculate compliance. In all cases the lumped model for compliance,  C  P=Q-Ax  lumped  (Equation (25) ), was used as a baseline for comparison due to its simplicity. Calculations were made for both the rod side and head side of the actuator and for both cases where the head side was "pushing" (connected to the pump) and rod side "pushing" as shown in Figure (23). The equations in Figure (23) also show the values used in calculating flow in the different cases. Using subscripts / and o to distinguish the side that fluid goes "in" and is "pushing" the piston, from the side that fluid goes "out" of and is "being pushed" by the piston, the relevant relations for compliance are  cf, = Q, ~ ? A  Cj  = Ax -  0  o  Q  o  and the equations for the flow through the main valve are  fi, = Q ^ 2 / p ( P -P.)  (44)  Qo - C/i /2/p(P -P )  (45)  o]  o  e  where P and P indicating the pump and tank pressure and P, and P indicating the "in" and "out" p  e  a  pressures that could be the head side or rod side pressure depending on which side was connected pump or tank. Similarly C„ C , fi„ Q , A a  0  h  and A indicate the compliance, valve flow and piston area of the 0  "in" and "out" sides of the actuator, which could be the head side or rod side depending on the spool position (which determines which side is connected to the pump or tank). The lower bounds for the actuator pressure derivatives (P and P ) magnitudes were determined using a method which will be h  87  described later. When the derivative magnitudes were below the lower bound, previously calculated compliance values were maintained. To ensure that noise was not mistaken for a true change in pressure, limits were also placed on the average magnitude of the pressure derivative for a moving window of past points. However, due to the low cut-off frequency of the filter used, this was not really necessary in this case. A number of data processing methods were used to test different models and methods for compliance calculation. To evaluate the results, the compliance of the system, C  , was simulated (calculated for  cffeclive  each time step) based on the system parameters and operating conditions for each experiment and the following compliance model (Equation (33) in Section (4.4.3))  effect  L  V  V  ~ J~  J~  V ~p~ •  ( °) 4  Values for V , V , and V were calculated at each step using the equations described in Table (1). To h  g  t  calculate ft„, the relation -2P  P =(6UD-2AS)P J- (\.n-e -), s  h  P  B  was used since the volume expansion coefficient provided by the manufacturer did not represent the change in fi , with pressure. The actual calculation required that the different diameters of the two types h  of hoses used be taken into account (see Appendix (C)). To calculate V at each step, a basic air content g  volume VgQ at atmospheric pressure was assumed which was dependent on the type of experiment.  Special issues regarding experimental methods used in individual experiment are discussed below.  Experiment 1 (Fault-free System) Method The method described above was used for this experiment. The system effective compliance C ff e  88  eclive  was  simulated using a 2% value for the air content volume at atmospheric pressure (V ) which is a typical air g0  content value for a system as suggested by Merritt [19].  Experiment 2 (System With Large Compliance) Method In this experiment a bladder type accumulator was installed on the  Cylinder  head side of the actuator as shown in Figure (24) to provide a large source of compliance. The accumulator volume was 0.946 liters (1  -'Accumulator  Quart) with a pre-charge pressure of 1.72x10 Pa (250 psi). 6  Using Equation (5) describing the accumulator compliance, the accumulator's pre-charge pressure and volume, and the head side  X Ma in Valve  pressure at any time, the accumulator compliance on the head side Pump  x ""N  which was the dominant source of compliance on the head side was simulated. The rod side compliance was simulated with a 2% air as in Tank  Experiment 1.  Figure 24 Simplified Circuit for Experiment 2  Experiment 3 (Air in the System) Method A large volume of air was added to the head side of the actuator in this experiment. Due to the physical arrangement of the test-bed, once the system was opened, it was not possible to control the volume of air entering the system accurately. The best method found was to empty the entire fluid contents on the head side, replacing it with air, and to re-introduce the fluid through the valve after closing the system. In this way the known valve flow was used to calculate the volume of fluid entering the head side to calculate the air content. Using this method a 31% air content was calculated.  Experiment 4 (Leakage in the System) Method In this experiment an artificial leak was introduced to study the problems caused by a leak to the outside. 89  This was achieved by adding an extra hose from the head side of the actuator to the tank. This hose was terminated with a terminator that could be replaced with a terminator with a hole of known diameter. The hose was kept in place during all experiments to minimize the changes to the basic system form experiment to experiment. The hole size used for this experiment was 0.5mm in diameter. To simulate the compliance of the system, as in Experiment 1, a 2% air content (at atmospheric pressure) was used.  5.2.3.3 Results The preliminary results for all experiments are shown in Figures (25 to 28). These figures show plots of important system properties with respect to time. The time scale shown for all plots (except for experiment 3) is for one full cycle of the input signal. During the first second, the head side of the actuator is connected to the pump and is "pushing" the piston while the fluid exits from the rod side. During the 2 second, the rod side is "pushing" and the fluid exits from the head side. For experiment 3, nd  two full cycles are shown since due to the injected air the system performance changed while the air compressed. It is worth mentioning that during Experiment 3 the air in the system also changed the characteristic operating sound of the system. The values shown here are the results after filtering the data. These figures have the following format: • Part (a) shows the pressure of the head and rod side of the actuator. The pump and tank pressures were about lxl0 Pa and 3.3xl07 Pa respectively for all the experiments. Note that in Experiment 1, at 7  5  about 2.5 seconds, when the rod side is connected to the pump (or "pushing"), the head side pressure increases (Figure (25a)) before the actuator begins to move (Figure (25g)). See Section (5.3.1). • Part (b) shows the flow going into the head side through the main valve (solid line) and the flow "leaving" the original volume of the head side of the actuator due to piston motion (broken line). Positive direction for flow is that of flow entering the head side from the valve. • Part (c) is similar to part (b), with the exception that the positive direction is that of flow entering into the rod side through the main valve. 90  • Part (d) shows the pressure derivatives for the head and rod sides. • Part (e) shows the net flow into the head side of the actuator. This net flow is the difference between the values shown in part (b). • Part (f) shows the net flow into the rod side of the actuator, which is the difference between the values shown in part (c). • Part (g) shows the spool position (scaled up by a factor of 50) and the actuator length as calculated using the arm angle and geometry (offset by 0.5 meters to fit in the plot). • Part (h) shows (solid line) the head side compliance as calculated directly from the net flow into the head side (shown in part (e)) and the head side pressure derivative (shown in part(d) ) using the lumped compliance model. The broken line shows the compliance value simulated for the head side, using the known parameters of the system and the conditions of the experiment. • Part (i) shows (solid line) the rod side compliance as calculated directly from the net flow into the rod side (shown in part (f) ) and the rod side pressure derivative (shown in part(d) ) using the lumped compliance model. The broken line shows the compliance value simulated for the rod side, using the known parameters of the system and the conditions of the experiment.  In parts (h) and (i) the constant compliance values (flat profile in the plots) indicate that the pressure derivative magnitude was below the "lower bounds". The "lower bounds" were calculated by using the expected compliance range for each case to determine the minimum actuator pressure derivative required to ensure that the transient flow induced by compliance is higher than the flow measurement error. This is further discussed in Section (5.2.3.4). The lower bounds for these calculations were: • experiment 1 (Fault-free System) 1.5 MPa/s • experiment 2 (System With Large Compliance) 0.2 MPa/s • experiment 3 (Air in the System) 1.0 MPa/s • experiment 4 (Leakage in the System) 1.5 MPa/s 91  i  x  6 0  ( ) Head and Rod Pressure a  x  x  <,-" (b) H e a d S i c e : Q , A . d X / d t  x  X Q  - > (c) R o d S i d i : Q , A . d X / d t  1.5  7 -Head; 6.5  rs/s  I  c Q O <U  -Rod  6  £ a  P 3 5.5 V)  E o  t/i a  &  n  s  5  o u.  4.5 1.5  4  T i m e (: x  1  0  '  :  S i t e Net Flow  R o d dP/dt  x 10"" < > e  H  e  a  d  x  1  0  -«(h)Head  A A i _  -0.5  V/  -1  : T/ 2.5 T i m e (sleconds) .  1  0  -«  ( f ) R o d Sic e Net Flow  T i m e (s a c o n d s )  S 3 C 0 nds) T i m e (i Side Compliance  ....'A...  0  S 3 C 0 nds) T i m e (i  T i m e (seconds)  £  I  1.5  (g) A c t u a t o r L e n g t h & S p o o l  0.04  0.5  2.5  sconds)  (d) H e a d 8  "•\ A  :  • -A.dX/dt  -1.5 1  0.06  1. — Q  x 10""W  R o d s i d e  —Actuator -0.5rfi  Compliance  - L u m p e d Model  .•.•Spopi. P o s i t i o n x 5 0  -Simulated  0.02  0 j= o> c -0.02 _J -0.04 -0.06 1.5  2.5  1.5  T i m e (seconds)  Head Side Pushing  T i m e (sBconds)  Rod Side Pushing >-  Head Side Pushing  Rod Side Pushing  <  Head Side Pushing -<  Figure 25 Basic Results for Experiment 1 (a) Actuator head and rod pressure versus time (b) Flow into and out of head side versus time (c) Flow into and out of rod side versus time (d) Head and rod pressure derivative versus time (e) Head side net flow versus time (f) Rod side net flow versus time (g) Actuator length and spool versus time (h) Head side simulated and lumped model compliance versus time (i) Rod side simulated and lumped model compliance versus time 92  2.5 T i m e (seconds)  Rod Side Pushing  io() Head and Rod Pressure 6 a  5.6 5.4  • 1 • -.  £5.2  I  I ;  1V .j  0  S  4.4 4.2  '  (d)Head  8 Rod dP/dt  2  I  —Q •-A.dX/dt  -1.5  1.5 25 Time (:S 3C0Inds)  1 x 1 0  f  Q)  j.  E  1 x  li  !j  1.5 I! 2.5 Time (seconds) 1 0  5 0 u. 3  0.5  A  d  X  /  d  t  \y t  -1 -1.5  '  f  V  -0.5  Q  A \\\  —A.dX/dt 0  :  1  2.5 1.5 SB Time (seconds)  1 x 10" (0 Rod Sice Sii Net Flow  - i (e) Head Si|de Net Flow  1'  i'  < 1  S  0  :  •' - '/p!j^*' • i T ": • i! : v i! : 1!  :  i' :  )..;  •  '  a Ii. ]  '•-.  :  — Head ;-Rod  1.5 2.5 Time (ssconds) 0.06  4  0.04 0  I  to <B  2  r  g -0.02 -ADtuator -0.5m •Spool Position x50  -0.04 1  2  Rod Side Pushing  3  E"  Time (spconds)  x 10"'°W  R  o  d S i d e  Compliance  J  r  2  — Lumped Mode -Simulated  1.5 2.5 Time (seconds)  Head Side Pushing  2.5  ide Compliance  E o 0  0  -0.06  x lo" ()  Time (seconds)  10 n H e a d S i l  (g) Actuator L ength & Spool  £  d s i d  '!: ii:  >  I  o  1  ni  0  F '•  B  0 J3 13  -1  0  li.  R  1 O O  n -0.5  4.6  f  c  0  I 4.8  x 10" <)  1.5  0.5  E  w  „-« (b) Head Sic e: Q, A.dX/dt  1  OJ 0  i  5  •a c 0 Q a> S/SJ  i ;  ....... ^7.  -Head -Rid  1  1.5  rs/s  x  1.5  -Lumpid Model -Sirriul kled""  Time (seconds)  Head Side Pushing  2.5  3  Rod Side Pushing  •<  1.5  Time (seconds)  Head Side Pushing  2.5  Rod Side Pushing >•  Figure 26 Basic Results for Experiment 2 (a) Actuator head and rod pressure versus time (b) Flow into and out of head side versus time (c) Flow into and out of rod side versus time (d) Head and rod pressure derivative versus time (e) Head side net flow versus time (f) Rod side net flow versus time (g) Actuator length and spool versus time (h) Head side simulated and lumped model compliance versus time (i) Rod side simulated and lumped model compliance versus time 93  -|o ( ) H 6 a  x  e a d  a  n  d  Rod Pressure  x  1 0  - " (b) Head Side: Q, A.dX/dt  x 10~" <) c  1.5  R  o  d S i d e :  Q  '  A  d  x  /  d  t  ! • <u  | 0.5 o>  1 o 0  §-0.5  1 2  Time (seconds) x 1 r  y  (<l)Head& Rod dP/dt  x  -1 -1.5  3 4 Time (seconds)  (e) Head Side Net Flow  |  1  I 0.5 % E B 3 1-0.5 u.  —  0  Time (seconds)  —Actuator.-C.5m •Spdbrposif on x50  0.04 I o £  0  0  2  0  o -0.02  _j  -0.04 -0.06  1  Time (seconds)  Rod Side Head Side Rod Side Head Side Pushing Pushing Pushing Pushing >•  Figure 27 Basic Results for Experiment 3 (a) Actuator head and rod pressure versus time (b) Flow into and out of head side versus time (c) Flow into and out of rod side versus time (d) Head and rod pressure derivative versus time (e) Head side net flow versus time (f) Rod side net flow versus time (g) Actuator length and spool versus time (h) Head side simulated and lumped model compliance versus time (i) Rod side simulated and lumped model compliance versus time 94  R o dS l d eN e F t | 0 W  I  l A  A ...  n Ji  (h) Head Side Compliance  0.06  3 4 Time (seconds)  x 10" O  -1  3 4 Time (seconds)  (g) Actuator Length & Spool  2 4  1.5  2  1  2  3 4 Time (seconds)  I  i g (a) Head and Rod Pressure 6  x  Q- (b) Head Sic e: Q, A.dX/dt 4  X  1  x  1 0  - " (c) Rod Side: Q, A.dX/dt  Time (s jcond:Is)  Time (; x  1  0  '  (d)Head  X  1  0  - 5  (f) Rod Siiid aNet Flow  15 ?  5  oi  4  I  ;  - H ead od  H  I  | \  1  V  v>  i \  £ o 1.5  Qi  i  18 10 <D  Vyyy  E  o  ~-0.5  5  ^V%r^-  3 o  J  DL  1  1.5  nds) x 10""( )  (g) Actuator Lbngth & Spool  0.06  h  H  2.5  3  1.5  Time (s sconds) e  a  d  Side Compliance  x 10""'')  S i l  R  Time (s sconds) o  d  s  i  d  e  2.5  Compliance  —Actuator -0.5rr •SpooliPosition <50  0.04 &  £  '  0  0.5  2.5 Time ((  B E  C  8 i  0.02 ~T 0  0  ~T 0  1  CD  5-0.02 -0.04  1  E o' O 1.5  2.5 Time (saconds)  Head Side Pushing -<  Rod Side Pushing  -Lumped Model -Simutated  -Lumped Model -Simulated"  1.5  2.5 Time (seconds)  Head Side Pushing  Rod Side Pushing  Figure 28 Basic Results for Experiment 4 (a) Actuator head and rod pressure versus time (b) Flow into and out of head side versus time (c) Flow into and out of rod side versus time (d) Head and rod pressure derivative versus time (e) Head side net flow versus time (f) Rod side net flow versus time (g) Actuator length and spool versus time (h) Head side simulated and lumped model compliance versus time (i) Rod side simulated and lumped model compliance versus time 95  1.5  2.5 Time (ssconds'  Head Side Pushing -<  Rod Side Pushing  5.2.3.4 Discussion  Leakage i n the Actuator Preliminary study of the results indicated that there was a systematic error affecting all the experiments. As it can be seen in Figures (25 to 28), based on calculations with the lumped compliance model, very large compliance values with both positive and negative sign were observed for all experiments. Since negative compliance is not possible, results were carefully studied to find possible sources of error. It was observed that in some cases the net flow into the actuator head or rod side appeared to be negative even though the pressure was rising, or the net flow was positive while the pressure was dropping. This can be observed by studying Figure (25) of Experiment 1. Furthermore, it was observed that when the rod side was "pushing" (connected to the pump), flow measurements appeared to indicate a large negative net flow into the head side of the actuator as shown in Figure (25e), and a large positive net flow of about the same magnitude into the rod side as shown in Figure (25f). This lead to the theory that a considerable volume of fluid leaked across the piston in the actuator. This theory was confirmed with flow calculation methods described in Appendix (D.2). Since the actuator flow was used to compute the flow through the valve (Appendix (D.2), Method 1), the leakage in the actuator meant that the results of the flow calculations were incorrect. Therefore, manual measurements had to be used to correctly calibrate the valve (Appendix (D.2) Method 3 ).  It was also observed that the leak flow appeared to be very small or negligible when the head side of the piston was pushing. As shown in Appendix (D.2), the manual flow measurements also confirmed this. This seemed to indicate that since the pressure difference between the head side and rod side was less when the head side was pushing, the slip leakage across the piston (which is dependent on this pressure difference) was also lower. Since the leakage in the actuator could not be eliminated for the experiments, it was not possible to 96  isolate individual faults in the experiments as originally planned. Therefore, attempts were made to model and simulate this leakage. The drag and slip leakage were simulated according to Equations (6) and (8) and based on reasonable values for the parameters of the actuator. Considering the large flow error, it was difficult to verify values obtained for the slip leakage, but it appeared that the actual leakage was less than the simulated value as the piston speed was reduced. This was reasonable since according to the manufacturer, the seals in the piston were supposed to be leak-free when stationary. Although, it appeared that due to damaged seals, leakage continued even when piston did not move, it seemed that the leakage magnitude was dependent on the piston speed. It was also found that the drag leakage was negligible compared to the slip leakage for the same conditions.  Determining the Required Flow Measurement Accuracy Using the compliance values predicted by simulation, the expected range of compliance values for each experiment was determined. The calculated actuator pressure derivative was then used to determine the expected range of transient flow due to compliance, C  dP/dt.  effecllve  This showed that for the average  values in the operating range of the fault-free system, a flow measurement accuracy of better than 0.1 cm /s (10" m /s) was required to get a signal-to-noise ratio of about 10. Had it not been for the relatively 3  7  3  low stiffness and large volume of the hoses used, the compliance of the system would have been about ten times lower and a flow measurement accuracy of better than 0.01 crnVs would have been required. This also implies that any leak of this magnitude can distort the compliance calculation. To see if this accuracy was possible some detailed error analysis was required.  Determining the Available Flow Measurement Accuracy To determine if the flow measurements were accurate enough to yield correct compliance values, the accuracy of the flow measurements Q„ Q , A x, and A x used in the relation C 0  {  o  analyzed. 97  P=Q-Ax  effeclive  was  The main source of error in this study was the measurement of flow through the valve (Q, and Q ) from a  equations (44) and (45), because of lack of accurate calibration curves as described in Appendix (D.2). The magnitude of this error was estimated to be potentially as high as l x l 0' m /s in the worst case, and 5  3  about lxl0" m /s in general. It should be noted that since this error is "built in" to the spool-flow curve 6  3  (derived here), it is likely to cause a systematic (as opposed to random) error in all the flow measurements. Furthermore, as shown in Appendix (D.l), using the range of values used in this study, even the resolution limit of the Analog to Digital (AD) converter used in measuring the pressure resulted in a flow error of about 0 to lxlO" m /s depending on the orifice size and pressure range. 6  3  The error in the flow measurement due to piston motion was calculated from the error in angle measurement and was estimated to be at most 2.5xl0" m /s as shown in Appendix (D.l). Thus, the error 6  3  in the net flow into the actuator, Q-Ax, which in the worst case is the sum of the errors in Q and Ax, is in the order of 10" m /s at best, but could at times be as high as 10" m /s. This level of error is 10 to 100 6  3  5  3  times has high as the maximum error required for a signal to noise ratio of 10.  Furthermore, some possible sources of error were not even taken into account in the above discussion. These are un-modeled dynamics due to flow forces in the main valve, main valve leakage, leakage due to the supporting circuitry (such as relief valves), and errors due to the fact that the valve steady state characteristics were used to represent its dynamic behavior. Considering the high flow accuracy required, even the discrepancy or leakage noticed in the valve flow that was originally was considered to be negligible may be a source of problem. As can be seen in Appendix (D.2), this valve leakage seems to grow to as much as 10% of the flow, as the flow grows . It seems that not only is the flow measurement inaccuracy a large obstacle to compliance calculation in this study, but using the spool position and valve characteristics may also not provide accurate enough flow measurements in general. 98  Two Simplified Compliance Models to Adapt to Leakage and Poor Accuracy Due to the low accuracy in the flow measurements, there was little hope that the parameters of a complex multi-parameter model such as the one described in Equation (35) or (36) could be estimated correctly and it was necessary to develop a simpler model. Since the actuator leakage was found to be a major source of problem, any model considered would have to at least distinguish between compliance and leakage. The following two methods for obtaining simpler models were proposed. a) Model 1 Starting from a simpler version of the model as described in Equation (32), and ignoring all leakage terms except the slip leakage across the piston, Equation (32), which is re-written below,  v  ,  /  •  C „ , P. = —P = Q. - Ax + K.x + KJP -P)p  effective I  i  I  *w  2  o  V  -  KJ(P-P  i  3y  f  v  i  /  , )+K'(P-P atm  J  3  V  ;  , ) aim'  was reduced to  Model 1:  Q. - Ax - C „ . P - KJP -P) , z-'i  where C  effeclive  effective t  o  2*-  i'  (47)  '  v  7  and K were the only parameters to identify. 2  b) Model 2 As previously discussed, the dynamics of the seals in the actuator made it difficult to use the simple slip leakage model of equation(8) to predict the actuator leakage. Therefore another model to get around the problem of estimating slip leakage was proposed. Since the leakage flow is confined to the actuator, if the actuator is considered as a whole system, then a relation for the net flow into the actuator can be obtained that is independent of the leakage. Equation (47) of Model 1 proposed above can be written for both the head side and the rod side as Head Side: Rod Side:  Q - A x = CP h  h  h  h  - Q,  AJ -Q = CP + Q, r  99  r  where without loss of generality it is assumed that the head side is "pushing" (connected to the pump). Summing the two equations gives  Model 2:  Q -Q h  r  - (A -A )x h  r  (48)  = C P *C P h  h  r  r  which describes the net flow into the entire actuator and is both linear in the parameters and independent of leakage. The two parameters to be estimated are the head side compliance, C and the rod side h  compliance C . r  c) Important Observation Study of the head side and rod side pressure for all the experiments (see Figures (25 to 28)) indicates that except for small regions, the pressures on both sides of the actuator are very closely related. This is due to the fact that as soon as the system reaches an equilibrium where the net force on the piston is close to zero, then the pressure on both sides of the piston is balanced by the ratio of the surface areas,  F  h  - F  r  =0 -  PA h  h  = PA  -  r  P  h  - P ±  .  This means that the pressures and their derivatives on both sides are linearly dependent which makes identification of the parameters of Model 2 difficult, since the measurement matrix becomes rank deficient. Therefore, the data should be monitored closely, to ensure only portions that are suitable are used by monitoring the condition number of the measurement matrix (see Appendix (A.l)).  Discussion for Experiment 1 (Fault-free System) It can be seen from Figures (25h and i) that the compliance calculated from the Lumped Model was not near the simulated value (based on a 2% air content assumption). Considering that the magnitude of the actuator pressure derivative, P, was on the order of 10 Pa/s, and the expected range of C 6  effeaive  100  for this  experiment was calculated to be in the order of 10' m /Pa, the expected transient flow due to 12  3  compressibility was only in the order of 10 Pa/s x 10" m /Pa= 10" m /s. Since even the AD bit6  12  3  6  3  resolution for the pressure measurement resulted in flow error of this order of magnitude, it was clear that the compliance could not be measured in this case, unless flow accuracy was significantly improved.  However, one interesting observation was made from this experiment. When the head is pushing, the head side pressure rises before the piston starts to move (Figure (25a)), as expected. However, when the rod side is pushing, the head side pressure follows the rod side pressure closely before the piston moves, indicating that the fluid flows across the piston. This provides a sufficient, but not necessary sign of leakage across the actuator piston.  Discussion for Experiment 2 (System With Large Compliance) The most noticeable feature in the results shown in Figure (26) is that due to the large compliance of the accumulator, the head side pressure (Figure (26a)) barely rose. The large pressure difference between the rod and head sides meant that there was a large slip leakage across the piston. It can be seen in Figure (26h) that the compliance by the Lumped Model did not match the accumulator compliance. In this experiment, the simulated accumulator compliance was close to 10"'° m /Pa. Since the total value of 3  compliance sources in a hydraulic circuit add in a similar way to parallel capacitors in an electrical circuit, the other sources of compliance should be added to the accumulator compliance to get the total compliance. In this case the compliance of the rest of the components, which was only about 10" m /Pa, 12  3  was negligible in comparison to the accumulator compliance. With flow errors in the order of 10" to 10" , to get a transient flow {C s  5  accumulator  x dP /dt) of twice the hea  magnitude of the worst-case flow-error, a head side pressure derivative of about 2x10 Pa/s would 5  suffice. While such low signal to noise ratio was far from desirable, it at least indicated that a small portion of the collected data, where the head side pressure derivative was between 2x10 and 5x10 Pa/s 5  101  5  could be worth further analysis. Models 1 and 2 (Equations (47) and (48) above) which were aimed at handling compliance and slip leakage were used to improve the estimate.  a) Compliance Estimation Using Model 1: To determine compliance from Model 1, C  effective  and K , the parameters of model 1 (Equation (47)) 2  Q - Ax = C „ .. P - KJP 2  effective i  V  -P) o  ,  i> '  were estimated using the Recursive Least Squares (RLS) algorithm described in Appendices (A.l and A.2) for the head side of the actuator. Since C ^, , is pressure dependent and K is dependent on the e  c lve  2  seal's dynamic performance, any estimated value would be only an average. Best results were obtained with a forgetting factor, X of 0.9. This meant that T, the "memory horizon" (number of measurements that carry more than 0.3 weight as explained in Appendix (A.l)) was about 10 data points. As can be seen in Figure (29a), which shows the estimate of the compliance versus the accumulator compliance, this estimate was only reasonably close over a very short period of time near 2.55 seconds (the shaded area indicates the significant portions of time where the algorithm was not active). This is despite the fact that Figure (29b) which shows the measured flow (the left hand side of Equation (47) ) versus the estimated flow (the right hand side of Equation (47)) values, indicates that the estimate was a good fit between the time of 2.4 to 2.72 seconds when the algorithm was active. This was particularly true for the time between 2.55 and 2.72 seconds. Possible explanations could be a systematic (nonrandom with mean other than zero) error in the flow, such as leakage in the valve, error in the valve spool-flow curve, or the effects of the un-modeled dynamics of the seal. As discussed in Appendix (A.l), such errors that do not have zero mean result in a biased estimate. Examining Figure (29d) shows another possible problem with this estimation. It shows that the head side pressure derivative and the "rod-head" pressure values (orP and P -P.from Equation (47)), which o  were the "measurements" used to estimate C g e  eclne  and K , do not vary much in the 2.45-2.7 seconds 2  102  region. While the head side pressure derivative seems to vary, the change is too small to induce enough transient flow to be detectable above the flow measurement errors. This means that the measurement matrix <£as defined in Appendix (A.l) could be less than full rank. It also means that the excitation condition as described in Equation (52) in Appendix (A.l) is not met or &(t) <P(t) is not invertible. Since the common way to check this for Least Squares problems is to T  compute the condition number of the measurement matrix <?(see Appendix (A.l)), for the RLS where the old measurements are discounted, the  a  I  (a) C o m p l i a n c e : M o d e l 1 & A c c u m u l a t o r  x 10 i  2  1 i  1  1  IT fl) _ ? •c .3  INI  Head Accum.  ~a.  memory horizon was used to  1  I-*  O  2.3  2.35  2.4  2.45  . n-5  approximately determine how many of  ZS  2.55  2.6  2.65  (b) E s t i m a t e d & M e a s u r e d  2.7 T  l  m  2.75  2.8  (seconds)  B  the measurements to consider in computing the condition number. 2.3  2.35  2.4  2.45  2.5  2.55  2.6  2.65  (c) C o n d i t i o n N u m b e r  2.7 T  i  m  e  T  i  m  e  2.75  2.8  (seconds)  Although according to the discussion in Appendix (A.l) the condition as seen in Figure (29c) is not unacceptable, some 2.3  2.35  of the peaks in the condition number as  2.4  2.45  ZS  2.55  2.6  (d) d ( H e a d ) / d t & R o d - H e a d  2.65  2.7  shown in Figure (29c) correspond to poor results in Figure (29a). A good example of this is the estimated value (e) N e t F l o w & L e a k  at about 2.5 seconds in Figure (29a)  Row  which deviates sharply as the condition number increases. The likely explanation for this is that while the measurements (Figure (29d)) are not  Figure 29 Results for Model 1 in Experiment 2  103  2.75  (seconds)  2.8  too close from a numerical point of view, they are too close from an information point of view due to the low resolution in the measurements. In this context, the condition number should only be considered as a relative measure of how close the consecutive measurements are.  b) Compliance Estimation Using Model 2: To determine compliance from Model 2 (Equation (48)) Q  h  -Q, ~ (A -A )x h  r  =C P h  h  C  +  r  P  r  the two parameters to be estimated were the head side compliance, C and the Rod side compliance C . h  Using a similar RLS algorithm as before with a forgetting factor, X of 0.9 these values were estimated. In this case the actuator pressure derivatives of both sides were used as the measurements for the estimation. This meant that finding a range where both derivatives were high enough to provide transient flows that were higher than the flow error was even more difficult. As before, the change in head side pressure was only high enough between about 2.4 to 2.72 seconds of the data as shown in Figure (30d). However, the rod side pressure which should have been at least 2xl0 Pa/s, 6  to even induce a transient flows of about twice the average flow error, was not high enough in most of the region (Figure (30d)). Since no other data was available, this data from this region was used as the best Figure 30 Results for Model 2 in Experiment 2 104  r  compromise. A lower bound of 2xl0 Pa/s was selected for the pressure derivatives and the shadowed 5  area in Figure (30a) shows where algorithm was not active. As with the previous model, due to the combination of poor condition and measurement inaccuracy, good results could not be obtained. From Figure (30b) which shows the estimated portion of the model C P +C P versus the measured portion Q -Q h  h  r  r  h  r  - (A -A )x , it can be seen that the estimation h  r  performance was not very good. However, comparing Figures(30a) and Figure(26a), shows that this method at least produced significantly better results than the lumped compliance model in the 2.5-2.6 seconds region where algorithm was active and the condition was not too high. Model 2 distinguished the order of magnitude of the head side compliance from the rod side compliance, even though the values were not estimated accurately.  c) Comments: Neither method was very successful in estimating the accumulator compliance correctly. The main reason for this was the lack of sufficient accuracy in the flow measurements. In Model 1, this lack of accuracy meant that the change in pressure was not high enough to produce a sufficiently rich input. For Model 2, this lack of accuracy meant that the transient flow on the rod side was mostly too low to be detectable for use in estimating rod side compliance. Since the noise or the error in the measurements was systematic as opposed to random error with zero mean, the RLS estimation method were not able to compensate for it.  Discussion for Experiment 3 (Air in the System) As with the previous experiment, the additional compliance from the air, allowed a better signal to noise ratio so that some data analysis could be performed. As shown in Figure (27a), the pressure difference between the head and rods side was low during the time up to 2.6 seconds, which means the slip leakage was low. Thus, this time period was the most suitable for determining compliance with the lumped 105  model. During the first second, the head side was filled with fluid which is not shown in the plots. The actual or expected value of compliance was simulated using a 30% air content for the head side. It is interesting to note that although air was only intentionally introduced in the head side, it seems that some air also entered the rod side, since as shown in Figure (27a), pressure did not build up in either side during the first two seconds. The correct air content of the rod side was unknown, but trial and error indicated that a 10% value seemed to match the measured values. When the pressure did change, it also changed the air volume, the compliance, and the transient flow due to compliance. To determine the lower bound for the pressure derivatives of each actuator side to produce a transient flow of at least 2xl0' m /s (twice as much as the worst measurement error) different 5  3  time periods were considered separately: In the 1.4-1.7 second region: • Head side: about 2x10 Pa/s near 1.4 seconds and 2x10 Pa/s near 1.7 seconds. 5  6  • Rod side (for 10% air content): about 7xl0 Pa/s near 1.4 seconds and 5xl0 Pa/s near 1.7 seconds 5  6  In the 2.4 to 2.7 seconds region: • Head side: about 2x10 Pa/s 6  • Rod side (for 10% air content): about lxlO Pa/s 7  Very few points in the collected data were actually above the required lower bounds so the value of lxlO Pa/s was used as the best compromise. It can be seen that near the 2.5 seconds regions where the 6  head and rod side pressures are close (see Figure (27a)) (lower slip leakage) and the pressure derivatives are higher than the lower bound (see Figure (27d)), the lumped model resulted in values that are very close to the simulated values for both the head side and rod side (see Figure (27h and i)). For the head side, these conditions were also valid near the 1.6-1.7 seconds region, and again, the lumped model and the simulation result in close values for compliance. As in the previous experiment, Models 1 and 2 were used again to try to improve the results.  106  a) Compliance Estimation Using Model 1: Model 1 was used with the RLS algorithm and a forgetting factor X of 0.95 (determined by experimentation) to estimate the head side compliance and leakage. The "measurements" were the pressure derivatives of the head and the pressure difference between the head side and rod side. In the 1.5-1.7 seconds region, the estimated compliance value starts very close to the simulated value (see Figure (31a)) but the results become worse towards 1.7 seconds. The reason for this is that the head side pressure derivative is much larger than the required lower bound (2x10 Pa/s) at first, but the 5  required lower bound increases as the compliance rapidly decreases in this region. Therefore, the transient flow due to compliance is high enough to be measurable despite the low measurement accuracy near 1.4 seconds. However, as the air compresses and the compliance is reduced, the transient flow is reduced to the point that it is not enough to be correctly detectable. In the 2.4-2.8 seconds region the estimated value for compliance is not very close to the simulated value at the beginning (see Figure (31b)) , but in the 2.5 - 2.7 seconds region, the estimated and simulated values are much closer. Before 2.4 and after 2.7 seconds, the poor estimate is probably due to low head side pressure derivative (see Figure (3 lj)). The poor estimate between 2.4 and 2.5 seconds may be explained by considering condition number for the measurement matrix in Figure (31h), which is quite high in this region. Comparing head side compliance in this region for Model 1 (Figure (31b)) and the Lumped Model (Figure (27h)) indicates that the results are somewhat improved, especially in the 2.7-2-8 seconds region. In this region, the pressure difference between the head and rod sides increases and the Lumped Model produces negative compliance values.  107  x 10~*  1.3  x  () * e  1.4  1 0  »  r , o w :  Estimated & Measured  1.5 1.6 Time (seconds)  1.7  x  1.8  j -s n  2.3  ([) Flow: Estimated & Measured  2.4  2.5 2.6 Time (seconds)  (g) Condition Number  (h) Condition Number  1.5 1.6 Time (seconds)  2.5 2.6 Time (seconds)  (i) d(Head)/dt & Rod - Head x  Figure 31 Results for Model 1 in Experiment 3  10'  (i) d(Head)/dt i Rod - Head  2.5 26 Time (seconds)  108  2.7  2.7  2.8  b) Compliance Estimation Using Model 2:  Model 2 was used with the RLS algorithm and a forgetting factor X of 0.97 (determined by experimentation). As seen before the parameters to estimate were the compliance of the head side and rod side. The "measurements" used in the RLS algorithms, were the pressure derivatives of the head and rod side. Model 2 was not very successful in this experiment. The main reason for this is that not only was it necessary for the pressure derivatives of both sides to be above the lower bound (to induce enough transient flow to be detectable by the low measurement resolution available), but the difference between the derivatives on both sides also had to be above the lower bound. If the difference between the derivative values is not high enough, as far as the RLS algorithm is concerned the values are not distinguishable and the rows of the "measurement matrix" are close to linearly dependent. This is because the transient flow induced by them is not distinguishable by the low measurement resolution.  In the 1.5-1.7 second region, for most of the data points, the rod side pressure derivative magnitude was too low to induce high enough transient flow to be measurable. Also the difference between the pressure derivatives of the head and rod sides was below the minimum lower bound for pressure derivative (see Figure (32g)). Only near 1.55 seconds, the results somewhat follow the simulated values for the head side (see Figure (32a)). In the 2.4-2.8 region the difference between pressure derivatives (Figure (32h)) was again below the required lower bound (especially the lower bound for the rod side). It can be seen that between 2.4 and 2.5 seconds, as the condition number is lower, the estimated compliance values converge towards the simulated ones, but as the condition number peaks (Figure (32f)), the values diverge (Figure (32b and 32d)). Here the condition number is not too high from a numerical point of view, but it is only used to keep track of the relative closeness of the measurement values. Since the condition number is not too high, if the flow measurement accuracy were not so poor, this algorithm might have been considerably more successful.  109  (b) H e a d C o m p l i a n c e  (a) H e a d C o m p l i a n c e  '-'"'4! i  "  I  •-  Model 2  —  Simulated  X  E  o O0  \  • \ • \.'  '>>.} \ 1.4  1.5  1.6  2.4  1.7  Z 5  2.6  Time (seconds)  Time (seconds)  (c) R o d C o m p l i a n c e  (d) R o d C o m p l i a n c e  - Model 2 -Simulated  3  % < 2 E  8  t  I  o  C  o O  -1 2.4  Z 5  2.6  Time (seconds)  (e) C o n d i t i o n N u m b e r  1.3  1.4  1.5  1.6  (f) C o n d i t i o n N u m b e r  1.7  Time (seconds)  ! 10°  (9) d ( H e a d ) / d t & d ( R o d ) / d t  x  Figure 32 Results for Model 2 in Experiment 3 110  1  0  '  (h) d ( H e a d ) / d t & d ( R o d ) / d t  2.7  c) Comments: In this experiment, Model 1 was somewhat successful. In both models, the poor accuracy of the estimates could be traced back to the lack of flow measurement accuracy. As discussed in the previous experiment, the main sources of error in the flow measurements were systematic (non-random with nonzero mean) and could not be over-come by the RLS algorithm.  Discussion for Experiment 4 (Leakage in the System) Although this experiment was originally designed to study the effects of leakage to outside, the problems with flow measurement and the already existing leakage in the system meant that the usefulness of this test for its original objective was diminished. However, looking at the results of this experiment, one interesting phenomena was observed. When the main valve was closed, due to the artificial leakage introduced the flow to the outside on the head side of the cylinder caused the head side pressure to drop rapidly and the piston to move from the rod side towards the head side. This can be seen by noticing that in Figure (28g) at about 1 and 3 seconds the spool is at zero position but in Figure (28e) the actuator length is decreasing. The pressure on both sides remained very close as they dropped rapidly as shown in Figure (28a). This situation provided the best opportunity so far to study compliance for several reasons: 1- Since the leakage across the piston was significantly reduced (or non-existent) when the pressures on both sides were very close, this provided a very good chance to observe the compliance in the absence of leakage. This meant that even the simple lumped compliance model of Equation (25) could be used. 2- Since the main valve was closed, there was no need to calculate the flow through the valve, thus removing the largest source of error from the calculations. 3- The large pressure drop, increased the magnitude of actuator pressure derivatives reducing the effect of any remaining flow error. When the valve was closed, the transient flow was estimated to be in the order of 5x10" m /s which is about twice the flow error due to error in arm angle measurement. 6  3  Ill  The value of the Rod side compliance of the rod side  a) Simulated & Estimated  was estimated using Equation (25) for the lumped model and a one-parameter RLS algorithm as described in Appendix (A.l) with a forgetting factor X of 0.95. For comparison, actual value of compliance was also simulated with a 2% air content as in experiment. From Figure (33a) it can be seen that:  0.5  _8  1.5  (b) Head & Rod Pressure  x10'  ra p_  " V "  3  In the 2.4-3 seconds region, when the rod side is  <i>4  Q_  \ \ \\ \ / \\..../.  O 1—  "pushing", there is no agreement between the  <S2  simulated and estimated compliance values. In this  H  \  10  1  0  ~  7  1.5  2  i  :  V  f:  V :  u  j  \\: :  < 0.5  region a large difference between the head and rod  3 3.5 Time (seconds)  Rod Head  —  £6  2.5  2  v -  2.5  I  V  \\. J  f: 3  3.5  (c) Rod Pressure Derivative"^'" ( 19  seconc  ' ) s  side pressures (Figure (33b)) indicates that the slip leakage is not negligible. In this case, the valve flow measurement error also distorts the measurements. In the 1.4-2 seconds region, when the head side is 2.5  "pushing" and the head side and rod side pressures  3 3.5 Time (seconds)  Figure 33 Compliance estimation for Experiment 4  are close, the simulated and estimated compliance values are not too far. In this case there is less slip  leakage, but the same flow measurement error due to valve flow. Near 1 and 3 seconds region, when the main valve is closed and the head and rod side pressures happen to be very close, the simulated and estimated compliance values are in agreement. For example in the 2.900-3.340 seconds range, the average magnitude of the deviation of the estimated compliance from the simulated values is only about 20% of the simulated values. In this case there is little or no slip leakage and no flow measurement error due to valve flow. 112  Overall, this experiment provided the best confirmation that: 1- The linear compliance model of Equation (46) predicted the compliance and its change due to pressure reasonably correctly, even for the very low compliance values of the rod side. 2- The valve flow calculations and leakage were the main sources of error, since in their absence, the results were much improved. 3- Any valve leakage does not appear to affect the system when the valve is closed. This is reasonable, since the leakage flow is likely to go from the pump to the lowest pressure point, which is the tank.  Additional Experiments a) Blocking One Side of the Cylinder In this experiment the head side of the cylinder was closed and the piston was moved completely to the end of the cylinder on the head side. The purpose of this test was to try to collect some data without the influence of the slip leakage. However, the flow values obtained were so small that they were within the error range and no useful compliance measurements were obtained. b) Trying to Use a Pulse and the Sonic Bulk Modulus The actuator was fully contracted and it was attempted to produce a pulse in the fluid to determine if the time it takes for the pulse to travel to the piston and back could be measured. The idea behind this experiment was to explore the possibility of using the Sonic Bulk Modulus of the fluid to obtain the Isothermal Bulk Modulus. This idea will be further explained in Section (5.3.3). However, there were many ripples in the pressure profile. Since due to the protective circuitry, there were more than one fluid path, the task of determining which path the pulse reflected from required some careful modeling.  113  5.2.3.5 Using Genetic Algorithms The discussion in Chapter (3) showed that Genetic Algorithms (GA) are most suitable for estimating parameters of models that are nonlinear and complex, where simple methods such as RLS cannot be used. In this study, low measurement accuracy resulted in such coarse results that any subtle improvements that might have been gained by using nonlinear models could not be measured. Thus, none of the models used in the experiments were suitable candidates for using GA. However, before relying on GA to solve a complex problem, it may be wise to understand its basic performance characteristics by using it on a simple problem and comparing the results with a well known method such as RLS. The simplest possible test is to use GA for direct calculation of compliance from the lumped model. For this purpose a portion of the data collected for Experiment 2 was used and the results were compared to the compliance values directly calculated from the lumped model. Applying Equation (25) of the lumped model for the case of head side "pushing", the following error function was derived E = Q - Ax - CP h  h  h  h  where Q , P ,C , and A are the flow into, the pressure, the compliance and the piston area on the head h  h  h  h  side. The fitness function was defined as Fitness = Bias-E  2  which allowed setting a threshold for maximum error as well as convenient method to maximize fitness, while minimizing the error. In this GA calculation, a population of 300 randomly generated compliance values with a chromosome length of 30 was used. The range selected for the compliance values was -10' to +10" , which gives a resolution of 1.86xl0" with the 30 bit parameters. The Bias was chosen to 9  9  18  be 1,000 based on the expected maximum error. The termination criteria used for the GA search for each data point was 40 generations or 99% of the population matching the fittest individual (whichever occurs 114  first). In this case GA is really being used as a very  Compliance: GA and Direct Calculation  expensive (computationally) alternative to simple division. However, this is a good test for the most basic performance of GA and as it can be seen in Figure (34), the results matched the values from direct calculation. This shows that the GA performs correctly and given the correct population size, chromosome length and termination criteria can give very accurate results.  The same data was used for the estimating the parameters of Model 2 developed in Section (5.2.3.4). Figure 34 Compliance calculation with GA  In this case, the head and rod side compliances, C and h  C„ were the parameters to be estimated. The chromosome length was selected to be 30 as before, but it was split into two parameters of 15 bits each for the head and rod side compliance. With a parameter range of-10" to +10", this gives a resolution of 6.1xl0" for each parameter. The population size and 9  9  14  termination criteria were also selected as before. From Equation (48) of Model 2, the error function was defined as E =Q  h  - Q - (A - A )x - CAP ' J, h  r  h  h  h  CP r  r  To use GA for parameter estimation, a moving window of historical data was used. Old data was discounted in a similar way to RLS with the same forgetting factor (0.9) used in Experiment 2, to make comparison with RLS results possible. Since GA is not inherently a recursive algorithm, the window size had to be explicitly determined. A window size of 20 which is twice as long as the memory horizon was used to have a similar influence of past points as in Experiment 2. Thus, the fitness function  Fitness  ]P  = Bias-  j-Window-\  115  XEj  was defined and the Bias was set to 20,000  Compliance: GA With Two Parameters  based on maximum acceptable error. The result of this analysis is shown in Figure (35). Comparing these results with those obtained from RLS shown in Figure (30), it can be seen that the estimated values follow the same general trend as those obtained by RLS, but the individual values are very volatile. This could be due to the random nature of GA search.  Figure 35 Estimating two compliances with GA  Experimentation with population size, bias value, and termination criteria produced different but equally volatile results. This demonstrate an inherent problem in GA regarding the determination of population size and termination criteria.  Comparing the results obtained by using GA for single parameter estimation to the result obtained by using it for estimating two parameter, it appears that GA performed much more reliably for single parameter estimation. This performance difference was also observed when GA was used with other data (not shown here). This is in agreement with the observation made in Section (3.3.4) that strings that contain multiple parameters are difficult to search for due to their relatively long defining length.  In summary GA was found to be simple to implement and use, but, as expected, computationally expensive (no formal cost analysis was performed). The results obtained using GA for estimation were always correct in the single parameter case, but volatile in the two parameter case. In general, it seems that GA should only be used as a last option if simpler and better known methods are not usable.  116  5.2.4 Answers to Some of the Questions Posed  Question 1: (Using Flow and Pressure to Determine Compliance) Contrary to expectations, it was found that the biggest obstacle to using flow and pressure in compliance calculation was not the fluid inertance, but it was the inability to obtain accurate flow measurement, as well as the persistent piston leak that was not as much the result of a major fault as it was part of the normal system operation that could not be eliminated. Fluid Inertia To determine if compliance can be estimated from pressure and flow measurements, the effects of fluid inductance and resistance on the actuator pressure should be determined to ensure that they are negligible compared to the effects of compliance. Here, the maximum pressure change due to these elements was found to be small enough to ignore. The details of how these values were roughly estimated are in Appendix (D.4). Similar methods can be used for any hydraulic machine to decide if flow and pressure based compliance calculation can be used in fault-detection for that machine. For mobile hydraulic machines, these properties are probably not a problem, since generally the fluid transmission lines are not too long and designers ensure the effects of resistance are minimized.  Measurement Errors In this study, flow measurement errors were of the same order of magnitude as the compliance range of the fault-free system. In general, the more indirect the measurement method, the more flow calculation steps are involved, and the more the inaccuracies are compounded. It is not clear if in general the required accuracy can be achieved. For example, the fact that the test-bed system can be operated entirely in a region that according the manufacturer specifications has negligible flow, indicates that even the manufacturer's operation curves may not be a reliable source of information for accurate flow calculation. Here, even the resolution of the Analog to Digital converter was not high enough for 117  accurate enough flow measurements. Using a flow meter would have made the task of compliance estimation much easier. While, the flow meter would need to be very accurate to be suitable for this task a very high response (more than 100Hz) is not necessary. Since industrial machines are substantially larger than the test-bed system used here, it is possible that due to higher volume of fluid and higher compliance values, the measurement errors can be acceptable in comparison. In particular, if the system has large hose volume the range of compliance is even higher and the relative magnitude of measurement errors is reduced.  Question 2: (How to ensure meaningful results and a robust on-line algorithm?) Question 2.1: (How can usable data portions be found systematically?)  The most important criteria for selecting usable data portions was found to be the lower bound for the actuator pressure derivative magnitude. The lower bound should be high enough to ensure that the required signal to noise ratio is available. The signal to noise ratio is the ratio of transient flow due to compliance to the flow error due to measurement (and leakage) errors. To determine the lower bound for actuator pressure derivative magnitude, the accuracy of the flow measurements and the expected compliance range should be known beforehand. With this information the lower bound can be used on line to systematically ensure that any data portions used can lead to meaningful results. Question 2.2: (How should noise be removed)  The FIR (finite impulse response) low-pass filter used to remove noise was generally found to be satisfactory in all the experiments. In a few instances, a higher cut-off frequency than 10Hz might have been more suitable. In building a robust on-line fault detection algorithm for a system, the designers should study the system beforehand to determine a suitable cut-off frequency for the operating range of interest. In general, filtering the data did not appear to have any significant negative impact on the calculations. In fact a low-pass filter with a relatively low cut-off frequency seems to be a good way of eliminating the pressure transients that are too short to be due to compliance. 118  Question 2.3: (Can the use of pressure derivative be avoided?) Since fdtering the data was successful, differentiating the actuator pressure was not as difficult as expected. The method suggested in Section (5.2.1) for using integration rather than differentiation, was not very satisfactory with unfiltered data, since only two pressure points were used in calculations and the noise in the pressure data manifested itself as large "jumps" in the results. When the data was filtered, for small integration time intervals similar results to the differentiation method was obtained. For large integration time intervals, the method is hard to use since it is difficult to manage the accumulated leakage and measurement errors.  Question 3: (What is a suitable model for compliance?) Question 3.1: (Lumped Compliance Model) In general the lumped parameter model (Equation (25)) can lead to misleading results. However, if the results are considered together with other information from the system, the results from the lumped model can be used to gain some understanding of the system. Question 3.2: (Multi-parameter Compliance Model) • The multi parameter models (Equations (35) and (36)) developed in Section (4.4.3) could not be fully tested due to lack of measurement accuracy, but a simplified version of the model (Equation (47)) showed some promise in separating leak from compliance. • The linear model of the effective compliance described in Equation (33), was adequate in describing the overall compliance of the system whenever, measurement errors were eliminated. This was a good way of comparing the estimated compliance with the expected value based on various assumptions. • The experience with trying to estimate compliance and leak indicates that the differences in the pressure dependencies of various elements may be too small to be observable in the multi-parameter model. The best strategy might be to try to first analyze the data to determine the most dominant elements of the model and reduce the number of parameters for each case. 119  • Even if the multi-parameter model can only be used to determine the most likely source of problem, it is a success from a fault detection point of view. Question 3.3: (Need for Nonlinear Models) Due to lack of accuracy in the measurements, the experiments performed here did not provide an opportunity to compare linear and nonlinear models. Furthermore, the experience here indicates that in general not only may accurate measurements not be available, but the main challenge in designing a suitable model for fault detection may be in separating leakage from compliance. Thus, any values obtained for compliance may not be very accurate. Since the nonlinearity in the model derived in Section (4.4.3) (Equation (36)) was for an accurate representation of the hose volume change, without accurate compliance values, there may be no point in this additional accuracy.  Question 4: (How does air solution and evolution affect compliance estimation?) The effects of air solution in compliance estimation could not be tested due to lack of accuracy in the measurements. However, experiments 3 and 4 showed that even without taking air solution into account, the linear model of the effective compliance described in Equation (33) resulted in values that were reasonably close to the observed values. Although the values were not accurate, large and nominal air contents could be distinguished. Since this distinction is the main aim of a fault detection system, this indicates that air solution may not be a big obstacle to using compliance in fault detection.  120  5.3 Proposed New Experiments and Methods  5.3.1 A Few Simple Observations for Detecting Leak Simple observations in the actuator pressure, piston motion, and flow can provide some practical and useful information for fault detection. If the type of leakage can be determined by these simple observations, then the appropriate number of components from Equation (35) can be selected to reduce the number of parameters to be identified. Signs of Leak 1- Pressure on either side of the actuator changes when the control valve is closed. 2- Negative compliance values that cannot be attributed to measurement errors (e.g. pressure rising even when it looks like the net flow is negative or pressure dropping when net flow seems positive). Signs of Leak Across the Piston 1- When the valve opens, the pressure on the exit side increases before the piston moves (this was observed in Experiment 1 at about 2.5 seconds and can be seen by noting the time for head side pressure rise in Figure (25a), and the actuator movement in Figure (25g) ). 2- Flow measurements indicate that the magnitude of the net flow into the two sides of the actuator match closely, while the signs are opposite. This shows that the flow that seems to be "missing" on one side matches the "excess" flow on the other side and is a good indicator of leakage across the piston.  121  5.3.2 Proposed Simple Method for Compliance Identification The experiments performed on the hydraulic test-bed indicated that very accurate flow measurements are essential to compliance estimation. If the flow measurement method is indirect and involves a large number of sensors, the inaccuracies are rapidly compounded. Furthermore, any leak in the actuator makes it very difficult to separate compliance from leak. The actuator leak may not be avoidable even in high performance equipment. It was also seen that, while it is desirable to know the volume of air in the system, it is very difficult to determine when system is pressurized as the relative volume of air is reduced and some air dissolves in the fluid.  Here, a simple addition to the system is proposed that may Cylinder  provide an inexpensive and practical method for compliance identification using flow and pressure measurements. It is proposed that a "Bleed Valve" be installed from the line that goes from each actuator to the tank as shown in Figure (36). These  "Bleed" Valve  valves should have fixed size orifices that can be electrically  4 X  opened and closed. When the system is not operating or is in rest position, depending on the system, one of these valves can be  Main Vplve Pump  used to let some fluid leave the actuator to depressurize and test the system. Using the known valve orifice size, the flow out of the system can be measured accurately. As the pressure is  "Bleed" Valve  PT  Tank  Figure 36 Using "Bleed Valves" to Depressurize the Actuator  lowered, the compliance can be calculated. The choice of which valve to open depends on each system, because depending on the difference in the forces on the two sides of the piston, the piston may or may not move. If the piston does not move, only the compliance of the side with the open valve can be calculated. If the piston moves, the net flow in/out of each side of the actuator can be calculated in a similar method as before. The main difference is that on the actuator side with the closed "bleed valve", 122  the flow measurement is dependent only on the actuator position sensor. On the side with the open valve, the flow calculation is dependent on the actuator position sensor and the fixed size orifice. If the cylinder is equipped with a good position sensor such as a linear position transducer, a position accuracy of up to 0.001 inches or 2.54 xlO" meters can be achieved. Given a piston area of about 10-20 cm , this 5  2  indicates a flow accuracy in the order of 10' m /s can be achieved. 7  3  The advantage of depressurizing the system is twofold. By reducing the pressure, any leaks to outside are reduced and this will allow a better accuracy for compliance estimation. On the other hand, as the pressure is reduced both the volume of air and the compliance of air are increased and the magnitude of the compliance is also increased. These factors ensure a more robust estimation of compliance. Furthermore, at lower pressures, dissolved air can come out of solution and a better measure of air content may be calculated using Equation (35).  If the load on the actuator is not large when the system is in rest position, the pressure difference on the two sides of the actuator is likely to be relatively small. In this case when the valve on the side with higher pressure is opened, the actuator is likely to begin to drift as the pressure on that side drops. This case provides a unique operating condition where not only is the compliance increasing, but due to the free-drifting motion of the actuator, the pressure difference and consequently the leak across the piston is also at its minimum. This was true in the test-bed system used in this research, as seen in Experiment 4, and is likely to be true in the Stick and Bucket actuators of an excavator. However, due to its large mass, this is not likely to be true for the excavator Boom actuator. If the load on the actuator is high while in rest position and slip leakage is detected, one of the simplified models developed in Section (5.2.3.4) can be used to separate compliance and leakage. These models are likely to perform much better when high flow accuracy is available. An additional practical advantage of this method is that the operating conditions are fixed and a close to 123  steady expansion of the fluid can be assured by design. This simplifies the task of any on-line algorithm since the suitability of the flow and pressure ranges and the number of available data points can be assured by design. While this method has not been explicitly tested, the results of experiment 4 indicated that this method is likely to perform much better than any method that relies on flow measurements from the main valve.  5.3.3 Compliance Identification Using the Fluid Sonic Bulk Modulus As was seen before, any compliance identification method that relies on flow measurement is susceptible to sensor errors and leakage. One option for avoiding this problem may be to use the sonic bulk modulus of the fluid in a similar way that was used by Jinghong et al [11], which was discussed in Section (4.1.2). As previously discussed, the sonic bulk modulus is defined as pc , where p is the fluid density at the 2  temperature and pressure of interest and c is the velocity of sound at the same temperature and pressure. Using manufacturers' data the density can be determined for the operating pressure and temperature. The speed of sound for a pulse traveling in the fluid, can be determined by two pressure sensors. If one sensor is placed immediately after the main valve and one before the actuator, then the time of travel of a pressure pulse from the first sensor to the second sensor and the fixed distance between them can be used to estimate c, the speed of pulse travel in the fluid. This pulse can be created by rapidly closing the valve. Since the sonic bulk modulus is equivalent to the isentropic tangent bulk modulus (see Section (2.2.2)), it can be used to calculate the isothermal tangent bulk modulus using the fact that P, , sen  thermal  where Y =  mpjc  = y  C,/C . v  However, for this method to work, it should be ensured that no reflections from various peripheral fluid paths can reach the second sensor in such a way that the pulse cannot be detected. Furthermore, tests should be performed to determine if the presence of air bubbles in the fluid affects its sonic bulk modulus homogeneously and if reliable effective bulk modulus values can be obtained with this method.  124  6 Conclusions and Suggestions for Future Research  6.1 Conclusions In this thesis, the hydraulic compliance and its application to fault detection in mobile hydraulic machines were investigated and the following conclusions were reached.  1- It was found that as long as there is no leakage and flow and pressure can be measured accurately, compliance can be calculated from a very simple lumped model (Equation(25)). However, these conditions are often not met and naive applications of this simple model can lead to meaningless results.  2- Particular emphasis was placed on using flow and pressure measurements in compliance calculations, since previous researchers had reported success with this method. However, this study found that using flow and pressure measurements to determine compliance is complicated due to three important issues: a) Leakage Since the transient flow due to compliance is very small compared to the flow into the system, even a leakage as low as 1% of the flow (which may generally be negligible from a performance point of view) affects compliance estimation. Therefore, in general it is important to use a model that takes leakage into account. However, leakage is difficult to model correctly, since not only are there many possible sources of leakage, but they are also likely to exist simultaneously. Furthermore, despite the fact that all types of leakage distort compliance measurement by "appearing" to increase or decrease it, they require different types of models. For example, leakage flow through a hole to the outside may be laminar or turbulent depending on the size of the hole, but it only depends on the pressure difference on both sides of the hole. On the other hand, leakage between moving surfaces depends on the relative motion of the surfaces as well as the size of the gap between and pressure across them. If the surfaces are sealed, then  125  the leakage is also dependent on the seal performance. Leakage in a valve is even more difficult to model since depending on the valve type, there are a number of moving surfaces and pressures involved. A given hydraulic subsystem may have a number of valves, moving parts and connectors that may each contribute a small amount to the leakage. Distinguishing these sources of leakage is difficult because, the individual contributions may be too small to detect, and they are all slightly varying functions of the actuator pressure. If these sources of leakage cannot be distinguished and correctly modeled, the transient flow due to compliance can not be measured correctly. b) Flow Measurement Errors Very accurate flow measurements are essential to correct compliance calculation, but since the transient flow is very small compared to the flow into the system, common flow measurement techniques, such as those relying on the valve characteristics may not provide the required accuracy. In the experiments performed, it was shown that even the resolution of the 12-bit Analog to Digital converter generated errors as large as the range of compliance of the trouble-free system. Analysis done on the data from the CAT 215 B excavator, indicated the same level of error for that machine. However, in large hydraulic machines, the fluid volume may be sufficiently high to make this level of error acceptable. c) Inertia and Turbulence in the Fluid Other sources of impedance in the hydraulic system (flow resistance and inertance) may result in pressure change in the fluid path. In that case, the pressure measurement at one point may not be an accurate representation of the pressure in the entire fluid path.  3- A literature review and study of the system components revealed that compliance of a hydraulic subsystem is composed of a number of pressure dependent elements. The elements relevant to fault detection are, the fluid volume and bulk modulus, the air volume in the fluid, the flexible hose volume and hose bulk modulus. All of these elements are dependent on pressure in one way or another.  126  4- A major contributor to compliance is the entrapped air in the fluid. However, despite years of studying the behavior of air in the fluid, researchers do not fully understand the details of the interaction between air and the fluid. There are no analytical models available to describe the air solution into and evolution out of the fluid or the pressure dependence of the hose bulk modulus.  5- Existing models that attempt to accurately describe even a subset of the pressure dependent elements of compliance are complex and nonlinear. However, it was shown here that when the compliance of hydraulic machines is studied from a fault detection point of view, the model can be significantly simplified by focusing on those elements that are relevant to fault detection and using actual measurements instead of model predicted values as much as possible. A model that includes compliance and the most likely sources of leakage was proposed (Equation (34)). It was shown that even after introducing the pressure dependence of components in compliance into the model, the model was still linear in the parameters in most cases (see Equation (35)). This indicated that if the parameters are identifiable, in most cases a simple and inexpensive RLS algorithm may be used to for parameter identification. However, in a few variations of the proposed model, nonlinear estimation may be required. If the volume of entrapped air in the system under isothermal conditions is to be estimated, Equation (35) can be used.  6- Identifiability of the parameters of the model could not be fully tested here due to very low measurement accuracy. In this study, lack of measurement accuracy was a fundamental problem and all estimation and calculation methods were limited by this lack of accuracy. During short time periods when accuracy permitted, some success was achieved with a simplified version of the original model which only attempted to estimate the slip leakage across the piston and the effective compliance. However, due to the dynamic properties of the seals in the actuator, using simple models for leakage did not seem to be always reliable. 127  7- To eliminate the need for modeling the leakage across the piston, a novel method of estimating compliance of both sides of the actuator was proposed. This method was found to be more sensitive to the flow measurement errors than a simple "leak plus compliance" model, but it seemed to be a good way of avoiding problems with leakage.  8- The compliance values calculated based on a simple linear model that did not take air solution into account, were in reasonable agreement with measured values. However, only small portions of the data, where measured values were not corrupted by measurement errors, could be used. Nevertheless, the results seem to indicate that air solution does not prevent obtaining "reasonable" compliance values. This is because in the context of a fault detection system, "reasonable" values do not need to be very accurate as long as the order of magnitude of the important elements can be determined correctly.  9- While this study encountered particularly difficult problems in using the valve characteristics for flow calculations in the hydraulic test-bed, the experience with the excavator also indicated that using the valve for flow calculations may be a problem. Not only did the flow calculations in the excavator lead to discrepancies, but the need for using the pump and two valve flows also indicated that several sources of error are likely to add-up and make the flow measurements particularly unreliable. This indicates that flow measurement errors may be even larger in more complex machines. Therefore, a practical method was proposed for a static test that avoided the need for using the valve characteristics. Another method was proposed that used the fluid sonic bulk modulus to avoid flow measurements altogether.  10- GA is a very powerful, but computationally expensive method that can be used for parameter estimation. Hybrid methods can be developed to combine the strength of GA for global search with the speed of local search. For example, GA can be used to find the initial search point near a global minimum and a hill climbing method can be used to find the minimum accurately and rapidly. While the 128  high computational requirements of GA is not ideal for on-line estimation, more efficient algorithms, parallel computing, and the increasingly powerful computers are likely to overcome this problem. However, selecting suitable operational parameters and termination criteria for GA can make the task of developing practical and robust GA based on-line estimation methods difficult. In particular, the random nature of GA search can result in volatility in the output for fixed operational parameters and termination criteria. Furthermore, preliminary results from this research indicate that the GA performance is reduced, when estimating more than one parameter.  6.2 Suggestions for Future Research 1- A survey of a number of heavy-duty mobile hydraulic machines should be performed to determine: • the expected range of compliance, operating pressure, and measurement accuracy in these machine to indicate if the required accuracy is practically available. • the likely failure modes of the machine and their impact on the machine compliance to identify the components of the proposed model that are really necessary. For example the change in the bulk modulus of a hydraulic hose due to wearing out should be studied to determine which of the constants in -K P  the model P =K ( 1.11 —e A  4  5  ) should be treated as time varying parameters.  • the likely sources and magnitude of leakage under normal operating conditions to indicate if using flow and pressure measurements can be practically used in compliance calculations. • the general levels of resistance and inertance in the fluid under normal conditions and likely failure modes to indicate if compliance can be calculated correctly from flow and pressure measurements.  2- The possibility of using a flow meter should be explored to determine if some of the problems with accurate flow measurement can be avoided. The flow meter accuracy of at least O.lcmVs would ensure that the flow meter does not add to the measurement errors in the system. However, the results of this work indicate that a very fast flow meter (higher than 100 Hz response) may not be necessary. 129  3- There are some basic issues regarding modeling that were only partially answered in this study due measurement inaccuracies that can only be answered if a problem free system is studied. For example: • Determine if the air solution and evolution is an obstacle to estimating the entrained air content. • Determine if using any nonlinear model produces better results than the linear models proposed. • Determine if the parameters of the proposed multi-parameter model can be identified, that is if various sources of compliance and leak can be distinguished from each other. If a problem-free system can not be obtained, it may be possible to use the same experimental setup as used in Experiment 4 with a prolonged period for main valve closure. A small accumulator, a known volume of air, or long and old hoses can then be used to further study the compliance models.  4- Further explore the idea of using "bleed valves" for a static test as explained in Section (5.3.2). The basic concepts for this idea were already partially tested in Experiment 4, but can be easily tested further by a similar setup. The two "bleed valves" may be simulated by a manually controllable, fixed, and known source of flow. In this case, uneven loads on the actuator, air in the system, and old hoses can be used to determine if these problems can be detected by this simple system.  5- Further explore the possibility of using the fluid sonic bulk modulus as explained in Section (5.3.3) to determine if this method is a practical alternative to using flow and pressure measurements in on-line measurement of compliance.  6- Due to numerous problems with using flow and pressure measurements in compliance calculations, other alternatives should be explored. For example, it was noted that during Experiment 3, there was an audible change in the characteristic sound of the system as the actuator moved. Considering that the compliance changes the characteristic (resonant) frequency of the system, there might be a possibility of using acoustic emissions of the system as a way of detecting major changes to its compliance. For 130  example, a contact microphone attached to a metallic pipe may be used to compare the acoustic emissions for different conditions. However, considering the complexities of the whole structure, it is possible that the changes in hydraulic compliance may be too subtle to be identifiable. Nevertheless, it is possible that the emissions can be at least used to monitor and record the number of cavitation (or small cavitations) which in itself may be an indication of system performance.  131  7 References [I] Astrom Karl J. and Bjorn Wittenmark. Computer Controlled Systems Theory and Design, PrenticeHall, Inc., 1984. [2] Astrom Karl J. and Bjorn Wittenmark. Adaptive Control, second edition, Addison-Wesley, 1995. [3] Blackburn, John F. Fluid Power Control, Technology Press of M.I.T., Cambridge, Mass. ,1960. [4] Bowers, E.H. et al "Electrically Modulated Actuator Control" Proc. Fluid Power International Conference, London, England, pp01-09, June 1972. [5] Brady, James E and Gerard E. Humiston. General Chemistry principles, and structure, third edition, John Wiley &Sons, 1982. [6] Davis, L.D. and M. T. Streenstrup. "Genetic Algorithms and Simulated Annealing: An Overview", Genetic Algorithms and Simulated Annealing, Morgan Kaufmann Publishers, 1987. [7] Goldberg, David E. . Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-  Wesley, 1989. [8] Hayward, A.T.J. "Aeration in Hydraulic Systems", Conf. On Oil Hydraulics, Inst, of Mech. Engr, London, pp. 216-224, 1961. [9] Hayward, A.T.J. Combating a problem, Air bubbles in oil, National Engineering Laboratory, East Kilbride, Glasgow, Scotland. [10] Holland, John H . Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence, The University of Michigan Press, 1975.  [II] Jinghong, Yu , Chen Zhaoneng and Lu Yuanzhang. "The Variation of Oil Effective Bulk Modulus With Pressure in Hydraulic Systems", Transactions ofASME, Vol. 116, March 1994. [12] Korn, J. Hydrostatic Transmission Systems , Intertext Books London, 1969. [13] Ljung, Lennart and Torsten Soderstrom. Theory and Practice of Recursive Identification,. MIT Press, Cambridge, Mass, 1983. [14] Ljung, Lennart. System Identification Toolbox Users Guide For Use With Matlab, The Mathworks  Inc., 1993. [15] Magorien Vincent G. "How hydraulic fluids generate air", Hydraulics and Pneumatics, June 1968 pp. 104-108. [16] Magorien Vincent G. "Effects of Air on Hydraulic Systems", Hydraulics and Pneumatics, October 1967 pp. 128-131. [17] Martin, Hugh. The Design of Hydraulic Components and Systems, Ellis Horwood, 1995.  132  [18] Martin, Hugh, "Effects of change in transmission line (hose length) on dynamic performance of hydraulic systems", National Conference on Fluid Power, 1981. [19] Merritt, Herbert E.  Hydraulic Control Systems,  [20] Press, William H. Numerical Recipes Press, Cambridge, New York, 1992.  John Wiley &Sons, New York, 1967.  in C: the art of scientific computing,  [21] Rendel, D. and G.R. Allen. "Air in Hydraulic Transmission Systems", 23, p. 337, 1956.  Cambridge University  Aircraft Engineering,  Vol.  [22] Schetz Joseph A. and Allen E. Fuhs. Handbook of Fluid Dynamics One, Fundamentals of Fluid Dynamics, John Wiley &Sons, 1996.  and Fluid Machinery,  Volume  [23] Schetz Joseph A. and Allen E. Fuhs. Handbook of Fluid Dynamics Three, Applications of Fluid Dynamics, John Wiley &Sons, 1996.  and Fluid Machinery,  Volume  [24] Sepehri, Nariman, G.A.M. Dumont, P.D. Lawrence and F. Sassani "Cascade Control of Hydraulically Actuated Manipulators", Robotica Volume 8, pp 207-216, 1990. [25] Sepehri, Nariman. Dynamic Simulation and Control of Teleoperated Heavy-Duty Manipulators, PhD thesis, Univ. Of British Columbia, Vancouver, B.C., 1990.  Hydraulic  [26] Songnian, Li, Ge Sihua, Shi Weixang, Xi'an Jiaotong. "The Technique for the On-line Measurement of the Bulk Modulus of Hydraulic Fluid Pe", Chinese Journal of Mechanical V.25, n3, Sep. 1989. [27] Sorenson, Harold W. Parameter  Estimation Principles  [28] Stearns, Samuel D., and Ruth A. David. Upper Saddle River, NJ, 1996. [29] Sullivan, James A. Fluid Power: Englewood Cliffs, N.J. ,1989.  and Problems,  Signal Processing  Algorithms  Theory and Applications,  Engineering,  M. Dekker., New York, 1980. in Matlab.  Prentice Hall,  Third Edition, Prentice Hall,  [30] Wan, F.L.K. . Genetic Algorithms, their Applications and Models in Nonlinear System Identification, Master's thesis, University of British Columbia, Vancouver, B.C. 1991. [31] Weber, Harold C. and Herman P. Meissner. Thermodynamics edition, John Wiley &Sons, Inc , New York, 1957.  for Chemical Engineering,  second  [32] Wright, W.A. "Prediction of Bulk Moduli and Pressure-Volume-Temperature Data for Petroleum Oils", ASLE Transactions, Volume 10, pp 349-356, 1967. [33] Yeaple, Franklin D. York, New York, 1996.  Fluid Power Design Handbook.  Third Edition, Marcel Dekker Inc., New  [34] The Student Edition of MATLAB High-Performance Numeric Computation and Visualization Software, Version 4 User's Guide, Prentice Hall, Eaglewood Cliffs, NJ, 1995. 133  Appendix A  A . l  Recursive Least  Squares  The following is a discussion of the recursive least squares (RLS) method and some of its properties that are interesting to this study.  Suppose a system has input signal fu(t)} and output fy(t)} sampled at time intervals / =1,2,3,4,... where the sampled values may be related by a model that may be described with the difference equation y(t) + a,y(t-l) + .... + aj(t-n) = b,u(t-l) +b u(t-2) +...+ b u(t-m) + v(t), 2  m  where v(t) represents an unknown noise or disturbance. If ff' =  (a,....a„b,...bj  and ¥ (t) = (-y(t-i)  -y(t-n) u(t-l) ... u(t-m)) ,  then the above equation can be rewritten as y(t) = ff <p(t) + v(t).  The vector ff is the parameter vector which is to be estimated from the sampled values (measurements) of the input and output signals. In the Least Squares Method the parameters are estimated by determining the vector 0such that the sum of the squares of the difference between the output value estimated for y(t) based on the values of #and the measured values oiy(t) is minimized over all the N measurements. This sum of squares can be expressed mathematically as the criterion function  N  i  The values a are used to weight different observations and may or may not be used depending on the t  134  application. Minimization of V(6) leads to  9(A0=[E ^(t)^ {t)Y'J2 a^(t)y(t) T  (=1  (=1  if the inverse exists. For  the recursive version of the above relationship leads to  8(o=8a-i) -l-/?-(o<p(oa [j<o-e 'a-i)cp(0] 7  ,  t  +  <  where -1) l[a,(p(0(p (0 -/?(/ -1)]. t  /?(/)  r  +  A variation of the above method that avoids inverting the R matrix at each step can be derived by allowing  P  r  t )  A -\ t R  = [ Ea <P(*)<P (*) ]"' k=\ R  t )  T  which leads to 6(7)=6(/-1) L(t)[y(t)-6V l)(p(0], +  l/a p (/)?(/-l)(p(0 r  (+(  />(/-i)<p(0<p (W-i) ^(0=^-1)l/a (p W-l)(P(/) r  r  (+  135  If the parameters to be estimated are changing with time and it is desirable to discount the effect of old measurement a "forgetting factor" A may be introduced to the criterion function so that  ^0)=E^"'«,[y(O-0 <P(O] i r  2  According to Ljung [14] with a forgetting factor of X, the measurements older than the "memory horizon" T = 1/(1-1) carry a weight of less than 0.3.  Minimizing the criterion function as before leads to the following set of equations or algorithm  B(/) =6(t-1) *K(l)bil) -<P (')6(( -1)],  (49)  r  *M-  •  JW-rL[*M)X(t)  (=0)  ft'-'ttfl^W-') ]. X(t)/a+<p (t)P(t-\)(p(t)  (  5  1  )  r  The measurement weights a, can be omitted by letting a, =1 for all t without loss of generality. The details for the derivation of the above results can be found in [13].  If the system has more than one input, it may be described as y(i) = 6,<p,(0 6 <p (/)+...+0 (p (z) where +  2  2  n  n  y(i) is the observed variable or output for the time step /' (z could also be a general index in regression models, but for dynamic system identification it usually denotes time step), (p,, (p , .... (p„ are known 2  functions which may be themselves functions of other known variables , and 6,, d ,... 6„ are the 2  parameters to be estimated. In vector form the inputs at time step / and parameters may be described as (f(i)=[  <p,(i) <p (i) ... <p„(i) ] 2  and  136  6 = [6, 6 ... 2  dj.  Astrom et al [2] derive the same RLS algorithm as the Ljung et al [13] algorithm (the weights a, are omitted) in which #and cp are the above vectors instead of scalars, with cp '(l) 7  *(0  \y(\)y(2)...y(t)Y  , P(t) = [ 0 ( / ) O W ] - ' , and Y(t) = r  In this format the least-squares criterion function is 1^  ^(e,o4ELv(o-(P(0 'e] 7  2  2 ;=1 and the LS estimate is  e = (O'ordyr which is only valid if  Pit)  =[*(/)*(0]" 7,  1  = [ £ q^VO')]"  1  (52)  (=1  exists or <F& is non-singular. This is the condition for the existence of a unique set of parameters to minimize the criterion function. This condition is called the excitation condition. From the equation above it can be seen that if t<n then & & is singular, so to start the calculation in a RLS algorithm the starting step t has to be larger than n for which he initial conditions can be expressed as 0  P(t ) = [d> (tMQY T  0  l  and  d(t ) = P(tMt )Y(t ) Q  0  0  .  Some of the important issues regarding LS and RLS estimations are stated here without the derivations.  Astrom et al [2]state that if the initial condition P(0) = P , where P is positive definite, then for a large 0  0  P the estimate can be made to be close to the true value. Furthermore, the condition for an unbiased 0  estimate for the parameters of a LS estimator is that the error in the model v(i) are independent randomly 137  distributed with zero mean and variance a and given this assumption the covariance of the estimate is 2  cov(6(0) = o ($ '$)- = 2  7  1  —  .  If the error v(i) are correlated or have nonzero mean the LS estimate will be biased. They point out that the above relation shows that the precision of the estimate depends on the rate of growth of P~' and if the observations cp(i) do not grow faster than lift  , then the estimate variance does not go to zero with  additional observations. The matrix P is also known as the covariance matrix since it is related to the covariance of the estimated parameters.  Intuitively, this algorithm starts from an initial estimate for the parameter 6(7-1) for time step t-1. Then, based on the error, y(t)-(p (t)Q(t-l) , or the difference between the measured output and that calculated T  from the estimated parameter, it improves that estimate with a gain factor of K(t) which is proportional to the input <p(t) and the P. If the parameter changes abruptly, the method recommended by Astrom et al [2] to make the RLS algorithm adapt faster to the parameter changes is to use covariance resetting which is letting resetting covariance P to al where  is a large number and / is the identity matrix.  According to Sorenson [27], if (5 is not deterministic and is an explicit function of the measurements, then <£and v are not independent and the LS estimate for the parameters is biased unless E[ (&R 0)'' 1  v]  = 0, where R is the covariance matrix for the noise v. This is the orthogonality  condition which according to Sorenson [27] is not only difficult to prove, but generally untrue and thus the estimator is to be expected to show satisfactory asymptotic properties only under specific conditions that will not be discussed here. A sufficient condition for obtaining a unique solution for the parameter unbiased estimate is that the error v and measurements <3> are independent.  138  One practical way of determining if the excitation condition is met for a given set of measurements is to look at the "condition number" of the measurement matrix <P(t). This condition number is also the square root of the condition number of <P(t) &(t) which should be inverted to get the covariance matrix P(t) = T  [<p(t) 0(t) ]'' . Sometimes, although the measurements appear to be changing with time, they may be r  actually linearly dependent so that the measurement matrix is close to being rank deficient from a numerical point of view. The condition number of a matrix, which is a ratio of its largest to smallest singular values is a measure of how close a matrix is to being ill-conditioned. The larger the condition number the closer the matrix is to being ill-conditioned. The singular values of a square matrix M can be found by using a "singular value decomposition" of M such that M = UDVfov a diagonal "singular value" D matrix and unitary U (unitary :: LfMJ" ) and ^matrices. The singular values are then, the 1  diagonal elements of D. A more detailed explanation of the "condition number" can be found in [20] and [28]. While the condition number is a relative value that depends on the machine precision, [28] suggests that 10 is a suitable limit for unacceptably high condition number. 4  A . 2 The R L S Algorithm Used in This Study To estimate parameters of the proposed models in various experiments, the standard Recursive Least Squares (RLS) algorithm described above with equations (50), (51), and (49) was used with all observation weights a, set to one. The forgetting factor X was set depending on how quickly the algorithm was required to track the parameters of interest. The algorithm was created in Matlab [34] and its correctness was tested by creating an example problem. The results were also compared to those obtained from standard Matlab [34] functions such as the arx() function.  139  A.3 The Effect of Genetic Operators on Schemata The following discussion demonstrates the effect of the individual genetic operators on the schemata in the population. The Effect of Reproduction on Schemata  Let n denote the population size, m(H,t) denote the number instances of schema Hat time t, f(H) denote the average fitness of all strings that contain the schema H and Yfi denote the sum of fitness values of all population members. Since the reproduction probability of a string with fitness / is/?, =//£/; then m(H,t+l)=m(Ht).n.f(H)lYf . i  If average population fitness is  / = E//»,  then  m(H,t+l) = m(H,t)£Q. f This shows that a specific schema grows at a rate proportional to the ratio of its "average fitness" to the population average fitness. Therefore the number of schemata with above average fitness will grow over the generations and that of schemata with below average fitness will be reduced. Since all schemata within all strings are treated with the same operations, the strings with the fittest schemata will dominate the population over the generations. Goldberg [7] argues that if a schema remains above average by a constant c f , then the above equation can be written as m(H,t+\) = m(H,t)^t = (i + ).m(h,i) f  which leads to  c  m(H,t)=m(H,0).(\ +c)'.  This indicates that the instances of fit schemata grows exponentially as a geometric progression. However, Goldberg does not justify why c remains constant, since one would expect that as the population becomes fitter, the average fitness £f/n should grow and c should become lower. The implicit assumption might be that the average fitness of strings containing this schema grows as well.  140  T h e Effect o f C r o s s - o v e r on S c h e m a t a  Cross-over splits and re-combines randomly selected pairs of strings that survived reproduction with a probability p at random positions to produce new strings with new genetic sequences. The idea behind c  this is to search new parts of the parameter space, but this search is not entirely random. For example, as cross-over splits up chromosomes, it also splits up and unless two identical strings are crossed, destroys some of the schemata contained in the string. However, unless the population is converging, the chances of identical strings being crossed is low. Since the cross-over position is selected randomly, one can conclude that schemata with higher defining length are more likely to be split or destroyed by cross-over. Consider the schemata HI, H2, and H3 and the individual A of string length / in a population A. Let /=8 and the individual and the schemata be A  =00101011  H2  =  *oi*****  H3 = *01*1**1. It can be seen that all three schemata are contained in the string A. There are l-\ = 8-1 = 7 possible positions for cross-over to split the string. While, only a cross-over at position 6 destroys HI, H3 will be destroyed by crossover at positions 2 to 7. Thus the probability p that HI, H2 and H3 are destroyed are d  1/7, 2/7 and 6/7 respectively or p = 6(H)/(l-l). The survival probability of the schema is therefore d  p =l-p = 1- 6(H)/(l-l). With the cross-over probability of p , the survival probability of a any schema s  d  c  will be given by b(H)  l-\  From the above it can be seen that the combined effect of reproduction and cross-over on the number of instances of the schema H from generation to generation is m(H,t+\) z m(H,t).£Q[\-p  f  . M  This means that schemata with above average fitness and low defining length grow exponentially 141  through reproduction and cross-over.  The Effect of Mutation on Schemata Mutation changes (mutates) the allele of each genes of each individual with a mutation probability p . m  Thus for a schema to survive mutation all 0(H) relevant alleles (bits) must survive mutation. Thus for mutation the survival probability of a schema is p = (l-pJ . 0<H>  s  For p « m  1, p can be approximated s  by l-0(H).p . Mutation probability is generally kept low to avoid destabilizing the population. m  A.4 Description of the G A Selected for Implementation The GA algorithm used here is essentially the same as the algorithm suggested by Goldberg [7] with some small modifications. Goldberg's algorithm has a few features that are not common to all other algorithms. The mutation probability of 0.003 and a crossover probability of 0.6 are selected as suggested by Goldberg. In the GA, there are many parameters that may be modified to alter or improve performance. In this work every effort has been made to select a reliable variation of the genetic algorithm, but large amounts of experimentation to attempt to improve performance is avoided. The GA program was written using the C programming language and was tested by using it to find the maximum value of simple algebraic functions with multiple local maxima.  Selection of Parents (the Slotted Roulette Wheel Method) Two parents are selected by multiplying the sum of thefitnessvalues of the entire population by a random number between 0 and 1. The fitness values of all the individuals are then accumulated sequentially and compared to the product obtained before, until the accumulated value is greater than or equal to the product. The last individual who's value was being accumulated is selected. Since the larger the individual's fitness, the higher probability it has to "fall" in the range that would make the accumulated sum exceed the product, this method selects the fitter individuals with a higher probability. 142  This method is shown graphically in Figure (37). It can be seen form the above process  £ f = Sum of fitness values for all population members  Lf-o  that although the total sum of the finesses Rand'0m_number x £ f  is calculated only once per generation, the second accumulation sum is performed once per selection.  n  fl  o  f4 fS f6 n i i i i i  U M I  1  re  f9 i i  II  Individual 3 with fitness fi is selected  Mutation, Reproduction and Cross Over  Figure 37 The Slotted Roulette Wheel Method After the parents are selected, a bit position is randomly selected for cross over to produce the offsprings. The cross over has a given probability of occurring. As the cross over is performed every bit is also mutated with a given probability. Other methods only select a single bit position randomly to mutate. Cross over and reproduction are done simultaneously in this algorithm. This is different from ranking methods where the number of the offsprings of an individual is directly proportional to its fitness.  Scaling the Fitness Values During the first few generations, it is possible that a "super individual" may dominate the entire population very rapidly, without giving the algorithm a chance to search. On the other hand, at later generations, the average fitness may be close to the best fitness and individuals with average fitness may be given the same chance to reproduce as the best individual. A simple scaling mechanism of the fitness is used to ease this problem by scaling each fitness/to /'such that f = af + b, where: f'ave  =  f'min  >  f max  fme>  since the average fitness should not change.  0 , since only positive finesses are used in this algorithm. ^favei  C is the ratio of the expected offsprings of the individual with maximum fitness to an individual with 143  average fitness. Statistically, the member with average fitness should produce one offspring. Goldberg suggests C=1.2 for populations of about 50. (This value is kept constant for this implementation.) With some algebraic manipulation a, and b are derived to be: f  (C-iy v  a =  /  "ave  -/  -'max •'ave  ,  i  b =  (f  -Cf  -'ave^max  /  )  J ave'  -/  .  -'max •'ave  Algorithm Termination Criteria There is no commonly defined termination criteria for the genetic algorithm. Since the population is intentionally randomly modified each generation, the entire population will never converge. One possible heuristic criteria to determine convergence is to wait for the generation when a large portion of the population has become close to the fittest individual (e.g. 80% of the population reaching 90% of the fittest individual). On the other hand if it is desired to control the maximum run time of the algorithm, it could simply be run for a fixed number of generations. For parameter estimation purposes, where the algorithm is run on a "moving window" of data, it is easier to use a fixed number of generation.  Additions to Goldberg's Algorithm To ensure that the fittest individual from a generation is not lost, after the new generation is formed, the fittest individual of the old generation is compared to the fittest of the new and if the old individual is fitter, it is used to replace the least fit individual of the new generation. To make the implementation versatile the fitness values are sorted after each generation. This makes selection of any number of the fittest individuals, replacement of any of the least fit individuals and determination of convergence quite easy. Sorting is an integral part of GA with ranking method, but in Goldberg's algorithm it is not necessary. Thus, from an execution speed point of view, adding'this additional step is not desirable.  144  Appendix B Additional Details on the Bulk Modulus Models Studied  1- Model by Rendel et al [21] Rendel et al [21] use Boyle's law and  n __ o v  dp  dV  , total volume V = V, + V', and the isothermal values  for P and V to get = az*  V  [l--^^],  v  +  which after differentiating gives dV  — = dP  P  n  V  i  <  v  +  v  —  or  P,  P  2  ,  v  a  p  p = p, [—- + 1] / [—- + — P.], V  V e  2'  l  y  g  h  P  where the subscript 0 indicate the vales at atmospheric pressure.  2- Model by Hayward [8] Hayward [8] points out that Rendel et al [21] introduce an error since their definition of  a He  _  V  °  d P  dV  does  not match either the tangent or secant definition of bulk modulus. Using the volume at atmospheric pressure V , and neglecting the volume change due to the change in pressure may be valid for the fluid, 0  but is not valid for the gas portion of V . Using p =-  v d p  0  al [21], except for integrating p - ^ V  /=  -P/Q  time accurately as V,=V. e  '  P  dV,  dV  to get an accurate expression for the liquid volume at a given  ' and letting  V = V.e-**' '  and a similar method to that used by Rendel et  >o  Oil  +  p  he arrives at 145  k =  P/ C  V,e -W, 'o  g  P  g  2  Since the exponential term is small compared to others, this can also be written as  3- Model by Jinghong et al [11] Jinghong et al [11] derive a comprehensive model for the effective bulk modulus of the combined air and fluid which includes a coefficient to estimate the portion of the air that goes in and out of solution. They assume that the container is not flexible and V= V = V,+ V so that equation (11) above becomes t  g  Using the isentropic compression relation for gases and adding the atmospheric pressure (10 in Pascals) 5  explicitly to the gage pressure P, they use the relations (P+10 ) V = k, 5  Y  g  and  fl  = (P+10 ) y 5  as well as a coefficient c, for air bubble volume variation due to solution (evolution) of air bubbles. Thus if V is the volume of entrained air at P = 0, the change in entrained air volume at pressure P will g0  be- c, V P and the volume change must obey the gas law so that g0  (v - v pyio  = k(P+io )  s  g0 Cl  or  vy  5  g0  g  V = V (1-c, P)/ (1 + 10r*P)''r. g  Using these relations for V and  g0  J3 and letting R= V /V, they get  g  g  g0  p/1 +10" P) 5  1+,/Y  J  P= E  (1 + 10- P) 5  1+1/Y  +10- R( 1 -C,P)(P/Y -10 -P) 5  5  146  where R, y, c , and /?/ are parameters to be estimated. Since this model is highly nonlinear, Jinghong et l  al [11] report that the estimation method was selected to avoid differentiating the objective function. Assuming that the change in the volume of the entrained air with the change in pressure can be described as -c V P, where c is another air bubble volume variation coefficient, they derived a simpler estimate 2  g0  2  of gas volume V =V g  -(c, + c,)V P  g0  g0  and a simpler model than equation (53) for fi  e  _\0\\+\Q- P)  p  5  AP +BP+C  e  where A= (c, + c^R/fi,  2  B=—(1 -R)+(—~-)( , c  P,  P/  +c )R  , and  7  C=—(1 -/?)+—  P,  Y  Y  However, the assumption that the change in air volume due to pressure change is proportional to the product of the pressure and the original volume does not seem to be reasonable since considering Boyle's law, Vgo P = V P, 0  g  and the change in gas volume Vgp P = (V 0  g0  gives  AV = V -V g0  - AV) P  or AV=  g  V  g0  (P-PJ  /P,  -c V^ P as it was assumed above.  which is not a close approximation to  2  An even simpler version of the model from equation (53) can be obtained by assuming the air bubble volume does not change or A= c = c = 0. The ultimate simplification of the model assumes the I  2  volume of air bubbles is negligible or R = 0 which gives fl = 1/B = /?,. e  147  Appendix C The Geometrical Properties of the Hydraulic Test-bed The geometrical configuration of the actuator and the arm it moves as well as some of the dimensions in the test-bed system is shown in Figure (38). Here the side of the piston attached to the rod is called the rod  side  and the other side  is called the head  side.  In the  diagram the distance between the rod side of the piston and the end of the chamber is labeled L , the distance R  between the head side of the piston and the end of the other chamber is labeled L , the length of the piston and extended rod is labeled b, the angle that is measured H  as the arm moves is labeled 6, and the angle required to solve for b is labeled a. When the arm is at vertical or zero position c?is zero and a = 60.3 °. When the piston is at the far end inside the cylinder and L  H  = 0 ,  the length  b  is 41.27  cm,  so in general L =  b-  H  41.27  and L = R  Stroke - L = H  66.67  - b.  The following table (Table (2)) shows geometrical dimensions relevant to calculating the volume of the fluid and various portions of the fluid containers. For a typical bulk modulus value of 1.517  x 10 Pa, and the volume of fluid shown in the table for the 9  head side, the expected compliance of the pure fluid (which is a good basis for comparison) is COM  =  V JP fl  fluid  =  258.34  cm 71.517x 3  10 Pa 9  =  1.7x  ICr m /Pa.  148  13  3  '.  Head Side  Head Side +  Rod Side  Accumulator  Piston area  11.4 cm  11.4 cm  9.42 cm  Length of piston fluid column for 6 = 0  L = 7.52 cm  L = 7.52 cm  L = 17.8*8 cm  Piston volume on each side for 0 = 0  85.73 cm  85.73 cm  168.43 cm  Length of fluid path in metallic connectors  65.5 cm'  Volume of fluid in the metallic connectors  2  2  H0  2  H0  3  R0  3  15.5 cm *  3  68.5 cm*  15.5 cm *  54 cm *  46.67 cm * 2.76 cm *  48.84 cm *  2.76 cm *  38.48 cm *  Length of hose  71 cm*  71 cm*  73 cm*  69 cm *  Volume of fluid in the hose (unpressurized)  50.62 cm  12.99cm  50.62 cm  12.99cm  49.17 cm  Additional hose volume (pressurized)  2.33 cm *  2.40 cm *  2.33 cm *  2.40cm *  2.26 cm *  Hose Volume (pressurized)  52.95cm *  15.39cm *  52.95cm *  15.39cm *  51.43 cm  Volume of fluid in hose (pressurized)  68.35 cm  Total fluid volume (pressurized)  117.77+L x 11.4 cm  3  3  73 cm* 3  3  3  3  3  3  3  3  3  68.35 cm  3  H  cm 203.50 cm  KC  3  3  89.91+L x9.42 R  cm  3D  3  248.61 cm +V -V  3  3  3  117.77+L x 11.4 +V  3  3  51.43 cm  3  H  Total fluid volume for 0 = 0 (pressurized)  3  3  3  3  3  3  acc  258.34 cm  3  accair  Table 2 Measurements for fluid volume calculations in the test-bed * Hose 1 : Hose type PARFLEX 540N-3; internal diameter = 0.476 cm; internal area = 0.178 cm  2  expansion coefficient = 1.8 cc/ft at working pressure. Maximum working pressure 75.57 x 10 Pa 6  * Hose 2 : Hose type PARFLEX 540N-6; internal diameter = 0.95 cm; internal area = 0.713 cm  2  expansion coefficient = 4.1 cc/ft at working pressure. Maximum working pressure 17.24 x 10 Pa 6  ° Accumulator volume V  is 0.946 liters and is full with gas at 250psi or 1.72 x 10 Pa. 6  acc  Using Boyle's Law : V P 0  0  = V, P,, and V  = 946x 1.72 x 10 IP cm at pressure P s  accai  3  Equivalent volume at atmospheric pressure is 16.27x10" m which represents a percentage of 3  3  gas content of 16.27xl0"7 (16.27xl0" +248.61xl0" ) or 98.5% 3  3  -IP  Hose bulk modulus using  P =(614D-2.18)/' /i  15 max  (1.11 -e ™), P  -2P  hose7  p =45.38(l.ll-e w  1551  )  MPa  -2P  hose2 P =261.28(l.ll-e w  1724  ) MPa 149  for pressure P in MPa  Appendix D  D . l Data Collection A sampling frequency of 1000 Hz and a 12-bit analog to digital converter (A/D) (1 sign bit) was used for data collection. Pressure Measurement  Pressure transducers measure various pressure values and the voltage output (± 10 Volts) from the transducer was fed to the A/D. P = pump pressure: range of 5000 psi over 11 bits or 4.888 psi or 33,701 Pa resolution. P  P = tank pressure: range of 500 psi over 11 bits or 0.4888 psi or 3,370.1 Pa resolution. T  P = pressure at the head side of the piston, P = pressure at the rod side of the piston: range of 3000 psi H  R  over 11 bits or 2.933 psi (20,222 Pa) resolution. Some of the error in flow calculation is due to pressure measurement errors. Since the difference of the pressure across the orifice has to be calculated to obtain flow, any error in pressure measurement also results in errors in flow. In the worse case that these errors add, an error of about A 1SS,2T1 + '/ 33,701 X  2  = 2.7 x 10" Pa in pressure should be used in the relationship  Q=C A ^j-(P -P ) ej  0  ]  2  to determine the  flow error caused by pressure. Using the range of values used in this study, this results in a flow error of about 0 to lxl0' m /s depending on the orifice size and pressure range. 6  3  Spool Position Measurement A Linear Variable Displacement Transformer (LVDT) is used to measure the spool displacement on the valve with a range or 2.54 cm over 12 bits or resolution of 0.0124 mm. Angle Measurement A potentiometer (POT) was used to measure the joint angle of the arm with range of ±350 degrees over 12 bits (350 bits over 11 bits and one sign bit) which is a resolution of 0.17 degrees or 0.003 radians. Error in the pot itself should also be taken into account. According to [23], potentiometers can be a 150  major source of poor performance in electro-hydraulic systems that use potentiometers for linear or angular position feedback. Thus, a calibration curve for potentiometer was derived manually by other researchers using the test-bed and the error was determined as a deviation of range 0.25 to 0.3 degrees. In worst case the error in angle is then 0.47 degrees. Using the geometry of the actuator, flow errors of as high as 2.5x10" m /s can be calculated. 6  3  D.2 Flow Calculation Generally the manufactures provide an operation curve to show the relationship between the spool position of the valve (or the input voltage signal to the pilot) and the flow through the valve under steady state conditions. After analyzing some of the data collected on the test-bed and the manufacturer's specifications, it was noticed that the range of the input signal used (0-4 Volts with most of the actuation in the 2.8 - 4 Volt range) was very close to the valve's dead-band zone (0-3 Volts) according the manufacturer's data. Thus, according to the manufacturer's data, the flow through the main valve should have been near zero. This meant that the operation curve provided by the manufacturer could not be used in flow calculations. Since the equipment could not be altered and no flow meters were available, the only option available was to try to determine the valve characteristics experimentally. Thus, much experimental time was spent on trying to devise methods of obtaining a practical operation curve for the valve used. To determine the flow through a valve with a sharp edged orifice and for a given pressure drop across the valve, the valve characteristic equation  Q=C^^-(P^  -P ) 2  , can be used [19] where  the orifice area of the valve A and the discharge coefficient C should be known. 0  d  First Method To determine the valve characteristics experimentally, the value C A d  equation  Q=CjA ^-{P -P ) 0  x  2  0  can be determined from the  by measuring the flow through the valve and pressure drop across the  valve for a range of spool positions. In this case according to the manufacturer's data, the orifice size for 151  all ports of the valve are identical for a given spool position. Therefore, only one set of measurements was necessary to get the required relationship for all ports. The collected data could then be fitted experimentally with a polynomial. Judging from the manufacturer's curve and other flow-spool relationship curves found in the literature a third to fifth order polynomial was deemed suitable for this task. A simple method to determine the flow through the valve for a valve that is connected to the actuator is to measure the movement of the actuator under steady state conditions. This means that the relationshipC j P=Q-Ax can be used for the case where there is no pressure change ef ecljve  and  C  P  effecljve  = Q-Ax  = 0 which means  Q = Ax .  This relationship only holds if there is no leak  across the piston in the actuator. If there is a leak across the piston, then the correct relationship is Q  ±  Qi Ax, where Q, denotes leak in the cylinder which is a value dependent on the pressure on both =  sides of the piston and the velocity of the piston as described in Section (2.8.2.2).  Under the assumption that the equipment being used was free of defect, the leak-free relationship was used to derive a spool-flow curve. Based on the curve derived, some of the data collected for the experiments above were analyzed and some anomalies were found. For example, it was Tank  noticed that in the case when the rod side of the piston was "pushing", which is shown in Figure (39), the net calculated flow Head Side  Qi  Rod Side  into the head side of the cylinder was negative for both the case of , )<T \  Q,  increasing and decreasing pressure. Form Figure (39), which Pump  simply shows the case of Figure (23a) seen before, this means that  Figure 39 Rod side pushing  the flow that appeared to leave the piston on the head side was larger than the flow entering that side or  C P Rod  0  = Ajc  - Q  o  <  0 for P  o  >  0. Since the compliance  has to be a positive number, the only way to explain this result is that the assumption of zero leak was 152  incorrect and the correct relationship was  C P RoJ  o  = A^pc - Q  o  ± Q  t  .In this case the leak flow had to  be into the head side to arrive at a positive compliance, which meant leakage from the rod side to the head side. This is reasonable since if there is a leak, it should be from the side with higher pressure to the side with lower pressure and the pressure on the rod side was observed to be much larger than that of the head side. This is due to the fact that when the rod is pushing the force on the rod side should be larger than that of the head side and since the piston area on the rod side is smaller than the head side, then the pressure on the rod side should be much larger than that of the head side.  A further confirmation for leak from rod to head side was the observation that on the rod side a cylinder an excess flow of about the same magnitude of the deficiency flow on the head side was observed, or mathematically  A ^ - Q - g . - A^x - Q o  (  t  . Furthermore, such a difference in flow or leak was not  observed for the case where the head side was pushing. This is attributed to the fact that in that case the pressure on the head side and rod side of the piston is not very different, although due to the higher area on the head side, the pressure is higher on the rod side, even when the net force on the head side is larger and the head side is pushing.  Second Method To further verify the existence of the leak experimentally and to try to find a new method to calculate the flow through the valve in the presence of the leak, the procedure for finding the spool-flow curve under steady state conditions was repeated for each valve port. Figures (23a and 23b) show the four valve ports and their orifice area as a, and a for the cases of head pushing and rod pushing. The Figure also a  shows the respective pressures that should be used on both sides of the orifice area in the valve equation Q-CjA^-'P^  -P ) . This experiment indeed yielded four different spool-flow curves for the four 2  ports. The main difference was between the curves produced from the experiments where the head side was 153  pushing and experiments where the rod was pushing. As previously mentioned due to the symmetrical design of the valve, according to the manufacturer these curves should have be identical. The only explanation for the difference is the fact that the relationships used to determine the flow through the valve  Q.r iF A  .  A  R  *  =  Q  0  .  Qr i? A  >  = A  H  *  Qo  where incorrect since the leakage flow was not taken  into account. Thus, the four spool-flow curves produced, implicitly incorporated the leakage in the cylinder. It was hoped that if these four spool-flow curves were produced under pressure levels that were close to the pressure levels of the rest of the experiments, the implicit leakage incorporated in the valve curves for each ports would be sufficient to provide close to correct values for flow under the test conditions. However, further analysis of the test data with these spool-flow curves lead to similar anomalies as previously observed. The reason for this was that the experimental spool-flow curves could not take into account the dependance of leakage on pressure and piston velocity and the implicit leak built into them was too large or too little depending on the conditions.  Third Method Having exhausted all "easy" experimental options and not having access to a flow-meter, the only option left to determine a correct spool-flow relationship was to manually measure the volume of the fluid exiting the valve with the pump pressure on one side and atmospheric pressure on the other side for a variety of spool positions. This was achieved by opening the exit  Pump  LVDT  X  Spool position  Flow side of the actuator and capturing and measuring the fluid exiting  atmospheric  under steady state conditions as shown in Figure (40). To achieve as much accuracy as possible, the fluid exiting the valve was Scale collected for as long as possible and the temperature and weight of the collected fluid was measured and used to determine the volume,  154  ^3  Figure 40 Determining valve characteristics manually  This method yielded flow calculation accuracy of up to l x l 0' m /s for low flow rates which allowed 9  3  fluid collection for several minutes. However, for high flow rates, the container and scale limits only allowed data collection time of order of seconds which resulted in measurement errors of up to lxlO"  5  m /s in the worst case. Data was collected in 0.1 Volt intervals in the expected operating range, which 3  resulted in 17 points in the spool range of 0-1.2 mm. A polynomial was fitted to the data in a similar way as before which has an averaging effect on the error in the experimental data. The "spool versus orifice area curve" obtained in this way was compared to the previous attempts. It agreed quite well with the curves obtain in the case of the "head side pushing" as shown in (a) Spool vs Orifice area x Dischage Coefficient !  Figure (41a and b). In fact this curve  O +  Exit Pressure  1  was between the curve for the pressure side and the exit side. The  Pushing  fact that the values obtained are so  \  close to the "head side pushing" case indicated that the leakage across the  Pushing  |  i  i  i  i  ! i Spool (meters)  piston should be zero or negligible in that case. This is reasonable since as  (b) Spool vs Orifice area x Dischage Coefficient  x 10  discussed previously, the leakage is dependent on the pressure difference across the piston and in the "head side pushing" case, the pressure on the head side and the rod side of the  -6  Spool (meters)  cylinder are close as compared to the "rod side pushing" case.  Figure 41 Main valve characteristics obtained experimentally (a) Main valve characteristic from Method 2 (b) Main valve characteristic from Method 3 compared to Method 2 155  Figure (41a) also shows that there is a discrepancy between the values for the case when the valve port is connected to the pressure (pump) side and when it is connected to the exit side. This could be a sign of leakage or simply a sign that the pressures used for the valve equations are not exactly correct. For example, since the excess flow from the pump is leaving the valve in the same hose as the exit flow from the cylinder, it is possible that in practice, the pressure inside the hose on the exit side is higher than the tank pressure. This means that when calculating the orifice area of the port for the exit side, the pressure difference is over-estimated which leads to an orifice area which is under-estimated. Regardless of the reason, this discrepancy is an indication of additional measurement errors due to using the valve flow characteristics. Studying the deviation of pressure and exit side from the manually derived curve, in Figure (41b), shows that for the high flow rates, this error could be as high as 10% of the flow. Finally, since the valve is being used outside of its manufacturer's expected operating range, the extent to which the steady-state behavior of the valve agrees with dynamic in this range is not known.  D.3 Selecting a Suitable Filter to Remove Noise A suitable filter had to be selected to remove the noise without losing important information. To select the filter cut-off frequency, it was assume that any transients faster than the response time of the system are likely to be noise or not relevant to compliance calculation. However, the circuit characteristics were not known well enough to calculate the response time. The only reference found to the response time in the literature was in [17]. In talking about the pressure rise due to flow entering a container, Martin [17] writes "The time required to build up pressure during compressibility can be taken as between 0.03 and 0.05 s". This implies that any pressure ripples with higher frequencies are not due to compliance. It also implies that any frequency components of 1/0.05 s = 20 Hz and above in data used to calculate flow will not affect the actuator pressure. Thus if this type of ripple is not filtered from the flow data, it will simply manifest itself as a "meaningless" fluctuation in the compliance value obtained from C ~ . =—. P  To decide if there was useful information in the pressure data, the frequency spectrum of the data was 156  analyzed and a 280-290Hz component was found in the pump data which was attributed to the gear pump characteristics of 29 cycles/sec revolution speed and 10 teeth gear ( 290/29=10). However, since there was no indication that there was a corresponding ripple in the flow data, it was concluded that no relevant information would be lost by filtering out this ripple. After experimentation with various cut-off frequencies, fdters, and least squares polynomialfitsand differentiation methods , an FIR (finite impulse response) low pass digital filter with 10Hz cut off was found to be the most suitable method for removing noise from, and differentiating the data. The same cut-off frequency was applied to all experiments for convenience, although careful study of the data indicated that in some cases, a higher cut-off frequency might have been more suitable. The "remez" function in Matlab [34] was chosen for this task. The linear phase response characteristic of the FIR filter was desirable since the time delay due to the filter could be compensated for by introducing a time delay in the non-filtered data and the relative time of events in the data which is very important in this study could be preserved. While the compliance estimation should be performed on-line, it does not necessarily need to be performed in real time and introducing a time delay was acceptable.  D.4 Pressure Change Due to Flow Restrictions, Inertia Forces Determining the net resistance is not easy since the nature of flow (turbulent or laminar) and the type of all restrictions should be known. Determining inertance is not difficult, since it depends on values that are easily available. Flow Restrictions Depending on the type of flow different relations are used to determine the pressure loss in the pipes. The Reynolds Number is used to describe the range of flow velocities in the pipe where the flow nature changes from turbulent to laminar. The Reynolds Number, N , is described in Sullivan [29] with the R  relation N = vD/v, where D is the pipe diameter, v is the flow velocity and v is the kinematic viscosity R  of the fluid. According to Sullivan [29] the flow is laminar for N <2000 and turbulent for N z4000 and R  157  R  neither laminar nor turbulent in between these two values. According to Martin [17], normally for design purposes values larger than 2000 indicate turbulent flow. For the system used in this study v is about 3.2 x 10" m /s and Reynolds Numbers in the range of about 600-1300 was calculated which 5  2  indicate laminar flow. Right angle bends and T connectors in the circuit introduce a further pressure drop. According to Martin [17] the loss coefficients are difficult to obtain for many components and may have to be determined experimentally. Based on examples in [17] and the ranges of values used here, the loss due to resistive forces is expected to be in the order of 10 Pa which is small compared to the 10 Pa operating range. 4  6  Inertia Forces According to Martin [17], inertance is described with AP = L dQ/dt where L = Ip/a is the fluid h  h  inertance, p is fluid density, / is pipe length, a is the pipe area, Q is the flow into the system, and AP is the pressure drop across the pipe. Using some of the typical data from the test-bed system used L is h  expected to be about 1.8 x 10 Kg/m . For the flow ranges used in this machine pressure changes in the 7  4  order of about 10 with peaks of about 6 x 10" were calculated. Since the pressure range used in this 3  study was in the order of 10 , this effect was neglected. However, the magnitude of inertance related 6  pressure change was not low enough to indicate that it can always be ignored.  158  

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