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Fault monitoring in hydraulic systems using unscented Kalman filter Sepasi, Mohammad 2007-12-03

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 Fault Monitoring in Hydraulic Systems using Unscented Kalman Filter   By Mohammad Sepasi B.Sc., Sharif University of Technology, Iran, 2005       A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES  (Mechanical Engineering)       The University of British Columbia  November 2007 ?Mohammad Sepasi, 2007   ii   Abstract  Condition  monitoring  of  hydraulic  systems  is  an  area  that  has  grown substantially  in  the  last  few  decades.  This  thesis  presents  a  scheme  that automatically generates the fault symptoms by on-line processing of raw sensor data from  a  real  test  rig.  The  main  purposes  of  implementing  condition  monitoring  in hydraulic  systems  are  to  increase  productivity,  decrease  maintenance  costs  and increase safety. Since such systems are widely used in industry and becoming more complex  in  function,  reliability  of  the  systems  must  be  supported  by  an  efficient monitoring and maintenance scheme. This  work  proposes  an  accurate  state  space  model  together  with  a  novel model-based  fault  diagnosis  methodology.  The  test  rig  has  been  fabricated  in  the Process Automation and Robotics Laboratory at UBC. First, a state space model of the  system  is  derived.  The  parameters  of  the  model  are  obtained  through  either experiments or direct measurements and manufacturer specifications. To validate the model, the simulated and measured states are compared. The results show that under normal operating conditions the simulation program and real system produce similar state trajectories.  For  the  validated  model,  a  condition  monitoring  scheme  based  on  the Unscented Kalman Filter (UKF) is developed. In simulations, both measurement and process  noises  are  considered.  The  results  show  that  the  algorithm  estimates  the   iiisystem states with acceptable residual errors. Therefore, the structure is verified to be employed as the fault diagnosis scheme.  Five  types  of  faults  are  investigated  in  this  thesis:  loss  of  load,  dynamic friction load, the internal leakage between the two hydraulic cylinder chambers, and the external leakage at either side of the actuator. Also, for each leakage scenario, three levels of leakage are investigated in the tests. The developed UKF-based fault monitoring scheme is tested on the practical system while different fault scenarios are  singly  introduced  to  the  system.  A  sinusoidal  reference  signal  is  used  for  the actuator  displacement.  To  diagnose  the  occurred  fault  in  real  time,  three  criteria, namely  residual  moving  average  of  the  errors,  chamber  pressures,  and  actuator characteristics,  are  considered.  Based  on  the  presented  experimental  results  and discussions, the proposed scheme can accurately diagnose the occurred faults.     iv CONTENTS ABSTRACT........................................................................................................................ii CONTENTS....................................................................................................................... iv LIST OF TABLES...........................................................................................................viii LIST OF FIGURES .........................................................................................................viii   CHAPTER 1 INTRODUCTION 1.1 PRELIMINARY REMARKS ............................................................................................ 1 1.2 HYDRAULIC POWER SYSTEMS.................................................................................... 4 1.3 POTENTIAL FAULTS IN HYDRAULIC SYSTEMS ............................................................. 7 1.4 RESEARCH OBJECTIVES.............................................................................................. 9 1.5 ORGANIZATION OF THE THESIS ................................................................................ 11   CHAPTER 2 SYSTEM MODELING AND SET-UP CONFIGURATION 2.1 INTRODUCTION......................................................................................................... 13 2.2 EXPERIMENTAL TEST RIG ........................................................................................ 14 2.3 SYSTEM MODELING ................................................................................................. 21 2.3.1 Governing Equations ....................................................................................... 21 2.3.2 State Space Model............................................................................................ 27 2.4 MODEL VALIDATION................................................................................................ 29 2.4.1 Experimental Results ....................................................................................... 30 2.4.2 Frequency Range ............................................................................................. 32   v 2.4.3 Repeatability of the Experiments ..................................................................... 33   CHAPTER 3 IMPLEMENTATION OF A FAULT MONITORING TECHNIQUE 3.1 INTRODUCTION......................................................................................................... 35 3.2 CLASSIFICATION OF FAULTS .................................................................................... 36 3.3 OVERVIEW OF FAULT DIAGNOSIS APPROACHES....................................................... 37 3.3.1 Model-based Fault Diagnosis Approaches...................................................... 38 3.3.1.1 Quantitative Model-based Methods...................................................... 39 3.3.1.2 Qualitative Model-based Methods........................................................ 40 3.3.2 Model-free Fault Diagnosis Approaches......................................................... 41 3.3.2.1 Quantitative Model-free Methods......................................................... 42 3.3.2.2 Qualitative Model-free Methods........................................................... 44 3.4 KALMAN FILTERING IN FAULT DIAGNOSIS............................................................... 46 3.5 UNSCENTED KALMAN FILTER .................................................................................. 51 3.5.1 Unscented Transformation............................................................................... 52 3.5.2 UKF Algorithm ................................................................................................ 54 3.6 UKF APPLICATION IN THE HYDRAULIC SYSTEM...................................................... 57 3.6.1 The UKF Propagation ..................................................................................... 58 3.6.2 Simulation Results............................................................................................ 61 3.7 CONCLUDING REMARKS........................................................................................... 63   CHAPTER 4 ONLINE MONITORING OF THE HYDRAULIC SYSTEM 4.1 INTRODUCTION......................................................................................................... 64 4.2 EXPERIMENTAL RESULTS AND DISCUSSION ............................................................. 65   vi 4.2.1 Leakage Faults................................................................................................. 66 4.2.1.1 Actuator External Leakage at Chamber 1............................................. 66 4.2.1.2 Actuator External Leakage at Chamber 2............................................. 70 4.2.1.3 Actuator Internal Leakage..................................................................... 74 4.2.2 Load Faults...................................................................................................... 77 4.2.2.1 Dynamic friction load ........................................................................... 77 4.2.2.2 Loss of Load ......................................................................................... 79 4.3 FAULT DIAGNOSIS AND DISCUSSION........................................................................ 82         4.3.1 Leakage at Chamber 1..................................................................................... 82         4.3.2 Leakage at Chamber 2..................................................................................... 84         4.3.3 Dynamic Friction Load.................................................................................... 84         4.3.4 Loss of Load..................................................................................................... 85         4.3.5 Internal Leakage.............................................................................................. 87   CHAPTER 5 CONCLUSION 5.1 SUMMARY................................................................................................................ 89 5.2 CONTRIBUTIONS....................................................................................................... 91 5.3 SUGGESTIONS FOR FUTURE WORKS ......................................................................... 92  REFERENCES................................................................................................................ 94    vii List of Tables   Table 2.1: The numeric values of the hydraulic set-up parameters .................................. 27 Table 3.1: The MAEs of the pressures and actuator movement....................................... 59 Table 4.1: Multilevel leakage coefficients........................................................................ 67 Table 4.2: Increase in MAEs due to the external leakage at chamber 1........................... 70 Table 4.3: Increase in MAEs due to the external leakage at chamber 2........................... 73 Table 4.4: Increase in MAEs due to the internal leakage ................................................. 77 Table 4.5: Increase in MAEs due to the dynamic friction load ........................................ 79 Table 4.6: Increase in MAEs due to the load lost............................................................. 81 Table 4.7: Increase in MAEs due to the occurred fault .................................................... 83   viiiList of Figures    Figure 2.1: Schematic diagram of the experimental set-up ..............................................15 Figure 2.2:  Servo-actuator functional diagram ................................................................ 16 Figure 2.3: Schematic diagram of the control units.......................................................... 17 Figure 2.4: Hydraulic test rig............................................................................................ 18 Figure 2.5: Hydraulic test rig in more detail, (a) Hydraulic circuit,................................. 19 Figure  2.6:  The  test  rig  characteristics,  (a)  actuator  position  reference  signal  and resulting  (b)  end-effector  displacement,  (c)  Pressure  in  chamber  1,  and  (d) Pressure in chamber 2............................................................................................... 20 Figure 2.7: Experimental results in actuator friction investigation................................... 27 Figure  2.8:  (a)  Simulated  and  measured  values  of  the  pressure  in  chamber  1,  (b) MAE between simulated and measured values ........................................................ 31 Figure  2.9:  (a)  Simulated  and  measured  values  of  the  pressure  in  chamber  2,  (b) MAE between simulated and measured values ........................................................ 31 Figure  2.10:  (a)  Simulated  and  measured  values  of  the  actuator  displacement,  (b) MAE between simulated and measured values ........................................................ 32 Figure 2.11: Error signal between measured and simulated pressure of chamber 1......... 33 Figure 2.12: Error signal between measured and simulated pressure of chamber 2......... 33 Figure 2.13 Error signal between measured and simulated displacement........................ 33 Figure 2.14: Pressure repeatability ................................................................................... 34 Figure 3.1: Classification of fault diagnosis methods....................................................... 38 Figure 3.2: Kalman filter algorithm.................................................................................. 50 Figure 3.3: Compression of the UT and linear approximation......................................... 53 Figure 3.4: UKF structure block diagram......................................................................... 56 Figure 3.5: Actuator position estimation error.................................................................. 62 Figure 3.6: Estimation error of the pressure in chamber 1 ............................................... 62 Figure 3.7: Estimation error of the pressure in chamber 2 ............................................... 62   ixFigure 4.1: Actuator displacement while medium level leakage occurred in chamber 1................................................................................................................................. 67 Figure  4.2:  Low  level  leakage  at  chamber  1;  (a)  Residual  errors  of  pressure  in chamber 1, (b) pressure in chamber 2, and (c) actuator position.............................. 68 Figure 4.3: Medium level leakage at chamber 1; (a) Residual errors of pressure in chamber 1, (b) pressure in chamber 2, and (c) actuator position.............................. 69 Figure  4.4:  High  level  leakage  at  chamber  1;  (a)  Residual  errors  of  pressure  in chamber 1, (b) pressure in chamber 2, and (c) actuator position.............................. 69 Figure 4.5: Actuator displacement while medium level leakage occurred in chamber 2................................................................................................................................. 71 Figure  4.6:  Low  level  leakage  at  chamber  2;  (a)  Residual  errors  of  pressure  in chamber 1, (b) pressure in chamber 2, and (c) actuator position.............................. 72 Figure 4.7: Medium level leakage at chamber 2; (a) Residual errors of pressure in chamber 1,  (b) pressure in chamber 2, and (c) actuator position............................. 72 Figure  4.8:  High  level  leakage  at  chamber  2;  (a)  Residual  errors  of  pressure  in chamber 1, (b) pressure in chamber 2, and (c) actuator position.............................. 73 Figure 4.9: Actuator displacement while medium level internal leakage occurred.......... 75 Figure 4.10: Low level internal leakage; (a) Residual errors of pressure in chamber 1, (b) pressure in chamber 2, and (c) actuator position............................................. 75 Figure  4.11:  Medium  level  internal  leakage;  (a)  Residual  errors  of  pressure  in chamber 1, (b) pressure in chamber 2, and (c) actuator position.............................. 76 Figure 4.12: High level internal leakage; (a) Residual errors of pressure in chamber 1, (b) pressure in chamber 2, and (c) actuator position............................................. 76 Figure 4.13: Actuator displacement while the dynamic friction load is applied .............. 78 Figure 4.14: Dynamic friction load; (a) Residual errors of pressure in chamber 1, (b) pressure in chamber 2, and (c) actuator position ...................................................... 78 Figure 4.15: Actuator displacement while the load is removed........................................ 80 Figure 4.16: Load lost; (a) Residual errors of pressure in chamber 1, (b) pressure in chamber 2, and (c) actuator position......................................................................... 81 Figure  4.17:  Pressure  characteristics  in  dynamic  friction  load  occurrence;  (a) pressure in chamber 1, and (b) pressure in chamber 2.............................................. 85   xFigure 4.18: (a) Pressure in chamber 1 while the load is removed and (b) the close-up plot ....................................................................................................................... 86 Figure 4.19: (a) Pressure in chamber 2 while the load is removed and (b) the close-up plot ....................................................................................................................... 86 Figure 4.20: Pressure characteristics in internal leakage occurrence; (a) pressure in chamber 1, and (b) pressure in chamber 2................................................................ 87   xiAcknowledgments  I would like to express my deepest appreciation to my advisor, Dr. Farrokh Sassani, for his inspiration, friendship, encouragement, patience and unconditional support. I am extremely fortunate for having the opportunity to work with him and having the freedom to explore science in his laboratory. I am really thankful for his support over the last two years. I am indebted to Dr. Yusuf Altintas for sharing his lab equipment and Dr. Ryozo Nagamune for being generous with his time and knowledge and providing intellectual support and invaluable comments. I  would  like  to  convey  my  gratitude  to  our  kind  graduate  secretary  of Mechanical Engineering Department, Ms. Yuki Matsumura. In addition, I would like to extend my sincere thanks to Mr. Glenn Jolly, Mr. Gord Wright, and Mr. Sean Buxton for their essential technical assistance during this research. Additionally, I would like to thank Mr. Perry Yabuno for timely acquisition of electro-mechanical devices required for the fabrication of the set-up.  . I  am  grateful  to  my  dear  colleagues  in  Process  Automation  and  Robotics Laboratory: Dr. Mohammad Mallakzadeh and Mr. Amin Karami. Their friendship and support made the lab a motivating environment during the period of my work. I am so lucky to have such supportive friends: Hamidreza Yamini, Vahid Bazargan, and Pirooz Darabi. Lastly, I wish to express my genuine gratitude to my wonderful family for their never-ending love, support, and guidance. My father?s perfectionism and seek   xiifor  the  truth  and  my  mother?s  energy,  encouragement  and  moral  support  have always been a great motivation for me. I also thank my brother, my durable source of inspiration, for his support in every step of my life.   xiiiDedication    To my parents as an inadequate but sincere expression of appreciation and love                1 CHAPTER 1    Introduction        1.1 Preliminary Remarks The advent of hydraulic systems with their speed and reliable force accelerated a rapid development in the modern industry such as heavy-duty industrial robots, mining, material  handling  and  press,  manufacturing  and  construction.  Hydraulic  systems developed and accompanied by automatic controls allow flexibility in various types of operations, and significantly speed up the processes involved.  Employing hydraulic systems rather than other power transmission systems (e.g., electrical motor) provides a number of relatively important advantages, some of which are the following ([1] Section 4.4 and [2] Chapter 1): 1.  Hydraulic  fluids  carry  away  heat  generated  from  moving  parts  in  the systems, as well as they act as superb lubricants. 2.  Hydraulic actuators can be employed in random and periodic operations without any considerable suffering.   2 3.  Hydraulic actuators can apply large forces with high load stiffness. 4.  Load effects are insignificant in hydraulic systems, compared with those in other power transmission systems. 5.  Hydraulic  systems  have  long  operating  lives  even  if  employed  in  harsh environments. 6.  Hydraulic systems have considerably less weight/ power ratio. However, there are a number of disadvantages associated with hydraulic power systems ([1] Section 4.4 and [2] Chapter 1): 1.  Generating hydraulic power is not so readily achievable as is with other forms of powers, e.g. electrical, and mechanical. 2.  Hydraulic systems are relatively expensive. 3.  Fire and explosion hazards exist during operation. 4.  Hydraulic power transmissions produce significant amount of loud noise. 5.  Hydraulic system designers face several complex characteristics such as non-linearity. Wide  industrial  applications,  along  with  the  design  challenges,  make  hydraulic systems one of the most interesting topics in engineering research [3-5]. The  main  objective  of  this  thesis  is  to  analyze  and  select  the  best  modeling approach, compromising between model accuracy and model complexity regarding real-time  applications  and  response  effectiveness  followed  by  developing,  verifying  and   3 integrating  modules  for  the  purpose  of  online  monitoring  and  diagnosis  of  faults  in hydraulic power systems . In  the  process  engineering  and  mechanical  machinery,  diagnosing  the  occurred faults  in  the  system  is  an  essential  issue.  Undoubtedly,  in  some  applications  such  as airplane power systems, faults should be detected and recovered immediately while the plant  is  still  operating  to  prevent  catastrophic  system  failure  and  loss  of  human  lives. Fault  diagnosis  in  some  other  applications,  such  as  off-highway  machinery  or manufacturing,  may  not  be  as  critical  but  nevertheless  early  diagnosis  of  plant  faults helps enhance the productivity  and avoid fault progression leading almost certainly to plant failure. Thereby, there has been considerable interest in this field from practitioners as  well  as  researchers  for  a  number  of  decades.  Based  on  physical  redundancy  and analytical  redundancy  ideas,  a  wide  variety  of  methodologies  have  been  developed  to improve  the  fault  diagnosis  in  dynamic  systems  [6].  In  the  first  approach,  redundant sensors and actuators are installed in the systems; therefore in the event of a failure, these surplus  devices  are  employed  instead  of  the  faulty  one.  The  analytical  redundancy approach  is  founded  on  the  accurate  model  of  the  dynamic  system.  If  a  malfunction occurs, the difference between the real plant and model behavior will evolve. Once this difference  exceeds  a  set  threshold  value,  it  is  concluded  that  the  system  operates undesirably and tends to function in uncontrollable regions.    4 1.2 Hydraulic Power Systems A  servo-system  is  a  feedback  system  consisting  of  at  least  the  following  three essential elements; sensor, servomotor, and controller. As for hydraulic systems, the vast majority  of  these  types  of  power  systems  in  industrial  applications  are  servo-systems. Beside the mentioned elements, a number of hydraulic components are interconnected to form  a  practical  hydraulic  servo-system  that  provides  the  desired  operational characteristics. The descriptions of fundamental components of a hydraulic servo-system are the following: Actuator A Hydraulic actuator (cylinder) employed in a hydraulic system is the motor side of the system as opposed to a pump, which is the generator or driver side of the system. Hydraulic actuators are activated by hydraulic pressure. They transform the fluid's energy to a linear work, which can be either rotational or translational. The actuator comprises a cylinder  tube,  in  which  a  piston  connected  to  a  rod  slides.  The  rod  slides  out  of  the cylinder through seals from either one or both sides. The piston, which consists of the sliding  rings  and  seals,  divides  the  inside  of  the  cylinder  into  two  chambers.  If  the effective areas of both chambers are identical, the cylinder is called symmetric, otherwise it is asymmetric. By pumping hydraulic oil to one side of the hydraulic cylinder, the rod begins sliding toward the other. Hence, the oil is pushed back through the return line to the reservoir from the other chamber.     5 Pump The pump is the heart of the hydraulic power system and, as such, is fundamental for an efficient system operation. Typically, the pump supplies the hydraulic power of the system, which is essential to activate the actuator. Most of the hydraulic pumps used in power systems are positive displacement pumps. Positive displacement means that the flow  is  directly  proportional  to  the  speed  of  the  pump.  Regarding  the  internal  pump volume, two types of pumps are designated: fixed displacement and variable (adjustable) displacement. Depending on the required pressure and flow, as well as the efficiency and life expectancy, either fixed displacement pump (low budget) or variable displacement pump (high quality) may be chosen. Fixed displacement pumps derive a constant amount of  fluid  in  each  revolution  of  the  rotor.  To  achieve  the  desired  characteristics  from hydraulic power system, several kinds of pumps have been developed (e.g., screw pumps, gear pumps, gerotor pumps, vane pumps, and piston pumps). Servo valve  Servo valves are employed in hydraulic systems to manage the flow of the fluid in the  hydraulic  circuit  pipelines.  They,  along  with  the  feedback  controllers,  generate continuously  regulated  outputs  as  functions  of  the  electrical  inputs.  Commonly,  the feedback signal is produced by the comparison between the position, velocity, or force of the actuator (end-effector) and reference signal sent to the control valve. Servo valves are designed in single-stage, two-stage, and three-stage models. The single-stage servo-valve, the simplest and least cost, consists of a spool valve whose position is controlled by a torque  motor  via  a  direct  connection.  Multi-stage  servo-valves  intensify  the  applied torque of the torque motor using one or two hydraulic amplifiers. Thereby, multi-stage   6 servo-valves can be employed for higher flow rates and pressures, as well as, they can overcome high friction.  Other components In  addition  to  the  above,  several  other  components  are  required  to  complete  a hydraulic  system,  e.g.  sensors  and  pipelines.  Sensors  must  be  installed  in  the  proper locations to measure the demanded data; on the contrary, the measurand should not be affected. The feedback controller may need this data to generate the input signal to servo-valve.  Several  valves,  such  as  needle  valves,  pressure  relief  valves,  check  valves,  are employed to control the pressure, rate of flow, and direction of fluid inside of the circuit. As an instance, the pressure relief valve reduces and controls the pressure not to exceed a certain level. To do so, it connects the high pressure flow to the reservoir. Mostly, the common  usage  is  in  the  protection  of  a  hydraulic  element  from  the  unwanted  high pressure.  Generally, a hydraulic circuit is a closed-loop system including a tank as a fluid reservoir.  A  pump  is  always  necessary  to  pressurize  the  fluid,  which  ordinarily  is hydraulic oil. Meanwhile, a servo-valve regulates this high pressure fluid to control the behavior of the end-effector, which could follow a complex trajectory  with a required force.  For this sake, a controller should be developed. For instance, Dutton and Groves [7] developed an adaptive controller based on a pole-placement algorithm.  Thus far, primary structure of a hydraulic system has been explained. Moreover, regarding a sophisticated hydraulic system, the couplings or interactions emerge from the   7 existence of signal connections or common elements between the different lines in the circuit [8].  1.3 Potential faults in hydraulic systems In hydraulic systems, a wide range of faults and failures may occur, which have been extensively investigated by researchers. Chen and Saif [9] developed an iterative learning observer (ILO) for estimation, fault detection and compensation. Their proposed methodology  can  diagnose  multiple  faults.  Three  most  common  faults  in  industrial hydraulic  systems  are  fluid  contamination  [10], supply  pressure  malfunction  [11],  and leakage [12]. Heron  and  Huges  [13]  developed  a  novel  contaminant  monitoring  scheme  to examine  the  cleanliness  level  of  fluid  in  a  hydraulic  system.  Fine  solid  contaminant emanating from the moving parts in the hydraulic circuit accumulates around the small clearances  of  spool  valves.  Consequently,  the  friction  in  the  valve  increases  and  the system operates erratically. To simulate this fault in their study, the oil was added with various level of contaminants and was passed through the hydraulic system and finally through the contaminant monitor. Leakages  whose  effects  include  a  reduced  pressure,  degraded  stability,  and decreased  efficiency  are  the  most  common  faults  occurring  in  the  practical  hydraulic power systems.  Leakages are classified into two categories:  external and internal. The greatest concern among the related topics is the leakage in Pumps and actuators as they are two essential components in all hydraulic power systems as well as they both contain moving parts accelerating the wear.   8 Actuator external leakages mostly occur at the location of the connecting hoses and the cylinder. An and Sepehri [14] modeled the external leakages through two bleed valves, each mounted on a bypass to each side of a cylinder. The amount of leakages was tuned  by  opening  these  bleed  valves.  However,  they  did  not  quantify  the  leakage.  In 2006,  An  and  Sepehri  [15]  developed  a  sequential  analysis  (SA)  and  addressed  a quantification scheme. The importance of this scheme is in recognizing the progression of the leakage fault. Crowther et al. [12] built a neural network model for a hydraulic actuation system in which the lack of supply pressure, internal leakage in the actuator, and dynamic friction load were investigated. The actuator internal leakage was modeled by a cross-line bleed valve. The internal leakage, which is relatively more common in practice rather than the external one, arises as a consequence of superfluous clearance between the inner wall of the cylinder and the piston. The seal may wear after being in operation  for  a  while.  Both  types  of  leakages  lead  to  the  demand  of  higher  flow  and pressure supply for the recovery intention. Furthermore, leakage in the hydraulic pump is another aspect of leakage faults. Leakage faults regularly lead to decrease in both the flow rate and pressure in discharge ports. It may happen by virtue of Excessive wear of pump internal clearances or poor sealing. Skormin and Apone [16] developed a failure prediction procedure, detecting and utilizing trends exhibited by parameter estimation. The investigated faults included the leakage of the hydraulic pumps. In addition, there are several minor reasons causing this decline in pressure and flow rate. Two most common of them are a) a pipe, impeller, or suction strainer that may be blocked, and b) The characteristics of the liquid may vary   9 from  the  manufacturer  specifications,  as  examples,  higher  viscosity,  or  density  than expected. Despite major development, researchers are constantly working on more accurate and  fast  algorithms,  which  are  suitable  for  real  time  applications  and  more  complex systems.  1.4 Research Objectives As a consequence of all above, over time, operational problems or faults due to factors  such  as  wear,  misuse  and  lack  of  proper  maintenance  develop  in  hydraulic systems.  Such  trends  if  not  prevented  can  lead  to  catastrophic  failure.  It  is  therefore, desirable to quickly resolve these problems and recover from their adverse consequences. This may be achieved with the use of sensors, signal processing and artificial intelligent (AI) techniques. This proposed research focuses on design, and fabrication of a hydraulic experimental set-up that allows fault emulation, and development of a model-based fault detection,  diagnosis  and  real-time  control  scheme.  Specifically,  the  objectives  of  the research are to: ?  Analyze and select the best modeling approach, compromising between model accuracy,  model  complexity,  and  real-time  detection  and  response effectiveness. ?  Develop  algorithms  to  characterize,  isolate  and  correctly  associate  various faults to causes. ?  Design  an  experimental  set-up,  which  gives  the  flexibility  to  induce  the proposed faults at any desired location in the system, and simultaneously does   10 not  produce  any  superfluous  affect  due  the  arrangement,  other  than  that induced by the simulator algorithm. ?  Experimentally verify the theoretical propositions. ?  Develop,  verify  and  integrate  fault  detection  and  diagnosis  module  for hydraulic systems. ?  Perform fine-tuning to obtain a trade-off between false alarms and detection rate. In  this  thesis,  Unscented  Kalman  Filter  (UKF),  a  novel  methodology  inspired from Kalman filters, is utilized for health monitoring of the system. This methodology is a recursive estimator. This means that only the current measurement and the estimated state from the previous  time step are  required to compute the estimate  for the  current state. The significant characteristic of the UKF is that they are suitable for highly non-linear systems. Faults  affecting  the  reliability  of  hydraulic  systems  are  the  most  critical  in industrial applications. Four of the common faults in such systems will be studied. These are: 1)  Dynamic friction due to the load. (Emulated by resistance from a pneumatic cylinder.) 2)  Sudden loss of load. 3)  Internal leakage of the actuator. (Emulated with a needle valve.) 4)  External leakage of the actuator. (Emulated with two needle valves.) Based  on  the  model  developed,  the  control  command  executed,  and  the measurements obtained on the response, Unscented Kalman filtering technique will be   11 used to determine the state errors or residuals. Devising practical and statistically viable thresholds,  the  residuals  will  be  used  to  isolate  a  possible  fault  and  to  identify  the probable cause. Once a degree of certainty is established, corrective actions can be taken. This may constitute a) modifying the control command, b) issuing a warning, or c) safely halting the operation if possible.  1.5 Organization of the Thesis This thesis, which is conducted to meet the outlined scope, is organized in the following  order.  Chapter  1:  The  hydraulic  systems  along  with  their  applications  are introduced and also potential faults that may occur in the practical hydraulic systems are discussed. The objectives of the current research are also explained. Chapter  2:  The  experimental  hydraulic  set-up  is  introduced  and  a  state  space model is developed and explained in detail. Simulations and tests are carried out in order to validate the developed state space model. Also, the frequency range of the system is discussed. Chapter 3: fault diagnosis methods are surveyed and a basis for understanding the Kalman filter is provided in this chapter. Moreover, the Unscented Kalman filter, as the base of the fault diagnosis methodology in this research, is studied in detail. The fault monitoring  scheme  applied  to  the  hydraulic  test  rig  is  developed  and  the  simulation results are shown to confirm the satisfactory operation of the proposed algorithm. Chapter  4:  Condition  monitoring  of  the  hydraulic  test  rig  is  investigated.  The results  of  the  real-time  state  estimation  are  shown  followed  by  the  fault  diagnosis discussions.    12 Chapter 5: The conclusions drawn from the investigation are presented, and the prospects for applications and further developments are discussed.   13 CHAPTER 2    System Modeling and Set-up Configuration        2.1 Introduction This chapter describes the experimental set-up on which the condition monitoring is conducted. The investigated test rig is typical of the systems used in many industrial applications.  It  is  composed  of  a  group  of  essential  hydraulic  components  including  a pump,  servo-valve,  solenoid-valve,  cylinder,  needle-valve,  relief-valve,  and  filter. Component specifications along with the non-linear dynamic equations are analyzed to develop  a  mathematical  model  of  the  system.  A  LabViewTM  interface  is  developed  to issue  control  commands  based  on  sensory  data  and  perform  real-time  condition monitoring and fault diagnosis. In  Section  2.2  the  test  rig  configuration  is  illustrated  and  its  subsystems  along with  their  operations  are  explained  in  detail.  Then,  all  five  potential  faults  that  are emulated  by  the  current  scheme  are  discussed.  Subsequently,  the  state  space  model   14 corresponding to the explained set-up is derived in Section 2.3. This mathematical model is  further  elaborated  with  the  details  of  the  various  governing  equations,  function modeling, and all related parameters. In Section 2.4, the actual data acquired from the set-up is compared with the simulation results in order to validate the developed state space model. 2.2 Experimental Test Rig The schematic diagram of the experimental set-up is shown in Figure 2.1. The entire system is divided into two subsystems: Pneumatic and Hydraulic subsystems. The hydraulic part is the main part of the system while the pneumatic subsystem is designed to emulate the dynamic friction load. The pneumatic subsystem consists of an SMC five-way solenoid valve connected to an air pressure supply. This supply provides the constant pressure of 750 KPa. The solenoid valve receives the control signal from a PC equipped with a PCI-6024E data acquisition board. The asymmetric pneumatic cylinder controlled by this solenoid valve applies a force to the hydraulic actuator as an occurred fault for the hydraulic subsystem. The direction of this force is always opposite of the hydraulic actuator movement. To do so,  as  the  hydraulic  servo-valve  pressurizes  the  retracting  chamber  (chamber  2  as illustrated  in  Figure  2.2),  the  pneumatic  solenoid  valve  connects  the  supply  line  to retracting  chamber  of  pneumatic  cylinder  and  vice  versa.  This  process  emulates  the dynamic  friction  load,  which  is  common  in  practical  servo-actuator  systems.  The magnitude of this force is 1162 N during the hydraulic actuator extraction period while it is 1042 N during the hydraulic actuator retraction period.   15  Figure 2.1: Schematic diagram of the experimental set-up  The hydraulic subsystem is powered by a pump supplying high-pressure hydraulic fluid  to  the  actuator.  In  order  to  investigate  a  more  general  system  modeling,  an asymmetric cylinder rather than a symmetric one is employed as the actuator. As shown in Figure 2.2, chambers are referred to as chamber 1 and chamber 2, and are connected to an Atchley Controls? servo-valve. Its control signal range is from -100 mA to 100 mA,   16 which is received from a computer equipped with the LabViewTM software. Analog and digital electronic units are shown schematically in Figure 2.3. Three sensors are installed in the set-up to measure the required states. Two pressure transducers are mounted to measure the pressure in both chambers. A position encoder measures the displacement of the  actuator.  All  these  signals  are  transmitted  to  the  PC  via  the  PCI-6024E  data acquisition board, and subsequently the command signals are sent to servo-valve from the PC. Points of leakages and external forces as potential faults are illustrated in Figure 2.2.  Flows  of  q1  and  q2  to  chambers  1  and  2  respectively  are  controlled  by  the  input current Ia of the servo-valve. Since the servo-valve has a symmetric spool, the positive direction for the spool movement is arbitrarily defined. As for the asymmetric cylinder, the positive direction for the actuator movement is outward to the left.   Figure 2.2:  Servo-actuator functional diagram   17  Figure 2.3: Schematic diagram of the control units  The set-up is designed to produce common faults in the hydraulic systems. Figure 2.4 shows a photograph of the experimental test rig. All three leakages are emulated by means  of  three  needle  valves  with  the  same  characteristics.  These  needle  valves  are shown in Figure 2.5a. On each chamber, there is a pressure transducer along with a visual gage for safety purposes. One needle valve is mounted as the bypass for the cross-port leakage emulation. As shown in Figure 2.5b, a pneumatic cylinder, a position encoder, an LVDT, and an external load are connected to the hydraulic end-effector. The LVDT is used  to  mark  the  origin  of  the  actuator,  while  the  position  encoder  is  employed  to measure the displacement from this origin. The actuator stroke is 10 cm and the origin of yx   is  located  at  the  point  where  the  ram  is  fully  retracted.  Since  the  external  load  is emulated by means of two suspended weights,  due to the  gravity, its  effect is a  force applied on the actuator asymmetrically. This force resists the piston displacement during the retracting period while it accords with the movement during the extracting period. Sudden loss of these weights is a scenario that will be discussed as another occurred fault in the system in section 4.2.2.2.  Server Computer    PCI-6024 E Signal Conditioning  Servo Valve Pressure Transducers & Position Encoder   18  Figure 2.4: Hydraulic test rig  The closed-loop controller regulates the characteristics of the hydraulic test rig. To validate the performance of  the proposed closed-loop system, the set-up is excited with  a  sinusoidal  reference  signal  that  the  end-effector  position  should  follow.  A  low frequency  sinusoidal  signal,  )   meter,  is  considered  as  the reference signal for a period of 50 seconds. Due to the digital computing pace limits, the sampling time is chosen as 20 ms in this research. Figure 2.6 shows the reference signal along with the measurement signals. With a comparison between Figures 2.6a and 2.6b, it can be observed that the end-effector tracks the reference signal satisfactorily; however, there is a phase difference of around  4pi , as well as a 4% amplitude offset. Additionally, Figures 2.6c and 2.6d show that the pressure in chamber 2 is much higher than that at chamber 1. The  reference signal is symmetrical  around the origin therefore the  servo-valve transmits higher pressure to chamber 2 because this chamber has less effective area than  that  of  chamber  1.  Moreover,  with  reference  to  the  hydraulic  system  schematic Hydraulic Cylinder Piston Weights Pneumatic Cylinder   19 diagram shown in the Figure 2.1 the asymmetric external load intensifies the pressure in chamber 2.     (a)       (b) Figure 2.5: Hydraulic test rig in more detail, (a) Hydraulic circuit, and (b) Pneumatic circuit and end-effector attachments   Needle Valves Pressure Transducers LVDT Position Encoder Pneumatic Cylinder Piston   20 00.020.040.060.080.10 10 20 30 40 50Reference Signal (m)    .  00.020.040.060.080.10 10 20 30 40 50Actuator Position (m)   . 02460 10 20 30 40 50Pressure (MPa)   . 02460 10 20 30 40 50Time(s)Pressure (MPa)  . Figure 2.6: The test rig characteristics, (a) actuator position reference signal and resulting (b) end-effector displacement, (c) Pressure in chamber 1, and (d) Pressure in chamber 2   (a)  (b) (c) (d)   21 2.3 System Modeling 2.3.1 Governing Equations The dynamics of almost all practical hydraulic components can be appropriately described by the equations presented by Merritt [2]. In this research, understanding the dynamic features of the actuator and the servo-valve are imperative. The flow through the servo-valve is proportional to the square root of the pressure drop across the port and the area  of  the  valve  opening.  The  valve  opening  area  is  proportional  to  the  spool displacement and therefore by applying a linear orifice area gradient related to the spool displacement, the following expression can be used to represent the flow equations of the servo-valve: 0)2)20)2)22121<-parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp--parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp-braceexbraceexbraceleftbtbraceexbraceexbraceleftmidbracelefttpgreaterequal-parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp--parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp-braceexbraceexbraceleftbtbraceexbraceexbraceleftmidbracelefttpvesvsexPpxpPrhoomegarhoomegarhoomegarhoomega                     (2.1) where: q1, q2: flow from the servo-valve to chamber 1 and chamber 2     Ps, Pe: supply and return line pressures      p1, p2: pressures in chamber 1 and chamber 2   22     xv: the servo-valve spool displacement     Cd, omega:  orifice coefficient of discharge and orifice area gradient     rho: density of the hydraulic oil It is assumed that the rod and the piston of the hydraulic cylinder are rigid, and the oil on either side of the piston is compressible. As this compressibility is proportional to  the  volume  in  which  pressure  acts,  the  rate  of  the  change  of  volumes  within  the actuator may be expressed as: ( ) )( ) )2max202111ppyy&&--braceexbraceexbraceleftbtbraceexbraceexbraceleftmidbracelefttpbetabeta                                 (2.2) where:     beta: effective bulk modulus of the hydraulic fluid     A1, A2: effective piston areas of chamber 1 and chamber 2     xy: actuator position     Xmax, Xmin: positions when the piston is fully extended and fully retracted V0:  volume  of  the  fluid  trapped  in  the  supply  pipes  connected  to  each chamber (it is assumed that this volume is identical for both chambers) The  continuity  equations  for  hydraulic  flows  of  the  actuator  are  applied  to determine  a  relationship  between  chamber  flows,  chamber  pressures,  and  the  piston velocity.  By  considering  the  compressibility  issue  from  the  preceding  equation,  the following descriptions are held for the flows of the chambers.   23   ( ) )( ) )2max2022111ppyy&&--braceexbraceexbraceleftbtbraceexbraceexbraceleftmidbracelefttpbetabeta                         (2.3) Equations 2.1 and 2.3 are combined to eliminate the flow parameters from the equations. By solving the obtained relationship for  1  and  2 , we achieve: braceexbraceexbraceleftbtbraceexbraceexbraceleftmidbracelefttp<bracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftbtbracketlefttp ---greaterequalbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftbtbracketlefttp ---=braceexbraceexbraceleftbtbraceexbraceexbraceleftmidbracelefttp<bracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftbtbracketlefttp ---greaterequalbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftbtbracketlefttp ---=02)(02)(02)(02)(2max2max21min1min1vyvyvyvyxxXAVxxXAVpxXxAVxXxAVp&&&&&&rhoomegabetarhoomegabetarhoomegabetarhoomegabeta        (2.4) The position of the flow controlling spool valve as a function of the drawn current can be characterized by a quadratic system: vxxdxuK2omegaomega ++= &&&                                    (2.5) where:     Ksp: spool valve positioning gain     omegan: natural frequency of the spool dynamics     u: valve input current      dm: damping ratio The dynamics of the actuator is modeled based on the summation of the forces acting on the piston of the cylinder. The following equation can be expressed:   24 extFfpApAxMf +--== 2&&                            (2.6) where:     Me: combined effective mass of the objects moving with the ram     ff, Fext: actuator friction and external forces As  illustrated  in  Figure  2.1,  the  ram  pulls  up  a  mass,  Ma  =  25  kg,  during  its retraction. For the hydraulic system this weight acts as an external force (Fext = 245 N). Moreover, it is assumed that this weight moves along with the ram and therefore its mass should be added to the effective mass of the moving objects. Also, the pneumatic cylinder ram and the position encoder are attached to the end-effector. Then as a result Me = 33 kg.  Friction  is  an  unavoidable  factor  in  moving  machinery  and  thus  it  has  to  be studied  as  a  part  of  dynamics.  Friction  is  a  non-stationary  and  non-linear  force  that depends on many physical parameters and even environmental situations. As an instance when two sliding materials are lubricated, different sliding speed causes different film thickness  of  the  lubricant  and  therefore  friction  characteristics  may  change.  Not  only because of the challenges of the friction modeling but also due to its significant impact on the dynamic of the system, there has been considerable interest in this field. Karnopp [17] proposed a typical stick-slip friction model in mechanical dynamic systems. Laval [18] improved the Karnopp model and his proposition is adopted in this study to model the friction inside of the hydraulic cylinder. His model is expressed as: braceexbraceleftbtbraceexbraceleftmidbracelefttp=notequal= -00||yyxslf xfxf y&&&    (2.7)            25 where:     fst, fsl: static and kinetic friction forces     C: lubrication coefficient     d: effective damping ratio In order to produce reliable numerical results, all above parameters and physical quantities should be known initially. The parameters of the current system are classified into two sets. Parameters of the first set such as effective bulk modulus of the fluid are those that are either specified by the manufacturer or measurable directly. On the other hand,  some  parameters,  which  are  categorized  as  the  second  group,  are  neither measurable  directly  nor  specified  by  the  manufacturer,  such  as  the  static  friction,  and hence must be determined by a set of experiments and measurements through the system states.  To  accurately  estimate  the  friction  of  the  actuator,  unloaded  set-up  is  run  in different scenarios to evaluate both static and kinetic frictions. To determine the kinetic friction parameter, fsl, the ram is run in several certain velocities. With reasonably good precision, it is assumed that these movements have no acceleration (except at start and stop) and hence the calculated actuation forces during the ram motions give an indication of the kinetic friction. The results of these experiments are shown in Figure 2.7, where actuator friction forces are seen to stay reasonably constant for speeds over 0.015 m/s in either  direction.  As  an  instant  of  experimental  result,  Figure  2.6  shows  the  actuator movement  from  which  it  can  be  derived  that  the  average  velocity  of  the  actuator  is around 0.02 m/s. Because this is more than 0.015 m/s, the steady friction force can be   26 chosen as the kinetic friction value. From the Force-Velocity curve (Figure 2.7) the actual magnitude of the kinetic friction can be appraised as 0.140 kN, which is the average of the actuation force values at the velocity of 0.015 m/s in either direction.  To recognize the static friction, fst, the input current is regulated to make the ram start moving. To do so, the current is increased slowly until the ram starts to move. For the sake of simplicity, since the ram starts moving slowly, the acceleration is ignored. This  experiment  is  carried  out  in  three  different  locations  along  the  cylinder  for  both retracting and extracting directions and the applied actuation force is calculated in each trial  run.  Subsequently,  the  average  of  these  six  force  values  is  taken  to  represent  the approximate static friction, which is obtained as 0.295 kN.  There are two other parameters in the friction formulation that should be dealt with:  the  effective  damping  ratio  (d)  and  the  lubrication  coefficient  (C).  The mathematical friction model (equation 2.7) is plotted for different values of d and C (not all calculations are reproduced here). Both values are positive and also in all surveyed literature in which this model of the friction is investigated, the inverse of the lubrication coefficient is a positive value less than one tenth. We hereby try to fit the mathematical model to the experimental data (Figure 2.7) in a trial and error approach. The values of d and C from the best fitted curve are chosen for the state space model parameters. The damping ratio, d, is 250 N.s/m, while the lubrication coefficient, C, is 28 s/m.  The  following  table  shows  all  actual  parameters  employed  in  the  state  space model.   27 -0.3-0.2-0.100.10.20.3-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.02 5Velocity (m/s)Force (kN) Figure 2.7: Experimental results in actuator friction investigation   Table 2.1: The numeric values of the hydraulic set-up parameters       2.3.2 State Space Model The  state  vector  of  the  state  space  model  consists  of  six  variables,  which  are defined as the following: [ ] [ ]TvyyvT x21654321 =          (2.9) A1 =31.7 ? 10-4 m2  fst = 295 N  v0 = 0.001 m/s A2 =26.6 ? 10-4 m2  fsl = 140 N  omega = 0.02 m2/m Me = 33 kg  C = 28 s/m  beta = 6.89 ?108 Pa Xmin = 0 m  ksp = 3.003 ? 10-5 V/m  rho = 857 kg/m3 Xmax = 0.1 m  omegan = 600 rad/s  Ps = 6.5 MPa d = 250 N.s/m  dm = 0.7  Pe = 0  MPa Cd = 2.605 ? 10-2  V0 = 18.09 ? 10-6 m3  Fext = 245 N   28 In order to acquire the state space model,  x should be derived. Having noted the definition of x1 and x6, the following expression can be written:   61 x                                                           (2.9) In order to determine functions for 2 and 3 , equation 2.4 can be recast as the following: braceexbraceexbraceleftbtbraceexbraceexbraceleftmidbracelefttp<bracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftbtbracketlefttp ---greaterequalbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftbtbracketlefttp ---=braceexbraceexbraceleftbtbraceexbraceexbraceleftmidbracelefttp<bracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftbtbracketlefttp ---greaterequalbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftbtbracketlefttp ---=02)(02)(02)(02)(141431min1min2xxXAVxxXAVxxXxAVxXxAVxesserhoomegabetarhoomegabetarhoomegabetarhoomegabeta&&         (2.10) and 4 can be easily expressed as: 5xx =&                                                           (2.11) To derive an expression for 5 , ff from equation 2.7 can be substituted in the dynamic model  of  the  actuator  (equation  2.6)  and  then  solved  to  obtain  an  equation  for  y . Generally,  a  threshold  is  considered  for  zero  in  numerical  computations,  therefore  a threshold  as  v0  is  designated  in  the  expression  for  ff.  The  following  shows  the  5  formulation in the state space model:    )1 32215 extfaFM +                                       (2.12) in which:   29 braceexbraceleftbtbraceexbraceleftmidbracelefttp<=<>= -00| )()]1)(([ 5vvfstxslf                     (2.13) By solving the equation 2.5 for v , the following expression holds for 6 : uspnnm +1266 omega                                      (2.14) 2.4 Model Validation Any  mathematical  model  should  be  validated  in  order  to  ascertain  that  the differences  between  the  measured  states  from  the  practical  system  and  the  simulated states from the state space modal do not exceed a certain threshold for a similar input signal.  For  hydraulic  systems,  these  thresholds  are  approximately  10%  of  the  actual measurements according to most literature such as [19-21]. These errors are considered as  model  uncertainties  presented  by  the  process  noise  in  the  proposed  fault  diagnosis algorithm that will be explained in Chapter 3.  In  the  current  research  the  moving  average  method  is  used  to  study  the  error signals. In this method, the average of a number of data is taken at each discrete time increment.  By  applying  this  algorithm  to  error  data,  the  effects  of  extra  noises  and transient errors on the error signal diminish. The moving average of the errors (MAEs) at kth time step is calculated by the following expression: meekmiksummation-=                                                      (2.15) in which m is the number of previous error data from which the average is calculated at each time step. ei is the error data at the ith time step. In the current study, m is chosen as   30 250 considering the sampling time and the reference signal frequency. As explained in section 2.2, the sampling time is 20 ms and the frequency of the reference signal is  pi  rad/sec. Thus by choosing m as 250, the MAE is calculated for a complete motion period, a full retraction and extraction, at each time step.  2.4.1 Experimental Results In  order  to  validate  the  state  space  model,  a  low  frequency  sinusoidal  signal, )   meter,  is  applied  to  both  mathematical  model  and  the experimental test rig as the position reference signal for a period of 60 seconds. Figures 2.8 and 2.9 illustrate the characteristics of the pressures in chamber 1 and chamber 2, respectively.  Both  measured  and  simulated  pressures  are  shown  and  also  the  error between the model and the system for the corresponding state is represented. It can be observed visually that the simulated and the measured pressures are so close to each other and the errors converge to satisfactory values. The MAE is 0.12 MPa for chamber 1 and 0.14 MPa for chamber 2, which are within 10% of the actual measurements. Figure 2.10 depicts both actuator displacement trajectories from sensor signal and state space model computation  as  well  as  the  corresponding  error  signals.  This  figure  shows  that  the simulated trajectory adequately matches the measured one. The MAE signal, which has a steady value around 2 mm, stays within 10% of the actual measurements of the actuator movement.     31    02460 20 40 60Time (S)Pressure (MPa)     .      Figure 2.8: (a) Simulated and measured values of the pressure in chamber 1, (b) MAE between simulated and measured values        02460 20 40 60Time (S)Pressure (MPa)     .    Figure 2.9: (a) Simulated and measured values of the pressure in chamber 2, (b) MAE between simulated and measured values  Simulated Measured Simulated Measured (a) (b) (a) (b)   32   Figure 2.10: (a) Simulated and measured values of the actuator displacement, (b) MAE between simulated and measured values  2.4.2 Frequency Range To compare the frequency responses of the practical system and the state space model, a sinusoidal position reference signal with a time varying frequency and constant amplitude is applied to the servo-valve and the state space model. The reference signal is )4 tt   meter.  The  error  signals  between  measured  and simulated results are shown in Figures 2.11 to 2.13. All three measurands are studied in the  frequency  range  between  0.05  to  0.8  Hz,  which  is  a  typical  range  of  practical operating  frequencies  for  hydraulic  systems.  The  simulation  errors  of  the  state  space model increase by increasing the input frequency as it can be observed from the figures. Simulated Measured (b) (a)   33 Nevertheless, the error signals still stay within the satisfactory boundary. In this study, all experiments are carried out at the frequency of 0.2 Hz.  -0.40-0.200.000.200 20 40 60Time (S)Error (MPa)     . Figure 2.11: Error signal between measured and simulated pressure of chamber 1  -0.40-0.200.000.200.400 20 40 60Time (S)Error (MPa)     . Figure 2.12: Error signal between measured and simulated pressure of chamber 2 -0.01-0.00500.0050.010 20 40 60Time (S)Error (m)     . Figure 2.13 Error signal between measured and simulated displacement  2.4.3 Repeatability of the Experiments It is essential that the system measurements are repeatable within an acceptable margin  in  any  experimental  study.  This  is  necessary  in  interpreting  the  results  and   34 drawing conclusions. In all the tests, the operating conditions such as fluid temperature, supply  pressure,  operating  scheme,  etc  remain  almost  constant.  The  operating  supply pressure  was  approximately  6.50  MPa  and  the  laboratory  temperature  was  set  to  be around 20?C. Repeatability  of  the  pressure  in  chamber  1  is  illustrated  in  Figure  2.14  as  an example. The test  rig was run  at different times and three sets of measurements were collected for the same operating conditions. It can be verified visually that the data of this particular state are very similar from test to test. All four data sets show similar trends, time periods and magnitudes. Variation between the traces is very small. It is perceived that  the  values  of  five  performance  indicators,  minimum,  maximum,  mean,  standard deviation, and range, for these three data sets are very close, all varying within 0.8%. These indicators suggest that the data repeatability is acceptable. 012340 10 20 30 40 50Time(s)Pressure (MPa)      .Series1Series2Series3 Figure 2.14: Pressure repeatability   35 CHAPTER 3    Implementation of a Fault Monitoring Technique        3.1 Introduction A fault is a deviation of a desirable characteristic property leading to the inability to fulfill an intended purpose. Practically, it is assumed that the system is healthy and no fault is present at the beginning but takes place some time, with magnitude, type and time of occurrence being unknown [22]. Generally, faults occur in two different manners: a) step  functions  such  as  when  the  end-effector  is  obstructed  by  an  object  or  b)  ramp functions such as leakage or deterioration of a device. It is essential to distinguish the difference  between  noise  and  fault.  Customarily,  any  noise  is  considered  as  a  random zero-mean signal, which could originate from any element of the plant especially from sensors; meanwhile, any non-zero-mean disturbance is treated as a fault. In this chapter the development and implementation of the condition monitoring technique for the current study are investigated. The material in this chapter is organized   36 as follows; initially, in Section 3.2, an overview of fault categories in dynamic systems is given. Afterwards, fault diagnosis approaches are classified and explained in Section 3.3. A  general  Kalman  filter  algorithm  is  explained  in  Section  3.4.  Sections  3.5  and  3.6 outline the Unscented Kalman filter and its application in the present system.  3.2 Classification of Faults Faults may be classified into the following groups [23]: 1.  Additive  faults:  they  are  unknown  inputs,  which  vary  the  plant  outputs independently of the known inputs. The existence of an external force or friction load as a fault  is  a  certain  instance  of  this  class  of  faults.  Additive  faults,  known  as  Error-In-Variable (EIV), [24] do not depend on whether other faults have occurred or not therefore they can easily be detected.  Sensor output signals are generally affected by non-zero-mean signals and hence they differ from the actual values of measurands. These faults are usually considered as additive faults (even though some sensor faults such as complete failure may be better designated  as  another  class,  which  is  ?multiplicative  faults?).  The  following  model describes the general sensor signals from the healthy system: braceleftbtbraceleftmidbracelefttp++)~)~tt          (3.1) where  ) is the system input signal and y(t) is the actual system output,  )  and )   are  fault-free  sensor  signals,  )~ t and  )~ t are  corresponding  noises.  Moreover, sensor signals from faulty systems can be modeled as the following:   37 braceleftbtbraceleftmidbracelefttp++)~)~ttyu                                     (3.2) in which fu(t) and fy(t) are the models of the fault situations.                2.  Multiplicative  faults:  they  may  turn  up  as  parameter  deviations  within  the systems,  which  lead  to  the  system  output  changes  [25,  Chapter  1].  These  deviations depend  on  the  magnitudes  of  the  plant  control  inputs.  Deteriorations  of  the  system elements such as loss of power are categorized as multiplicative faults. This type of faults can be modeled as the following [26]:  braceexbraceleftbtbraceexbraceleftmidbracelefttpdeltadelta))tt                           (3.3) in which A, B, C, and D are the state space model matrices and increments show the deviations from the healthy system matrices resulting from the parameter changes. 3.3 Overview of Fault Diagnosis Approaches There  is  an  abundance  of  literature  on  process  fault  diagnosis  ranging  from artificial  intelligence  to  analytical  methods  and  statistical  approaches.  Fault  diagnosis methods are usually classified due to the type of the usage of a priori process knowledge. The  a  priori  knowledge  is  the  relationship  between  the  failures  and  the  observations, which may be obtained either explicitly or from any source of domain knowledge. Based on the knowledge that requires a priori, fault diagnosis techniques surveyed in [27-29] can be categorized into two general classes; model-based and model-free (data driven) methods.   38 From  a  modeling  perspective,  there  are  methodologies  that  require  accurate process models called quantitative models. At the other end of the spectrum, there are methodologies  that  do  not  require  any  form  of  model  information  and  rely  only  on previous process data [30].  Given the process knowledge, there are various methods that can be employed to perform diagnosis. A taxonomy of major fault diagnosis techniques is shown in Fig. 3.1.     Figure 3.1: Classification of fault diagnosis methods    3.3.1 Model-based Fault Diagnosis Approaches All  model-based  methodologies  can  be  categorized  into  two  major  groups: qualitative and quantitative. The models of the systems are usually developed based on the fundamental understanding of the physics of the processes. In a quantitative approach Fault Diagnosis  Methods Model-based  Model-free Quantitative  Qualitative  Quantitative  Qualitative  Fuzzy  Qualitative Trend Analysis (QTA)  Statistical  Neural Network  Qualitative Physics Fault Tree Diagraph Frequency Domain Observer  Parity Space  Wavelet Analysis   39 the fundamentals of the dynamic systems are demonstrated in a mathematical model in which some functions show the relationship between the system inputs and outputs. On the other hand, in a qualitative model the fundamental system relationships are expressed by qualitative functions. 3.3.1.1 Quantitative Model-based Methods Most  studied  reported  in  literature  working  on  quantitative  model-based approaches  develop  a  state  space  model  of  the  system  where  residual  errors  can  be generated. These techniques can be broadly classified into frequency domain, observer, and parity space classes.  Frequency Domain Approach The basic idea behind the frequency domain approaches is to generate residuals via factorization of the transfer functions of the dynamic systems. Frank and Ding [31] studied a methodology in frequency domain for robust residual generation.  Observer Approach Residuals  can  also  be  generated  by  estimation  of  the  system  outputs  from  the sensor data by using appropriate observers. Observer-based residual generators for linear systems were developed by Chen  and Patton in [32], they also studied  the correlation between  factorization-based  and  observation-based  residual  generators.  Regarding  the observer criteria, these approaches may be classified as a) fixed (Chapter 13 of [33]) or adaptive [34], b) reduced [35] or full-order [36], and c) linear [37] or non-linear [38].     40 Parity Space Approach Parity  space  methods  rearrange  the  model  of  the  system  and  examine  the consistency  of  the  model  with  sensor  data  and  known  system  inputs.  Willsky  [39] introduced dynamic parity relations for failure detection.  3.3.1.2 Qualitative Model-based Methods Based  on  various  forms  of  qualitative  knowledge  used  in  fault  diagnosis, qualitative  model-based  approaches  can  be  classified  into  fault  trees,  digraphs  and qualitative physics classes. Qualitative Physics Approach Qualitative physics is an area of artificial intelligence, which is concerned with reasoning  about  the  behavior  of  physical  systems.  In  fault  diagnosis  applications, qualitative  physics  knowledge  is  represented  in  two  main  groups.  In  the  first  group, qualitative equations (behavior) are derived from the differential equations. Umeda et al. [45] developed a fault detection methodology whose knowledge for diagnosis is based on qualitative physics. In the second group, the qualitative equations (behavior) are derived from  ordinary  differential  equations  (ODEs).  Sacks  [46]  was  the  first  who  examined piece-wise  linear  approximations  of  non-linear  differential  equations  by  means  of  a qualitative mathematical regulator to infer the qualitative properties of a dynamic system.  Fault Tree Approach Fault  trees  are  commonly  formed  of  layers  of  nodes  and  usually  applied  in analyzing of the system reliability. These logic trees propagate faults to the next level of layers. Different logic operations such as ?AND? and ?OR? are performed at each node   41 for propagation. He et al. [40] developed a methodology to accurately assess the system reliability with limited statistical data. They used fault tree analysis based on the fuzzy logic. Fault trees have been used in a variety of risk assessment and reliability analysis. Tartakovsky  [41]  estimated  the  probabilities  of  the  system  failures  by  means  of uncertainty quantification techniques. Then, applied fault tree analyses to combine these probabilities in order to estimate the risk of the system failures. Digraph Approach Generally, a digraph is a graph, which includes directed arcs between nodes. And a signed diagraph (SDG) is a digraph in which these directed arcs are designated with either a positive or a negative sign. SDGs have been widely used to represent qualitative models or cause-effect relationships and also in the form of causal knowledge, they are employed  to  diagnose  the  process  faults.  The  first  development  of  the  SDG  for  fault diagnosis was reported by Iri et al. [42]. They derive a cause-effect graph (CE graph) from  SDG.  Vedam  and  Venkatasubramanian  [43]  developed  an  SDG  approach  for multiple fault detection. Han et al. [44] incorporated the fuzzy set theory into SDGs to provide an accurate resolution of fault origin.  3.3.2 Model-free Fault Diagnosis Approaches Model-free  Fault  diagnosis  approaches  are  based  on  the  historical  process  data rather than the models of the systems. From another point of view, these methodologies extract  the  feature  characteristics  from  the  previous  data.    Based  on  whether  the knowledge  about  process  characteristics  is  required  or  not,  one  can  perform  either qualitative or quantitative feature extraction.    42 3.3.2.1 Quantitative Model-free Methods There are a notable number of methods, which are employed in decision making by the use of quantitative information without any necessity for system modeling. These can be categorized into two main classes: non-statistical methods such as neural network, and statistical methods such as Partial Least Squares (PLS). Statistical Approach Essentially,  faults  can  be  diagnosed  by  considering  the  combination  of  the instantaneous estimates of the pattern recognition over time. Historical information about the properties of the system failure mode is used and hence fault diagnosis can be treated in a statistical pattern recognition framework. The  extensive  improvement  in  computing  power  of  softwares  has  had  an increasing impact on the practical applications of statistical science. The applications of statistical development in fault diagnosis area have been comprehensively studied in the literature. Kresta et al. [47] gave an overall overview of statistical monitoring in process analysis.  They  introduced  a  basic  technique  of  using  the  PLS  and  PCA  (Principal Component  Analysis)  to  efficiently  monitor  the  fulfillment  of  large  processes  and  to quickly detect process changes. Qin and Li [48] studied a methodology based on the PCA for  sensor  fault  detection.  A  non-linear  PCA  method  for  batch  processes  has  been developed by Dong and McAvoy [49]. Neural Network Approach A  neural  network  consists  of  several  processing  modules  (the  number  of  these modules  depends  on  the  problem  complexity).  These  modules  are  connected  to  each   43 other by means of several elements that store information along with the programming functions.  Classification  processes  including  decision  making,  pattern  recognition,  and fault diagnosis are the most common tasks to which neural networks are applied.  Considerable attention to the application of neural networks  for fault diagnosis has appeared in the literature. The practicality of neural networks for fault diagnosis in chemical  engineering  was  demonstrated  for  the  first  time  in  the  1980s  by  researchers such as Venkatasubramanian and Chan [50] and Unger et al [51]. Later, the hierarchical neural network architecture, successful in the detection of multiple faults, was proposed by Watanabe et al. [52].  Back-propagation  strategy,  first  described  by  Werbos  in  1974,  is  the  most conventional  supervised  learning  technique  for  training  neural  networks.  Back-propagation  neural  networks  have  been  vastly  investigated  in  engineering  research  in which problems of fault diagnosis are addressed. The idea of feature presentation, which has  been  demonstrated  by  Farell  and  Roat  [53]  and  Tsai  and  Chang  [54],  is  the foundation of the performance progress of the primary back-propagation neural networks in fault diagnosis applications. Recently, to overcome the slow convergence of the back-propagation  algorithm  a  number  of  techniques  such  as  the  diagonal  recurrent  neural network have been proposed. Wang and He [55] introduced an adaptive dynamic back-propagation  algorithm  to  determine  the  optimum  number  of  the  hidden  layer  neurons. They  employed  two  diagonal  recurrent  neural  networks  to  detect  stator  winding  turn fault.  One  determines  the  fault  intensity  and  the  other  is  used  to  estimate  the  exact number of fault turns.   44 In addition to back-propagation, various network architectures have been studied in the fault diagnosis area. For example, Adaptive Resonance Theory 2 Neural Network (ART2  NN)  is  a  self-organizing  neural  network  structure,  which  is  a  prosperous methodology in fault diagnosis. Lee et al. [56] proposed an algorithm composed of three parts: parameter estimation, fault detection, and fault isolation. Once a fault is detected in the system, in order to isolate the occurred fault the estimated parameters are transmitted to the ART2 NN structure.. Wavelet Analysis Wavelet analysis is a common methodology for analyzing localized variations of signal.  In this method, to determine the dominant modes of variability  and how those modes vary in time, a time series is decomposed into time?frequency space. The wavelet transform  has  been  used  for  numerous  studies,  for  example,  Tafreshi  et  al.  [57]  used wavelet packet to recognize different conditions of one cylinder in a 12-cylinder engine. The  combustion  malfunctions  in  the  cylinder  were  detected  using  the  wavelet  packet providing a useful data analysis structure for extracting features. Wang et al. [58] and Chen et al. [59] discuss the integration of ART networks with wavelets to develop fault diagnosis algorithms 3.3.2.2 Qualitative Model-free Methods Qualitative  model-free  methodologies,  which  extract  qualitative  historical information, are studied in two major classes: a) expert systems and b) Qualitative Trend Analysis (QTA).    45 Fuzzy Approach Zadeh [60] was the first to recognize the importance of the concept of the fuzzy theory. Rule-based feature extraction methodology has been broadly employed in expert systems for fault diagnosis applications. Important efforts to apply expert systems in fault diagnosis applications can be seen in the works of Hakami and Newborn [61] and Stewart [62].  The  idea  of  using  task  framework  in  knowledge-based  diagnostic  systems  was developed by Ramesh et al. [63]. Tarifa and Scenna [64] proposed a hybrid system that uses  the  combination  of  fuzzy  logic  and  signed  directed  graphs  (SDG).  Scenna  [65] discussed an expert system approach for fault diagnosis in batch processes. Leung and Romagnoli  [66]  described  an  implementation  of  a  probabilistic  model-based  fault diagnosis  expert  system.  Ghodsi  and  Sassani  [67]  demonstrated  an  adaptive  fuzzy algorithm, which continuously adapts to variations in the input data. They demonstrated the  efficiency  of  the  structure  in  minimizing  wood  waste  cutting  process.  Later,  they improved  the  adaptive  fuzzy  algorithm  using  a  recursive  sub-algorithm  to  designate  a preferred cut patterns among all possible patterns [68].  Qualitative Trend Analysis Approach In process monitoring and supervisory control, qualitative trend analysis (QTA) is an  important  component,  which  is  utilized  to  describe  important  events  in  a  dynamic process to predict future condition and diagnose system faults. Even though the QTA is robust  and  accurate  in  system  monitoring,  its  real-time  application  to  very  large-scale plants is prohibitive due to its computational complexity.  Initial formal framework for the representation of process trends can be found in Cheung and Stephanopoulos?s work [69]. Vedam and Venkatasubramanian [70] proposed   46 an adaptive trend analysis framework based on the wavelet theory. Subsequently, they proposed a dyadic B-spline-based trend analysis algorithm achieving data compression by removing noise from the sensor data. Maurya et al. [71] resolved the shortcoming of the  QTA  in  real-time  applications  by  applying  the  QTA  on  the  principal  components rather than on the sensor data. The mathematical model-based approach adopted in the present thesis falls into the observer category, which is described in Section 3.3.1.1. The approach is based on the Kalman Filter algorithm developed to monitor the highly non-linear dynamic systems.    3.4 Kalman Filtering in Fault Diagnosis The Kalman Filter algorithm developed by R. Kalman is a recursive filter using noisy and even incomplete measurements to estimate the states of a linear system in the time domain. It is applied to the states of the discretised system to estimate the new states at each discrete time increment. Inputs of the algorithm are the system information and    optionally some knowledge from the controls on the plant if it is known. The existence of two  independent  noises  is  considered;  one  perturbs  the  information  from  system (measurement noise) and the other is mixed with linear operator (process noise). A  general  linear  discrete-time  system  is  represented  by  two  equations;  system equation: kwBuAxx ++=                                             (3.4) and output equation: kvHxy +=                                                     (3.5)   47 where A is the state matrix, B is the input matrix, H is the output matrix, x is the state vector, y is the system output, u is the system input, w is the process noise and v is the measurement noise. Both process and measurement noises are assumed to have a mean of zero,  and  be  Gaussian  (normal  distribution).  There  are  process  noise  matrix  Q  and measurement  noise  matrix  R,  which  are  related  to  process  noise  vector  w  and measurement noise vector v, respectively. The noise matrices are the expected values (the sum of the probability of each possible consequence of the experiment multiplied by its value) of corresponding vectors described mathematically as ]T                                                     (3.6) and   ]T                                                      (3.7) where Q and R are the covariance matrices of the measurement noise and process noise, respectively. The noise level during measurements and the accuracy of sensors together with the modeling uncertainties (such as what is discussed in Section 2.4.1) are essential to derive the covariance noise matrices. The  Kalman  filter  performs  the  estimation  in  a  predictor-corrector  approach. Using the given system model, the a priori (predicted) state estimate vector at step k is defined from previous trajectory of x and calculated by the ?Time-update? equation:   kkk Bu-- 1                                                 (3.8)   48 where  1-k  is the previous time step a posteriori (corrected) state estimate vector and uk is  the  known  system  input.  The  residual  is  defined  as  the  difference  between  the measured and predicted output: --kxHy ?                                                       (3.9) noting that when the Kalman filter is implemented in a real system,  k  is information from sensors, but in a simulation  k  is calculated from equation 3.5. The  a  posteriori  state  estimate  vector  uses  the  information  in  the  current observation and is calculated by the ?Measurement-update? equation: )-- - kkkkk x                                          (3.10) where  -k  is the a priori state estimate vector from Equation 3.8 and Kk is the Kalman gain. Note that if the residual ( -kk x ) is zero, the a priori state estimate vector will equal  the  a  posteriori  state  estimate  vector.  A  function  for  the  Kalman  gain  must  be defined  to  minimize  the  a  priori  and  a  posteriori  estimate  errors  and  thus  accurate estimation of the system states is achieved. The a priori and a posteriori estimate error covariances that we try to minimize are defined as: ]T---                                        (3.11) and ]T                                          (3.12) respectively. Referring to [72] Chapter 4, the resulting function for the Kalman gain is 1)( -+= RHHPHPK TkTk                                      (3.13)   49 and the minimized a priori estimate error covariance is found as QTkk +-- 1                                              (3.14) and the minimized a posteriori estimate error covariance is  --=kPHKIP )(                                           (3.15) The Kalman algorithm should be initialized at first with the a posterior estimate error covariance and the state variable values. The recursive relations of the predictor-corrector structure for the Kalman filter can be presented by the block diagram as shown in Figure 3.2.  This  structure  evolves  into  a  computational  scheme,  which  is  run  recursively  in parallel with a sampled-data system to acquire the real-time state estimates. Sampled-data systems such as the one used in the present investigation and described in Chapter 2 are continuous-time dynamic systems controlled by digital devices.  Early attempts at the application of Kalman filter for fault diagnosis can be found in the works of Dalle Molle and Himmelblau [73] and Bergman and Astrom [74]. Tsuge et al. [75] introduced a hierarchical methodology consisting of the signed directed graph and the extended Kalman filter. Pirmoradi et al. [76, Chapter 23] investigated a Kalman filter methodology to track conditions of the spacecraft attitude control systems.     50  Figure 3.2: Kalman filter algorithm  Although  the  Kalman  filter  is  a  broadly  used  filtering  strategy  in  system monitoring  applications  attributable  to  its  optimality,  robustness,  and  simplicity,  the employment of the Kalman filter to non-linear systems is quite difficult. To overcome this  shortcoming,  the  Extended  Kalman  Filter  (EKF)  [77]  can  be  used  instead.  This method simply linearizes the non-linear state space model at each time step around the last states, then the ?Time-update? and the ?Measurement-update? phases are performed sequentially  as  shown  in  Figure  3.2.  The  EKF  propagates  the  estimations  through  the first-order linearization; therefore it provides biased estimates when the system equations are highly non-linear. Although the EKF can be readily extended to incorporate higher Initial Estimates  ])00TxxxxEPx-=   Time Update  kBuxAx += --1??  QTkk +-- 1  Measurement Update  1)( -+= RHHPHPK TkTk  )-- - kkkkk x  --=kPHKIP )(    51 order  terms,  it  leads  to  high  computational  costs.  There  are  a  number  of  weaknesses associated with the EKF algorithm such as [78]: 1.  The  EKF  performance  depends  heavily  on  the  time-step  interval.  For successive estimation it should be sufficiently small, especially for highly non-linear  models,  otherwise  the  linearization  leads  to  unstable  filter performance. 2.  Providing  Jacobian  matrices  makes  the  EKF  not  suitable  for  large dimension systems because of the calculation of the derivatives. 3.  Since  the  EKF  algorithm  is  based  on  linearization  to  propagate  the covariance and mean of the system states, it gives unreliable estimates and is difficult to be tuned provided that the system is highly non-linear. Although the EKF has been one of the most widely used algorithms for parameter estimation  and  tracking  for  forty  years,  it  has  led  to  a  general  agreement  within  the control community that the EKF is difficult to implement and because of the linearization error (as mentioned above) it is only reliable for systems, which are not highly non-linear on  the  time  scale  of  the  update  intervals.  In  the  next  section,  another  non-linear transformation for the mean and covariance will be introduced to handle the linearization issue. 3.5 Unscented Kalman Filter Generally, the basic difference between the EKF and the Unscented Kalman Filter (UKF) emerges from the manner in which the non-linear model states and parameters are approximated. The UKF introduced by Julier [79] employs the unscented transformation,   52 which is a non-linear transformation. In this algorithm, the state probability distribution is represented by a minimal set of data points, referred to as sampled sigma points.  3.5.1 Unscented Transformation Julier  and  Uhlmann  [77]  calculated  the  mean  and  covariance  by  the  use  of  a sampling  approach,  called  the  Unscented  Transformation  (UT),  instead  of  an  arbitrary non-linear function. Consider an n-element variable,  n , with known covariance, Pxx, and mean value,  x . To statistically approximate the mean and covariance of the non-linear transformation, y = h(x) in which  m , a vector of 2n sigma points is formed according to the following: nnniTiiTii2~,...,~,...,~)))===+                                (3.16) where  i is the ith row of the  ) ;  )  , which is called the matrix square root of nP, is defined as follows: nPT =                                               (3.17) Then  the  calculated  sigma  point  vectors  are  simply  propagated  through  the  h function, which can be either linear or non-linear: nii 2)()( =                                       (3.18) note that no linearization is applied for propagation. The estimated mean and covariance of y are determined as follows:   53  summationsummation==-=niTuiuiuniiuynyn21)21))2121                                    (3.19) Simon in Chapter 14 of [80] investigated the UT.  A typical problem was studied to  make  a  comparison  between  UT  and  EKF  performances.  The  non-linear transformations examined by Simon are as the following:   thetathetasincos21rr==                                                       (3.20) He  calculated  the  covariance  and  mean  of  300  stochastic  points,  which  are  dispersed uniformly in the ranges of -0.01<r<0.01 and -0.35<theta <0.35 through both the UT and EKF.  Then  the  unscented  (from  the  UT)  and  linearized  (from  the  EKF)  results  were compared with the exact (reference) data. Figure 3.3 depicts a summarized illustration of the approach for both UT and EKF along with the reference resolution.   Figure 3.3: Compression of the UT and linear approximation  (From Optimal State Estimation, Dan Simon, Copyright ? 2006, reprinted with permission of John Wiley & Sons, Inc.)     54  With the use of the UT algorithm the estimation of mean and covariance are closer to the real  values  and  that  holds  because  this  approach  leads  to  a  third  order  accuracy  for Gaussian  inputs  [81].  Contrarily,  the  linearization  technique  used  in  the  EKF  scheme results in the first order accuracy for similar inputs. 3.5.2 UKF Algorithm The UKF algorithm is an extension of the UT to the recursive estimation in non-linear filtering problems. The UKF algorithm is summarized next [80]: The  following  expressions  form  a  general  model  of  non-linear  discrete-time systems to which the UKF may be applied at step k: kkvw+++))1                                        (3.21) in which f is the non-linear system function and h is the non-linear measurement function and the other parameters are the same as those defined in Section 3.4. The set of sigma points of the augmented state is constructed as: nnnikikTikiTii2~,...,~,...,~)1)11)1)===--+-                          (3.22) then  the  UKF  can  be  proceeded  by  the  predictor-corrector  step  in  the  Kalman  filter algorithm. The a priori state estimate,  -k , and its predicted error covariance,  -k , are calculated from the combination of the transformed sigma points as follows: ))( 1)( kkikik t-                                              (3.23)   55 summation=- =niikxnx21)?21?                                                  (3.24) 1)21) )??()??(21--=- +--= summationkTkiknikik QxxxxnP                             (3.25) Similarly  the  predicted  observation  vector  k   and  its  predicted  covariance  y   are calculated as: ))()( kikik t                                                 (3.26) summation==niikyny21)?21?                                                 (3.27) kTkiknikik RyyyynP +--= summation=)21 )(21)                                 (3.28) and the cross covariance matrix between  -k  and  k  is obtained as: Tkiknikik yyxxnP )??()??(21 )(21) --= summation=-                                    (3.29) The filter gain  k , the updated state estimate  k , and the covariance  k  are computed as:   1-yxyk P                                                    (3.30) )kkkkk y-                                           (3.31) TkKPKPP -=-                                              (3.32) The structure of the UKF can be presented by the schematic block diagram as shown in Figure 3.4.    56  Figure 3.4: UKF structure block diagram Initialization  ])00TxxxxEPx-=   Time Update (Propagation)  ))( 1)( kkikik t-  summation=- =niikxnx21)?21?  1)21) )??()??(21--=- +--= summationkTkiknikik QxxxxnP   ))()( kikik t  summation==niikyny21)?21?  kTkiknikik RyyyynP +--= summation=)21 )(21)   Tkiknikik yyxxnP )??()??(21 )(21) --= summation=-  Time Update (Sigma Points)  nnnikikTikiTii2~,...,~,...,~)1)11)1)===--+-  Measurement Update  1=yPPK  )kkkkk y-  TkKPKPP -=-  Iteration   57 In contrast with the noise in real systems, both process and measurement noises are considered as additive and hence the explained UKF algorithm is not rigorous. Wan [82] augmented the noise onto the state vector as: bracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp=kkkakvwxx )(                                               (3.33) then the augmented state  )(ak  is estimated. The initialization is as the following: bracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp=00)?00xxa                                             (3.34) bracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp -=0000000RQxPTa                        (3.35) Now the presented UKF can be used except that as we estimate the augmented mean and covariance, the terms  1-k  and  k  should be removed. 3.6 UKF Application in the Hydraulic System The UKF algorithm has been applied in non-linear control applications through the state estimation. The state space models are assumed known in these applications. In this  section,  we  demonstrate  the  use  of  the  UKF  to  the  dynamic  model  explained  in Chapter 2, and illustrate the results to show the performance of the developed algorithm.   58 3.6.1 The UKF Propagation Since the UKF is a discrete-time  algorithm and  also in order to be  compatible with real time digital computing, the state space equations should be discretised prior to using the UKF. Equation 3.36 shows the discretised state space model, which is derived by the use of Forward Difference method. Note that all parameters in this model are time invariant.  braceexbraceexbraceexbraceexbraceexbraceexbraceexbraceexbraceexbraceexbraceexbraceexbraceexbraceleftbtbraceexbraceexbraceexbraceexbraceexbraceexbraceexbraceexbraceexbraceexbraceexbraceexbraceexbraceleftmidbracelefttp+++braceexbraceexbraceleftbtbraceexbraceexbraceleftmidbracelefttp<bracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftbtbracketlefttp --->bracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftbtbracketlefttp ---=braceexbraceexbraceleftbtbraceexbraceexbraceleftmidbracelefttp<bracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftbtbracketlefttp --->bracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftbtbracketlefttp ---=+)))02))((02))((...02))((02))((...)62654141431min1min21kkMTkkkxXAV TkkxXAV TkkXkxAV TkXkxAV Tkkspextaesseomegarhoomegabetarhoomegabetarhoomegabetarhoomegabeta             (3.36)  in which T is the sampling time and k denotes the process step number. Due to the digital computing chrematistics of the computer used, the sampling time was chosen as 20 ms in this research. The discretised actuator friction is calculated from:   59 braceexbraceleftbtbraceexbraceleftmidbracelefttp<=<>=-00|)))5vvkstkslf         (3.37) Since the system has three outputs (measurements), two chamber pressures (x2, x3) along with the actuator displacement (x4), the measurement matrix, which is linear here, can be written by inspection as: bracketrightexbracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp=000H                                           (3.38) For the sake of simplicity, the noise characteristics of different states are assumed to be independent of each other. Therefore, the noise covariance matrices are diagonal. With  reference  to  Figures  2.8  to  2.10,  the  MAEs  between  model  output  and  test  rig measurements are included in Table 3.1. Each MAE reflects the model uncertainty due the corresponding state. Therefore, the process noise matrix component in accord with the  pressure  in  chamber  1  is  1010  Pa2,  the  pressure  in  chamber  2  is  1010  Pa2,  and  the actuator movement is 10-6 m2.  Table 3.1: The MAEs of the pressures and actuator movement State Pressure in Chamber 1  Pressure in Chamber 2   Actuator Position Model MAE  0.12 (MPa)  0.14 (MPa)  2.0 (mm)     60 Other components of Q matrix are determined regarding the physical attributes of the  corresponding  state.  Then,  these  values  are  tuned  to  achieve  the  satisfactory  UKF performance. The following matrix is defined as the process noise matrix in this system in each of the experiment:  bracketrightexbracketrightexbracketrightexbracketrightexbracketrightexbracketrightexbracketrightexbracketrightexbracketrightexbracketrightexbracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftexbracketleftexbracketleftexbracketleftexbracketleftexbracketleftexbracketleftexbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp=----20661010201000000Q                           (3.39) in which all quantities are in the SI system.  The measurement noise matrix is defined regarding the accuracy of the sensors. The  two  msisensors?  pressure  transducers  have  the  accuracy  of  Pa5   and  the accuracy of the Mitutoyo? position encoder is calculated as  m-6 . Since there is no significant noise source near the set-up, it is assumed that the noise levels lie in the sensor  accuracies.  Therefore,  the  measurement  noise  matrix  can  be  represented  as  the following in each of the experiment: bracketrightexbracketrightexbracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp=-1110101000R                                             (3.39)  in which all quantities are in the SI system.   61 The preceding matrices should be defined before the UKF algorithm can be built. Now the UKF must be initialized; regarding the UKF structure block diagram (Figure 3.4)  0 and 0  should be defined.  0  is a 1?6 matrix whose second and third components, the chamber pressures, should be within the range of [Pe, Ps] and its fourth component, the actuator displacement, should be within the range of [Xmin , Xmax]. Generally, 0 can be defined as a positive-definite matrix.  [ ][ ]braceexbraceleftbtbraceexbraceleftmidbracelefttp=?-10060100diagx T                      (3.40) in which all quantities are in the SI system. After setting the initial conditions as well as the covariance matrices the recursive part of the UKF algorithm can be performed in a series of steps. At first, k is unity and the  sigma  points  are  calculated.  Following  the  UKF  structure,  these  sigma  points  are propagated and the ?measurement update? block is performed. Then to proceed in the recursive algorithm to the next time step, we return to sigma point calculations and k is incremented by one. 3.6.2 Simulation Results After describing the UKF recursive algorithm as applied to the system, sample numerical simulations are run in this part. In order to simulate the measurement noise we call the Gaussian random number generator once every time-step interval and therefore the output is calculated via equation 3.5. By applying the UKF to the state space model a sequence  of  estimated  state  vectors  is  available  and  hereupon  the  estimation  residual   62 vector  can  be  computed  by  applying  equation  3.9.  Due  to  the  measurement  noise,  R matrix, and the modeling uncertainties, Q matrix, the residual errors can not be ignored but still should remain at relatively low levels depending on the Q and R matrices? values. To show the performance of the developed UKF, the residual errors in the state estimations  of  the  system  are  presented  in  Figures  3.5  to  3.7.  Figures  3.5  and  3.6 demonstrate  the  residual  errors  in  the  estimates  of  the  chamber  pressures.  Figure  3.7 illustrates the residual error in the estimation of the actuator position.  -0.100.10 20 40 60 80 100Time (s)Residual (MPa)  . Figure 3.5: Estimation error of the pressure in chamber 1 -0.100.10 20 40 60 80 100Time (s)Residual (MPa)  . Figure 3.6: Estimation error of the pressure in chamber 2 -0.00400.0040 20 40 60 80 100Time (s)Residual (m) Figure 3.7: Actuator position estimation error    63 In the preceding  graphs, dashed lines display the plus and minus values of the theoretical  standard  deviations  of  the  corresponding  measurements.  Theoretically,  the residual  error  results  should  fall  within  the  sigma   bounds  at  least  68%  of  the  time approximately. Accordingly, we can visually verify that the experimental and theoretical results appear to be in agreement. Note that in these simulations, the noise-added model represents the experimental data. This agreement should be held in order to show that the UKF algorithm is performing appropriately. 3.7 Concluding Remarks In this chapter, fault diagnosis methods were surveyed and the overall structures of the Kalman filter and UKF schemes were described. The recursive part of the UKF consists of two phases, Time Update and Measurement Update. In ?Time Update?, the sigma points are calculated to be used in mean and covariance calculations, and then the a priori state estimates and covariances are obtained, while in ?Measurement Update?, the  a  posteriori  state  estimates  and  covariances  are  calculated.  The  purpose  of  this scheme is to estimate the system states and generate the residual errors, which is used in fault detection by threshold testing. To verify the proposed UKF algorithm performance, it was applied to the state space model. The UKF used the information from the dynamic of the system along with the  sensor  measurements  to  predict  future  sensor  outputs.  The  results  showed  that  the difference  between  the  actual  and  predicted  states  stayed  in  the  acceptable  bound. Therefore,  the  investigated  UKF  scheme  is  reliable  to  be  employed  in  condition monitoring of the experimental set-up.   64 CHAPTER 4    Online Monitoring of the Hydraulic System        4.1 Introduction Online condition monitoring through examining certain system state relationships and  comparing  the  states  with  known  nominal  values  is  the  most  practical  method  to predict  the  occurred  faults  in  hydraulic  systems.  This  prediction  increases  the  system efficiency by decreasing the chance of the further degradation beside the maintenance cost. To do so, an accurate state space model (as discussed in Chapters 2) along with a robust and reliable fault diagnosis methodology (as discussed in Chapters 3) is required.  In this chapter, the terminology used in [25] is employed to address the functions within the fault monitoring scheme: Fault Detection By the ?Fault Detection? process, fault occurrence is determined along with its arising time.     65 Fault Diagnosis By  the  ?Fault  Diagnosis?  process,  which  includes  fault  detection,  the  size, location, kind and time of fault occurrence is determined.  Monitoring By the ?Monitoring? process, the conditions of the system are surveyed in real-time continuously.  In this chapter, the experimental results of artificially introduced faults into the hydraulic  circuit  are  presented  followed  by  the  discussions  pertaining  to  the  fault diagnosis scheme performance.  4.2 Experimental Results and Discussion Similar  to  the  previous  experiments  and  simulations,  the  sinusoidal  signals  are used as the position reference signals to control the test rig activities in this section. All the  experiments  are  carried  out  under  the  UKF  observation  and  a  closed-loop  control regulation.  The  reference  signal  is  given  as  ).  Simulation results of previous chapters show that in the normal operational conditions the state space model  follows  the  experimental  test  rig  characteristics  in  a  satisfactory  precision  and moreover  the  UKF  scheme  estimates  the  system  states  precisely.  All  proposed  faults occur in the set-up approximately 50 seconds after the test starts and are kept until the end  of  the  tests,  which  last  100  seconds.  Note  that  faults  exist  in  a  multiple-fault environment but occur singly.    66 The matrices of the process noise, Q, measurement noise, R and initial covariance, P0 along with the initial state vector are defined the same as what is derived in Section 3.6.1: [ ][ ][ ][ ]braceexbraceexbraceexbraceleftbtbraceexbraceexbraceexbraceexbraceleftmidbracelefttp===?---1120100601010100diagdiagdiagx T                      (4.1) The  results  of  the  experiments  are  reported  in  the  following  sections.  The experiments were intended to diagnose the occurred faults in real-time  operations. All five introduced faults are studied in two groups: leakage and load faults. 4.2.1 Leakage Faults This  section  is  developed  to  study  the  affects  of  leakages  on  the  system. Therefore, different types of leakage faults in various levels are investigated. 4.2.1.1 Actuator External Leakage at Chamber 1 In this section, we introduce the fluid loss in the first chamber?s connecting hoses. In real systems, this form of leakage may occur because of the poor pipe connections or hose  ruptures.  As  a  result  of  this  fault,  the  pressure  reduces  in  chamber  1  during  an extraction stroke when this chamber is connected to the pressure supply line through the servo-valve.  The amount of the leakage is regulated manually by tuning the adjusting knob of the  needle  valve  mounted  on  chamber  1.  The  bypass  is  illustrated  in  Figure  2.5.  The   67 experiments are carried out for three different levels of the leakages as shown in Table 4.1. Table 4.1: Multilevel leakage coefficients Leakage level  Effective Control Flow coefficient of the needle valve Low  Cv = 0.0143 (Valve open one turn) Medium  Cv = 0.0612 (Valve open two turns) High  Cv = 0.112 (Valve open three turns)   For brevity, since the system response is similar for all levels of leakages only the medium  level  is  discussed  in  more  detail  here.  Figure  4.1  shows  the  actuator displacement while medium level external leakage occurs at chamber 1. It can be verified visually that the actuator shifts slightly toward chamber 1 after the leakage occurs. This happens because the leakage at chamber 1 reduces the pressure in this chamber during the extracting  period  and  as  a  result  there  is  not  sufficient  power  to  identically  push  the piston toward chamber 2. Note that due to the closed-loop control, after about two cycles the piston stops shifting, but a bias error remains in the actuator referenced-movement.   00.020.040.060.080.10 20 40 60 80 100Time (S)Displacement (m)   Figure 4.1: Actuator displacement while medium level leakage occurred in chamber 1 Leakage  initiated   68 All three residual error signals due to the low level external leakage at chamber 1 are  shown  in  Figure  4.2.  It  can  also  be  observed  visually  that  residual  error  of  the corresponding  state  (pressure  in  chamber  1)  increases  slightly  more  than  that  of  the pressure in chamber 2. This distinction is more obvious in the next two experiments in which  medium  and  high  level  leakages  are  considered.  The  results  are  illustrated  in Figures 4.3 and 4.4, respectively.  00.10.20.30.40 20 40 60 80 100MAE (MPa)      . 00.10.20.30.40 20 40 60 80 100MAE (MPa)    . 051015200 20 40 60 80 100Time (S)MAE (mm)     . Figure 4.2: Low level leakage at chamber 1; (a) Residual errors of pressure in chamber 1, (b) pressure in chamber 2, and (c) actuator position             a) b) c)   69 00.10.20.30.40 20 40 60 80 100MAE (MPa)   . 00.10.20.30.40 20 40 60 80 100MAE (MPa)  . 051015200 20 40 60 80 100Time (S)MAE (mm)   . Figure 4.3: Medium level leakage at chamber 1; (a) Residual errors of pressure in chamber 1, (b) pressure in chamber 2, and (c) actuator position  00.10.20.30.40 20 40 60 80 100MAE (MPa)   . 00.10.20.30.40 20 40 60 80 100MAE (MPa)   . 051015200 20 40 60 80 100Time (S)MAE (mm)    . Figure 4.4: High level leakage at chamber 1; (a) Residual errors of pressure in chamber 1, (b) pressure in chamber 2, and (c) actuator position a) b) c) a) b) c)   70 The  following  table  shows  the  corresponding  variation  of  residual  MAEs.  It  is intuitive that by opening of the needle valve, the leakage increases much more when the chamber 1 is activated, and therefore, the residual MAE increases more for higher level of leakage. Table 4.2: Increase in MAEs due to the external leakage at chamber 1                           Leakage level Measurand  Low  Medium  High Pressure in Chamber 1 (MPa)  0.026  0.05  0.073 Pressure in Chamber 2 (MPa) 0.016  0.026  0.034 Actuator Position (mm)  1.2  2.35  3.85   4.2.1.2 Actuator External Leakage at Chamber 2 The fluid loss in connecting hoses of chamber 2 is introduced in this section. As the system has same characteristics for different levels of leakages, the medium level of leakage is discussed in more detail here. Figure 4.5 illustrates the actuator displacement while medium level external leakage occurs in chamber 2. It is intuiuive that the actuator shifts toward chamber 2 accordingly after the occurrence of the leakage. The reason is that the leakage at chamber 2 reduces the pressure in this chamber during the retracting period and as a result there is not sufficient power to identically push the piston toward chamber  1.  The  same  as  previous  section,  after  less  than  two  cycles  the  piston  stops shifting as a result of the closed-loop control, but a bias error remains. With reference to Figures  4.1  and  4.5,  it  does  seem  almost  certain  that  the  actuator  malfunctions  more   71 aggressively in the presence of the leakage at chamber 2 rather than that at chamber 1. This happens because the pressure is much higher in chamber 2 and therefore connecting this chamber to the tank causes more pressure loss, which leads to further defection in the system.  00.020.040.060.080.10 20 40 60 80 100Time (S)Displacement (m)  . Figure 4.5: Actuator displacement while medium level leakage occurred in chamber 2  Figures 4.6 to 4.8 show the results of introducing leakage into chamber 2. Graphs indicate that the residual error of the pressure in chamber 2 increases much more than that  in  chamber  1.  Similar  to  the  previous  section,  these  experiments  are  carried  out considering three different leakage levels as shown in Table 4.1.      Leakage  initiated   72 00.10.20.30.40 20 40 60 80 100MAE (MPa)   . 00.10.20.30.40 20 40 60 80 100MAE (MPa)   . 051015200 20 40 60 80 100Time (S)MAE (mm)    . Figure 4.6: Low level leakage at chamber 2; (a) Residual errors of pressure in chamber 1, (b) pressure in chamber 2, and (c) actuator position     00.10.20.30.40 20 40 60 80 100MAE (MPa)  . 00.10.20.30.40 20 40 60 80 100MAE (MPa)  . 051015200 20 40 60 80 100Time (S)MAE (mm)  . Figure 4.7: Medium level leakage at chamber 2; (a) Residual errors of pressure in chamber 1,  (b) pressure in chamber 2, and (c) actuator position     a)  b)  c) a) b) c)   73 00.10.20.30.40 20 40 60 80 100MAE (MPa)  . 00.10.20.30.40 20 40 60 80 100MAE (MPa)    . 051015200 20 40 60 80 100Time (S)MAE (mm)    . Figure 4.8: High level leakage at chamber 2; (a) Residual errors of pressure in chamber 1, (b) pressure in chamber 2, and (c) actuator position   What is noticeable in the results is that the residual MAE of the faulty chamber increases proportionally with the increase of the leakage. As expected, the residual MAE of the pressure in chamber 2 increases much more aggressively than that in chamber 1 as shown in Table 4.3. Table 4.3: Increase in MAEs due to the external leakage at chamber 2                    Leakage level     Measurand  Low  Medium  High Pressure in Chamber 1 (MPa)  0.012  0.015  0.015 Pressure in Chamber 2 (MPa) 0.045  0.074  0.094 Actuator Position (mm)  2.75  5.4  8.5    a) b) c)   74 4.2.1.3 Actuator Internal Leakage In this section, experiments are carried out in order to assess the performance of the UKF algorithm in diagnosing of the actuator internal leakage. There are a number of potential causes that lead to internal leakage in a hydraulic cylinder, each of which may result in a set-up that fails to operate appropriately. For instance, if the cylinder?s piston seal is impaired, fluid may leak internally between chambers. Furthermore, the internal leakage may occur as a result of the improper operation of a bypass valve. All of which may result in the differential force between the extension and retraction chambers be out of balance and hence the end-effector characteristics become unreliable.  In the experiments, the internal leakages are adjusted manually by tuning the knob of the cross-over needle valve shown in Figure 2.5. Tests are carried out for the same three  leakage  levels  used  earlier.  Figure  4.9  illustrates  the  variation  in  actuator characteristic due to the medium level internal leakage occurrence. The point is that the actuator shifts toward the chamber with higher pressure (chamber 2). That is because the internal  leakage  decreases  the  pressure  difference  between  two  chambers  and consequently the chamber with higher effective area (chamber 1) can apply more force to the piston.     75 00.020.040.060.080.10 20 40 60 80 100Time (S)Displacement(m)  . Figure 4.9: Actuator displacement while medium level internal leakage occurred  The residual errors due to the internal leakage are illustrated in Figures 4.10 to 4.12.  However,  the  residual  MAEs  of  all  measurements  increase,  the  increments  in pressure  MAEs  are  less  than  that  in  actuator  displacement  MAE.  Moreover,  these variations increase in amount for higher level of leakages.  00.10.20.30.40 20 40 60 80 100MAE (MPa)   . 00.10.20.30.40 20 40 60 80 100MAE (MPa)   . 051015200 20 40 60 80 100Time (S)MAE (mm)   . Figure 4.10: Low level internal leakage; (a) Residual errors of pressure in chamber 1,  (b) pressure in chamber 2, and (c) actuator position  a) b) c) Leakage  initiated   76 00.10.20.30.40 20 40 60 80 100MAE (MPa)   . 00.10.20.30.40 20 40 60 80 100MAE (MPa)   . 051015200 20 40 60 80 100Time (S)MAE (mm)     . Figure 4.11: Medium level internal leakage; (a) Residual errors of pressure in chamber 1, (b) pressure in chamber 2, and (c) actuator position  00.10.20.30.40 20 40 60 80 100MAE (MPa)   . 00.10.20.30.40 20 40 60 80 100MAE (MPa)   . 051015200 20 40 60 80 100Time (S)MAE (mm)   . Figure 4.12: High level internal leakage; (a) Residual errors of pressure in chamber 1, (b) pressure in chamber 2, and (c) actuator position a) b) c) a) b) c)   77 Table 4.4 shows that as expected, the residual MAEs corresponding pressures are almost equal to each other.   Table 4.4: Increase in MAEs due to the internal leakage                    Leakage level     Measurand  Low  Medium  High Pressure in Chamber 1 (MPa)  0.01  0.028  0.057 Pressure in Chamber 2 (MPa) 0.008  0.025  0.046 Actuator Position (mm)  1.8  4.95  11.15    4.2.2 Load Faults As opposed to the leakage faults, load faults are not quantified. Therefore, each set of test is carried out for a certain level of fault. 4.2.2.1 Dynamic friction load This section is developed to study the affects of the dynamic friction load on the hydraulic  system.  This  force  is  applied  to  the  hydraulic  actuator  by  means  of  the pneumatic cylinder as explained in Section 2.2. All experiments are carried out for the dynamic friction of 1162 N as the actuator is extracting and 1042 N as the actuator is retracting.  These  applied  forces  vary  in  amount  because  of  utilizing  the  asymmetric pneumatic cylinder. The  following  Figure  shows  the  actuator  displacement.  There  is  no  significant variation in the sinusoidal movement of the actuator after fault occurs. The reason is that   78 the controller compensates by inputting current with higher amplitude to the servo-valve, and therefore the capability of the actuator in making up for the dynamic friction load improves.   00.020.040.060.080.10 20 40 60 80 100Time (S)Displacement(m)   . Figure 4.13: Actuator displacement while the dynamic friction load is applied  Figure 4.14 shows the residual error signals due to the dynamic friction load. It can be observed visually that residual error of the actuator position increases much more than that of the pressures in chambers 1 and 2.  00.10.20.30.40 20 40 60 80 100MAE (MPa)   . 00.10.20.30.40 20 40 60 80 100MAE (MPa)   . 051015200 20 40 60 80 100Time (S)MAE (mm)   . Figure 4.14: Dynamic friction load; (a) Residual errors of pressure in chamber 1,  (b) pressure in chamber 2, and (c) actuator position a) b) c) Fault initiated   79 The  following  table  presents  the  variation  of  residual  MAEs.  Both  pressure residual errors increase equably. Table 4.5: Increase in MAEs due to the dynamic friction load Measurand  Pressure in Chamber 1 (MPa) Pressure in  Chamber 2 (MPa) Actuator Position  (mm) Increase in MAE  0.061  0.05  13.0   4.2.2.2 Loss of Load The goal of this section is to examine the system characteristics subject to loss of the  load.  This  type  of  faults  is  mostly  related  to  the  industrial  robotics  and  crane machinery.  In  this  experiment  two  weights  are  disconnected  from  the  actuator simultaneously. Similar to the previous section result, removing the load does not cause significant disturbance to the end-effector movement as shown in the following figure. With  a  comparison  between  the  actuator  characteristics  in  this  and  previous experiments, it can be observed that the actuator movement diverges from the reference signal  by  occurrence  of  any  leakage  fault.  Although,  it  follows  the  reference  signal satisfactorily by occurrence of faults regarding the load. The reason is that by applying any disturbance on the healthy system the regulator starts compensating via the current input to the servo-valve. In leakage scenarios the servo-valve increases the pressures of the corresponding chambers. Since the leakage is proportional to the pressure difference, the amount of the leakage intensifies, but it is not a case of load faults. The following equation expresses the leakage through the needle valve:   80 pvv delta                                               (4.2) in  which  v is  the  flow  coefficient, v   is  the  valve  spool  position,  and  incrementp  shows  the pressure difference between valve ports.    00.020.040.060.080.10 20 40 60 80 100Time (S)Displacement(m)   . Figure 4.15: Actuator displacement while the load is removed  Experimental  results  are  illustrated  in  Figure  4.16.  All  residual  MAEs  increase just as the fault occurs and also this variation for actuator displacement residual is much more than that for pressure residuals.         Fault initiated   81 00.10.20.30.40 20 40 60 80 100MAE (MPa)   . 00.10.20.30.40 20 40 60 80 100MAE (MPa)    . 051015200 20 40 60 80 100Time (S)MAE (m)   . Figure 4.16: Load lost; (a) Residual errors of pressure in chamber 1, (b) pressure in chamber 2, and (c) actuator position  Corresponding  variation  of  the  residuals  are  shown  in  Table  4.6.  The  same  as previous section, pressure residual errors increase equably. Table 4.6: Increase in MAEs due to the load lost Measurand  Pressure in  Chamber 1 (MPa) Pressure in  Chamber 2 (MPa) Actuator Position  (mm) Increase in MAE  0.007  0.006  1.3   a) b) c)   82 4.3 Fault Diagnosis and Discussion The performance of the designed fault monitoring scheme in fault detection has been discussed in Section 4.2. The fault can be diagnosed reliably in a case that fault signatures are distinguishable satisfactorily. Table 4.7 provides the variations of residual MAEs of the three measurements corresponding to all five studied faults. In  order  to  diagnose  the  occurred  fault,  there  are  three  criteria  that  should  be concerned;  residual  MAEs,  chamber  pressures,  and  actuator  characteristics.  The influences of each fault on the system, which lead to the fault diagnosis, are described individually.  4.3.1 Leakage at Chamber 1 In  a  case  that  the  residual  MAE  of  pressure  in  chamber  1  increases  almost  to twice or more than that in chamber 2, it can be concluded that external leakage occurs at chamber 1 regardless of any other criterion variation. Since the residual errors increase proportional  to  the  amount  of  the  leakage  at  chamber  1,  the  intensity  of  the  external leakage can also be estimated by observing the amounts of the residual error increments.   83 Table 4.7: Increase in MAEs due to the occurred fault                    Measurand     Fault Pressure in  Chamber 1 (MPa) Pressure in  Chamber 2 (MPa) Actuator Position  (mm) Leakage at Chamber 1  (Low) 0.026  0.016  1.2 Leakage at Chamber 2 (Low)0.012  0.045  2.75 Internal Leakage  (Low) 0.01  0.008  1.8 Leakage at Chamber 1  (Medium) 0.05  0.026  2.35 Leakage at Chamber 2 (Medium)0.015  0.074  5.4 Internal Leakage  (Medium) 0.028  0.025  4.95 Leakage at Chamber 1  (High) 0.073  0.034  3.85 Leakage at Chamber 2 (High)0.015  0.094  8.5 Internal Leakage  (High) 0.057  0.046  11.15 Dynamic friction load  0.61  0.5  15.5 Load Rupture 0.007  0.006  1.3      84 4.3.2 Leakage at Chamber 2 The difference between the pressure residual error increments is much higher in the presence of leakage at chamber 2 rather than that at chamber 1 for a certain leakage level.  As  an  instance  for  medium-level  leakage  occurrence  in  chamber  1,  the  residual MAE of pressure in chamber 1 increases 24 kPa more than that in chamber 2, but for a medium-level leakage at chamber 2 the residual MAE difference between chamber 1 and chamber 2 is 59 kPa.  However, still we follow the regulation regarding the leakage at chamber 1. This imitation is decided for the sake of simplicity of the real-time practical fault monitoring scheme.  Therefore, in a case that the residual MAE of pressure in  chamber 2 increases almost to twice or more than that in chamber 1, it can be concluded that external leakage occurs at chamber 2 regardless of any other criterion variation. Since the residual errors increase  proportional  to  the  amount  of  the  leakage  at  chamber  2,  the  intensity  of  the external leakage can also be estimated by  observing the amounts of the residual  error increments. 4.3.3 Dynamic Friction Load It can be observed that by occurrence of either internal leakage or load faults all residual MAEs grow, however, as opposed to the presence of external leakage, residual errors of pressure in chamber 1 and chamber 2 increase almost identically. To distinguish the dynamic friction load, it is essential to consider the characteristics of pressures and actuator movement. Figure 4.17 shows the pressure attributes as the dynamic friction load takes place.    85 02460 20 40 60 80 100Pressure (MPa)   . 02460 20 40 60 80 100Time (S)Pressure (MPa)    . Figure 4.17: Pressure characteristics in dynamic friction load occurrence; (a) pressure in chamber 1, and (b) pressure in chamber 2  It  can  be  observed  that  to  compensate  the  external  force,  higher  pressures  are applied on the actuator, which leads to the agreement between the end-effector movement and the reference signal (as illustrated in Figure 4.13).  In summary, when the dynamic friction load fault occurs, the pressure residual errors grow almost equably along with the increase in the actuator displacement residual error. Moreover, pressure transducers measure higher pressures at both chambers as the opposite of the position encoder, which shows no variation in the actuator movement. 4.3.4 Loss of Load Again, all three criteria should be considered to diagnose the load disconnection. The residual error increment trends are consistent with those of the dynamic friction load experiment. As illustrated in Figure 4.15 there is no significant variation in the actuator referenced-movement while the load is removed. To study the pressure characteristics, pressure transducer signals are shown in the following figures.  a) b)   86 02460 20 40 60 80 100Pressure (MPa)   . 11.522.533.540 50 60 70Time (S)Pressure (MPa)  . Figure 4.18: (a) Pressure in chamber 1 while the load is removed and (b) the close-up plot       02460 20 40 60 80 100Pressure (MPa)    . 11.522.533.540 50 60 70Time (S)Pressure (MPa)  . Figure 4.19: (a) Pressure in chamber 2 while the load is removed and (b) the close-up plot   The  close-up  plots  show  the  pressure  variations  better  at  the  time  of  load disconnection, from which it can be seen that pressure in chamber 1 increases slightly while  the  pressure  in  chamber  2  decreases  a  little.  This  happens  because  of  the asymmetry in the load attached to the actuator as discussed in Section 2.2. In scenarios in which  symmetric  loads  are  applied  to  the  actuator,  due  to  the  acceleration  issue,  by a) b) a) b)   87 removing the load the pressure transducers show less pressures in compare with the time that the fault is not introduced to the system.  In  summary,  when  the  load  is  disconnected,  the  pressure  residual  errors  grow almost  equably  along  with  the  increase  in  the  actuator  displacement  residual  error. Moreover, at least one of the pressures decreases in amount while the position encoder displays no variation in actuator referenced-movement. 4.3.5 Internal Leakage By introducing the internal leakage to the actuation system the most significant characteristic variation pertains to the actuator movement as shown in Figure 4.9. This variation makes the internal leakage fault distinguishable from faults related to the load. However, the pressures in chambers decrease in this experiment. Figure 4.20 illustrates the pressures while the high level of internal leakage occurs in the system.   02460 20 40 60 80 100Pressure (MPa)  . 02460 20 40 60 80 100Time (S)Pressure (MPa)   . Figure 4.20: Pressure characteristics in internal leakage occurrence; (a) pressure in chamber 1, and (b) pressure in chamber 2 a) b)   88 Pressures reduce because both chambers are connected to the tank continuously after the internal leakage is introduced. For instance, chamber 1 is connected to the return line directly during the retraction period while it is connected to the return line through the  needle  valve,  which  emulates  the  internal  leakage,  and  chamber  2  during  the extraction period.  In summary, when the internal leakage occurs, the pressure residual errors grow almost  equably  along  with  the  increase  in  the  actuator  displacement  residual  error. Moreover, pressure transducers show decreases in pressures. Another point that should be notified is the actuator movement variation from the reference signal. Since the residual errors increase proportional to the amount of the leakage as it can be observed from Table 4.7,  the  intensity  of  the  cross-port  leakage  can  also  be  estimated  by  observing  the amounts of the residual error increments.   89 CHAPTER 5    Conclusion        5.1 Summary The  goal  of  this  research  was  to  develop  an  on-line  condition  monitoring technique based on the Unscented Kalman Filter (UKF). This scheme was applied to a hydraulic  test  rig  to  diagnose  artificially  induced  faults.  Since  the  UKF  requires  an accurate model of the system to give good estimation of the system states, the set-up was inspected carefully to derive its state space model.  First, the investigated test rig configuration was explained with the aid of pictures and schematic diagrams. Two subsystems, pneumatic and hydraulic, together with their essential  components  were  described  in  detail.  All  five  potential  faults,  which  were emulated by the current scheme, were discussed, namely:  1, 2.  External leakages at both hydraulic cylinder chambers 3.  Dynamic friction load 4.   Sudden loss of load   90 5.  Leakage inside of the hydraulic cylinder Then, the linear time-invariant state space model corresponding to the set-up was derived. Six  state  variables  characterized  the  system:  the  servo-valve  spool  displacement  and velocity,  the  hydraulic  cylinder  chamber  pressures,  and  the  actuator  displacement  and velocity. This mathematical model was further elaborated with the details of the various governing  equations,  function  modeling,  and  all  related  parameters.  A  number  of experiments were carried out to determine the parameters, which were not measurable directly such as the parameters of the actuator friction model. Then, to validate the state space model, a sinusoidal position reference signal was applied to the closed-loop system. The results verified that the system model characteristics satisfactorily converged to the corresponding set-up features as it operated in normal conditions. Fault  detection  and  diagnosis  techniques  were  explained,  and  a  number  of valuable previous works were outlined for each of the methodologies. The Kalman filter and EKF algorithms were studied in more detail, and their advantages and disadvantages when applied to dynamic systems were discussed. Then, the UKF was introduced and its capabilities  in  handling  the  highly  non-linear  systems  were  described.  Its  satisfactory performance in state estimation was tested on the model of the set-up. The measurement data were generated by adding white noise to the output of the mathematical model of the hydraulic  test  rig.  The  estimation  results  of  three  states,  actuator  displacement  and chamber pressures, were compared with the measurement data to generate residual errors.  Since all the errors were satisfactorily within the acceptable bandwidth, it was concluded that the UKF methodology was reliable to be employed in condition monitoring of the experimental set-up.   91 The developed UKF-based fault monitoring scheme was tested on the physical system  while  different  fault  scenarios  were  singly  introduced  to  the  system.  Again,  a sinusoidal reference signal was used for the actuator displacement. The fault diagnosis scheme  estimated  the  system  states  and  generated  residual  errors  in  real  time.  To diagnose  the  occurred  fault,  three  criteria,  residual  MAEs,  chamber  pressures,  and actuator characteristics, were considered. Based on the presented experimental results and discussions, the proposed scheme could reliably diagnose the occurred faults. Also, for each leakage scenario, three levels of leakages were introduced to the test rig. However, it was impracticable to evaluate quantity of the leakage from residual errors, the proposed algorithm can be employed to qualitatively assess the leakage level. 5.2 Contributions This thesis consists of a number of contributions, which have been made to both fields of hydraulics and fault diagnosis. The major contributions of this thesis are outlined below: ?  A  fully  operational  hydraulic  test  rig  capable  of  emulating  faults  was constructed. ?  A novel UKF application in hydraulic systems for on-line diagnosis of faults was  proposed.  This  structure  accepted  raw  sensor  data  as  input  and automatically generated fault symptoms. ?  Five  of  the  most  common  faults  in  industrial  hydraulic  systems  were investigated. Two faults were related to the load and three were pertinent to   92 the external leakage on either side and internal to the actuator. The leakage faults  were  quantified  and  the  proposed  fault  monitoring  scheme  was successful in indicating the leakage fault level. ?  A  comprehensive  mathematical  model  for  accurate  simulation  of  a  servo-actuator system was developed and validated using the experimental data. The asymmetry of the system (the asymmetric load was carried by the asymmetric cylinder)  was  the  major  challenging  aspect  in  this  test  rig.  The  model calculations  converged  to  the  sensor  data  with  an  MAE  within  10%  of  the actual measurements. 5.3 Suggestions for Future Works Model-based fault diagnosis for an uncertain non-linear system is an extremely rich area for research both in terms of theoretical problems and practical implementation issues.  There  are  many  ways  to  improve  the  proposed  technique  or  introduce  new methods.  In  future  works,  the  proposed  strategy  can  be  extended  in  the  following direction: ?  The  state  space  model  acquired  for  fault  monitoring  can  be  extended  to incorporate other hydraulic and even pneumatic components. To do so, it is required to obtain good representative mathematical models and appropriate estimation for their parameters. As a result, the system states may be predicted more accurately, which makes the system more reliable.   93 ?  An  expert  system  can  diagnose  the  occurred  fault  by  considering  the  UKF estimations,  residual  errors  and  sensor  signals,  in  real-time.  Designing  and implementing  a  graphical  user  interface  to  show  decisions  of  the  expert system is also desirable. ?  More  common  faults  can  be  introduced  to  the  system  to  examine  the effectiveness  of  the  proposed  condition  monitoring  scheme  in  diagnosing them.  ?  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