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A new inference method for multivariable fuzzy systems with application in industrial control 2001

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A NEW INFERENCE METHOD FOR MULTIVARIABLE FUZZY SYSTEMS WITH APPLICATION IN INDUSTRIAL CONTROL B y Roya Rahbari B . A . S c . K.N.Toosi University of Technology, Tehran, Iran M . A . S c . Middle East Technical University, Ankara, Turkey A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T FOR T H E D E G R E E O F D O C T O R OF P H I L O S O P H Y in T H E F A C U L T Y OF G R A D U A T E S T U D I E S M E C H A N I C A L E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A April 2001 © Roya Rahbari, 2001 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purpose may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mechanical Engineering The University of British Columbia 2324 Main Mall Vancouver, Canada V6T 1Z4 DATE: Abstract Fuzzy logic has been proven to be a useful technique for information fusion in intelligent (or expert) systems as well as both direct control and supervisory control of industrial processes. In this technique, knowledge or expertise of humans are formulated in the form of if-then rules. Usually a human can correlate two to three input variables (in the if part) to an output variable (in the then part). However, complex practical systems are typically multi-input-multi-output (MIMO) systems where there are many input variables that relate to one or more output variables. In such a system, it would be rather unrealistic for a human to keep track of a large number of interacting variables and to express the operational expertise as a set of coupled rules consisting of all input variables and output variables. This thesis develops a new inference method to address the decision making of multivariable fuzzy systems. The developed inference method is based on the concepts of fuzzy measure and the fuzzy integral. The method first separates a coupled rulebase into several single-input-single- output (SISO) modules or, dual-input-single-output (DISO) modules. Then, it assigns a degree of importance to these SISO modules. Finally the inferences of these modules are combined (fused) to obtain an overall inference, through the use of a fuzzy integral. The technique was originally motivated by the requirements of an industrial fish cutting machine that has been developed in the Industrial Automation Laboratory, University of British Columbia. This inference method can be embedded in any commercial expert system, any direct fuzzy logic controller or a supervisory fuzzy control system that deals with multiple fuzzy variables in a coupled rulebase. The validity of this technique is illustrated by implementing it on hydraulic manipulator of the prototype fish cutting machine and through thorough experimentation. Two inference methods—individual — rulebased method and composition-based method, are implemented on a direct proportional (P)-type fuzzy controller in real time on the prototype hydraulic manipulator. ii A proportional-plus-derivative (PD)-type fuzzy controller is developed and implemented in real time on the prototype hydraulic manipulator. These various implementations are used to carry out an extensive experimental investigation and study the significance of selecting and tuning of such components as the membership functions, rulebase, inference method, and the defuzzification technique in practical fuzzy logic systems. A supervisory fuzzy control system is developed and implemented integral with an intelligent fuzzy tuner for the hydraulic manipulators of the prototype fish cutting machine. This control system monitors, the performance of the machine and automatically tunes the parameters of the PD controllers of the hydraulic actuators. The control system has a three-layer hierarchical structure. The middle layer monitors the performance parameters of the PD-controllers, based on a step-input response. The top layer infers the tuning actions for the PD servo units. The knowledge base for tuning the low level controller is developed using expert knowledge and represented as fuzzy rulebase modules. The investigation presented in this thesis makes important contributions for design, development and implementation of fuzzy knowledge-based systems, particularly control systems, in practical applications. In particular, the present work concentrates on inference techniques, fuzzy rulebase, membership functions, and defuzzification in these systems. i i i Table of Contents Abstract i i List of Tables vii List of Figures viii Acknowledgement xi 1. Introduction 1 1.1 Research Objectives 2 1.2 Motivation ; 2 1.3 Thesis Outline 7 2. Experimental System 8 2.1 Novel Prototype Machine for Fish Cutting 8 2.2 Controller Considerations 9 2.3 The Hydraulic Manipulator 11 2.4 Modeling of the Electrohydraulic System 13 3. PD-Type Fuzzy Control on a Hydraulic Manipulator 16 3.1 Introduction 16 3.2 PD-like Fuzzy Logic Controller 16 3.3 Choice of Membership Functions 20 3.4 Effect of Tuning the Membership Functions 23 3.4.1 Tuning the membership functions of output 24 3.4.2 Tuning the membership functions of input 1 27 3.4.3 Tuning the membership functions of input 2 27 3.5. Effect of Tuning the Rulebase 29 3.6. Comparison of Defuzzification Methods 32 4. P-Type Fuzzy Control on a Hydraulic Manipulator 36 4.1 P-Type Fuzzy Logic Controller 36 4.2 Effect of Tuning the Input Membership Functions 38 4.3 Effect of Tuning the Output Membership Functions 44 4.4 Effect of Rulebase Tuning 47 5. Application of Composition Based Inference to P-Type Fuzzy Control 51 iv 5.1 Inference Based on Rule Composition 51 5.2 Discussion 65 5.2.1 Effect of number of discretized points of input membership functions 66 5.2.2 Effect of number of discretized points of output membership functions 66 5.2.3 Effect of inference method 67 6. Development of an Intelligent Hierarchical Control System for a Hydraulic Manipulator 69 6.1 Introduction 69 6.2 Development of the Hierarchical Intelligent Control System 71 6.3 Experimental Results 78 6.4 Discussion 80 7. A New Inference Method Based on Fuzzy Integral 81 7.1 Introduction 81 7.1.1 Purpose of the development 82 7.1.2 Motivation 82 7.1.3 Rationale 82 7.1.4 Related literature 83 7.2 Theoretical Background 84 7.2.1 Fuzzy measure 85 7.2.2 A - Fuzzy measure 85 7.2.3 Fuzzy integral 87 7.2.4 Information fusion using "fuzzy integral" 88 7.3 Development of the inference technique 95 7.3.1 Universal approximation theorem 104 7.3.2 Stone-Weirstrass theorem 105 7.4. Experimental Investigation 106 7.5 Discussion 110 8. Conclusion 114 8.1 Major Contributions 114 8.2 Significance of the Research 115 8.3 Directions for Further Research 116 Bibliography 117 Appendices 125 Appendix A 125 Appendix B 137 vi List of Tables 2.1. Physical parameters of the hydraulic manipulator (one of two axes) 13 3.1. The rule base of a PD-like fuzzy knowledge-based controller in tabular form 19 3.2. Rulebase in Case 5 31 3.3. Rulebase in Case 6 32 6.1. Rulebase for the roughness of the response 75 7.1 Calculation of the values of g' 92 7.2. Computation of the values of the fuzzy integral for class t, 93 7.3. Computation of the values of the fuzzy integral for class t2 94 7.4. Memberships of the outputs for case 1 100 7.5. Comparison of the methods of computing inference in case 1 101 7.6. Memberships of the outputs for case 2 103 7.7. Computation of inference in case 2 103 7.8. Computation of the conventional inference in case 2 104 7.9. Output membership functions 109 vii List of Figures Figure. 1.1 .a The automated Iron Butcher (research prototype) 3 Figure. 1. l.b The cutter assembly of the research machine 4 Figure. 1.1.c A schematic diagram of the cutter assembly 4 Figure. 1.2 Hierarchical monitoring and supervisory control system of the prototype machine.... 6 Figure 2.1 .The fish cutting machine (industrial prototype) in the Industrial Automation Laboratory 10 Figure 2.2.The observer for velocity estimation 11 Figure 2.3. The electrohydraulic positioning system (for one axis) 12 Figure 3. 1. Block diagram of the fuzzy logic controller 18 Figure 3.2. Membership functions for easel 21 Figure 3.3. Membership functions for case2 22 Figure 3.4. Membership functions for case3 22 Figure 3.5. Membership functions for case4 23 Figure 3.6. Effect of tuning the membership functions of output (the position response) 25 Figure 3.7. Effect of tuning the membership functions of output (the control action) 26 Figure 3.8. Effect of tuning the membership functions of inputl 28 Figure 3.9. Effect of tuning the membership functions of input2 29 Figure 3.10. Effect of tuning the rulebase 30 Figure 3.11. Membership functions of case 5 and 6 31 Figure 3.12. Response of the system with centre of gravity defuzzification method 33 Figure 3.13. Response of the system with centre of average defuzzification method 35 Figure 4.1 Block diagram of the fuzzy logic controller 37 Figure 4.2 Inference based on individual rules for the P-type controller 38 Figure 4.3 Membership functions of the input 39 Figure 4.4 Membership functions of the output 40 Figure 4.5 Rulebase of the P-type fuzzy controller 40 viii Figure 4.6 Comparison of the responses of the system in the four cases, with the same initial conditions 41 Figure 4.7 Response of the system in case 1 41 Figure 4.8 Response of the system in case 2 42 Figure 4.9 Response of the system in case 3 42 Figure 4.10 Response of the system in case 4 43 Figure 4.11 Membership functions of input in three cases 43 Figure 4.12 Membership functions of output in three cases :44 Figure 4.13 The rulebase of the system in the three cases 44 Figure 4.14 Response of the system in easel 45 Figure 4.15 Response of the system in case 2 46 Figure 4.16 Response of the system in case 3 46 Figure 4.17 Rulebase in case 1 47 Figure 4.18 Rulebase in case 2 47 Figure 4.19 Rulebase in case 3 48 Figure 4.20 Membership functions of input and output in all three cases 48 Figure 4.21 Rulebase in easel 49 Figure 4.22 Rulebase in case2 49 Figure 4.23 Response of the system in case3 50 Figure 5.1 Discretized membership functions of the input 58 Figure 5.2 Discretized membership functions of the output. 58 Figure 5.3 Membership functions of the input 59 Figure 5.4 Rulebase of the fuzzy system 60 Figure 5.5 The rulebase in the form of fuzzy relation 60 Figure 5.6 Transformation of crisp input signal to fuzzy singleton 61 Figure 5.7 Inference engine of the fuzzy system 61 Figure.5.8 Defuzzification for the fuzzy system 62 Figure.5.9 Response of the system in the Case N=7 and M=9 63 Figure.5.10 Response of the system in the Case N=13 and M=9 63 Figure.5.11 Response of the system in the Case N=7 and M=17 64 Figure.5.12 Response of the system in the Case N=13 and M=17 64 ix Figure 6.1 Schematic representation of the intelligent hierarchical controller 71 Figure 6.2 The observer for velocity estimation 72 Figure 6.3 The flow chart of the algorithm in the second layer 73 Figure 6.4 The response of system with different overshoots 74 Figure 6.5 The response of system with different degrees of oscillations 75 Figure 6.6 Examples of different speeds of response of the system 76 Figure 6.7 Knowledgebase of the top level of the hierarchy 77 Figure 6.8 Membership functions in the knowledgebase 78 Figure 6.9.a Experimental results (overshoot improvement due to tuning) 79 Figure 6.9.b Experimental results (improved oscillations due to tuning) 79 Figure 6.9.c Experimental results (improved speed of response due to tuning) 80 Figure 7.1. The decomposition of the MEVIO rulebase into two SISO modules 96 Figure.7.2.(a). Rulebase modules for case 1 99 Figure.7.2.(b). Schematic representation of inference in case 1 100 Figure.7.2.(c). Rulebase modules for case 2 101 Figure 7.2.(d). Schematic representation of the inference in case 2 102 Figure 7.3. Block diagram of the fuzzy logic controller 106 Figure 7.4. Membership functions of input and output variables 107 Figure 7.5. Modules of the rulebase in three cases of investigation 108 Figure 7.6. Module 1 and module 2a (easel) 111 Figure 7.7. Module 1 and module 2b (case2) 112 Figure 7.8. Module 1 and module 2c (case3) 112 Figure 7.9. Comparison of the responses (gl=l, g2=0) 113 x Acknowledgements I wish to express my sincere gratitude to my supervisor, Prof. Clarence W. de Silva, for his support, guidance and invaluable patience. His wisdom, patience and proactive attitude toward new ideas, as well as his graceful and professional manner have made him a true role model for his graduate students as well as a leader in technology. The Funding for my research was provided by supervisor through Natural Sciences and Engineering Research Council of Canada, B.C. Packers,Ltd., and Gartfield Weston Foundation for the Chair of Industrial Automation, which is gratefully appreciated. I would like to thank members of my supervising committee, Prof. Sander Calisal and Prof. Farokh Sassani for their kind and careful attention to the progress of my research. I would also appreciate Prof. Elizabeth Croft as well as the members in Industrial Automation Laboratory for their cooperation. I gratefully acknowledge the strong support of the Faculty of Applied Science, as well as Department of Mechanical Engineering, particularly the heads Prof. Bob Evans, Prof. Dale Cherchas and Prof. Nimal Rajapakse , in addition to my supervisor, Prof. Clarence de Silva, for their true and strong support and encouragement for advancement of women in engineering. I wish to express my special gratitude to Prof. Ian Gartshore for his true and sincere encouragement of all graduate students. His wisdom and vision have made him a true professional and role model for maintaining peace and integrity in the department. The kind assistance of Mr. Alan Steeves, Mr. Gerry Rohling, and Mr. Albert Sawchuk to all graduate students is greatly appreciated. Mr. Jon Mikkelsen and Mr. Gordon Wright with their wonderful communication skills have always been very helpful to all teaching assistants and graduate students, this is greatly appreciated as well. I greatly appreciate Mr.Sean Bygrave and Mr.Doug Yuen, for their kind assistance everytime and anytime in the shop facilities. Cooperation of the office staff, Ms. Christine Black, Ms. Barb Murray, Ms. Jan Marsden and Ms. Louise Soga is greatly appreciated. I thank god who gave me power to overcome the barriers to reach my goal. Last, and definitely not least, I wish to thank my loving, understanding, and cultured parents who have always supported and encouraged me to reach my goals. xi To my parents who gave me wings to fly, held me when I fell, & their love is the greatest gift of all. To my supervisor, Prof. Clarence W. de Silva The gift from heaven And To Anxious Patience Xll Chapter 1. Introduction Chapter 1 Introduction Fuzzy sets were first introduced by Zadeh [6,10,92-104]. Following that Mamdani used the concept of fuzzy logic for applications in knowledge-based control [42,51,52,61]. Since then, numerous applications of fuzzy logic have been successfully implemented in such areas as control, image processing, object recognition, and automation. Fuzzy logic has been demonstrated to be useful in information fusion in intelligent systems [4,77], expert systems used in process control [51,52,63], telecommunication networks [73], and many other applications. Expert systems have been developed for a wide range of applications [1,2,40,47,49,63,64,69,71]. In the context of the present research, an important application of fuzzy logic is in expert systems. The name expert system is used interchangeably with the term knowledge-based system. An expert system is a computer program which embodies human knowledge and understanding in a way that imitates a human expert who can solve specific types of problems [54]. Fuzzy logic is used to capture expertise of human experts and represent in the form of a rulebase of fuzzy variables. Another application area of fuzzy logic is in information fusion. Sugeno, first developed the concept of a fuzzy integral based on fuzzy logic and fuzzy measures [72]. Following that fuzzy logic has been used in information processing based on the concept of the fuzzy integral [77]. In these cases relevant knowledge or expertise of humans in the form of linguistic IF-THEN rules needs to be developed. Commercial expert system shells are available, which make the process of development of a rulebase easier. Specifically, the knowledge engineer would extract knowledge of humans (or, domain experts) and represent it as appropriate IF-THEN fuzzy rules. The knowledge includes heuristic and rule-of-thumb information. The expert system processes these IF-THEN rules using a built in inference method. 1 Chapter 1. Introduction 1.1 Research Objectives The main objective of the research reported in this thesis is to develop an efficient method for handling multiple fuzzy variables in a rulebase for the purpose of information fusion and decision making. The technique will be particularly suitable for systems that use fuzzy logic for information representation (fusion) and combination. Hence, the research will aim at developing a practical inference method for handling fuzzy systems with multiple variables. A related objective is to implement the technique in an industrial prototype machine for fish cutting and to evaluate the performance of the method through experimentation. 1.2 Motivation This research is motivated by a novel and automated fish-cutting machine which Industrial Automation Laboratory is developing. The machine that is commonly used in the fish processing industry for head removal of fish is known as the "Iron Butcher". This machine has been designed and built at the turn of the century and has seen little modification or improvement since then. It is a purely mechanical device, which employs friction to detect the gill of a fish and a tactile engagement mechanism to align the collarbone of the fish with the cutter blade. Due to lack of active sensing and feedback control, this simple mechanism is known to cause considerable wastage by frequently over-feeding the fish into the cutter. In the salmon processing industry of British Columbia, Canada, for example, the wastage of useful meat (due to the poor cutting accuracy of the Iron Butcher) is estimated to be 25 million dollars annually, which corresponds to a wastage of 5% of the useful fish meat. Also, the machine causes under- feed errors, caused when the mechanical positioning foot slides over the collarbone of the fish. In this case the processed fish has to be reprocessed, resulting in a reduced throughput rate. The Industrial Automation Laboratory has undertaken research aimed at addressing deficiencies of the Iron Butcher by employing modern sensing and actuation 2 Chapter 1. Introduction technologies to detect and position the fish accurately at the cutter in order to minimize the wastage [17]. In particular, two automated machines have been developed for head removal of fish [20,21,22,84]. The research prototype as shown in Figure 1.1 represents an attempt toward developing an autonomous and intelligent machine that is able to remove the head of a fish with minimal wastage. The machine has been designed to operate with minimal intervention of human operators, by taking advantage of robotic technology and knowledge based intelligent systems. This machine would be able to learn and reason based on information that is received from various sensors and actuators, which is then processed and analyzed to determine the performance, similar to the way that a human does and then take control actions automatically in order to improve the performance of the system automatically, without intervention of human operators. Chapter 1. Introduction Figure. 1. 1(b) The cutter assembly of the research machine. Figure 1.1(a) and Figure 1.1(b) show two views of the developed research prototype. Figure 1.1(c) shows a schematic diagram of the cutter assembly of the machine. Induction Motor for Top Blade Platform- Servomotor for platform^ positioning Optical Encoder. Flexible Shaft Cutter Blades \ • Lead Screw 1 oupling \ Guideway Induction Motor for Bottom Blade Optical Encoder Servomotor for Cutter Positioning Figure. 1.1(c) A schematic diagram of the cutter assembly. Chapter 1. Introduction The conveyor moves the fish continuously from the feeding-end to the cutter. The fish are separated by means of the push lugs which are attached to a set of chains around the conveyor. The conveyor has a linear speed of 2 lugs per revolution of the drive shaft, while the distance between two adjacent lugs is 23 cm. The conveyor shaft is driven by a DC motor through a belt drive. For a typical throughput rate of 2 fish/s, the linear speed of the conveyor is 46 cm/s. By the time a fish reaches the cutting zone, the cutter assembly has to be moved to the correct cutting location horizontally. Simultaneously, the feeding platform is adjusted in the vertical direction so as to present the fish to the cutter. The horizontal adjustment is made in order to align the collarbone of the fish with the cutter blades. The vertical adjustment of the platform is necessary so that each fish is delivered symmetrically into the cutter, while compensating for the thickness of the fish in order to minimize the meat wastage. After the fish leaves the cutting zone, it enters the secondary vision system which analyzes the image of the cut fish, in order to determine the quality of the cut. This will provide high level information for the purpose of monitoring the system performance and for machine control. The machine consists of the following components: 1. A conveyor, which carries fish from the feeding point to the cutting location. 2. A hold down mechanism, which is necessary to make sure that the fish are held properly throughout the processes of imaging and cutting. 3. An ultrasonic displacement sensor, which measures the thickness of the fish near the area of the collarbone. This measurement is needed to control the positioning platform in the vertical direction so that each fish is delivered properly into the cutter blades. 4. A primary C C D camera, which grabs the image of each fish on the conveyor, when triggered by an object-detection sensor on the conveyor table. The low-level vision system analyzes the image and determines the cutting locations [31,65]. 5. Cutter assembly, which consists of two circular blades arranged in a V-configuration and driven by two independent high-speed induction motors through flexible 5 Chapter I. Introduction coupling. Motion of the cutter assembly in the horizontal x direction is digitally controlled and actuated by a DC servo motor. 6. An adjustable platform adjacent to the cutter assembly, which places each fish symmetrically with respect to the pair of cutter blades. The position of this platform is adjusted by a DC servomotor, which is digitally controlled. 7. A secondary CCD camera at the exit end of the cutting zone, which provides images to a high-level vision system, in order to determine visual information related to the quality of the processed fish and in turn the performance of the machine. This prototype machine is a complex system, which consists of many sensors, actuators, controllers, and mechanical devices. The overall performance of the machine depends on improving the meat recovery and the quality of the cut fish, which are subjective measures. Autonomous operation of the machine relies on a high level control i architecture. It needs to be designed so as to process the complex, low-level information from various sensors and actuators, and properly combine them to make a proper decision for product quality assessment and for improvement of performance of the system. Figure 1.2 shows the hierarchical architecture of monitoring and control of the research prototype machine. Level 3 Data Base Monitoring/Control Knowledge Base Mach ine Status i Fuzzy Inferences Level 2 Preprocessing of Sensory Data Context Generat ion Defuzzification Sensory Information Parameter Updates, Tuning/Reference C o m m a n d s Level 1 Primary Thickness Load C a m e r a Sensor Sensors Secondary Inspection Conveyor Camera Controls Cutter Controls Platform Controls Figure. 1.2 Hierarchical monitoring and supervisory control system of the prototype machine. 6 Chapter 1. Introduction Since the automated machine consists of many components (sensors, actuators controllers, signal conditioners, etc.) which make it necessary to deal with large amounts of information, a suitable structure for handling the information from various sources is required to be developed. 1.3 Thesis Outline As a precursor to the development of the new inference scheme for multivariable fuzzy systems, this thesis first studies the real time implementation of different methods of fuzzy logic control; specifically the proportional plus derivative type (PD-type) and proportional type(P-type). A comparison of two practical inference methods is given with experimentation results on an industrial hydraulic manipulator. This foundation of development, implementation, and experimentation with conventional fuzzy decision making and control, using an industrial prototype machine for fish cutting, leads to a new inference method. Chapter 2 explains the industrial hydraulic manipulator (industrial fish cutting machine), which is used for the implementation of the direct low level fuzzy controllers. Chapter 3 describes the development of a PD-type fuzzy controller, which is implemented on the experimental system. Chapter 4 presents the development of a P-type fuzzy controller using "individual rulebase firing". Chapter 5 describes the development of a P-type fuzzy controller using "composition based inference". Chapter 6 gives the development and implementation of fuzzy logic control at high level in the form of an intelligent hierarchical control system for tuning the parameters of a conventional PD type controller, which has been implemented in the experimental machine. Chapter 7 presents the novel inference method that is developed, which is an original contribution of the thesis. Chapter 8 concludes the thesis by summarizing the contributions of this thesis, indicating their significance, and providing directions for future research. 7 Chapter!. Experimental System Chapter 2 Experimental System The algorithms and control schemes developed and/or investigated in this research are implemented in the hydraulic manipulator of an industrial prototype of an automated fish cutting machine which has been developed in the Industrial Automation Laboratory. This chapter explains this prototype machine which is used as the test bed for this research, particularly for the direct fuzzy controllers developed and investigated in this research. This is the second generation industrial prototype, which is an outgrowth of the original research prototype outlined in Chapter 1. 2.1 Novel Prototype Machine for F i sh Cut t ing The Industrial Automation Laboratory of the University of British Columbia has undertaken research aimed at addressing the main deficiencies of the Iron Butcher. The Laboratory has been established primarily for research and development of advanced technology in automation of the fish processing industry [17]. The industrial prototype machine that is used in the present work is an outgrowth of an activity of the laboratory, which first developed the research prototype outlined in Chapter 1. Now the industrial prototype which is the test bed for the present research, is outlined. The machine consists of several parts: 1. An intermittent-motion conveyor system to move the fish towards the cutting zone. 2. A guillotine type blade that operates by a pneumatic actuator in the vertical direction. 3. Horizontal X-Y table that carries the cutting blade assembly, which is driven by means of two hydraulic actuators. 4. A digital CCD camera that takes the image of each fish as it enters the cutting zone. A view of the prototype machine is shown in Figure 2.1. The conveying mechanism has several parallel arms which are equally spaced along the belt. The fish are pushed 8 Chapter!. Experimental System forward intermittently, by means of these arms. Also, there is a matrix of retaining pins which point vertically upwards. These pins can fold in one direction only, thereby restraining any backward motion of the fish. The conveyor pushes a fish forward every 1.24s, which corresponds to the period of the cyclic motion of the mechanism. Accordingly, the rate of processing is about 48 fish/minute. During the first half period the conveyor pushes the fish forward, and in the next half period while the fish are stationary, the imaging of one fish and cutting of the previous fish are carried out. Also during this half period, the conveying arms move backward in order to get ready to push the fish forward in the next cycle. There is an optimum position at which the blade should be located with respect to a fish so that the meat wastage would be minimum. This optimum position is determined by using a computer vision system. While a fish is stationary an image of the fish is taken by the CCD camera and the corresponding digital information is acquired by a frame grabber board of the vision computer. Image processing routines automatically determine the optimum position of the cutter, according to the gill position of the fish [31,65]. This point is used as the reference input (set point) for the hydraulic controller, which moves the cutting blade assembly. After the cutter assembly is moved to the desired position (optimum point), the pneumatic actuator is activated and as a result, the cutter blade is released vertically. The operations of the system are synchronized such that, by the time a fish arrives at the cutting zone, the blade has already been positioned at the desired position. 2.2 Controller Considerations As mentioned earlier, the period of cyclic motion of the conveyor is 1.24 seconds. This means that image capture of a fish by the computer vision system and the cutting of the previous fish are carried out during the return half cycle of 620ms of the conveyor. Image processing and determination of the desired cutting location is carried out soon after an image is captured and it should be completed during the return half cycle. Positioning of the cutting blade assembly to the desired position needs to be completed during the forward half cycle of 620ms. This means that the hydraulic positioning system has to reach the steady state at the end of the forward half cycle. If the controller fails to reach the steady state by this time, the blade will be released while the cutting blade unit 9 Chapter!. Experimental System is still moving, but the fish would be in the stationary position. This may lead to such problems as poor-quality cuts or even complete entanglement of a fish with the cutter. In view of this, it is crucial that the cutting blade unit reaches the set point in a stable manner at the end of the forward half cycle of the conveyor motion. The time required for image processing with the available hardware system is 170ms, which is well within the available time of 620ms for image capture, processing, and the generation of the set point for the cutter position. Note that the cutter cannot be positioned for the next fish before the present fish is cut. The cutting time for one fish using the pneumatic actuator is limited to about 240ms, which is more than the time needed for image capture and set point determination for the next fish. Then a time duration of about 1.0 second would be available for positioning the cutter. Accordingly, the following specifications are used for the low-level controller: 1. Reach the desired position in steady state within 1.0 s. 2. Allowed offset for the cutter position (cutting accuracy) is 5mm. Figure 2.1. The fish cutting machine (industrial prototype) in the Industrial Automation Laboratory. 10 Chapter!. Experimental System 2.3 The Hydraulic Manipulator In the prototype system, the manipulator ( X-Y table ) which positions the cutter blade has two control inputs, which are the currents applied to the electrohydraulic proportional valves corresponding to the X and Y directions of motion. Since the two degrees of freedom in the manipulator are quite similar and also independent, only the X direction is discussed here. Figure 2.2 shows a simplified schematic diagram of the manipulator, for the X axis. The servovalve consists of a torque motor, a flapper pilot stage, and a double sliding spool boost stage. It is balanced by the centering springs which produce a spool position proportional to the differential pressure and therefore to the input current. For measuring the position of the piston, a Temposonics linear displacement transducer is placed at the head of the hydraulic cylinder. This industrial magnetostrictive "sensor precisely senses the position of an external magnet, which is connected to the piston. The sensor in the prototype machine has a resolution of 0.025 mm . In this research, the actuator velocity is estimated using a linear observer which has been developed in [75,76]. The dynamics of the observer is shown in Eq. (2.1) and the block diagram is shown in Figure 2.2, where kv = 500. v = zv +kvx t, =-kJ (2.1) Position Estimated i 1 Velocity Figure 2. 2. The observer for velocity estimation. 11 Chapter!. Experimental System Two gage pressure transducers are also installed on the head and the rod sides of the cylinder, in order to measure the fluid pressures P] and P2, respectively. Figure 2.3 shows the schematic diagram of setup. A sampling frequency of / v = 1 kHz has been used to implement the digital controllers. A 12 bit PC-based data acquisition system is used for data collection. The computer system that has been used for the experiment setup is an Intel 486, 33 MHz processor with DOS operating system, and the code is written in W A T C O M C. The physical parameter values of the prototype manipulator are listed in Table 2.1. Check Valve Hydraulic Piston and Cylinder Figure 2.3. The electrohydraulic positioning system (for one axis). 12 Chapter!. Experimental System Table 2.1. Physical parameters of the hydraulic manipulator (one of two axes). Parameter Value Measured moving mass M 32.7 Kg Head-side piston area Al 1.14xl0~3 m 2 Rod-side piston area A2 0.633 xlCT 3 m 2 Maximum piston displacement L 5 0 x l 0 - 3 m Volume of each hose between the servovalve and the cylinder Vh 8.9xlO~ 5m 3 2.4 Modeling of the Electrohydraulic System A dynamic model is derived in this section for the electrohydraulic manipulator [23,29,55]. This analysis is primarily intended to emphasize the nonlinear and asymmetric nature of the actuator dynamics. There are two types of nonlinearities in the servovalve system; namely, the hysteresis resulting from the electromagnetic characteristics of the torque motor and the flow forces on the valve spools. Generally, servovalve dynamics can be ignored in comparison to the actuator dynamics. Displacement of the valve spools from their null position creates a pressure difference across the single-rod hydraulic actuator, and the resulting fluid flow causes a change in the position of the piston. The force applied by the actuator (given in Eq.(2.2)) can be calculated from the cylinder pressures. Newton's second law applied to the actuator yields: F = PtAt - P2A2 = Mx + Ff (2.2) where F is the actuator force, Ff is the opposing frictional force, x is the piston displacement, and M is the moving mass in the X direction. The amount of fluid flow 13 ) Chapter2. Experimental System to the head-side (g,) and from the rod-side (Q2) of the cylinder is a function of the valve spool position and cylinder pressures. Assuming that the valve is critical center type with matched and symmetrical ports, the relationship can be expressed in the following form : Gi =K.xv.[s(xv)JP,-Pl +s(-xV).V>7] Q2 = K.xv. [s(xv )VP7 + s(-xv )JPS - P2 ] where xv is the valve spool displacement, K is a fixed gain, Ps is the supply pressure, and s(xv) is a switching function defined as in Eq (2.4): s(xv) xv>0 x<0 (2.4) Using the continuity principle, and taking fluid compressibility into account, we can write: A . Ax+Vh • Ql=Aix+ ' hP Ql — ^2X A2(L-x) + Vh. P (2.5) where fi is the effective bulk modulus of the fluid and Vh is the volume of the fluid inside each of the hoses that connect the servovalve to the actuator. The state vector is defined as: x = [x, x2 x3 x4]T=[x x P, P2Y (2.6) By combining equations (2.2), (2.3), and (2.5), the following nonlinear state-space model of the system is obtained: 14 Chapter!. Experimental System [AJXJ A2X4 Ff ] ^ _- (*(*v w^v - * 3 + ^(-^v )V^T)~ A *2 ] (2.7) 1*4 = A 2 ( L - x , ) + V( ^ — — [bcv (y(xv )Jx~A + *(-JC„ )VPv ~ * 4 )- A2*2 ] As can be shown in state space dynamics of the system in Eq (2.7), the actuator dynamics depends on position and velocity of the piston. Considering this information, next chapter develops a fuzzy controller based on information received from position and velocity of the cylinder. 15 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator Chapter 3 PD-Type Fuzzy Control on a Hydraulic Manipulator 3.1 Introduction The application of fuzzy logic control in hydraulic manipulators is rare, particularly in the research community. An application of a fuzzy logic controller has been reported in [107] in which a linear state controller combined with a fuzzy rule evaluation is used to produce what has been termed as fuzzy state controller. Their proposed method indirectly detects the presence of a load, based on the fuzzy rule evaluation of actuator velocity and decides on the gains of the state controller accordingly, in order to make the controller insensitive to load variations. Application of fuzzy logic controller to a class of hydraulically actuated industrial robots is investigated in [13,70]. This chapter studies the development and experimental analysis of a proportional plus derivative (PD)-type fuzzy logic controller, as implemented on the planar hydraulic manipulator which was outlined in Chapter 2. The performance of the fuzzy logic controller is investigated with respect to the control requirements for this machine, explained in Section 2.2, and the effect of tuning of the membership functions is illustrated using experimental results. This foundation will provide an insight into the experimental system and the practical implementation of a fuzzy logic controller in such a system. This investigation will lead to the implementation of another controller, as presented in Chapter 4 and 5. The inference techniques used in these controllers have provided the motivation for the original development given in Chapter7. 3.2 PD-Type Fuzzy Logic Controller In conventional control, the PD control law is written in the form: u = KP.e + KD.e (3.1) 16 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator where e and e are the error and the rate of error, respectively, and Kp and KD are the associated proportional and differential gain coefficients. In fuzzy logic control, the control law is presented as "if-then" statements; for example, if X is A and Y is B then Z is C "X is A " and "7 is B" are called the rule antecedents and "Z is C" is called the rule consequent. In a fuzzy rulebase, X and F are the inputs to the fuzzy system and Z is the output of the fuzzy system. Then a PD-type fuzzy logic controller consists of fuzzy rules, with the inputs error and rate of error, and the output control action. The symbolic description of each rule can be given as [28]: if e(k) is <a fuzzy state> and e(k) is < a fuzzy state > then u(k) is < a fuzzy state > where < a fuzzy state > is the symbolic name of a fuzzy linguistic value. The natural language equivalent of the above symbolic rule reads as follows: For each sampling time k, if the value of the error is <linguistic value> and the value of the change-of-error is <linguistic value> then the value of the control output is <linguistic value> For position control of the prototype hydraulic actuator by means of a fuzzy controller, the inputs are position error and velocity , and the output is current to the servovalve. First the membership functions have to be designed for these variables. Velocity is estimated by a linear observer, as outlined in Chapter 2 and shown in Figure 2.2 . A block diagram of the fuzzy knowledge-based controller (FKBC) that is used in the present application is shown in Figure 3.1. 17 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator Fuzzy Controller Reference position *-Cl Position error e Position. Velocity Estimator Velocity ( Control Signal ^ Curr en I lo the Servovarve ) Plant Hydraulic Actuator of Cutter Assembly Actual position — — • Figure 3. 1. Block diagram of the fuzzy logic controller. There are several types of inference methods. If we consider the rule: IPjc isA A N D y i s B THEN z is C There are two operations that are commonly used for " A N D " operation; "min" and "algebraic product". There are many types of implication methods that have been introduced for implication operation "THEN" [78]; min, algebraic product, bounded product, drastic product, Lukasiewicz, Boolean logic, Goedel logic, Goguen. There are two types of operations used for rule aggregation; "max" and "sum". In the present research "min" is chosen for " A N D " operation, "min" is chosen for implication T H E N and "max" is chosen for rule aggregation. This type of inference is known as Mamdani's inference method. However, there are two different inference methods from "mechanism of output generation" point of view; "individual rule-based inference" and "composition based inference" [28]. In "individual rule-based inference" rules are fired individually and then the output of those fired rules are aggregated. This method is explained in Chapter 4, in more details. However, in "composition based inference", first all the rules are combined into a fuzzy relation, and then the input is composed with the fuzzy relation to compute the output with "composition operation". This method is explained in Chapter 5 in more details. In the present application, the inference engine of the fuzzy controller uses individual rule-based inference, as opposed to composition based inference, because of the lower memory requirement of the former in real time processing. The fuzzy sets of the inputs (position error and velocity) and the output (current to the servovalve) all are chosen to be equal in number and contain the same linguistic expressions for the 18 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator magnitude part of the linguistic values; i.e.,{NL, N M , NS, ZER, PS, P M , PL} where N L denotes negative large, N M denotes negative medium, NS denotes negative small, ZER denotes zero, PS denotes positive small, P M denotes positive medium, and PL denotes positive large. The rule base is shown in Table 3.1 in which e - x - x_des is the position error and vel is the estimated velocity . Table 3.1. The rule base of a PD-like fuzzy knowledge-based controller in tabular form. vel N L N M NS ZER PS P M PL N L PL PM PM PS PS PS ZER N M P M P M PS PS PS ZER ZER NS PM PS PS PS ZER ZER NS ZER PS PS PS ZER ZER NS NS PS PS PS ZER ZER NS NS N M P M PS ZER ZER NS NS N M N M PL ZER ZER NS NS N M N M N L The rules in the center of the table are related to the steady-state behavior of the process. This means that the current value of the process output variable (position) has deviated from the set-point but is still close to it. When both the position error and the velocity are negative, the position is below the set-point and the cylinder is moving further away. In response the control action should be positive such that it will reduce the position error. When the velocity is positive, while the position error is negative, the cutter is moving towards the set-point and control action should be low enough to slow down the approach to the set-point. Furthermore, note the asymmetry of the rulebase in Table 3.1. This is justified in view of the directional asymmetry of the hydraulic actuator with unequal effective areas on the two sides of the piston, as clear from the formulation in Chapter 2. 19 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator 3.3 Choice of Membership Functions For computational effectiveness, efficient use of memory, and the relative ease of performance analysis, a uniform representation of the membership functions would be desirable. Such a uniform representation has been realized in the present application by using membership functions with uniform shape and parametric functional definition. The most popular choice for membership functions is the triangular shape, which can be represented by a parametric functional description. Furthermore, these membership functions can be stored with a minimal use of memory and manipulated efficiently, with respect to real-time requirements, by the inference engine. As can be seen in figures 3.2 through 3.5, there is an overlap between every pair of adjacent membership functions. This means that every crisp value of an input to the controller will belong to at least one membership function, with the degree of membership strictly greater than zero. If this were not the case, then there would be a crisp input value which could not be matched to a rule-antecedent during the fuzzification phase, and none of the rules would fire and hence a value for the control action would not be computed. This situation is what is known as discontinuity in the control output. The present system has been designed in such a way that two adjacent membership functions have a cross-point level of 0.5 and cross-point ratio of 1, (i.e., symmetric 50% overlap). In this way, for each crisp input, two rules will fire and the control action would be continuous. There is an important point that has been considered with regard to the width of membership functions of the controller inputs (rule-antecedents). The membership functions of the nine central rules in the table of rulebase are responsible for the steady state response of the controller. It follows that the peaks of these membership functions are among the parameters that need to be tuned carefully. The membership functions of the controller output (rule- consequent) is symmetrical as well (constant width) so that peak and the centre of gravity of each membership function are identical. In the present controller in this chapter an individual rule-based inference method, as outlined before, is employed. The implication is based on Mamdani's method (Min-Max). The "centre of gravity" method is used for defuzzification. With the form of the membership functions that are used for the controller inputs, at each sampling period only two rules will fire for each input value. 20 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator Since there are two inputs, the computer processor will deal with only four rules at each sampling period (1 ms). This choice of inference and membership functions significantly reduces the complexity of computation of the fuzzy control law. Membership Functions in case 1 Input Current to Servovalve (mA) Figure 3.2. Membership functions for easel. 21 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator Membership Functions in case 2 -60 -40 -20 0 20 40 60 Input Current to Servovalve (mA) Figure 3.3. Membership functions for case2. Membership Functions in case 3 -50 -40 -30 -20 -10 0 10 20 30 40 50 Position error (mm) Input Current to Servovalve (mA) Figure 3.4. Membership functions for case3. 22 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator Membership Functions in case 4 -50 -40 -30 -20 -10 0 10 20 30 40 50 Position error (mm) -60 -40 -20 0 20 40 60 Input Current to Servovalve (mA) Figure 3.5. Membership functions for case4. 3.4 Effect of Tuning the Membership Functions For further studying the designed fuzzy logic controller, the system is now analyzed with regard to the effect of tuning the membership functions of the input and output variables. In all the cases, the response of the system is analyzed for four experiments (a,b,c,d). Experiments (a) and (b) are those where the initial position error is less than zero which corresponds to the cases where the piston is moving from left to right. Experiments (c) and (d) are those where the cylinder is moving from right to left. Note that the set points (desired reference positions) are chosen differently in the four experiments. 23 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator 3.4.1 Tuning the membership functions of output Figure 3.6 shows the performance of the system in the four experiments of different initial conditions under two cases of membership choices. In these two cases, the membership functions of the two inputs, the rulebase (shown in Table 3.2), the inference method, and the defuzzification method are the same. The only difference between the two cases is the different choice of membership functions of the output. Figures 3.2 and 3.3 show the membership functions in those two cases. Specifically, Figure 3.6 illustrates effect of tuning the membership functions of output. The membership functions of the output in both cases are uniform and symmetric. However, in Case 1 the universe of discourse is [-60 60] and in Case 2 is [-40 40]. Figure 3.6 shows that the response in Case 2 it is slower than in Case 1. The reason is that the larger universe of discourse provides a stronger control action. Accordingly, we can conclude that tuning the universe of discourse of the output membership functions has a direct effect on the speed of response of the system. Figure 3.7 shows the control action (the input signal to the actuator) for these two cases, in the four experiments (a)-(d). 24 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator Figure 3.6. Effect of tuning the membership functions of output (the position response). (Four experiments (a),(b),(c), and(d) with different initial conditions) Casel: Membership functions in Figure 3.2 Case2: Membership functions in Figure 3.3 25 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator < E, CD > CC > o £ CD CO o Z -20 c CD 60 40 20 0 -40 -60 Initial Position Error>0 easel case2 (a ) if if Initial Position Error<0 0 200 400 600 800 1000 60 40 20 0 -20 -40 -60 (c) — f • _ 0 200 400 600 800 1000 60 _ 40 CD 75 20 > o % 0 w o Z -20 c CD | - 4 0 -60 (b ) ^ • — • if 200 400 600 Time (ms) 800 1000 60 40 20 0 -20 -40 -60 (d) ~'— ^—.. 200 400 600 Time (ms) 800 1000 Figure 3.7. Effect of tuning the membership functions of output (the control action). 26 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator 3.4.2 Tuning the membership functions of input 1 Figure 3.8 shows the effect of tuning the membership functions of input 1 (position error) in four experiments (a)-(d) under two cases: Casel and Case3. Membership functions for Case 1 and Case3 are given in figures 3.2 and 3.4, respectively. Both sets of membership functions are uniformly distributed and symmetric. As can be seen in Figure 3.8, the response is much slower in Case 3, and the system does not reach the desired position within the allowed time of 1 second. Comparing figures 3.2 and 3.4, we can see that the support set of input 1 is much larger in Case 3 than in Case 1. 3.4.3 Tuning the membership functions of input 2 Figure 3.9 shows the effect of tuning of the second input (velocity of the cylinder) by comparing the response of the system in four experiments (a)-(d) under two cases, Case 1 and Case 4. Figures 3.2 and 3.5 show the membership functions of these two cases. Membership functions of input 1 and the output are common for the two cases, but the membership functions of input 2 (velocity) are different. In Case 1, the membership functions of input 2 are chosen to be symmetric and uniformly distributed, but in case 2 it is not . The effect of this tuning on the response is shown in Figure 3.9. Note that the nonuniform membership functions of the velocity do not provide a desirable response, which is too slow and never reaches the steady state. 27 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator Figure 3.8. Effect of tuning the membership functions of input 1. (Four experiments (a)-(d) with different initial conditions) Case 1: Membership functions in Figure 3.2. Case 3: Membership functions in Figure 3.4. 28 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator 0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 Time (ms) Time (ms) Figure 3.9 . Effect of tuning the membership functions of input 2. (Four experiments (a)-(d) with different initial conditions) Casel: Membership function in Figure 3.2. Case 4: Membership function in Figure 3.5. 3.5. Effect of Tuning the Rulebase The effect of tuning the rulebase is shown in Figure 3.10. Again four experiments (a)-(d) with different initial conditions are carried out under two choices of rulebase, denoted by Case 5 and Case 6. As can be seen, the speed of response in Case 6 is faster than that in Case 5. The membership functions in cases 5 and 6 are given in tables 3.2 and 3.3, respectively. The membership functions of inputs and output are common for the two 29 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator cases, and shown in Figure 3.11. The present comparison shows the difference that the tuning of a rulebase can make in order to achieve an improvement in performance. 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 Time (ms) Time (ms) Figure 3.10. Effect of tuning the rulebase. (Four experiments (a)-(d) with different initial conditions) Case 5: Rulebase of Table 3.2. Case 6: Rulebase of Table 3.3. 30 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator Membership Functions in case 5 and 6 -400 -300 -200 -100 0 100 200 300 400 Velocity (mm/s) Input Current to Servovalve (mA) Figure 3.11. Membership functions for case 5 and case 6. Table 3.2. Rulebase for case 5. V N L N M NS ZER PS P M PL N L PL PM P M PS PS ZER ZER N M P M P M PS PS ZER ZER ZER NS P M PS PS ZER ZER ZER NS ZER PS PS ZER ZER ZER NS NS PS ZER ZER ZER ZER NS NS N M P M ZER ZER ZER NS NS N M N M PL ZER ZER NS NS N M N M N L 31 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator Table 3.3. Rulebase for case 6. V N L N M NS ZER PS P M PL N L PL PL PM P M P M PS ZER N M P M P M P M P M PS ZER NS NS P M P M P M PS ZER NS N M ZER P M P M PS ZER NS N M N M PS P M PS ZER NS N M N M N M PM PS ZER NS N M N M N M N L PL ZER NS N M N M N M N L N L 3.6. Comparison of Defuzzification Methods In this section we compare the performance of the system with common membership functions for inputs and output and a common rulebase, but with two different commonly used defuzzification methods. Case 5 is chosen for this comparison, and two experiments withdifferent initial conditions are carried out. Membership functions of the inputs and the output are shown in Figure 3.11 and the rulebase is given in Table 3.2. The performance of this fuzzy logic controller is compared with two different defuzzification methods. Figure 3.12 shows the performance of the system with the centre of gravity defuzzification method. Figure 3.13 gives the system response when the defuzzification method is changed to the "centre average" method given in Eq. (3.3). These two defuzzification methods are more common because they are easier to compute. In the defuzzification method using the "centre-of-gravity" given in Eq.(3.2), a crisp output ycJ'sp is chosen using the centre-of-area and area of each implied fuzzy set, and is given by; 32 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator Position Error>0 Position Error<0 50 E 4 0 -§-30 o : i 2 0 o °- 10 0 ~ 200 0 E > -200 50 < E, ¥ ° o -50 0 200 400 600 800 1000 . , , V 200 400 600 800 1000 200 400 600 800 1000 Time (ms) 0 200 400 600 800 1000 200 0 -200 i 50 -50 0 200 400 600 800 1000 r— 0 200 400 600 800 1000 Time (ms) ;ure3.12. Response of the system with the centre of gravity defuzzification method. (Two experiments with different initial conditions) crisp (3.2) where R is the number of rules, bf is the centre of area of the membership function of Bp associated with the implied fuzzy set k'q for the i'h rule (j,k,...,l;p,q){, and I JU-fr (ya)dya denotes the area under (y ) . In the defuzzification method using the centre-average, a crisp output ycnsp is chosen using the centres of each of the output 33 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator membership functions and maximum certainty of each of the conclusions represented with the implied fuzzy sets, and is given by; £ > f s u p {//.,. (y?)} 1 V R r / m K } L-,SUP*</Vy,)} Observe that the position of the cylinder reaches the desired position within the specified time in Figure 3.12, but in Figure 3.13 the response is slower and cylinder does not reach the desired position in the specified time. However, the membership functions and the rulebase had already been tuned for the case where the defuzzification is by the centre of gravity method. It follows that if we change the defuzzification method, the system needs to be tuned accordingly. However, Figure 3.12 (Defuzzification by the centre of gravity method) shows that the control signal is much smoother than the case with the centre average method. 34 Chapter 3. PD-Type Fuzzy Control on a Hydraulic Manipulator Figure 3.13. Response of the system with the centre of average defuzzification method. (Two experiments with different initial conditions) The present comparison shows that tuning of the membership functions of input 1, input 2, and output, and of the rulebase, and also proper selection of the defuzzication method can have a significant effect on the performance of the system response. The actual experimental results using an industrial hydraulic actuator have shown that the designed PD-type fuzzy logic controller is able to meet the specifications of the specific industrial application, assuming that the membership functions, the rulebase, and the defuzzification method are chosen and tuned properly . Since the velocity of the piston is estimated based on the designed observer in Eq 2.1, it takes a few samples (about 50 samples) to converge and make a correct estimation. This is the reason that the velocity of the first 50 ms in the Figures 3.12 and 3.13 are not showing the correct estimation. 35 Chapter 4. P-Type Fuzzy Control on a Hydraulic Manipulator Chapter 4 P-Type Fuzzy Control on a Hydraulic Manipulator Implementation of a proportional-plus-derivative (PD)-type fuzzy logic controller on the prototype hydraulic manipulator was presented in Chapter 3. Issues related to the inference scheme rulebase membership functions, and the defuzzification method were investigated through experimentation. In that controller, there are two inputs. The maximum number of rules in the control rulebase is given by nm , where n is the number of inputs and m is the number of fuzzy states that each input variable can have. It follows that, depending on the number of fuzzy states, the number of rules can be quite large. In the particular example, given in Table 3.1, the number of developed rules for the PD-type fuzzy controller was 7 x 7=49 rules. Clearly, it is useful to see whether a comparable performance to the PD controller could be achieved with a simple controller and correspondingly a simpler rulebase. This chapter explores the use of a proportional (P)-type controller for the hydraulic manipulator with the advantage of simplicity. 4.1 P-Type Fuzzy Logic Controller In this section, the proportional (P)-type fuzzy logic controller, as implemented in the hydraulic manipulator is described. The conventional proportional (P) control law is given in Eq (4.1) by; u = KP.e (4.1) where e is the response error and Kp is the proportional control gain. Then a P-type fuzzy logic controller, where the input (context) is the response error and the output is the control action, consists of fuzzy rules [28] of the form if e(k) is <a fuzzy state> then u(k) is < a fuzzy state > The corresponding linguistic form is: // the value of the response error is <linguistic value> then the value of the control output is <linguistic value> for each sampling period k. 36 Chapter 4. P-Type Fuzzy Control on a Hydraulic Manipulator In the present problem of position control of the prototype hydraulic actuator, the input to the fuzzy decision making system is the position error e and the output is the current applied to the servovalve. A block diagram of the fuzzy knowledge-based controller (FKBC) is shown in Figure 4.1. Reference position Position error > ® — " Figure 4.1 Block diagram of the fuzzy logic controller. The P-type fuzzy logic controller is implemented on the prototype hydraulic manipulator (Chapter 2, and Figure 2.3) in the same manner as with PD-type controller (Chapter 3). The rulebase used for two specific experimental investigations will be given in subsequent sections. The fuzzy states of the input and the output are all chosen to be equal in number and use the same linguistic descriptors:(NL, N M , NS, ZR, PS, P M , PL}; where N L = negative large, N M = negative medium, NS = negative small, ZR = zero, PS = positive small, P M = positive medium, and PL= positive large. The membership functions of these variables, as used in the experimental investigations presented here will be given in subsequent sections. The inference method is based on individual rules [28] as outlined in Chapter 3. This inference method, as applied in the present implementation, is shown in Figure 4.2 and can be simply described as follows: 1. Compute the degree of match between the crisp input and the fuzzy states of the rule- antecedent (position error). 2. "Clip" the fuzzy state of the rule-consequent (control action) by the membership degree to which the rule-antecedent is matched by the crisp input. Fuzzy Controller Q ( Control Signal | Current to the Servovalve I Plant Cutter Assembly Actual position • 37 Chapter 4. P-Type Fuzzy Control on a Hydraulic Manipulator 3. Aggregate the "clipped" values of the control output of each rule, to form the value of the overall control output. 4. Determine the centroid of the aggregated control inference, giving a crisp control action. Membership Functions of the Input Positio ~7\ h Error (mm) Membership Functions of the Output Current Input to the Servovalve (mA) NS Z E R /yyyyy^yyy^. PS -40 -30 -20 -10 PM J 20 3 Fuzzy Inference 50 Crisp Inference PS Z E R -40 -30 -20 -10 0 10 20 30 40 Figure 4.2 Inference based on individual rules for the P-type controller. 4.2 Effect of Tuning the Input Membership Functions The effect of tuning the input membership function for the P-type controller is investigated by implementing the controller in four different cases. Membership functions of the input to the fuzzy system in the four cases are shown in Figure 4.3. The common membership function of the output for all the four cases is shown in Figure 4.4. The comparison has been conducted under the common rulebase as given in Figure 4.5. A comparison of the response of the hydraulic actuator for the same setpoint and the same initial condition is shown in Figure 4.6. Note that Case 1 which has the highest input resolution, provides the fastest response. Case 4 which has a nonuniform resolution, with a low resolution for large errors and a finer resolution for small errors gives the next best response. Figures 4.7 through 4.10 show position of the cylinder and the input current to the servovalve with two different initial conditions and setpoints 38 Chapter 4. P-Type Fuzzy Control on a Hydraulic Manipulator giving two directions of motion, as denoted by (a) and (b). Comparing Figures 4.3 through 4.6, we can see that case 1 shows the fastest speed of response and Figure 4.8 shows that the membership functions of Case 1 is finer in the centre of the universe of discourse. Also, as before, Case 3 gives a very good response in view of the nonuniform resolution of the input membership function. This shows how tuning the membership functions can improve the speed of response of the system. Input Membership Functions -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 Posit ion Error (mm) Figure 4.3 Membership functions of the input. 39 Chapter 4. P-Type Fuzzy Control on a Hydraulic Manipulator Output Membership Functions in case 1,2,3,4 50 - 4 0 - 3 0 - 2 0 - 1 0 0 10 20 30 40 50 Input Current to Servovalve (mA) Figure 4.4 Membership functions of the output. IF Pos. IF Pos. IF Pos. IF Pos. IF Pos. IF Pos. IF Pos err err err err err err en- is NL THEN is NM THEN isNS THEN is ZR THEN is PS THEN isPM THEN is PL THEN control control control control control control control signal is PL signal is PM signal is PS signal is ZR signal is NS signal is NM signal is NL Figure 4.5 Rulebase of the P-type fuzzy controller. 40 Chapter 4. P-Type Fuzzy Control on a Hydraulic Manipulator Effect of Tuning the Input Membership Functions 1 i 45 40 35 £ E 30 = 20 O 15 10 100 300 400 500 600 Time(ms) 900 1000 gure 4.6 Comparison of the responses of the system in the four cases, with the same initial conditions. Position Error>0 Position Error<0 200 400 600 800 1000 200 400 600 800 1000 50 < E. <a 2. ra > o -50 200 400 600 800 1000 Time (ms) 50 -50 0 200 400 600 800 1000 Time (ms) Figure 4.7 Response of the system in Case 1. 41 Chapter 4. P-Type Fuzzy Control on a Hydraulic Manipulator Position Error>0 Position Error<0 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 Time (ms) Time (ms) Figure 4.8 Response of the system in Case 2. Position Error>0 Position Error<0 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 Time (ms) Time (ms) Figure 4.9 Response of the system in Case 3. 42 Chapter 4. P-Type Fuzzy Control on a Hydraulic Manipulator Position Error>0 Position Error<0 600 800 1000 200 400 600 800 1000 50 r - 5 0 L 200 400 600 800 1000 Time (ms) 200 400 600 800 1000 Time (ms) Figure 4.10 Response of the system in Case 4. Input Membership Functions in case 1,2,3 -10 0 10 Position Error (mm) Figure 4.11 Membership functions of input in three cases. 43 Chapter 4. P-Type Fuzzy Control on a Hydraulic Manipulator Output Membership Functions in case 1,2,3 -20 0 20 Input Current to Servovalve (mA) Figure 4.12 Membership functions of output in three cases. IF Pos_err is NL THEN control signal is PL IF Pos_err is NM THEN control signal is PM IF Pos_err is NS THEN control signal is PS IF Pos_err is ZR THEN control signal is ZR IF Pos_err is PS THEN control signal is NS IF Pos_err is PM THEN control signal is N M IF Pos_err is PL THEN control signal is NL Figure 4.13 The rulebase of the system in the three cases. 4.3 Effect of Tuning the Output Membership Functions Next, a set of experiments is conducted to study the effect of tuning the output membership functions. The experiments are carried out in three different cases. The common membership function of the input variable for all three cases is shown in Figure 4.11. The output membership functions for the three cases are shown in Figure 4.12. The common rulebase for all 44 Chapter 4. P-Type Fuzzy Control on a Hydraulic Manipulator three cases is shown in Figure 4.13. The performance of the system in three cases is shown in figures 4.14 through 4.16 for two different initial conditions and set points giving two directions. It is clear that the response in Case 1 (Figure 4.14) is the best among the three. By examining the associated membership functions of output, it is seen that Case 1 has a finer and uniform resolution. This case has resulted in a fast speed of response with a setting time of less than 600ms. Figure 4.16 demonstrates the case where only one rule is fired and that is "IF the position error is large T H E N control signal should be large". Referring to output membership functions in case 3, the output membership function of " L A R G E " corresponds to signal more than 45 mA. In the code, the signals higher than 45mA is set to 5 mA for showing this particular case. So even though the the control decision is set to be large, it actually sends a low current to the servovalve, therefore always the position error is large and always the same rule "IF the position error is large T H E N control signal should be large" is fired. This shows that hardware limitations need to be considered while designing the output membership functions. < E > o i w o 200 400 600 800 1000 Time (ms) 0 200 400 600 800 1000 Time (ms) Figure 4.14 Response of the system in Case 1. 45 Chapter 4. P-Type Fuzzy Control on a Hydraulic Manipulator Position Error>0 Position Error<0 200 400 600 800 1000 200 400 600 800 1000 200 400 600 800 1000 Time (ms) -50 0 200 400 600 800 1000 Time (ms) Figure 4.16 Response of the system in Case 3. 46 Chapter 4. P-Type Fuzzy Control on a Hydraulic Manipulator 4.4 Effect of Rulebase Tuning A final set of experiments is conducted to investigate the effect of tuning the rulebase. Here three different rulebases, as given in figures 4.17 through 4.19, are used. The membership functions of the input to the fuzzy system (position error) and the control output (input current to the servovalve) are shown in Figure 4.20. The system responses for the three cases of rulebase are shown in figures 4.21 through 4.23 where two sets of initial conditions and set points are used, giving two directions of motion as denoted by (a) and (b). Rulebase in Case 3 (Figure 4.23) provides a stronger control signal, which results in a fast speed of response. The rulebase in Case 2 provides a milder control signal, which results in a slower speed of response as shown in Figure 4.22. IFPos. IFPos. IFPos. IFPos. IFPos. IFPos. IFPos err err err err err err err is NL THEN is NM THEN isNS THEN is ZR THEN is PS THEN isPM THEN is PL THEN control control control control control control control signal is PL signal is PM signal is PS signal is ZR signal is NS signal is N M signal is NL Figure 4.17 Rulebase in Case 1. IF Pos_err is NL THEN control signal is PM IF Pos_err is N M THEN control signal is PS IF Pos_err is NS THEN control signal is PS IF Pos_err is ZR THEN control signal is ZR IF Pos_err is PS THEN control signal is NS IF Pos_err is PM THEN control signal is NS IF Pos_err is PL THEN control signal is NM Figure 4.18 Rulebase in Case2. 47 Chapter 4. P-Type Fuzzy Control on a Hydraulic Manipulator IF Pos. IF Pos. IF Pos. IF Pos. IF Pos. IF Pos. IF Pos err err err err err err en- is NL THEN is NM THEN isNS THEN is ZR THEN is PS THEN isPM THEN is PL THEN control control control control control control control signal is PL signal is PL signal is PL signal is ZR signal is NL signal is NL signal is NL Figure 4.19 Rulebase in Case3. Membership Functions of Input and Output 50 - 4 0 - 3 0 - 2 0 - 1 0 0 10 20 30 40 50 Input: Position Error (mm) 50 - 4 0 - 3 0 - 2 0 - 1 0 0 10 20 30 40 50 Output: Input Current to the Servovalve (mA) Figure 4.20 Membership functions of input and output in all three cases. 48 Chapter 4. P-Type Fuzzy Control on a Hydraulic Manipulator Position Error>0 Position Error<0 200 400 600 800 1000 200 400 600 800 1000 50 200 400 600 800 1000 Time (ms) -50 0 200 400 600 800 1000 Time (ms) Figure 4.21 Response of the system in Casel. Position Error>0 Position Error<0 1000 200 400 600 800 1000 - 5 0 1000 400 600 800 1000 Time (ms) Figure 4.22 Response of the system in Case2. 49 Chapter 4. P-Type Fuzzy Control on a Hydraulic Manipulator Position Error>0 Position Error<0 200 400 600 800 1000 200 400 600 800 1000 50 200 400 600 800 1000 Time (ms) - 5 0 0 200 400 600 800 1000 Time (ms) Figure 4.23 Response of the system in Case3. In summary, the present comparisons show the importance of tuning of the membership functions and the rulebase in order to obtain a satisfactory performance. The experiments also showed that the simplest type of fuzzy control (P-type) can provide a performance that is comparable to what is obtained with a PD-type fuzzy controller in an industrial prototype of a hydraulic manipulator. 50 Chapter 5. P-Type Fuzzy Control on a Hydraulic Manipulator using Composition Based Inference Chapter 5 Application of Composition Based Inference to P-Type Fuzzy Control In the thesis, thus far, two specific types of control (PD-type and P-type) have been implemented on a prototype hydraulic actuator and investigated with regard to such aspects as the membership functions of inputs and outputs and the rulebase. Experience has been gained as well concerning the generation of the control inference. The specific inference technique, which has been used is the individual-rule based approach. Since inference generation in a knowledge- based system is a major focus of the present research, it is useful to investigate another common inference technique. In this chapter, the same P-type fuzzy control (with the same rulebase and the same membership functions) as investigated in Chapter 4 is implemented with the composition based inference method. The performance of the hydraulic actuator with the P-type fuzzy controller will be compared, under the two inference methods. Such a practical comparison of the performance of a fuzzy controller under these two types of inference techniques is not found in the literature. 5.1 Inference Based on Rule Composition The inference method based on rule composition [28,78], also called Compositional Rule of Inference, is defined in [96] as: If R is a fuzzy relation (see Appendix A) from U to V, and x is a fuzzy subset of U, then the fuzzy subset y of V which is induced by x is given by the composition of R and x, given in Eq (5.1); y = x°R (5.1) in which x plays the role of a unary (one dimensional) relation. Consider the generalized modus ponens: Premise 1: IF x is A THEN y is B (5.2) Premise 2 : x is A' Consequence: y is B' 51 Chapter 5. P-Type Fuzzy Control on a Hydraulic Manipulator using Composition Based Inference where A, A', B, B' are fuzzy sets. First we need to convert the IF-THEN rule in premise 1 into the fuzzy relation RA_>B . Then the conclusion 5'is inferred from the fuzzy relations RA^B and A'of premise 2, applying the compositional operation given in Eq (5.3): The reasoning process based on (5.2) is called the compositional rule of inference. Note that the rule A — > B is converted to a fuzzy relation. The conversion methods depend on how the implication A — > B is interpreted. Implication based on Mamdani's method is commonly used to compose the fuzzy relation/?, particularly in control applications given in Eq (5.4), and is defined as follows : This conversion is based on the definition of the Cartesian product of two fuzzy sets given in Eq B' = A'oR (5.3) (5.4) (5.5): a — » b = a Ab (5.5) Implication A — > B based on Mamdani's method can be summarized as Eq (5.6): juR(x,y) = jUA(x)AjUB(y) (5.6) This is the case where there is one input and one output or, single-input-single-output (SISO) fuzzy system. Now consider a fuzzy system with two inputs and one output, as: IF x is A and y is B THEN z is C (5.7) 52 Chapter 5. P-Type Fuzzy Control on a Hydraulic Manipulator using Composition Based Inference where A, B and C are the fuzzy subsets of X, Y, and Z, respectively. The rule (5.7) can be simplified as: A and B -> C (5.8) The cylindrical extension [Appendix A. 16 ] is applied as follows: "x is A and y is 5" <=>(*, y) is (AxY)n(XxB) (5.9) (x,y) is the cross product A x B (5.10) Therefore, the fuzzy relation "A and B" can be expressed as Eq (5.11): A a n d f l = A x f l = j(jiA(x)A/iB(y))/(x,y) (5.11) XxY Hence, the rule " A and B —> C " can be translated as: R = A and B = AxBxC (5.12) = L v 7 ( ^ W A / / B ( y ) A / / c ( z ) ) / ( x , y , z ) (5.13) flR(x,y,z) = MA(x)AflB(y)Afilc(z) (5.14) When there are more than one rule to process, the foregoing approach can be easily extended. For example, consider the following two rules: Rule 1: IF x is A, T H E N y is B, Rule 2: IF x is A2 T H E N y is 5 2 53 Chapter 5. P-Type Fuzzy Control on a Hydraulic Manipulator using Composition Based Inference These two rules may be rewritten as: A, -> B, else A2 -4 B2 (5.15) These two rules are converted into two fuzzy relations as follows: A,->B, (5.16) A2 -> B2 => R2 In Mamdani's method "else" is interpreted as an "OR" operation. Hence the compiled fuzzy relation R is given as: R = R,(jR2 (5.17) Example: X = {x,,x2,x3} and A},A2 c X Y = {y„y2,y3} and B„B2aY Suppose that the fuzzy sets A,,A2,B, and B2 are given by: , 1.0 0.6 0.0 A, = + + A 0 0 0 - 8 L 0 A2-— + — + — (5.18) „ 1.0 0.6 0.1 B, = + + — yi y2 y3 „ 0.2 0.8 0.9 B2= + + yj y2 y3 First we should convert the rule: Rule 1 A, —> B, into the fuzzy relation R, . With the "min" operation, we get: 54 Chapter 5. P-Type Fuzzy Control on a Hydraulic Manipulator using Composition Based Inference (5.19) MB,(y,) 1.0 MBl(y2) 0.6 MBl(y3) 0.1 1.0 1.0 0.6 0.1 0.6 0.6 0.6 0.1 0.0 0.0 0.0 0.0 or, 1.0 0.6 0.1 0.6 0.6 0.1 0 0 0 (5.20) Next we convert the rule: Rule 2 A2 —> B2 into the fuzzy relation R 2 • R2 = MB2(yj) 0.2 MB2(y2) 0.8 MB2 (y3) 0.9 0.0 0.0 0.0 0.0 0.8 0.2 0.8 0.8 MAMS) 1.0 0.2 0.8 0.9 (5.21) or, R2 = 0.0 0.0 0.0 0.2 0.8 0.8 0.2 0.8 0.9 (5.22) Finally, we form the overall rulebase relation R as: '1.0 0.6 0.1 R = R,uR2 = 0.6 0.8 0.8 0.2 0.8 0.9 (5.23) Once R is formed in this manner, for a given input of fuzzy set A ' on X , the output fuzzy set B' on Y can be obtained as: 55 Chapter 5. P-Type Fuzzy Control on a Hydraulic Manipulator using Composition Based Inference B'=A'oR (5.24) where " o " is the composition operation. This is the principle of composition-based inference. The composition by " o " is quite similar to the multiplication of matrices. Specifically, if the multiplication and the addition in a matrix multiplication are replaced by the min-operation and the max-operation, respectively, then the composition operation is realized. As an illustrative example, suppose that the input fuzzy set A' on X is , / °-8 0 3 A= + — (5.25) The outcome (inference) of the reasoning, which is B' on Y , is given by B' = A'oR (5.26) y, y2 y3 xi 1.0 0.6 0.1 [0.8 0.3 0] o 0.6 0.8 0.8 x3 0.2 0.8 0.9 (5.27) Then, B' = [(0.8 A 7.0) v (0.3 A 0.6) v (0 A 0.2), (0.8 A 0.6) v (0.3 A 0.8) v (0 A 0.8), (5.28) (0.8 A 0.1) v (0.3 A 0.8) v (0 A 0.9)] or, B' = [0.8v0.3v0 0.6v0.3v0 O.lvOJvO] (5.29) which gives, B' = [0.8 0.6 0.3] (5.30) This composition-based inference method is implemented in real time on the prototype hydraulic manipulator. In order to be able to compare the results of this method with the individual rule- based inference method used in Chapter 4, we implement the P-type fuzzy logic controller with 56 Chapter 5. P-Type Fuzzy Control on a Hydraulic Manipulator using Composition Based Inference the same membership functions shown in Figure 4.20 and the rulebase given in Figure 4.17. In this manner, responses shown in Figure 4.21 can be compared with the results obtained in the present chapter. The main steps of the present implementation are given below: Step 1: The membership functions of the input and the output (Figure 4.20) are discretized (Figure 5.1 and 5.2). Pseudo code of the algorithm of this method, formulates the memberships of the input in the form of a two dimensional array, where one element of the pair indicates the fuzzy state of the membership function (NS,ZR,PS,...) and the other element indicates the number of the discretized point. For example, the membership functions of the input can be represented in the form : Mf Unput [ I ] [j ] (5.31) where i represents the fuzzy state and j represents the point number. N = number of points of the Input membership function M - number of points of the Output membership function and i =0,1,..., 6 j =0,1,....,7V It follows that, for example, Mf l_input [ 0 ] [ j ] represents the input membership function of Negative Large (NL), and so on. Accordingly, the membership functions of the input variable are as given in Figure 5.3. 57 er 5. P-Type Fuzzy Control on a Hydraulic Manipulator using Composition Based Inference (a) M=9 - t — + — • — t - A , , fj4 4 ' -* + + •*- _4 (- /——\ -* » 1 f if.— 1- -*—•—*—+•——1— -40 -20 0 20 40 Current to the Servovalve (mA) (b) M=17 > \ t * % t • • % % -*-*-*-*' W -» -» -» -4 - - A - 4 V " 7 Y NL NM NS ZR PS PM -+-t-*-t~+~-*--t-t-(f-t--+-»̂ PL -40 -20 0 20 40 Current to the Servovalve (mA) Figure 5.2 Discretized membership functions of the output. 58 Chapter 5. P-Type Fuzzy Control on a Hydraulic Manipulator using Composition Based Inference Mfl_input[0] [ j ] is NL Mfl_input[l][j]is NM Mf l_input [ 2 ] [ j ] is NS Mfl_input [ 3 ] [j] is ZER Mfl_input [ 4 ] [j]isPS Mfl_input[5] [j] isPM Mfl_input[6] [j] isPL Figure 5.3 Membership functions of the input. Figure 5.1 shows two cases of discretized membership functions of the input variable, where the universe of discourse is divided into ./V points with N=7 or 13. Figure 5.2 shows two cases of discretized membership functions of the output variable, where the universe of discourse is divided to M points with M=9 or 17. The effect of the number of discretization points on the performance of the fuzzy system is in this chapter, through experimentation. Step 2 : The rulebase is generated in the form of a fuzzy relation, based on the discretized input and output membership functions as determined in Stepl. This relation is generated based on the Mamdani min-max implication method. Note that, N = number of discretized points in the universe of discourse of an input membership function M = number of discretized points in the universe of discourse of an output membership function mf 1= two dimensional array of the input membership functions mf3= two dimensional array of the output membership functions The rulebase is given in Figure 5.4. This rulebase is the same as what is given in Figure 4.13 (Case 1). Figure 5.5 shows the pseudo code for generating the rulebase, in the form of a fuzzy relation. 59 Chapter 5. P-Type Fuzzy Control on a Hydraulic Manipulator using Composition Based Inference pos_erris NL (mfl[0]) THEN output is PL (mf3[6]) pos_erris N M (mfl[l]) THEN output is PM(mf3[5]) pos_erris NS (mfl[2]) THEN output is PS (mf3[4]) pos_err is ZR (mf 1 [3]) THEN output is ZR(mf3[3]) pos_erris PS (mfl[4]) THEN output is NS (mf3[2]) pos_erris PM (mfl[5]) THEN output is NM(mf3[l]) pos_erris PL (mfl[6]) THEN output is NL(mf3[0]) Figure 5.4 Rulebase of the fuzzy system. MAKERULE (mfl[mfl_states][ N], mO[mf3_states][ M], N, M) for (i=0; i<=N ;i++) for(k=0; k<=M;k++) i I rulel =min(mfl[0][i],mf3[4][k]); //pos. .err is NL (mfl[0]) THEN output is PS(mf3[6]) rule2 = min(mfl[l][i],mf3[4][k]); //pos. .err is NM (mflfl]) THEN output is PS(mf3[5]) rule3 = min(mfl[2][i],mf3[4][k]); //pos. .err is NS (mfl[2]) THEN output is PS(mO[4]) rule4 = min(mfl[3][i],mf3[3][k]); //pos. .err is ZR (mfl[3]) THEN output is ZR(mf3[3]) rule5 = min(mfl[4][i],mf3[2][k]); // pos. err is PS (mfl[4]) THEN output is NS(mf3[2]) rule6 = min(mfl[5][i],mf3[2][k]); // pos. .err is PM (mfl[5]) THEN output is NS(mf3[l]) rule7 = min(mfl[6][i],mf3[2][k]); //pos_ err is PL (mfl[6]) THEN output is NS(mf3[0]) max 1 =max(rule 1 ,rule2); max2=max(maxl,rule3); max3=max(max2,rule4); max4=max(max3 ,rule5); max5=max(max4,rule6); max6=max(max5 ,rule7); rulebasefi] [k]=max6; \ I return (rulebase[i][k]); } Figure 5.5 The rulebase in the form of fuzzy relation. Step 3: The crisp input is changed to a fuzzy singleton shown in Figure 5.6 (where it has membership value of 1.0 at the representative point of the universe of discourse, all the other points having membership values of 0.0). In this manner the crisp input is fuzzified and becomes an array of N elements where all the elements are zero except for one which has the membership grade of 1.0. The fuzzy singleton is rounded to the closest discretized point in the universe of discourse. 60 Chapter 5. P-Type Fuzzy Control on a Hydraulic Manipulator using Composition Based Inference Discretized Universe of Discourse of Input N =7 ( A ) • • • -15 -10 -5 T 0 5 10 15 ft ( B ) • • • • -10 0 5 10 15 (C) -15 -10 -5 0 5 10 15 Position Error (mm) Figure 5.6 Transformation of crisp input signal to a fuzzy singleton. Step 4: The fuzzified input (fuzzy singleton) is composed with the rulebase, using the compositional rule of inference. The result of this operation is a fuzzy decision (control inference). The fuzzy inference is in the form of a vector with M number of discretized output values. The inference engine of this fuzzy system is given in Figure 5.7. N= number of points of the input membership function M- number of points of the output membership function INFERENCE( input [N], rulebase [N][M], N , M) for (i=0; i<=mfl_points;i++) for(k=0; k<=mf3_points ;k++) { m=min(input[i], rulebase[i][k]); infer[k]=max(infer[k],m); return(infer[k]); Figure 5.7 Inference engine of the fuzzy system. 61 Chapter 5. P-Type Fuzzy Control on a Hydraulic Manipulator using Composition Based Inference Step 5: The crisp decision as a signal to the actuator is generated from a fuzzy decision (inference). The "centre of gravity" is used as the defuzzification method. The associated pseudocode is shown in Figure 5.8. DEFUZ( infer[mf3_points], mf3_points, X3[mf3_points]) { for (k=0; k<=mf3_points; k++) { weighted_sum+=X3 [k] *infer[k]; sum+=infer[k]; } signal=weighted_sum/sum; return(signal); } Figure 5.8 Defuzzification scheme for the fuzzy system. In implementation of the "Composition Based Inference" in real time, steps 1 and 2 are carried out off line. However, the steps 3, 4 and 5 are performed at each sampling period. Figures 5.9 through 5.12 show the response of the system for four different cases of discretization levels for the input and output membership functions, where; N = number of discretized points in the universe of discourse of input membership function. M = number of discretized points in the universe of discourse of output membership function. In each case, for experiments (a)-(d) are conducted with different initial conditions and set points, giving two directions of actuator motion. 62 Chapter 5. P-Type Fuzzy Control on a Hydraulic Manipulator using Composition Based Inference Figure 5.9 Response of the system in the case N=l and M=9. (Four initial error conditions (a)-(d)). Figure 5.10 Response of the system in the case At 13 and M=9. (Four initial error conditions (a)-(d)). 63 Chapter 5. P-Type Fuzzy Control on a Hydraulic Manipulator using Composition Based Inference Figure 5.11 Response of the system in the case /V=7 and M=17. (Four initial error conditions (a)-(d)). Figure 5.12 Response of the system in the case N=\3 and M=17. (Four initial error conditions (a)-(d)). 64 Chapter 5. P-Type Fuzzy Control on a Hydraulic Manipulator using Composition Based Inference 5.2 Discussion The fuzzy controller based on the "compositional rule of inference" operates in two phases, one offline and the other processed at each sampling time. Steps 1 and 2, which are discretization of the membership functions and generation of the rulebase, are processed offline. Depending on the number of discretization points, the dimension of the fuzzy relation can be quite large, requiring a large memory. The effect of the number of discretization points on the performance of the system has not been discussed in the literature. However, in the present experiments, two different discretization levels were used for the input membership functions and two different discretization levels were used for the output membership functions as well. The reason that the effect of the number of discretization points would be of interest is the fact that the size of the created fuzzy relation is influenced by number of these points, which can be shown using Figure 5.5 and Eq (5.20) and (5.22). The higher the resolution of the points on the universe of discourse, the greater the memory requirement for storing the fuzzy relation. Two sets of discretization levels are compared for input and output membership functions. The number of discretization points for the input membership functions is chosen as N=l or 13 the number of discretization points for the output membership functions is chosen as M=9 or 17. Figures 5.1 and 5.2 show the rationale for choosing the particular numbers for N and M . The universe of discourse is divided evenly so that at least one point has the membership grade of 1.0 for each state of (NL, NM, NS, ZR, PS, PM, PL). In other words, all the membership functions are designed to be normal, meaning the height of the set is 1.0 (see Appendix A). As shown in Figure 5.1(a), the universe of discourse is [-30,30] mm. Figure 5.1(b) shows the universe of discourse [-30,30] mm, which has been divided evenly into smaller segments of 5mm. In this case, we have more than one point that has a membership grade greater than zero and has small enough resolution (5mm) to realize the effect of changing the number of discretization points for the input membership function. It is not necessary to divide the universe of discourse into equal segments. However, for the sake of comparison and to study the effect of resolution of the universe of discourse in terms of the number of discretization points, the universe of discourse is divided into equal segments. The resulting membership functions, as shown in figures 5.1 and 5.2 are exactly the same as the membership functions shown in Figure 4.20 . In figures 5.1 and 65 Chapter 5. P-Type Fuzzy Control on a Hydraulic Manipulator using Composition Based Inference 5.2 , each state ( NL, N M , NS, ZR, PS, P M , PL) is plotted separately, while in Figure 4.20 all the states are plotted on a common universe of discourse.In particular, in this manner the membership function associated with a particular discretized point on the universe of discourse can be shown with improved clarity. 5.2.1 Effect of number of discretized points of input membership functions The effect of the input membership function when the number of discretized points of the output is M=9, can be examined through the comparison of figures 5.9 and 5.10. The effect of the input membership function when the number of discretized points of the output is M=17 can be examined through the comparison of figures 5.11 and 5.12. The membership functions of the input are shown in Figure 5.1. With regard to the speed of response and the offset, the two levels of discretization both provide the same performance. Now compare the following response pairs: Figure 5.9(a) and Figure 5.10(a); Figure 5.9(b) and Figure 5.10(b); Figure 5.9(c) and Figure 5.10(c); and Figure 5.9(d) and Figure 5.10(d). The performance in the two cases shown in Figure 5.9 and Figure 5.10 appears to be quite similar. A comparison of the performance shown in Figures 5.11 and 5.12 indicates that no consistent difference exists between the two cases. In Figure 5.12, all the four cases (a),(b),(c),(d) show a stable response where it reaches a steady state with an offset, but Figure 5.11(c) shows that the system is not stable yet and has not reached a steady state. The foregoing experiments demonstrate that the number of discretization points of the input membership function does not affect the performance of the system. If the higher resolution doesn't significantly improve the performance of the system, then it is not necessary to increase the size of the fuzzy relation. 5.2.2 Effect of number of discretized points of output membership functions The effect of the output membership function when the number of discretized points of the input is N=7 can be examined through the comparison of figures 5.9 and 5.11. The effect of the output membership function when the number of discretized points of the input is 7V=13 can be examined through the comparison of figures 5.10 and 5.12. The membership function of the output is shown in Figure 5.2. Figure 5.11(a) shows a stable response when compared with 66 Chapter 5. P-Type Fuzzy Control on a Hydraulic Manipulator using Composition Based Inference Figure 5.9(a). However, the comparison of Figure 5.10(d) and Figure 5.11(d) indicates that increasing the number of discretized points of the output would not necessarily improve the performance of the system. Also, the comparison of figures 5.10 and 5.12 indicates that increasing the number of discretized points of the output does not affect the performance of the system drastically. A l l four cases (a),(b),(c),(d) in Figure 5.12 seem to have reached the steady state with an offset of 5mm in less than 200ms. However, this stable performance has not occurred in case (d) of Figure 5.11; case (a) of Figure 5.10; and cases (a) and (d) of Figure 5.9. It follows that, the case 7Y=13 and M=17 in Figure 5.12 has shown the best performance among all cases. 5.2.3 Effect of inference method The main objective of the present chapter is to compare the practical performance of the two inference methods: (individual rule-based inference and composition-based inference), through experimentation on the prototype industrial hydraulic manipulator. To accomplish this, we compare the performance of the system, while keeping the membership functions and the rulebase the same but using two different inference methods; "individual-rule-based inference" and "composition-based inference". If these two methods produce practically the same results, then in view of the analytical developments in the area of rulebase decoupling [18,25,32,33,45] and related analytical concepts of "composition based inference", one may proceed to employ the "individual-rule-based inference" instead, since the development of a fuzzy controller based on the "individual-rules" inference is significantly simpler. For this reason, a fuzzy controller based on the composition-based inference is developed for the P-type fuzzy controller,(with membership functions as given in Figure 4.20 and the rulebase). The performance of the composition-based inference as shown in Figure 5.12 is now compared with that of the individual-rule-based inference as shown in Figure 4.21. Figure 4.17(a) is compared with Figure 5.12(b) and Figure 4.17(b) is compared with Figure 5.12.(c). As can be seen, in all cases of composition-based inference, there is a significant offset of 5 mm; however, in individual rule- based inference, there is no offset and the response has reached the steady state, in less than 600ms. 67 Chapter 5. P-Type Fuzzy Control on a Hydraulic Manipulator using Composition Based Inference The present experimental results show that the two inference methods do not produce the same performance. Furthermore, the composition-based inference method requires more memory. It follows that, if the composition-based inference method presents practical limitations in the development of a fuzzy controller, due to the large memory that it occupies and the rather inconvenient form of discretization of the universe of discourse, it would be advantageous to use the individual-rule- based inference method . 68 Chapter 6. Development of an Intelligent Hierarchical Control System Chapter 6 Development of an Intelligent Hierarchical Control System for a Hydraulic Manipulator Previous chapters investigated two types of low level direct fuzzy logic controllers with respect to such considerations as the membership functions, rulebase, inference technique, and the defuzzified approach, as applied to a hydraulic manipulator of a prototype fish processing machine. This chapter presents the development of a hierarchical intelligent control system for the same prototype hydraulic manipulator. This supervisory control system is designed to be an integral part of the prototype machine. Even though the hierarchical control system possesses some characteristics that are not common with the low-level direct controller, there are other considerations that can benefit from the knowledge of developing the low-level controllers that occupy the lowest level of the overall hierarchical control system. 6.1 Introduction The concept of intelligent control was introduced by Fu [30] as an attempt to formalize in a systematic way, an engineering discipline that deals with knowledge-based control. Within this framework, a hierarchical intelligent control architecture was proposed by Saridis [35,68] as a unified theoretic model of cognitive and control systems methodologies. Since an automated machine may consist of many components (sensors and actuators) that will be required to deal with a large amount of information, an efficient structure for handling information from various sources would be required. In the literature, one finds the concept of "intelligent hierarchical control", which may be particularly suitable for complex industrial machine such as the prototype fish cutting machine as described in Chapter 1. Saridis [4,35,68] introduced intelligent Control as a combination of three areas; Artificial Intelligence (AI), Operation Research (OR) and Control theory. He then presented the concept of "Hierarchical Intelligent Control" as a unified theoretic approach of cognitive and control 69 Chapter 6. Development of an Intelligent Hierarchical Control System systems methodologies. Inspired by Saridis' idea, a hierarchical structure for processing the information from various sensors and actuators, based on principle of "Decreasing precision with Increasing Intelligence" [56], has been proposed and developed on the prototype fish cutting machine [21,84]. This hierarchical structure has been designed in such a way that the lowest level of the hierarchy is occupied by the various controllers and sensors that provide feedback information for process control. The information generated at this level is mainly visual data from the C C D cameras, position and speed data from the encoders and cutter load data from the induction motors. In this level all the information is present in large quantities and of high resolution. The second level of the hierarchy is a data processing level where high-resolution crisp data from the sensors are filtered. This filtered data (pre-processed data) provides an abstract representation of the current state of the machine. The filter operations include averaging the signal, peak detection, pattern recognition, and computation of performance attributes such as rise time, damping ratio, and settling time. These parameters are interpreted according to some subjective criteria of performance evaluation and then transcribed into fuzzy sets. The top level of the hierarchy is the knowledgebase and the inference engine which makes decisions based on the current fuzzy information about the process. The knowledgbase can be considered as a collection of expert instructions on how to improve the yield while maintaining the product requirements and keeping the individual components at their optimum performance. The information at this level, as well as the inference, are exclusively represented by fuzzy linguistic variables. The control decisions are made based on various monitored fuzzy variables, in order to improve the performance of the machine by means of adjustments made on the parameters of the process actuators. Some related works is presented in [3,9,14,46]. Based on this hierarchical concept, de Silva and MacFarlane [19,20] developed a hierarchical knowledge-based control system for a two-degree-of-freedom robot. The system consists of a robot simulator, two servo experts for the joints of the robot, and a fuzzy controller to generate the tuning commands for the PID controllers of the joints. Extending this work, Barlev [5] implemented a hierarchical intelligent tuner on a servomotor system in real-time. Wickramarachchi [21,84] developed a knowledge-based hierarchical control structure for process automation and applied it to a prototype fish-processing machine in the Industrial Automation Laboratory, University of British Columbia. This machine has been designed to 70 Chapter 6. Development of an Intelligent Hierarchical Control System automatically cut fish, taking advantage of many sensors and actuators available in this machine. Goulet [34] implemented an intelligent hierarchical controller for a deployable manipulator, which may be used in both space and ground-based applications, where the direct controller is a proportional-integral-derivative (PID) controller for the servomotor. The chapter presents the development of an intelligent hierarchical controller for the prototype industrial fish cutting machine (see Chapter 2) in the Industrial Automation Laboratory. In this system, the lowest level consists of conventional proportional-plus-derivative (PD) controllers for the hydraulic actuators. The middle level monitors the performance of the manipulator, preprocesses the response signal and extracts the performance parameters, based on a step-input response. The top level makes inferences giving appropriate tuning actions for the PD controllers. 6.2 Development of the Hierarchical Intelligent Control System The general structure of the developed intelligent hierarchical control system is shown in Figure 6.1. Note that at the lowest level is conventional PD controller. The second level acts as an interface between the direct controller at the bottom layer, and the knowledge-based monitoring controller at the top layer. The second layer handles such tasks as performance specification, response processing, and computation of performance indices. This stage involves, for example, averaging or filtering of the data points and computation of the rise time, overshoot, and steady state offset [24]. Tuned Parameters of Kd & Kp Intelligent Supervisor (Knowledgebase) Level 3 Actual posit ion The Plant Figure 6.1 Schematic representation of the intelligent hierarchical control system. 71 Chapter 6. Development of an Intelligent Hierarchical Control System In the first layer, a conventional PD controller uses as the control signal the current to the servovalve. This is determined using, the error (e) between the actual position (x) and the desired position (xd) of the cylinder of the hydraulic actuator. The parameters Kp and Kd are the proportional gain and the derivative gain, respectively. The control law may be expressed as i = Kp (xd -x) + Kdv + Ai (6.1) with Ai = -Id sgn(xd - x) (6.2) Equation (6.2) represents a correction term which is added to the control signal, to compensate for the deadband behavior of the servovalve [75,76]. The variable v is the velocity estimated by an observer that has been designed in [75,76] and represented by the transfer function in Figure '6.2. Position Estimated i 1 Velocity Figure 6.2 The observer for velocity estimation. In this system, the second layer is designed to monitor the response of the system, which extracts three parameters: Overshoot in the response, oscillation in the response, and speed of response. 72 Chapter 6. Development of an Intelligent Hierarchical Control System Level 2 Level 3 Figure 6.3 The flow chart of the algorithm of information abstraction and decision making. As shown in the flow chart of Figure 6.3, the second layer monitors the response. The sequence of monitoring the performance has been prioritized based on the degree of importance of the parameters (Overshoot, Oscillations, Speed of Response) for the system. This system has been designed to first check for the overshoot, because the overshoot in the response is the most critical parameter and may be damaging for the system. The overshoot is calculated by determining the first peak after crossing the setpoint. The second stage in the algorithm shown in the flow chart checks for oscillations in the response, while the response does not have an overshoot. Once it detects an overshoot, the system operation immediately goes to the top level 73 Chapter 6. Development of an Intelligent Hierarchical Control System where the knowledgebase (A) deals with unsatisfactory overshoot. The knowledge base at the top level matches with the existing information about the system, arriving from the lower level (second layer). Consequently it makes a decision for changing (tuning) the parameters of the direct controllers. After defuzzification, the resulting crisp values for changing the parameters of the controllers are sent to the corresponding direct controllers. Figure 6.4 shows the response of the system with different overshoots. An example of responses with oscillations is shown in Figure 6.5. Different Qualities of Overshoot (Large,Medium,Small) 50 45 40 35 £ E.30 c o o 25 Q. <5 f 20 O 15 10 5 0 0 100 200 300 400 500 600 700 800 900 1000 Time (ms) Figure 6.4 The response of system with different overshoots. Different Qualities of Roughness in the Response (Large.Medium,Small) 50 45 40 35 E E . 3 0 c o I25 Q. CU TJ f20 o 15 10 5 0 0 100 200 300 400 500 600 700 800 900 1000 Time (ms) Figure 6.5 System response with different degrees of oscillations. 74 Chapter 6. Development of an Intelligent Hierarchical Control System In the second layer, the algorithm detects the location and the magnitude of the local peaks on the response curve and describes the monitored system in a qualitative form, by the label "roughness of the response". The algorithm detects a peak and the immediately next minimum point of the response and the separation in position between these two points. This magnitude is considered as an index for the depth of the peaks. If the roughness of the response is unsatisfactory, then the control immediately goes to the top level knowledge base (B). It deals with the oscillations in response, by commanding a change to the parameters of the controller. Table 6.1 shows the rulebase for determining the roughness of the response in the second layer of the structure. Table 6.1. Rulebase for the roughness of the response. IF T H E N Peak AI TO Depth Roughness 1 Large Large 1 Small Small More than 1 Large Large More than 1 Small Moderate The third stage in the flow chart in Figure 6.3 is the second layer, which monitors the speed of response. Figure 6.6 shows three responses having different speeds. Different Qualities of Speed of Response (Slow.Medium.Fast) 700 800 900 1000 Time (ms) Figure 6.6 Examples of different speeds of response of the system. 75 Chapter 6. Development of an Intelligent Hierarchical Control System The speed of response is determined by the time in which the response first reaches the desired position. The algorithm computes the index for the speed of response using this data. This index is transmitted to the top level of the knowledgebase (C), The knowledgebase in the top layer is shown in Figure 6.7 and the membership functions in the knowledgebase are shown in Figure 6.8. Knowledgebase (A) IF Overshoot is large THEN decrease Kp considerably IF Overshoot is medium THEN decrease Kp slightly IF Overshoot is small THEN do not change Kp Knowledgebase (B) IF Oscillations are large THEN increase Kd considerably IF Oscillations are medium THEN increase Kd slightly IF Oscillations are small THEN do not change Kd Knowledgebase (C) IF Speed of Response is slow THEN increase Kp considerably IF Speed of Response is medium THEN increase Kp slightly IF Speed of Response is fast THEN do not change Kp Figure 6.7 Knowledgebase of the top level of the hierarchy. 76 Chapter 6. Development of an Intelligent Hierarchical Control System Small Medium Large ML NS ZR PS PL Overshoot (mm) m m Small Medium Large NL NS ZR PS PL -300 - 200 - 100 0 100 200 300 Oscillations 0 300 450 600 1 0 0 0 ' m o "4000 - 2000 0 2000 4000 6000 Speed of Response (msec) A KP "mm Figure 6.8 Membership functions in the knowledgebase. 6.3 Experimental Results Figure 6.9(a) shows the response of the system before tuning where the response has a medium size overshoot, and the response after tuning where there is no overshoot. The knowledgebase (A) makes a decision to decrease the proportional gain slightly. Figure 6.9(b) shows the response of the system before tuning where the response is oscillatory. The knowledge base (B) makes a decision to increase the derivative gain slightly which improves the response, as shown in the same figure. Figure 6.9(c) shows the response of the system where the tuning action has improved the speed of response by slightly increasing the proportional gain. 77 Chapter 6. Development of an Intelligent Hierarchical Control System Experimental Resuls of Auto tuning _J I I L_ before tuning after tuning _ l L_ 0 100 200 300 400 500 600 700 800 900 1000 Time (ms) Figure 6.9.a Experimental results (overshoot improvement due to tuning). 50 45 40 35 E £ 30 o 25 i 20 15 10 Experimental Resuls of Auto tuning 1 1 1 1 1 before tuning after tuning i . . 1 / / _ / . 1 / J 1 j 11 1 1 0 100 200 300 400 500 600 700 800 900 1000 Time (ms) Figure 6.9.b Experimental results (improved oscillations due to tuning). 78 Chapter 6. Development of an Intelligent Hierarchical Control System Experimental Resuls of Auto tuning Figure 6.9.c Experimental results (improved speed of response due to tuning). 6.4 Discussion The period of the cyclic motion of the conveyor is 1. 24 seconds. The fish are stationary during half the period (620 ms) and moving during the other half period. While a fish is moving (i.e., being pushed forward by the conveyor), the manipulator is at rest for half a cycle (620 ms). The tuning action at the higher level takes place during this 620ms. In the control system of the manipulator, the performance of the direct controller is monitored after completing each cutting task. Therefore, the computation of the information abstraction in the second level and the decision making in the knowledge base at the top level, have to be completed in 620 ms. Knowledge base modules at the top level are designed in the form of SISO fuzzy rules where they make tuning decisions regarding one parameter of the controller at a time. This means that in each step of tuning, only one parameter is tuned based on the priority of the performance 79 Chapter 6. Development of an Intelligent Hierarchical Control System indices. Since the membership functions are designed in such a form that each crisp value can represent at most two fuzzy variables, at each time instant a maximum of two rules would be fired and processed. Consequently, the overall computations can be carried out in 620ms or less. The rulebase modules are based on both common knowledge of control and extensive experimentation on the setup. Mamdani's method of fuzzy reasoning is used and the center of gravity method is used for defuzzification. 80 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral Chapter 7 A New Inference Method Based on Fuzzy Integral From the investigation that has been carried out in the previous chapters it is clear that the method of decision making, or inference generation, plays a crucial role in a fuzzy expert system, particularly a fuzzy logic controller. Two common approaches of inference generation—the individual-rule-based method and the composition-based method are investigated and experimentally evaluated in Chapter 2 through 5, giving their respective advantages and shortcomings. Clearly, there is room for improvement in these inference methods. The present chapter develops a new fuzzy inference method, which is particularly useful for multivariable fuzzy systems. This method is developed using the concept of fuzzy integral. In the first section, the purpose of the development, the motivation, and the rationale behind the new inference method are explained, followed by literature review from different view points. Then the required theoretical background for understanding the basis of the new inference method is explained in section 7.2. The detailed development of the proposed new inference method is explained in section 7.3, followed by the experimental results and discussions. 7.1 Introduction In complex processes, often many variables have to be monitored by the control system and many parameters need to be changed accordingly. In a multi-input-multi-output (MIMO) system, it is difficult to expect a human to keep track of a large number of interacting variables and to express the expertise as a set of coupled rules consisting of all input and output variables. Even if it is possible to build a multi-input-multi-output (MIMO) rulebase system, the issue of speed of processing needs to be addressed. In view of this, it is quite appealing to consider the possibility of decoupling the MIMO rulebase into a set of single-input-single-output (SISO) rules. This chapter develops a novel method of inference which is able to deal with multivariable fuzzy systems. 81 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral 7.1.1 Purpose of the development This chapter presents an approach for processing large numbers of fuzzy variables that are present in complex knowledge-based expert systems. Motivated by the practical problems that are present in complex industrial machines, the present chapter focuses on processing linguistic fuzzy information from many different sources, as used in practical expert systems and supervising control systems. The associated problem may be interpreted as both an inference method for MIM O fuzzy rulebases that are employed in knowledge-based systems such as control expert systems, and information fusion in complex systems that use expert systems. The method introduces a structured way to organize the variables and build a manageable rulebase. It is based on the concept of fuzzy measure and fuzzy integral, which has been introduced by Sugeno [72,74], and has been inspired by the work done on information fusion in computer vision using the fuzzy integral [77]. 7.1.2 Motivation The need to develop an effective inference method for multivariable fuzzy systems is motivated by the control needs of an automated fish cutting machine, two prototypes of which have been developed at the Industrial Automation Laboratory of the University of British Columbia (see Chapter 1). The industrial prototype is a complex machine, which consists of many sensors, actuators, controllers, and mechanical devices. The overall performance of the machine depends on improving the meat recovery and the quality of the cut fish, which are subjective measures. Autonomous operation of the machine relies on an intelligent supervisory control system which is based on fuzzy logic. The associated fuzzy systems are multivariable. The realtime operation of such control expert systems will considerably benefit from fast and accurate inference methods. 7.1.3 Rationale As noted before in this thesis, in order to control a process using a fuzzy knowledge- based technique, the control decisions have to be made based on monitored process variables. For this purpose, a knowledgebase in the form of fuzzy rulebase has to be developed. During the system operation, the monitored process variables are matched with the knowledgebase and the 82 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral control decisions are made. There are many variables in a "multi-input-multi- output (MIMO) systems". Since each fuzzy rule in the knowledgebase is developed based on the expertise of a human expert, expecting a human to keep track of many variables at a time is rather unrealistic, since a human may typically react to a single response variable before turning attention to another response variable. However, in a complex system, the number of process variables (number of information sources) can be quite large. Even when the M I M O system can be built, still the speed of processing would be low. To overcome this problem, one needs to find a way to separate the system into modules of SISO rulebases. 7.1.4 Related literature The problem of dealing with large fuzzy rule-based systems (MIMO fuzzy rulebases) has been investigated from the point of view of number of rules in [62]. The main issue addressed in that work is to reduce the number of fuzzy rules in a control system. In a complete fuzzy rulebase, the number of rules increases exponentially with the system variables. Specifically, a complete set of rules in a rulebase can have mn different rules, where m is the number of fuzzy states in a variable and n is the number of system variables in the rulebase. The work in [62] has proposed a method to reduce the number of rules so that it would have a linear relation with number of system variables. Their method is based on a hierarchical structure in which the most influential parameters are chosen as the system variables in the first level, the next most important parameters are chosen as the system variables in the second level, and so on. In the hierarchy, the first level gives an approximate output, which is then modified by the rule set in the second level. The variables in the second level include the approximate output of the first level and system variables. This process is repeated in succeeding levels of hierarchy. The total number of rules is equal to the sum of the rules at each level. Apart from reducing the number of rules, their approach introduces an architecture where the large number of system variables can be organized in a hierarchical manner. In a simulation example, they have shown that this method would be useful when the system is such that it is possible to construct the rulebase in a form where the output of one set in the low level of the hierarchy can be treated as one of the input variables of the set of rules in the next level. They have applied their fuzzy control architecture to control the feedwater flow to a steam generator of a power plant. Their simulation 83 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral results show that the hierarchical fuzzy controller has a superior performance over a conventional proportional-integral-derivative (PID) controller. Large rulebases have also been studied from another point of view—the decomposition of MIMO fuzzy rulebases. There are two primary methods of making inferences using a fuzzy rulebase [28]; namely, the "compositional rule of inference" of Zadeh[102] and "direct fuzzy inference" of Mamdani [52]. One group of researchers have approached the problem of simplifying the MTMO fuzzy rulebase by decomposing it into fuzzy relations [25,26,32,33,36,45,48,67,81,82,88,89,90]. Assuming that the fuzzy relations of the M I M O rulebase can be easily constructed, they have proposed methods to manipulate and simplify the fuzzy relations so that the number of real-time calculations required for inference making can be reduced and as a result the control signal could be computed faster. . It has been shown that complex rules can be broken up into simple rules when the rules are represented with any S- implication or any R-implication [16]. Based on the method presented in [16], a method has been introduced for parallel processing of fuzzy rules [53]. Some related works are presented in [39,41,80,91]. A method for decomposing a MIMO fuzzy rulebase has been proposed in [105] where the input space is divided into sub-input spaces so that a general fuzzy system can be decomposed into several simpler sub-fuzzy systems. That paper argues that in practice a fuzzy rulebase is constructed based on the information of human experts. Since it is rather difficult for a human to keep track of many variables; say, more than two variables at a time, forming a large rulebase incorporating many input fuzzy variables may be practically somewhat unrealistic. Decomposition of a MIMO system has been approached in [38] through a structure called "Network of Fuzzy Adaptive Nodes" (NetFAN) which is a structured network of simple, interacting fuzzy or neuro-fuzzy systems. Its main characteristic is that it propagates numerical values as intermediate signals. Another problem with large rulebases is the speed of processing. Methods have been introduced, however to handle this problem [11,15]. 7.2 Theoretical Background Fuzzy measures and fuzzy integrals have emerged in the context of fuzzy sets. These concepts were envisioned by Sugeno [74] in his efforts to compare membership functions of fuzzy sets with probabilities. Since no direct comparison is possible, Sugeno conceived the 84 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral generalization of classical measures to fuzzy measures as analogous to the generalization of classical (crisp) sets to fuzzy sets [44,50,58,59,60,79,85,106]. Specifically, fuzzy measures are obtained by replacing the additivity requirement of classical measures with weaker requirements of monotonicity and continuity. 7.2.1 F u z z y m e a s u r e Fuzzy measures are generally considered as subjective scales for fuzziness [27,74]. These are set functions that satisfy monotonicity but do not necessarily possess additivity. Let X be a nonempty set, £ be a nonempty class of subsets of X , and / / : £ —» [0,°o] be a nonnegative, extended real-valued set function defined on £ . Definition 1. (Sugeno [85]): JU is called a fuzzy measure on {X, £) iff (if and only if) 1) fiiff)) = 0 when (/> e £ (vanishing at (/)); 2) Ee£, Fe£, and EczF imply / / (£) < //(F) (monotonicity); 3) {En }a£, ExazE2cz...., and U~=i En e £ imply \imju(En) = ju n=l (continuity from below); 4) { £ „ } c £, EX^E2 3..., n(Ef>< °°, and f) En e £ imply n=\ r HmJu(En) = ju n n=\ (continuity from above); 7.2.2 X - F u z z y m e a s u r e Definition 2: (Sugeno [85]): JU is called a A -fuzzy measure on £ iff it satisfies the a - A - rule on £ and there exists at least one set Es £ such that fi{E) < °°. Definition 3: (Sugeno [85]): (a) JU satisfies the "A- rule" (on £ ) iff there exists 1 sup// 85 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral where sup// = supEe iju(E), such that ji(EuF)=/i(E)+/i(F)+A.fi(E)fi(F) (7.1) whenever Ee£,Fe£, EuFei, and EnF = 0. (b) JU satisfies the "finite A - rule " (on £ ) iff there exists the above-mentioned A such that -|n[i + A . / z ( £ j - i j , for 2 * 0 , (7.2) for 2 = 0, for any finite disjoint class {£1,...,£'„}of sets in £ whose union is also in £ . (c) ju satisfies the "finite a—A — rule" (on £ ) iff there exists the above-mentioned A such that I k V'=' 7 2 n[i + A. / / (£ ; ) ] - i , for 2 * 0 , (7.3) 1.=/ for 2 = 0, for any disjoint sequence {£„} of sets in £ whose union is also in £ . Usually the A -fuzzy measure is denoted by g%- When £ is a o - algebra (Appendix B) and g^(X) = 1, the A -fuzzy measure g^ is called a Sugeno measure. Theorem 1: (Sugeno [85]): If g^ is a A -fuzzy measure on a class ^containing the empty set 0 , then gA (<p) = 0, and satisfies the finite A -rule. If X ={JC, , . . . ,*„} is a finite set, £ consists of X and all singletons of X , and fd is defined on £ such that //({jc t}<ju(X)<°o for i=l,2,...,n, and there are at least two points and x ;-2 86 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral satisfying //(j*,-. j)> 0, j=l, 2, then such a set function JU is always a X -fuzzy measure on £ for some parameter X. When fJ,{X) = Z " = i / ^ ( { ^ } ) > the parameter X=0; otherwise A can be determined by the following equation : 1 n(i+^//({^}))-i i= l (=1 (2) A = 0when £//({*,• })=//(*). i= i (3) 7 ^ 7 < ^ < 0 when £//({*,-})>//(X). M X ) i=l When JU(X) = 1, we can determine A by solving the following equation: (7.4) Theorem 2: (Sugeno [85]): Under the condition mentioned above, the equation 1 + X.ju(X) = ft [ l + M f c } ) ] (7-5) i = i determines the parameter X uniquely. In particular, (1) X > 0 when £ ju({x{ })< / / (Z ). l + l = n[lrl//fe})] (7.6) 1=1 7.2.3 Fuzzy integral Definition 4. (Sugeno [85]): Let A e F , / e F . The/wzzy integral of / on A with respect to / / , is denoted by : jAfdfi= sup [otAfi(AnFa)]. (7.7) ae[0,°°] 87 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral When A = X , the fuzzy integral may also be denoted by Ifdju. The fuzzy integral is also called Sugeno integral. It is sufficient to define a fuzzy integral for all a e [0,l]. Suppose that (X , F) is a measurable space, where X e F , ju : F —> [0,°°] is a fuzzy measure, and that F is the class of all finite nonnegative measurable functions defined on (X , F). For any given / £ F, we write Fa = { X | / ( J C ) > a} where a e [0,°°]. The set Fa is the well-known a-cut of / . 7.2.4 Information fusion using "fuzzy integral" Consider the problem of information fusion in computer vision [12,43]. An object can be recognized and classified by integrating information from a variety of sources. This information can be used to increase the confidence in object classification. Many types of features such as shape measures, texture measures, and statistical measures may be determined through sensing and signal processing. Since, usually there is no single feature that can identify an object of interest, it would be useful to utilize several features. There is always uncertainty inherent in the recognition problem. Each feature or group of features can be considered as evidence in the identification of an object. Obviously, each of these features or group of features would have a degree of importance in the identification process of the object. Fuzzy integral has been used in the recognition problem and classification integration. For improved clarity we now present the definition of fuzzy integral using the notation given in [77]: Definition 5. (Sugeno [37,77]) Let (X ,Q) be a measurable space and let h : X —> [0,1] be a im- measurable function. The fuzzy integral over A c X o f the function hwith respect to a fuzzy measure g is defined by !Ah(x)og(.)= sup [mm(a,g(AnFa))] (7.8) ae[0,l] where Fa = {x : h(x) > a}. (7.9) 88 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral In [77] X has been interpreted as a set of sources of information (sensors); h(x)e [0,1] as the decision for the object when source xe X is considered; and the fuzzy measure g({x}) (density function) as the degree of importance of this source (sensor). Corresponding to each x{, there is a degree of importance gl, which gives how important xt is in the recognition of the class tk, where tkeT = {*i,*2>—>*m}, and T is a set of classes of interest. These densities can be subjectively assigned by an expert, or can be generated from training data. The method for "evidence fusion" [77] combines objective evidence (in the form of a fuzzy membership function), with subjective evaluation of the worth of the sources with respect to the decision. The following example illustrates the approach. Example 1: Consider the two classes: T = feeautiful, mediocre] as applied to the appearance of a person. Keller [77] interprets X as a set of features and h(x)e [0,1] as the decision for the person's appearance when feature XG X is considered. For class one: = {beautiful} the feature set is A = ̂ eyes,nose,lips} with X\ = eyes, x2 =nose, x^-lips. Then, hx (xj )=0.8, h\{x2}=§n /ZI(JC3)=0.68 correspond to the class f] = {beautiful}. Note that hk : X —»[0,l] is a partial evaluation of a person for class tk . In other words, hk ( X j ) is an indication of how certain one is in the classification of the person (the object) to be in class tk. Using the feature xi, where a value of 1 indicates absolute certainty that the object (the person) is in class tk and a value of 0 implies absolute certainty that the object is not in class tk, Keller interprets the fuzzy measure g({*;}) as the degree of importance of the feature xi (density function) for evaluating the person. These densities g({.x.})can be subjectively assigned by an expert, or can be generated from training data. Suppose that we are given; g({Xl})=0.9 g({x2})=0.6 g({x3})=0.85 89 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral In order to make a decision at this level, the class tk with the largest integral value, is chosen. The fuzzy integral can be realized as follows [77]: e = max [min(/i(^-), g(At))] (7.10) j=i for the given example, minfo (x, ) g{xx ))= min(0.8,0.9)= 0.8 min(/i] (x2 ), g(x2 ))= min(0.7,0.6)= 0.6 min (ft, (x3 ) g(x 3 ))= min (0.68,0.85)= 0.68 It follows that e = max(0.8,0.6,0.68)= 0.8 Accordingly, it may be concluded that the appearance of person is "beautiful" with a fuzzy integral value e = 0.8. Next, suppose that / i 2 (xi)=0.3 / i 2 (x 2 )=0.35 /i i(x 3 )=0.2 corresponding to class two t2 - mediocre. Then, with the same density values g({xj) as before, we have in in^fa) , g{xx))= min(0.3,0.9)= 0.3 mm (h2 (x2 ) g(;c2 ))= min(0.35,0.6)= 0.35 mm(h2 (x3 ) g(x 3 ))= min(0.2,0.85)= 0.2 we get e = max(0.3,0.35,0.2)= 0.35 It follows that the appearance of the person is considered mediocre with a fuzzy integral value e = 0.35. Note that if the goal is to make a decision at this level, then class tk with the largest integral value is chosen. In this simple example, the largest value of fuzzy integral is e = 0.8. Consequently, the class corresponding to the value "beautiful" is chosen. In the present example, the subsets of A = fel,x2,x3} given by {xi}{x2}{x3}are used. In other words, the effect of each 90 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral individual feature alone is considered. The fuzzy measure corresponding to each individual feature is given by: gl=g(M)=o.9 * 2 = s(fe})=0.6 g3 = g({x3})=0.85 This is called the fuzzy measure g . However, in the case where all the subsets are considered, as given by: {x]}{x2}{x 3},{r ],X2}{x:],X3},{x2,X3}{x 1,X2,X3} then, all combinations of these features have to be used. In this case, the g^ - fuzzy measure is calculated as follows: Step 1: Determination of X X can be found according to Theorem 2, Eq(7.5) and (7.6) as given below : ;i+i=n(i+V) (7.ii) (=1 / For the present example then, X + \ = ^ + Xg{\ + Xg2\ + Xg*) or, X +1 = (1 + 0.9A X l + 0 ' . 6 / lX l + 0.85^) we get X = 0,-0.995,-2.955 Now, since X > -1 and X 0, we pick X = -0.995 91 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral Step 2: Comparison of the density values g' This step is carried out as given in Table 7.1. Table 7.1 Calculation of the values of gl. Subset A fe} gl = 0.9 fe} g 2 =0 .6 fe} g 3 =0.85 fe >  x2 } gl+g2+Aglg2 =0.96 fe' x3 } g2+g3+Ag2g3 =0.94 fe' x3 } gl+g3+lglg3 =0.99 1.0 92 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral Step 3: Computation of the fuzzy integral e for class tx This step is carried out as given in Table 7.2. Table 7.2. Computation of the values of the fuzzy integral for class tx Subset A 8x(A) For the class tx hx min(/i, (Xi),gx'Ai)) fe} g 1 =0.9 0.8 0.8 fe} g 2 =0 .6 0.7 0.6 fe} g 3 =0.85 0.68 0.68 fe ' x2 } gl+g2 + Xglg2 =0.96 0.85 0.85 fe. x3 } g2+g3+Ag2g3 =0.94 0.9 0.9 fe ' x3 } gl+g3+Aglg3 =0.99 0.95 0.95 {c],x 2 x3\ 1.0 0.88 0.88 n e = maxtminCfyCK.O.g^A,-))] = 0.95 «'=1 93 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral Step 4: Computation of the fuzzy integral e for class t2 This step is carried out as given in Table 7.3. Table 7.3. Computation of the values of the fuzzy integral for class t2 • Subset A *A0*) For the class f 2 h2 min(/i 2 (x,) ,g A (A ( )) fe} gl=0.9 0.3 0.3 fe} g2=0.6 0.35 0.35 fe} g 3 =0.85 0.2 0.2 fe' x2 } g1+g2+Ag1g2 =0.96 0.51 0.51 fe' x3 } g 2 + g 3 + Z g 2 g 3 =0.94 0.35 0.35 fe' x3 } gl+g3+Aglg3 =0.99 0.1 0.1 1.0 0.2 0.2 n e = max (min(/z2 (x{ ),gPi (A,-)) = 0.51 It follows that the class "beautiful" is recognized with e = 0.95 and the class "mediocre" is recognized with e = 0.51. Note that, if the goal is to make a decision at this level, the class tk with the largest integral value should be chosen. Keller used this technique for both object classification and multisource integration [77]. 94 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral 7.3 Development of the inference technique Inspired by Keller's technique for evidence fusion [77], a new inference method is proposed for knowledge-based decision making in multi-input-multi-output (MIMO) systems. The present approach uses the concepts of fuzzy measure and fuzzy integral as defined in the previous section. As mentioned in the beginning of the chapter, the development of the present technique was originally motivated by the requirements of an industrial fish cutting machine that has been developed in the Industrial Automation Laboratory [17] of the University of British Columbia ( see Chapter 1, Section 1.1). A hierarchical structure has been developed for this machine, in order to deal with large amount of information that is generated [21, 84]. In the developed system, the low level information is processed and transformed into a high level linguistic form. At the top level of this hierarchy, a knowledgebase receives the preprocessed (fuzzy) information from the lower levels. In order to improve the performance of the system, the top level knowledgebase should combine the information from various sources and make a control decision, in fuzzy form. The status of the machine is monitored through fuzzified response measurements from various sensors. The rules in the knowledge base are extracted from human knowledge and expertise. However, expecting a human to be able to relate more than two variables to one control action at a time is not realistic. To explain the basis of the present method in general terms, consider a multi-input-multi-output (MIMO) system with the four input variables: xl,x2,x3,x4 and the two control actions (output variables): yx,y2- If there are 3 fuzzy states : (small, medium, large) for each input variable, then in a complete rulebase there will be mn = 3 4 = 81 rules, where m is the number of fuzzy states in each input variable and n is the number of fuzzy input variables [24]. Instead of building these 81 rules using domain experts, which may not be practical, the present approach builds the knowledgebase in the form of several single-input-single-output (SISO) fuzzy rulebases and then fuse them using the fuzzy integral. 95 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral First, the knowledge base is divided to two (number of outputs) sections, as shown in Figure 7.1. The modules of rules in sections 1 and 2 run independently, since the outputs are different. Section 1 gives all the SISO rules corresponding to one output (y ; ) and Section 2 gives all the SISO rules corresponding to the second output (y 2 ) . Each section consists of a number of modules which depends on the number of input fuzzy variables. In our example there are four input variables, and accordingly, there are four modules in each section to relate each of the input variables to the particular output. Each module contains the rules that relate just one input fuzzy variable to only one output. Each module consists of several rules which depend on the number of fuzzy states in the particular input variable. For the case of three fuzzy states (for example: low, medium, high) for each variable, there will be at most three single-input-single-output rules in a module. In most practical cases, these rules are based on heuristic decisions of human experts. Since a human usually finds it difficult to make decisions simultaneous for more than three states, assigning three states for each fuzzy variable is quite reasonable and practical. Section 1 Section 2 Module 1: IF JC. is , THEN y\ is .... Module 2: IF x2 is . THEN y\ is .... Module 3: JFx3 is ,. THEN y\ is .... Module 4: IF x4 is . THEN y\ is .... Module 5: IF x1 is ,, T H E N yi is Module 6: IF x2 is . T H E N y2 is .... Module 7: IF x3 is , T H E N yi is .... Module 8: IF x4 is , T H E N yi is .... Figure 7.1. The decomposition of the MIMO rulebase into two SISO modules. The foregoing discussion indicates that, instead of dealing with 81 rules in the form of a MIMO system, it is desirable to deal with the 8 modules shown in Figure 7.1, each module of which containing just 3 rules (for a fuzzy variable having three fuzzy states), giving a total of 24 SISO rules. Note that in the present example, four (number of modules) different decisions are made for each output. Combining these decisions and arriving at the final decision is an important consideration of the new inference method developed in this chapter. In particular, an 96 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral approach is needed to decouple the MIMO (4 inputs and 2 outputs in the present example) knowledge base into several sections (which depends on the number of outputs) of SISO modules. The fuzzy integral is realized through the standard max-min composition of the two series {h(xt)} and {g(x,)} for each i,i = 1,2,..., n [77]; specifically, e = max[minW*,-), <?(4)) ] (7-12) j=i where At = {xx,x1,...,xi} and g is a g%-fuzzy measure. The values of g(A,)are determined recursively as: g(Ai) = g'{xl}) = gl (7.13) g(Ai) = gi + g(Ai_l) + Agig(Ai_x) for 1< i < n (7.14) According to Keller's interpretation of fuzzy integral (see Example 1), if one considers the fuzzy input variables , x 2 > *3 > x 4 as the features or sources of information and y,, y2 as the required changes for the input signals to the actuators, then density g can be interpreted as the degree of influence of the contribution of each module in achieving the overall goal of the system. Degree of influence is a subjective decision of a human expert. Specifically, a knowledge engineer (KE) would ask a human expert (HE): " i f we vary the input signal to the actuator y • by some amount, what would be its influence in improving the response variable xt ? In the context of a complex control system (MIMO), the density function g is taken to be the degree of influence of the contribution of each SISO module in achieving the overall goal of the system. It may be possible to build MISO sub-rulebases (say, with two inputs and one output). Then the degree of influence of the contribution of each MISO module in achieving the overall goal of the system, for this module will be a g^ - fuzzy measure, where A is computed using Eq (7.11) and g^ is computed using Eq (7.14), as in Step 3 of Example 1. 97 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral Example 2: Consider a process with three response variables X\,x2,x3 , which are fuzzified. A human expert relates each fuzzy variable to a control action yj in a decoupled (SISO) manner with a degree of importance gl for each modules. Two cases are examined. Case 1 as shown in Figure 7.2(a) is an example of where the human expert is able to relate all the variables of the knowledgebase in a SISO manner. Case 2 as shown in Figure 7.2(c) gives a situation where in addition to the SISO modules, the expert uses a rule of thumb for the two inputs and the single output, which will contribute to the general inference. Figure.7.2 (b) shows the schematic diagram of the inference in Case 1 and Figure.7.2 (d) shows the schematic diagram of the inference in Case 2. Table 7.4 presents the membership of the output for Case 1 and Table 7.5 presents the comparison of the methods of computing inference for this case. Table 7.6 presents the membership functions of the outputs for Case2 and Table 7.7 presents the computation of inference for this case. This example clearly demonstrates the application of the inference method in the two cases. Note that the modules shown in the two cases only show the rules that are fired, not all the rules in the module. For instance, if one of the variables has three fuzzy states, three rules have to be considered for that module; but, depending on the design of the membership functions and the way they overlap different number of rules may be fired. In the present example, each crisp value can be translated to two fuzzy variables; therefore, only two rules would be fired. 98 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral Module 1 IF x\ is x\ T H E N y\ is yf IF x\ 2 IS X] T H E N y, is yl Module 2 IF 2 x2 is x2 T H E N y\ ^ yf IF x2 is x\ T H E N y\ is yf Module 3 IF x3 is x\ T H E N y\ is y\ IF x3 2 is x3 T H E N y\ 3 is yx Figure 7.2(a). Rulebase modules for case 1. 99 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral Figure 7.2(b) Schematic representation of inference in Case 1. Table 7.4. Membership functions of the outputs for Case 1. Output State 1 y\ Output State 2 yl Output State3 yl Module 1 8l tl(y\) = (l(x}) M(yi) = fUl) Module 2 82 M(y?) = M<4) M(yl) = M(4) Module 3 83 Overall Conclusion => yl yl yl 100 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral Table 7.5. Comparison of the methods of computing inference in case 1. New Inference Method Conventional Inference Method yl max [min(g 1, //(xf)), min( g3, ju(x\))] max[n(x?),ju(x\)] yl m a x C m i n ^ 1 , / / ^ 1 ) ) ^ ^ ^ 2 , / / ^ ! ) ) ] max[ju(xl),ju(x2)] yl max[min(g 2,//(x|)),min(g3,//(x32))] max[//(^),//(x2)] Case 2: Module 1 IF X\ is l x \ THEN y\ 2 is y, IF Xi is x\ THEN y\ is y\ Module 2 IF 2 x2 1 S x 2 THEN y\ is y, 2 IF 3 x2 is x2 THEN y\ 3 is yx Module 3 IF x3 is x\ THEN y \ i s i IF x3 2 IS X 3 THEN ?1 is yl Module 4 IF X] is AND x2 2 is x2 THEN y\ is IF X] is A AND x2 3 is x2 THEN yi is Figure 7.2 (c). Rulebase modules for case 2. 101 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral Figure 7.2 (d ). Schematic representation of the inference in case 2. 102 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral Table 7.6. Membership functions of the outputs for case 2. Output State 1 y\ Output State 2 Output State 3 Module 1 g 1 M(yf) = M(4) Module 2 82 ju(yf) = ju(xl) M y ? ) = M * i ) Module 3 g 3 M y ? ) = M ^ 3 ) Module 4 8x ju(yf) = mm(ju(xf),ju(xi)) {i(y3) = mm(p:(xl),ju(xl)) Overall Conclusion => A yl yl Table 7.7. Computation of inference in case 2. New Inference Method 1 yc max [min(g1, ju(xf)), min( g3, fi(x\))] 2 yc max[min(g1, M*i)), min(g 2 , M * 2 ))> min(g A , min(//(x 2), //(*f ))] yl max[min(g 2,//(^)),min(g 3,^(x 3 2)),min(gA,min(//(x 1 1),//(^))] In Table 7.6 , g A is calculated using definition 3 and Eq (7.14). Equation(7.11) is used to determine A. 103 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral Table 7.8. Computation of the conventional inference in case 2. Conventional Inference Method yl max[ju(xf),ju(x\)] yl max[ju(xl),ju(xl),mm(ju(x2),ju(xl))] yl max[n(x2),H(x2),mm(ll(xl),il(xl))] The two inference methods given in tables 7.7 and 7.8 cannot be directly compared. Specifically, the inference method that is developed here uses a structured rulebase method for dealing with many fuzzy variables in the form of SISO modules. In summary, the new inference method may be represented in the general form : m julc = et = max[min(g J,/4)] (7.15) u=i^—.— (7.16) Mc where m is the number of module, j is a variable which represents various modules, and i is a variable which represents each state of the output. A crisp control action can be computed using the centroid method. 7.3.1 Universal approximation theorem Suppose that the input universe of discourse U is a compact set in Rn. Then, for any given real continuous function g(x) on U and arbitrary e >0, there exists a fuzzy system f(x) such that s u p | / « - g ( x ) | < £ (7.17) xeU 104 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral The proof relies on the Stone-Weirstrass theorem [ 66] . The universal approximation property of fuzzy systems guarantees the existence of a fuzzy system that can approximate any continuous function. 7.3.2 Stone-Weirstrass theorem Let Z be a set of real continuous functions on a compact set U. If (i) Z is an algebra, that is, the set Z is closed under addition, multiplication, and scalar multiplication; (ii) Z separates points on U, that is, for every x,y e U,x* y, there exists / e Z such that f(x) * f(y); and (iii) Z does not vanish at any point of U, that is, for each XG U there exists f e Z such that / ( j t ) * 0 ; then for any real continuous function g(x)on U and arbitrary e >0, there exists f e Z such that s u P*6<y|/(*) -£(*)| < 8 • The universal approximation property for fuzzy systems shows that; if one works sufficiently hard at tuning, one should be able to make the system perform in a desired manner[57]. It is seen that the universal approximation theorem guarantees that there exists a way to define a fuzzy system f(x), but it does not show how to find that system. Many fuzzy systems are known to satisfy the universal approximation property. For example, in [86,87] it has been proved that a fuzzy logic control (FLC) system with Gaussian membership functions, the product fuzzy implication and the center of area defuzzification, is a universal approximator. In [7] it has been proved that modified Sugeno type fuzzy controllers are universal approximators. In [8] it has been shown that approximation is possible for all FLC systems, with both fuzzy consequents and nonfuzzy consequents. Essentially, then all F L C systems are universal approximators. The analytical proof in [8], guarantees that a fuzzy system with the inference method developed here is also a universal approximator. 105 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral 7.4 Experimental Investigation A fuzzy logic control system with the new inference method developed in this chapter is implemented on the prototype hydraulic manipulator (see Chapter 2, Section 2.3). The knowledge base of this system has two input variables (position error and velocity) and one output variable (current input to the servovalve of the hydraulic actuator). Velocity is calculated according to Eq. (2.1), as shown in Figure 2.2. A block diagram of the fuzzy knowledge-based controller (FKBC) used in the control of the hydraulic manipulator is shown in Figure 7.3. Reference position - f x ) Position error e Position Velocity Estimator Velocity Fuzzy Controller Plant Cutter Assembly Actual position • Figure 7.3. Block diagram of the fuzzy logic controller. The goal of the control system is to control the position of the cylinder in the hydraulic actuator. The membership functions of the input and output variables are shown in Figure 7.4. The centroid method is used to defuzzify the control action. Three cases are investigated where three different rulebases for the variable "velocity" are tested, each case having three different degrees of importance g in the contribution of modules. The results are shown in Figure 7.6 through Figure 7.8. For Case 1 we have two modules: Module 1 and Module 2(a). For Case 2, we have two modules: Module 1 and Module 2(b). For Case 3, we have two modules: Module 1 and Module 2(c), as shown in Figure 7.5. 106 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral x1: Position Error (mm) Figure 7.4. Membership functions of input and output variables. 107 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral Module 1 IF xl is N L T H E N yis PL TF xl is N M T H E N y is P M TF xl is NS THEN yis PS IFxZ is ZR T H E N y is ZR IF xl is PS THEN yis NS IF xl is P M THEN y is N M TF xl is PL THEN yis N L Module 2(a): Case 1 IF x2 is N L T H E N y is PL IF x2 is N M T H E N y is P M IFjc2is NS T H E N y is PS IF x2 is ZR T H E N y is ZR TEjc2is PS T H E N y is NS IF x2 is P M T H E N y is N M TF;c2is PL T H E N yis N L Module 2(b): Case 2 IFx2is N L T H E N y is P M IF x2 is N M T H E N y is PS IFx2is NS THEN y is PS IF x2 is ZR THEN y is ZR IF x2 is PS T H E N y is NS IF x2 is P M T H E N y is NS IFjc2is PL THEN yis N M Module 2(c): Case 3 IFx2 is N L THEN yis PS IFx2 is N M T H E N y is PS IF x2 is NS THEN yis PS IF x2 is ZR THEN y is ZR JF x2 is PS THEN yis NS W x2 is P M THEN yis NS TF x2 is PL THEN yis NS Figure 7.5. Modules of the rulebase in three cases of investigation. 108 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral Table 7.9. Output membership functions. Output Membership Functions (Current to the servovalve) (mA) Module 8 N L N M NS ZR PS P M PL 1 (xl) S i Ml Ml Ml ft Ml Mt Ml 2 (x2) 82 M'I M\ M\ M2 M\ Mi M\ output M' M2 M3 Ml M5 M6 M1 The control membership functions are computed using jJc = ei = max[min( c? ;,// ' ,)] 7=1 (7.18) where i is the number of states in the output and j is the number of modules of rules in the knowledgebase. Specifically, ju1 =max[min(g],//11)min(g2,//2)J mm(g],jUi)mm(g2,/u2J min(g 1,// 1 3)min(g 2,//2)] [nm(guju?)min(g2,ju%J (7.19) \mm(gl,rf)mm(g2,ju52\ m i n ^ . / z f j i n i n ^ . ^ l } ( ? i» i " i 7 )min (? 2 , / ^ ) ju = max ju3 - max 4 max max ju = max ,7 m i n ju = max[ Next, the centroid method is used to defuzzify the control action, according to 2 > r / 4 (7.20) 109 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral 7.5 Discussion From the results given in figures 7.6 through 7.8, the effect of changing the rulebase (tuning the rulebase) of the second module on the performance of the system is observed. Note that the inference method has been able to produce a good response. It is assumed that the rulebases of the modules are tuned properly, which is a common situation with fuzzy systems. In each of the cases, the effect of the degree of importance of the second module g2 within the contribution of each module is studied. In all the cases, the importance of the first module is considered to be gl=\. This is because the first module is about the rules related to position variable. In Figure 7.9, the case where the effect of the second module has been eliminated (gl = 1, g2 = 0) is compared with the other three cases. This result shows the contribution of the second fuzzy input variable, velocity, to the fuzzy controller. The contribution of the second module in Case 1 (Figure 7.6) results in an overshoot in the response. The other two cases have produced a smooth and damped response; however, the case gl = 1, g2 = 0 where the effect of the second module has been eliminated, shows a faster response. This simply illustrates the damping effect of the velocity contribution to the inference. However, on relative terms, the degree of influence of the second module as determined by the value of g2, is rather small. The method developed here provides a generalized and organized way of handling the problem of fuzzy decision making in complex multivariable systems. The technique uses a particular way to arrange the variables into single-input-single-output modules. It follows that there is no limitation as to how many variables can be handled by this technique. An important aspect of the present technique is that it provides the potential of using fuzzy logic for controlling multivariable systems, with reduced number of rules and increased speed of processing. 110 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral 111 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral (gi=g2=i) (g1=1 g2=0.6) (g1=1 g2=0.2) 30 r < 20 E. I 1° 3 o -10 • 1000 I 1 3 0 ! i 1 3 0 i . 1 _ m i . 1 _ m l . 500 1000 0 500 1000 0 500 1000 Time (ms) Time (ms) Time (ms) Figure 7.7. Module 1 and module 2(b) (case2). 112 Chapter 7. A Novel Fuzzy Inference Method based on Fuzzy Integral Q\ 1 1 1 1 1 1 1 1 1 I 0 100 200 300 400 500 600 700 800 900 1000 Time (ms) Figure 7.9.Comparison of the responses (gl-l, g2=0) 113 Chapter 8. Conclusion Chapter 8 Conclusion The theme of the research reported in this thesis is the efficient practical implementation of fuzzy logic in knowledge-based decision making, particularly in industrial process control. Considerations such as the inference scheme, knowledge base, membership functions and the defuzzification approach are all important in these applications, and have been investigated in the thesis. Fuzzy logic is known to be useful for information fusion and control of plants that are complex and the overall goal is subjective. The thesis specifically considered the application of fuzzy logic in the control of an industrial prototype of an automated fish cutting machine that has been developed in the Industrial Automation Laboratory, University of British Columbia. The research was motivated by the complexity of this machine, which consists of many sensors and actuators that has made it hard to model and control. The application of fuzzy logic on a hydraulic manipulator of the prototype fish cutting machine as a direct fuzzy controller was investigated. As an original contribution to the knowledge, a new inference method was developed, implemented, and tested on this machine. This concluding chapter summarizes the primary contributions of the research, outlines the significance of these contributions, and indicates possible directions of further research in the area. 8.1 Major Contributions The major contributions of this research are summarized below: 1. A novel inference method for multivariable fuzzy systems was developed, implemented on an industrial machine, and evaluated. This inference method is based on the concept of fuzzy integral. The technique can be embedded in any commercial expert system, particularly any direct fuzzy logic controller that has to deal with many fuzzy variables in a coupled rulebase. The feasibility and effectiveness of this technique were demonstrated by implementing it on a hydraulic manipulator of a prototype industrial fish cutting machine. 114 Chapter 8. Conclusion 2. A low level proportional plus derivative (PD)-type fuzzy logic controller was designed and implemented in realtime on the prototype hydraulic manipulator. Choice of membership functions, effect of tuning of the membership functions of inputs and outputs the effect of tuning the rulebase, and the choice among common defuzzification methods, were studied through experimentation and comparative evaluation. 3. A low level proportional (P)-type fuzzy logic controller was designed and implemented in realtime on the prototype hydraulic manipulator. In particular, two common methods of fuzzy inference—individual-rule-based and composition-based, were studied using this controller. The performance of the prototype system was studied in detail under these schemes of control inference. 4. A supervisory fuzzy control system containing an intelligent tuner for low-level controllers was developed and implemented in the prototype hydraulic manipulator. This control system automatically tunes the parameters of the conventional PD controllers which drive the hydraulic actuators of the prototype machine. This control system has three-layer hierarchical structure. In the bottom layer of the hierarchy, a conventional PD controllers are used to position the two-degree-of-freedom cutting module using hydraulic actuators. The middle layer monitors the performance of the position response, preprocesses the response signals, and extracts the performance parameters, based on a step-input response. The top layer infers the tuning actions for (the PD servo units of the actuators. The knowledge base for tuning the low level controllers was developed using control experience and represented by fuzzy rulebase modules. 8.2 Significance of the Research While this research originated from an analytical perspective, its results have significant practical value, which can impact a wide range of applications. In particular, the new inference method which was developed can be embedded in any commercial expert system. Applications include process control, professional consulting (medical, 115 Chapter 8. Conclusion legal, business, etc.) or any other application of fuzzy logic such as pattern recognition. Such applications deal with multivariable systems with high complexity. The problem of knowledge-based decision making in such a system can be greatly facilitated by the approach developed in the thesis. This research also investigated the performance of two well-known inference methods in fuzzy logic. Observations were made as to which inference method has particular merits in practical applications. The considerations of the importance of proper design related to the membership functions, rulebase, and the defuzzifications are also of practical significance. An intelligent tuner for the hydraulic manipulators of an industrial prototype of an automated fish cutting machine was developed. A system of this type may be appropriate as an intelligent supervisory control system for industrial machines, and can relieve the burden of informed decision making on the product quality and the machine performance from dedicated operators. 8.3 Directions for Further Research No research performed can be complete. There are several issues and aspects that should be further investigated in the context of the research reported in this thesis. The following are some directions for possible future research: The new inference method that is developed in the present work can be implemented on a variety of systems with multiple fuzzy variables. In that regard, an analytical study can be conducted to explore the effect of number of variables, number of fuzzy states for each variable in the fuzzy rules, with degree of influence of each variable on the output variable and its effect on the performance of the system. In this research, the min-max implication was used; however, the effect of different implication methods can be studied as well. The proportional plus derivative (PD)-type controller was developed based on the individual rulebased method. This controller may be realized through the compositional rule of inference and fuzzy relations too. Further analytical studies may be conducted and experimental investigations maybe carried out on this issue. Comparison of two well-known inference methods was carried out in the present work. It was observed that these two methods do not generate the same performance. For further research, the level of equivalence of these two inference methods may be analytically explored. 116 References 1. Ahmed, M . and Ward, R. (1999) "An Expert System for General Symbol Recognition", Pattern Recognition Journal, June. 2. Ahmed, M . 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F but it can be the member of that set with a certain degree of membership that is assigned to every x e X a value from the unit interval [0,1] instead of the two-element set {1,2}. The set that is defined based on such an extended membership function is called a fuzzy set. Definition A . l : Fuzzy Set. A fuzzy set F in a universe of discourse X is defined by a membership function juF which takes values in the interval [0,1]. Specifically this may be represented by the mapping, /uF: X—>[0,1]. When the universe of discourse X is continous, a fuzzy set F is written concisely as, F= ^juF(x)/x (A. l) When U is discrete, a fuzzy set F is represented as, F = ̂ fiF(xi)/xi (A.2) i=i Definition A.2: Support set The support set of a fuzzy set F is the crisp set formed by the collection of all elements xt e X such that fiF(X;)>0. 125 Definition A.3: Fuzzy Singleton A fuzzy set whose support set is a single point (or element) in X is referred to as a. fuzzy singleton. Typically but not necessarily, the associated membership grade is unity; fi F=l.O. A.2 Operations on Fuzzy Sets Definition A.4: Two fuzzy sets are equal (A = B) if and only if \/xeX: LtAx)=uJx) (A.3) Definition A.5: A is a subset of B (A e B) if and only if VxeX: fiA(x)<nB(x) (A.4) Definition A.6: A' is the complement of the fuzzy set A if and only if V x e X : fiA.(x) = l-MA(x) (A.5) Definition A.7: Union (OR) A[jB Consider two fuzzy sets A and B in the same universe X. Their union is a fuzzy set A U B. The membership function is given by: / ^ M W = m a x [ / / A ( 4 / / B W ] V x e X (A.6) Definition A.8: Intersection (AND) AC\B Consider two fuzzy sets A and B in the same universe X. Their intersection is a fuzzy set A f l B. Its membership function is given by: / / A n B W = min [ / / A (4 / / 8 (x ) ] VxeX (A.7) 126 Definition A.9: A triangular norm or T-norm denotes a class of binary operations that can represent the generalized intersection operation, which has the following properties : T: [0,1] x [0,1]-> [0,1] 1 - xTy = yTx (A.8) 2 - (xTy)Tz = xT(yTz) (A.9) 3 - x<y and w<z implies xTw<yTz (A. 10) 4 - xTl = x and JCT0 = 0 ( A . l l ) Definition A.10: A Triangular Co-norm or S-norm denotes a class of binary operations that can represent the generalized union operation which has the following properties: the first three criteria of the T-norm are the same as S-norm but xSO = x . Definition A . l l : Cartesian Product of Two Fuzzy Sets Let A be a fuzzy subset of a universe of discourse U, and let B be a fuzzy subset of a possibly different universe of discourse V. The Cartesian product of A and B as denoted by Ax B is the set of all possible ordered pairs constructed by combining elements of A with elements of B such that the first element in each pair is a member of A and the second element is a member of B . Its membership function is defined by: MAxB(u>v) = min[/ilA(u),jLiB(v)] V « e £ / , V v e V (A. 12) A.3 Classical Relations A relation can be considered as a set of tuples, where a tuple is an ordered pair. A binary tuple is denoted as (JC, y); a ternary tuple is (x, y,z); and an n-ary tuple is ( x , , x 2 x n ) . Example A . l : Let X be the domain of men {Adam, Bernhard, Charles} and Y the domain of women {Diana, Eva, Francis}, then the relation "married to" on Xx Y is : {(Adam, Eva),(Bernhard, Francis), (Charles, Diana)}. 127 Consider the domain of natural numbers. Then < is a relation on N x 7Y defined as {(m,n)\ m<n}. Therefore, this relation contains pairs like, (2,3), and (5,18). But the pair(4,l) is not in this relation . A ternary relation on N3 is { (p,q,r) I p2 +q2 = r2}. This relation contains elements like (3,4,5) and (5,12,13). A.4 Fuzzy Relation The former section stated that a relation can be considered as a set of tuples. Extending this, a fuzzy relation is a fuzzy set of tuples, where each tuple has a membership degree between 0 and 1. Fuzzy Relations are fuzzy sets defined on universal sets which are Cartesian products. Definition A.12: Let U and V be continuous universes, and juR: U x V —» [0,1]. Then Example A.2: Let U = V ={1,2,3}. Then "approximately equal" is the binary fuzzy relation R (A.13) is a binary relation on U x V. If U and V are countable (discrete) universes, then R = ̂ juR(u,v) I (u,v). UxV 1/(1,1) + 1/(2,2) + 1/(3,3) + 0.8/(1,2) + 0.8/(2,3) + 0.8/(2,1)+ 0.8/(3,2) + 0.3/(1,3)+ 0.3/(3,1). (A. 14) The membership function juR of this relation can be described by 128 MR (x>y) 1 when x = y 0.8 when \x-y\=\ 0.3 when \x-y\=2 (A.15) In a matrix notation this can be represented as in Table A. 1. Table A . 1: Degree of membership of relation "R" Y 1 2 3 1 1 0.8 0.3 X 2 0.8 1 0.8 3 0.3 0.8 1 Example A.3: Consider a fuzzy rule: if e is PB and e is PS, then u is NM where PB, PS and NM are fuzzy sets, defined on the universes of discourse (domains) E,AE,AU, respectively. This rule is often represented by the ternary fuzzy relation (three dimensional relation) R given by R= J min(// PB (e), // PS (e), fj, NM («)) / (e, e, u) ExAExAU (A. 16) Each triple (e,e,u) has a membership degree equal to the minimum of juPB(e),jups(e) A N D MNMW- 129 A.5 Operations on Fuzzy Relations Fuzzy relations are very important in fuzzy control because they can describe interactions among fuzzy variables. This is quite important in the case of if-then rules. Let us first define two binary relations R and S (in tables A.2 and A.3), where as: R = " x is considerably larger than y " S - " y is very close to x " Table A.2: Degree of membership relation "R" yi y 4 0.8 l 0.1 0.7 *2 0 0.8 0 0 X3 0.9 1 0.7 0.8 Table A.3: Degree of membership relation "S" y\ yi V4 0.4 0 0.9 0.6 0.9 0.4 0.5 0.7 x3 0.3 0 0.8 0.5 Definition A.13: Let R and S be binary relations defined on X x Y. The intersection of R and S is defined by : V(x,y)e XxY: juRnS(x,y) = juR(x,y) T jus(x,y) (A.17) where T is the T-norm operation. Since minimum operation is the most common operation in the category of T-norm, the intersection operation may also be defined as: MRns (*> y) = m m ( ^ R U> y). Ms y)) (A-18) However, any other operation that satisfies the T-norm conditions, can be used as well. 130 Definition A.14: Let R and S be binary relations defined on X xY. The union of R and S is defined by : V(x ,y )e XxY: juRuS(x,y) = juR(x,y) S jus(x,yj (A.19) where S is the S -norm operation. Since maximum operation is the most common operation in the category of S -norm, the union operation can also be defined as: MRvs(x>y) = max(iLlR(x>y) ,Ms(x>y)) (A-2°) However, any other operation that satisfies the S-norm conditions, can be used as well. Definition A.15: In the binary case, let R be defined on X xY. The Projection of R on Fcan be defined as follows: Proj R on Y = Jsup^ juR(x,y)/y (A.21) Y Instead of the sup (Supremum) operation which is necessary when X and Y are continuous, it is usual to deal with discrete domain using the maximum operation. In fact the projection operation brings a ternary relation (three dimension) back to a binary relation or a binary relation to a fuzzy set, or a fuzzy set; to a single crisp value. Example A.4: Consider the relation R as given before (see Table A.4). Table A.4: Degree of membership "i?" yt ? 2 > 3 v4 x{ 0.8 1 0.1 0.7 X2 0 0.8 0 0 0.9 1 0.7 0.8 Then the projection on X means that; xx is assigned the highest membership degree from the tuples ( * i , y\), ( * i , y2 )> (*i > v 3 ) a n d ( * i , y 4 ) ; i.e., 1 is the maximum of the first row. 131 x2 is assigned the highest membership degree from the tuples (x2, y,), (x2, y2), (x2, y 3 ) and (x2, y 4 ) ; i.e., 0.8 is the maximum of the second row. x3 is assigned the highest membership degree from the tuples (JC3 , y,), (JC3 , y2), (jt3, y 3) and (x 3 , y 4 ) ; i.e., 1 is the maximum of the third row. So we can obtain the fuzzy set Proj R on X = 1/*,+ 0.8/x2+ l/x3 (A.22) In the same way, the projection on Y can be taken by searching for the maximum of the four columns. We have Proj R on F= 0.9/y,+0.8/y2+0.7/y3+0.8/y4 (A.23) Definition A.16: In the binary case, let F be a fuzzy set defined on Y. The cylindrical extension of F on X x Y is the set of all tuples (x,y)e X x Y with membership degree equal to juF(y): ce(F)= \juF(y)Kx,y) (A.24) XxY In fact, cylindrical extension extends fuzzy sets to fuzzy binary relations; fuzzy binary relations to fuzzy ternary relations; etc. If we consider A as a fuzzy set defined on X , and R as a fuzzy relation defined on X x Y, it is obviously not possible to take the intersection of A and R. But, when A is extended to XxY, it will make the cylindrical extension possible. Example A .5: Consider the fuzzy set A = Proj of R on X = 1 / + 0.8/*2 + 1/*3 (A.25) Its cylindrical extension on the domain X x Y is given in Table A.5. 132 Table A.5: Cylindrical extension of fuzzy set "A" , ce(A) y, yi y 3 >4 X, l l 1 1 x2 0.8 0.8 0.8 0.8 1 1 1 1 Consider the fuzzy set 73= Proj R on Y = 0.9/ y, +0.8/ y2 +0.7/ y 3 +0.8/ y4 (A.26) Its cylindrical extension on X x Y is given in Table A.6. Table A.6: Cylindrical extension of fuzzy set "B", ce(B) yt yi y 3 y* 0.9 l 0.7 0.8 x2 0.9 l 0.7 0.8 x3 0.9 l 0.7 0.8 Example A.6: Consider the relation R expressing "x is considerably larger than y" and suppose it is known that "x is small" by the fuzzy set: A = 0.3/*,+ Vx2+ 0.8/^3 (A.27) The combination of the fuzzy relation R and the fuzzy set A, expressed by "x is considerably larger than y A N D x is small" can be given by the intersection of the relation and the extension of A. The extension of A into X x Y is given in Table A.7. 133 Table A.7: Cylindrical extension of fuzzy set "A", ce(A) yi y-i > 3 ?4 X, 0.3 0.3 0.3 0.3 x2 1 1 1 1 X, 0.8 0.8 0.8 0.8 The intersection of R and ce(A) is given in Table A. 8. Table A.8: Intersection of relation "R" and the cylindrical extension of fuzzy set "A" , n ce(A) y\ yi y3 v 4 x{ 0.3 0.3 0.1 0.3 x2 0 0.8 0 0 x} 0.8 0.8 0.7 0.8 The combination of fuzzy sets and fuzzy relation with the aid of the cylindrical extension and the projection is called "Composition" and it is denoted by " ° " . Definition A.17: Let A be a fuzzy set defined on X and R be a fuzzy relation defined on 1 x 7 . Then the composition of A and R resulting in a fuzzy set B defined on Y is given by: B = Ao R = Proj(ce(A)nR) on Y (A.28) or, if intersection is performed with the minimum operation and projection with the maximum, then / / B (y) = max min(p:A(x),p:R(x,y)) (A.29) X This is called the max-min composition. 134 A.6 Implicat ion ( I F - T H E N ) : A - > fl Consider a fuzzy set A in a universe X and a second fuzzy set fl in another universe Y. The fuzzy implicat ion A —> B is a fuzzy relation in the Cartesian product space 1 x 7 . Different methods for determining the implication operation have been proposed in literature. Some examples are given below: 1. Mamdani's method (Logical Product), A —> B = AAB 2. Algebraic Product A -> B = AB 3. Bounded Product A -> fl = 0 v (A + fl -1) A asfl = l 4. Drastic Product A -> fl = < fl asA = l 0 otherwise 5. Zadeh's Method (Lukasiewicz's implication method),A—>fl = 1 A (1 - A + fl) 6. Boolean Logic Implication A—> B = ( l - A ) v f l fl A<B 7. Goedel Logic Implication A —> fl = ^ s F [fl A > f l fl A < f l 8. Goguen's Implication A —> fl = <̂ 5 F l f l / A A > f l The first four methods use " v " operation to aggregate the individual conclusions, however the second four methods use " A " operation to aggregate the individual conclusions. The first method is the most common one used to express the implication in fuzzy relations. A.7 The Composit ional Rule of Inference If fl is a fuzzy relation from U to V, and A is a fuzzy subset of U, then the fuzzy subset fl of V which is induced by A is given by the composition ofR and A ; that is fl=Aofl ( A . 3 0 ) 135 Therefore, we should take the cylindrical extension of A, take the intersection with R, and project the result onto the Y-axis. A.8 Modus Ponens (Deduction) If statement "IF A T H E N B" is true and we know that A is true too, then we can infer that B is true as well. A—>B A B (A.31) A.9 Approximate Reasoning The inferencing technique of fuzzy logic could be considered as a generalization of the classical rule of implication of modus ponens. A major difference from the modus ponens based on binary logic is that in the generalized modus ponens, we can use a different fuzzy set for A in premise 1 and A' (NOT A) in premise 2. In binary logic, crisp set A in premise 1 and premise 2 must be completely identical. In other words, in the generalized modus ponens based on fuzzy logic, fuzzy sets A and A ' i n premise 1 and premise 2 do not have to be precisely the same. We can still infer the conclusion that B'is true. For this reason, fuzzy reasoning is called "approximate reasoning". 136 Appendix B Definition.Bl. A family Q of complex functions defined on a set E is said to be an algebra i f : (i) f + g^Q (ii) fgeQ (iii) cfeQ for all / e Q , g e Q and for all complex constants c; that is, if Q is closed under addition, multiplication, and scalar multiplication. Definition.B2. For each positive integer k, let Rk be the set of all ordered k -tuples X= (xx,X2,.--,Xjc) , where real numbers, called the coordinates of x. The elements of R are called points, or vectors, especially when k >1. We denote vectors by boldfaced letters. If y = ( y i y k ) and if a is a real number, put x+y=(x1 + yl, ... ,xk + yk) a x=(a xj, ... ,a xk) so that x + y e Rk and ax e Rk. This defines the addition of vector, as well as multiplication of a vector by a real number (a scalar). These two operations satisfy the commutative, associative, and distributive laws and make Rk into a vector space over the real field. The "inner product" (or scalar product) of JC andy is given by k x.y= I ^ J i 1=1 and the norm of x by = (x.x )1/2 = 2>? v 1 J The structure defined in this manner (the vector space Rk with the above inner product and norm) is called Euclidean k -space. 137 Definition.B3. Let X be a metric space and E is a subset of X. £ is closed if every limit point of E is a point of E. E is open if every point of E is an interior point of E. Definition B4. A set X , whose elements we shall call points, is said to be a metric space if with any two points p and q of X there is associated a real number d(p,q) called the distance from p to q, such that (a) d(p,q) > 0 if p ± q; d(p,p) = 0; (b) d(p,q) = d(q,p); (c) d(p,q) < d(p,r) + d(r,q), for any re X . Definition B5. of a set £ in a metric space X we mean a collection { Ga } of open subsets of X such that £ c z U G e - ar Definition B6. A subset AT of a metric space X is said to be compact if every open cover of K contains a finite subcover. 138

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