Extensional fuzzy logic controllers for uncertain systems

EXTENSIONAL FUZZY LOGIC CONTROLLERS FOR UNCERTAIN SYSTEMS LAI JUNWEI NATIONAL UNIVERSITY OF SINGAPORE 2007 Acknowledgement First of all, I would like to thank to my project supervisor Dr. Tan Woei Wan for her great guidance and assistance along the difficult research road. Her trust and patience are truly appreciated when I encountered difficulties in my research. Her insight into different aspects of control engineering and fuzzy logic theories has helped to solve many problems and fine-tune many important ideas. I have also learned a lot from her since joining the university. I would also like to express my sincere and heartfelt gratitude to my wife and my son Elwin. During the long time of thesis revision, I may not be able to perform my husband role very well to take care of my wife when she was pregnant. She always gives me a good environment to concentrate on my thesis writing, even in the first month after my baby was born. I am forever grateful to my loving parents, I have to thank to their consistent support and endless love. Thanks for their assistance in taking care my wife and my son, I can settle down to concentrate on my research and thesis writing during the recent year. It is my immense pleasure to dedicate this small accomplishment to my family. Last but definitely not least, I would like to take this opportunity to express my gratitude to my colleagues for their camaraderie and friendship. Over the four years, we have shared together and this is always one of the most enjoyable and impressionable period in my life. Contents List of Figures x List of Tables xi Summary xii Introduction 1.1 Uncertainty in the Real World . . . . . . . . . . . . . . . . . . . . . . 1.2 Historical Review on Fuzzy Control . . . . . . . . . . . . . . . . . . . 1.3 Extension to Type-1 Fuzzy Logic Theory . . . . . . . . . . . . . . . . 1.3.1 Non-singleton type-1 fuzzy logic systems . . . . . . . . . . . . 1.3.2 Type-2 fuzzy logic systems . . . . . . . . . . . . . . . . . . . . 1.3.3 Recent research in type-2 fuzzy controllers . . . . . . . . . . . 1.4 Aims and Scope of the Work . . . . . . . . . . . . . . . . . . . . . . . 1.5 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 11 Theories on Extensional Fuzzy Logic 13 2.1 Singleton Type-1 Fuzzy Logic Systems . . . . . . . . . . . . . . . . . 13 2.2 Realization of PID Control Using Type-1 FLSs . . . . . . . . . . . . . 17 2.3 Non-singleton Type-1 Fuzzy Logic Systems . . . . . . . . . . . . . . . 20 2.4 Type-2 Fuzzy Logic Theories . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.1 Type-2 membership functions . . . . . . . . . . . . . . . . . . 24 2.4.2 Embedded type-2 and type-1 sets . . . . . . . . . . . . . . . . 27 i Contents ii 2.4.3 Operations of type-2 fuzzy sets . . . . . . . . . . . . . . . . . 30 2.4.4 Centroid of type-2 fuzzy sets . . . . . . . . . . . . . . . . . . . 31 2.4.5 Properties of the centroid for an interval type-2 set . . . . . . 33 2.4.6 Type reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.7 Interval type-2 fuzzy logic systems . . . . . . . . . . . . . . . 38 Non-singleton Type-1 Fuzzy Controller for Noise Rejection 3.1 3.2 42 Properties of Symmetric Triangular Non-singleton Fuzzifier . . . . . . 43 3.1.1 Case I: Support of X partially overlaps the support of S1 . . . 45 3.1.2 Case II: Support of X is a subset of the support of S1 . . . . 48 3.1.3 Case III: Support of S1 is a subset of X . . . . . . . . . . . . 49 Non-singleton Type-1 PI Fuzzy Controller . . . . . . . . . . . . . . . 50 3.2.1 Structure of non-singleton PI controller . . . . . . . . . . . . . 50 3.2.2 Structure of inference engine . . . . . . . . . . . . . . . . . . . 51 3.2.3 Characteristics of fuzzy PI controller using symmetric nonsingleton fuzzifier . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3 Non-symmetric non-singleton Fuzzifier . . . . . . . . . . . . . . . . . 58 3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.4.1 pH process in CSTR . . . . . . . . . . . . . . . . . . . . . . . 60 3.4.2 Performance of proposed controller . . . . . . . . . . . . . . . 63 3.5 Case Study: Thermal chamber . . . . . . . . . . . . . . . . . . . . . . 71 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Type-2 Fuzzy PI Controller with Adjustable Type-reduced Output 76 4.1 Realization of Type-2 Fuzzy PI Controller . . . . . . . . . . . . . . . 78 4.2 Analysis of Type-2 Fuzzy PI Controller . . . . . . . . . . . . . . . . . 82 4.3 Theorems on Properties of Centroids . . . . . . . . . . . . . . . . . . 83 4.4 Adaptive Algorithm for Type-reduction . . . . . . . . . . . . . . . . . 88 4.4.1 Switch point adjustment algorithm . . . . . . . . . . . . . . . 88 4.4.2 Derivatives of centroid with respect to switch points . . . . . . 92 Contents 4.4.3 iii Algorithm initialization . . . . . . . . . . . . . . . . . . . . . . 94 4.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.6 Comparison with Fuzzy PI Gain-scheduling Control . . . . . . . . . . 102 4.6.1 Uncertain parameters for pH neutralization process . . . . . . 105 4.6.2 Simulation results for pH neutralization process with uncertain parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.7 Case Study: Thermal chamber . . . . . . . . . . . . . . . . . . . . . . 110 4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 On-line Learning Algorithm for Type-2 Fuzzy-Neural Controller 115 5.1 Type-1 and Type-2 Fuzzy-Neural Systems—General Background . . . 116 5.2 Architecture of type-2 FNC . . . . . . . . . . . . . . . . . . . . . . . 118 5.3 Control Scheme of Type-2 Fuzzy-Neural Control System . . . . . . . 122 5.4 On-line Self-learning Algorithm for MF Variables and Weights . . . . 124 5.5 5.4.1 Weight update rules . . . . . . . . . . . . . . . . . . . . . . . 126 5.4.2 MF variables update rules . . . . . . . . . . . . . . . . . . . . 130 Case Study: pH Neutralization Process . . . . . . . . . . . . . . . . . 138 5.5.1 Performance of type-2 FNC with online weights and MF variables update . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.5.2 5.6 Performance of type-1 FNC . . . . . . . . . . . . . . . . . . . 155 Case Study: Thermal chamber . . . . . . . . . . . . . . . . . . . . . . 162 5.6.1 Performance of type-2 FNC . . . . . . . . . . . . . . . . . . . 163 5.6.2 Performance of conventional PI controller and type-1 FNC with 12 rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Conclusions and Future Work 178 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.2 Suggestions for Future Work . . . . . . . . . . . . . . . . . . . . . . . 181 Contents Appendix A iv Relationship between FOU and control surface 183 A.1 Control surface using type-2 triangles with uncertain base . . 186 A.2 Control surface using parallel type-2 triangles . . . . . . . . . 189 A.3 Control surface using type-2 triangles with uncertain peak . . 192 Appendix B Update rules for lower MF variables 196 Author’s Publications 199 Bibliography 200 List of Figures 2.1 Examples for type-1 fuzzy set and singleton . . . . . . . . . . . . . . 14 2.2 The structure of type-1 FLS . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 The fuzzy sets of fuzzy PID controller . . . . . . . . . . . . . . . . . . 17 2.4 Structure of a Type-2 rule-based FLS . . . . . . . . . . . . . . . . . . 24 2.5 Type-2 membership functions . . . . . . . . . . . . . . . . . . . . . . 25 2.6 Example of interval type-2 membership function . . . . . . . . . . . . 26 2.7 Upper or lower membership function and embedded fuzzy set . . . . . 28 2.8 Example of embedded type-2 fuzzy set . . . . . . . . . . . . . . . . . 29 2.9 Switch points for calculating the centroid . . . . . . . . . . . . . . . . 35 3.1 The structure of fuzzy PI controller 3.2 The antecedents of PD-like FLSs . . . . . . . . . . . . . . . . . . . . 52 3.3 Triangular non-singleton fuzzifier with small spread for e . . . . . . . 53 3.4 Triangular non-singleton fuzzifier with small spread for e˙ . . . . . . . 54 3.5 Triangular non-singleton fuzzifier with large spread . . . . . . . . . . 57 3.6 Rectangular nonsymmetric non-singleton fuzzifier . . . . . . . . . . . 59 3.7 Titration curve for a weak acid, strong base reaction . . . . . . . . . 62 3.8 The CSTR configuration with two influent streams . . . . . . . . . . 63 3.9 The control scheme for CSTR . . . . . . . . . . . . . . . . . . . . . . 64 . . . . . . . . . . . . . . . . . . 51 3.10 The details of e and e˙ of the proposed nonsymmetric non-singleton fuzzy PD plus integrator fuzzy controller at the steady state pH=8.5 v 65 List of Figures vi 3.11 Comparison of singleton type-1 PI controllers with moving average filters and non-singleton fuzzy PD plus integrator controller . . . . . 66 3.12 The pH responses of singleton PI controller and proposed non-singleton fuzzy controllers at different setpoints . . . . . . . . . . . . . . . . . . 67 3.13 The responses of proposed non-singleton fuzzy controllers with different v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.14 The responses of proposed non-singleton fuzzy controllers with different αf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.15 The responses of proposed non-singleton fuzzy controllers with different Bv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.16 Diagram of a thermal chamber . . . . . . . . . . . . . . . . . . . . . . 72 3.17 The responses of proposed non-singleton controller and conventional singleton controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.18 Control signals of proposed non-singleton controller and conventional singleton controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.1 The input and output fuzzy sets . . . . . . . . . . . . . . . . . . . . . 79 4.2 Lower and upper bounds of type-reduced output set . . . . . . . . . . 82 4.3 An example of type-2 fuzzy set . . . . . . . . . . . . . . . . . . . . . 89 4.4 An example of two Theorems for the particular type-2 fuzzy set . . . 90 4.5 The standard fuzzy set used in this chapter . . . . . . . . . . . . . . . 92 4.6 Illustration of equivalent gains for type-2 PI using the algorithm . . . 96 4.7 ITAEs of type-2 fuzzy PI controller in Monte Carlo uncertainty analysis 99 4.8 Histogram of ITAEs of type-2 fuzzy PI controller . . . . . . . . . . . 100 4.9 ITAEs of type-1 fuzzy PI controller in Monte Carlo uncertainty analysis100 4.10 Histogram of ITAEs of type-1 fuzzy PI controller . . . . . . . . . . . 101 4.11 Responses of three control systems when K = 0.9 , τ = 4.5. (a) The first step response; (b) Step response after adaptation. . . . . . . . . 103 Appendix B Update rules for lower MF variables Similar to the procedures for upper MF variables, the partial derivatives of q l , q l , q r and q r with respect to U Fj,i can be derived as: ∂q lh ∂U Fj,i ∂q lh ∂U Fj,i ∂q rh ∂U Fj,i ∂q rh ∂U Fj,i = = = = Nl2 Nl2 Nr2 Nr2 ∂f h ∂U Fj,i ∂f h ∂U Fj,i ∂f h ∂U Fj,i ∂f h ∂U Fj,i ∂Nl Nl − f h ∂U Fj,i ∂Nl Nl − f h ∂U Fj,i Nr − f h Nr − f h ∂Nr ∂U Fj,i ∂Nr ∂U Fj,i , h ∈ [1, · · · , L] (B-1) , h ∈ [L + 1, · · · , M ] (B-2) , h ∈ [R + 1, · · · , M ] (B-3) , h ∈ [1, · · · , R] (B-4) But now U Fj,i is associated only with the lower firing level f , the partial derivatives can be simplified as: ∂q lh ∂U Fj,i ∂q lh ∂U Fj,i ∂q rh ∂U Fj,i ∂q rh ∂U Fj,i = Nl2 = Nl2 = Nr2 = Nr2 M −f h ∂f k , h ∈ [1, · · · , L] Fj,i k=L+1 ∂U M ∂f h ∂U Fj,i Nl − f h R −f h k=1 ∂f h ∂U Fj,i ∂f k Fj,i k=L+1 ∂U ∂f k , h ∈ [L + 1, · · · , M ] (B-6) , h ∈ [R + 1, · · · , M ] ∂U Fj,i R Nr − f h k=1 196 ∂f k ∂U Fj,i (B-5) , h ∈ [1, · · · , R] (B-7) (B-8) Appendix B 197 Hence, ∂q lh ∂U Fj,i ∂q lh ∂U Fj,i ∂q rh ∂U Fj,i ∂q rh ∂U Fj,i = = = = ∂µF M Nl2 −f h Nl2 ∂µF n j,k ∂U Fj,i k=L+1 ·( µF ) ∂U Fj,i ·( Nr2 −f h Nr2 ∂µF µF N,h ) · Nl − f h k=L+1 N =1, N =j ∂µF R ∂U k=1 ∂U Fj,i n j,k ∂U Fj,i ·( µF ) , N,k N =1, N =j n j,k Fj,i ·( µF ) , N,k N =1, N =j R n j,h ∂µF M n j,h , N,k N =1, N =j ·( µF N,h ) · Nr − f h k=1 N =1, N =j ∂µF n j,k ∂U Fj,i ·( µF ) N,k N =1, N =j The notations of derivative vectors are also defined as:  ΨXoj   =    µF o j,1 .  ∂µF o  j,i  , µ o (Xj ) = o  Fj,i Fj,i ∂U  (B-9) µF o j,M and  ΨXj   µFj,1  . . =  .  µF  ∂µF  j,i  , µ (Xj ) =  Fj,i Fj,i ∂U  (B-10) j,M To revise them into concise forms, ∂q lh ∂U Fj,i ∂q lh ∂U Fj,i n X X = , s ∈ [L + 1, M ], IU j (s) = IU j (i) ΨXN (s) −f (h) ΨXj (s) · Nl s N =1, N =j  n    (h) · ΨXj (s)· ΨXN (h) · Nl − f (h) Ψ   Xj  Nl2  s N =1, N =j    n   X X X X   , IU j (h) = IU j (i); s ∈ [L + 1, M ], IU j (s) = IU j (i) ΨXN (s) = N =1, N =j  n     (h) (s) · , Ψ ΨXN (s) −f   Xj  Nl2  s N =1, N =j     X X j j  I (h) = I (i); s ∈ [L + 1, M ], I Xj (s) = I Xj (i) U U U U Appendix B ∂q rh n Nr2 = ∂U Fj,i 198 ΨXj (k) · −f (h) ΨXN (k) , N =1, N =j k X X k ∈ [1, R], IU j (k) = IU j (i)  n    ΨXN (h) · Nr − f (h) ΨXj (k)· ΨXj (h) ·    Nr2  N =1, N =j k    n   X X X X   , I j (h) = I j (i); k ∈ [1, R], I j (k) = I j (i) Ψ (k) ∂q rh ∂U Fj,i U U U U XN = N =1, N =j  n     ΨXN (k) −f (h) ΨXj (k) ·    Nr2  N =1, N =j k      I Xj (h) = I Xj (i); k ∈ [1, R], I Xj (k) = I Xj (i) U U U U , Finally, ∂ul ∂U Fj,i = Nl + L ∂ur wa = ∂U Fj,i a=1 M ∂q la ∂U Fj,i + wb b=L+1 R c=1 N =1, N =j n ΨXj (k) · = Nl ur · − Nr ΨXN (k) + N =1, N =j Nr n ws · ΨXj (s) · s ΨXN (s) − N =1, N =j n ΨXj (k) · k ΨXN (k) + N =1, N =j d=R+1 ∂q rd ∂U Fj,i M · · (B-11) n wk · ΨXj (k) · k ΨXN (k) N =1, N =j n ΨXj (s) · s ΨXN (s) N =1, N =j n wk · ΨXj (k) · k ΨXN (k) N =1, N =j X X X wd wc f (c) wd f (d) + Nr2 Nr2 d=R+1 ul · Nl Nr ∂U Fj,i + M − ΨXN (s) M ∂q rc wa f (a) wb f (b) + Nl Nl2 b=L+1 a=1 n s c=1 − ΨXN (s) N =1, N =j wc + L ws · ΨXj (s) · ΨXj (s) · k ∂U Fj,i n s R ∂q lb X (s ∈ [L + 1, M ], IU j (s) = IU j (i); k ∈ [1, R], IU j (k) = IU j (i)) Also, the gradient descent rule for updating U Fj,i is: U Fj,i (t) = U Fj,i (t − 1) + δ · e(t) · F F ∂ul ∂U Fj,i + ∂ur ∂U Fj,i , δ= η 2γ (B-12) where U Fj,i is either U Lj,i or U Rj,i which is determined by the value of input Xj . 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[...]... neural networks to learn the fuzzy control rules and the membership functions of a fuzzy logic control system The combination brings the low-level computational power and learning ability of neural networks into fuzzy logic systems to automate and realize the design of fuzzy logic control systems; it also provides the high level IF-THEN rule thinking and reasoning of fuzzy logic systems into neural networks... uncertain or complex relationship and examine the advantage of the extra freedom in type-2 fuzzy sets This thesis seeks to develop controllers utilizing extensional fuzzy logic theories, namely non-singleton fuzzy logic and type-2 fuzzy logic and evaluate these controllers performance on handling different kinds of uncertainty In view of the above discussion, the specific objectives are as follows: 1 To... Type-2 fuzzy logic and neural networks[56] Lee and Lin applied type-2 fuzzy neural systems with adaptive filter to nonlinear uncertain systems[ 39] Singh and et al also proposed a type-2 fuzzy neural model based controller for a nonlinear system[85] Wang, Chen and Lee developed a type2 fuzzy neural network to handle uncertainty with dynamical optimal learning[91] Excellent results were obtained for the... reasoning of fuzzy logic systems into neural networks 1.3 Extension to Type-1 Fuzzy Logic Theory In spite of the many applications utilizing type-1 fuzzy controllers, type-1 fuzzy set and fuzzy logic system (FLS) is not adequate for handling all kinds of uncertainty when constructing rule-based FLS[57] It is known that the uncertain knowledge used to construct a FLS may arise from the following sources:... type-2 fuzzy controller in handling uncertainty[83] Lin and et al designed a type-2 fuzzy logic controller for buck DC-DC converters[48] There are some other works that utilized neural based system to learn the parameters of type-2 fuzzy controllers since type-1 fuzzy neural systems have been successfully developed and applied in last decade Melin and Castillo designed an adaptive controller for non-linear... since the birth of fuzzy controllers for real systems in 1975[92] Mamdani and Assilian first established the basic framework of fuzzy controller based on Mam- Chapter 1 Introduction 3 dani fuzzy logic system (FLS) and applied the fuzzy controller to control a steam engine[52] Control of cement kilns was another early industrial application[21] Since the first consumer product using fuzzy logic was marketed... may also possibly be lacking Fuzzy sets, the foundation of fuzzy theory, were introduced forty years ago as a way of expressing non-probabilistic uncertainties[97] Since then, fuzzy theory has been applied to construct different kinds of fuzzy controllers to control systems where tradition methods may not have good results 1.2 Historical Review on Fuzzy Control Zadeh proposed fuzzy theory more than 40... improve the performance on minimizing the effect of uncertain information in the input 2 To develop an adaptive type-reduction method based on properties of centroid for an interval type-2 fuzzy logic controller to obtain a variable control surface To evaluate the performance of such a type-2 fuzzy logic controller with variable control surface to track a reference trajectory when the system are uncertain. .. type-2 fuzzy- neuro controller (FNC) using BP algorithm for updating the consequent and antecedent parameters online To evaluate the online performance of a type-2 fuzzy- neuro controller when it is applied to a nonlinear and uncertain systems Chapter 1 Introduction 11 Fuzzy neural network (FNN) system can be tuned both for neuron parameters and the structure of network, but the uncertain information... controller The fuzzy PID controller is now actually non-linear version of conventional PID controller The fuzzy PID controllers are generally superior to the conventional ones, particularly for higher-order, time-valued, and nonlinear systems, and for those systems that have only vague mathematical models which are difficult, if not impossible, for a conventional PID to handle Such nonlinear fuzzy PID controllers . Type-1 Fuzzy Logic Theory In spite of the many applications utilizing type-1 fuzzy controllers, type-1 fuzzy set and fuzzy logic system (FLS) is not adequate for handling all kinds of uncertainty when. distributions” [96]. Later, Zadeh formalized these ideas into the paper Fuzzy Sets”. The fuzzy logic theory introduced by Zadeh is also termed type-1 fuzzy logic. Since then, fuzzy logic theory has developed. Non-singleton type-1 fuzzy logic systems . . . . . . . . . . . . 5 1.3.2 Type-2 fuzzy logic systems . . . . . . . . . . . . . . . . . . . . 5 1.3.3 Recent research in type-2 fuzzy controllers . .