New Approaches to Fuzzy Modeling and Control

by Margaliot, Michael; Langholz, Gideon

• ISBN13: 9789810243340
• ISBN10: 9810243340
• Format: Hardcover
• Copyright: 2000-06-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

Fuzzy logic has found applications in an incredibly wide range of areas in the relatively wide range of areas in the relatively short time since its conception. It was invented by Lotfi Zadeh, a leading systems expert, so it is perhaps not surprising that system theory is one of the areas in which fuzzy logic has made a profound impact. Fuzzy logic combined with the paradigm of computing with words allows the use and manipulation of human knowledge and reasoning in the modeling and control of dynamical systems.This monograph presents new approaches to the construction of fuzzy models and to the design of fuzzy controllers. The emphasis is on developing methods that allow systematic design on the one hand and mathematical analysis of the resulting system on the other. In particular, the methods described allow rigorous analysis of the stability and robustness of the systems, which are crucial issues in control theory.The first theme of the book is a new approach to the system design and analysis of fuzzy controllers, given linguistic information concerning the plant and the control objective. The new approach, fuzzy Lyapunov synthesis, is a computing-with-words version of the well-known (classical) Lyapunov synthesis method.The second theme of the book is to show that fuzzy controllers are in fact solutions to a nonlinear optimal control problem. The authors formulate a novel nonlinear optimal control problem, consisting of a new state-space model -- referred to as the hyperbolic state-space model -- and a new cost functional and show that its solution is a fuzzy controller. This leads to a new framework for fuzzy modeling and control that combines the advantages of the fuzzyworld, such as linguistic interpretability, and of classical optimal control theory, such as guaranteed stability and robustness.

Table of Contents

Preface	vii
Prologue	1	(16)
Background	1	(7)
Preview	8	(7)
Fuzzy Lyapunov synthesis	9	(3)
Hyperbolic optimal control	12	(3)
Notes and References	15	(2)
Fuzzy Lyapunov Synthesis	17	(30)
Introduction	17	(2)
The Design Method	19	(1)
Putting it to Work: Design Examples	20	(24)
Stabilizing the inverted pendulum	20	(3)
Inverted pendulum tracking	23	(3)
Designing a fuzzy PID controller	26	(3)
Designing a dynamic fuzzy scheduler	29	(9)
Designing a TS-type fuzzy controller	38	(6)
Summary and Discussion	44	(1)
Notes and References	45	(2)
Fuzzy Lyapunov Synthesis and Stability Analysis	47	(12)
Introduction	47	(1)
Classical Lyapunov Stability Analysis	48	(5)
Fuzzy Lyapunov Stability Analysis	53	(3)
Summary and Discussion	56	(1)
Notes and References	57	(2)
Adaptive Fuzzy Controller Design	59	(14)
Introduction	59	(2)
Designing the Rule-Base	61	(1)
Adaptation	61	(2)
Example: Inverted Pendulum Control	63	(8)
Designing the controller's rule-base	65	(4)
Adapting the controller	69	(2)
Summary and Discussion	71	(1)
Notes and References	71	(2)
Inverse Optimality for Fuzzy Controllers	73	(30)
Introduction	73	(1)
Review of Optimality and Inverse Optimality	74	(4)
Stability and robustness of optimal controllers	76	(2)
The Scalar Case	78	(4)
N-th Order Systems	82	(6)
Stability	88	(1)
Robustness	89	(3)
Summary and Discussion	92	(2)
Notes and References	94	(1)
Appendix	94	(9)
Hyperbolic Approach to Fuzzy Modeling	103	(24)
Introduction	103	(2)
The Hyperbolic State-Space Model	105	(1)
Approximation Capabilities	106	(2)
The Modeling Process	108	(5)
On-Line Parameter Estimation	113	(13)
Summary and Discussion	126	(1)
Notes and References	126	(1)
Fuzzy Controllers for the Hyperbolic State-Space Model	127	(24)
Introduction	127	(1)
Diagonal Stabilization	128	(4)
Nonlinear H2 Optimal Control	132	(4)
Nonlinear H∞ Optimal Control	136	(7)
Linguistic Interpretation	143	(6)
Comparing the diagonally stabilizing and H2 controllers	144	(1)
Fuzzy game strategies	145	(2)
Fuzzy Lyapunov stability analysis	147	(2)
Summary and Discussion	149	(1)
Notes and References	150	(1)
Epilogue	151	(2)
Appendix A Mathematical Background	153	(24)
A.1 Introduction	153	(1)
A.2 The Lipschitz Condition	154	(2)
A.3 Lyapunov Stability	156	(5)
A.4 Passivity	161	(4)
A.5 Optimal Control via the HJB Equation	165	(2)
A.6 Optimal Control via Circuit Analysis	167	(7)
A.6.1 Linear passivity and optimality	167	(3)
A.6.2 Nonlinear passivity and optimality	170	(4)
A.7 The Integral-Invariance Principle	174	(2)
A.8 Notes and References	176	(1)
References	177	(6)
Index	183