New Approaches to Fuzzy Modeling and Control

The following sections are included:

• Preface

• Contents

Our introductory chapter provides some background material in fuzzy logic and fuzzy control. We concentrate on the issues relevant to set the stage for the following chapters. We also give a brief preview of the main topics considered in this monograph.

In this chapter we present the fuzzy Lyapunov synthesis method. Basically, it is an extension of the classical Lyapunov synthesis method to the domain of computing with words. Given only fuzzy knowledge about the plant to be controlled, and the control objective, the proposed method enables us to systematically derive the rule-base for a suitable fuzzy controller.

In classical Lyapunov synthesis the controller is designed through an attempt to make some candidate function V a Lyapunov function. Closed-loop stability is automatically guaranteed and can be proven using the same V used in the design. In this chapter, we extend the same idea to the case of fuzzy controllers designed using our fuzzy Lyapunov synthesis method by presenting two alternatives to stability analysis. The first combines fuzzy Lyapunov synthesis with classical Lyapunov stability analysis. The second is concerned with performing Lyapunov analysis in the domain of words.

In this chapter we describe a systematic approach to the design of adaptive fuzzy controllers. The controller's rule-base is determined through fuzzy Lyapunov synthesis. The closed-loop system is monitored and the controller's parameters are adapted based on the error between the desired and actual behavior of the Lyapunov function associated with the system.

In this chapter we pose and solve an inverse optimal problem for fuzzy controllers. This leads to a new type of state-space model and a new form of cost functional. Once the fuzzy controller is shown to be an optimal controller, analytical properties, such as closed-loop global stability and robustness, are easy to prove.

We consider a new approach to fuzzy modeling which is based on the formulation of a novel state-space representation consisting of hyperbolic functions. The new representation, designated the hyperbolic state-space model, combines the advantages of fuzzy system theory and classical control theory. On the one hand, it is easily derived from a set of Mamdani-type fuzzy rules. On the other hand, as we will show in the next chapter, given a hyperbolic state-space model it is possible to design for it a fuzzy controller guaranteeing closed-loop stability and robustness.

In this chapter we develop methods for designing fuzzy controllers for the hyperbolic state-space model. Specifically, we present three methods: diagonal stabilization, nonlinear H₂ control, and nonlinear H_∞ control that can be utilized to design nonlinear fuzzy controllers for the hyperbolic state-space model. The resulting controllers guarantee closed-loop stability and robustness. Because they are fuzzy, each is equivalent to a set of Mamdani-type fuzzy rules. We use this linguistic interpretability to linguistically compare and analyze the controllers. In the H_∞ setting, this allows a linguistic interpretation of the optimal controller and optimal disturbance.

Please refer to full text.

The following sections are included:

• Appendix A Mathematical Background
  • Introduction
  • The Lipschitz Condition
  • Lyapunov Stability
  • Passivity
  • Optimal Control via the HJB Equation
  • Optimal Control via Circuit Analysis
    • Linear passivity and optimality
    • Nonlinear passivity and optimality
  • The Integral-Invariance Principle
  • Notes and References

• References

• Index