Stability of Fuzzy Control Systems
FKBC has been proven to be a powerful tool when applied to the control of processes which are not amenable to conventional, analytic design techniques. The design of most of the existing FKBC has relied mainly on the process operator’s or control engineer
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6.1 Introduction FKBC has been proven to be a powerful tool when applied to the control of processes which are not amenable to conventional, analytic design techniques. The design of most of the existing FKBC has relied mainly on the process operator's or control engineer's experience based heuristic knowledge. Hence, the controller's performance is very much dependent on how good this expertise is. Thus, from the control engineering point of view, the major effort in fuzzy knowledge based control has been devoted to the development of particular FKBC for specific applications rather than to general analysis and design methodologies for coping with the dynamic behavior of control loops. The development of such methodologies is of primary interest for control theory and engineering. In particular, stability analysis is of extreme importance, and the lack of satisfactory formal techniques for studying the stability of process control systems involving FKBC has been considered a major drawback of FKBC. Fuzzy control systems are essentially nonlinear systems. For this reason it is difficult to obtain general results on the analysis and design of FKBC. Furthermore, the knowledge of the dynamic behavior of the process to be controlled is normally poor. Therefore, the robustness of the fuzzy control system must be studied to guarantee stability in spite of variations in process dynamics. In this chapter we consider several existing approaches for stability analysis of FKBC. In the fuzzy control literature this type of analysis of FKBC is usually done in the context of the following two views of the system under control: • Classical nonlinear dynamic systems theory: The system under control is a "non-fuzzy" system, and the FKBC is a particular class of nonlinear controller. • Dynamic fuzzy systems. The second view is associated with Zadeh's Extension Principle [237] (see Section 2.2.4) and so far is only of theoretical interest. Research in this area includes stability criteria based on the concept of energy or the controllability offuzzy systems [76, 117]. The work presented in this chapter corresponds to the first view. We use the control structure shown in Fig. 6.1, where the FKBC is represented by means of D. Driankov et al., An Introduction to Fuzzy Control © Springer-Verlag Berlin Heidelberg 1993
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6. Stability of Fuzzy Control Systems
u
PROCESS
x
I
Set of RUlesl (x)
Fig. 6.1. FKBC in a closed-loop.
a nonlinear function u = tP(x) . As it has been shown in Garcia-Cerezo et al. [73] and in Section 4.2.2, the FKBC is a nonlinear transfer element represented by the function tP(x). Then, the structure of Fig. 6.1 can be used to analyze the dynamic behavior of the closed-loop system. One of the first works dealing with FKBC closed-loop analysis is that of Tong [215]. The analysis is based on the "relation matrix," which is a discrete version of the fuzzy relation , R, or fJR representing the meaning of the rule base. The nonlinear function tP( x) can be computed from R by means offuzzification, composition-base
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