Inference And Stability Analysis Of Fuzzy Systems

In 1965, Zadeh, an expert of control theory in American, createdfuzzy set theory and proposed fuzzy control technology. From then on,the power of fuzzy theory has been displayed. Fuzzy control theory isunder development and meanwhile, fuzzy control technology has beensucceed in applying to industry process control and new-type householdelectrical appliances. Unfortunately, the fuzzy theory itself still exits somedrawbacks that give rise to much controversy. Consequently, it is veryimportant to further study and perfect the fuzzy theory.In the paper, three main contributions of fuzzy theory are listed asfollows.1) Fuzzy reasoning. The reason, "why the compositional rule ofinference does not have the reducibility and cannot conduct the fuzzyreasoning correctly when Rz operator is used", is analyzed andmeanwhile, the conditions of using CRI correctly are pointed. Both thedrawback of the triple-I algorithm and the reason "why that algorithm hasthe reducibility" are analyzed. In addition, all kinds of implicationoperators of fuzzy reasoning are compared and accordingly, the reasons,"why some operators are unreasonable and others can be reasonable", areanalyzed and summarized. As a result, the correct inference algorithm canbe obtained. Meanwhile, the contradiction between the theory and theapplications of fuzzy reasoning can be explained. To thoroughly analyzethe fuzzy reasoning is very significant for the research of fuzzy theory,the design and applications of fuzzy controllers and the guidance of fuzzytheory for fuzzy controllers.2) Study on approximation ability of nonlinear function. TheStone-Weierstrass theorem is utilized to prove that any continuousnonlinear function can be approximated by fuzzy systems with arbitrarymembership functions, rules of inference and center average defuzzifier.That is theoretical basic to apply fuzzy systems to system identification.In addition, its correctness is validated by simulation examples.3) Stability analysis of fuzzy systems. Local stability of the T-Sfuzzy systems is clearly analyzed and then the deficiency of sufficientcondition of global asymptotic stability of fuzzy system is found.Meanwhile, the condition is proved not to be necessary condition. Themembership functions of fuzzy sets are substituted in fuzzy systemexpressions, a special nonlinear discrete system will be obtained. Thestability region is determined from three aspects: the analysis region,stability criterion condition and stability range. The instability criterion ispresented with regard to instable systems. Simulation example verifiedthe validity of the method. Whether the system is stable at infinity isirrelevant to practical applications, so the local stability criterion is ofgreat value around the stability point of systems.