Abstract: While the design methodology for fuzzy controllers has proven itself in certain commercial and industrial applications, there is a significant need to perform mathematical analysis of fuzzy control systems prior to implementation: (i) to verify and certify their behavior so that, for example, instabilities can be avoided for applications demanding highly reliable operation such as aircraft and nuclear reactor control, and (ii) to provide insight to the expert on how to modify the fuzzy controller to guarantee that performance specifications are met (e.g., to guarantee a specified rise-time or the absence of steady state tracking error). In this paper we provide a survey of, and an introduction to the area of nonlinear analysis of fuzzy control systems. We begin by overviewing several approaches to stability analysis including Lyapunov's Direct and Indirect Methods, and the Circle Criterion. We provide examples to illustrate how to design stable fuzzy control systems and test for stability, including an application of Lyapunov's direct method to Takagi–Sugeno fuzzy systems. Next, we introduce the idea of analyzing the steady state tracking error of a class of fuzzy control systems and provide examples of how to predict and reduce steady state error. Finally, we provide an introduction to the use of the describing function technique for the prediction of the existence, frequency, amplitude, and stability of limit cycles. We provide examples of limit cycle analysis and show how to design fuzzy controllers to avoid limit cycles. While our primary objective is to provide a control-theoretic introduction to, and survey of approaches to nonlinear analysis of fuzzy control systems where we utilize several existing results and provide useful tutorial examples, in the process we actually make contributions by providing, for example, the first results that show how to analyze steady state tracking error for fuzzy control systems.
