Fuzzy adaptive control of a certain class of SISO discrete-time processes

In this manuscript, we address the problem of the stability of a certain class of SISO discrete-time processes controlled by an adaptive fuzzy controller, by using Lyapunov stability theory. These results were recently obtained for adaptive neural controllers, and are extended here to adaptive fuzzy controllers of Sugeno's type. In order to achieve tracking of a reference signal with this kind of fuzzy system, we allow both the membership functions and the consequent part of the rules to be adjusted by a parameter adaptation law. We first present the gradient-based (steepest descent) adaptation law, and we argue that this gradient-based adaptation law can be simplified dramatically. Thereafter, we show the asymptotic stability of the overall system (the convergence of the tracking error to zero) when using this simplified parameters adjustment law. Unfortunately, this result can only be proved when the outputs of the fuzzy controller can be expanded to the first order around the optimal parameter values that allow perfect tracking; that is, when the parameters are initialized not too far from their optimal values (local stability). However, when the set of tunable parameters is restricted to the set appearing in the linear consequent part of the rules (i.e. the membership functions of the premises are not modified) and when the reference signal is the delayed desired output, the stability result is strictly valid: the parameters do not have to be initialized around the perfectly tuned values. In this case, the algorithm can be simplified further by only considering the sign of the derivative of the output of the process in terms of its last influential input.