Stable controller design for T-S fuzzy systems based on Lie algebras

In this paper, we study the stability of fuzzy control systems of Takagi-Sugeno-(T-S) type based on the classical theory of Lie algebras. T-S fuzzy systems are used to model nonlinear systems as a set of rules with consequents of the type x˙(t)=Alx(t)+Blu(t). We conduct the stability analysis of such T-S fuzzy models using the Lie algebra LA generated by the Al matrices of these subsystems for each rule in the rule base. We first develop our approach of stability analysis for a commuting algebra LA, where all the consequent state matrices Al's commute. We then generalize our results to the noncommuting case. The basic idea here is to approximate the noncommuting Lie algebra with a commuting one, such that the approximation error is minimum. The results of this approximation are extended to the most general case using the Levi decomposition of Lie algebras. The theory is applied to the control of a flexible-joint robot arm, where we also present the decomposition procedure.