Nonconvex stabilization criterion for polynomial fuzzy systems

Abstract

This paper presents a nonconvex criterion represented in terms of bilinear sum of squares (SOS) conditions for the stabilization of polynomial fuzzy systems. In the existing literature dealing with the stabilization of polynomial fuzzy control systems, the construction of Lyapunov function candidates is restricted in order to transform the nonconvex SOS conditions into convex SOS conditions. Moreover, the transformation from nonconvex conditions into convex conditions itself also induces some conservativeness for the stabilization of polynomial fuzzy control systems. This study proposes a stabilization criterion for polynomial fuzzy control systems remaining in the nonconvex form of bilinear SOS conditions. Therefore, the proposed nonconvex SOS stabilization criterion, in some way, can provide more relaxed results than the existing convex SOS stabilization criteria. Moreover, the restriction on the construction of Lyapunov function candidates in literature does not exist in the proposed nonconvex stabilization criterion. An iterative algorithm is utilized to obtain solutions of the nonconvex stabilization criterion represented in terms of bilinear SOS conditions. The iterative algorithm does not guarantee either to obtain the global optimum of the nonconvex criterion or even to converge. However, this algorithm is easy to implement and can yield feasible solutions efficiently in the design example of this work.