Abstract
It is known that a fuzzy modeling approach is effective for constructing a fuzzy controller based on some fuzzy models. The asymptotic stability of such a fuzzy system can be assured, when exsisting a common Lyapunov solution to the Lyapunov inequalities for each subsystem. In addition, the stability check of A_iA_j is conventionally used as a necessary condition for searching a common Lyapunov solution, where A_i is a stable system matrix for the i-th subsystem. We sometimes can not conclude whether the fuzzy system is globally asymptotically stable or not, because a common Lyapunov solution does not exist, though A_iA_j is stable.In this paper, we specialize the fuzzy system as a mixed system or a periodically time-varying system, where in the former case the rule confidences p_i(k) are assumed to have 0 < p_i(k) < 1,and in the latter case they are assumed to have p_i(k)=1 or p_i(k)=0 with the fixed configuration of a period L. It is shown that if the fuzzy system is asymptotically stable as a mixed system and a periodically time-varying system with a relatively long period, then the fuzzy system provides a stable behavior in the computer simulation, even though there exist no common Lyapunov solutions. Of course, if there exists a common Lyapunov solution, then the fuzzy system is said to be asymptotically stable as both special systems; in particular, it is asymptotically stable as a periodically time-varying systems with any period L.
