Relaxed stability condition for T-S fuzzy discrete system

It is well known that the stability condition, based on Lyapunov stability criterion, for a T-S fuzzy discrete system is to find a common P to satisfy all Lyapunov's inequalities of rules of the system. If the number of rules r of a fuzzy system is large, the problem for finding the common P to satisfy r inequalities is not easy, even using linear matrix inequality (LMI). In practical, when inputs are singletons, the fuzzy system can be represented by a set of local state space models, and the number of fired rules in a local region is always less than (at most equal to) r. Thus, using only one fixed common matrix P for satisfying all rules is not necessary. However some boundary problem will exist between local stability and global stability. This paper tries to relax the stability condition for T-S fuzzy discrete system and to conquer the boundary problem also.