Relaxed stability conditions for T-S fuzzy systems

It is well known that the general stability condition for a T-S fuzzy system is A_i^T P + PA_i < O (for continuous systems) or A_i^T PA_i - P < O (for discrete systems), i=1,2,...,r, where r is the number of system's rules. If rules' number r of the fuzzy system is large, the problem for finding the common P to satisfy r inequalities is not easy, even Linear Matrix Inequality (LMI) is used. However, in practical, when inputs are singletons, the number of the fired rules at the instance is always very less than (at most equal to) r. Those rules, which are not fied, have zero fired grade membership values. Therefore it is not necessary to consider them into the system's stability condition. This paper will investigate the problem to relax the stability condition, that is, the common P only needs to satisfy h inequalities instead of r inequalities, where h(≤r) is the number of the fired rules by each input sets. Thus, the new and relaxed stability condition is established.