Stability criterion for Takagi-Sugeno models

This paper provides a conducive criterion to investigate the stability of unforced continuous-time T-S fuzzy models in case of having second-order state matrices. According to the Tanaka and Sugeno theorem, the stability of a Takagi-Sugeno (T-S) fuzzy model can be proved by finding a common symmetric positive-definite matrix P for all the subsystems satisfying set of Lyapunov inequalities. But in many cases, especially by increasing the number of fuzzy rules, seeking such a common matrix seems to be a daunting task. In this paper, a simple sufficient stability criterion in terms of Lyapunov function candidate is proposed to investigate the global asymptotic stability of continuous-time T-S fuzzy models. In this case, the region of existence for the common positive-definite matrix P is investigated. It will be shown that choosing any arbitrary point from the obtained common region leads to a matrix P which can fulfill the relevant set of Lyapunov inequalities. In other words, finding any common region for such a matrix concludes the asymptotic stability of the system. This approach has the merit of specifying the existence or non-existence of the common matrix P for a fuzzy system. Several examples are given to illustrate the effectiveness of the proposed method.