The Exponential Stability and Asynchronous Stabilization of a Class of Switched Nonlinear System Via the T-S Fuzzy Model

In this paper, we investigate the problem of exponential stability and asynchronous stabilization for a class of switched nonlinear systems. The Takagi and Sugeno (T-S) fuzzy model is employed to approximate the subnonlinear dynamic systems. With two-level functions, namely, crisp switching functions and local fuzzy weighting functions, we introduce continuous-time switched fuzzy systems, which inherently contain the features of the switched hybrid systems and T-S fuzzy systems. By the use of delicately constructed piecewise Lyapunov-like functions (PLFs) and minimum dwell time method, we obtain the exponential stability of the switched fuzzy systems, which allows us to have stable and unstable nonlinear subsystems. In practice, for the control problem, it inevitably takes some time to identify the system modes and apply the matched controller, the asynchronous phenomena between the system modes switching and the controllers switching generally exists. Based on the result of stability, the fuzzy state feedback controller under asynchronous switching is proposed for switched fuzzy systems. In addition, the lower bound of minimum dwell time can be obtained using convex optimization such that the switched fuzzy system can be exponentially stabilized if its minimum dwell time is larger than the bound. The stability results and control laws of the switched fuzzy systems are formulated in the form of linear matrix inequalities that are numerically feasible. Finally, two illustrated numerical examples are presented to show the effectiveness of the obtained theoretical results.