Membership-function-dependent stability analysis and local controller design for T–S fuzzy systems: A space-enveloping approach

Abstract

For Takagi–Sugeno (T–S) fuzzy systems, relaxed stability conditions are obtained by more effectively enveloping the trajectory of the membership functions (MFs) in a unified space of MF. Considering an open-loop T–S fuzzy system, the system premise variables operation domain can be divided into a series of subdomains. Based on an MF unified space extremum calculation technique, the MFs extremum values in each premise variable corresponding to the unified space subdomain are calculated. With these extremes, a tight local convex polyhedron enveloping the MF trajectory is constructed in each subdomain. Thus, linear matrix inequality (LMI) stability conditions are derived via a piecewise Lyapunov function. Then, a state-feedback local controller is designed to close the system loop. From a geometric viewpoint, MF extremum enveloping and piecewise linear-approximation methods are both utilized to achieve relaxed stability and robustness conditions. Finally, several examples are adopted to illustrate the metrics of the proposed approaches.