On Matrix Norms, Stability and Stabilization of a Class of Discrete Takagi–Sugeno Fuzzy Systems

Abstract

Based on a result of stability for linear time varying systems, a sufficient condition for global stability that does not use Lyapunov theory has been established for a class of discrete Takagi–Sugeno fuzzy systems. The condition involves testing the norms of products of the matrices of the consequences. The approach seeks to determine if at an instant of time the system becomes a contraction mapping. Because the computation burden associated with these tests may become prohibitive, a necessary and sufficient condition for local stability is derived. In a way, this decreases the computational burden and allows to customize the test. A two-step iterative stabilization algorithm is then introduced. The problem involves norm minimization and a posteriori stability test. Numerical examples are presented to demonstrate the applicability of the approach.