Ultimate boundedness sufficient conditions for nonlinear systems using TS fuzzy modelling

Abstract

Using Takagi–Sugeno (TS) fuzzy modelling, sufficient conditions to ensure ultimate boundedness of solutions of nonlinear switched systems are given. The sufficient conditions are given in terms of properties of invariant sets and of an auxiliary system formed by a convex combination of the switching subsystems. By exploring the results of this paper, estimates of the attractor and domain of attraction can be found even when (i) the derivative of an auxiliary function V, which plays the same role of a Lyapunov function, attains positive values in some sets and (ii) the solutions of each subsystem of the switched system are not necessarily ultimately bounded. The sufficient conditions are formulated as a problem of checking the feasibility of linear matrix inequalities (LMIs). Indeed, these LMIs provide a systematic procedure that can help to find auxiliary scalar Lyapunov-like functions for a class of switched nonlinear systems. A numerical example illustrates the effectiveness of the proposed approach in estimating attractors of nonlinear dynamic switched systems.