Sufficient conditions in terms of linear matrix inequalities for guaranteed ultimately boundedness of solutions of switched Takagi-Sugeno fuzzy systems using the S-procedure

Abstract

• LMIs sucient conditions guarantee ultimate boundedness of the solutions of switched TS fuzzy systems. • A constrained switching law plays the role of the controller to assure a bounded solution of a class of non-linear systems. • The derivative of the used Lyapunov-like energy function can assume positive values in a bounded set described as level sets. • The LMIs explore the S-procedure to reduce the conservativeness of the results. • Estimates of the upper bounds on the derivative of the membership functions are used to obtain bounded sets. In this paper, sufficient conditions to ensure the existence of a switching law that makes the solutions of switched Takagi-Sugeno (TS) fuzzy systems ultimately bounded are developed by means of linear matrix inequalities (LMIs). These LMIs are based on the existence of a scalar function, which plays a role similar to Lyapunov energy functions for an auxiliary system formed by a convex combination of all subsystems of the switched system. A feature of the developed results is that the derivatives of the scalar function can assume positive values in a bounded set described as level sets. The LMIs explore the S-procedure to obtain low levels of conservativeness and do not require the calculation of the derivative of the membership functions, which facilitates their application to switched TS fuzzy systems with many rules. Exploring the proposed conditions, we estimated the attractor and basin of attraction of some examples of switched TS fuzzy systems under a measurable switching law. These numerical examples showed the effectiveness of the proposed approach in maximizing the estimation of the bounded attraction domain.