Relaxed Controller Design Conditions for Takagi–Sugeno Systems with State Time-Varying Delays

Abstract

This paper deals with the design of fuzzy controllers for Takagi–Sugeno (T-S) fuzzy models with state time-varying delays. New relaxed delay-dependent conditions for the stabilization purpose are proposed in terms of linear matrix inequalities (LMIs), including the knowledge of the bounds of the time-varying delay and its rate of variation. The conservatism improvement is brought through three points: (1) the choice of a convenient augmented Lyapunov–Krasovskii functional candidate, (2) the application of an extension of the Jensen’s inequality, and (3) the Finsler’s lemma. In this context, a parallel distributed compensation control law, which includes both memoryless and delayed state feedbacks, is considered. To apply such control law, it is required to assume that the time-varying delay is available online. Under this assumption, it is highlighted that the proposed LMI-based conditions are significantly relaxed for high rate of variation of the time delay. On the other hand, when this assumption cannot be guaranteed, straightforward corollaries are proposed. A numerical example is provided to illustrate the effectiveness of the proposed LMI-based conditions and their conservatism improvement regarding to previous results.