Stability Analysis and Control of Two-Dimensional Fuzzy Systems With Directional Time-Varying Delays

Abstract

This paper is concerned with the problem of stability and stabilization of two-dimensional (2-D) Takagi–Sugeno fuzzy systems described by the Roesser model with state delays. The directional state delays are unknown and time-varying, which belong to certain ranges with known upper and lower bounds. The common center-average defuzzifier approach is first utilized to present the defuzzified output dynamics of 2-D fuzzy rule-based systems. Then, in light of a 2-D Lyapunov-like scheme, we construct a suitable augmented fuzzy parameter-dependent Lyapunov functional candidate to establish stability conditions for the open-loop system. Based on some novel 2-D summation inequalities, which generalize the ones based on Wirtinger inequality, Abel lemma, or Cauchy–Schwarz inequality, and by utilizing a reciprocally convex combination technique, delay-dependent stability conditions are derived in terms of tractable linear matrix inequalities. The obtained results on stability analysis are then utilized to address the problem of designing a rule-based state feedback controller that makes the 2-D fuzzy closed-loop system stable. The effectiveness and advantage of the obtained results is illustrated through numerical examples.