Delay-dependent local stabilization of nonlinear discrete-time system using T-S models through convex optimization

In this paper we develop convex delay-dependent conditions in terms of linear matrix inequalities (LMIs) for the synthesis of fuzzy stabilizing feedback controllers. The condition is developed from a novel Lyapunov-Krasovskii fuzzy function. We consider that the T-S fuzzy model represents the nonlinear system only inside a region of validity. Because of this, we determine a domain of stability inside the region of validity, such that the trajectories of the nonlinear system in closed-loop starting from this domain converge asymptotically to origin. The domain of stability is characterized through a Cartesian product of two sets, where the first one is used to treat the initial state vector at the sample k = 0, and the second set is used to treat the delayed state vectors and the difference between two sampling of the delayed state vectors. We also develop a convex optimization problem to compute the gains of the fuzzy controllers to maximize the domain of stability. Finally, we show an example to demonstrate the developed conditions.