Local ℓ2-stabilization of nonlinear discrete-time systems with delayed states through T-S fuzzy models

Abstract

We propose the synthesis of local l2 stabilizing fuzzy state feedback controllers for nonlinear discrete-time systems with time-varying delay. The nonlinear system is described by Takagi-Sugeno (T-S) fuzzy models in a subset of the state space. The proposed conditions are formulated in terms of linear matrix inequalities (LMIs) concerning two fundamental issues: the disturbance effects in the nonlinear systems and the local domain of evolution of the nonlinear system, here called region of validity. In consequence, this proposal ensures that the closed-loop nonlinear system is input-to-state stable (ISS) in l2-sense and the resulting trajectories evolve only inside a contractive region. Therefore, we can guarantee that these trajectories never leave the region of validity used to construct the T-S fuzzy model. A fuzzy Lyapunov-Krasovskii (L-K) function candidate is used to develop the synthesis conditions. The closed-loop performance can be investigated through three optimization procedures: a) one that maximizes the allowable disturbance energy affecting the nonlinear system in closed-loop; and b) two procedures to minimize the disturbance effect on the closed-loop nonlinear system for a given size of disturbance. A numerical example is given to illustrate the proposal.