Local stabilization of nonlinear discrete-time systems with time-varying delay in the states and saturating actuators

Abstract

Although most of the control design methods assume unbounded control signals, real systems do have saturating actuators, which may degenerate closed-loop performance or even lead to unstable behavior. Additionally, the delay is generally almost ubiquitous in processes, that is also imposing performance and stability constraints. Our main contribution is to provide a controller design methodology for the stabilization of delayed systems under saturating actuators. Specifically, we address the design of non-Parallel Distributed Compensation (non-PDC) state feedback fuzzy control laws that locally stabilize a class of nonlinear discrete-time systems with state time-varying delay and saturating actuators. The proposed non-PDC control law depends on the current state xk and the state delayed by d¯ samples. Based on the Lyapunov–Krasovskii approach, we characterize the safe region of initial conditions through two sets: an ellipsoidal one for the current state vector, and another set for the delayed state vectors. Through two convex optimization procedures, we can maximize the estimate of the region of attraction of the closed-loop control system. Additionally, a relaxation method inspired by the Frank-Wolfe algorithm is introduced, yielding better estimates of the region of attraction. The achievements are compared with other finds in the literature, illustrating the efficiency of this proposal.