Stabilization of Linear Switched Systems with Long Constant Input Delay via Average or Averaging Predictor Feedbacks

TL;DR

This paper introduces average predictor-based feedback laws to stabilize linear switched systems with unknown future switching signals and delays, ensuring exponential stability under small parameter variations.

Contribution

It develops novel average predictor feedback laws for linear switched systems with delays, overcoming the challenge of unavailable future switching information.

Findings

Closed-loop systems are exponentially stable under the proposed laws.

Stability depends on small differences in system matrices and controller gains.

Numerical simulations confirm the effectiveness of the average predictor approach.

Abstract

We develop delay-compensating feedback laws for linear switched systems with time-dependent switching. Because the future values of the switching signal, which are needed for constructing an exact predictor-feedback law, may be unavailable at current time, the key design challenge is how to construct a proper predictor state. We resolve this challenge constructing two alternative, average predictor-based feedback laws. The first is viewed as a predictor-feedback law for a particular average system, properly modified to provide exact state predictions over a horizon that depends on a minimum dwell time of the switching signal (when it is available). The second is, essentially, a modification of an average of predictor feedbacks, each one corresponding to the fixed-mode predictor-feedback law. We establish that under the control laws introduced, the closed-loop systems are (uniformly) exponentially stable, provided that the differences among system's matrices and among (nominal stabilizing) controller's gains are sufficiently small, with a size that is inversely proportional to the delay length. Since no restriction is imposed on the delay, such a limitation is inherent to the problem considered (in which the future switching signal values are unavailable), and thus, it cannot be removed. The stability proof relies on multiple Lyapunov functionals constructed via backstepping and derivation of solutions' estimates for quantifying the difference between average and exact predictor states. We present consistent numerical simulation results, which illustrate the necessity of employing the average predictor-based laws and demonstrate the performance improvement when the knowledge of a minimum dwell time is properly utilized for improving state prediction accuracy.