Fixed-Time and Arbitrarily Fast Exponential Stabilization of Discrete-Time Switched Linear Systems

TL;DR

This paper develops conditions and constructive methods for fixed-time and arbitrarily fast exponential stabilization of discrete-time switched linear systems, applicable in both mode-dependent and mode-independent scenarios.

Contribution

It introduces a geometric approach and structural decomposition techniques to analyze and design controllers ensuring fixed-time stabilization regardless of switching sequences.

Findings

Conditions for fixed-time stabilizability derived

Constructive procedures for feedback gain computation provided

Fixed-time stabilizability shown to be equivalent to arbitrarily fast exponential stabilizability

Abstract

In this paper we first study the fixed-time stabilizability of discrete-time switched linear control systems. Using a geometric approach, we derive conditions under which such systems can be stabilized within a prescribed number of steps, independently of the switching sequence. We address both the mode-dependent case, where the controller has access to the active mode, and the mode-independent case, where a common feedback law must be employed. For each setting, we present constructive procedures to compute the stabilizing state-feedback gains. Building on these results, we then introduce a structural decomposition of switched systems, which serves to simplify stabilizability analysis and controller design. This allows us to establish the equivalence between fixed-time stabilizability and arbitrarily fast exponential stabilizability. The effectiveness of the proposed methods is illustrated through a numerical example.