A Finite‐Time Criterion for Non‐Stabilizability of Discrete‐Time Switched Linear Autonomous Systems

ABSTRACT Existing methods based on state space covering often fail to determine the non‐stabilizability of discrete‐time switched linear autonomous systems within a finite number of steps. To address this limitation, this article proposes a novel approach that establishes a finite upper bound for the verification time of stabilizability. First, we rigorously define the convergence exponent for switched linear systems and establish its correspondence between the original system and a scaled‐down system. Second, we derive the relationship between the contraction regions induced by any switching path in both the original and the scaled‐down systems. This leads to an estimate of a lower bound on the maximum path length for a global norm‐contracting path set in the original system. By integrating this bound with the derived properties of the convergence exponent, we develop a verifiable criterion for determining the stabilizability of the original system. Furthermore, to facilitate the analysis of the scaled‐down system, we propose an efficient method to verify the stabilizability of systems with minimal non‐contraction regions. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed approach.