Reduced‐Order Stabilizability Verification of Discrete‐Time Switched Linear Systems

ABSTRACT The stabilization problem for switched linear systems is a fundamental and long‐standing open problem that has attracted considerable attention in the literature. Previous works have shown that the problem is solvable if and only if there is a finite set of switching paths whose norm compressed regions cover the entire state space. In this work, we propose a reduced‐order verification of the state space coverage. First, we show that each compressed region is either connected or dual‐connected. Then, we show that the switched system is exponentially stabilizable if and only if the boundary intersection points of the compressed regions are norm compressed, which simplifies the verification from ‐dimensional to at least ‐dimensional. Next, we propose a computational method to determine the exact boundary intersection states via determined or underdetermined systems of quadratic equations. Finally, we present an illustrative example to demonstrate the advantages of the proposed method.