Tree-Based Algorithms for the Stability of Discrete-Time Switched Linear Systems Under Arbitrary and Constrained Switching

We present a direct approach to study the stability of discrete-time switched linear systems that can be applied to arbitrary switching, as well as when switching is constrained by a switching automaton. We explore the tree of possible matrix products, by pruning the subtrees rooted at contractions and looking for unstable repeatable products. Generically, this simple strategy either terminates with all contracting leafs—showing the system's asymptotic stability—or finds the shortest unstable and repeatable matrix product. Although it behaves in the worst case as the exhaustive search, we show that its performance is greatly enhanced by measuring contractiveness w.r.t. sum-of-squares polynomial norms, optimized to minimize the largest expansion among the system's modes.