We introduce a novel framework for the stability analysis of discrete-time linear switching systems with switching sequences constrained by an automaton. The key element of the framework is the algebraic concept of multinorm, which associates a different norm per node of the automaton, and allows to exactly characterize stability. Building upon this tool, we develop the first arbitrarily accurate approximation schemes for estimating the constrained joint spectral radiusρ, that is the exponential growth rate of a switching system with constrained switching sequences. More precisely, given a relative accuracy r > 0, the algorithms compute an estimate ofρ within the range [ρ, (1 + r)ρ]. These algorithms amount to solve a well defined convex optimization program with known time-complexity, and whose size depends on the desired relative accuracy r > 0.

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A Path-Complete Lyapunov Function is an algebraic criterion composed of a finite number of functions, called its pieces, and a directed, labeled graph defining Lyapunov inequalities between these pieces. It provides a stability certificate for discrete-time switching systems under arbitrary switching. In this paper, we prove that the satisfiability of such a criterion implies the existence of a Common Lyapunov Function, expressed as the composition of minima and maxima of the pieces of the Path-Complete Lyapunov function. The converse, however, is not true even for discrete-time linear systems: we present such a system where a max-of-2 quadratics Lyapunov function exists while no corresponding PathComplete Lyapunov function with 2 quadratic pieces exists. In light of this, we investigate when it is possible to decide if a Path-Complete Lyapunov function is less conservative than another. By analyzing the combinatorial and algebraic structure of the graph and the pieces respectively, we provide simple tools to decide when the existence of such a Lyapunov function implies that of another.