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We present a stability analysis framework for the general class ofdiscrete-time linear switching systems for which the switching sequences belongto a regular language. They admit arbitrary switching systems as special cases.  Using recent results of X. Dai on the asymptotic growth rate of such systems,we introduce the concept of multinorm as an algebraic tool for stabilityanalysis.  We conjugate this tool with two families of multiple quadratic Lyapunovfunctions, parameterized by an integer T >= 1, and obtain converse LyapunovTheorems for each.  Lyapunov functions of the first family associate one quadratic form per stateof the automaton defining the switching sequences. They are made to decreaseafter every T successive time steps. The second family is made of thepath-dependent Lyapunov functions of Lee and Dullerud. They are parameterizedby an amount of memory (T-1) >= 0.  Our converse Lyapunov theorems are finite. More precisely, we give sufficientconditions on the asymptotic growth rate of a stable system under which one cancompute an integer parameter T >= 1 for which both types of Lyapunov functionsexist.  As a corollary of our results, we formulate an arbitrary accurateapproximation scheme for estimating the asymptotic growth rate of switchingsystems with constrained switching sequences.

Converse Lyapunov theorems for discrete-time linear switching systems with regular switching sequences