  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 radius r, that is the exponential growth rate of a switching system
with constrained switching sequences. More precisely, given a relative accuracy
a > 0, the algorithms compute an estimate of r within the range [r; (1 + a)r].
These algorithms amount to solve a well defined convex optimization program
with known time-complexity, and whose size depends on the desired relative
accuracy a > 0.
