3 research outputs found
Converse Lyapunov theorems for discrete-time linear switching systems with regular switching sequences
We present a stability analysis framework for the general class of
discrete-time linear switching systems for which the switching sequences belong
to 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 stability
analysis.
We conjugate this tool with two families of multiple quadratic Lyapunov
functions, parameterized by an integer T >= 1, and obtain converse Lyapunov
Theorems for each.
Lyapunov functions of the first family associate one quadratic form per state
of the automaton defining the switching sequences. They are made to decrease
after every T successive time steps. The second family is made of the
path-dependent Lyapunov functions of Lee and Dullerud. They are parameterized
by an amount of memory (T-1) >= 0.
Our converse Lyapunov theorems are finite. More precisely, we give sufficient
conditions on the asymptotic growth rate of a stable system under which one can
compute an integer parameter T >= 1 for which both types of Lyapunov functions
exist.
As a corollary of our results, we formulate an arbitrary accurate
approximation scheme for estimating the asymptotic growth rate of switching
systems with constrained switching sequences