A new methodology for the stability analysis of pairwise triangularizable and related switching systems
Abstract
We consider the asymptotic stability of the time‐varying dynamic system ष : x˙ = A ( t ) x , A ( t ) ∈ R n × n , A ( t ) ∈ A = { A 1 , …, A m }, where A i is Hurwitz and where a set of non‐singular matrices T i j exist such that any pair of matrices { T i j A i T i j −1 , T i j A j T i j −1 }, i, j ∈ {1, …, m }, are upper triangular. Switching systems of this form are referred to as pairwise triangularizable switching systems. It can be established that (a) pairwise triangularizability is not sufficient to guarantee the existence of a common quadratic Lyapunov function for the linear time‐invariant dynamic systems ऱ A i : x˙ = A i x ; (b) additional conditions can be specified which guarantee asymptotic stability of the switching system ष. In this paper we also show that pairwise triangularizability is not even sufficient to guarantee asymptotic stability of the switching system ष. We also show that the method of proof of stability in (b), which does not assume the existence of a common quadratic Lyapunov function, can be used to prove the asymptotic stability of more general switching systems (systems that are not pairwise triangularizable). Finally, we show that our results can be used as the basis for the design of practical control systems; namely, for the design of an automobile speed switched controller with guaranteed stability properties.