Stability of discrete-time switching systems with constrained switching sequences

Abstract

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.

Invariant Sets Analysis for Constrained Switching Systems

“…Several methods exist for computing the scalars Γ ≥ 1, ρ ∈ (0, 1) in (9), see, e.g., [11], [12], [27].…”

“…Taking into account Lemma 1, we can establish the corollary convex versions of Theorems 1 and 2. (12), (13). Then, the multi-set {conv( …”

“…Corollary 2 Consider the System (1)- (6) [12] it is shown thatρ(A, G) < 1 while in [34]ρ(A, G) is computed exactly. We consider different state constraint sets and dis- turbance sets …”

“…In what follows, we leverage the results above and compute in the worst case the maximal invariant set for a number of iterations proportional to the square root of k. The method exploits the T-iterated dynamics of the system, which has been proven useful before for assessing stability, see e.g. [35], [12]. (19), a pair (Γ, ρ) satisfying (21) and k as in (20).…”

“…Most importantly, by utilizing recent, practicable, results concerning approximations of the exponential stability metrics [11], [12], [27], we establish a priori upper bounds on the maximum number of iterations required for computing the multi-sets. Preliminary results are in [28].…”