Minimax joint spectral radius and stabilizability of discrete-time linear switching control systems

Abstract

To estimate the growth rate of matrix products An · · · A1 with factors from some set of matrices A, such numeric quantities as the joint spectral radius ρ(A) and the lower spectral radiusρ(A) are traditionally used. The first of these quantities characterizes the maximum growth rate of the norms of the corresponding products, while the second one characterizes the minimal growth rate. In the theory of discrete-time linear switching systems, the inequality ρ(A) < 1 serves as a criterion for the stability of a system, and the inequalityρ(A) < 1 as a criterion for stabilizability.For matrix products AnBn · · · A1B1 with factors Ai ∈ A and Bi ∈ B, where A and B are some sets of matrices, we introduce the quantities µ(A, B) and η(A, B), called the lower and upper minimax joint spectral radius of the pair {A, B}, respectively, which characterize the maximum growth rate of the matrix products AnBn · · · A1B1 over all sets of matrices Ai ∈ A and the minimal growth rate over all sets of matrices Bi ∈ B. In this sense, the minimax joint spectral radii can be considered as generalizations of both the joint and lower spectral radii. As an application of the minimax joint spectral radii, it is shown how these quantities can be used to analyze the stabilizability of discrete-time linear switching control systems in the presence of uncontrolled external disturbances of the plant.

On Convergence of Infinite Matrix Products with Alternating Factors from Two Sets of Matrices

“…Taking into account that by virtue of (11), for each m, the estimate k m ≤ k * m is fulfilled, from here and from (13) we obtain for the number p a lower estimate: p ≥ n k * − 1. And then from the estimates established earlier for D * , D 1 , .…”

“…The presence of alternating factors in the products of the matrices (1) substantially complicates the problem of convergence of the corresponding matrix products for all possible sequences of matrices {A i ∈ A} due to a suitable choice of sequences of matrices {B i ∈ B} in comparison with the problem of convergence of matrix products A n · · · A 1 for all possible sequences of matrices {A i ∈ A}. A discussion of the arising difficulties can be found, e.g., in [13]. One of the applications of the results obtained in this paper for analyzing the new concept of the so-called minimax joint spectral radius is also described there.…”

The finiteness conjecture holds in (SL2Z⩾0)2 *

“…Research burst after the publication of a series of paper by Daubechies and Lagarias in the nineties, and connections with fields such as wavelet regularity, optimization, control theory, combinatorics, Lyapunov exponents and ergodic theory rapidly emerged. We refer to [4,12,15,17,21] and references therein for a broad panorama. In this paper we are concerned with a further connection, namely to dynamics on the hyperbolic plane.…”