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

TL;DR

This paper introduces the minimax joint spectral radius for matrix products involving two sets of matrices, providing new tools to analyze the stability and stabilizability of discrete-time linear switching systems under disturbances.

Contribution

It defines the minimax joint spectral radii as a generalization of existing spectral radii and applies them to assess stabilizability in control systems with disturbances.

Findings

Defined the lower and upper minimax joint spectral radii.

Showed how these quantities relate to system stabilizability.

Demonstrated applications in control system analysis.

Abstract

To estimate the growth rate of matrix products $A_{n}\cdots A_{1}$ with factors from some set of matrices $\mathcal{A}$, such numeric quantities as the joint spectral radius $\rho(\mathcal{A})$ and the lower spectral radius $\check{\rho}(\mathcal{A})$ 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 $\rho(\mathcal{A})<1$ serves as a criterion for the stability of a system, and the inequality $\check{\rho}(\mathcal{A})<1 $ as a criterion for stabilizability. For matrix products $A_{n}B_{n}\cdots A_{1}B_{1}$ with factors $A_{i}\in\mathcal{A}$ and $B_{i}\in\mathcal{B}$, where $\mathcal{A}$ and $\mathcal{B}$ are some sets of matrices, we introduce the quantities $\mu(\mathcal{A},\mathcal{B})$ and $\eta(\mathcal{A},\mathcal{B})$, called the lower and upper minimax joint spectral radius of the pair $\{\mathcal{A},\mathcal{B}\}$, respectively, which characterize the maximum growth rate of the matrix products $A_{n}B_{n}\cdots A_{1}B_{1}$ over all sets of matrices $A_{i}\in\mathcal{A}$ and the minimal growth rate over all sets of matrices $B_{i}\in\mathcal{B}$. 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.