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

To estimate the growth rate of matrix products $A_{n}··· A_{1}$ with factors from some set of matrices $\mathscr{A}$, such numeric quantities as the joint spectral radius $ρ(\mathscr{A})$ and the lower spectral radius $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \rho } (\mathscr{A})$ 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 $ρ(\mathscr{A})<1$ serves as a criterion for the stability of a system, and the inequality $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \rho } (\mathscr{A})<1 $ as a criterion for stabilizability.

Given a set $\mathscr{A}$ of $N×M$ matrices and a set $\mathscr{B}$ of $M×N$ matrices. Then, for matrix products $A_{n}B_{n}··· A_{1}B_{1}$ with factors $A_{i}∈\mathscr{A}$ and $B_{i}∈\mathscr{B}$, we introduce the quantities $μ(\mathscr{A},\mathscr{B})$ and $η(\mathscr{A},\mathscr{B})$, called the lower and upper minimax joint spectral radius of the pair $\{\mathscr{A},\mathscr{B}\}$, respectively, which characterize the maximum growth rate of the matrix products $A_{n}B_{n}··· A_{1}B_{1}$ over all sets of matrices $A_{i}∈\mathscr{A}$ and the minimal growth rate over all sets of matrices $B_{i}∈\mathscr{B}$. 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.