Stability Testing of Matrix Polytopes

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 radius r, th...

We give sufficient conditions for stability of a continuous-time linear switched system consisting of finitely many subsystems. The switching between subsystems is governed by an underlying graph. The results are applicable to switched systems having some or all non-Hurwitz subsystems. We also present a slow-fast switching mechanism on subsystems comprising simple loops of underlying graph to ensure stability of the switched system.

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 l...

Stability analysis of discrete-time switched systems under minimum dwell-time is studied using a new type of LMI conditions. These conditions are convex in the matrices of the system and shown to be equivalent to the nonconvex conditions proposed by Geromel and Colaneri. The convexification of the conditions is performed by a lifting process which introduces a moderate number of additional decision variables. The convexity of the conditions can be exploited to extend the resu...

We investigate the stabilizability of discrete-time linear switched systems, when the sole control action of the controller is the switching signal, and when the controller has access to the state of the system in real time. Despite their apparent simplicity, determining if such systems are stabilizable appears to be a very challenging problem, and basic examples have been known for long, for which the stabilizability question is open. We provide new results allowing us to bo...

Univariate polynomials with only real roots -- while special -- do occur often enough that their properties can lead to interesting conclusions in diverse areas. Due mainly to the recent work of two young mathematicians, Julius Borcea and Petter Br\"and\'en, a very successful multivariate generalization of this method has been developed. The first part of this paper surveys some of the main results of this theory of "multivariate stable" polynomials -- the most central of the...

We consider the following problem: given $d \times d$ rational matrices $A_1, \ldots, A_k$ and a polyhedral cone $\mathcal{C} \subset \mathbb{R}^d$, decide whether there exists a non-zero vector whose orbit under multiplication by $A_1, \ldots, A_k$ is contained in $\mathcal{C}$. This problem can be interpreted as verifying the termination of multi-path while loops with linear updates and linear guard conditions. We show that this problem is decidable for commuting invertible...

A multidimensional nonnegative matrix is called polystochastic if the sum of entries in each line is equal to $1$. The set of all polystochastic matrices of order $n$ and dimension $d$ is a convex polytope $\Omega_n^d$. In the present paper, we compare known bounds on the number of vertices of the polytope $\Omega_n^d$ and prove that the number of vertices of $\Omega_3^d$ is doubly exponential on $d$.

This paper treats the global stabilization problem of continuous-time switched affine systems that have rank-deficient convex combinations of their dynamic matrices. For these systems, the already known set of attainable equilibrium points has higher dimensionality than in the full-rank case due to the existence of what we define as singular equilibrium points. Our main goal is to design a state-dependent switching function to ensure global asymptotic stability of a chosen po...

In this thesis, we focus on some of the NP-hard problems in control theory. Thanks to the converse Lyapunov theory, these problems can often be modeled as optimization over polynomials. To avoid the problem of intractability, we establish a trade off between accuracy and complexity. We develop a sequence of tractable optimization problems - in the form of LPs and SDPs - whose solutions converge to the exact solution of the NP-hard problem. However, the computational and memor...