Matthew Philippe

Stability of discrete-time switching systems with constrained switching sequences

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 ź ź , 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 of ź ź within the range ź ź , ( 1 + r ) ź ź . 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 .

Converse Lyapunov theorems for discrete-time linear switching systems with regular switching sequences.

We present a stability analysis framework for the general class of discrete-time linear switching systems for which the switching sequences belong to a regular language. They admit arbitrary switching systems as special cases. Using recent results of X. Dai on the asymptotic growth rate of such systems, we introduce the concept of multinorm as an algebraic tool for stability analysis. We conjugate this tool with two families of multiple quadratic Lyapunov functions, parameterized by an integer T ≥ 1, and obtain converse Lyapunov Theorems for each. Lyapunov functions of the first family associate one quadratic form per state of the automaton defining the switching sequences. They are made to decrease after every T successive time steps. The second family is made of the path-dependent Lyapunov functions of Lee and Dullerud. They are parameterized by an amount of memory (T - 1) ≥ 0. Our converse Lyapunov theorems are finite. More precisely, we give sufficient conditions on the asymptotic growth rate of a stable system under which one can compute an integer parameter T ≥ 1 for which both types of Lyapunov functions exist. As a corollary of our results, we formulate an arbitrary accurate approximation scheme for estimating the asymptotic growth rate of switching systems with constrained switching sequences.

A sufficient condition for the boundedness of matrix products accepted by an automaton

We study the boundedness of products of matrices associated with words in a regular language. This question naturally arises in the stability analysis of switching systems with constrained switching sequences. Our main contribution is a sufficient condition for the boundedness of all the possible products of matrices that may occur in a marginally unstable system. We show that this condition can be expressed in terms of products of finite lengths, and is therefore algorithmically checkable. We then compare our condition with a second one, inspired by a lifting procedure introduced by Kozyakin, and prove that our condition is at least as powerful as this second one.

The minimum achievable stability radius of switched linear systems with feedback

We present a scheme for estimating the minimum achievable decay rate of a switched linear system via path-dependent feedback control, when the switching signal belongs to a language generated by a strongly connected graph. This growth rate is characterized by the constrained joint spectral radius (CJSR) of the system, a generalization of the joint spectral radius to account for the switching constraint. Our key tool in analyizing the CJSR is the multinorm, a collection of mode-indexed norms which demonstrate contractiveness along admissible modal trajectories. We may approximate the CJSR to any desired accuracy by computing quadratic multinorms as solutions to a system of LMIs, using an estimation scheme presented in [15]. These LMIs are of similar form to those which characterize the stability of the switched system in [6]. The feasiblity of any one of these LMIs allows the construction of a suitable controller; we use the infeasibility of such an LMI to provide a lower bound on the closed-loop decay rate achieved by any path-dependent controller.

Path-Complete Graphs and Common Lyapunov Functions

A Path-Complete Lyapunov Function is an algebraic criterion composed of a finite number of functions, called pieces, and a directed, labeled graph defining Lyapunov inequalities between these pieces. It provides a stability certificate for discrete-time arbitrary switching systems. In this paper, we prove that the satisfiability of such a criterion implies the existence of a Common Lyapunov Function, expressed as the composition of minima and maxima of the pieces of the Path-Complete Lyapunov function. the converse however is not true even for discrete-time linear systems: we present such a system where a max-of-2 quadratics Lyapunov function exists while no corresponding Path-Complete Lyapunov function with 2 quadratic pieces exists. In light of this, we investigate when it is possible to decide if a Path- Complete Lyapunov function is less conservative than another. By analyzing the combinatorial and algebraic structure of the graph and the pieces respectively, we provide simple tools to decide when the existence of such a Lyapunov function implies that of another.

On Path-Complete Lyapunov Functions: Geometry and Comparison

We study optimization-based criteria for the stability of switching systems, known as path-complete Lyapunov functions, and ask the question “can we decide algorithmically when a criterion is less conservative than another?”. Our contribution is twofold. First, we show that a path-complete Lyapunov function, which is a multiple Lyapunov function by nature, can always be expressed as a common Lyapunov function taking the form of a combination of minima and maxima of the elementary functions that compose it. Geometrically, our results provide for each path-complete criterion an implied invariant set. Second, we provide a linear programming criterion allowing to compare the conservativeness of two arbitrary given path-complete Lyapunov functions.

Extremal storage functions and minimal realizations of discrete-time linear switching systems

We study the Lp induced gain of discrete-time linear switching systems with graph-constrained switching sequences. We first prove that, for stable systems in a minimal realization, for every p ≥ 1, the Lp-gain is exactly characterized through switching storage functions. These functions are shown to be the pth power of a norm. In order to consider general systems, we provide an algorithm for computing minimal realizations. These realizations are rectangular systems, with a state dimension that varies according to the mode of the system. We apply our tools to the study on the of Lp-gain. We provide algorithms for its approximation, and provide a converse result for the existence of quadratic switching storage functions. We finally illustrate the results with a physically motivated example.