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

Minimally Constrained Stable Switched Systems and Application to Co-simulation

We propose an algorithm to restrict the switching signals of a constrained switched system in order to guarantee its stability, while at the same time attempting to keep the largest possible set of allowed switching signals. Our work is motivated by applications to (co-)simulation, where numerical stability is a hard constraint, but should be attained by restricting as little as possible the allowed behaviours of the simulators. We apply our results to certify the stability of an adaptive co-simulation orchestration algorithm, which selects the optimal switching signal at run-time, as a function of (varying) performance and accuracy requirements.Comment: Technical report complementing the following conference publication: Gomes, Cl\'audio, Beno\^it Legat, Rapha\"el Jungers, and Hans Vangheluwe. "Minimally Constrained Stable Switched Systems and Application to Co-Simulation." In IEEE Conference on Decision and Control. Miami Beach, FL, USA, 201

Optimal Switching Synthesis for Jump Linear Systems with Gaussian initial state uncertainty

This paper provides a method to design an optimal switching sequence for jump linear systems with given Gaussian initial state uncertainty. In the practical perspective, the initial state contains some uncertainties that come from measurement errors or sensor inaccuracies and we assume that the type of this uncertainty has the form of Gaussian distribution. In order to cope with Gaussian initial state uncertainty and to measure the system performance, Wasserstein metric that defines the distance between probability density functions is used. Combining with the receding horizon framework, an optimal switching sequence for jump linear systems can be obtained by minimizing the objective function that is expressed in terms of Wasserstein distance. The proposed optimal switching synthesis also guarantees the mean square stability for jump linear systems. The validations of the proposed methods are verified by examples.Comment: ASME Dynamic Systems and Control Conference (DSCC), 201

Issues in the design of switched linear systems : a benchmark study

In this paper we present a tutorial overview of some of the issues that arise in the design of switched linear control systems. Particular emphasis is given to issues relating to stability and control system realisation. A benchmark regulation problem is then presented. This problem is most naturally solved by means of a switched control design. The challenge to the community is to design a control system that meets the required performance specifications and permits the application of rigorous analysis techniques. A simple design solution is presented and the limitations of currently available analysis techniques are illustrated with reference to this example

A Gel'fand-type spectral radius formula and stability of linear constrained switching systems

Using ergodic theory, in this paper we present a Gel'fand-type spectral radius formula which states that the joint spectral radius is equal to the generalized spectral radius for a matrix multiplicative semigroup \bS^+ restricted to a subset that need not carry the algebraic structure of \bS^+. This generalizes the Berger-Wang formula. Using it as a tool, we study the absolute exponential stability of a linear switched system driven by a compact subshift of the one-sided Markov shift associated to \bS.Comment: 16 pages; to appear in Linear Algebra and its Application