Stabilizing Randomly Switched Systems

This article is concerned with stability analysis and stabilization of randomly switched systems under a class of switching signals. The switching signal is modeled as a jump stochastic (not necessarily Markovian) process independent of the system state; it selects, at each instant of time, the active subsystem from a family of systems. Sufficient conditions for stochastic stability (almost sure, in the mean, and in probability) of the switched system are established when the subsystems do not possess control inputs, and not every subsystem is required to be stable. These conditions are employed to design stabilizing feedback controllers when the subsystems are affine in control. The analysis is carried out with the aid of multiple Lyapunov-like functions, and the analysis results together with universal formulae for feedback stabilization of nonlinear systems constitute our primary tools for control designComment: 22 pages. Submitte

A graph theoretic approach to input-to-state stability of switched systems

This article deals with input-to-state stability (ISS) of discrete-time switched systems. Given a family of nonlinear systems with exogenous inputs, we present a class of switching signals under which the resulting switched system is ISS. We allow non-ISS systems in the family and our analysis involves graph-theoretic arguments. A weighted digraph is associated to the switched system, and a switching signal is expressed as an infinite walk on this digraph, both in a natural way. Our class of stabilizing switching signals (infinite walks) is periodic in nature and affords simple algorithmic construction.Comment: 14 pages, 1 figur

Approximately bisimilar symbolic models for incrementally stable switched systems

Switched systems constitute an important modeling paradigm faithfully describing many engineering systems in which software interacts with the physical world. Despite considerable progress on stability and stabilization of switched systems, the constant evolution of technology demands that we make similar progress with respect to different, and perhaps more complex, objectives. This paper describes one particular approach to address these different objectives based on the construction of approximately equivalent (bisimilar) symbolic models for switched systems. The main contribution of this paper consists in showing that under standard assumptions ensuring incremental stability of a switched system (i.e. existence of a common Lyapunov function, or multiple Lyapunov functions with dwell time), it is possible to construct a finite symbolic model that is approximately bisimilar to the original switched system with a precision that can be chosen a priori. To support the computational merits of the proposed approach, we use symbolic models to synthesize controllers for two examples of switched systems, including the boost DC-DC converter.Comment: 17 page

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