Incremental stability of hybrid dynamical systems

International audienceThe analysis of incremental stability typically involves measuring the distance between any two solutions of a given dynamical system at the same time instant, which is problematic when studying hybrid dynamical systems. Indeed, hybrid systems generate solutions defined with respect to hybrid time instances (that consists of both the continuous time elapsed and the discrete time, which is the number of jumps experienced so far), and two solutions of the same hybrid system may not be defined at the same hybrid time instant. To overcome this issue, we present novel definitions of incremental stability for hybrid systems based on graphical closeness of solutions. As we will show, defining incremental asymptotic stability with respect to the hybrid time yields a restrictive notion, such that we also investigate incremental asymptotic stability notions with respect to the continuous time only or the discrete time only, respectively. In this manner, two (effectively dual) incremental stability notions are attained, called jump-and flow incremental asymptotic stability. To present Lyapunov conditions for these two notions, in both cases, we resort to an extended hybrid system and we prove that the stability of a well-defined set for this extended system implies incremental stability of the original system. We can then use available Lyapunov conditions to infer the set stability of the extended system. Various examples are provided throughout the paper, including an event-triggered control application and a bouncing ball system with Zeno behaviour, that illustrate incremental stability with respect to continuous time or discrete time, respectively

Symbolic models for nonlinear control systems without stability assumptions

Finite-state models of control systems were proposed by several researchers as a convenient mechanism to synthesize controllers enforcing complex specifications. Most techniques for the construction of such symbolic models have two main drawbacks: either they can only be applied to restrictive classes of systems, or they require the exact computation of reachable sets. In this paper, we propose a new abstraction technique that is applicable to any smooth control system as long as we are only interested in its behavior in a compact set. Moreover, the exact computation of reachable sets is not required. The effectiveness of the proposed results is illustrated by synthesizing a controller to steer a vehicle.Comment: 11 pages, 2 figures, journa

Contraction analysis of switched Filippov systems via regularization

We study incremental stability and convergence of switched (bimodal) Filippov systems via contraction analysis. In particular, by using results on regularization of switched dynamical systems, we derive sufficient conditions for convergence of any two trajectories of the Filippov system between each other within some region of interest. We then apply these conditions to the study of different classes of Filippov systems including piecewise smooth (PWS) systems, piecewise affine (PWA) systems and relay feedback systems. We show that contrary to previous approaches, our conditions allow the system to be studied in metrics other than the Euclidean norm. The theoretical results are illustrated by numerical simulations on a set of representative examples that confirm their effectiveness and ease of application.Comment: Preprint submitted to Automatic