Definitions of incremental stability for hybrid systems

Abstract

Abstract-The analysis of incremental stability properties typically involves measuring the distance between any pair of solutions of a given dynamical system, corresponding to different initial conditions, at the same time instant. This approach is not directly applicable for hybrid systems in general. Indeed, hybrid systems generate solutions that are defined with respect to hybrid times, which consist of both the continuous time elapsed and the discrete time, that is the number of jumps the solution has experienced. Two solutions of a hybrid system do not a priori have the same time domain, and we may therefore not be able to compare them at the same hybrid time instant. To overcome this issue, we invoke graphical closeness concepts. We present definitions for incremental stability depending on whether incremental asymptotic stability properties hold with respect to the hybrid time, the continuous time, or the discrete time, respectively. Examples are provided throughout the paper to illustrate these definitions, and the relations between these three incremental stability notions are investigated. The definitions are shown to be consistent with those available in the literature for continuous-time and discrete-time systems.

Synchronization of impacting mechanical systems with a single constraint

“…(26)- (30) and let the quotient distance function d in (13) be defined using the impact mapZ in (24). Then,…”

“…□ Lemma 2-4 show that the Lyapunov function (26)- (30) is positive definite in the quotient distance function, it is non-increasing at discontinuities and the densityV in the different cases takes a simple form. Therefore, the proposed Lyapunov function is suitable to study the synchronization problem of the mechanical system (22)- (24).…”

“…This approach has been extended in [24] towards incremental stability. These approaches, however, are not applicable if the time between state jumps can be arbitrarily small, as occurs generally in unilaterally constrained mechanical systems (especially in Zeno events) since the hybrid system framework employed is not suitable to investigate solutions after the occurrence of Zeno behavior.…”

Incremental Stability of Hybrid Dynamical Systems

“…The objective of this paper is to investigate incremental stability for hybrid systems, for which results in the literature are rare. Exceptions are the recent works in [16]- [19], where incremental stability is studied for a class of hybrid systems in the formalism of [20]. Results on convergence for a class of measure differential inclusions can be found in [21], [22].…”

“…The characterisation of flow incremental stability in terms of set-stability reminds of the approach in [18], when the assumptions in [18] are satisfied, even though we use a different extended system. Focusing on systems with complete solutions, in our preliminary work in [19], we have advertised the proposed definitions for incremental stability. In the present paper, we generalise these definitions to systems with non-complete solutions, and, in addition, provide a characterisation of flow and jump incremental stability using extended hybrid systems and setstability results, not given in [19].…”