Incremental stability of hybrid dynamical systems

J.J.B. Biemond, Romain Postoyan, W.P.M.H. Heemels, Nathan van de Wouw

Research output: Contribution to journalArticleAcademicpeer-review

10 Citations (Scopus)
14 Downloads (Pure)


The 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 this paper, including an event-triggered control application and a bouncing ball system with Zeno behavior, that illustrate incremental stability with respect to continuous time or discrete time, respectively.

Original languageEnglish
Article number8350276
Pages (from-to)4094 - 4109
Number of pages16
JournalIEEE Transactions on Automatic Control
Issue number12
Publication statusPublished - Dec 2018


  • Hybrid systems
  • Incremental stability
  • Lyapunov stability
  • incremental stability


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