Distance function design and Lyapunov techniques for the stability of hybrid trajectories

Benjamin Biemond, W.P.M.H. Heemels, R.G. Sanfelice, N. van de Wouw

Research output: Contribution to journalArticleAcademicpeer-review

14 Citations (Scopus)
8 Downloads (Pure)


The comparison between time-varying hybrid trajectories is crucial for tracking, observer design and synchronisation problems for hybrid systems with state-triggered jumps. In this paper, a generic distance function is designed that can be used for this purpose. The so-called “peaking phenomenon”, which occurs when using the Euclidean distance to compare two hybrid trajectories, is circumvented by taking the hybrid nature of the system explicitly into account. Based on the proposed distance function, we define the stability of a trajectory and present sufficient Lyapunov-type conditions for hybrid system with state-triggered jumps. A constructive Lyapunov function design is presented for hybrid systems with affine flow and jump maps and a jump set that is a hyperplane. The stability conditions can then be verified using linear matrix conditions. Finally, for this class of systems, we present a tracking controller that asymptotically stabilises a given hybrid reference trajectory and we illustrate our results with an example.

Original languageEnglish
Pages (from-to)38-46
Number of pages9
Publication statusPublished - 1 Nov 2016


  • Hybrid systems
  • Lyapunov stability
  • Stability analysis
  • Tracking control


Dive into the research topics of 'Distance function design and Lyapunov techniques for the stability of hybrid trajectories'. Together they form a unique fingerprint.

Cite this