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

Journal article (2016)

Authors

J. J Benjamin Biemond Katholieke Universiteit Leuven
W. P.M.H. Heemels Eindhoven University of Technology
Ricardo G. Sanfelice University of California
N. van de Wouw Team Bart De Schutter - Mechanical, Maritime and Materials Engineering , University of Minnesota, Eindhoven University of Technology
Research Group
Team Bart De Schutter (Mechanical, Maritime and Materials Engineering) (TU Delft)
Stability analysis Hybrid systems Lyapunov stability Tracking control
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Published Date

2016

Language

English

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Faculty

Mechanical, Maritime and Materials Engineering

Department

Delft Center for Systems and Control

Research Group

Team Bart De Schutter

Abstract

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.