L2-gain analysis for a class of hybrid systems with applications to reset and event-triggered control: A lifting approach

W.P.M.H. Heemels, G.E. Dullerud, A.R. Teel

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57 Citations (Scopus)
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In this paper we study the stability and L2-gain properties of a class of hybrid systems that exhibit linear flow dynamics, periodic time-triggered jumps and arbitrary nonlinear jump maps. This class of hybrid systems is relevant for a broad range of applications including periodic event-triggered control, sampled-data reset control, sampled-data saturated control, and certain networked control systems with scheduling protocols. For this class of continuous-time hybrid systems we provide new stability and L2-gain analysis methods. Inspired by ideas from lifting we show that the stability and the contractivity in L2-sense (meaning that the L2-gain is smaller than 1) of the continuous-time hybrid system is equivalent to the stability and the contractivity in l2-sense (meaning that the l2-gain is smaller than 1) of an appropriate discrete-time nonlinear system. These new characterizations generalize earlier (more conservative) conditions provided in the literature.We show via a reset control example and an event- triggered control application, for which stability and contractivity in L2-sense is the same as stability and contractivity in l2-sense of a discrete-time piecewise linear system, that the new conditions are significantly less conservative than the existing ones in the literature. Moreover, we show that the existing conditions can be reinterpreted as a conservative l2-gain analysis of a discretetime piecewise linear system based on common quadratic storage/ Lyapunov functions. These new insights are obtained by the adopted lifting-based perspective on this problem, which leads to computable l2-gain (and thus L2-gain) conditions, despite the fact that the linearity assumption, which is usually needed in the lifting literature, is not satisfied.

Original languageEnglish
Article number7332744
Pages (from-to)2766-2781
Number of pages16
JournalIEEE Transactions on Automatic Control
Issue number10
Publication statusPublished - Oct 2016


  • Periodic event-triggered control (PETC)
  • Piecewise affine (PWA)
  • Piecewise linear (PWL)


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