L2-gain Analysis for a Class of Hybrid Systems with Applications to Reset and Event-triggered Control: A Lifting Approach

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 `2-sense (meaning that the `2-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 `2-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 `2-gain analysis of a discrete-time 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 `2-gain (and thus L2-gain) conditions, despite the fact that the linearity assumption, which is usually needed in the lifting literature, is not satisfied. Maurice Heemels is with the Control System Technology Group, Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands, e-mail: [email protected]. Geir Dullerud is with the Mechanical Science and Engineering Department, and Coordinated Science Laboratory, University of Illinois, Urbana, IL 61801, e-mail: [email protected]. Andrew Teel is with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA, USA, e-mail: [email protected]. Maurice Heemels is supported by the Innovational Research Incentives Scheme under the VICI grant “Wireless control systems: A new frontier in automation” (No. 11382) awarded by NWO (The Netherlands Organisation for Scientific Research) and STW (Dutch Technology Foundation). Geir Dullerud is supported by grants NSA SoS W911NSF-13-0086 and AFOSR MURI FA9550-10-1-0573. Andrew Teel is research supported in part by AFOSR grant numbers FA9550-12-1-0127 and FA9550-15-1-0155, and NSF grant numbers ECCS-1232035 and ECCS-1508757.