Stabilizing Dynamic Control Design for Hybrid Systems: A Hybrid Control Lyapunov Function Approach

This paper proposes a dynamic controller structure and a systematic design procedure for stabilizing discrete-time hybrid systems. The proposed approach is based on the concept of control Lyapunov functions (CLFs), which, when available, can be used to design a stabilizing state-feedback control law. In general, the construction of a CLF for hybrid dynamical systems involving both continuous and discrete states is extremely complicated, especially in the presence of non-trivial discrete dynamics. Therefore, we introduce the novel concept of a hybrid control Lyapunov function, which allows the compositional design of a discrete and a continuous part of the CLF, and we formally prove that the existence of a hybrid CLF guarantees the existence of a classical CLF. A constructive procedure is provided to synthesize a hybrid CLF, by expanding the dynamics of the hybrid system with a specific controller dynamics. We show that this synthesis procedure leads to a dynamic controller that can be implemented by a receding horizon control strategy, and that the associated optimization problem is numerically tractable for a fairly general class of hybrid systems, useful in real world applications. Compared to classical hybrid receding horizon control algorithms, the proposed approach typically requires a shorter prediction horizon to guarantee asymptotic stability of the closed-loop system, which yields a reduction of the computational burden, as illustrated through two examples. Stefano Di Cairano is with Mitsubishi Electric Research Laboratories, Cambridge, Massachusetts. E-mail: [email protected]. Maurice Heemels and Mircea Lazar are with the Dept. of Mechanical Eng. (W.P.M.H. Heemels) and the Dept. of Electrical Eng. (M. Lazar), Eindhoven Univ. of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mails: [email protected], [email protected]. Alberto Bemporad is with the IMT Institute for Advanced Studies, Lucca, Italy, E-mail: [email protected]. Maurice Heemels, Mircea Lazar, and Alberto Bemporad were partially supported by the European Commission under project FP7-INFSO-ICT-248858 ‘MOBY-DIC Model-based synthesis of digital electronic circuits for embedded control’. Preliminary results related to this research were presented in [1], [2]. This paper provides new results on the stability properties of the proposed control strategy, full detailed proofs, a novel and detailed procedure for the synthesis of the control law, and an example of practical interest demonstrating the proposed approach. January 15, 2014 DRAFT