Stabilizing Model Predictive Control of Hybrid Systems

TLDR

The authors investigate the stability of hybrid systems under model predictive control and develop novel techniques for computing terminal costs, constraint sets, and low‑complexity invariant sets for constrained piecewise affine prediction models. They derive a priori sufficient Lyapunov stability conditions in terminal cost/constraint set form, translating them into linear matrix inequalities for quadratic costs and norm inequalities for infinity‑norm costs, and present computational methods for low‑complexity piecewise polyhedral invariant sets. An illustrative example demonstrates the applicability of the proposed theory.

Abstract

In this note, we investigate the stability of hybrid systems in closed-loop with model predictive controllers (MPC). A priori sufficient conditions for Lyapunov asymptotic stability and exponential stability are derived in the terminal cost and constraint set fashion, while allowing for discontinuous system dynamics and discontinuous MPC value functions. For constrained piecewise affine (PWA) systems as prediction models, we present novel techniques for computing a terminal cost and a terminal constraint set that satisfy the developed stabilization conditions. For quadratic MPC costs, these conditions translate into a linear matrix inequality while, for MPC costs based on 1, infin-norms, they are obtained as norm inequalities. New ways for calculating low complexity piecewise polyhedral positively invariant sets for PWA systems are also presented. An example illustrates the developed theory