Stability of Hybrid Model Predictive Control

In this paper we investigate the stability of hybrid systems in closed-loop with Model Predictive Controllers (MPC) and we derive a priori sufficient conditions for Lyapunov asymptotic stability and exponential stability. A general theory is presented which proves that Lyapunov stability is achieved for both terminal cost and constraint set and terminal equality constraint hybrid MPC, even though the considered Lyapunov function and the system dynamics may be discontinuous. For particular choices of MPC criteria and constrained Piecewise Affine (PWA) systems as the prediction models we develop novel algorithms for computing the terminal cost and the terminal constraint set. For a quadratic MPC cost, the stabilization conditions translate into a linear matrix inequality while, for an ∞-norm based MPC cost, they are obtained as ∞-norm inequalities. It is shown that by using ∞-norms, the terminal constraint set is automatically obtained as a polyhedron or a finite union of polyhedra by taking a sublevel set of the calculated terminal cost function. New algorithms are developed for calculating polyhedral or piecewise polyhedral positively invariant sets for PWA systems. In this manner, the on-line optimization problem leads to a mixed integer quadratic programming problem or to a mixed integer linear programming problem, which can be solved by standard optimization tools. Several examples illustrate the effectiveness of the developed methodology.