Lazar, Mircea and Heemels, W.P.M.H. and Weiland, Siep and Bemporad, Alberto
Stability of hybrid model predictive control.
In: 43rd IEEE Conference on Decision and Control, December 14-17, 2004 , Atlantis, Paradise Island, Bahamas
pp. 1-40.
(Unpublished)
(2004)
Abstract
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 1-norm based
MPC cost, they are obtained as 1-norm inequalities. It is shown that by using 1-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.
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