Non-smooth Model Predictive control: Stability and applications to Hybrid Systems

In this report 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.

[1]  J. P. Lasalle.  The stability of dynamical systems , 1976 .

[2]  Eduardo Sontag.  Nonlinear regulation: The piecewise linear approach , 1981 .

[3]  Gene H. Golub,et al.  Matrix computations , 1983 .

[4]  M. Cwikel,et al.  Convergence of an algorithm to find maximal state constraint sets for discrete-time linear dynamical systems with bounded controls and states , 1985, 1985 24th IEEE Conference on Decision and Control.

[5]  Franco Blanchini,et al.  Ultimate boundedness control for uncertain discrete-time systems via set-induced Lyapunov functions , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[6]  Manfred Morari,et al.  Robust constrained model predictive control using linear matrix inequalities , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[7]  James B. Rawlings,et al.  Discrete-time stability with perturbations: application to model predictive control , 1997, Autom..

[8]  Anders Rantzer,et al.  Computation of piecewise quadratic Lyapunov functions for hybrid systems , 1997, 1997 European Control Conference (ECC).

[9]  E. Gilbert,et al.  Theory and computation of disturbance invariant sets for discrete-time linear systems , 1998 .

[10]  Jamal Daafouz,et al.  Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach , 2002, IEEE Trans. Autom. Control..

[11]  D. Mayne,et al.  Optimal control of constrained, piecewise affine systems with bounded disturbances , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[12]  S. Joe Qin,et al.  A survey of industrial model predictive control technology , 2003 .

[13]  M. Morari,et al.  Stability analysis of hybrid systems with a linear performance index , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[14]  Mato Baotic,et al.  Multi-Parametric Toolbox (MPT) , 2004, HSCC.

[15]  Alberto Bemporad,et al.  Efficient conversion of mixed logical dynamical systems into an equivalent piecewise affine form , 2004, IEEE Transactions on Automatic Control.

[16]  Pascal Grieder,et al.  Efficient computation of feedback controllers for constrained systems , 2004 .

[17]  D. Mayne,et al.  Computation of invariant sets for piecewise affine discrete time systems subject to bounded disturbances , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[18]  Alberto Bemporad,et al.  Stabilizing receding horizon control of PWL systems: An LMI approach , 2004 .

[19]  Alberto Bemporad,et al.  Stabilization conditions for model predictive control of constrained PWA systems , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[20]  Graham C. Goodwin,et al.  Constrained Control and Estimation: an Optimization Approach , 2004, IEEE Transactions on Automatic Control.