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Optimal Controllers for Hybrid Systems: Stability and Piecewise Linear Explicit Form


A. Bemporad, F. Borrelli, M. Morari

IEEE Conference on Decision and Control, Sydney, Australia, no. 39

In this paper we propose a procedure for synthesizing piecewise linear optimal controllers for hybrid systems and investigate conditions for closed-loop stability. Hybrid systems are modeled in discrete-time within the mixed logical dynamical (MLD) framework, or, equivalently, as piecewise affine (PWA) systems. A stabilizing controller is obtained by designing a {em model predictive controller} (MPC), which is based on the minimization of a weighted $ell_1/infty$-norm of the tracking error and the input trajectories over a finite horizon. The control law is obtained by solving a {em mixed-integer linear program} (MILP) which depends on the current state. Although efficient branch and bound algorithms exist to solve MILPs, these are known to be ${cal NP}$-hard problems, which may prevent their on-line solution if the sampling-time is too small for the available computation power. Rather than solving the MILP on line, in this paper we propose a different approach where all the computation is moved off line, by solving a {em multiparametric} MILP (mp-MILP). As the resulting control law is piecewise affine, on-line computation is drastically reduced to a simple linear function evaluation. An example of piecewise linear optimal control of an heat exchange system shows the potentiality of the method.


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% Autogenerated BibTeX entry
@InProceedings { BemBor:2000:IFA_281,
    author={A. Bemporad and F. Borrelli and M. Morari},
    title={{Optimal Controllers for Hybrid Systems: Stability and
	  Piecewise Linear Explicit Form}},
    booktitle={IEEE Conference on Decision and Control},
    address={Sydney, Australia},
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