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Optimization-Based Verification and Stability Characterization of Piecewise Affine and Hybrid Systems


A. Bemporad, F.D. Torrisi, M. Morari

International Workshop on Hybrid Systems: Computation and Control, Pittsburgh, USA, vol. 1790, no. 3, pp. 45-58, B.H. Krogh, N. Lynch (Eds.), Lecture Notes in Computer Science

In this paper, we formulate the problem of characterizing the stability of a piecewise affine (PWA) system as a verification problem. The basic idea is to take the whole $R^n$ as the set of initial conditions, and check that all the trajectories go to the origin. More precisely, we test for semi-global stability by restricting the set of initial conditions to an (arbitrarily large) bounded set $XX(0)$, and label as ``asymptotically stable in $T$ steps'' the trajectories that enter an invariant set around the origin within a finite time $T$, or as ``unstable in $T$ steps'' the trajectories which enter a (very large) set $XXinst$. Subsets of $XX(0)$ leading to none of the two previous cases are labeled as ``not classifiable in $T$ steps''. The domain of asymptotical stability in $T$ steps is a subset of the domain of attraction of an equilibrium point, and has the practical meaning of collecting initial conditions from which the settling time of the system is smaller than $T$. In addition, it can be computed algorithmically in finite time. Such an algorithm requires the computation of reach sets, in a similar fashion to what has been proposed for verification of hybrid systems. In this paper we present a substantial extension of the verification algorithm presented in~(Bemporad and Morari, 1999, LNCS 1569) for stability characterization of PWA systems, which is based on linear and mixed-integer linear programming. As a result, given a set of initial conditions we are able to determine its partition into subsets of trajectories which are either asymptotic stable, or unstable, or not classifiable in $T$ steps.

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% Autogenerated BibTeX entry
@InProceedings { BemTor:2000:IFA_32,
    author={A. Bemporad and F.D. Torrisi and M. Morari},
    title={{Optimization-Based Verification and Stability
	  Characterization of Piecewise Affine and Hybrid Systems}},
    booktitle={International Workshop on Hybrid Systems: Computation and
    address={Pittsburgh, USA},
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