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

  • Alberto Bemporad
  • Fabio Danilo Torrisi
  • Manfred Morari
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1790)


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 IR 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 X(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 set X inst of (very large) states. Subsets of X(0) leading to none of the two previous cases are labeled as “non-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 the initial conditions from which the settling time to a specified set around the origin 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 as what has been proposed for verification of hybrid systems. In this paper we present a substantial extension of the verification algorithm presented in [6] for stability characterization of PWA systems, 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 asymptotically stable, or unstable, or non-classifiable in T steps.


Hybrid System Outer Approximation Hybrid Automaton Switching Sequence Multiple Lyapunov Function 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Alberto Bemporad
    • 1
  • Fabio Danilo Torrisi
    • 1
  • Manfred Morari
    • 1
  1. 1.Automatic Control LaboratorySwiss Federal Institute of TechnologyZürichSwitzerland

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