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

## Abstract

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.

## Keywords

Hybrid System Outer Approximation Hybrid Automaton Switching Sequence Multiple Lyapunov Function## Preview

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## References

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