A piecewise ellipsoidal reachable set estimation method for continuous bimodal piecewise affine systems

TL;DR

This paper introduces a novel piecewise ellipsoidal method for estimating reachable sets in continuous bimodal piecewise affine systems, leveraging piecewise quadratic Lyapunov functions to exploit system structure for simpler analysis.

Contribution

It proposes a new, less complex approach using piecewise quadratic Lyapunov functions tailored for bimodal systems, improving reachability estimation.

Findings

Derived conditions for positivity of quadratic functions on half spaces

Exploited continuity of bimodal quadratic functions on hyperplanes

Developed linear matrix characterizations for reachable set estimates

Abstract

In this work, the issue of estimation of reachable sets in continuous bimodal piecewise affine systems is studied. A new method is proposed, in the framework of ellipsoidal bounding, using piecewise quadratic Lyapunov functions. Although bimodal piecewise affine systems can be seen as a special class of affine hybrid systems, reachability methods developed for affine hybrid systems might be inappropriately complex for bimodal dynamics. This work goes in the direction of exploiting the dynamical structure of the system to propose a simpler approach. More specifically, because of the piecewise nature of the Lyapunov function, we first derive conditions to ensure that a given quadratic function is positive on half spaces. Then, we exploit the property of bimodal piecewise quadratic functions being continuous on a given hyperplane. Finally, linear matrix characterizations of the estimate of the reachable set are derived.