Geometric control of hybrid systems

TL;DR

This paper introduces a geometric method for computing controlled invariant sets in hybrid control systems, extending beyond ellipsoidal approximations to more general convex sets with polynomial or piecewise quadratic support functions.

Contribution

It reformulates invariance conditions as inequalities on support functions, enabling the analysis of more complex convex sets in hybrid systems.

Findings

Provides algebraic conditions for invariance of polynomial support function sets

Extends invariance analysis beyond ellipsoidal approximations

Applicable to constrained or switched linear systems

Abstract

In this paper, we present a geometric approach for computing controlled invariant sets for hybrid control systems. While the problem is well studied in the ellipsoidal case, this family is quite conservative for constrained or switched linear systems. We reformulate the invariance of a set as an inequality for its support function that is valid for any convex set. This produces novel algebraic conditions for the invariance of sets with polynomial or piecewise quadratic support function.