Geometric control of algebraic systems

TL;DR

This paper introduces a geometric method for computing controlled invariant sets in continuous-time systems, offering less conservative results than traditional ellipsoidal approaches, especially for constrained or switched systems.

Contribution

It develops a novel algebraic framework using support functions for invariance, applicable to convex sets with polynomial or piecewise quadratic support functions, improving over existing methods.

Findings

The new approach is less conservative than ellipsoidal methods.

It provides algebraic conditions for invariance of polynomial and piecewise quadratic sets.

Comparison shows significant improvement over traditional algebraic approaches.

Abstract

In this paper, we present a geometric approach for computing the controlled invariant set of a continuous-time control system. While the problem is well studied for 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. We compare it with the common algebraic approach for polynomial sublevel sets and show that it is significantly more conservative than our method.