A Novel Unified Approach to Invariance for a Dynamical System

TL;DR

This paper introduces a unified approach to determine invariance conditions for various convex sets in linear dynamical systems, extending traditional methods to nonconvex and unbounded sets using the S-lemma.

Contribution

It presents a novel, unified framework for invariance analysis applicable to multiple convex sets, utilizing the S-lemma instead of Lyapunov methods, and extends to nonconvex and unbounded sets.

Findings

Derived invariance conditions for polyhedra, cones, ellipsoids, and Lorenz cones.

Extended invariance conditions to nonconvex and unbounded sets.

Linked discrete and continuous system invariance conditions via Euler methods.

Abstract

In this paper, we propose a novel, unified, general approach to investigate sufficient and necessary conditions under which four types of convex sets, polyhedra, polyhedral cones, ellipsoids and Lorenz cones, are invariant sets for a linear continuous or discrete dynamical system. In proving invariance of ellipsoids and Lorenz cones for discrete systems, instead of the traditional Lyapunov method, our novel proofs are based on the S-lemma, which enables us to extend invariance conditions to any set represented by a quadratic inequality. Such sets include nonconvex and unbounded sets. Finally, according to the framework of our novel method, sufficient and necessary conditions for continuous systems are derived from the sufficient and necessary conditions for the corresponding discrete systems that are obtained by Euler methods.