A novel unified approach to invariance conditions for a linear dynamical system

摘要

In this paper, we propose a novel, simple, and unified approach to explore sufficient and necessary conditions, i.e., invariance conditions, under which four classic families of convex sets, namely, polyhedra, polyhedral cones, ellipsoids, and Lorenz cones, are invariant sets for a linear discrete or continuous dynamical system.For discrete dynamical systems, we use the Theorems of Alternatives, i.e., Farkas lemma and S-lemma, to obtain simple and general proofs to derive invariance conditions. This novel method establishes a solid connection between optimization theory and dynamical system. Also, using the -lemma allows us to extend invariance conditions to any set represented by a quadratic inequality. Such sets include nonconvex and unbounded sets.For continuous dynamical systems, we use the forward or backward Euler method to obtain the corresponding discrete dynamical systems while preserves invariance. This enables us to develop a novel and elementary method to derive invariance conditions for continuous dynamical systems by using the ones for the corresponding discrete systems.Finally, some numerical examples are presented to illustrate these invariance conditions.