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A Novel Unified Approach to Invariance in Control

Zoltán Horváth (horvathz***at***szu.hu)
Yunfei Song (yus210***at***lehigh.edu)
Tamás Terlaky (terlaky***at***lehigh.edu)

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.

Keywords: Invariant Set, Dynamical System, Polyhedron, Lorenz Cone, Farkas Lemma, S-Lemma

Category 1: Convex and Nonsmooth Optimization

Category 2: Linear, Cone and Semidefinite Programming

Category 3: Applications -- Science and Engineering (Control Applications )

Citation: Department of Mathematics and Computational Sciences, Széchenyi István University, Győr, Hungary Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, USA 05/2014

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Entry Submitted: 05/20/2014
Entry Accepted: 05/20/2014
Entry Last Modified: 05/21/2014

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