Positively invariant sets for continuous-time systems with the cone-preserving property

Abstract

The main objective of this paper is to determine positively invariant and asymptotically stable polyhedral sets for a linear continuous-time system [xdot](t) = Ax(t) for which matrix e ^1A is a cone-preserving matrix, that is, e ^1A K ⊂ K, for some proper cone K. Necessary and sufficient conditions guaranteeing that some bounded sets are positively invariant and contractive are given. These sets are obtained by means of the intersection of shifted cones. First, some results presented under a geometrical form and also in algebraic form allow characterization of systems having the cone-preserving property. Finally, as an application, the proposed results are used to determine a stability domain for a state feedback regulator with constraints on either or both states and controls.