Improved reachable set estimation for positive systems: A polyhedral approach

Abstract

This paper revisits the problem of estimating the set of states of a positive system, that are reachable from the origin, by constrained exogenous inputs. More specifically, we seek the smallest polyhedron that bounds the reachable set of a positive system, subject to inputs whose column sums have a bounded peak value. By resorting to a particular auxiliary function, the maximum of linear functions of state variables, a sufficient condition is given such that the reachable set is bounded by a prescribed polyhedron. This auxiliary function is not necessarily positive definite even for nonnegative state variables and, therefore, the face normals of the resulting bounding polyhedron may not be positive vectors. This greatly improves the existing results using linear copositive Lyapunov functions, and also paves the way for the reachable set estimation of unstable positive systems. Moreover, we provide an approach to compute the bounding hyper-rectangle, which turns out to be nonconservative for the single-input case. The obtained bounding polyhedron depends on the time delays, while the bounding hyper-rectangle does not. On the other hand, for a positive system subject to inputs with the integral of their column sums bounded by a prescribed value, we give an estimate for the reachable set using the convex hull of hyper-rectangles, as well as a bounding polyhedron whose outward face normals are not necessarily positive vectors. These two categories of bounding sets are both delay independent. Finally, we show via numerical examples the superiority of the developed results.