External positivity of linear systems: approximate characterization via convex polytopes

Abstract

A linear system is said to be externally positive if the system output is non-negative for all time when driven by a non-negative input from the zero initial state. External positivity is well known to be equivalent to non-negativity of the impulse response and hence monotone nondecreasing step response. Despite the apparent simplicity of characterizing systems with non-negative impulse response, the determination of necessary and sufficient conditions for external positivity from a given transfer function is a long-standing open problem. In this paper, we propose a method which approximately characterizes the true region capturing the numerator polynomials of all (strictly proper) externally positive linear systems whose specified poles are assumed to satisfy a known necessary condition for external positivity. We compute an (outer) approximation of the true region via the construction of a convex polytope, each facet of which is contained in a supporting hyperplane of the true region. The proposed method requires only modest computational effort; has an accuracy which can be increased readily; applies to systems having orders n ≥ 4 for which no general characterizations of external positivity are currently known; and handles systems with complex poles (possibly repeated). Numerical examples illustrate the effectiveness of the proposed method.