Robust stability and transient behaviour of positive linear systems

After a brief review of available results the main focus of the paper is on the transient behaviour of positive systems and their stability radii with respect to highly structured perturbations. Simple upper bounds for the transient gain of positive systems are obtained by means of linear Lyapunov functions on the positive orthant. The minimization of these bounds is discussed and algorithms for computing optimal Lyapunov vectors are presented. By means of linear Lyapunov functions we get new formulae for the stability radii of positive linear systems with respect to structured and time-varying perturbations of Gershgorin-Brualdi type. With every time-invariant linear system we associate a corresponding positive system and this correspondence allows to transfer some of the results to non-positive linear systems.