Stability radius of linear parameter-varying systems and applications

In this paper, we present an unifying approach to the problems of computing of stability radii of positive linear systems. First, we study stability radii of linear time-invariant parameter-varying differential systems. A formula for complex stability radius under multi perturbations is given. Then, under hypotheses of positivity of the system matrices, we prove that the complex, real and positive stability radius of the system under multi perturbations coincide and they are computed via a simple formula. As applications, we consider problems of computing of (strong) stability radii of linear time-invariant time-delay differential systems and computing of stability radii of positive linear functional differential equations under multi perturbations. We show that for a class of positive linear time-delay differential systems, the stability radii of the system under multi perturbations are equal to the strong stability radii. Next, we prove that the stability radii of a positive linear functional differential equation under multi perturbations are equal to those of the associated linear time-invariant parameter-varying differential system. In particular, we get back some explicit formulas for these stability radii which were given by Ngoc et al., (2005). For sake of space, exposure is kept to minimum in this paper