Positive stabilization with maximum stability radius for linear time-delay systems

This paper considers the problem of positive stabilization of uncertain linear time-delay systems by state feedback such that the resulting closed-loop system attains maximum stability radius with positivity constraint. First, we focus on the class of linear continuous-time positive delay systems (Metzlerian delay systems) and outline its interesting properties. Using the stability properties associated with this class, we formulate a constrained stabilization problem for the linear time delay system and provide conditions for the existence of controllers satisfying the stability and Metzlerian constraints. The Metzlerian stabilization is solved using Linear Matrix Inequality (LMI) or Linear Programming (LP). Next, we characterize the uncertainties associated with the positive delay systems and define the stability radius associated with this class which can be expressed in a closed form. Finally, we combine Metzlerian stabilization with maximum stability radius with the aid of bounded real lemma (BRL) and provide a complete solution using LMI. Examples are included for the purpose of illustration.