Stability Radius Characterization of Lσ-Gain for Positive Systems and its Significance in Stabilization with Optimal Performance

Abstract

Abstract This paper provides a connection between stability radius and Lσ-gain of positive systems. The L1-, L2-, and L∞-gains of an asymptotically stable positive system are characterized in terms of stability radii and useful bounds are derived. We show that the structured perturbations of a stable matrix can be regarded as a closed-loop system with unknown static output feedback, which makes it possible to obtain the main results of this paper. In particular, we use the closed-form expressions for stability radii of positive systems to compute the Lσ-gains without resorting to solve optimization problems. We also show that positive stabilization with maximum stability radius can be considered as an L2-gain minimization, which can be solved by LMI. This inherently achieves performance criterion and establishes a link to the reported LP formulations. A significant result of this paper is the unique commonality among the optimal state feedback gain matrices in obtaining Lσ-gains of the stabilized system. Numerical examples are provided to support the theoretical results.