ℓ1ℓ1-induced norm and controller synthesis of positive systems

In this paper, the problem of ℓ1ℓ1-induced controller design for discrete-time positive systems is investigated with the use of linear Lyapunov function. An analytical method to compute the exact value of ℓ1ℓ1-induced norm is first presented. Then, a novel characterization for stability and ℓ1ℓ1-induced performance is proposed. Based on the characterization, a necessary and sufficient condition for the existence of desired controllers is derived, and an iterative convex optimization approach is developed to solve the condition. In addition, the synthesis of the state-feedback controller for single-input multiple-output (SIMO) positive systems is investigated. For this special case, an analytic solution is established to show how the optimal ℓ1ℓ1-induced controller can be designed, and some links to the spectral radius of the closed-loop systems are provided. Finally, the theoretical results are illustrated through a numerical example.