$\ell _{1}$-Induced Controller Design for Positive Systems

Abstract

In this chapter, the problem of $\ell _{1}$-induced state-feedback controller design is investigated for discrete-time positive systems based on linear Lyapunov functions. For a given positive linear system, the objective is to design a state-feedback controller such that the closed-loop system is positive, asymptotically stable and satisfies a prescribed $\ell _{1}$-induced performance. In detail, the exact value of $\ell _{1}$-induced norm is firstly computed with the use of an analytical method. A novel stability and $\ell _{1}$-induced performance characterization is then proposed. Based on such a characterization, necessary and sufficient conditions are derived for the existence of desired controllers. To solve the conditions, a corresponding iterative convex optimization algorithm is developed. Secondly, the controller synthesis problem for single-input multiple-output (SIMO) positive systems is investigated. Specifically speaking, an analytical method is established to obtain the optimal $\ell _{1}$-induced controller for the special case. Moreover, some links to the spectral radius of the closed-loop systems are provided. Finally, the effectiveness of the obtained theoretical results in this chapter are illustrated through a numerical example.