$L_{1}$-Induced Output-Feedback Controller Synthesis for Interval Positive Systems

Abstract

This chapter is concerned with the design of $L_{1}$-induced output-feedback controller for continuous-time positive systems with interval uncertainties. A necessary and sufficient condition for stability and $L_{1}$-induced performance of positive linear systems is proposed in terms of linear inequalities. Based on this, conditions for the existence of robust static output-feedback controllers are established and an iterative convex optimization approach is developed to solve the conditions. The problem of controller synthesis is completely solved for single-input-multiple-output (SIMO) positive systems with the help of an analytical formula for $L_{1}$-induced norm. The result is further extended to the multiple-input-multiple-output (MIMO) case by employing a structured controller. Moreover, the design of $L_{1}$-induced sparse controller is investigated for continuous-time interval positive systems. In addition, a dynamic output-feedback controller is designed for interval positive systems under $L_1$ performance. Three illustrative examples are provided to show the effectiveness and applicability of the theoretical results.