Stability and -Gain Control of Positive Switched Systems with Time-Varying Delays via Delta Operator Approach

This paper investigates the problems of stability and <svg style="vertical-align:-3.27605pt;width:12.15px;" id="M2" height="17.424999" version="1.1" viewBox="0 0 12.15 17.424999" width="12.15" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.625,12.725)"><use xlink:href="#x1D459"/></g> <g transform="matrix(.012,-0,0,-.012,5.2,16.8)"><use xlink:href="#x31"/></g> </svg>-gain controller design for positive switched systems with time-varying delays via delta operator approach. The purpose is to design a switching signal and a state feedback controller such that the resulting closed-loop system is exponentially stable with <svg style="vertical-align:-3.27605pt;width:12.15px;" id="M3" height="17.424999" version="1.1" viewBox="0 0 12.15 17.424999" width="12.15" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.625,12.725)"><use xlink:href="#x1D459"/></g> <g transform="matrix(.012,-0,0,-.012,5.2,16.8)"><use xlink:href="#x31"/></g> </svg>-gain performance. Based on the average dwell time approach, a sufficient condition for the existence of an <svg style="vertical-align:-3.27605pt;width:12.15px;" id="M4" height="17.424999" version="1.1" viewBox="0 0 12.15 17.424999" width="12.15" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.625,12.725)"><use xlink:href="#x1D459"/></g> <g transform="matrix(.012,-0,0,-.012,5.2,16.8)"><use xlink:href="#x31"/></g> </svg>-gain controller for the considered system is established by constructing an appropriate copositive type Lyapunov-Krasovskii functional in delta domain. Moreover, the obtained conditions can unify some previously suggested relevant methods in the literature of both continuous- and discrete-time systems into the delta operator framework. Finally, a numerical example is presented to explicitly demonstrate the effectiveness and feasibility of the proposed method.