Weighted ℒ1-gain observers for discrete-time switched positive linear systems with time-varying delay

This paper investigates the £₁-gain observation problem for discrete-time switched positive linear systems with time-varying delay. An extended Luenberger positive observer is designed to efficiently estimate the nonnegative states of the underlying system. First, we analyze the stability with £₁-gain of the associated error system with average dwell time switching under the positivity assumption by means of a piecewise linear copositive Lyapunov-Krasovskii functional. Then, based on the obtained result, we propose a sufficient condition, which guarantees that not only the designed observer is positive but also the error system is globally uniformly exponentially stable with a weighted £₁-gain bound for a class of switching signals with average dwell time (ADT) greater than a certain positive constant. The derived condition is formulated as a set of linear inequalities, by solving which the desired observer gain matrices and the corresponding minimal ADT are easily gotten resorting to existing linear programming toolboxes. Finally, a numerical example is given to illustrate the effectiveness of the presented approach.