L1-gain Analysis and Control of Positive Linear Systems with Delays

In this paper, the problem of L-1-gain analysis and control is investigated for positive linear systems with delays. Due to the nonnegative property of positive systems, it is natural to evaluate the system performance via the L-1-gain. Firstly, the necessary and sufficient condition is given to guarantee that the positive linear delay system is asymptotically stable and satisfies the prescibed L-1-gain. Then, based on the above condition, for the single-input system, by using the singular value decomposition (SVD) approach, the necessary and sufficient condition is provided for the existence of the L-1-controller. This condition can be described as a linear programming (LP) problem. The proposed approach can be extended to the multi-input case and the existence condition is provided for the controller with a special form. Finally, a numerical example is provided to show the effectiveness and correctness of the theoretical results.