Positive Systems with Distributed Delays

Abstract

This chapter is concerned with the $L_{\infty }$-gain analysis problem for positive linear systems with distributed delays. Inspired by the results in Chap. 2, for positive systems with distributed delays, it is natural to ask how distributed delays affect the $L_\infty$-gain. In this chapter, we provide a complete answer to this question. It turns out that unlike time-varying discrete delays, distributed delays do have impact on the $L_\infty$-gain of positive systems. Different from the existing techniques for the analysis of positive systems, in this chapter, a simple approach is provided which only needs to look into the monotonicity of the distributed delay system. By virtue of the positivity and linearity of the system, it is shown that the $L_\infty$-gain of an asymptotically stable positive system with distributed delays is the same as that of a corresponding delay-free system.