Persistence analysis of interconnected positive systems with communication delays

This is a continuation of our preceding studies on the analysis of interconnected positive systems. Under mild conditions on positive subsystems and a nonnegative interconnection matrix, we showed that the state of the interconnected positive system converges to a positive scalar multiple of a prescribed positive vector. As a byproduct of this property, called persistence, it turned out that the output converges to the positive right eigenvector of the interconnection matrix. This result is effectively used in the formation control of multi-agent positive systems. The goal of this paper is to prove that the essential property of persistence is still preserved under arbitrary (time-invariant) communication delays. In the context of formation control, this preservation indicates that the desired formation is achieved robustly against communication delays, even though the resulting formation is scaled depending upon initial conditions for the state. From a mathematical point of view, the key issue is to prove that the delay interconnected positive system has stable poles only except for a pole of degree one at the origin, even though it has infinitely many poles in general. To this end, we develop frequency-domain (s-domain) analysis for delay interconnected positive systems.