Asymptotic Stability and Decay Rates of Homogeneous Positive Systems With Bounded and Unbounded Delays

TL;DR

This paper investigates the stability and decay rates of nonlinear positive systems with time-varying, possibly unbounded delays, providing conditions for delay-independent stability and explicit decay rate formulas for continuous and discrete systems.

Contribution

It establishes necessary and sufficient conditions for delay-independent stability in positive systems with homogeneous vector fields, extending stability analysis to unbounded and time-varying delays.

Findings

Delay-independent stability conditions derived for continuous-time positive systems.

Explicit decay rate formulas for various classes of time delays.

Discrete-time positive systems also analyzed with stability and decay bounds.

Abstract

There are several results on the stability of nonlinear positive systems in the presence of time delays. However, most of them assume that the delays are constant. This paper considers time-varying, possibly unbounded, delays and establishes asymptotic stability and bounds the decay rate of a significant class of nonlinear positive systems which includes positive linear systems as a special case. Specifically, we present a necessary and sufficient condition for delay-independent stability of continuous-time positive systems whose vector fields are cooperative and homogeneous. We show that global asymptotic stability of such systems is independent of the magnitude and variation of the time delays. For various classes of time delays, we are able to derive explicit expressions that quantify the decay rates of positive systems. We also provide the corresponding counterparts for discrete-time positive systems whose vector fields are non-decreasing and homogeneous.