A biology-inspired approach to the positive integral control of positive systems -- the antithetic, exponential, and logistic integral controllers

TL;DR

This paper develops a biological-inspired theory for positive integral control using nonnegative controls, introducing antithetic, exponential, and logistic integral controllers, with stability analysis and practical examples.

Contribution

It introduces novel biologically inspired positive integral controllers and analyzes their stability properties, extending control theory to biological systems.

Findings

Antithetic controller's stability is independent of the coupling parameter with proper gain.

Exponential controller's stability depends only on its exponential rate.

Local stability conditions are established for all proposed controllers.

Abstract

The integral control of positive systems using nonnegative control input is an important problem arising, among others, in biochemistry, epidemiology and ecology. An immediate solution is to use an ON-OFF nonlinearity between the controller and the system. However, this solution is only available when controllers are implemented in computer systems. When this is not the case, like in biology, alternative approaches need to be explored. Based on recent research in the control of biological systems, we propose to develop a theory for the integral control of positive systems using nonnegative controls based on the so-called \emph{antithetic integral controller} and two \emph{positively regularized integral controllers}, the so-called \emph{exponential integral controller} and \emph{logistic integral controller}. For all these controllers, we establish several qualitative results, which we connect to standard results on integral control. We also obtain additional results which are specific to the type of controllers. For instance, we show an interesting result stipulating that if the gain of the antithetic integral controller is suitably chosen, then the local stability of the equilibrium point of the closed-loop system does not depend on the choice for the coupling parameter, an additional parameter specific to this controller. Conversely, we also show that if the coupling parameter is suitably chosen, then the equilibrium point of the closed-loop system is locally stable regardless the value of the gain. For the exponential integral controller, we can show that the local stability of the equilibrium point of the closed-loop system is independent of the gain of the controller and the gain of the system. The stability only depends on the exponential rate of the controller, again a parameter that is specific to this type of controllers. Several examples are given for illustration.