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

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