Constrained Control of Positive Systems with Delays

This technical note addresses controller design for continuous-time delayed linear systems. The control is under positivity constraint, which means that the resulting closed-loop systems are not only stable, but also positive. Bounded control is also considered, further requiring that the trajectory of the closed-loop system is bounded by a prescribed boundary, if the initial condition is bounded by the same boundary. The contribution lies in three aspects. First, some necessary and sufficient conditions are presented for the existence of controllers satisfying the positivity constraint. Second, a necessary and sufficient condition is established, determining whether the trajectory of a positive system can be bounded by a prescribed boundary, provided that the initial condition is bounded by the same boundary. Third, a sufficient condition is presented for bounded control, and it is shown to be also necessary if asymptotic stability is not required.