Scalable Control of Positive Systems

In this article, we first present some foundational results about the stability and positive stabilization of continuous-time positive systems. Necessary and sufficient conditions for achieving stability are provided, together with some desired performance in terms of disturbance attenuation. These conditions are expressed in terms of linear programming and scale well with the system size. We then discuss the interconnection of positive subsystems by means of a static output feedback that preserves positivity, and propose conditions to achieve both stability and the asymptotic alignment of the closed-loop output to a desired vector. Finally, we describe some results for a class of parameterized positive systems. The second part of the article presents some interesting applications of the results presented in the first part. Specifically, control problems for heating networks, formation control, power control in wireless communication, and the evolutionary dynamics of cancer and HIV are formalized and solved as optimal control problems for positive systems.