A Tutorial on Positive Systems and Large Scale Control

In this tutorial paper we first present some foundational results regarding the theory of positive systems. In particular, we present fundamental results regarding stability, positive realization and positive stabilization by means of state-feedback. Special attention is also paid to the system performance in terms of disturbance attenuation. Under the asymptotic stability assumption, such performance can be measured in terms of Lp-gain of the positive system. In the second part of the paper we propose some recent results about control synthesis by linear programming and semidefinite programming, under the positivity requirement on the resulting controlled system. These results highlight the value of positivity when dealing with large scale systems. Indeed, stability properties for these systems can be verified by resorting to linear (copositive) or diagonal Lyapunov functions that scale linearly with the system dimension, and such linear functions can be used also to design stabilizing feedback control laws. In addition, stabilization problems with disturbance attenuation performance can be easily solved by imposing special structures on the state feedback matrices. This is extremely valuable when dealing with large scale systems for which state feedback matrices are typically sparse, and their structure is a priori imposed by practical requirements.