Optimal Balancing of Tropical Discrete-Event Systems Through Feedback Control

Dynamical Tropical systems are described by means of Tropical Algebra (for instance, Min- or Max-plus ones), which is a kind of idempotent semifield. For such systems, we are interested in the study of general algebraic properties ensuring optimal balancing through feedback control. By balancing, we mean that all events, or transitions, occur at the same rate, meaning that there is no sub-product accumulation inside the system. In this context, after formulating the problem for Tropical Semifields, the first result is the development, thanks to Residuation Theory, of the expression of the maximum feedback matrix expressed in terms of a vector parameter, ensuring that the closed-loop matrix has a desired eigenvalue. Under the assumption of controllability and boundedness of the controllability matrix, we develop a method to properly choose this maximum feedback matrix. In order to illustrate the method, we present a solution for the problem of balancing two unconnected networks by means of feedback control.