Modeling and control for (max, +)-linear systems with set-based constraints

We consider (max, +)-linear systems (i.e., discrete event systems ruled by conditions of the form: “for all k ≥l, occurrence k of event β2 is at least τ units of time after occurrence k-l of event e₁” with τ, l ϵ N₀) with the additional restriction that certain events can only occur at time instants in predefined ultimately periodic sets of integers. Such phenomena frequently occur in transportation networks (e.g., a vehicle passes a traffic light only when it is green). In this paper, we extend transfer function matrix models and model reference control developed for (max, +)-linear systems to the above class of systems, which is more general than (max, +)-linear systems. To illustrate these results, we consider a road traffic example with traffic lights.