Modeling and Control of Resource Sharing Problems in Dioids Soraia Moradi 1,2,3 , Laurent Hardouin 3 , and J¨ org Raisch 1,2 Abstract— The topic of this paper is the modeling and control of a class of timed Petri nets with resource sharing problems in a dioid framework. We first introduce a signal which denotes the number of resources available for each competing subsystem at each instant of time. Based on this signal, the overall system is modeled in min-plus algebra. Using residuation theory, an optimal control policy is developed, where optimality is in the sense of a lexicographical order reflecting the chosen prioritization of subsystems. I. I NTRODUCTION Timed event graphs (TEG)s are a subclass of timed Petri nets where each place has exactly one upstream and one downstream transition and all arcs have weight 1. The time/event behavior of TEGs, under the earliest functioning rule (i.e., transitions fire as soon as they are enabled), can be expressed linearly over some dioids [1]. TEGs can only model synchronization but not concurrency or choice. In many applications, like railway networks and manufacturing systems, there are only limited resources, which are shared among different users. For example, in a railway network, there may be single track segments which are used by multiple trains, but, at each instant of time, only one train can occupy the track. This problem is called the ”Resource Sharing” (RS) problem. Systems with RS problems can be modeled by timed Petri nets but not by TEGs, as they contain choice or conflict. In the literature, various methods have been investigated to deal with the RS problem. In [2], systems with RS are modeled by switching max-plus linear systems, where a system can switch between different modes of operation and in each mode is modeled by a linear max-plus system. Using model predictive control (MPC), the optimal switching sequence is obtained. In [3], modeling and control of switching max-plus-linear systems with random and deterministic switching have been discussed. In [4], the just in time control problem of switching max-plus linear systems where the switching variable on the study horizon is given is considered. In [5], the model consists of a TEG and some additional inequalities which model the limited availability of shared resources. In [6], conflicting time event graphs are modeled in the max-plus algebra and an approach to calculate the cycle time is proposed. In 1 Fachgebiet Regelungssysteme, Technische Universit¨ at Berlin, Einstein- ufer 17, 10587 Berlin, Germany, 2 Control Systems Group, Max Planck Institute for Dynamics of Complex Technical Systems , Magdeburg, Germany, 3 Laboratoire d’Ing´ enierie des Syst` emes Automatis´ es, Universit´ e d’Angers. 62, Avenue Notre Dame du Lac, 49000 Angers, France. moradi@control.tu-berlin.de, laurent.hardouin@univ-angers.fr, raisch@control.tu-berlin.de [7], modeling and performance evaluation of timed Petri nets with different levels of priority are investigated. Three possible place/transition patterns are considered, namely, conflict, synchronization and priority configurations. In [8], systems with RS problems are modeled in the max-plus algebra. A method to detect conflicts by checking the time line overlaps of processes is introduced. In order to solve conflicts, the schedule is changed to move up the process with low priority. In this paper, a method to model and control the RS problem in the min-plus algebra is proposed. The main con- tribution of this work is as follows. First, a signal denoting the number of resources available for competing subsystems, at each instant of time is introduced. The definition of this signal incorporates a predefined prioritization policy. Based on this signal, the overall system is modeled in min- plus algebra. Using residuation theory, an optimal control policy is developed, where optimality is in the sense of a lexicographical order reflecting the chosen prioritization of subsystems. In essence, we are aiming at firing input transitions of subsystems as late as possible, while making sure that the firing of output transitions is not later than specified in given reference signals. Moreover, the control of lower-priority subsystems may not degrade the performance of higher-priority subsystems. The paper is organized as follows. Section II recalls the necessary algebraic tools. In Section III, modeling of a system with RS problem over the dioid Z min is discussed. Section IV addresses the optimal control problem, and Sec- tion V provides some conclusions. II. ALGEBRAIC PRELIMINARIES The following is a summary of basic results from dioid theory and residuation theory. The interested reader is invited to peruse [1], [9], and [10] for more details. A. Dioid Theory A dioid D is a set endowed with two internal operations denoted ⊕ (addition) and ⊗ (multiplication), both associative and having a neutral element denoted ε (zero element) and e (unit element), respectively. Moreover, ⊕ is commutative and idempotent (∀a ∈D,a ⊕ a = a), ⊗ distributes over ⊕, and ε is absorbing for ⊗ (∀a ∈D,ε ⊗ a = a ⊗ ε = ε). By convention, multiplication is often expressed by juxtapo- sition, i.e., a ⊗ b = ab. The operation ⊕ induces an order relation  on D, defined by: ∀a, b ∈D,a  b ⇔ a ⊕ b = b. A dioid is said to be complete if it is closed for infinite sums and if multiplication distributes over infinite sums. In this case, the greatest (in the sense of the above order) element Proceedings of the 13th International Workshop on Discrete Event Systems, Xi'an, China, May 30 - June 1, 2016 We_1_B.4 978-1-5090-4190-9/16/$31.00 ©2016 IEEE 410