## Section: New Results

### Optimal control of systems

Participants : Nidhal Rezg, Alexandre Sava.

The research activity that our team develops on this topic concerns discrete event systems modeled by Petri nets. In the past, our team has developed control synthesis techniques based on the theory of regions. These techniques provide efficient controllers to enforce General Mutual Exclusion Constraints on discrete event systems modeled by bounded Petri nets. Furthermore, a particular class of Petri nets, called marked graphs, triggered our attention due to its ability for modeling and performance evaluation of flexible manufacturing system. We proposed an efficient control synthesis algorithm for marked graphs not necessarily bounded. This approach does not take into account the liveness of the closed loop system. However, deadlock avoiding is an important specification for industrial applications. Therefore, we explored this problem while including progressively real specifications like the uncontrollable and/or unobservable nature of some events and the time information on event occurrence.

Thus, we developed a control synthesis approach for partially controllable marked graphs subject to General Mutual Exclusion Constraints. We prove that the risk of deadlock is a consequence of the existence of a particular structure that we call risky transitions. Then, we proposed a sufficient condition to avoid deadlocks and developed a deadlock free control synthesis method for marked graphs not necessarily bounded [46] . This approach was extended in [64] by taking into account the unobservable nature of some events.

Some control synthesis problems are subject to specifications which consist in avoiding given values for the marking of Petri net places. In order to handle these problems, we propose a new type of specifications called Marking Exclusion Constraints (MEC). The main advantage of MEC specification is an increased modeling power regarding General Mutual Exclusion Constraints (GMEC). We define three types of MEC: MEC-OR and MEC-AND and M-MEC and we proposed a technique to build the controller to enforce MEC specifications for discrete events systems modeled by partially controllable marked graphs [29] , [28] . The control synthesis problem for partial observable marked graphs subject to marking Exclusion Constraints has been studied in [27] .

The principle of control synthesis is to authorize or forbid the occurrence of controllable events according to the state of the system in order to prevent the evolution of the system to states that are not desirable. Time information on the occurrence of the events can be used to compute more permissive control laws as the controller does no longer need to completely forbid the execution of an event. Time introduces a new dimension of considerable interest in DES control, but also of significant complexity. Actually our research work deals deadlock free control synthesis for timed marked graphs subject to GMEC and MEC specifications. Some preliminary work has been done on control synthesis for partially controllable timed marked graphs subject to MEC constraints [59] . However, the deadlock avoidance problem is to be solved.

Another research direction deals with building optimal control laws which aim to optimize given performance criteria while respecting specifications on resource availability and security. The applications concerned by this research activity are the air traffic control systems. The approach we are working on combines time Petri nets and Binary Decision Diagrams to obtain a reasonable size representation for the space state of an air traffic network. A cost function is associated to each state. This cost function takes into account the waiting time before take off, the cost of flight canceling and the cost of carburant burned by the airplane during a flight. This cost function allows the evaluation of given flight plans. Our aim is to build a decision making tool to help air traffic controllers to find the best flight plan while being faced to variations of the capacity of the airways due to weather perturbations. To overcome these perturbations, different flying scenarios are generated which include: 1) delay the flight; 2) using other airways and 3) cancel the flight. The approach that we developed during this year consists in the following steps: 1) build the time Petri net model for the air traffic network; 2) generate the BDD model for the solutions space; 3) associate a cost value to each BDD node and 4) calculate the optimal flight plan based on a Branch and bound approach [62] [60] .

Furthermore, we focus on using ordinal optimization techniques to generate good enough flight plans in a reasonable time. We are also concerned with extending this problem to air traffic management systems with multiple airports and dynamic allocation of resources.