Section: New Results
Optimal control of systems
Participants : Zied Achour, Nidhal Rezg, Alexandre Sava.
The research work that our team is doing in this domain deals with discrete event systems modeled by Petri nets. Specifications considered are modeled by General Mutual Exclusion Constraints. In the past out team has developed control synthesis techniques which based on the theory of regions. These techniques provide efficient controllers for discrete event systems modeled by bounded Petri nets. Furthermore, we proposed an efficient control synthesis algorithm for marked graphs not necessarily bounded. However, this approach does not take into account the liveness of the closed loop system.
This year we continued our research work on developing a deadlock free controller for discrete event systems modeled by live marked graphs subject to General Mutual Exclusion Constraints. We show that the restriction imposed by controller may generate deadlocks even if the marked graph to be controlled is live. Thus, we prove that the risk of deadlock is a consequence of the existence of a particular structure that we call risky transitions. Furthermore, we proposed a sufficient condition to avoid deadlocks and developed a suboptimal deadlock free control synthesis method for marked graphs not necessarily bounded [24] . Then we improved this technique by providing a necessary and sufficient condition to avoid deadlocks and we proposed a maximal permissive deadlock free controller for the same class of control problems [61] . However, the occurrence of some events can not be observed during the evolution of the system. Therefore, the next step of our research work is to integrate the events observability and propose a deadlock free control synthesis for partially observable marked graphs.
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 Constraint (MEC). The main advantage of MEC specification is an increased modeling power regarding General Mutual Exclusion Constraints (GMEC). We define two types of MEC: MEC-OR and MEC-AND and we propose a technique to build the controller which enforces MEC specifications for discrete events systems modeled by marked graphs [52] .
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 which 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.
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 proposed approach uses time Petri net modeling tool to represent an air traffic management system. Then we built the time reachability graph and we associate a given cost to each state. The 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. We also take into account the perturbations in the capacity of airways which may vary according to climatic conditions. 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 proposed allows computing the optimal flying scenario for an air traffic system made of two airports and several airways with variable capacity [56] . Actually, we are concerned with extending this problem to air traffic management systems with multiple airports and dynamic resources allocation.
