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    SMax+

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    Research project SMax+

    Max-Plus Cyclic Systems (non-stationary)

     

    Team: Dynamic and discrete events and Optimization

     

    Labeling: none


    Term: 1 year (2018-2019)

     

    Funding: Atlanstic RFI 2020

     

    LARIS staff involved: Bertrand Cottenceau, Laurent Hardouin

     

    Project partners: Jörg Raisch (TU Berlin)

     

     

    Description

     Max-plus systems are families of timed discrete event systems for which the evolution of the state can be described by recurrent equations on max-plus algebra. This concerns, for example, manufacturing production systems where synchronization phenomena dominate (assembly, rendez-vous), but also some networks of communication ("network calculus") or transport networks (bus, train) where connections generate synchronization phenomena.

     In recent years, we are also interested in systems where grouping / unbundling phenomena appear [5], as well as partial synchronization phenomena [4]. These phenomena correspond for example to palletizing in manufacturing systems or traffic control by means of traffic lights. In terms of dynamics, the systems obtained are then richer than the max-plus systems traditionally considered. The main difference being that they are then non-stationary max-plus systems: their behavior is dependent on the time even of the events occurred. Nevertheless, their behavior can still be described by means of cyclic transfer functions. The response of the system is not stationary but it is repeated with cyclical event and temporal cyclicity, hence the name.

     The work presented in [5] has shown that it is possible to describe some cyclic non-stationary max-plus systems by means of formal series consisting of a finite number of elementary operators. This is the case, for example, of a subclass of Petri nets called Weight-Balanced Timed Event Graphs. Their common point with stationary max-plus systems is that they can be represented by means of rational series with a Kleene's single star. In other words, this work is in line with the work of INRIA's Max-Plus team (Cohen, Gaubert, Quadrat), which showed how stationary max-plus systems could be reduced to formal series in two gamma operators and delta. Moreover, for stationary max-plus systems, a C ++ calculation library has been developed within the Laris' SDO team (MinMaxGD library). This library is essential to carry out the computation of certain control laws (feedback, precompensator, or observer based) on systems max-plus.

     With regard to the cyclic non-stationary max-plus systems, the main difficulty lies in the fact that the algorithms known to manipulate the max-plus systems do not extend immediately. Even though the work done jointly with TU Berlin [1] [2] [3] [4] has led to a better understanding of the manipulation of formal series for cyclic max-plus systems, there is still a lot of development work to be done to have a computational library comparable to the MinMaxGD library.

    This project therefore aims to develop a such a computational tool for cyclic max-plus systems. The development of this library of computation will require to carry out works jointly with TU Berlin and to organize work stays in Angers or Berlin. The purchase of high performance PC computers for the calculation would also be an asset to carry out the mission.

     

    Bibliography

    [1] J. Trunk, B. Cottenceau, L. Hardouin, J. Raisch, "Output Reference Control for Weight-Balanced Timed Event Graphs", CDC'17, Melbourne, Australia, December 2017.

    [2] J. Trunk, B. Cottenceau, L. Hardouin, J. Raisch, "Model Decomposition of Weight-Balanced Timed Event Graphs in Dioids: Application to Control Synthesis" , IFAC World Congress, Toulouse, France, July 2017.

    [3] B. Cottenceau, L. Hardouin, J. Trunk, "Weight-Balanced Timed Event Graphs to Model Periodic Phenomena in Manufacturing Systems" , IEEE TASE, 14(4):1731-1742, October 2017.

    [4] X. David-Henriet, J. Raisch, L. Hardouin, B. Cottenceau, "Modeling and Control for Max-Plus Systems with Partial Synchronization",WODES 2014, Paris, May 2014.

    [5] B. Cottenceau, L. Hardouin, J.L. Boimond, "Modeling and Control of Weight-Balanced Timed Event Graphs in Dioids",IEEE Trans. Automatic Control,  59(5):1219-1231, May 2014.