Modelling and control of periodic time-variant event graphs in dioids

Timed Event Graphs (TEGs) can be described by time invariant (max,+) linear systems. This formalism has been studied for modelling, analysis and control synthesis for decision-free timed Discrete Event Systems (DESs), for instance specific manufacturing processes or transportation networks operating under a given logical schedule. However, many applications exhibit time-variant behaviour, which cannot be modelled in a standard TEG framework. In this paper we extend the class of TEGs in order to include certain periodic time-variant behaviours. This extended class of TEGs is called Periodic Time-variant Event Graphs (PTEGs). It is shown that the input-output behaviour of these systems can be described by means of ultimately periodic series in a dioid of formal power series. These series represent transfer functions of PTEGs and are a convenient basis for performance analysis and controller synthesis.

[1]  S. Gaubert.  Theorie des systemes lineaires dans les dioides , 1992 .

[2]  Bertrand Cottenceau,et al.  Modeling and control for (max, +)-linear systems with set-based constraints , 2015, 2015 IEEE International Conference on Automation Science and Engineering (CASE).

[3]  Isabel Demongodin,et al.  Max-Plus Control Design for Temporal Constraints Meeting in Timed Event Graphs , 2012, IEEE Transactions on Automatic Control.

[4]  M. Plus.  Second Order Theory of Min-Linear Systems and its Application to Discrete Event Systems , 1998 .

[5]  J. Boimond,et al.  Just-in-time control of time-varying discrete event dynamic systems in (max,+) algebra , 2008 .

[6]  Bertrand Cottenceau,et al.  Modeling and Control of Weight-Balanced Timed Event Graphs in Dioids , 2014, IEEE Transactions on Automatic Control.

[7]  Eric Thierry,et al.  An Algorithmic Toolbox for Network Calculus , 2008, Discret. Event Dyn. Syst..

[8]  J. Loiseau,et al.  Model matching for timed event graphs , 1996 .

[9]  Bertrand Cottenceau,et al.  Observer-Based Controllers for Max-Plus Linear Systems , 2017, IEEE Transactions on Automatic Control.

[10]  Bart De Schutter,et al.  Model predictive control for max-plus-linear discrete event systems , 2001, Autom..

[11]  Ronald L. Graham,et al.  Concrete mathematics - a foundation for computer science , 1991 .

[12]  Bertrand Cottenceau,et al.  Optimal closed-loop control of timed EventGraphs in dioids , 2003, IEEE Trans. Autom. Control..

[13]  Bertrand Cottenceau,et al.  Control and State Estimation for Max-Plus Linear Systems , 2018, Found. Trends Syst. Control..

[14]  Bertrand Cottenceau,et al.  Weight-Balanced Timed Event Graphs to Model Periodic Phenomena in Manufacturing Systems , 2017, IEEE Transactions on Automation Science and Engineering.

[15]  Laurent Hardouin,et al.  Model Decomposition of Timed Event Graphs under Partial Synchronization in Dioids , 2018 .

[16]  Geert Jan Olsder,et al.  Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications , 2005 .

[17]  Bertrand Cottenceau,et al.  Modeling of Time-Varying (max, +) Systems by Means of Weighted Timed Event Graphs , 2014, WODES.

[18]  V. Garg.  A (max,+) Algebra for Non-stationary Periodic Timed Discrete Event Systems , 1998 .

[19]  Stéphane Gaubert,et al.  Rational computation in dioid algebra and its application to performance evaluation of Discrete Event Systems , 1991 .

[20]  Second Order Theory of Min-Linear Systems Discrete Event Systems , 1991 .

[21]  Geert Jan Olsder,et al.  Synchronization and Linearity: An Algebra for Discrete Event Systems , 1994 .

[22]  Laurent Hardouin,et al.  Modeling and Control for Max-Plus Systems with Partial Synchronization , 2014, WODES.

[23]  A. Forbes.  Modeling and control , 1990, Journal of Clinical Monitoring.

[24]  Vijay K. Garg,et al.  A Generalized Max-Algebra Model for Performance Analysis of Timed and Untimed Discrete Event Systems , 1993, 1993 American Control Conference.