Max-algebraic hybrid automata: Modelling and equivalences

This article introduces the novel framework of max-algebraic hybrid automata as a hybrid modelling language in the max-plus algebra. We show that the modelling framework unifies and extends the switching max-plus linear systems framework and is analogous to the discrete hybrid automata framework in conventional algebra. In addition, we show that the framework serves as a bridge between automata-theoretic models in max-plus algebra and switching max-plus linear systems. In doing so, we formalise the relationship between max-plus automata and switching max-plus linear systems in a behavioural sense. This also serves as another step towards importing tools for analysis and optimal control from conventional time-driven hybrid systems to discrete-event systems in max-plus algebra.

[1]  Jean Mairesse,et al.  Modeling and analysis of timed Petri nets using heaps of pieces , 1997, 1997 European Control Conference (ECC).

[2]  Bart De Schutter,et al.  Modeling and control of switching max-plus-linear systems with random and deterministic switching , 2012, Discret. Event Dyn. Syst..

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

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

[5]  G. S. Y. Koelemeijer,et al.  ON THE BEHAVIOUR OF CLASSES OF MIN-MAX-PLUS SYSTEMS , 2003 .

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

[7]  Christos G. Cassandras,et al.  Introduction to Discrete Event Systems , 1999, The Kluwer International Series on Discrete Event Dynamic Systems.

[8]  John Lygeros,et al.  Verified hybrid controllers for automated vehicles , 1998, IEEE Trans. Autom. Control..

[9]  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 .

[10]  Alberto Bemporad,et al.  HYSDEL-a tool for generating computational hybrid models for analysis and synthesis problems , 2004, IEEE Transactions on Control Systems Technology.

[11]  Bart De Schutter,et al.  Properties of MPC for Max-Plus-Linear Systems , 2002, Eur. J. Control.

[12]  S. Gaubert.  Performance evaluation of (max, +) automata , 1995, IEEE Trans. Autom. Control..

[13]  Bart De Schutter,et al.  Equivalence of hybrid dynamical models , 2001, Autom..

[14]  Jean-Louis Boimond,et al.  Max-plus algebra in the history of discrete event systems , 2018, Annu. Rev. Control..

[15]  A.J. van der Schaft,et al.  Bisimulation as congruence in the behavioral setting , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

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

[17]  J. Quadrat,et al.  Max-Plus Algebra and System Theory: Where We Are and Where to Go Now , 1999 .

[18]  Ton J.J. van den Boom,et al.  Model predictive scheduling of semi-cyclic discrete-event systems using switching max-plus linear models and dynamic graphs , 2020, Discret. Event Dyn. Syst..

[19]  A. J. van der Schaft.  Equivalence of dynamical systems by bisimulation , 2004, IEEE Transactions on Automatic Control.

[20]  Bart De Schutter,et al.  MPC for continuous piecewise-affine systems , 2004, Syst. Control. Lett..

[21]  Jeremy Gunawardena,et al.  Min-max functions , 1994, Discret. Event Dyn. Syst..

[22]  John Lygeros,et al.  Controllers for reachability specifications for hybrid systems , 1999, Autom..

[23]  Geert Jan Olsder,et al.  Eigenvalues of dynamic max-min systems , 1991, Discret. Event Dyn. Syst..

[24]  J. Lygeros,et al.  AN INTRODUCTION TO HYBRID SYSTEM MODELING ANALYSIS AND CONTROL JOHN LYGEROS GEORGE PAPPAS AND SHANKAR SASTRY , 1999 .

[25]  L.C.G.J.M. Habets,et al.  Book review: Introduction to mathematical systems theory, a behavioral approach , 2000 .