Max-Plus $(A,B)$-Invariant Spaces and Control of Timed Discrete-Event Systems

The concept of (A,B)-invariant subspace (or controlled invariant) of a linear dynamical system is extended to linear systems over the max-plus semiring. Although this extension presents several difficulties, which are similar to those encountered in the same kind of extension to linear dynamical systems over rings, it appears capable of providing solutions to many control problems like in the cases of linear systems over fields or rings. Sufficient conditions are given for computing the maximal (A,B)-invariant subspace contained in a given space and the existence of linear state feedbacks is discussed. An application to the study of transportation networks which evolve according to a timetable is considered

[1]  S. Gaubert,et al.  THE DUALITY THEOREM FOR MIN-MAX FUNCTIONS , 1997 .

[2]  J. G. Braker,et al.  Algorithms and Applications in Timed Discrete Event Systems , 1993 .

[3]  Jean Assan.  Analyse et synthèse de l'approche géométrique pour les systèmes linéaires sur un anneau , 1999 .

[4]  Jean-François Lafay,et al.  On feedback invariance properties for systems over a principal ideal domain , 1999, IEEE Trans. Autom. Control..

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

[6]  Wilton R. Abbott,et al.  Network Calculus , 1970 .

[7]  A. M. Perdon,et al.  Problems and Results in a Geometric Approach to the Theory of Systems over Rings , 1994 .

[8]  M. L. J. Hautus.  Disturbance rejection for systems over rings , 1984 .

[9]  Laurent Hardouin,et al.  Just-in-time control of timed event graphs: update of reference input, presence of uncontrollable input , 2000, IEEE Trans. Autom. Control..

[10]  Stéphane Gaubert,et al.  Rational semimodules over the max-plus semiring and geometric approach to discrete event systems , 2004, Kybernetika.

[11]  Bertrand Cottenceau,et al.  Synthesis of greatest linear feedback for timed-event graphs in dioid , 1999, IEEE Trans. Autom. Control..

[12]  Christos G. Cassandras,et al.  A new approach to the analysis of discrete event dynamic systems , 1983, Autom..

[13]  J. Quadrat,et al.  Algebraic tools for the performance evaluation of discrete event systems , 1989, Proc. IEEE.

[14]  Christos G. Cassandras,et al.  Introduction to the Modelling, Control and Optimization of Discrete Event Systems , 1995 .

[15]  Gaetano Borriello,et al.  Idempotency: A general linear max-plus solution technique , 1998 .

[16]  Vijay K. Garg,et al.  Modeling and Control of Logical Discrete Event Systems , 1994 .

[17]  Stéphane Gaubert,et al.  Methods and Applications of (MAX, +) Linear Algebra , 1997, STACS.

[18]  B. Schutter,et al.  On max-algebraic models for transportation networks , 1998 .

[19]  Imre Simon,et al.  Limited subsets of a free monoid , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[20]  Ines Klimann,et al.  A solution to the problem of (A, B)-invariance for series , 2003, Theor. Comput. Sci..

[21]  Peter Butkovic,et al.  The equation A⊗x=B⊗y over (max, +) , 2003, Theor. Comput. Sci..

[22]  Didier Dubois,et al.  A linear-system-theoretic view of discrete-event processes , 1983 .

[23]  Jean Cochet-Terrasson.  A constructive xed point theorem for min-max functions , 1999 .

[24]  P. Ramadge,et al.  Supervisory control of a class of discrete event processes , 1987 .

[25]  Ricardo D. Katz,et al.  Reachability and Invariance Problems in Max-plus Algebra , 2003, POSTA.

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

[27]  Subiono,et al.  On Large Scale Max-Plus Algebra Models in Railway Systems , 1998 .

[28]  Mehdi Lhommeau.  Etude de systèmes à événements discrets dans l'algèbre (max,+) : synthèse de correcteurs robustes dans un dioi͏̈de d'intervalles : synthèse de correcteurs en présence de perturbations , 2003 .

[29]  J. Pin.  Tropical Semirings , 2005 .

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

[31]  G. Basile,et al.  Controlled and conditioned invariants in linear system theory , 1992 .

[32]  M. Hautus.  Controlled invariance in systems over rings , 1982 .

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

[34]  J. Pearson.  Linear multivariable control, a geometric approach , 1977 .

[35]  Anna Maria Perdon,et al.  The Disturbance Decoupling Problem for Systems Over a Ring , 1995 .

[36]  Stéphane Gaubert,et al.  Duality of Idempotent Semimodules , 2001 .