Previous |  Up |  Next

# Article

Full entry | PDF   (0.9 MB)
Keywords:
max-plus algebra; control; monotonicity; positive invariance; residuation; duality
Summary:
Haar's Lemma (1918) deals with the algebraic characterization of the inclusion of polyhedral sets. This Lemma has been involved many times in automatic control of linear dynamical systems via positive invariance of polyhedrons. More recently, it has been used to characterize stochastic comparison w.r.t. linear/integral ordering of Markov (reward) chains. In this paper we develop a state space oriented approach to the control of Discrete Event Systems (DES) based on the remark that most of control constraints of practical interest are naturally expressed as the inclusion of two systems of linear (w.r.t. idempotent semiring or semifield operations) inequalities. Thus, we establish tropical version of Haar's Lemma to obtain the algebraic characterization of such inclusion. As in the linear case this Lemma exhibits the links between two apparently different problems: comparison of DES and control via positive invariance. Our approach to the control differs from the ones based on formal series and is a kind of dual approach of the geometric one recently developed. Control oriented applications of the main results of the paper are given. One of these applications concerns the study of transportation networks which evolve according to a time table. Although complexity of calculus is discussed the algorithmic implementation needs further work and is beyond the scope of this paper.
References:
[1] Ahmane M., Ledoux, J., Truffet L.: Criteria for the comparison of discrete-time Markov chains. In: 13th Internat. Workshop on Matrices and Statistics in Celebration of I. Olkin’s 80th Birthday, Poland, August 18-21, 2004
[2] Ahmane M., Ledoux, J., Truffet L.: Positive invariance of polyhedrons and comparison of Markov reward models with different state spaces. In: Proc. Positive Systems: Theory and Applications (POSTA’06), Grenoble 2006 (Lecture Notes in Control and Information Sciences 341), Springer–Verlag, Berlin, pp. 153–160 MR 2250251 | Zbl 1132.93333
[3] Ahmane M., Truffet L.: State feedback control via positive invariance for max-plus linear systems using $\Gamma$-algorithm. In: 11th IEEE Internat. Conference on Emerging Technologies and Factory Automation, ETFA’06, Prague 2006
[4] Ahmane M., Truffet L.: Sufficient condition of max-plus ellipsoidal invariant set and computation of feedback control of discrete events systems. In: 3rd Internat. Conference on Informatics in Control, Automation and Robotics, ICINCO’06, Setubal 2006
[5] Aubin J.-P.: Viability Theory. Birkhäuser, Basel 1991 MR 1134779
[6] Baccelli F., Cohen G., Olsder G. J., Quadrat J.-P.: Synchronization and Linearity. Wiley, New York 1992 MR 1204266 | Zbl 0824.93003
[7] Bertsekas D. P., Rhodes I. B.: On the minimax reachability of target sets and target tubes. Automatica 7 (1971), 233–247 MR 0322648 | Zbl 0215.21801
[8] Blyth T. S., Janowitz M. F.: Residuation Theory. Pergamon Press, 1972 MR 0396359 | Zbl 0301.06001
[9] Braker J. G.: Max-algebra modelling and analysis of time-table dependent networks. In: Proc. 1st European Control Conference, Grenoble 1991, pp. 1831–1836
[10] Butkovic P., Zimmermann K.: A strongly polynomial algorithm for solving two-wided linear systems in max-algebra. Discrete Appl. Math. 154 (2006), 437–446 MR 2203194
[11] Cochet-Terrasson J., Gaubert, S., Gunawardena J.: A constructive fixed point theorem for min-max functions. Dynamics Stability Systems 14 (1999), 4, 407–433 MR 1746112 | Zbl 0958.47028
[12] Cohen G., Gaubert, S., Quadrat J.-P.: Duality and separation theorems in idempotent semimodules. Linear Algebra Appl. 379 (2004), 395–422 MR 2039751 | Zbl 1042.46004
[13] Costan A., Gaubert S., Goubault, E., Putot S.: A policy iteration algorithm for computing fixed points in static analysis of programs. In: CAV’05, Edinburgh 2005 (Lecture Notes in Computer Science 3576), Springer–Verlag, Berlin, pp. 462–475 Zbl 1081.68616
[14] Cuninghame-Green R. A., Butkovic P.: The equation $A \otimes x= B \otimes y$ over $(\max ,+)$. Theoret. Comp. Sci. 293 (2003), 3–12 MR 1957609 | Zbl 1021.65022
[15] Vries R. de, Schutter, B. De, Moor B. De: On max-algebraic models for transportation networks. In: Proc. Internat. Workshop on Discrete Event Systems (WODES’98), Cagliari 1998, pp. 457–462
[16] Farkas J.: Über der einfachen Ungleichungen. J. Reine Angew. Math. 124 (1902), 1–27
[17] Gaubert S., Gunawardena J.: The duality theorem for min-max functions. Comptes Rendus Acad. Sci. 326 (1999), Série I, 43–48 MR 1649473
[18] Gaubert S., Katz R.: Rational semimodules over the max-plus semiring and geometric approach of discrete event systems. Kybernetika 40 (2004), 2, 153–180 MR 2069176
[19] Golan J. S.: The theory of semirings with applications in mathematics and theoretical computer science. Longman Sci. & Tech. 54 (1992) MR 1163371 | Zbl 0780.16036
[20] Haar A.: Über Lineare Ungleichungen. 1918. Reprinted in: A. Haar, Gesammelte Arbeiten, Akademi Kiadó, Budapest 1959
[21] Hennet J. C.: Une Extension du Lemme de Farkas et Son Application au Problème de Régulation Linéaire sous Contraintes. Comptes Rendus Acad. Sci. 308 (1989), Série I, pp. 415–419 MR 0992520
[22] Hiriart-Urruty J.-B., Lemarechal C.: Fundamentals of Convex Analysis. Springer–Verlag, Berlin 2001 MR 1865628 | Zbl 0998.49001
[23] Katz R. D.: Max-plus (A,B)-invariant and control of discrete event systems. To appear in IEEE TAC, 2005, arXiv:math.OC/0503448
[24] Klimann I.: A solution to the problem of (A,B)-invariance for series. Theoret. Comput. Sci. 293 (2003), 1, 115–139 MR 1957615 | Zbl 1025.68050
[25] Ledoux J., Truffet L.: Comparison and aggregation of max-plus linear systems. Linear Algebra. Appl. 378 (2004), 1, 245–272 MR 2031795 | Zbl 1052.15014
[26] Lhommeau M.: Etude de Systèmes à Evénements Discrets: 1. Synthèse de Correcteurs Robustes dans un Dioide d’Intervalles. 2. Synthèse de Correcteurs en Présence de Perturbations. PhD Thesis, Université d’Angers ISTIA 2003
[27] Lhommeau M., Hardouin, L., Cottenceau B.: Optimal control for (Max,+)-linear systems in the presence of disturbances. In: Proc. Positive Systems: Theory and Applications (POSTA’03), Roma 2003 (Lecture Notes in Control and Information Sciences 294), Springer–Verlag, Berlin, pp. 47–54 MR 2019300 | Zbl 1059.93090
[28] Muller A., Stoyan D.: Comparison Methods for Stochastic Models and Risks. Wiley, New York 2002 MR 1889865
[29] Dam A. A. ten, Nieuwenhuis J. W.: A linear programming algorithm for invariant polyhedral sets of discrete-time linear systems. Systems Control Lett. 25 (1995), 337–341 MR 1343218
[30] Truffet L.: Monotone linear dynamical systems over dioids. In: Proc. Positive Systems: Theory and Applications (POSTA’03), Roma 2003 (Lecture Notes in Control and Information Sciences 294), Springer–Verlag, Berlin pp. 39–46 MR 2019299 | Zbl 1059.93093
[31] Truffet L.: Some ideas to compare Bellman chains. Kybernetika 39 (2003), 2, 155–163. (Special Issue on max-plus Algebra) MR 1996554
[32] Truffet L.: Exploring positively invariant sets by linear systems over idempotent semirings. IMA J. Math. Control Inform. 21 (2004), 307–322 MR 2076223 | Zbl 1098.93025
[33] Truffet L.: New bounds for timed event graphs inspired by stochastic majorization results. Discrete Event Dyn. Systems 14 (2004), 355–380 MR 2092597 | Zbl 1073.93039
[34] Wagneur E.: Duality in the max-algebra. In: IFAC, Commande et Structures des Systèmes, Nantes 1998, pp. 707–711

Partner of