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Stabilising feedback in Max-Plus linear models of discrete processes

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PL
Stabilizacyjne sprzężenie zwrotne w Max-Plus liniowych modelach procesów dyskretnych
Języki publikacji
EN
Abstrakty
EN
This article relates to a synthesising output feedback that is used to control a network of discrete events. The feedback stabilises the system without reducing its initial throughput and its synthesis is mainly based on the theory of residues and the Kleene operator. This article suggests some theoretical results and mathematical foundations of max-plus algebra theory, and in particularly, discusses various other aspects of controlling discrete processes and their modelling in the context of a linear max-plus system.
PL
Artykuł dotyczy syntezy sprzężenia zwrotnego w sterowaniu siecią zdarzeń dyskretnych. Sprzężenie zwrotne służy do stabilizacji systemu bez zmniejszenia jego początkowej przepustowości i jego synteza opiera się głównie na wynikach teorii residuów i operatora Kleene’a. W artykule zasygnalizowano pewne wyniki teoretyczne i wprowadzono matematyczne podstawy max-plus algebry. Omówiono także inne aspekty sterowania procesami dyskretnymi oraz ich modelowanie w kategoriach liniowego systemu max-plus.
Rocznik
Strony
49--60
Opis fizyczny
Bibliogr. 38 poz., wz., il.
Twórcy
autor
  • Institute of Computer Science, Cracow University of Technology
autor
  • Institute of Computer Science, Cracow University of Technology
Bibliografia
  • [1] Applebaum D., Markov processes, semigroups and generators, Cambridge Studies in Advanced Mathematics, No. 93, p. 120-189.
  • [2] Bacceli F., Cohen G., Olsder G., Quadrat J., Synchronization and Linearity. An Algebra for Discrete Event Systems, John Wiley & Sons Ltd, 1992.
  • [3] Bezem M., Nieuwenhuis R., Hard problems in max algebra, control theory, hypergraphs and other areas, Information Processing Letters, 110(4), 2010, p. 133-138.
  • [4] Blyth T., Janowitz M., Residuation Theory, Pergamon press 1972.
  • [5] Brunsch T., Hardouin L., Raisch J., Formal Methods in Manufacturing: Modelling Manufacturing Systems in a Dioid Framework, Campos J. ed., Part 1, Ch. 2 2, p. 29-94.
  • [6] Cohen G., Gaubert S., Quadrat J., Max-plus algebra and system theory: Where we are and where to go now, Annual Rev. Elsevier-IFAC, 1999, Vol. 23, No. 1, p. 207-219.
  • [7] Cohen G., Residuation and applications. Algèbres Max-Plus et applications en informatique et automatique, Ecole de printemps d’informatique théorique, 1998.
  • [8] Cohen G., Dubois D., Quadrat J., Viot M., A linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing, IEEE Trans. on Automatic Control, 1985, AC–30, p. 210-220.
  • [9] Commault C., Feedback stabilization of some event graph models, IEEE Transaction. on Automatic Control, 43(10), October 1998, p. 1419-1423.
  • [10] Cottenceau B., Hardouin L., Boimond J.-L., Ferrier J.-L., Synthesis of greatest linear feedback for Timed Event Graphs in dioid, IEEE Transactions on Automatic Control, 1999, Vol. 44, No. 6, p. 1258-1262.
  • [11] Davey B., Priestley H., Introduction to Lattices and Order, Cambridge University Press 1990.
  • [12] D’Souza D. Shankar P., Modern Applications of Automata Theory, World Scientific Publishing Co Pte Ltd, 2012.
  • [13] El Hichami O., An algebraic method for analysing control flow of BPMN models, International Journal of Applied Mathematics & Computer Science (in revision 2015).
  • [14] Gaubert S., Resource Optimization and (min, +), IEEE Transactions on Spectral Theory Automatic Control, Vol. 40, Issue 11, 1995, p. 1931-1934.
  • [15] Gaubert S., Theorie des systemes lineaires dans les dioides. These de doctorat, Ecole des Mines de Paris, Paris 1992.
  • [16] Hardouin L. Sur la Commande des Systèmes (max,+) Linéaires, DEA Automatique et Informatique Appliquée – Angers, Décembre 2004.
  • [17] Hardouin L., Menguy E., Boimond J.-L., Ferrier J.-L., SISO Discrete Event Systems Control in Dioids Algebra, Special issue of Journal Européen des Systèmes Automatisés (JESA), Vol. 31, No. 3, 1997, p. 433-452.
  • [18] Heidergott, B., Olsder, G.J., van der Woude, J., Max Plus at Work, Modeling and Analysis of Synchronized Systems, Princeton University Press, 2006.
  • [19] Jamroż L., Raszka J., Applying methods supporting project management information system process, Automatyzacja Procesów Dyskretnych, Vol. I, Gliwice 2010, p. 57-64.
  • [20] Jamroż L., Raszka J., Simulation method for the performance evaluation of system of discrete cyclic processes, Proceedings of the 16-th IASTED International Conference on Modelling, Identification and Control, Innsbruck, Austria, 17-19.02.97, p. 190-193.
  • [21] John M., Smith S., Application-Specific Integrated Circuits, Addison-Wesley Publishing Company, VLSI Design Series, June 1997.
  • [22] Karatkevich A., Wisniewski R., Polynomial Algorithm for Finding a State Machine Cover of Petri Nets, Applied Mathematics & Computer Science ( in revision 2015).
  • [23] Katz R., Schneider H., Sergeev, S., Commuting matrices in max-algebra (Preprint 2010/03). University of Birmingham, School of Mathematics, 2010.
  • [24] LeBoudec J.-Y., Thiran P., Network Calculus, Springer Verlag 2002.
  • [25] Lhommeau M., Etude de systèmes à événements discrets dans l’algèbre (max,+): 1. Synthèse de correcteurs robustes dans un dioïde d’intervalles. 2. Synthèse de correcteurs en présence de perturbations, Thèse, LISA - Université d’Angers, 2003.
  • [26] Lhommeau M., Hardouin L., Cottenceau B., Optimal control for (max,+) ‒ linear systems in the presence of disturbances. Positive Systems: Theory and Applications, POSTA, Springer LNCIS 294, 2003, 47-54.
  • [27] Lhommeau M., Hardouin L., Ferrier J.-L., Ouerghi I., Interval analysis in dioid: Application to robust open loop control for timed event graphs. Decision and Control, 2005, European Control Conference, CDC-ECC’05, 44th IEEE, p. 7744-7749.
  • [28] Lotito P., Mancinelli E., Quadrat J.P., A minplus derivation of the fundamental car-traffic law, Report 324, INRIA 2001.
  • [29] Maia C.A., Hardouin L., Santos-Mendes R., On the Model Reference Control for Max-Plus Linear Systems, In 44th IEEE CDC-ECC’05 Sevilla, 2005, p. 7799-7803.
  • [30] Maia C., Identification et Commande de systèmes à événements discrets dans l’algèbre (max,+), Thèse, LISA ‒ Université d’Angerse, Université de Campinas ‒ Brésil 2003.
  • [31] Maia C.A., Hardouin L., Optimal closed-loop control of Timed Event Graphs in Dioid, IEEE Transactions on Automatic Control, 48 (12), 2003, p. 2284-2287.
  • [32] Menguy E., Boimond J.-L., Hardouin L., Ferrier J.-L., Just in time control of timed event graphs: update of reference input, presence of uncontrollable input, IEEE Transactions on Automatic Control, 45(11), 2000, p. 2155-2159.
  • [33] Merlet G., Semigroup of matrices acting on the max-plus projective space, Elsevier, Linear Algebra and Its Applications, 432, 2010, p. 1923-1935.
  • [34] Subiono O.G., Gettrick, M.M., Course notes: On large scale max-plus algebra model in railway systems. Algèbres Max-Plus et applications en informatique et automatique, Ecole de printemps d’informatique théorique.
  • [35] Farhi Q.N., Goursat M., Derivation of the fundamental traffic diagram for two circular roads and a crossing using minplus algebra and Petri net modeling, Decision and Control, 2005, European Control Conference, CDC-ECC’05, 44th IEEE.
  • [36] Schutter B.D., van den Boom T., Model predictive control for max-plus-linear discrete event systems, Automatica, 37(7), 2001.
  • [37] Shang Ying, Sain M.K., On zero semimodules of systems over semirings with applications toqueueing systems, [in:] American Control Confer., 2005, p. 225-230.
  • [38] Szpyrka M., Sieci Petriego w modelowaniu i analizie systemów współbieżnych, WNT, Warszawa 2008.
Uwagi
PL
Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.
Typ dokumentu
Bibliografia
Identyfikator YADDA
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