On the control of max-plus linear system subject to state restriction

This paper deals with the control of discrete event systems subject to synchronization and time delay phenomena, which can be described by using the max-plus algebra. The objective is to design a feedback controller to guarantee that the system evolves without violating time restrictions imposed on the state. To this end an equation is derived, which involves the system, the feedback and the restriction matrices. Conditions concerning the existence of a feedback are discussed and sufficient conditions that ensure the computation of a causal feedback are presented. To illustrate the results of this paper, a workshop control problem is presented

Duality and interval analysis over idempotent semirings

In this paper semirings with an idempotent addition are considered. These algebraic structures are endowed with a partial order. This allows to consider residuated maps to solve systems of inequalities $A \otimes X \preceq B$. The purpose of this paper is to consider a dual product, denoted $\odot$, and the dual residuation of matrices, in order to solve the following inequality $A \otimes X \preceq X \preceq B \odot X$. Sufficient conditions ensuring the existence of a non-linear projector in the solution set are proposed. The results are extended to semirings of intervals

An integrated control strategy to solve the disturbance decoupling problem for max-plus linear systems with applications to a high throughput screening system

International audienceThis paper presents the new investigations on the disturbance decoupling problem (DDP) for the geometric control of max-plus linear systems. The classical DDP concept in the geometric control theory means that the controlled outputs will not be changed by any disturbances. In practical manufacturing systems, solving for the DDP would require further delays on the output parts than the existing delays caused by the system breakdown. The new proposed modified disturbance decoupling problem (MDDP) in this paper ensures that the controlled output signals will not be delayed more than the existing delays caused by the disturbances in order to achieve the just-in-time optimal control. Furthermore, this paper presents the integration of output feedback and open-loop control strategies to solve for the MDDP, as well as for the DDP. If these controls can only solve for the MDDP, but not for the DDP, an evaluation principle is established to compare the distance between two output signals generated by controls solving for the MDDP and DDP, respectively. This distance can be interpreted as the number of tokens or firings that are needed in order for the controls to solve for the DDP. Moreover, another alternative approach is finding a new disturbance mapping in order to guarantee the solvability of the DDP by the same optimal control for the MDDP. The main results of this paper are illustrated by using a timed event graph model of a high throughput screening system in drug discovery.</p

Commande sous contraintes temporelles des réseaux de graphes d'événements temporisés en conflit

International audienceDans ce papier, nous abordons le problème de modélisation et de commande des systèmesàévénements discrets avec des ressources partagées représentés par une classe particulière des réseaux de Petri temporisés. Précisément, nous considérons des Réseaux de Graphes d'Evénements Temporisés en Conflit (RGETC) soumisà des contraintes tem-porelles strictes. Premièrement, une formalisation algébrique en termes de systèmesà com-mutation Max-Plus est proposée pour décrire le comportement dynamique des RGETCs. Deuxièmement, des lois de commande en boucle fermée sont calculées pour garantir le respect de ces contraintes de temps imposéesà certaines places du réseau. Des conditions suffisantes pour l'existence de telles lois de commande ontété fournies. Finalement, nous appliquons les résultats théoriques développés précédemment pour contrôler un système ferroviaire de croisement de trainà temps critique

Control Problem in Max Plus Linear Model with Temporal Constraints

[ES] Este artículo trata del control de sistemas de eventos discretos sujetos a sincronización y fenómenos de retraso, descritos por un modelo max plus lineal. Definimos y caracterizamos el conjunto de condiciones iniciales admisibles, las cuales originan soluciones no decrecientes. Restricciones temporales son impuestas al espacio de estado del sistema. Estas restricciones son descritas en el cono max plus definido por la imagen de la estrella de Kleene de la matriz asociada a las restricciones temporales. Propiedades geométricas de este cono max plus, para garantizar que la evolución del sistema en lazo cerrado satisface las restricciones, son estudiadas. Condiciones suficientes concernientes a la existencia y cálculo de una retroalimentación de estado son presentadas. Para ilustrar la aplicación de este enfoque, dos problemas de control son discutidos, para los cuales un controlador es diseñado con el objetivo de garantizar la satisfacción de las restricciones temporales.[EN] This article deals with the control of discrete event systems subject to synchronization and delay phenomena, described by a plus max linear model. The temporal constraints are imposed on the state space of the system. These constraints are described in the max plus cone defined by the image of the Kleene star of the matrix associated with the temporal constraints. In consequence, the problem of determining a control that force the satisfaction of time constraints, is formulated in terms of the invariance of the cone. Sufficient conditions for the existence of a solution to this problem have been established. Our approach allows the design of a satisfactory control of the form of a static state feedback. We emphasize that our solution takes into account two aspects which are the initialization of the control law, and its causality, important for its implementation. To illustrate the application of this approach, two control problems are presented.Cárdenas, C.; Cardillo, J.; Loiseau, J.; Martínez, C. (2016). Control de Modelos Max Plus Lineales con Restricciones Temporales. Revista Iberoamericana de Automática e Informática industrial. 13(4):438-449. https://doi.org/10.1016/j.riai.2016.07.001OJS438449134Allamigeon, X., Gaubert, S., Goubault, E., 2010. "The tropical double descrip- ' tion method", in Proc. Symp. Theor. Aspects Comp. Sci., Nancy, France, pp. 47-58.Allamigeon, X., Gaubert, S., Goubault, E., 2012. Computing the ' vertices of tropical polyhedra using directed hypergraphs, Discrete Comput. Geom.Amari, S., Demongodin, I., Loiseau, J. J., Martinez, C., 2012. Max-plus control design for temporal constraints meeting in timed event graphs, IEEE Trans. Automatic Control, Vol. 57, No. 2, pp. 462-467.Atto A., Martinez C., Amari S., 2011. Control of discrete event systems with respect to strict duration: supervision of an industrial manufacturing plant. Comput Inf Syst 61(4):1149-1159.Baccelli, F., Cohen, G., Olsder, G.-J., Quadrat, J.-P., 1992. Synchronization and Linearity. John Wiley & Sons, New York.Cohen, G., Gaubert, S., Quadrat, J. P.,1999. "Max-plus algebra and system theory: where we are and where to go now,"Annu. Rev. Control, vol. 23, pp. 207-219.Cohen, G., 2001. Analisis ' y Control de sistemas de eventos discretos: De redes de Petri temporizadas. Argentina: ENPC & INRIA (Francia).Gaubert, S., Katz, R., 2007. The Minkowski theorem for max-plus convex sets. Linear Algebra and Appl., 421:356-369.Gaubert, S., Katz, R., 2009. The tropical analogue of polar cones. Linear Algebra and Appl., 431:608-625.Gaubert, S., Katz, R., 2011. Minimal half-spaces and external representation of tropical polyhedra, Journal of Algebraic Combinatorics 33, no. 3, 325348.Katz, R. D., 2007. Max-plus (A,B)-invariant spaces and control of timed discrete-event systems, IEEE Trans. Automatic Control, Vol. 52, No. 2, pp. 229-241.Kim, J. H., Lee, T. E. 2003. Schedule stabilization and robust timing control for time-constrained cluster tools. In IEEE international conference on robotics and automation, pp. 1039-1044. Taipei, Taiwan.Libeaut, L., Loiseau, J., 1995. Admissible initial conditions and control of timed event graphs, 34th Conference on Decision and Control, New Orleans, Louisianna.Maia, C., Andrade, C., Hardouin, L., 2011. On the control of max plus linear system subject to state restriction. Automatica 47(5): 988-992.Murata, T., 1989. Petri nets: Properties, analysis and applications. IEEE, Proc 77(4), 541-580.Wonham, W. M., Linear Multivariable Control: A Geometric Approach, 3rd ed. New York: Springer-Verlag.Wu, N., Chu, C., Chu, F., Zhou, M. 2008. A Petri net method for schedulability and scheduling problems in single-arm cluster tools with wafer residency time constraints, IEEE Trans. Semiconduct. Manuf., vol. 21, pp. 224-237