The concept of (A, B)-invariant subspace is the fundamental concept of the geometric approach of control design. It has been extended by many authors to that of (A, B)-invariant module or semimodule, for the sake of extending the solution of various control problems to the case of systems over rings or semi rings. In this paper is discussed the use of dynamic feedback control laws for systems over semirings, and it is shown that an (A, B)-invariant semimodule over a commutative semiring can be made invariant for the closed-loop system by dynamic feedback.

International audienceThis 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.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