Control of Time-Constrained Dual-Armed Cluster Tools Using (max, +) Algebra

International audienceThe problem studied in this paper is the control of discrete event systems subject to strict temporal constraints using (max, +) algebra. Initially we sought necessary and sufficient conditions for the existence of a causal control law guaranteeing the respect of the temporal constraints. Subsequently, a method for calculating the control law, if any, is proposed. The application which we are interested in is the control of a manufacturing semiconductor wafers process subject to strict temporal constraints

SIZING, CYCLE TIME AND PLANT CONTROL USING DIOID ALGEBRA

Using an industrial process from the car sector, we show how dioid algebra may be used for the performance evaluation, sizing, and control of this discrete-event dynamic system. Based on a Petri net model as an event graph, max-plus algebra and min-plus algebra permit to write linear equations of the behavior. From this formalism, the cycle time is determined and an optimal sizing is characterized for a required cyclic behavior. Finally, a strict temporal constraint the system is subject to is reformulated in terms of inequalities that the (min, +) system should satisfy, and a control law is designed so that the controlled system satisfies the constraint

Control of Discrete Event Systems with Respect to Strict Duration: Supervision of an Industrial Manufacturing Plant

26 pagesInternational audienceIn this paper, we propose a (max,+)-based method for the supervi- sion of discrete event systems subject to tight time constraints. Systems under consideration are those modelled as timed event graphs and repre- sented with linear (max,+) state equations. The supervision is addressed by looking for solutions of constrained state equations associated with timed event graph models. These constrained state equations are derived by reducing duration constraints to elementary constraints whose con- tributions are injected in the system's state equations. An example for supervisor synthesis is given for an industrial manufacturing plant subject to a strict temporal constraint, the thermal treatment of rubber parts for the automotive industries. Supervisors are calculated and classified ac- cording to their performance, considering their impact on the production throughput

On Max-Plus Linear Dynamical System Theory: The Regulation Problem

A class of timed discrete event systems can be modeled by using Timed-Event Graphs, a class of timed Petri nets that can have its firing dynamic described by using an algebra called “Max-plus algebra”. For this kind of systems it may be desirable to enforce some timing constraints in steady state. In this paper, this problem is called a “max-plus regulation problem”. In this context we show a necessary condition for solving these regulation problems and in addition that this condition is sufficient for a large class of problems. The obtained controller is a simple linear static state feedback and can be computed using efficient pseudo-polynomial algorithms. Simulation results will illustrate the applicability of the proposed methodology

Discrete events: Perspectives from system theory

Systems Theory;differentiaal/ integraal-vergelijkingen

Observer design for weighted timed event graphs in a dioid framework

An observer is used in a feedback control system to compensate for the fact that not all states can be measured. This work refers to systems that can be modeled by timed Petri nets, more precisely, Weighted Timed Event Graphs (WTEGs). The WTEG allows modeling of problems without conflicts of resources. In the context of manufacturing systems, WTEGs are appropriate to design complex assembly lines, where dynamic behavior is described by the synchronization and saturation effects. We propose the design of an observer for WTEGs that, which consists of: (i) an Optimal Observer that is based on the conversion of the WTEG to the Timed Event Graph(TEG), which has a linear representation in the mathematical structure of the dioids; (ii) a Simulator, which is a copy of the system without disturbances; (iii) and an Interface that is used to connect the WTEG to the Optimal Observer and to the Simulator.Um observador é utilizado em um sistema de controle por realimentação, para compensar o fato de que nem todos os estados podem ser medidos. Esse trabalho aborda sistemas que podem ser modelados por redes de Petri temporizadas, mais precisamente, por Grafos de Eventos Temporizados Ponderados (WTEGs). A estrutura do WTEG permite a modelagem de problemas sem conflitos de recursos. No contexto de sistemas de manufatura, os WTEGs são apropriados para reproduzir linhas de montagem complexas, em que o comportamento dinâmico é descrito pelos efeitos de sincronização e saturação. Neste trabalho é proposto o projeto de um observador para WTEGs que consiste na construção: (i) do Observador Otimo que se baseia na conversão do WTEG para o Grafo de Evento Temporizado (TEG), o qual possui uma representação linear na estrutura matemática dos dióides; (ii) do Simulador, que é uma cópia do sistema sem os distúrbios; (iii) e da Interface que é utilizada para conectar o WTEG ao Observador Otimo e ao Simulador

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

Optimalsteuerung zeitbehafteter Synchronisationsgraphen mit Ressourcenkonkurrenz und Aktualisierung von Referenzsignalen

Timed event graphs (TEGs) are a subclass of timed Petri nets that model synchronization and delay phenomena, but not conflict or choice. We consider a scenario where a number of TEGs share one or several resources and are subject to changes in their output-reference signals. Because of resource sharing, the resulting overall discrete event system is not a TEG. We propose a formal method to determine the optimal control input for such systems, where optimality is in the sense of the widely adopted just-in-time criterion. Our approach is based on a prespecified priority policy for the TEG components of the overall system. It builds on existing control theory for TEGs, which exploits the fact that, in a suitable mathematical framework (idempotent semirings such as the max-plus or the min-plus algebra), the temporal evolution of TEGs can be described by a set of linear time-invariant equations.Zeitbehaftete Synchronisationsgraphen (ZSGen) bilden eine spezielle Klasse zeitbehafteteter Petri-Netze. Sie können Synchronisations- und Verzögerungsphänomene modellieren, nicht aber Konflikte. Wir untersuchen ein Szenario, in dem sich mehrere ZSGen eine oder mehrere Ressourcen teilen und die Referenzsignale der ZSGen unvorhersehbaren Änderungen unterworfen sind. Da die beteiligten ZSGen um Ressourcen konkurrieren, ist das Gesamtsystem kein ZSG. Wir beschreiben eine formale Vorgehensweise zur Bestimmung des im just-in-time Sinne optimalen Stellsignals für dieses Gesamtsystem. Unser Ansatz basiert auf einer vorab festgelegten Priorisierung der einzelnen ZSGen. Er baut auf der existierenden Regelungstheorie für ZSGen auf und nutzt die Tatsache, dass sich die zeitliche Entwicklung von ZSGen in einem geeigneten mathematischen Rahmen (idempotente Halbringe wie beispielsweise die max-plus- oder die min-plus-Algebra) durch lineare zeitinvariante Gleichungen beschreiben lässt