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

Modeling and control of nested manufacturing processes using dioid models

Manufacturing systems are frequently adapted to changing customer demands. However, every extension with respect to the hardware requires a modification of the corresponding model and controller. In this paper we propose a dioid model of manufacturing processes in which parts may visit the same resource more than once. The proposed model can be used to determine a controller that maintains the throughput while starting activities just-in-time. Furthermore, the model can easily be adapted in case of hardware modifications. Also nested processes, i.e., processes in which some activities of part k may be executed prior to activities of part (k - 1), can be modeled with the proposed approach

Modeling and control of high-throughput screening systems

In previous work it has been shown how a max-plus algebraic model can be derived for cyclically operated high-throughput screening systems and how such a model can be used to design a controller to handle unexpected deviations from the predetermined cyclic operation during run-time. In this paper, this approach is extended by modeling the system in a general dioid algebraic setting. Then a feedback controller can be computed using residuation theory. The resulting control strategy is optimal in the sense of the just-in-time criterion, which is very common in scheduling practice

Stock Reduction for Timed Event Graphs Based on Output Feedback

In timed event graphs (TEG), stock represents the number of tokens present inthe system. It is very similar to work-in-process for manufacturing systems. In general, when choosing a feedback controller, a compromise is sought between fastness of the system and stock size. The classical choice in the (max,+) literature consists in reducing the stock as much as possible without delaying the output. In this paper, the constraint is weakened: the response of the controlled system to a specific, predefined reference input w must be as fast as the one of the uncontrolled system, but may be slower for other inputs. In return, lower stock is expected. After formally defining this new controller, its performance in terms of stock is compared with the one of the classical controller. This last part relies heavily on second order theory for TEG