Timed Event Graph Model Research Articles

A New Class of Sequences Without Interferences for Cluster Tools With Tight Wafer Delay Constraints

Robotized cluster tools for semiconductor wafer fabrication may have a wafer wait within a processing chamber after processing there until the wafer is unloaded from the chamber by a robot. Such wafer delays cause wafer quality degradation or variability due to residual gases and heats within the chamber. There have been numerous works on characterizing wafer delays and scheduling under upper limits on wafer delays while presuming that the tools are operated by well-known simple robot task sequences such as swap or backward sequences. However, when the wafer delay constraints are tight, there may not be feasible schedules for such sequences. We wish to know whether there can be alternative robot task sequences which can satisfy such tight wafer delay constraints. In this paper, we identify a new class of robot task sequences that can better satisfy tight wafer delay constraints than the conventional swap or backward sequences while keeping the same minimum tool cycle time. By examining the circuits of timed event graph (TEG) models for the tool operation behaviors in many different robot task sequences, we identify that such robot task sequences do not make interferences between the work cycles of the resources such as the robot and chambers. The resource interference can cause delays in the work cycles, and hence increase the wafer delays or the tool cycle time. To prove this, we examine circuits in an extended TEG model, a negative event graph, which incorporates time constraints as negative places and tokens. From this, we derive closed-form conditions for which such sequences are feasible against given wafer delay constraints. By experiments, we show that the proposed new sequences have shorter wafer delays, and hence better satisfy tight wafer delay constraints than conventional sequences. Note to Practitioners —As circuit widths shrink down to several nanometers, cluster tools for semiconductor fabrication require extreme process quality control. Even wafer delays within processing chambers of cluster tools can cause wafer quality degradation and variability. Therefore, it is desirable for cluster tools to have much shorter wafer delays. However, conventional sequences such as the swap and backward sequences, which are being prevalently used in practice, may not satisfy tight wafer delay constraints. Our proposed sequences, which are as simple as the conventional sequences, have much shorter wafer delays while keeping the same minimum tool cycle time.

Control of Petri nets subject to strict temporal constraints using Max-Plus algebra

ABSTRACTIn this paper, we treat the control problem of timed discrete event systems under temporal constraints. This type of constraint is very frequent in production systems, transportation network and in networked automation systems. Precisely, we are interested in the validation of strict temporal constraints imposed on the paths in a timed event graph (TEG) by using Max-Plus algebra. Not all the transitions of the considered TEG model are controllable, i.e. only the input transitions are controllable. An analytical approach for computing state feedback controllers is developed. Sufficient condition is given for the existence of causal control laws satisfying the temporal constraints. In the first, a TEG with observable transitions is considered. Then, the proposed approach is extended to the partially observable TEG. The synthesised feedback can be interpreted by places of control connected to the TEG to guarantee the respect of the time constraints. The proposed method is illustrated in the assembly system example.

Characterizing Token Delays of Timed Event Graphs for $K$- Cyclic Schedules

A timed discrete event system, which repeats identical work cycles, has task delays due to synchronization between work cycles. Real such systems tend to operate mostly in a K-cyclic timing regime, where a sequence of identical timing patterns is repeated for every K cycles. Therefore, the task delays fluctuate and repeat a sequence of K different values, and hence have higher risk of violating an upper limit. Task delays correspond to token delays at the system's timed event graph model. We therefore examine token delays in K-cyclic schedules of a timed event graph, an essential class of Petri nets. We first identify all possible K-cyclic schedules and define their initial phases. We then develop a closed-formula on the token delays on a path for a K-cyclic schedule, which can be computed by the longest path lengths between the nodes in an associated directed graph. We also present a formula for 1-cyclic schedules. The formulae can be used for computing statistics on K-different token delays, maximizing or minimizing the token delays with regard to all possible initial phases, and verifying task delay constraints, if any.

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

This 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.

Control of discrete event systems with respect to strict duration: Supervision of an industrial manufacturing plant

In this paper, we propose a (max, +)-based method for the supervision of discrete event systems subject to tight time constraints. Systems under consideration are those modeled as timed event graphs and represented 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 contributions 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 according to their performance, considering their impact on the production throughput.

Timed event graph-based cyclic reconfigurable flow shop modelling and optimization

The manufacturing process of a part involves sequential steps and each step could be viewed as the part being manufactured by a process module with some specific function. The module must be placed on a machine and connected to the machine via standard interfaces. The machine considered here is a carrier or general platform that can hold one or several different modules simultaneously. Based on the idea that modules are independent of machines and different combinations of modules and machines result in different configurations, the cyclic reconfigurable flow shop is proposed for the new manufacturing paradigm—Reconfigurable Manufacturing System (RMS). The cyclic reconfigurable flow shop can be modelled as a timed event graph. Different cases of cyclic reconfigurable flow shops are discussed and the optimal configuration can be obtained by solving the corresponding mixed-integer program derived from the timed event graph model.