Double Exponentially Weighted Moving Average Research Articles

A Comment on “Sample-Size Determination for Achieving Asymptotic Stability of a Double EWMA Control Scheme”

Tseng and Hsu developed an approximate sample size formula for achieving asymptotic stability of a double EWMA control scheme. In this Comment, the corresponding exact expression is derived in terms of elementary functions. The numerical performance of the proposed approximation is assessed

Design and investigation of EWMA and double EWMA with quadratic process model in R2R controllers

In the last few years, “Run-to-Run” (R2R) control techniques have been developed and used to control various processes in industries. These techniques combine control engineering and statistical process control techniques. The R2R controller is basically consists of a linear regression model that relates the input variables to the output variables using Exponentially Weighted Moving Average (EWMA) or double EWMA (dEWMA) scheme. In this paper, we have developed a quadratic R2R controller model as an extension of Box, Luceno and Del Castillo models. The validity and performance of the developed models were tested in the magnetic ceramic process. By this study we have found that the quadratic model improves capability of process and applying dEWMA procedure decreases variance of the response variables. The results of the study shows that the quadratic R2R control models outperform linear R2R controls in the certain manufacturing process.

An Investigation of the Optimal Discount Factors of the Double EWMA and DIA Controllers

Motivated by the question of how to reduce the time for a complete search technique to find the optimal discount factors for commonly used run to run (R2R) controllers, we study properties concerning the discount factors of the double exponentially weighted moving average (dEWMA) and its variation, the initial intercept iteratively adjusted (IIIA), controllers for a linear process. We show that the search time can be reduced more than 99.9 percent compared to the benchmark complete search technique.

An Extended EWMA Mean Chart

In this paper, we extend the exponentially weighted moving average (EWMA) technique to double exponentially weighted moving average (DEWMA) technique. We show that DEWMA mean charts perform better than EWMA mean charts in detecting small mean shifts ranging from 0.1 to 0.5 of the process standard deviation, and that the two types of charts perform similarly when mean shifts are larger than 0.5 standard deviation. The design of DEWMA mean charts is also discussed.

AN ANALYSIS AND MIMO EXTENSION OF A DOUBLE EWMA RUN-TO-RUN CONTROLLER FOR NON-SQUARED SYSTEMS

The double EWMA (exponentially weighted moving average) control method is a popular algorithm for adjusting a process from run to run in semiconductor manufacturing. Until recently, the dEWMA controller had been applied only for the single controllable factor (or input), single quality charcteristic (or output) case. Recently, Del Castillo and Rajagopal4 propose a multivariate double EWMA controller for squared multiple-input, multiple-output (MIMO) processes, where there is an equal number of inputs and outputs. This paper extends the MIMO dEWMA controller for non-squared systems. Two different MIMO dEWMA controllers are presented and their performance studied with application to a Chemical-Mechanical Polishing (CMP) process, a critical semiconductor manufacture processing step that exhibits non-linear dynamics.

Poisson DEWMA Control Chart

We introduce a double exponentially weighted moving average (DEWMA) control chart for monitoring Poisson data. Using simulation, we compare average run lengths and standard deviations of run lengths of the Poisson DEWMA chart with those of the Poisson EWMA chart. It is shown that the Poisson DEWMA chart improves upon the Poisson EWMA chart.

A multivariate double EWMA process adjustment scheme for drifting processes

The “predictor-corrector” feedback controller, a process adjustment scheme proposed for semiconductor manufacturing run-to-run processes that drift, is extended to the multiple-input-multiple-output case. The controller is based on two coupled multivariate Exponentially-Weighted-Moving-Average (EWMA) equations, thus its performance depends on the choices of EWMA weight matrices. Stability conditions are given for a pure gain process adjusted with a MIMO double EWMA (EWMA) controller. It is shown that the stability conditions are invariant with respect to various realistic drift disturbance models. Recommendations on how to choose the EWMA weight matrices are given. An analysis is conducted to assess the impact of errors in the estimates of the process gains. The proposed MIMO EWMA feedback controller is compared to the common practice of using multiple single-input-single-ouput dEWMA controllers running in parallel.

SISO run-to-run feedback controller using triple EWMA smoothing for semiconductor manufacturing processes

During the past decade, a variety of 'run-to-run' (R2R) control schemes have been proposed and extensively investigated under various assumed models of process dynamics and disturbances whilst applied to the setpoints (recipe) adjustment between batches for the production of semiconductor manufacturing. The R2R control mechanism of this kind represents an interdisciplinary methodology combining response surface modelling, engineering process control (EPC) and statistical process control (SPC), of which the main objective is to fine-tune the recipe so as to maintain the process output of each run as close to the nominal target as possible. In this paper, the single-input single-output (SISO) case is addressed. To overcome the shortcomings in published R2R controllers, an EWMA feedback controller through a strategy of triple smoothing is proposed (in line with the construction of EWMA statistics) which, in the standard R2R practice, compensates process shifts, a deterministic process drift, disturbance of coloured noises, as well as positive autocorrelation (up to the autoregressive order of two) present in the observed response. It is shown via an experimental study that the triple EWMA filter can provide better control performance for R2R applications as compared with those of the control actions of double EWMA and self-tuning (ST). At last, a relevant application to chemical mechanical planarization (CMP) in semiconductor manufacturing, a critical step in chip fabrication, is used to illustrate the proposed controller.

Some Properties of EWMA Feedback Quality Adjustment Schemes for Drifting Disturbances

Control schemes that are simple enough for operators to implement and that are robust with respect to the underlying model assumptions are usually sought in the area of process adjustment. In the last few years, process adjustment schemes based on the Exponentially-Weighted-Moving-Average (EWMA) statistic have gained popularity in the manufacturing of semiconductors. In the literature, the analysis of this type of adjustment technique has concentrated on a single EWMA controller applied to a system that experiences a deterministic trend disturbance. Although other types of drift models can provide a better description of the disturbance, the performance analysis of EWMA controllers under such conditions has not been undertaken in the past. In this paper we present practical stability and robustness results for single and double EWMA controllers under a variety of process disturbances acting on a “responsive” system. Drift disturbances considered include a random walk with drift, IMA(1,1), and an ARMA disturbance with drift. The effects of errors in the model gain estimate are studied for single EWMA controllers applied to these disturbances. The optimal weight parameter of a single EWMA controller is derived for the case of a random walk with drift disturbance. It is shown that the conditions required for obtaining a stable quality characteristic under the adjustments of the controller are invariant with respect to a large family of drift disturbances. A method is presented for choosing the weights in a double EWMA controller under the assumption that the process gain is known.

Age-based double EWMA controller and its application to CMP processes

In the originally proposed run-by-run control scheme, the EWMA statistic is used as an estimate of the process deviation from its target. However, the controller based on the EWMA statistic is not sufficient for controlling a wearing out process. The PCC controller has been thus proposed to enhance the run-by-run controller capability. In this paper, we first reexamine the fundamentals of the PCC formulations and propose an adjustment that is advantageous in controlling processes subject to both random shifts and drifts. The adjusted PCC controller is then further refined to take into account the process age. This age-based double EWMA scheme is then applied to the CMP process, which is known in the semiconductor industry to be rather unstable.

Development and Evaluation of Control Charts Using Double Exponentially Weighted Moving Averages

The double exponentially weighted moving average (DEWMA), which is known in the literature as Brown′s one‐parameter linear method for forecasting is proposed as a control tool for process monitoring and detecting shifts in the process mean. Obtains a closed‐form expression for the asymptotic standard deviation of the proposed DEWMA control statistic and discusses the determination of its average run length. Provides examples and comparisons between the proposed DEWMA and the standard EWMA. The results reveal that the proposed DEWMA control scheme performs much better than a Shewhart scheme for small and moderate shifts in the process mean and it has average run length properties similar to those for EWMA control schemes. However, DEWMA has smaller variability and it allows more smoothing of the data with no compromise in the sensitivity of detecting shifts in the process mean. It also shifts the range of the design parameters for optimal ARL to larger values as compared with EWMA schemes. Such properties are more desirable for some industrial processes.