Some Discussions Of EWMA Feedback Controller

The exponentially weighted moving average(EWMA) feedback controller is a process algorithm based on the EWMA statistic which integrates and combines the strengths of statistical process control(SPC) and automatic/engineering process control(APC/EPC). These control techniques have been applied to many semiconductor manufacturing processes. Although the EWMA controller with a small fixed discount factor can gaurantee a long-term stability(under fairly regular conditions), it usually requires a moderately large number of runs to bring the output of a process to its target. The output deviations are usually very large in the first few runs and the output may be out of process specifications. This is impractical for process with small batches as the process output possibly does not converge to its target when the production is finished. Tseng et al(2003) proposed a variable discount factor to tackle the problem and its results are potentially useful to practical manufactures. In order to reduce a possibly high rework rate and bring the process output closer to its target as soon as possible we discuss the variable discount factor in SISO systems on the basis of previous work. The predetermined criterion we choose is to find the optimal varialbe discount factor by minimizing the total mean square error (MSE) within the first N runs. We derive an explicit form for the optimal variable discount factor under the situation of IMA(1,1) process disturbance and numerical optimal solutions for other general process disturbances. Moreover, a heuristic is proposed to simplify the computation of variable discount factor when the process disturbance follows MA(1).As the EWMA feedback control scheme consists of the process, the predicted model, and the feedback control scheme, all the study of the long-term stability conditions, performances and the optimal discount factor of the EWMA feedback control scheme are obtained only when the predicted model is prefixed. In practice, since the predicted model is established by a random sample from the population of I-O variables, the strength of the linear relationship between I-O variables plays an important role in determing the validation of these stability condions. For SISO systems, by controlling a guaranteed probability, the study derives the formula for an adequate sample size required to construct the predicted model if the value of discount factors can be limited with some prior information, and we also consider how to control the discount factors if the sample size we can choose is smaller than n_min .