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.

Keywords: Exponential Weighted Moving Average, Feedback Control, Process Adjustment, Process Drift.

*by* **Enrique Del Castillo, The Pennsylvania State
University, University Park, PA 16802 **

*INTRODUCTION*

QUALITY engineers have concentrated on adjustment schemes that are simple enough for plant operators to implement yet are capable of efficiently compensating against a variety of realistic disturbances. These are some of the goals, for example, of discrete proportional-integral (PI) controllers (Box and Luceño (1995), (1997a), (1997b), and Tsung et al. (1998)). In the semiconductor industry, similar goals have led to the development of exponentially weighted moving average (EWMA) feedback controllers for compensating against disturbances that affect the batch-to-batch (or "run-to-run") variability in the quality characteristics of silicon wafers at a processing step (Sachs et al. (1995), Butler and Stefani (1994), and Del Castillo and Hurwitz (1997)). Although EWMA-based controllers have been in use in the semiconductor industry since the early '90s, analysis of these controllers is relatively recent. The Ingolfsson and Sachs (1993) paper generated much of the recent interest in analyzing these controllers (Smith and Boning (1997), Del Castillo (1999), Patel and Jenkins (1998), and Hamby et al. (1998)), but most of the past analysis has concentrated on a particular type of disturbance, namely, a deterministic trend or ramp disturbance with noise (see Equation 2 below). In this paper practical results related to EWMA-based controllers are presented for various process disturbances that model drift. Drift disturbances considered in this paper include the case of IMA(1,1) noise—for which a single EWMA controller can provide minimum mean square error (MMSE) control under certain conditions—and also unidirectional drifting disturbances such as a deterministic trend (DT) with noise and a random walk with Drift (RWD) disturbance.

It will be assumed in this paper that the dynamic properties
of the quality characteristic associated with batch *t*,
*Y _{t}*, are only due to the drift disturbance
according to the model

Two popular time series models for unidirectional drift (see Hamilton (1994, ch. 15)) are the deterministic trend (DT) model

and the random walk with drift (RWD) model In Equations (2-3), denotes a white noise sequence , and denotes the per time unit expected drift. A popular drift process model in the process industries is the IMA(1,1) (Integrated Moving Average) process with defined as before, and where is a moving average parameter (Box, Jenkins and Reinsel (1994)). The disturbances in Equations (2-4) are non-stationary, but the disturbances in Equations (2-3) always drift (in the long run) in the same direction, contrary to an IMA(1,1) disturbance that can drift in any direction with equal probability. Unidirectional drift models are probably more realistic in discrete manufacturing processes that exhibit "wear." Ingolfsson and Sachs (1993) analyzed the performance of a single EWMA controller for a deterministic trend and for a random walk (with no drift) disturbance.Figure 1 shows one realization of each disturbance in Equations (2-4) for the case when = 0.1, = 1, and = 0.3. The 3 realizations were computed with the same random numbers, shown by the series labeled "white noise."

Note that, contrary to the DT disturbance model, the RWD
disturbance does not "adhere" to the line defined
by a linear trend with slope , but
eventually it drifts in the same direction as given by the
slope . The IMA(1,1) model drifts
erratically, not "sticking" to the line defined
by the time axis. An EWMA controller under a more general
disturbance is studied in Del Castillo (2000) where a possibly
non-invertible IMA(1,1) disturbance
(with __<__ 1) is considered.
This disturbance includes models in Equations (2-4) as particular
cases (see also Jensen and Vardeman (1993) and Luceño
and González (1999), who consider related drifting
disturbances for process adjustment).

The EWMA controllers considered in this paper can be written as

from (Butler and Stefani (1994)), whereTuning the EWMA controller (Equations (5-7)) consists of selecting
weights _{1} and _{2}
that achieve some desired property for the process under control.
In the particular case when _{2}
= 0 and *D*_{0} = 0 in Equation (7), the so-called
single EWMA controller is obtained, which is a pure integral
(I) feedback controller. When both _{1}
and _{2} differ from zero,
the scheme will be referred to as a double EWMA feedback controller.
As shown in a later section, a double EWMA controller is not
of the proportional integral derivative (PID) type.