计量延迟下双EWMA控制器的稳定性与控制性能深度探讨

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本文是一篇研究论文，针对具有计量延迟的双指数加权移动平均（d-EWMA）控制器的稳定性和控制性能进行了深入探讨。d-EWMA控制器在半导体制造等领域广泛应用，特别是在过程控制中，其性能对生产效率和产品质量至关重要。文章首先提出了问题背景，指出在处理一般扰动时，控制器设计中的关键因素之一是计量延迟，它可能影响系统的稳定性。作者利用Routh-Hurwitz稳定性判据，对d-EWMA控制器在存在计量延迟的情况下进行了稳定性条件的推导。他们发现，当过程增益不匹配度超过4/3时，计量延迟的存在会导致双权重因子可行区域减小。换句话说，这意味着控制器的参数选择空间会受到限制，特别是随着计量延迟的增加，这个可操作的参数范围会进一步缩小。此外，作者还对不同类型的工艺扰动进行了分析，如过程噪声、设备故障等，以了解这些扰动如何影响控制器的控制性能。他们可能通过模拟和理论分析，量化了在不同条件下，控制器如何适应并抑制扰动，以及如何保持系统的稳定性和响应速度。论文的核心贡献在于为设计者提供了一种理论框架，以确定在考虑计量延迟的情况下，如何优化d-EWMA控制器的设计以确保系统稳定，并在实际应用中实现最佳的控制效果。这对于工程师在实际半导体制造过程中优化生产流程，提高产品质量和生产效率具有重要意义。总结来说，这篇论文不仅深化了我们对带有计量延迟的d-EWMA控制器的理解，而且还提供了实用的指导原则，帮助工程师们在面对实时变化的生产环境时，做出更明智的控制器参数选择，以保证系统的稳定性和控制性能。

0894-6507 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSM.2015.2504951, IEEE

Transactions on Semiconductor Manufacturing



Abstract—This paper mainly focuses on establishing the

stability conditions of the double exponentially weighted moving

average (d-EWMA) controller with metrology delay when the

process experiences a general disturbance and analyzing its

control performance under some typical types of process

disturbance. The necessary and sufficient conditions for stability

are established by Routh-Hurwitz criteria. It has been shown that

if the process gain mismatch is greater than 4/3, metrology delay

will make the feasible region of two weighting factors in the

d-EWMA controller decrease, and the larger metrology delay is,

the smaller the feasible region is. Moreover, given two weighting

factors, it is found that metrology delay will weaken the

robustness of the d-EWMA controller. The influence of metrology

delay on the asymptotical control performance is evaluated based

on asymptotical mean square error (AMSE). Optimal weighting

factors and AMSE can be solved based on analytical expression of

the AMSE, which is derived by using the Yule-Walker equations

to avoid function summing limit operation. Through numerical

simulations, it demonstrates that metrology delay may increase

the variability of the process output under certain conditions.

Finally, the Chemical Mechanical Polishing (CMP) process in

semiconductor manufacturing is used to illustrate the validity of

theoretical results.

Index Terms—control performance, double EWMA controller,

metrology delay, run to run control, semiconductor

manufacturing

I. INTRODUCTION

URING the past few decades, as an advanced process

control technique, Run-to-Run (R2R) control has drawn

considerable attention from both industry and academia all over

the world [1]-[3]. In R2R control, data generated from previous

runs are used to adjust the process recipe of the new runs so that

quality characteristics of products can achieve the desired

targets. Until now R2R control has been successfully applied to

Manuscript received May 29, 2015. This work was supported by the

National Natural Science Foundation of China under grant 61203178,

61290323, 61304214 and Samsung Electronics Company.

The first three authors are with the Key Laboratory of System Control and

Information Processing, Department of Automation, Ministry of Education,

Shanghai Jiao Tong University, Shanghai 200240, China (e-mail:

gqs1989@sjtu.edu.cn; gkyang@sjtu.edu.cn; pan_cc@sjtu.edu.cn). The first

and third authors is also with Shanghai Key Laboratory of Navigation and

Location based Services, Shanghai, 200240.

Moon-Sang Lee is with System Engineering Team, Samsung Electronics,

Korea, (e-mail: pantom@kaist.ac.kr).

many semiconductor manufacturing processes, such as CMP

process ([4], [5]), chemical vapor deposition process ([6], [15],

[21]), polysilicon gate etching ([7]).

Due to its simplicity and effectiveness, EWMA-based

controller has already become the most popular R2R controller

in semiconductor manufacturing. Ingolfsson and Sachs [8] first

proposed the basic control structure, which is known as single

EWMA (s-EWMA) controller. They analyzed the stability and

sensibility of the closed-loop process under different kinds of

process disturbance, and showed that the s-EWMA controller

was effective to eliminate shifts in the process. However, in the

presence of process drifts, there would be an offset existing in

the process output when the s-EWMA controller was applied.

Later Butler and Stefani [7] proposed the d-EWMA controller

to deal with this situation. Del Castillo [9] and Tseng et al. [10]

separately investigated the control performance of the

d-EWMA controller, including establishing the stability

conditions and selecting the optimal weighting factors. All the

above work was restricted to the single input single output

processes. In practice, many processes in semiconductor

manufacturing are multiple input multiple output (MIMO) by

nature. Tseng et al. [11] studied the control performance of

multivariate s-EWMA controller in terms of stability and

optimal weighting factors for a MIMO process with no drift.

For different kinds of drifting MIMO process, the stability

conditions of multivariate d-EWMA controller were derived by

Del Castillo and Rajagopal [12], Rajagopal and Del Castillo

[13], and Lee et al. [14], respectively. In addition, Chen and

Wang [15], Fan and Chang [16], Wu and Maa [17], and Pan et

al. [18] also made great endeavors to improve the control

performance of multivariate d-EWMA controller.

The metrology delay is a common phenomenon occurrence

in the feedback control loops. In R2R control environments, it

is very difficult to measure quality characteristics of product

in-situ by current tools. Usually each completed run of wafers is

taken to a metrology station for ex-situ measurement. Since this

post-process measurement strategy is relatively time

consuming, it leads to that measurement information of current

run cannot be used in time to adjust the process recipe for next

run, so metrology delay is inevitable in the feedback control

loops. Metrology delay leads to closed-loop process instability

and degrades the performance of controller. Therefore, it is

necessary to investigate the influence of metrology delay on the

control performance. Several researchers have discussed the

EWMA-based controller with metrology delay. For a MIMO

process with no drift, Good and Qin [19] investigated the

Stability and Control Performance Analysis of

Double EWMA Controller with Metrology

Delay

Qing-Song Gong, Gen-Ke Yang, Chang-Chun Pan and Moon-Sang Lee

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