Stability analysis and optimal tuning of EWMA controllers - Gain adaptation vs. intercept adaptation

Jin WANG*, Q. Peter HE, S. Joe QIN

*Corresponding author for this work

Research output: Journal PublicationsJournal Article (refereed)peer-review

15 Citations (Scopus)


Exponentially weighted moving average (EWMA) controllers are the most commonly used run-to-run controllers in semiconductor manufacturing industry. An EWMA controller can be implemented in two different ways. One way is to keep the process gain as its off-line estimate and update the intercept term at each run, which is termed EWMA with intercept adaptation; the other is to keep the intercept term as its off-line estimate and update the process gain at each run, which is termed EWMA with gain adaptation. Despite the fact that gain variation and adaptation is typical in semiconductor industry, most EWMA formulations are for intercept adaptation and few results exist on the stability and sensitivity of EWMA with gain adaptation. In this paper, we propose a general formulation to analyze the stability of both EWMA controllers. The proposed state-space representation not only reveals the similarities and differences between two types of EWMA controllers, but also explains why the stability conditions for both types of EWMA controllers are independent of process disturbances. In addition, we propose a general framework that unifies the analysis of the optimal control performance for both types of EWMA controllers. The proposed framework is different from existing approaches in that it decouples the state estimation from the control law, and derives the optimal weighting based on the state estimation performance. The proposed framework significantly simplifies the analysis procedure, especially for EWMA with gain adaptation. Using this framework, we derive the optimal EWMA weighting through solving the discrete-time algebraic Riccati equation (DARE) for various process disturbances that are encountered in semiconductor manufacturing industry. Simulation examples are given to illustrate the optimality of the EWMA weighting derived using the framework. Some practical aspects of controller tuning are also discussed based on the simulation results. © 2009 Elsevier Ltd. All rights reserved.
Original languageEnglish
Pages (from-to)134-142
Number of pages9
JournalJournal of Process Control
Issue number2
Early online date16 Jul 2009
Publication statusPublished - Feb 2010
Externally publishedYes

Bibliographical note

Financial support from NSF is gratefully acknowledged by JW under Grant CBET-0853983 and QPH under Grant CBET-0853748.


  • Exponentially weighted moving average (EWMA)
  • Optimal weighting
  • Run-to-run control
  • Stability
  • State-space representation
  • The Kalman filter


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