Properties of EWMA Controllers With Gain Adaptation

Exponentially weighted moving average (EWMA) controllers are the most commonly used run-to-run controllers in the semiconductor industry. Using a linear model, an EWMA controller can be implemented in two different ways: either process gain or process intercept can be updated using EWMA statistics at each run. The most commonly used EWMA controller formulation is to keep the process gain as its off-line estimate and update the intercept term at each run. We term this formulation as "EWMA controller with intercept adaptation (EWMA-I)", and its properties have been extensively studied and well understood. We term the other implementation, i.e., keeping the intercept term as its off-line estimate and updating the process gain at each run, as "EWMA with gain adaptation (EWMA-G)". Despite the fact that gain variation and adaptation is typical in the semiconductor industry, little research has been done to investigate the properties of an EWMA-G controller, such as its stability and sensitivity. In this work, the stability and sensitivity properties of an EWMA-G controller are investigated and compared with those of an EWMA-I controller. Both stationary and drifting processes are considered. In addition, the expressions of the process outputs are derived and the output variances for stochastic processes are evaluated. Simulation examples are given to illustrate these properties and the relevance of these properties to semiconductor process control is discussed. The analysis results will provide some useful guidance on the industrial applications of the EWMA controllers.