Due to their simplicity and robustness, exponentially weighted moving average (EWMA) controllers are the most commonly used run-to-run feed-back controllers in semiconductor manufacturing. EWMA controller usually employs a linear static model. Consider a single input single output process,

y[n] = α + βu[n] + ε[n]

where α and β are model parameters, ε[n] is the process disturbance sequence, u[n] is the input to the process at the beginning of run n and y[n] is the output of the process at the end of run n. The commonly used EWMA controller with intercept updating is defined as

u[n+1] = (T - a[n]) / b

a[n] = ω(y[n] - bu[n]) + (1 - ω)a[n - 1]

where b is the off-line estimate of the process gain β, a[n] is the on-line update of the intercept term αƒzƒnand ω is the update weighting. Ingolfsson and Sachs gave the stability and sensitivity analysis in [1] and it was shown that for EWMA controller with intercept update, the controller is stable if and only if 0 < (ωβ)/b < 2. Partially due to this guaranteed stability, EWMA controller gained wide applications in semiconductor manufacturing industry. However, in many cases the EWMA controller is implemented in the way that the gain, instead of the intercept, is updated using EWMA statistics in a run-to-run manner. The examples for EWMA controllers with gain updating include many processes with processing rate updating, such as chemical mechanical polishing process (CMP) where polishing rate is updated; low pressure chemical vapor deposition process (LPCVD) where deposition rate is updated; and etch process where etch rate is updated. In the case of gain updating, the EWMA control law and the state estimation become

u[n + 1] = (T - a) / b[n]

b[n] = ω(y[n] - a) / u[n] + (1 - ω) b[n-1]

where a is the off-line estimate of intercept α, and b[n] is the on-line update of β.

Although EWMA controller with gain updating has been applied in many processes, to the best of the author's knowledge, there hasn't been any analysis on the stability and sensitivity of the EWMA controller with gain updating. In this work, we present a stability and sensitivity analysis for the EWMA controller with gain updating for both stationary and drifting process. We show that for a stochastic first order stationary process, the stability condition and output variance of the EWMA controller with gain updating has the striking similarity with that of the EWMA controller with intercept updating. The stability condition for EWMA controller with gain updating is 0 < (T - α)ω / (T - a) < 2 which is in the same format as the stability condition for EWMA controller with intercept updating. The output variance is Var(y[n]) = σ²(1 + (1 - θ) / (1 + θ)), where σ is the standard deviation of the noise term, and &theta = 1 - (T - α)ω / (T - a). We further show that for drifting process, there are some difference between the property of EWMA controller with gain updating and intercept updating. For EWMA controller with gain updating, stability condition depends on the model structure, i.e., whether the gain is drifting or the intercept is drifting. Several simulation examples are given to illustrate the analysis results.

Reference

[1] Ingolfsson A. and E. Sachs (1993), Stability and sensitivity of an EWMA controller, Journal of Quality Technology, Vol. 25, No. 4, 271-287

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