Robust forecasts and run-to-run control for processes with linear drifts

In semiconductor manufacturing, processes that experience a linear drift over time are forecasted using a double exponentially weighted moving average (d-EWMA) filter. The forecast is frequently used for run-to-run control where processing condition for next batch is adjusted based on the result of current and past batches. This filter has also been claimed as an optimal filter for the IMA(2,2) process, which is the stochastic equivalent of a process with linear drifts. We show that the optimal filter for the IMA(2,2) process has a different structure from the d-EWMA filter but can be put in a form similar to it with same effective tuning parameters. In addition, we address the situation where the process gain varies from batch to batch but within some a priori known bound. We propose a robust run-to-run control algorithm, which uses the d-EWMA or optimal filter but solves a min-max problem to find the input adjustment that minimizes the worst-case predicted error. Through a simple example, we demonstrate that the min-max formulation of run-to-run control allows for more aggressive choice of the filter tuning parameters without sacrificing robustness and therefore leads to better overall control performance.