Compensating for the Initialization and Sampling of EWMA Run-to-Run Controlled Processes

The exponentially weighted moving average (EWMA) filter is commonly used for state estimation of run-to-run controllers in semiconductor manufacturing. It is widely known that, when at steady state, the EWMA filter provides the minimum mean square error (MSE) forecast for an integrated moving average (IMA) process. The forecast, however, is optimal if and only if every output of the IMA process is measured. If an EWMA controller is implemented utilizing sampled process data, then it is necessary to retune the controller to maintain optimal performance. Furthermore, in practice, the complex interacting selection criteria of advanced sampling applications often cause measurement frequency to be highly irregular. In this paper, a sampling compensation algorithm (SCA) is derived based on the minimum-norm IMA (MNIMA) forecast. The algorithm provides the minimum mean-square-error (MMSE) forecast of an IMA process for irregularly sampled processes and is robust to initialization.