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

Processes experiencing linear drift over time are usually forecasted using a double exponentially weighted moving average (d-EWMA) filter. d-EWMA has incorrectly been claimed as an optimal filter for the integrated moving average (2,2) (IMA(2,2)) process, which is a stochastic equivalent of a process with linear drifts (ramps). It is shown that the optimal filter for such a process has a different structure but can be put in a similar form with same effective tuning parameters. The problem of batch-to-batch process gain variation (with known bounds) has been addressed by using a robust run-to-run control algorithm. This algorithm solves a minimax problem that determines the next run input adjustment by minimizing the worst-case predicted error. The conditions for equivalence of the minimax controller to a nominal model inverse based controller for a simple SISO system based on the type of model used and the nature of bounds have been investigated. An important implication of the equivalence result for nonlinear systems is pointed out. The proposed robust run-to-run controller formulation is tested on a number of examples including a chemical mechanical polishing (CMP) process model.