On the stability of MIMO EWMA run-to-run controllers with metrology delay

As production requirements in semiconductor manufacturing continue to escalate, it is rarely possible to perform quality measurements on a product before subsequent process operations are performed. This delay between manufacturing and metrology coupled with inaccurate process models can lead to closed-loop instabilities. This paper investigates the effect of metrology delay on the closed-loop stability of a multiple input-multiple output exponentially weighted moving average run-to-run controller. The generalized Routh-Hurwitz stability criteria are used to derive the necessary and sufficient conditions for stability. A sufficient condition for stability is then derived for systems with any length of metrology delay. This condition states that if all of the eigenvalues of a model-mismatch matrix fall inside a circle with unit radius and centered at {1,0} on the complex plane, then the closed-loop system is stable.