In order to generate an accurate estimated state, the state estimator should be able to track the process changes and to reject the measurement noises simultaneously. There are many sources of the process variations, such as tools, devices, layers etc. Because there is no direct measurement of each individual source, the traditional method is to segregate the historical data into different bins and use the ones that match the current event of the specific tool, device and layer, etc. for feedback control. This segregation is based on the idea that wafers with similar process history have similar characteristics, and the control based on the segregated data is also known as “threaded” control. The benefit of threaded control is that within the thread definition, one does not need to worry about individual state changes because the overall state will track all of them as a lumped parameter; while the disadvantage is that this narrowly defined process stream can result in insufficient data and a large amount of “send-ahead” wafers  when applied to low-volume high-mix manufacturing, especially for litho processes where many factors contribute significantly to the process variations.
Because many state-of-the-art fabs are operating with increasingly diversified product mixes, run-to-run state estimation methods in low-volume high-mix production have drawn considerable interest in the last few years, and several different methods have been reported [1-9]. Some of the methods have certain theoretical background, such as [1, 4], while others are largely ad hoc, such as [3, 7]. Among these methods, the general theme is trying to share information among different contexts, and different algorithms such as regression and Kalman filter are applied to sort out the contribution from different variation sources.
In this work, a theorem is first formulated and rigorous proof is given to show that for high-mix production the matrix that contains the context information is always rank deficient, therefore, unbiased estimates exist only for certain linear combinations of the unknown parameters (i.e. the linear combinations of the individual state of each component). Then a general singular Gauss-Markov model is used to represent the high-mix production processes. Based on this model, the best linear unbiased estimate (BLUE) [10-11] of the system states is derived and extended to a unifying framework, where the effect of initial estimates of individual state and associated error variance are explicitly represented. In this framework, the published analytical algorithms [1, 3] can be explained as special cases. Using this framework, the advantage and disadvantage of different algorithms are discussed and a new state estimation method is proposed. In the proposed method, recursive least squares with adaptive forgetting factor is applied to track system states, and a numerically stable algorithm  is applied to handle the singular Gauss-Markov model. Process changes are detected by a Bayesian algorithm , and the forgetting factor is adjusted once the process change is detected. Simulation and industrial examples are given to illustrate the performance of the proposed algorithm.
Key words: low-volume high-mix, regression, Gauss-Markov process, best linear unbiased estimate (BLUE), Kalman filter, threaded control.
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