Particle filtering of hidden Gamma processes for robust Predictive Maintenance in semiconductor manufacturing

Predictive Maintenance methods are aimed to obtain reliable estimates of the remaining life cycle of an equipment from time series of suitable process parameters, named “health factors”, typically exhibiting a monotone evolution associated with the equipment wear. The present study was motivated by the predictive maintenance of a dry etching equipment within a semiconductor manufacturing process. The optimal prediction of the health factor, represented by the cooling helium flow, must cope with noisy measurements of the health factor (possibly masking its monotonicity) and non uniform sampling times. The problem is formulated as a stochastic filtering problem in which a stochastic process has to be optimally predicted based on noisy and irregularly sampled observations. In particular, a hidden Gamma process model is proposed in order to capture all the features of the health factor, namely its nonnegativity and nonnegativity of its derivative. Since this filtering problem is not amenable to a closed form solution, a numerical Monte Carlo approach based on particle filtering is developed. Additionally, an adaptive parameter identification procedure is proposed to achieve the best trade off between promptness and noise insensitivity.