Particle Filter with Unknown Statistics to Estimate Liquid Level in the Silicon Single-Crystal Growth

In the process of silicon single-crystal growth, the detection of melt level is an important step to ensure that the grown silicon single crystal is of high quality. Since there exist many mechanical motions, chemical interactions, thermal convection, and so on, it is extremely difficult to obtain the noise statistics of the liquid-level data. To attain a high-accuracy detection of liquid level with unknown noise statistics, we present an extended set-membership-based particle filter (PF). The key points are as follows. We construct a nonlinear state-space model for liquid-level detection system based on the kinematic principle and laser triangulation measurement principle and cast the detection problem of the liquid level into a state estimation problem. Based on this model, we use the extended set-membership estimation theory to generate an ellipsoid set that contains the true value of the liquid-level state. To overcome the difficulty of drawing particles under the case of unknown statistics noise, we draw particles from a Gaussian distribution in the obtained ellipsoidal set. In order to resample particles when the measurement noise statistic is unknown, we introduce a cost function to obtain the weights of the particles, and the ultimate estimation of liquid level is calculated by the weighted particles after the resampling procedure. Comparing with the extended Kalman PF (EPF) and the cost-reference PF (CRPF), the proposed algorithm has higher estimation accuracy. Some simulation and experiment results are presented to illustrate the effectiveness of the proposed method.