Model-based Control of the Czochralski Silicon Crystal Pulling Process
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This study explores the use of mathematical modeling as an aid in automatic control of the standard Czochralski process (Cz). The aim of the modeling has been to develop a simple model, while covering the essential process dynamics, with the ability to update the model from measurements (on-line, during and between runs), rather than developing a highly accurate and detailed complex model. This should result in a model-based controller with acceptable performance, both in accuracy and calculation time. An overview of the process, with emphasis on aspects related to control theory, is presented. The most challenging aspects of the growth process are identified. The conventional method of control is described along with its limitations and disadvantages. A thorough review of existing publications related to modeling and control of the Czochralski process is presented, before introducing a mathematical description of the process. With proper choice of model parameters, the represented nonlinear model is able to express the main hydrodynamic-geometrical dynamics, as well as the essential thermal dynamics of the Czochralski process used for production of Si crystals with specific crystal diameters. Furthermore, the model is assessed based on properties such as local controllability and observability. For the purpose of estimating the actual crystal radius, standard Kalman filter (KF) and Extended Kalman filter (EKF) are explored. Based on simulations, the EKF gave better results in terms of estimation error when compared to the KF. This might indicate that describing the essential and complex dynamics of the Cz process acceptably accurately with a linear model can be a difficult task. Additionally, in order to evaluate the amount of model accuracy loss due to the linearization process in the EKF, Unscented Kalman Filter (UKF) was also applied. Most application studies on state estimation only have on-line, noise corrupted/uncertain measurements available. In this case, the crystal radius, as the main process variable, can be measured with much greater accuracy after the crystal is produced. Thus, based on collected data from actual plant operation, the estimators' performance is assessed, which shows a very similar estimation quality when using EKF and UKF. This might indicate that the nonlinearity in the model is not strong enough to benefit very much from using the UKF. However, the fact that the use of the posteriori radius measurements confirm the findings of the preceding simulation study, is interpreted as a confirmation that the model used represents the nonlinear effects in the meniscus/radius dynamics with reasonable accuracy. The tightly coupled dynamics of the Cz process and the existence of physical constraints on states and manipulators, motivate the application of a model-based control approach that can handle the nonlinearities, as well as the imposed operational constraints, in an optimal fashion. Hence, the nonlinear model is also used to apply Nonlinear Model Predictive Control (NMPC) method for automatic radius and melt temperature control. For an open-loop stable linear system, the stability can be guaranteed by the MPC, without the need for terminal constraints. However, if the system is unstable or has an integrating mode (which suffer from infinitely large time constants), the solution will not converge to a stable point at the end of the control horizon. Based on numerical analysis, the introduced nonlinear model for the Cz process has a close to integrating behavior. In order to effectively deal with this challenge, the widely known and established prestabilization technique is applied. What is involved is a kind of re-parametrization of the controller. The problem can be overcome by expressing predictions as perturbations on a stabilizing linear feedback law. Moreover, two types of system dynamics are identified during the controller tuning phase. The fast dynamics (quick response to input changes) are represented by the geometrical properties of the process such as the radius and the meniscus shape. On the other hand, the bulk part of the process is responsible for the slow dynamics due to delays in the heat transportation. Therefore, it is chosen to divide the control part into two separate controllers where the puller MPC handles the fast dynamics while the heater MPC deals with the slow dynamics. This structure makes it possible to choose different tuning parameters for each of the controllers based on their model dynamics, and to use different sampling rates and prediction horizons in the two MPCs. However, there exists a link of communication between them in terms of the melt temperature. As the simulation results show, the MPC controllers can follow their predefined reference trajectories acceptably accurately, while respecting the physical constraints on both the manipulators and the states. Since the trajectories are designed by experienced metallurgists, an adequate trajectory tracking should most probably result in crystals with less structural defects. Therefore, in addition to reducing the cut-offs from the crystal for wafer production, the enhanced control scheme also improves the crystal qualities required for solar cell productions. Similar to the separation of the controllers, two EKFs are applied for reconstruction of the noisy or unmeasured states. Each estimator has it's own specific sets of tasks that must be fulfilled. The first EKF estimates the real radius, growth angle and a parameter related to calculation of crystallization rate. The second EKF estimates the necessary temperatures at the hot-zone such as the melt temperature, in addition to a parameter required for calculating the entering heat energy from the heater elements into the melt. In order to evaluate the parameter estimation ability of these observers, model mismatch has been introduced in the simulations. That means the applied model for the MPC controllers and the EKF estimators are considerably different from the plant model, in terms of two important model parameter. Additionally, this is done to illustrate the performance ability of the MPC controllers in situations where the model cannot describe the plant dynamics accurately. As illustrated by the simulation results, the estimated parameters seem to converge towards their respective true values, which indicates the satisfactory performance of the applied observers in this study. For commercial Czochralski ingot pullers using the conventional control scheme, feedforward control is an essential part of the overall control system. For instance, the conventional pullers depend to a large extent on the feed-forward temperature reference trajectory to keep the desired pulling rate. A common method in practice is to determine the feedforward control by careful analysis of repeated growth runs, resulting in a trajectory for the control inputs which can then be used as part of the recipe. Although this method is widely accepted, it suffers from the fact that it is extremely time consuming, and thus expensive. Besides that, it works well if the same conditions are repeated and there are no significant variations from run to run. However, any changes in plant setup, process equipments or in desired crystal diameter, means repeating this procedure. Moreover, when using a simplified lumped-parameter model to express the Cz process, the model accuracy depends on updating the time-varying parameters (during, as well as between runs) used in the system equations. Therefore, adaptation of model quantities and trajectories to the real crystal-growing process is essential in terms of controller performance enhancement and ingot quality. Thus, Run-To-Run (R2R) control strategy is suggested to update the process model parameters in a recursive way when output from the previous run becomes available, and thereby help generate better recipes for future runs. Combining MPC, estimation and Run-To-Run control has enabled robust, effective control of the Czochralski crystallization process. Using this approach, the traditional feed-forward control of the growth rate is replaced with a more direct control of the melt temperature. Consequently, for each completed run, the growth rate reference tracking becomes tighter, which results in faster and better control of the crystal diameter. As suggestion for future research, better technique for melt temperature measurement is proposed. It is suggested that better measurement techniques and instrumentation could potentially increase the model accuracy, thus, improving the overall performance of the controllers.