## Model Based Control of Solidification

##### Abstract

The objective of this thesis is to develop models for use in the control of a solidification process. Solidification is the phase change from liquid to solid, and takes place in many important processes ranging from production engineering to solid-state physics. Often during solidification, undesired e¤ects like e.g. variation of composition, microstructure, etc. occur. The solidification structure and its associated defects often persist throughout the subsequent operations, and thus good control of the solidi.cation process is of great importance.
It is well known that the two most important quantities a¤ecting the quality of the solidified material is the solidification rate and the thermal conditions at the interface between liquid and solid phase. Neither the solidi.cation rate nor the thermal conditions at the interface can be measured in most industrial applications, and thus it is desirable to obtain good estimates of these quantities. The estimates may be achieved from a mathematical model of the solidi.cation process. For phase-change processes, two different model formulations exist, and the two formulations are presented and discussed. The numerical solution of phase-change models is non-trivial, because of the unknown moving boundary. Event location is used to simplify the implementation of the numerical solution of one of the model formulations.
A possible solidification process at Elkem Solar, a pilot plant is used as case study. A mechanistic model of the pilot furnace has been developed. The model has been implemented in a simple version for both model formulations, not accounting for the dynamics at the .xed boundaries. A revised version including the radiation dynamics at the boundaries has been implemented for one of the model formulations. The chosen formulation requires small discretization elements in space, and thus the model order is quite high. The revised model has been used for parameter identifiability analysis. The most identifiable parameters are estimated with the nonlinear least squares method using temperature measurements from experiments carried out on the furnace. The model fitting to the measured temperatures from the furnace is improved after parameter estimation is performed, and the results are acceptable. However, it is di¢ cult to decide whether the estimate of the interface position is improved, because no information about the interface position was available in the experiments.
Control of the solidification rate requires either a measurement or a reliable real-time estimate of the interface position. As the interface position most often can not be measured, state estimation from temperature measurements is implemented for di¤erent instrumentation settings. The iterative Extended Kalman Filter (IEKF) is used, and yields acceptable results for the estimation in the liquid phase. However, due to weak observability of the state undergoing phase-change, the estimation errors increase during phase-change. The Square Root Unscented Kalman Filter (SRUKF) is also implemented, but the SRUKF results in spiky temperature estimates during the phase change. The SRUKF also has a very high computation time because of the high model order, and is (at present time) not able to run in real time for the pilot furnace.
PI-control of the estimated interface position is tested for the simpli.ed model and the extended model. The current to the heaters above the crucible is the manipulated input; the valve for heat sink is not automated on the pilot plant. The PI-controller yields good performance for some chosen reference trajectories. However, for high solidi.cation rates the manipulated input gets saturated because of the constraints on the manipulated input, and a deviation between the reference and the estimated interface position is observed. The performance of the PI-controller thus depends on the reference trajectory. The optimal reference should be given by Elkem, after thorough experiments and succeeding analysis of the ingot quality are performed. During the work of this thesis, the optimal reference trajectory of the interface position was not available.
Input-output linearization of the pilot furnace was considered as an alternative to linear PI-control. However, for this case study the relative degree of the system is quite high and also changes as a function of the interface position: it decreases when the interface position increases. An implementation of input-output linearization for the pilot furnace result in a controller in which the relative degree is switching. Thus, for the pilot furnace process, input-output linearization is not considered as a feasible strategy. However, further investigation regarding the dynamic nature of the relative degree may be carried out in future work.
An advanced control strategy is tested by the use of linear MPC. The simulation results show that the performance of the MPC controller is satisfactory; the chosen reference trajectories are tracked well. For the MPC controller, the control deviation is smaller and reaches zero faster than for the PI-controller, but a more aggressive control action is observed. For some of the reference trajectories, the aggressive control action leads to rapid saturation in the manipulated input. Also, small oscillations on the control input is observed. By using .ner discretization in space and/or increasing the smoothing interval on the model, the oscillations are reduced.