## 4 Case Studies
### Case 2: RTCVD System  
This section takes the thermal process of a three-zone RTCVD reactor as an example to verify the effectiveness of the proposed method. The structure of the RTCVD is shown in Figure 10. The furnace body is designed with three zones of heating lamps (A, B, and C). Argon mixed with 10% silane enters the reactor from the top of the furnace body, and then the silane is decomposed into silicon and hydrogen in the reactor. The goal of this process is to uniformly deposit a 0.5-micron thick polycrystalline silicon film on the wafer surface. Since the temperature difference between the upper and lower surfaces of the wafer is very small and the wafer rotates slowly during heating, the wafer temperature is assumed to vary only in the radial direction. Based on the above observations and assumptions, the thermal dynamics of the wafer in the radial direction can be represented by the following partial differential equation:  
\[\frac{\partial T_{r}}{\partial t}=k_{0}\Big{(}\frac{1}{r}\frac{\partial T_{r} }{\partial r}+\frac{\partial^{2}T_{r}}{\partial r^{2}}\Big{)}+\sigma_{0} \Big{(}1-T_{r}^{4}\Big{)}+\omega_{r}q_{a}(r)u_{a}+\omega_{r}q_{b}(r)u_{b}+ \omega_{r}q_{c}(r)u_{c} \tag{34}\]  
Figure 8: TNAE comparison of the different methods in Case 1. Figure 9: RLNE comparison of the different methods in Case 1. They are subject to the boundary conditions as follows:  
\[\left\{ \begin{array}{c}\frac{\partial T_{r}}{\partial r}=\sigma_{ed}(1-T_{r})+q_ {ed}u_{b},\;\;r=1\\ \frac{\partial T_{r}}{\partial r},\;\;r=0\end{array} \right. \tag{35}\]  
\[k_{0}=0.0021,\sigma_{0}=0.0012,\sigma_{ed}=0.0037,q_{ed}=4.022,\omega_{r}=0.026\]  
where \(T_{r}=T_{r}^{\prime}/T_{amb}\) denotes the dimensionless wafer temperature, \(T_{r}^{\prime}\) is the actual wafer temperature, \(T_{amb}=300\) K is the ambient temperature, \(r=r^{\prime}/R_{w}\) is the dimensionless radial position of the wafer, \(r^{\prime}\) is the actual radial position of the wafer, and \(R_{w}=7.6\) cm is the wafer radius. \(q_{b}(r)\), \(q_{b}(r)\), and \(q_{c}(r)\) represent the amount of thermal radiation in the radial direction of the wafer for the three heating blocks, respectively. \(k_{0}\) is the thermal conductivity of the wafer, \(\sigma_{0}\) is the radiation coefficient of the quartz chamber, \(\sigma_{ed}\) is the radiation coefficient of the wafer, \(q_{ed}\) is the incident radiation flux at the edge of the wafer, and \(\omega_{r}\) is the density of the wafer. In order to fully obtain the dynamic information of the system, an interference signal with an amplitude not exceeding 10% was added to the system input signal. The input variable expression with the interference signal is shown as follows:  
\[u_{a}(t) =0.2028+0.1*0.2028*rand()*sign(rand()-0.5) \tag{36}\] \[u_{b}(t) =0.1008+0.1*0.1008*rand()*sign(rand()-0.5)\] \[u_{c}(t) =0.2245+0.1*0.2245*rand()*sign(rand()-0.5)\]  
where 0.2028, 0.1008, and 0.2245 are the steady-state inputs for \(u_{a}\), \(u_{b}\), and \(u_{c}\) at a furnace temperature of 1000 K.  
Eleven measurement sensors were placed along the radial position of the wafer, with each sensor providing a set of measurement data. To simulate the measurement noise, independent white Gaussian noise was added to each of the 11 measurement datasets. The noise was generated with a mean of 0 and a standard deviation of 0.2, which corresponds to an amplitude of 0.2. The noise signal was generated using a random number generator to produce independent samples at each time step, ensuring that the noise was uncorrelated across sensors and time steps. The sampling period was set to \(\Delta t=0.5\) s, with the offline simulation period lasting 500 s and a total of 1000 offline data samples being collected  
Figure 10: System structure of RTCVD. for the initialization model. The online simulation period lasted for 200s. Two cases were simulated to demonstrate the effectiveness of the proposed method: one representing external environmental changes and the other simulating internal environmental changes in the system.