# Early Fault Detection and Operator-Based MIMO Fault-Tolerant Temperature Control of Microreactor
## 4 Modeling
### Heat Absorption by Peltier Device  
In this study, the control target is cooled by heat absorption from the Peltier device. For the modeling of the heat spreader performed in Section 3, the control input is the heat absorption \(u_{d_{n}}\) from the Peltier device. In this section, the thermal model of the Peltier device is presented and the calculation method of the current input to the experimental device is described.  
The heat absorption of the Peltier device is defined as positive. The model of the heat absorption \(u_{\mathrm{d}}\)[W] is shown in Equation (9). Similarly, the model of heat dissipation \(u_{\mathrm{h}}\)[W] when heat dissipation is defined as positive is shown in Equation (10).  
\[u_{\mathrm{d}}=ST_{\mathrm{c}}i-K(T_{\mathrm{h}}-T_{\mathrm{c}}) -\frac{1}{2}Ri^{2} \tag{9}\] \[u_{\mathrm{h}}=ST_{\mathrm{h}}i-K(T_{\mathrm{h}}-T_{\mathrm{c}}) +\frac{1}{2}Ri^{2} \tag{10}\]  
The command value given to the experimental apparatus is the current \(i_{n}\). When expressed as above, we consider determining the current flowing through the Peltier device based on the heat absorption \(u_{d_{n}}\) to be referred to. In this case, the current \(i_{n}\) is obtained by solving the Equation (9) for \(i_{n}\) as follows.  
\[i=\frac{ST_{\mathrm{c}}\pm\sqrt{(ST_{\mathrm{c}})^{2}-R(2K(T_{\mathrm{h}}-T_{ \mathrm{c}})+2u_{\mathrm{d}})}}{R} \tag{11}\]  
Thermal Model Deformation  
Equivalent transformations are performed for the models shown in Section 3, and the transformation is made into an easy-to-handle form. Defining the temperature drop of the controlled object from the outside temperature \(y_{a_{n}}(t)=T_{0}-T_{a_{n}}(n=1,2,3)\), the model for the heat spreader is as follows.  
\[\mathrm{Part}\,\mathrm{A}_{n}:\frac{dy_{a_{n}}(t)}{dt}=\omega_{a_{n}}(t)+ \sum_{m=1}^{4}(-1)^{m}A_{a_{nm}}y_{a_{n}}^{m}(t) \tag{12}\]  
In this case, \(\omega_{a_{n}}(t)\) (\(n=1,2,3\)) is as follows. The \(A_{11}\) to \(A_{34}\) are constants obtained by Equation transformation.  
\[\omega_{a_{1}}(t)=\frac{2u_{d_{1}}(t)+\alpha_{w}S_{6}y_{1}(t)+\frac{\lambda_ {a}S_{3}}{dx}y_{a_{2}}(t)}{m_{a}c_{a}} \tag{13}\]  
\[\omega_{a_{2}}(t)=\frac{\alpha_{w}S_{9}y_{2}(t)+\frac{\lambda_{a}S_{3}}{dx}y_{ a_{1}}(t)+\frac{\lambda_{a}S_{3}}{dx}y_{a_{3}}(t)}{m_{a}c_{a}} \tag{14}\]  
\[\omega_{a_{3}}(t)=\frac{2u_{d_{1}}(t)+\alpha_{w}S_{6}y_{3}(t)+\frac{\lambda_{a }S_{3}}{dx}y_{a_{2}}(t)}{m_{a}c_{a}} \tag{15}\]Similar to the Section 4.2, the equation transformation is performed for the model for the tube Part. Defining the temperature \(y_{k}=T_{0}-T_{w_{k}}(k=1,2,3)\) of the drop from the outside temperature to be controlled, the model for the tube is as follows.  
\[\text{Part}\,\text{W}_{k}:\frac{dy_{k}(t)}{dt}=\omega_{w_{k}}(t)-A_{w_{k}}y_{k}(t) (k=1,2,3) \tag{16}\]  
In this case, \(\omega_{w_{k}}(k=1,2,3)\) are expressed as follows. \(A_{w_{1}}\) to \(A_{w_{3}}\) are constants obtained by equation transformation.  
\[\omega_{w_{1}}(t)=\frac{\alpha_{w}S_{6}y_{a_{1}}(t)}{m_{w_{1}}c_{w}}\qquad \quad\omega_{w_{2}}(t)=\frac{\alpha_{w}S_{9}y_{a_{2}}(t)}{m_{w_{2}}c_{w}}\qquad \quad\omega_{w_{3}}(t)=\frac{\alpha_{w}S_{6}y_{a_{3}}(t)}{m_{w_{3}}c_{w}} \tag{17}\]