# Early Fault Detection and Operator-Based MIMO Fault-Tolerant Temperature Control of Microreactor
## Appendix A One-Class SVM  
One-Class SVM, a one-class classification method [22; 23], is described. The algorithm is described below. First, the training data \(\mathbf{x}_{i\in[I]}\) are prepared, and the inner product of the projection data by \(\mathbf{\phi}\) is defined as a kernel function as in Equation (A1). In this study, the RBF kernel is used.  
\[k(\mathbf{x}_{i},\mathbf{x}_{j})=\mathbf{\phi}(\mathbf{x}_{i})^{\mathsf{T}}\mathbf{\phi}(\mathbf{x}_{ j})\] (A1)  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Description** & **Value/Unit** & **Description** & **Value/Unit** \\ \hline Mean absolute error & & Mean absolute error & \\ of Part W\({}_{1}\) & 0.0181 \({}^{\circ}\)C & of Part W\({}_{3}\) & 0.0173 \({}^{\circ}\)C \\ (Simulation) & & (Simulation) & \\ Mean absolute error & & Mean absolute error & \\ of Part W\({}_{1}\) & 0.0735 \({}^{\circ}\)C & of Part W\({}_{3}\) & 0.0700 \({}^{\circ}\)C \\ (Experiment) & & (Experiment) & \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of simulation and experiment.  
Assuming that the normal and abnormal data are linearly separable in the high-dimensional space after mapping, the separation plane is expressed as Equation (A2) using the weight coefficient \(\mathbf{\omega}\) and bias \(\rho\).  
\[\mathbf{\omega}^{T}\mathbf{\phi}(\mathbf{x})-\rho=0\] (A2)  
To allow for misclassification, a slack variable \(\xi_{i}\geq 0\) is introduced. In this case, we consider maximizing the distance from the origin with respect to the classification plane. Converting the maximization problem into a minimization problem, it is shown as follows, including the constraints using the normalization parameter \(\
u\in(0,1)\).  
\[\min_{\mathbf{\omega},\xi,i,\rho}\frac{1}{2}\|\mathbf{\omega}\|^{2}-\rho+ \frac{1}{I\
u}\sum_{i\in[l]}\xi_{i}\] (A3) \[\text{s.t.}\ (\mathbf{\omega}^{T}\cdot\mathbf{\phi}(\mathbf{x}_{i}))\geq( \rho-\xi_{i}),\quad\xi_{i}\geq 0,\forall i\in[l]\]  
Also, introducing the new variables \(\alpha_{i},\beta_{i}\geq 0,\quad i\in[l]\) and Equation for the minimization problem, the Lagrangian function is shown as follows.  
\[L(\mathbf{\omega},\mathbf{\xi},\rho,\mathbf{\alpha},\mathbf{\beta})= \frac{1}{2}\|\mathbf{\omega}\|^{2}-\rho+\frac{1}{I\
u}\sum_{i\in[l]} \xi_{i}\] \[-\sum_{i\in[l]}\alpha_{i}((\mathbf{\omega}^{T}\cdot\mathbf{\phi}(\mathbf{x}_ {i}))-\rho+\xi_{i})-\sum_{i\in[l]}\beta_{i}\xi_{i}\] (A4)  
The minimization problem becomes a \(\mathbf{\alpha}\)-only problem, as follows.  
\[\min_{\mathbf{\alpha}}\frac{1}{2}\sum_{i,j}\alpha_{i}\alpha_{j}k\big{(} x_{i},\mathbf{x}_{j}\big{)}\] (A5) \[\text{s.t.}\ 0<\alpha_{i}\geq\frac{1}{n\
u},\ \forall i\in[n],\sum_{i\in[l]} \alpha_{i}=1\]  
Using the final optimized Lagrange multiplier \(\alpha_{i}\), the decision function for the unknown data \(\mathbf{x}\) are as follows.  
\[f(\mathbf{x})=\text{sgn}\Bigg{(}\sum_{i\in[l]}\alpha_{i}k(\mathbf{x}_{i},\mathbf{x})-\rho \Bigg{)}\] (A6)  
The outlier-likeness of unknown input patterns mapped to a higher-dimensional space increases as the distance from the origin is shortened. Therefore, input patterns that are placed closer to the origin than the classification plane do not belong to a class and are judged to be outliers.