# Early Fault Detection and Operator-Based MIMO Fault-Tolerant Temperature Control of Microreactor
## 5 Control Design
### Fault-Tolerant Control System  
In this section, we design a control system that can follow a reference input even in the event of a tube-mounted temperature sensor fault by activating the sensor in the event of the fault. The fault-tolerant control system in this study consists of the fault detection system and the fault signal compensation control system. The control system described is the fault-tolerant control system that is extended from the one-input, one-output fault-tolerant control system of the two-partition model of the previous study [15] to a three-partition model. The fault signal compensation control system used in this study is shown in Figure 5. As in the Section 5.2, only \(n=k=(1,3)\) is explained. Here, FD in Figure 5 denotes the fault detection system, which switches to the fault-tolerant control system. Operator \(Q_{n}\) is an operator that outputs a signal to compensate for faults and is enabled by the fault detection system.  
Figure 4: Control system based on operator theory.  
In the case of a sensor fault, the fault signal \(f_{n}\) is shown as in Equation (30) using the fault element \(d_{\omega_{n}}\).  
\[f_{n}(t)=B_{n}(y_{n}+d_{\omega_{n}})(t)-A_{n}(u_{d_{n}})(t) \tag{30}\]  
Here, when \(Q_{n}\) is enabled, the output of the tracking compensation controller \(C_{w_{k}}\) is organized as in Equation (31).  
\[(S_{w_{k}}\tilde{N}_{w_{k}}-QB_{n}\tilde{N}_{w_{k}})\tilde{N}_{w_{ k}}^{-1}(y_{n})(t)\] \[= C_{w_{k}}(e_{k})(t)-(R_{w_{k}}D_{a_{n}}+QA_{n}D_{a_{n}})\tilde{N }_{w_{k}}^{-1}(y_{n}+d_{w_{n}})(t) \tag{31}\]  
In this case, design \(A_{n}\) and \(B_{n}\) so that the left side is \(\tilde{N}_{w_{k}}\) and the term including the fault element on the right side is \(O\) operator, respectively. Since the model equation for \(A_{n}\) is the same as for Part A\({}_{n}\), the variable \(A_{n}(u_{d_{n}})\) uses the temperature sensor measurements of Part A\({}_{n}\).  
\[A_{n}(u_{d_{n}})(t) =N_{a_{n}}D_{a_{n}}^{-1}(u_{d_{n}})(t) \tag{32}\] \[B_{n}(y_{n}+d_{w_{n}})(t) =P_{w_{k}}^{-1}(y_{n}+d_{w_{n}})(t) \tag{33}\]  
When operators \(A_{n},B_{n}\) satisfying the conditions are designed as in Equations (32) and (33), operator \(Q_{n}\) is shown as in Equation (34).  
\[Q_{n}(f_{n})(t)=-\tilde{N}_{\omega_{k}}^{-1}R_{w_{k}}D_{a_{n}}P_{w_{k}}(f_{n} )(t) \tag{34}\]  
Substituting Equations (32)-(34) into Equation (31) again, a relational expression such as Equation (35) is shown.  
\[(S_{w_{k}}\tilde{N}_{w_{k}}+R_{w_{k}}D_{a_{n}})\tilde{N}_{w_{k}}^{-1}(y_{n})( t)=C_{w_{k}}(e_{k})(t) \tag{35}\]  
The design of the operator as in Equation (A12) also shows the relationship as in Equation (36).  
\[\tilde{N}_{w_{k}}^{-1}(y_{n})(t)=C_{w_{k}}(e_{k})(t) \tag{36}\]  
The relationship in Equation (36) transforms the operator \(Q_{n}\) into an operator with a tracking compensation operator \(Cw_{k}\). However, to account for thermal disturbances caused by modeling errors of the control target and thermal disturbances caused by thermal transfer with the liquid flowing in the channel, it is transformed like an operator with \(Mw_{k}\) whose gain of \(Cw_{k}\) is adjusted separately.  
\[Q_{n}(f_{n})(t)=-M_{w_{k}}R_{w_{k}}D_{a_{n}}P_{w_{k}}(f_{n})(t) \tag{37}\]  
The relationship in Equation (36) removes the fault element from the output, allowing it to follow the reference input even in the event of a fault. Note that \(Q_{n}(f_{n})(t)=0\) when the sensor is determined to be normal.  
Figure 5: Fault-tolerant control system.