# The Design of 2 DOF IMC-PID Controller in Biochemical Reaction Networks
## 4 Conclusions  
In this paper, the first-order time-delay system is selected as the control object, and a 2 DOF IMC-PID controller is constructed by using CRNs as the programming language. The 2 DOF IMC-PID controller is designed according to the mathematical model of the controlled subject by incorporating the IMC principle and the 2 DOF control theory. With the 2 DOF IMC-PID control tactics, the set-point tracking and disturbance suppression characteristics are decided by two independent controllers, respectively. The performance of the controller is obtained only by adjusting the two filtering parameters \(\lambda_{1}\) and \(\lambda_{2}\), and the complexity of parameter tuning is greatly reduced compared with the 2 DOF PID controller that requires five parameters to be adjusted. The 2 DOF IMC-PID control strategy is verified to be effective in stabilizing biochemical reactions based on the DSD reaction network. The simulation results in Visual DSD show that the 2 DOF IMC-PID controller has better set-point following and disturbance rejection capabilities. Uncertainty in the calculation results is caused by the leak reaction in the division gate. For this reason, a 2 DOF IMC-PID division gate control system is established, and the leak reaction is  
Figure 16: Calculations of the division gate control system when the initial species \(A\) and \(B\) leak simultaneously.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**The Initial Species Concentrations** & **Leak Parameters** \\ \hline \([A]_{0}=16\) nM, \([B]_{0}=8\) nM & \(\gamma=0.2,\gamma^{\prime}=0.1\) \\ \([A]_{0}=36\) nM, \([B]_{0}=12\) nM & \(\gamma=\gamma^{\prime}=0.1\) \\ \([A]_{0}=15\) nM, \([B]_{0}=10\) nM & \(\gamma=0.1,\gamma^{\prime}=0.2\) \\ \hline \hline \end{tabular}
\end{table}
Table 9: The initial species concentrations and leak parameters \(\gamma\) and \(\gamma^{\prime}\).  
considered as the external disturbance signal. Although there are leak reactions in the division gate, the calculations of the 2 DOF IMC-PID division gate control system are still accurate, which proves that the 2 DOF IMC-PID controller can effectively suppress the leak reactions. In future work, we will continue to explore the possibility of constructing molecular control circuits in biochemical reaction networks by combining some advanced optimization algorithms in the field of control engineering.  
The following supporting information can be downloaded at: [[https://github.com/ly810342287/Ly](https://github.com/ly810342287/Ly)]([https://github.com/ly810342287/Ly](https://github.com/ly810342287/Ly)).  
Conceptualization, H.L. and Y.L.; data curation, Y.L.; formal analysis, Y.L.; investigation, Y.L.; methodology, Y.L.; project administration, H.L. and X.W.; resources, H.L.; software, Y.L.; supervision, H.L. and X.W.; validation, H.L., Y.L. and X.W.; visualization, Y.L.; writing--original draft, Y.L.; writing--review & editing, H.L. and X.W. All authors have read and agreed to the published version of the manuscript.  
This work is supported by 111 Project (No. D23006), the National Natural Science Foundation of China (Nos. 62272079, 61972266), the Liaoning Revitalization Talents Program (No. XLYC2008017), the Natural Science Foundation of Liaoning Province (Nos. 2020-KF-14-05, 2021-MS-344, 2021-KF-11-03, 2022-KF-12-14), the Scientific Research Fund of Liaoning Provincial Education Department (No. LJKZZ2020147), the State Key Laboratory of Synthetical Automation for Process Industries, the State Key Laboratory of Light Alloy Casting Technology for High-end Equipment (No. LACT-006), the Postgraduate Education Reform Project of Liaoning province (No. LNYG2022493), and the Dalian Outstanding Young Science and Technology Talent Support Program (No. 2022 RJ08).  
Not applicable.  
The data supporting the results reported here are available from the corresponding author upon request.  
The authors declare no conflict of interest.