# The Design of 2 DOF IMC-PID Controller in Biochemical Reaction Networks
## 3 Results
### Performance of the 2 DOF IMC-PID Biomolecular Controller
In addition, there are only two filtering parameters (\(\lambda_{1}\) and \(\lambda_{2}\)) to be adjusted in the 2 DOF IMC-PID controller; the complexity of the controller parameter tuning is reduced. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Controlled Object \(G_{p}\)** & **Control System \(F_{1}\)** & **Control System \(F_{2}F_{3}\)** \\ \hline \(\tau=10^{5}\) s & \(\lambda_{1}=1.75\times 10^{5}\) & \(k_{P}=2\) \\ \(K=1\) & \(\lambda_{2}=1\times 10^{5}\) & \(k_{1}=9.5\times 10^{-6}\) s\({}^{-1}\) \\ \(T=2\times 10^{5}\) s & \(k  
The parameter \(\lambda_{1}\) is adjusted while keeping the parameter \(\lambda_{2}\) constant in Figure 9a. As can be seen, the set-point following characteristic of the controller can be improved by reducing the filtering parameter \(\lambda_{1}\). Similarly, the parameter \(\lambda_{2}\) is adjusted while keeping the parameter \(\lambda_{1}\) constant. In Figure 9b, the disturbance rejection performance of the 2 DOF IMC-PID controller gets improved by increasing the filtering parameter \(\lambda_{2}\). The performance of the controller for the different filtering parameters \(\lambda_{1}\) and \(\lambda_{2}\) are shown in Tables 4 and 5. As a conclusion, the set-point following performance of the controller can be improved by keeping the parameter \(\lambda_{2}\) constant and reducing the parameter \(\lambda_{1}\). Similarly, the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Filter Parameter \(\lambda_{1}\) and \(\lambda_{2}\)** & **Rise Time** & **Settling Time** & **Overshoot** \\ \hline \(A:\lambda_{1}=1.5\times 10^{5},\lambda_{2}=10^{5}\) & 526,379 s & 665,496 s & 0.047\% \\ \(B:\lambda_{1}=2\times 10^{5},\lambda_{2}=10^{5}\) & 634,209 s & 778,493 s & 0.037\% \\ \(C:\lambda_{1}=3\times 10^{5},\lambda_{2}=10^{5}\) & 850,657 s & 1,049,520 s & 0.056\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Effect of filtering parameter \(\lambda_{1}\) on controller set-point tracking performance. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Control System**} & \multicolumn{3}{c}{\(\boldsymbol{r(t)}\)=**1(\(t\)),\(\boldsymbol{d(t)}\)=1(\(t\))**} \\ \cline{2-4}  & **Rise Time** & **Settling Time** & **Overshoot** \\ \hline
2 DOF IMC-PID Control System \(F_{1}\) & 573,587 s & 900,343 s & 0.024\% \\
1 DOF PID Control System \(F_{2}\)[29; 30] & 238,511 s & 979,059 s & 12.10\% \\
2 DOF PID Control System \(F_{3}\)[30] & 427,343 s & 990,834 s & 2.48\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance of three control systems when the disturbance signal is \(d(t)=1(t)\). Figure 9: Output response curves of the 2 DOF IMC-PID controller: (**a**) variation in controller set-point tracking performance; (**b**) variation in controller disturbance rejection performance. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Filter Parameter \(\lambda_{1}\) and \(\lambda_{2}\)** & **Rise Time** & **Settling Time** & **Overshoot** \\ \hline \(D:\lambda_{1}=2.8\times 10^{5},\lambda_{2}=0.75\times 10^{5}\) & 252,556 s & 276,659 s & 2.21\% \\ \(E:\lambda_{1}=2.8\times 10^{5},\lambda_{2}=1.5\times 10^{5}\) & 250,733 s & 775,082 s & 9.08\% \\ \(F:\lambda_{1}=2.8\times 10^{5},\lambda_{2}=2\times 10^{5}\) & 248,043 s & 1,014,084 s & 15.85\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Effect of filtering parameter \(\lambda_{2}\) on controller disturbance rejection performance. excellent disturbance suppression capability of the controller is attained by keeping the parameter \(\lambda_{1}\) constant and reducing the parameter \(\lambda_{2}\). To better demonstrate the rejection performance of the 2 DOF IMC-PID controller based on CRNs for different disturbance signals, the trigonometric signal \(d(t)=\xi\sin\omega t\) is selected as the disturbance input. The suppression effect of the 2 DOF IMC-PID controller based on the CRNs for the trigonometric signal \(d(t)=\xi\sin\omega t\) with different frequencies is simulated.