# The Design of 2 DOF IMC-PID Controller in Biochemical Reaction Networks
## 2 Materials and Methods
### Design of 2 DOF IMC-PID Control System Based on CRNs
#### 2.2.2 The 2 DOF IMC-PID Controller Based on CRNs  
By adopting the internal model controller design method, the internal model is decomposed into two portions (\(G_{m}(s)=G_{m+}(s)G_{m-}(s)\)). The expressions of 2 DOF IMC controllers are as follows. \[Q_{1}(s)=G_{m-}^{-1}(s)F_{1}(s) \tag{15}\]  
\[Q_{2}(s)=G_{m-}^{-1}(s)F_{2}(s) \tag{16}\]  
**Remark 2**. : _In Equations (15) and (16), \(G_{m-}(s)\) is the minimum-phase part of the internal model \(G_{m}(s)\), and \(F_{1}(s)\) and \(F_{2}(s)\) are low-pass filters._  
\[G_{m-}(s)=\frac{K}{T_{\rm S}+1} \tag{17}\]  
\[F_{1}(s)=\frac{1}{\lambda_{1}s+1} \tag{18}\]  
\[F_{2}(s)=\frac{1}{\lambda_{2}s+1} \tag{19}\]  
_where\(\lambda_{1}\) and \(\lambda_{2}\) are the filtering parameters. \(Q_{1}(s)=\frac{T_{\rm S}+1}{K(\lambda_{1}s+1)}\) and \(Q_{2}(s)=\frac{T_{\rm S}+1}{K(\lambda_{2}s+1)}\) are further obtained according to Equations (15)-(19). Thus, the reference-value-tracking characteristic of the 2 DOF IMC control system is regulated by the filtering parameter \(\lambda_{1}\), and the disturbance rejection capability is influenced by filtering parameter \(\lambda_{2}\). The expressions of 2 DOF controllers are further obtained via Equations (3) and (4)._  
\[C_{1}(s)=\frac{\lambda_{2}s+1}{\lambda_{1}s+1} \tag{20}\]  
\[C_{2}(s)=(K_{P}+\frac{K_{I}}{s}+K_{\rm D}s)\frac{1}{T_{\rm F}s+1} \tag{21}\]  
_where the PID control structure is adopted for controller \(C_{2}(s)\), and the controller parameters \(K_{P}=\frac{2T+\tau}{2K(\lambda_{2}+\tau)}\), \(K_{I}=\frac{1}{K(\lambda_{2}+\tau)}\), \(K_{D}=\frac{\tau\cdot T}{2K(\lambda_{2}+\tau)}\), and \(T_{\rm F}=\frac{\tau\lambda_{2}}{2(\lambda_{2}+\tau)}\). \(T,K,\tau\) are known parameters in the controlled object, so the controller parameters are determined by two filtering parameters \(\lambda_{1}\) and \(\lambda_{2}\)._  
In Figure 2, the 2 DOF IMC-PID controller is composed of the set-point filter \(C_{1}(s)\) and the PID controller \(C_{2}(s)\). Equations (22) and (23) are derived from Equations (20) and (21), which can be useful for constructing controllers within the framework of CRNs in the following section. In Equations (22) and (23), approximation is used instead of differential representation. Set-point filter \(C_{1}(s)\) and PID controller \(C_{2}(s)\) are modified to obtain a block diagram of the transfer function and the control structure in Figures 3 and 4. \[C_{1}(s)=\frac{\lambda_{2}s+1}{\lambda_{1}s+1}=k_{7}+\frac{k_{8}}{k_{9}+s} \tag{22}\]  
\[C_{2}(s)=(K_{P}+\frac{K_{I}}{s}+K_{\rm D}s)\frac{1}{T_{\rm F}s+1}=(k_{1}+\frac {k_{2}}{s}-\frac{k_{3}}{k_{4}+s})\frac{k_{5}}{k_{6}+s} \tag{23}\]In set-point filter \(C_{1}(s)\), \(k_{7}=\frac{\lambda_{2}}{\lambda_{1}}\), \(k_{8}=\frac{\lambda_{1}-\lambda_{2}}{\lambda_{1}!^{2}}\), and \(k_{9}=\frac{1}{\lambda_{1}}\). In PID controller \(C_{2}(s)\), \(k_{1}=K_{P}+\frac{2K_{0}}{\tau}\), \(k_{2}=K_{I}\), \(k_{3}=\frac{4K_{0}}{\tau^{2}}\), \(k_{4}=\frac{2}{\tau}\), and \(k_{5}=k_{6}=\frac{1}{T_{F}}\). According to Equations (22) and (23), the expressions of CRNs are given for set-point filter \(C_{1}(s)\) and PID controller \(C_{2}(s)\), respectively. The set-point filter \(C_{1}(s)\) can be represented by CRNs as follows:  
\[R^{\pm}\stackrel{{ k_{7}}}{{\rightarrow}}R^{\pm}+X_{4}^{\pm},X_{4 }^{\pm}\stackrel{{ k_{8}}}{{\rightarrow}}\varnothing,X_{4}^{+}+X_ {4}^{-}\stackrel{{ k_{8}}}{{\rightarrow}}\varnothing \tag{24}\]  
\[Q_{2}(s)=G_{m-}^{-1}(s)F_{2}(s) \tag{16}\]0  
\[Q_{2}(s)=G_{m-}^{-1}(s)F_{2}(s) \tag{16}\]1  
\[Q_{2}(s)=G_{m-}^{-1}(s)F_{2}(s) \tag{16}\]2  
The set-point filter \(C_{1}(s)\) shown in Figure 3 is established by Equations (24)-(27). Equations (24) and (25) represent the CRNs of the proportional gain \(x_{4}=k_{7}\tau\) and the first-order system \(x_{5}=-k_{9}x_{5}+k_{8}r\), respectively. Equations (26) and (27) refer to the summation result \(v\) of signals \(x_{4}\) and \(x_{5}\).