# A Two-Stage Method for Parameter Identification of a Nonlinear System in a Microbial Batch Process
## 2 Parameter Identification Model for the Microbial Batch Process
### Parameter Identification Model  
To identify the values of the parameters \(m_{\text{S}}\), \(m_{\text{PD}}\), \(m_{\text{HAc}}\), \(m_{\text{EOH}}\), \(Y_{\text{S}}^{m}\), \(Y_{\text{PD}}^{m}\), \(Y_{\text{HAc}}^{m}\), \(\Delta q_{\text{S}}^{m}\), \(\Delta q_{\text{PD}}^{m}\), \(\Delta q_{\text{HAAc}}^{m}\), \(\Delta q_{\text{EOH}}^{m}\), \(K_{\text{S}}^{*}\), \(K_{\text{PD}}^{*}\), \(K_{\text{HAc}}^{*}\), and \(K_{\text{EOH}}^{*}\) in Equations (1)-(12), we propose the following parameter identification model for the microbial batch process of glycerol to 1,3-PD:  
\[\min F=\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(X(t_{j})-X_{e}(t_ {j})\right)^{2}/X_{\text{max}}^{2}\Big{]}+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[} \left(\dot{X}(t_{j})-\dot{X}_{e}(t_{j})\right)^{2}/\dot{X}_{\text{max}}^{2} \Big{]} \tag{13}\] \[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{\text{S}}(t_{j})- C_{\text{S}e}(t_{j})\right)^{2}/C_{\text{Smax}}^{2}\Big{]}+\frac{1}{N}\sum \limits_{j=1}^{N}\Big{[}\left(\dot{C}_{\text{S}}(t_{j})-\dot{C}_{\text{S}e}(t_{j}) \right)^{2}/\dot{C}_{\text{Smax}}^{2}\Big{]}\] \[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{PD}(t_{j})-C_{PDe }(t_{j})\right)^{2}/C_{PD\max}^{2}\Big{]}+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[} \left(\dot{C}_{PD}(t_{j})-\dot{C}_{PDe}(t_{j})\right)^{2}/\dot{C}_{PD\max}^{2} \Big{]}\] \[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{HAAc}(t_{j})-C_{HA Ac}(t_{j})\right)^{2}/C_{HAc\max}^{2}\Big{]}+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[} \left(\dot{C}_{E\text{H}Ac}(t_{j})-\dot{C}_{HAce}(t_{j})\right)^{2}/\dot{C}_{HAc \max}^{2}\Big{]},\]subject to satisfying  
\[\frac{\mathrm{d}X}{\mathrm{d}t}=0.67\frac{XC_{\mathrm{S}}}{0.28+C_{\mathrm{S}}} \bigg{(}1-\frac{C_{\mathrm{S}}}{2039}\bigg{)}\bigg{(}1-\frac{C_{\mathrm{PD}}}{9 93.5}\bigg{)}\bigg{(}1-\frac{C_{\mathrm{HAC}}}{1026}\bigg{)}\bigg{(}1-\frac{C_{ \mathrm{EOH}}}{360.9}\bigg{)}, \tag{14}\]  
\[\frac{\mathrm{d}C_{\mathrm{S}}}{\mathrm{d}t}=-m_{\mathrm{S}}X-\frac{0.67}{Y^{ \mathrm{S}}_{\mathrm{S}}}\frac{C_{\mathrm{S}}X}{0.28+C_{\mathrm{S}}}\big{(}1- \frac{C_{\mathrm{S}}}{2039}\bigg{)}\Big{(}1-\frac{C_{\mathrm{PD}}}{939.5} \Big{)}\Big{(}1-\frac{C_{\mathrm{HAC}}}{1026}\Big{)}\Big{(}1-\frac{C_{\mathrm{ EOH}}}{360.9}\Big{)}-\Delta q^{m}_{\mathrm{S}}\frac{C_{\mathrm{S}}X}{C_{ \mathrm{S}}+K^{*}_{\mathrm{S}}}, \tag{15}\]  
\[\frac{\mathrm{d}C_{\mathrm{PD}}}{\mathrm{d}t}=m_{\mathrm{PD}}X+\frac{0.67Y^{ \mathrm{m}}_{\mathrm{PD}}C_{\mathrm{S}}X}{0.28+C_{\mathrm{S}}}\Big{(}1-\frac{ C_{\mathrm{S}}}{2039}\Big{)}\Big{(}1-\frac{C_{\mathrm{PD}}}{939.5}\Big{)}\Big{(}1- \frac{C_{\mathrm{HAC}}}{1026}\Big{)}\Big{(}1-\frac{C_{\mathrm{EOH}}}{360.9} \Big{)}+\Delta q^{m}_{\mathrm{PD}}\frac{C_{\mathrm{S}}X}{C_{\mathrm{S}}+K^{*}_ {\mathrm{EOH}}}, \tag{16}\]  
\[\frac{\mathrm{d}C_{\mathrm{EOH}}}{\mathrm{d}t}=m_{\mathrm{HAAC}}X+\frac{0.67Y^{ \mathrm{m}}_{\mathrm{HAAC}}C_{\mathrm{S}}X}{0.28+C_{\mathrm{S}}}\Big{(}1- \frac{C_{\mathrm{S}}}{2039}\Big{)}\Big{(}1-\frac{C_{\mathrm{PD}}}{939.5}\Big{)} \Big{(}1-\frac{C_{\mathrm{HAC}}}{1026}\Big{)}\Big{(}1-\frac{C_{\mathrm{EOH}}}{36 0.9}\Big{)}+\Delta q^{m}_{\mathrm{HAAC}}\frac{C_{\mathrm{S}}X}{C_{\mathrm{S}}+K ^{*}_{\mathrm{HAC}}}, \tag{17}\]  
\[\frac{\mathrm{d}C_{\mathrm{PD}}}{\mathrm{d}t}=m_{\mathrm{EOH}}X+\frac{0.67Y^{ \mathrm{m}}_{\mathrm{PD}}C_{\mathrm{S}}X}{0.28+C_{\mathrm{S}}}\Big{(}1-\frac{ C_{\mathrm{S}}}{2039}\Big{)}\Big{(}1-\frac{C_{\mathrm{HE}}}{939.5}\Big{)}\Big{(}1- \frac{C_{\mathrm{HE}}}{1026}\Big{)}\Big{(}1-\frac{C_{\mathrm{EOH}}}{360.9} \Big{)}+\Delta q^{m}_{\mathrm{EOH}}\frac{C_{\mathrm{S}}X}{C_{\mathrm{S}}+K ^{*}_{\mathrm{EOH}}}, \tag{18}\]  
\[t\in[t_{0},t_{N}], \tag{19}\]  
\[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{\text{S}}(t_{j})- C_{\text{S}e}(t_{j})\right)^{2}/C_{\text{Smax}}^{2}\Big{]}+\frac{1}{N}\sum \limits_{j=1}^{N}\Big{[}\left(\dot{C}_{\text{S}}(t_{j})-\dot{C}_{\text{S}e}(t_{j}) \right)^{2}/\dot{C}_{\text{Smax}}^{2}\Big{]}\]0  
\[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{\text{S}}(t_{j})- C_{\text{S}e}(t_{j})\right)^{2}/C_{\text{Smax}}^{2}\Big{]}+\frac{1}{N}\sum \limits_{j=1}^{N}\Big{[}\left(\dot{C}_{\text{S}}(t_{j})-\dot{C}_{\text{S}e}(t_{j}) \right)^{2}/\dot{C}_{\text{Smax}}^{2}\Big{]}\]1  
\[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{\text{S}}(t_{j})- C_{\text{S}e}(t_{j})\right)^{2}/C_{\text{Smax}}^{2}\Big{]}+\frac{1}{N}\sum \limits_{j=1}^{N}\Big{[}\left(\dot{C}_{\text{S}}(t_{j})-\dot{C}_{\text{S}e}(t_{j}) \right)^{2}/\dot{C}_{\text{Smax}}^{2}\Big{]}\]2  
\[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{\text{S}}(t_{j})- C_{\text{S}e}(t_{j})\right)^{2}/C_{\text{Smax}}^{2}\Big{]}+\frac{1}{N}\sum \limits_{j=1}^{N}\Big{[}\left(\dot{C}_{\text{S}}(t_{j})-\dot{C}_{\text{S}e}(t_{j}) \right)^{2}/\dot{C}_{\text{Smax}}^{2}\Big{]}\]3  
\[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{\text{S}}(t_{j})- C_{\text{S}e}(t_{j})\right)^{2}/C_{\text{Smax}}^{2}\Big{]}+\frac{1}{N}\sum \limits_{j=1}^{N}\Big{[}\left(\dot{C}_{\text{S}}(t_{j})-\dot{C}_{\text{S}e}(t_{j}) \right)^{2}/\dot{C}_{\text{Smax}}^{2}\Big{]}\]4  
\[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{\text{S}}(t_{j})- C_{\text{S}e}(t_{j})\right)^{2}/C_{\text{Smax}}^{2}\Big{]}+\frac{1}{N}\sum \limits_{j=1}^{N}\Big{[}\left(\dot{C}_{\text{S}}(t_{j})-\dot{C}_{\text{S}e}(t_{j}) \right)^{2}/\dot{C}_{\text{Smax}}^{2}\Big{]}\]5  
\(C_{\text{PDe}}(t_{j})\), \(C_{\text{HAc}}(t_{j})\), and \(C_{\text{EOHe}}(t_{j})\) are the measured data for biomass, glycerol, 1,3-PD, acetic acid, and ethanol at the sampling time \(t_{j}\) (\(j=1,2,\cdots,N\)), respectively; \(X(t_{j})\), \(C_{\text{S}}(t_{j})\), \(C_{\text{PD}}(t_{j})\), \(C_{\text{HAc}}(t_{j})\), and \(C_{\text{EOH}}(t_{j})\) denote the computed concentrations for biomass, glycerol, 1,3-PD, acetic acid, and ethanol at \(t_{j}\), respectively; \(X_{\text{max}}\), \(C_{\text{Smax}}\), \(C_{\text{PDmax}}\), \(C_{\text{HAcmax}}\), and \(C_{\text{EOHmax}}\) are the maximum measured concentrations of biomass, glycerol, 1,3-PD, acetic acid, and ethanol in \([t_{0},t_{N}]\), respectively; \(N\) is the number of sampled data points; \(\dot{X}_{e}(t_{j})\), \(\dot{C}_{\text{Se}}(t_{j})\), \(\dot{C}_{\text{PDe}}(t_{j})\), \(\dot{C}_{\text{HAc}}(t_{j})\), and \(\dot{C}_{\text{EOHe}}(t_{j})\) are the approximate experimental slopes for biomass, glycerol, 1,3-PD, acetic acid, and ethanol at \(t_{j}\), respectively; \(\dot{X}(t_{j})\), \(\dot{C}_{\text{PD}}(t_{j})\), \(\dot{C}_{\text{HAc}}(t_{j})\), and \(\dot{C}_{\text{EOHH}}(t_{j})\) are the computed slopes for biomass, glycerol, 1,3-PD, acetic acid, and ethanol at \(t_{j}\), respectively; \(\dot{X}_{\text{max}}\), \(\dot{C}_{\text{Smax}}\), \(\dot{C}_{\text{PDmax}}\), \(\dot{C}_{\text{HAcmax}}\), and \(\dot{C}_{\text{EOHmax}}\) are the maximum experimental slopes of biomass,  
\[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{\text{S}}(t_{j})- C_{\text{S}e}(t_{j})\right)^{2}/C_{\text{Smax}}^{2}\Big{]}+\frac{1}{N}\sum \limits_{j=1}^{N}\Big{[}\left(\dot{C}_{\text{S}}(t_{j})-\dot{C}_{\text{S}e}(t_{j}) \right)^{2}/\dot{C}_{\text{Smax}}^{2}\Big{]}\]6  
\[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{\text{S}}(t_{j})- C_{\text{S}e}(t_{j})\right)^{2}/C_{\text{Smax}}^{2}\Big{]}+\frac{1}{N}\sum \limits_{j=1}^{N}\Big{[}\left(\dot{C}_{\text{S}}(t_{j})-\dot{C}_{\text{S}e}(t_{j}) \right)^{2}/\dot{C}_{\text{Smax}}^{2}\Big{]}\]7  
\[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{\text{S}}(t_{j})- C_{\text{S}e}(t_{j})\right)^{2}/C_{\text{Smax}}^{2}\Big{]}+\frac{1}{N}\sum \limits_{j=1}^{N}\Big{[}\left(\dot{C}_{\text{S}}(t_{j})-\dot{C}_{\text{S}e}(t_{j}) \right)^{2}/\dot{C}_{\text{Smax}}^{2}\Big{]}\]8  
\[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{\text{S}}(t_{j})- C_{\text{S}e}(t_{j})\right)^{2}/C_{\text{Smax}}^{2}\Big{]}+\frac{1}{N}\sum \limits_{j=1}^{N}\Big{[}\left(\dot{C}_{\text{S}}(t_{j})-\dot{C}_{\text{S}e}(t_{j}) \right)^{2}/\dot{C}_{\text{Smax}}^{2}\Big{]}\]9  
\[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{PD}(t_{j})-C_{PDe }(t_{j})\right)^{2}/C_{PD\max}^{2}\Big{]}+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[} \left(\dot{C}_{PD}(t_{j})-\dot{C}_{PDe}(t_{j})\right)^{2}/\dot{C}_{PD\max}^{2} \Big{]}\]0  
\[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{PD}(t_{j})-C_{PDe }(t_{j})\right)^{2}/C_{PD\max}^{2}\Big{]}+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[} \left(\dot{C}_{PD}(t_{j})-\dot{C}_{PDe}(t_{j})\right)^{2}/\dot{C}_{PD\max}^{2} \Big{]}\]1  
where \(\mathbf{x}\in\mathbb{R}^{5},\mathbf{x}_{0}\in\mathbb{R}^{5},\mathbf{x}_{e}\in\mathbb{R}^{5 },\mathbf{x}_{\rm max}\in\mathbb{R}^{5},\mathbf{p}\in\mathbb{R}^{16},\mathbf{p}^{l}\in \mathbb{R}^{16}\), and \(\mathbf{p}^{u}\in\mathbb{R}^{16}\) have the following formulations:  
\[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{PD}(t_{j})-C_{PDe }(t_{j})\right)^{2}/C_{PD\max}^{2}\Big{]}+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[} \left(\dot{C}_{PD}(t_{j})-\dot{C}_{PDe}(t_{j})\right)^{2}/\dot{C}_{PD\max}^{2} \Big{]}\]2  
\[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{PD}(t_{j})-C_{PDe }(t_{j})\right)^{2}/C_{PD\max}^{2}\Big{]}+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[} \left(\dot{C}_{PD}(t_{j})-\dot{C}_{PDe}(t_{j})\right)^{2}/\dot{C}_{PD\max}^{2} \Big{]}\]3  
\[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{PD}(t_{j})-C_{PDe }(t_{j})\right)^{2}/C_{PD\max}^{2}\Big{]}+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[} \left(\dot{C}_{PD}(t_{j})-\dot{C}_{PDe}(t_{j})\right)^{2}/\dot{C}_{PD\max}^{2} \Big{]}\]4  
\[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{PD}(t_{j})-C_{PDe }(t_{j})\right)^{2}/C_{PD\max}^{2}\Big{]}+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[} \left(\dot{C}_{PD}(t_{j})-\dot{C}_{PDe}(t_{j})\right)^{2}/\dot{C}_{PD\max}^{2} \Big{]}\]5  
\[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{PD}(t_{j})-C_{PDe }(t_{j})\right)^{2}/C_{PD\max}^{2}\Big{]}+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[} \left(\dot{C}_{PD}(t_{j})-\dot{C}_{PDe}(t_{j})\right)^{2}/\dot{C}_{PD\max}^{2} \Big{]}\]6 \[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{PD}(t_{j})-C_{PDe }(t_{j})\right)^{2}/C_{PD\max}^{2}\Big{]}+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[} \left(\dot{C}_{PD}(t_{j})-\dot{C}_{PDe}(t_{j})\right)^{2}/\dot{C}_{PD\max}^{2} \Big{]}\]7  
\[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{PD}(t_{j})-C_{PDe }(t_{j})\right)^{2}/C_{PD\max}^{2}\Big{]}+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[} \left(\dot{C}_{PD}(t_{j})-\dot{C}_{PDe}(t_{j})\right)^{2}/\dot{C}_{PD\max}^{2} \Big{]}\]8 \[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{PD}(t_{j})-C_{PDe }(t_{j})\right)^{2}/C_{PD\max}^{2}\Big{]}+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[} \left(\dot{C}_{PD}(t_{j})-\dot{C}_{PDe}(t_{j})\right)^{2}/\dot{C}_{PD\max}^{2} \Big{]}\]9  
\[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{HAAc}(t_{j})-C_{HA Ac}(t_{j})\right)^{2}/C_{HAc\max}^{2}\Big{]}+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[} \left(\dot{C}_{E\text{H}Ac}(t_{j})-\dot{C}_{HAce}(t_{j})\right)^{2}/\dot{C}_{HAc \max}^{2}\Big{]},\]0 \[+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[}\left(C_{HAAc}(t_{j})-C_{HA Ac}(t_{j})\right)^{2}/C_{HAc\max}^{2}\Big{]}+\frac{1}{N}\sum\limits_{j=1}^{N}\Big{[} \left(\dot{C}_{E\text{H}Ac}(t_{j})-\dot{C}_{HAce}(t_{j})\right)^{2}/\dot{C}_{HAc \max}^{2}\Big{]},\]1  
The general method for solving a dynamic optimization problem depends on the discretization of dynamic system equations for the framework of the direct transcription [25, 26, 27].