# 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  
### Microbial Batch Process  
Based on previous work [2], the material balance relationships of the microbial batch process of glycerol to 1,3-PD by _Klebsiella pneumoniae_ can be expressed as follows:  
\[\frac{\mathrm{d}X}{\mathrm{d}t}=\mu X, \tag{1}\]  
\[\frac{\mathrm{d}C_{\mathrm{S}}}{\mathrm{d}t}=-q_{\mathrm{S}}X, \tag{2}\]  
\[\frac{\mathrm{d}C_{\mathrm{PD}}}{\mathrm{d}t}=q_{\mathrm{PD}}X, \tag{3}\]  
\[\frac{\mathrm{d}C_{\mathrm{HAC}}}{\mathrm{d}t}=q_{\mathrm{HAAC}}X, \tag{4}\]  
\[\frac{\mathrm{d}C_{\mathrm{EOH}}}{\mathrm{d}t}=q_{\mathrm{EOH}}X, \tag{5}\]  
\[t\in[t_{0},t_{\mathrm{N}}], \tag{6}\]  
\[X(t_{0})=X_{0},C_{\mathrm{S}}(t_{0})=C_{\mathrm{S0}},C_{\mathrm{PD}}(t_{0})=C _{\mathrm{PD0}},C_{\mathrm{HAC}}(t_{0})=C_{\mathrm{HAAC0}},C_{\mathrm{EOH}}(t _{0})=C_{\mathrm{EOH}0}, \tag{7}\]  
\[\mu=0.67\frac{C_{\mathrm{S}}}{0.28+C_{\mathrm{S}}}\bigg{(}1-\frac{C_{\mathrm{ S}}}{2039}\bigg{)}\bigg{(}1-\frac{C_{\mathrm{PD}}}{939.5}\bigg{)}\bigg{(}1- \frac{C_{\mathrm{HAAC}}}{1026}\bigg{)}\bigg{(}1-\frac{C_{\mathrm{EOH}}}{360.9 }\bigg{)}, \tag{8}\]  
\[q_{\mathrm{S}}=m_{\mathrm{S}}+\frac{\mu}{Y_{\mathrm{S}}^{m}}+\Delta q_{ \mathrm{S}}^{m}\frac{C_{\mathrm{S}}}{C_{\mathrm{S}}+K_{\mathrm{S}}^{*}}, \tag{9}\]  
\[q_{\mathrm{PD}}=m_{\mathrm{PD}}+\mu Y_{\mathrm{PD}}^{m}+\Delta q_{\mathrm{PD} }^{m}\frac{C_{\mathrm{S}}}{C_{\mathrm{S}}+K_{\mathrm{PD}}^{*}}, \tag{10}\]  
\[\frac{\mathrm{d}C_{\mathrm{S}}}{\mathrm{d}t}=-q_{\mathrm{S}}X, \tag{2}\]0\[\frac{\mathrm{d}C_{\mathrm{S}}}{\mathrm{d}t}=-q_{\mathrm{S}}X, \tag{2}\]1  
where \(t\) denotes the culture time (h); \(t_{0}\geq 0\) and \(t_{N}>0\) are the initial and terminal times of the culture process, respectively (h); \(X\) denotes the biomass (g/L); \(C_{\text{S}}\) denotes the substrate (glycerol) concentration (mmol/L); \(C_{\text{PD}}\) denotes the concentration of product 1,3-PD (mmol/L); \(C_{\text{HAc}}\) denotes the concentration of product acetic acid (mmol/L); \(C_{\text{EOH}}\) denotes the concentration of product ethanol (mmol/L); \(X_{0}\), \(C_{\text{S0}}\), \(C_{\text{PD0}}\), \(C_{\text{HAc0}}\), and \(C_{\text{EOH0}}\) are the initial values of \(X\), \(C_{\text{S}}\), \(C_{\text{PD}}\), \(C_{\text{HAc}}\), and \(C_{\text{EOH}}\), respectively; \(\mu\) denotes the specific growth rate of cells (1/h); \(q_{\text{S}}\) denotes the specific consumption rate of substrate glycerol (mmol/(g-h)); \(q_{\text{PD}}\), \(q_{\text{HAc}}\), \(q_{\text{EOH}}\) denote the specific formation rates of products 1,3-PD, acetic acid, and ethanol, respectively (mmol/(g-h)); \(m_{\text{S}}\) is the maintenance term of glycerol consumption under substrate-limited conditions (mmol/(g-h)); \(m_{\text{PD}}\), \(m_{\text{HAc}}\), and \(m_{\text{EOH}}\) denote the maintenance terms of corresponding product formation under substrate-limited conditions, respectively (mmol/(g-h)); \(Y_{\text{S}}^{m}\) denotes the maximum growth yield (g/mmol); \(Y_{\text{PD}}^{m}\), \(Y_{\text{HAc}}^{m}\), and \(Y_{\text{EOH}}^{m}\) are the product yield of the corresponding products (mmol/g); \(\Delta q_{\text{S}}^{m}\) is the maximum increment of the glycerol consumption rate under substrate-sufficient conditions (mmol/(g-h)); \(\Delta q_{\text{PD}}^{m}\), \(\Delta q_{\text{HAAc}}^{m}\), and \(\Delta q_{\text{EOH}}^{m}\) are the maximum increments of the corresponding product formation rate under substrate-sufficient conditions (mmol/(g-h)); and \(K_{\text{S}}^{*}\), \(K_{\text{PD}}^{*}\), \(K_{\text{HAc}}^{*}\), and \(K_{\text{EOH}}^{*}\) denote the saturation constants of glycerol and corresponding products in the excess terms, respectively (mmol/L).