# On Invertibility of an Interconnected System Composed of Two Dynamic Subsystems
## 5 Numerical Simulations
### Modeling of the Interconnected System
#### 5.1.2 High-Level Subsystem Modeling  
The HEX can be modeled based on the mass and energy balances that describe the evolution of the characteristic values--temperature, mass, composition, pressure, etc. From [16], the dynamic equations governing the heat balance of the process fluid and the utility fluid are given by  
\[\left\{\begin{array}{c}{\dot{{\rm{T}}}_{\rm{H}}}=\frac{{\rm{v_{H}}}}{{\rho_ {\rm{H}}}{\rho_{\rm{C}}}{\rm{v_{C}}}}{\rm{(T_{C}-T_{H})}}+\frac{{\rm{l_{H}}}}{ {\rm{l_{C}}}}{\rm{(T_{H}^{in}-T_{H})}}{\rm{Q_{H}}}\\ {\dot{{\rm{T}}}_{\rm{C}}}=\frac{{\rm{g_{v}}}}{{\rho_{\rm{C}}}{\rho_{\rm{C}}}{ \rho_{\rm{C}}}}{\rm{(T_{H}-T_{C})}}+\frac{{\rm{l_{C}}}}{{\rm{l_{C}}}}{\rm{(T_{C }^{in}-T_{C})}}{\rm{Q_{C}}}\end{array}\right. \tag{25}\]  
where \({\rho_{\rm{H}}},\;{\rho_{\rm{C}}}\) are the density of the process fluid and utility fluid (kg.m\({}^{-3}\)), \({V_{H}},\;{V_{C}}\) are volume of the process fluid and utility fluid (m\({}^{3}\)), \({C_{PH}},{C_{PC}}\) are specific heat of the process fluid and utility fluid (J.kg\({}^{-1}\).K\({}^{-1}\)), \({U}\) is the overall heat transfer coefficient (J.m\({}^{-2}\).K\({}^{-1}\).s\({}^{-1}\)), \({A}\) is the reaction area (m\({}^{2}\)), \({Q_{H}},\;{Q_{C}}\) are mass flowrate of the process fluid and utility fluid (kg.s\({}^{-1}\)), \({T_{H}}\) is the process fluid temperature, \({T_{H}^{in}}\) is the inlet temperature of the process fluid, \({T_{C}}\) is the utility fluid temperature, and \({T_{C}^{in}}\) is the inlet temperature of utility fluid.  
Define the state vector as \({\rm{x_{H}}}^{\rm{T}}=\left[{\rm{x_{H_{1}}}},{\rm{x_{H_{2}}}}\right]^{\rm{T}}= {\rm{[T_{H}}},{\rm{T_{C}}}]^{\rm{T}}\), the control input \({\rm{u_{L}}}^{\rm{T}}={\rm{[u_{L1}}},{\rm{u_{L2}}}\right]^{\rm{T}}={\rm{[Q_{H}}},{ \rm{Q_{L}}}\right]^{\rm{T}}\), the output vector of measurable variables \({\rm{y_{H}}}^{\rm{T}}={\rm{[y_{H1}}},{\rm{y_{H2}}}\right]^{\rm{T}}={\rm{[T_{H} }},{\rm{T_{C}}}]^{\rm{T}}\), then the above two equations can be rewritten in the following state-space form:  
\[\left\{\begin{array}{c}{\dot{{\rm{x}}}_{\rm{H}}}={\rm{f_{H}}}{\rm{(x_{H})}}+ \sum\limits_{{\rm{i=1}}}^{2}{\rm{g_{H{i}}}}{\rm{(x_{H})}}{\rm{u_{L}}}\\ {\rm{y_{H}}}={\rm{h_{H}}}{\rm{(x_{H},u_{L})}}\end{array}\right. \tag{26}\]  
where \({\rm{f_{H}}}{\rm{(x_{H})}}=\left(\begin{array}{c}{\rm{f_{H1}}}{\rm{(x_{H})}} \\ {\rm{f_{H2}}}{\rm{(x_{H})}}\end{array}\right)=\left(\begin{array}{c}\frac{{\rm{ h_{H}}}{\rm{A}}}{{\rho_{H}}{\rm{c_{PH}}}{\rm{(T_{H}-T_{C})}}}\\ \frac{{\rm{h_{H}}}{\rm{g_{C}}}}{{\rho_{\rm{C}}}{\rho_{\rm{C}}}{\rho_{\rm{C}}}{\rho_ {\rm{C}}}{\rho_{\rm{C}}}{\rho_{\rm{C}}}{\rm{(T_{C}-T_{H})}}}\end{array}\right)\), and \({\rm{g_{H}}}={\rm{(g_{H1}}},{\rm{g_{H2}}})=\left(\begin{array}{c}\frac{{\rm{(T_{H }^{in}-T_{H})}}}{{\rm{V_{H}}}}&0\\ 0&\frac{{\rm{(T_{H}^{in}-T_{C})}}}{{\rm{V_{C}}}}\end{array}\right)\), \({\rm{y_{H1}}}={\rm{x_{H1}}}\), \({\rm{y_{H2}}}={\rm{x_{H2}}}\), \({\rm{T_{H}^{in}}}\), \({\rm{T_{C}^{in}}}\) are measured and constant. An interconnected system is constituted by (26) and (27).