## 3 Trajectory Tracking Control of a Nonlinear Reactor: A Case Study  
### A Preliminary Simulation Analysis of the I/O FBL Control Design  
As previously investigated by [40; 41], the dimensionless modeling equations for a CSTR were considered in which an irreversible, first-order, exothermic reaction occurs:  
\[\left\{ \begin{array}{l}\dot{X}_{1}=-X_{1}+Da(1-X_{1})\exp\left(\frac{X_{2}} {\left(\frac{X_{2}}{\rho}\right)+1}\right)-D_{2}\\ \dot{X}_{2}=-X_{2}+T_{r}Da(1-X_{1})\exp\left(\frac{X_{2}}{\left(\frac{X_{2}}{ \rho}\right)+1}\right)-\delta(X_{2}-X_{2c})+\delta U+D_{1}\end{array} \right. \tag{41}\]  
\(D_{2}\) and \(D_{1}\) refer to the disturbances pertaining to the concentration and inlet temperature, respectively. Reference [41] can be referred to for the original nomenclature. These disturbances were considered as steps. Table 1 lists out this system's dimensionless parameters. \begin{table}
\begin{tabular}{c c} \hline \hline
**Description** & **Parameters** \\ \hline Activation energy & \(\sigma=E/RT_{0}\) \\ Adiabatic temperature rise & \(T_{r}=(-\Delta H)c_{Af_{0}}/Qc_{P}T_{f_{0}}\) \\ Damkohler number & \(Da=k_{0}\exp(-\gamma V)/F_{0}\) \\ Heat transfer coefficient & \(\delta=hA/Qc_{P}F_{0}\) \\ Dimensionless time & \(t=t^{\prime}(F_{0}/V)\) \\ Dimensionless conception & \(X_{1}=\left(c_{Af_{0}}-c_{A}\right)/c_{Af_{0}}\) \\ Dimensionless temperature & \(X_{2}=\left(T-T_{f_{0}}\right)/T_{f_{0}}\) \\ Dimensionless control input & \(U=\left(T_{c}-T_{c_{0}}\right)/T_{f_{0}}\) \\ Feed temperature disturbance & \(D_{1}=\left(T_{f}-T_{f_{0}}\right)/T_{f_{0}}\) \\ Feed composition disturbance & \(D_{2}=\left(c_{Af}-c_{Af_{0}}\right)/c_{Af_{0}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensionless parameters for the CSTR model. For the following set of the CSTR parameters: \(T_{r}=8\), \(\rho=0,3\), \(\sigma=20\), \(Da=0.078\) and \(X_{2c}=0\), the process displays several stable and unstable steady states as:  
* First stable equilibrium operating zone: \(\left(X_{1},X_{2}\right)_{A}=\left(0.14,\ 0.89\right)\);
* Unstable equilibrium operating zone: \(\left(X_{1},X_{2}\right)_{B}=\left(0.447,\ 2.75\right)\);
* Second Stable equilibrium operating zone: \(\left(X_{1},X_{2}\right)_{C}=\left(0.765,\ 4.704\right)\). The objective entails the development of a control input allowing the CSTR desired output to exhibit acceptable reference output trajectory tracking as well as disturbance elimination properties. Previously, [42] studied the control of the CSTR via feedback linearization that employed the concentration of the product as the available output. As the CSTR output, in this work, the temperature variable is investigated for two main reasons. First, industrial environments typically lack reliable concentration measurements, and second, the maximum admissible temperature is usually limited to avoid the occurrence of secondary reactions. Due to these constraints, the unstable equilibrium point is considered to conduct safe operating actions under control:  
\[y=H(X)=X_{2} \tag{42}\]  
For the selected output, the following parameters numerical values are considered \(q=1\), \(C_{2}=2\), and \(C_{1}=1\). Therefore, the control input that allows I/O FBL of the system from the I/O point of view as well as allows decoupling of the output from measurable disturbances can be defined as:  
\[U=\frac{-\sum\limits_{l\geq 1}\Lambda_{l}^{1}X^{[i]}-\Gamma_{1}\sum\limits_{i=1 }^{n}H_{i}X^{[i]}+v}{\sum\limits_{k\geq 1}\Pi_{k}^{l}X^{[k]}}+\frac{\sum \limits_{k\geq 1}\int_{k}^{l}X^{[k]}}{\sum\limits_{k\geq 1}\Pi_{k}^{l}X^{[k]}}D_{1} \tag{43}\]  
Substituting the Lie derivatives and considering the compensation for original settings because of the presence of unobservable ZD, Equation (43) gives:  
\[U=\Big{[}X_{2}-T_{r}Da(1-X_{1})\exp(X_{2}/((X_{2}/\sigma)+1))+\delta X_{2}-y_{ d}^{(1)}-\Gamma_{1}(X_{2}-y_{d})-B_{1}\Big{]}\delta^{-1} \tag{44}\]  
Since the ZD was seen as a minimum phase, confirmation of the asymptotic stability assumption is done, and thus, this method can be used.