## 2 Problem Statement  
### CSTR Process Modeling  
A continuous stirred tank reactor (CSTR) serves as a well-mixed vessel in industrial processes, with reactants continuously entering and products continuously exiting. Continuous stirring ensures uniform mixing and a consistent reaction rate. Figure 1 depicts a schematic diagram of the CSTR. In Figure 1, \(C_{M}\) and \(T_{f}\) represent the reactant concentration and reactor temperature at the inlet of the reactant, respectively. \(C_{A}\) and \(T\) denote the corresponding values at the outlet. \(F\) is the reactant flow rate. Similarly, \(T_{\phi f}\) and \(T_{c}\) represent the coolant temperature at the inlet and outlet, respectively. \(F_{c}\) is the coolant flow rate. CSTRs are known for their challenging control due to highly nonlinear and complex behavior. This study investigates a CSTR where an exothermic, first-order reaction (A\(\rightarrow\)B) takes place. Maintaining a constant internal temperature is critical for achieving optimal reaction conditions within the reactor. Assuming thorough stirring, a constant volume, identical input and output flow rates, and constant physical parameters, the following equations can be derived by applying the principles of mass and energy conservation in the reactor and jacket [31; 32; 33]:  
\[\text{Reactant material balance}:\ \frac{dC_{A}}{d\tau}=\frac{F}{V}(C_{Af}-C_{A}) -k_{0}exp(-\frac{E}{RT})C_{A}, \tag{1a}\] \[\text{Energy balance in the reactor}:\ \frac{dT}{d\tau}=\frac{F}{V}(T_{f}-T)- \frac{\Delta Hh_{0}C_{A}}{\mathcal{R}_{p}}exp(-\frac{E}{RT})+\frac{\text{U}A }{\mathcal{R}_{p}\mathcal{V}}(T_{c}-T),\] (1b) \[\text{Energy balance in the jacket}:\ \frac{dT_{c}}{d\tau}=\frac{F_{c}}{V_{ j}}(T_{cf}-T_{c})-\frac{ULA}{\rho_{j}C_{pj}V_{j}}(T_{c}-T), \tag{1c}\]  
The parameters are listed in Table 1.  
Figure 1: A schematic diagram of the CSTR process with a cooling jacket. In general, the jacket temperature dynamics respond swiftly to changes in the reactor's state. This implies that the time constant of the jacket temperature dynamics is much shorter than that of the reactor. Given this fact, the jacket's dynamics can be neglected. Then, a simple steady-state equation can easily be obtained between the coolant flow rate \(F_{c}\) and the coolant temperature \(T_{c}\):  
\[T_{c}=\frac{\rho_{j}C_{pj}F_{c}T_{cf}+UAT}{\rho_{j}C_{pj}F_{c}+UA}. \tag{2}\]  
The primary control objective for the CSTR process is to maintain a stable reactor temperature (\(T\)) at a desired setpoint. To achieve this, the coolant flow rate is manipulated. However, due to the inherent relationship between \(F_{c}\) and \(T_{c}\) as defined in (2), utilizing \(T_{c}\) as the control input can simplify controller synthesis. Following the approach of previous studies [1; 31; 32; 33], the coolant jacket temperature \(T_{c}\) is employed as the control input in this work. Then, Equation (1) becomes a dynamic equation governed by two variables \(C_{A}\) and \(T\) exclusively. To express them more succinctly, the dimensionless variables and parameters are introduced as  
\[\gamma=\frac{E}{RT_{f}},\ H=\frac{-\Delta HC_{Af}\gamma}{C_{p}T_{f}},\ \beta=\frac{UA}{FC_{p}}\ \text{and}\ D_{a}=\frac{Vk_{0}e^{\gamma}}{F}, \tag{3a}\] \[x_{1}=\frac{C_{f}-C_{A}}{C_{f}},\ x_{2}=\frac{T-T_{f}}{T_{f}}\gamma,\ u=\frac{T_{c}-T_{f}}{T_{f}}\gamma\ and\ t=\tau\frac{F}{V}, \tag{3b}\]  
where \(\gamma\) is the dimensionless activation energy, \(H\) is the dimensionless heat of the reaction, \(\beta\) is the dimensionless cooling rate, and \(D_{a}\) is the Damkohler number. \(x_{1},x_{2},u\), and \(t\) are the dimensionless reactant concentration, reactor temperature, jacket coolant temperature, and