## 4 Application Results
### The Case of Continuous Stirred-Tank Reactor Process  
The Continuous Stirred-Tank Reactor (CSTR) is a fundamental type of chemical reactor that is widely used in chemical and biochemical engineering [32]. It involves a mixing process where reactants are continuously fed into the reactor, and products are continuously removed. The CSTR is designed to maintain homogeneous conditions throughout the reactor, with key parameters such as the temperature, flow rate, and reactant concentrations being carefully controlled. The inherent mixing properties of the CSTR make it an ideal system for evaluating control strategies, particularly in the context of maintaining stability and optimizing production. The CSTR process is also well suited for studying fault detection and diagnosis due to its susceptibility to various operational faults. These faults may include disturbances in the temperature, changes in the flow rate, or sensor malfunctions, all of which can significantly impact the quality and yield of the reaction. By leveraging detailed measurements from the CSTR dataset, it is possible to analyze the reactor's behavior under both normal and faulty conditions, providing valuable insights for developing robust monitoring and fault detection methods that ensure process safety and reliability. Figure 10 shows a schematic of the CSTR with measurement locations and control strategy. The reactor temperature is regulated by adjusting the coolant flow rate. The controller parameters are set to \(K_{\mathrm{C}}=1.0\) and \(\tau_{I}=0.2\), with the flow rate restricted between 10 L/min and 200 L/min to simulate realistic conditions. This saturation helps model situations where faults become too severe for the control system. In normal operation, parameters \(a\) and \(b\) are both 1.00, and their gradual decay toward 0 simulates catalyst decay and heat-transfer fouling. Sensor drift also affects each of the seven measured variables, adding further complexity. More details on these fault conditions are given in Table 3.  
Figure 10: Flowchart of the CSTR process [32]. Fault-free and faulty datasets were generated from the CSTR simulation over a 20 h operation period. During this time, operating conditions were varied by perturbing the control inputs, such as the flow rates or reactant concentrations, around their nominal values every 60 min. The sampling interval for all variables was set to 1 min. These input disturbances highlighted the system dynamics, resulting in temporally correlated and non-Gaussian-distributed measurements due to process nonlinearity. In faulty datasets, faults were introduced after 200 min of normal operation. To ensure robust performance evaluation, 250 faulty datasets were generated for each fault scenario, with variations in the process noise, measurement noise, and input disturbances through different random seeds. In the CSTR process, the training dataset consisted of 1200 fault-free samples. This process monitors seven variables: \(C_{i}\), \(T_{i}\), \(T_{ci}\), \(Q_{c}\), \(T_{c}\), \(C\), and \(T\). Among these, \(C_{i}\), \(T_{i}\), \(T_{ci}\), and \(Q_{c}\) were designated as \(X\), while \(T_{c}\), \(C\), and \(T\) were assigned to \(Y\). Accordingly, the training data dimensions were 1200 \(\times\) 4 for \(X\) and 1200 \(\times\) 3 for \(Y\). The fault detection approach is applied to the CSTR system, focusing on Fault 3, Fault 4, Fault 5, Fault 6, and Fault 10. Among these faults, Fault 3 is classified as a Multiplicative fault, indicating that the fault affects system parameters through Multiplicative deviations, which are often associated with changes in process gains or coefficients. In contrast, Fault 4, Fault 5, Fault 6, and Fault 10 are categorized as Additive faults, which represent abrupt or gradual deviations added to the process variables. Additive faults are typically caused by sensor biases, external disturbances, or direct changes in the process states. Figure 11 presents the monitoring performance for all the methods applied to Fault 4 in the CSTR process.