## 1 Introduction  
As industrial systems expand, the interdependence between facilities is increasing [1]. The interconnectivity of these systems enhances productivity and efficiency by enabling real-time information sharing for process operation. However, this integration also introduces vulnerabilities. The tightly coupled nature of modern industrial systems increases the risk of fault propagation, where an issue in one component can spread across interconnected systems, amplifying its impact and potentially causing large-scale failures. Even minor faults can disrupt operations and, if not addressed, propagate through interconnected systems. Therefore, effective fault detection techniques are essential to mitigating risks and ensuring stable operation. For example, a power generation system consisting of various equipment such as turbines, compressors, and generators may fail due to equipment wear, failure in particle size control devices, or lack of lubrication. A malfunction in a turbine can cause a drop in power generation, which, in turn, may trigger an overload in connected compressors and generators. This can result in a reduction in overall system efficiency, increased wear on equipment, and potentially, a system-wide shutdown if not addressed promptly. In particular, circulating fluidized bed combustion (CFBC) boilers are prone to failure risks such as fouling, corrosion, erosion, scaling, and tube leakage due to erosive action in the circulating bed. Typically, faults develop gradually until they become catastrophic, making them difficult for operators to identify [2; 3; 4]. Furthermore, depending on the source of the fault, the rate of propagation of the fault variable can vary, with the fault variable rising rapidly before the power generation system shuts down. As a result, early fault detection technologies are essential to maintaining the reliability of industrial systems and minimizing potential losses. Timely fault detection plays a crucial role in ensuring reliability and cost-effectiveness across industries. Consequently, industries are placing significant emphasis on fault detection and diagnosis (FDD) [5; 6; 7; 8; 9]. As shown in Figure 1, a fault is defined as a state in which at least one characteristic of the target system deviates from its normal operating condition [10]. The types of faults are divided into driftwise, stepwise, malfunction, and failure [11]. Driftwise and stepwise faults occur in the form of a sudden or gradual increase in fault magnitude. These faults can develop into a malfunction or failure over time. In the case of a malfunction, specific variables exhibit intermittent failure characteristics, whereas failure means that the target system is completely offline. Therefore, FDD techniques capable of identifying early warning signs before the target system enters a state of failure are required. The main objective of FDD is to analyze sensor data patterns to assist field operators in taking appropriate actions. Fault detection methods are categorized into physical and data-driven approaches. Physical methods build models that describe the physical nature of failures based on hardware information such as design criteria, thresholds, and fault mechanisms. Widely used physical methods include parity equations [12], diagnostic observers [13], structural graphs, parameter estimation [14], and correlation coefficient method [15]. These methods can provide accurate results if they precisely capture the system's dynamics. However, their application in real industrial environments faces several challenges; first, industrial systems are often highly complex, incorporating numerous interconnected components with nonlinear and time-varying behavior. Accurately modeling such systems requires extensive knowledge of physical properties, material behaviors, and operational conditions, making the formulation of an accurate model challenging.