## 1 Introduction  
As control systems continue to grow in complexity, ensuring their safety and robustness becomes increasingly critical across a wide range of industrial processes. The reliability, robustness, availability, and environmental safety of such systems are often evaluated based on their sensitivity to faults, which may occur randomly during operation. This growing need has resulted in a distinct area within engineering theory and practice-fault detection and isolation (FDI) [1, 2, 3, 4, 5]. From a practical engineering standpoint, faults are commonly categorized into additive and multiplicative types. Additive faults are unexpected deterministic inputs (e.g., sudden load change or sensor biases) that affect the system output without altering its internal structure. These faults, which include actuator jamming or sensor drift, are typically absent under normal conditions but produce clear effects on outputs once present [6, 7, 8, 9, 10, 11, 12]. Multiplicative faults, by contrast, involve changes in system parameters, such as resistance, inductance, inertia, or friction coefficients. These faults may appear gradually (due to aging or wear) or abruptly (as in structural damage) and are more challenging to detect due to their parameter-driven, indirect, and often masked effects on system output [13, 14, 15, 16, 17]. In DC motors, multiplicative faults include misalignment of brushes, wear of the commutator, armature winding faults, variations in armature resistance, and changes in mechanical friction. They may also occur in the accompanying amplifier--making accurate real-time diagnosis a challenging but essential task [18, 19, 20, 21]. Various FDI methods have been developed to address these issues, which can be classified into four broad categories:  
1. Signal-based techniques rely on analyzing vibration or current signals, often using spectral, statistical, or wavelet-based methods [19, 22, 23, 24, 25]. While model-free and simple to implement, such methods are generally limited to offline analysis, and their performance deteriorates in low-power or noisy systems. They are also less suitable for detecting multiplicative faults, which often do not generate prominent signal features. 2. Model-based techniques exploit the system's dynamic model and include: * Parity equation methods, which compare measured and predicted outputs to generate residuals sensitive to specific faults [3, 10, 11, 26]. These methods are simple to implement and effective for additive fault isolation, but they perform poorly for multiplicative faults due to their reliance on accurate models and sensitivity to noise and parameter uncertainties. Moreover, they typically lack fault identification capability, as they do not quantify fault magnitude. * Observer and filter-based methods (e.g., Luenberger observers, Kalman filters, sliding-mode observers) that estimate system states and detect deviations due to faults [7, 14, 27, 28]. Such approaches provide good dynamic tracking and are suitable for real-time use, but they often require precise model tuning and can become unstable or inaccurate in the presence of model uncertainty or high noise. * Parameter estimation methods (e.g., Least Squares (LS), Recursive Least Squares (RLS), Extended Kalman Filters (EKF), Unscented Kalman Filters (UKF), Algebraic-Geometric Technique (AGT) [18, 26, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. 3. Knowledge and data-driven approaches such as fuzzy logic, expert systems, and neural networks can learn or encode relationships between faults and observable features [6, 12, 18, 39, 40]. While effective for nonlinear and complex systems, these methods require large training datasets and may lack interpretability. Optimization-based techniques such as Genetic Algorithms (GA), Particle Swarm Optimization (PSO), and Whale Optimization Algorithm (WOA)) that track deviations in estimated parameters, particularly relevant for multiplicative faults [41; 42].