Motorized systems are fundamental to a wide range of advanced applications, driving a diverse array of devices including industrial machinery, semiconductor wafer production, robotics, and electric vehicles. The crux of leveraging these systems lies in the development of computational or mathematical models that faithfully emulate their dynamic behavior. Such motor models are tailored to meet the varying degrees of sophistication and accuracy required by their application domains, empowering researchers and engineers to simulate, predict, and fine-tune system performance with high precision [1]. These models are informed by data sourced from an array of sensors and other inputs, translating into precise mathematical representations of the motors’ dynamics [2]. Through this modeling, intricate insights into the functioning of motorized systems under diverse conditions are unearthed [3].

The challenge becomes even more critical when dealing with perturbed motorized systems-those subjected to disturbances, noise, and environmental variations—which directly impact their efficiency, reliability, and overall performance. In such systems, precise system identification is essential, as it enables the detection, quantification, and modeling of perturbation effects. Accurate identification lays the foundation for developing robust and adaptive control strategies capable of anticipating and mitigating unpredictable operational disruptions [4]. Without a well-structured approach to system identification, control mechanisms may struggle to maintain stability and efficiency, particularly in dynamic environments.

This becomes especially crucial in high-precision industries such as aerospace, automotive engineering, and industrial automation, where even minor deviations can lead to significant performance degradation or safety risks. By integrating perturbation models into system identification, motorized systems can achieve enhanced resilience, stability, and adaptability, even under fluctuating operating conditions. Thus, advancing system identification techniques for perturbed motorized systems is not just an academic pursuit but a practical necessity for ensuring reliable and high-performance operation in increasingly complex and demanding applications.

Modeling approaches for dynamic systems can be broadly categorized into two principal types: data-driven modeling and first-principle modeling. First-principle modeling, also referred to as white-box modeling, relies entirely on fundamental physical laws, such as Ohm’s law and Newton’s laws of motion, to describe system dynamics without the need for experimental data [5]. In contrast, data-driven modeling constructs system representations using input-output data and includes black-box and grey-box approaches [6,7]. Black-box modeling develops models solely from input-output data without any knowledge of internal system structures, while grey-box modeling integrates both empirical data and partial system knowledge to improve model accuracy [8]. A thorough understanding of these modeling paradigms is crucial for building accurate system representations and designing effective control strategies.

In recent years, advanced nonlinear modeling techniques have been proposed to capture complex system behaviors with greater precision. Block-oriented models, including Hammerstein and Wiener models, are particularly prominent due to their capability to represent nonlinear systems through a combination of linear dynamic components and static nonlinearities [9–11]. Furthermore, neural networks and neuro-fuzzy systems have gained considerable attention owing to their powerful approximation capabilities and adaptive learning features, rendering them suitable for modeling highly nonlinear and time-varying systems [12]. These methodologies have been extensively investigated in the literature to enhance modeling accuracy and robustness.

The optimization of motor parameters through grey- and black-box modeling has been extensively studied in prior research, employing various algorithms. Least square (LS) serves as a foundational method for this purpose. An extension of LS, the Levenberg-Marquardt (LM) algorithm [13], enhances the traditional approach by incorporating mechanisms to improve convergence speed and robustness, especially in nonlinear optimization problems. Additionally, metaheuristic algorithms are pivotal in optimizing parameters based on selected cost functions. These algorithms are categorized into four main types: swarm intelligence, evolutionary, physics-based, and human-inspired algorithms. Swarm intelligence algorithms emulate cooperative behaviors among agents to tackle complex problems, drawing inspiration from social organisms such as ants and birds [14,15]. Notable examples include particle swarm optimization (PSO), where particles adjust their positions in the solution space based on personal and collective experiences [16]. Other prominent algorithms in this category include whale optimization algorithm (WOA), grey wolf optimization (GWO), ant colony optimization (ACO), artificial bee colony (ABC), bat algorithm (BA), and cuckoo search (CS) [17,18]. There are also novel swarm algorithms proposed including marine predator algorithm, artificial circulation system algorithm (ACSA), wolverine optimization algorithm (WoOA), and synergistic swarm optimization algorithm (SSOA) [19–21]. These techniques harness natural dynamics and social structures to enhance search efficiency effectively.

Evolutionary algorithms on the other hand imitate the evolutionary process to search for the optimal solutions in challenging situations. These algorithms, including differential evolution (DE) [22] and genetic algorithms (GA) maintain a population of potential solutions that evolve across generations through mutation, crossover, and selection methods. Human-based algorithms use human judgment and knowledge to address optimization challenges [23]. These algorithms often require the development of rule-based systems, manual optimization, and expert judgment, such as the jaya algorithm (JA), teaching–learning–based optimization (TLBO). Physics-based algorithms, in contrast, model the physical behaviors of systems through the application of physics and mathematics principles. These methods are crucial for precise simulation and analysis, commonly employing tools like computational fluid dynamics and finite element analysis [24]. Utilized extensively in engineering and scientific research, physics-based algorithms enable detailed investigation and optimization of complex physical phenomena, facilitating advanced developments in motor parameter optimization and other areas of applied mechanics.

Recent surveys in the field have predominantly focused on electromagnetic field modeling perspectives [25], laboratory testing methodologies [26], and parameter identification approaches tailored to specific types of motors such as permanent magnet synchronous motors (PMSM) [27], inverter-fed induction motors [28], three-phase induction motor drives [29], and permanent magnet DC motors [30]. While some surveys, such as [31], have addressed model structures, they often lack comprehensive discussions on model parameter optimization. None have systematically explored the relationship between different model structure types and parameter optimization methods concerning perturbations. A thorough exploration of parameter optimization methods and state-of-the-art techniques for managing perturbations remains underrepresented in the existing literature. This paper aims to fill these gaps with the following primary contributions:

Providing a comprehensive review of recent advancements in system identification methods for perturbed motorized systems, including data acquisition processes and setups, with a focus on brushed DC motors, brushless DC motors (BLDC), and PMSMs.

The remainder of the paper is structured as follows: Section 2 describes an organized research framework that help to ensure the transparency of systematic review reporting. Section 3 provides a comprehensive examination of motor mathematical models. Section 4 delves into the synthesized findings from the preceding systematic review of the literature concerning system identification techniques for motorized systems under various perturbations. Section 5 reviews various parameter optimization methods in optimizing motor model parameters and performance. Section 6 provides a systematic analysis and discussion of the literature reviewed in Sections 4 and 5. Section 7 presents a comparative performance analysis of several widely used optimization methods for system identification, applied to a perturbed DC motor under both open-loop and closed-loop settings. Section 8 outlines potential directions for future research. Finally, Section 9 concludes the study by summarizing the key findings and highlighting their significance.

In this section, an organized research framework was developed. Fig. 1 shows the flow diagram of preferred reporting items for systematic reviews and meta-analyses PRISMA 2020 as a graphical representation of the study screening and selection process. The flow diagram can be expressed in three major steps, namely, identification, screening, and included.

In the identification phase, research area and goal were first defined. The research areas are the system identification of perturbed motorized systems and parameter optimization methods. The research goal is to provide a systematic research scheme for existing modeling and optimization methods. The articles were gathered as references by using key terms, such as system identification, motor modeling, motor parameter optimization, LS parameter estimation, motor identification using PSO, etc. Online journal database used for searching research articles based on key terms as listed as Web of Science, Scopus, IEEE Xplore, ScienceDirect, Taylor & Francis, and Springer.

In screening phase, the articles are screened for eligibility and relevancy with the research area and goal. Inclusion and exclusion criteria are specified to ease the screening phase. Inclusion criteria such as relevance to system identification, motor modeling, and parameter optimization. Exclusion criteria such as low citation, irrelevant to motorized systems, reports before 2010, etc. In third phase which is the included phase, the remaining articles are selected for review and analysis.

This section describes modeling methods for brushed DC motors, BLDC, and PMSM based on first principles, which is also known as white-box modeling. The models of these motors, which detail their structural and functional aspects, are in some cases crucial for applying both grey-box and black-box modeling techniques. Grey-box modeling strikes a balance between theoretical physical models and empirical data, allowing for predictions that account for known behaviors and measured data simultaneously. This approach is particularly useful when some system parameters are unknown or difficult to measure directly but where some theoretical knowledge about the system’s behavior is available. The black-box modeling, on the other hand, which relies solely on input-output data without any presumption about the system’s internal workings, benefits from the detailed model structures to better interpret the data patterns and relationships that emerge during the modeling process.

In the context of a perturbed motorized system, where factors like friction, varying loads, and other disturbances introduce complexities that challenge traditional first principles modeling, system identification emerges as a powerful solution. By leveraging observed input-output data, system identification enables the creation of accurate models that capture the dynamic behavior of the system amidst perturbations. These models not only adapt to changing conditions but also provide insights into the underlying dynamics, facilitating the design of robust control strategies tailored to mitigate the effects of disturbances.

The diagram in Fig. 6 outlines a standard procedure in system identification, starting from “Experiment Design” which is informed by “Prior Knowledge” [35]. Here, data is gathered and used to “Choose Model Set” based on a selected “Criterion of Fit.” This stage ensures the model aligns well with the collected data and system dynamics. The motor model structures based on first principles as described in the previous section can usually serve as hypotheses or initial guesses on the model structure. Next, the “Calculate Model” step involves applying the chosen model to predict or simulate system behavior. If the model predictions do not align with the expectations or further data (“Not OK”), the process includes a “Revise” loop, indicating iterative refinement of the model. Upon successful validation (“Validate Model”), where the model sufficiently represents the system under study, it is deemed acceptable for use (“OK: Use it”). This structured approach emphasizes iterative refinement and validation, leveraging both empirical data and theoretical knowledge to develop an accurate model of the system.

In [36], the study employs system identification to model a perturbed BLDC motor to achieve precise speed control. It introduces an ideal back EMF for enhanced precision in speed control in the face of varying load torques ranging from 0 to 20 Nm. The work in [37] utilizes system identification to design a proportional-integral-derivative (PID) control with auto-tuning. In [38], the system identification method is used to estimate both static and dynamic characteristics of a DC motor using starting rheostat as the starting current limiter.

A recent comparative study in [39] demonstrates that AutoRegressive with eXogenous input (ARX) and second-order continuous-time TF models offer a more accurate representation of the mathematical model of a DC motor under varying payloads. Additionally, in a separate study [40], it was demonstrated that the Autoregressive Moving Average Exogenous (ARMAX) structure provides a superior representation of the mathematical model of a DC servo motor compared to the ARX structure. In [41], a subspace-aided closed-loop system identification method is introduced for a DC motor system which demonstrates superior performance in pole estimation compared to a class of subspace identification methods utilizing principal component analysis algorithms. The study in [42] employs system identification to establish a low-order TF model of a physiological system (PSys) using a Multi-Input Multi-Output (MIMO) ARMAX model. Fig. 7 illustrates the flow chart of the proposed method where the model obtained from the system identification is then used to optimize the control parameters.

Batool et al. [43] discussed three estimation techniques for multi-domain DC servo motor model parameters: the Compare Coefficient Method, MATLAB Parameter Estimation Toolbox, and System Identification Toolbox. Experimental data is used to compare the accuracy of these methods, with the parameter estimation technique proving to be the most accurate and computationally efficient, guiding researchers in preference for DC servo motor simulation model parameter estimation. The proposed method in [44] optimizes accuracy and reliability of permanent magnet DC (PMDC) motor models while considering dynamic characteristics and operational conditions. Eight parameters were identified, such as brush current constant, back EMF constant, motor resistance, brush voltage current, motor inertia, viscous friction, motor inductance, and stiffness constant.

The study in [45] utilized system identification to estimate load torque and moment of inertia in PMSM drives based on sliding mode observer (SMO), which offered precise estimation and quick convergence. In [46], various estimators for PMSM electrical parameter identification were compared where the finding revealed that the moving horizon estimation outperformed the rest in accuracy. The Kalman filter (KF) of extended KF (EKF) and unscented KF (UKF) performed similarly, while the model reference adaptive system (MRAS) was sensitive to disturbances and required extensive tuning recursive LS yielded inaccurate results.

The method proposed in [47] utilizes motor impedance characteristics for parameter estimation, suitable for both offline and online environments. It achieves high accuracy even at low or zero current and speed, minimizing the influence of identification errors on different parameters. A KF-based approach discussed in [48] enables continuous updates of motor parameters, effectively handling noisy measurements for improved accuracy. EKF and dual EKF (DEKF) are compared, with DEKF performing better in estimation quality and resilience to initial states. In [49], a deadbeat control is introduced for tracking reference currents with zero steady-state error, adapting PMSM parameters during control. It effectively eliminates control offsets caused by parameter errors. The study in [50] enhances control robustness, minimizes torque ripples, and improves disturbance rejection, mainly integrating deadbeat control while addressing torque oscillations and noise reduction.

The system identification method is employed in [51] to model the inertia of a PMSM drive. It uses a sinusoidal perturbation torque with a DC offset to drive the PMSM, and then measures the amplitudes of the torque and speed response to calculate the system’s inertia. This method is robust and provides accurate inertia values quickly compared to other approaches. The studies in [52–56] utilized a black-box modeling approach for unmanned aerial vehicles (UAV), PSys stabilizer, flexible maneuvering systems (FMS), ground robots, and fuel control unit, respectively. This approach allows for adaptability to system changes without requiring extensive recalibration.

In system identification, after selecting or obtaining the model structure, the subsequent step involves parameter optimization. This section explores diverse parameter optimization methods utilized in the literature, which have shown promising results. In optimization algorithms, the error-or more specifically, the fitness value, a standard term in metaheuristic algorithms-serves as a quantitative measure of how well the identified model represents the actual system dynamics. Commonly used metrics include Sum of Squared Errors (SSE), Mean Squared Error (MSE), Integral of Absolute Errors (IAE), and Integral of Time-weighted Absolute Error (ITAE). These metrics are calculated by comparing the observed outputs (yobs) to the predicted outputs (ypred) of the model. For instance, SSE is expressed as ∑i=1n(yobs,i−ypred,i)2, MSE as 1n∑i=1n(yobs,i−ypred,i)2, IAE as ∑i=1n|yobs,i−ypred,i|, and ITAE as n∑i=1n|yobs,i−ypred,i|. These error metrics guide the optimization process, ensuring accurate parameter estimation and reliable system representation by minimizing the discrepancies between observed and predicted outputs.

This section delves into the synthesized findings from the preceding systematic review of literature concerning system identification techniques for motorized systems under various perturbations. Tables 1 and 2 summarize the system identification techniques employed for perturbed DC and AC motors, respectively. Each table organizes the studies according to the modeling method, motor structure, parameter optimization method, source of perturbations, and their respective applications or use cases. The diverse range of studies documented in the tables has been graphically represented below to offer a clearer overview of prevalent trends and distinct methodologies in both DC and AC motor applications.

Such perturbations can significantly impact real-time control applications by affecting system stability, accuracy, and overall performance. Load variations cause sudden torque demands, leading to speed deviations that can destabilize the system if not properly managed. Speed fluctuations, often resulting from external disturbances or supply inconsistencies, introduce tracking errors that reduce the precision of motion control. Friction, particularly at low speeds, leads to nonlinear effects such as stick-slip motion and increased energy losses, making precise control difficult. Hysteresis, commonly found in magnetic materials and mechanical systems, results in lagging system responses and inconsistent behavior, which complicates accurate control. Additionally, noise, both mechanical and electrical, interferes with sensor measurements and control signals, leading to erroneous feedback and degraded system performance.

These perturbations collectively challenge the reliability and responsiveness of motorized systems, particularly in high-precision and high-speed applications. Variations in load and speed can cause oscillations, overshooting, or delays in response, reducing efficiency and increasing wear on components. Friction and hysteresis introduce unwanted nonlinearities that make modeling and prediction difficult, leading to unpredictable deviations in system behavior. Noise further compounds these issues by distorting control signals, potentially causing instability or reduced sensitivity to actual system dynamics. In critical applications such as robotics, industrial automation, and electric vehicles, these perturbations can lead to performance degradation, increased energy consumption, and even system failures if not properly accounted for in the control design.

Fig. 27 demonstrates the distribution of parameter optimization methods utilized in the reviewed articles, where LS and PSO emerge as the most frequently used techniques. As detailed in Table 3, LS is widely adopted due to its versatility in handling both linear and nonlinear functions and its adaptability to real-time applications, although it remains sensitive to data outliers which can affect parameter estimation. In addition, the preference for LS can be attributed to its widespread availability in system identification tools and its designation as a default technique in popular software applications like MATLAB, which enhances its accessibility and ease of use. PSO is favored for its robustness and efficient control parameter tuning but can struggle with exploring certain regions of the search space. Additionally, the table highlights other methods like GA and NN, which excel in exploring large search spaces and extracting relevant features from complex data, respectively. However, these methods also face challenges such as high computational demand and overfitting. The diversity of other population-based optimization strategies such as TLBO, GWO, DE, and CS, reflects ongoing innovation in the field to address the unique challenges posed by different motor configurations and perturbation types. Hybrid algorithms integrate the advantages of multiple optimization techniques, enhancing both exploration and exploitation capabilities, thereby achieving superior performance. This analysis underscores the importance of selecting an optimization method that not only aligns with the specific dynamics and challenges of motor system perturbations but also balances between computational efficiency and the ability to handle complex data structures.

Figs. 28 and 29 offer insights into the types of perturbations and their correlation with modeling methods for both DC and AC motors. From Fig. 28, it is clear that fluctuations in loads and speed are extensively studied in the literature concerning system identification. These variations present challenges in accurately modeling motor behavior due to their dynamic and unpredictable nature. System identification methods enable researchers to derive meaningful insights from data collected during perturbed operation, facilitating the development of models that capture the intricate interactions between motor dynamics and external influences. Understanding how motors respond to these perturbations is essential for optimizing performance across various operational scenarios, ranging from industrial applications to consumer electronics. Furthermore, the precise characterization and simulation of these behaviors contribute to advancements in control algorithms designed to mitigate the impacts of load and speed fluctuations, thereby enhancing overall motor efficiency and reliability in diverse environments.

The model structure selection procedure in system identification is a critical step that determines the accuracy and effectiveness of the identified model in representing system dynamics. The first step involves data analysis, where the system’s input-output behavior is examined to determine whether it follows a linear or nonlinear trend. This initial assessment guides the choice between simpler linear models, such as ARX, or more complex nonlinear models, such as NARX. A TF model is an appropriate choice for system identification when the system is linear time-invariant and its behavior can be effectively described by input-output relationships without explicitly modeling internal states. TF models are also beneficial when data availability is limited, as TF allow system characterization based solely on input-output measurements. To ensure model reliability, validation techniques such as Mean Squared Error (MSE), R-squared values, and residual analysis are employed to compare different models and identify the one that achieves the best balance between accuracy and generalization. Additionally, cross-validation is essential to assess the model’s performance on unseen data by dividing the dataset into training and validation sets. This step helps mitigate overfitting and enhances the model’s robustness for real-world applications.

Based on the plot in Fig. 29, it is clear that the TF structures of order two to four were the most commonly employed across different types of perturbations, particularly excelling in addressing speed and voltage fluctuations. This is followed by ARMAX models, which also show significant usage, particularly effective in handling noise and load perturbations. SS models, although not as prevalent, are notably utilized for handling friction and voltage disturbances. The ARX model appears to be favored for disturbances like load and voltage, whereas NARX models are selectively applied, mainly for noise and current perturbations. First-order TF models, while less common compared to higher-order TFs, are still relevant in scenarios involving heat and hysteresis perturbations. This distribution of modeling techniques illustrates a targeted approach where specific models are chosen based on their efficacy in handling particular disturbances in motorized systems, ensuring optimal performance and robustness under varying operational conditions.

Fig. 30 illustrates the distribution of research articles focused on the target applications for modeling perturbed motorized systems. Control development (CD) emerges as a primary area, underscoring the importance of capturing motor uncertainties and unmodeled dynamics for designing robust and adaptive controllers. Following closely is generic modeling (GM), which provides foundational insights into various motor types, offering substantial benefits for those seeking to deepen their understanding of motor behaviors. Additional target applications span a diverse range of fields, including UAV, wheeled mobile robots (WMR), robot manipulators (RM), speed control (SC), hybrid electric vehicles (HEV), artificial immune systems (AIS), underground mine lifting installations (UMLI), wheelchairs (WC), DC servomotors (DCS), PSys, FMS, and hybrid parallel kinematic machines tricept robots (HPKM TR). Each application highlights different challenges and requirements, demonstrating the extensive reach and relevance of system identification methods across varied motorized systems.

Fig. 31 complements this by presenting a comparative analysis of the model structures employed across various applications, indicating that different models are tailored to the unique challenges of each application. The standard forms of perturbed motor models, specifically TF and SS structures, are prevalently used in GM and CD. The figure highlights the dominance of 2nd–4th order TF models in CD, reflecting their capability to handle the complexities and dynamic behaviors inherent in control systems. SS models are also prominently used in CD and other applications such as DCS, demonstrating their versatility and effectiveness in capturing system dynamics.

Additionally, the figure reveals that other modeling techniques like ARMAX and NARX are applied across various applications, although less frequently. These methods are valuable for their ability to model nonlinear behaviors and handle a variety of perturbations. The diversity in modeling approaches underscores the necessity of selecting appropriate models based on the specific requirements of each application, such as the precision needed in SC, the robustness required in HEV, and the adaptability crucial for WMR. This tailored approach ensures that the unique challenges of each application are effectively addressed, optimizing the performance and reliability of motorized systems in different operational contexts.

Fig. 32 displays the distribution of parameter optimization methods used across various types of perturbations in motorized systems. From the figure, it is evident that parameter optimization methods are selectively utilized across various types of perturbations in motorized systems. LS and LM are commonly used for various perturbations including speed, noise, load, disturbance, and voltage, leveraging their respective capabilities in minimizing residuals and efficiently exploring parameter spaces.

Swarm intelligence algorithms, such as PSO, GWO, ACO, ABC, BA, WOA, CS, AOA, HHO, ROA, MPA, GSA, SCA, and SA are prominently employed for managing perturbations. PSO is favored for its adaptability in systems affected by speed, noise, load, disturbance, friction, current, and voltage, despite its slower convergence speed. GWO is effective in handling noise and friction-related challenges through iterative optimization processes. BA, AOA and HHO are specialized for speed-perturbed systems, while CS is applied to noise-related systems. These algorithms cater to specific perturbation types, adapting their strategies to optimize performance in motorized systems.

Evolutionary algorithms which include GA and DE find application across diverse perturbations such as speed, noise, load, friction, voltage, and current, demonstrating their adaptability in optimizing motorized systems. Human-based algorithms such as TLBO and JA are deployed for load and noise systems, respectively, offering streamlined parameter requirements and effective optimization strategies. NN excel in handling noise, speed, and load systems by learning from extensive data, providing robust solutions to complex perturbations. Finally, methods like FPA specialize in accurately estimating motor model parameters under speed-related perturbations. These trends illustrate a strategic use of optimization methods tailored to specific challenges encountered in perturbed motorized systems.

This section compares the performance of several widely used methods identified via the analysis in the previous section, which include grey-box modeling using LS, PSO, GWO, BA, GA, and black-box modeling using NN, to parameterize the DC motor shown in Fig. 3, which is perturbed by a time delay, τd. The parameters of the perturbed DC motor are set to Ra=2Ω, La=2 H, Jm=0.5 kgm2, Bm=1 Nm/rads−1, Kt=5 Nm/A, Ke=0.6 V/rads−1, and τd=1 s. The resulting plant can be expressed as P(s)=be−τds/(a2s2+a1s+a0) where b=Kt, a2=JmLa, a1=JmRa+BmLa, and a0=BmRa+KtRa. Two approaches are employed: the first using the standard open-loop setup, and the second using a closed-loop setup that incorporates a PID controller where the parameters are set as Kp=8, Ki=5, and Kd=2. The maximum number of iterations is set to 50, the population size is fixed at 30. Each algorithm is executed ten times to evaluate its performance metrics. In PSO, w, c1 and c2 are set to 0.7, 1.5 and 1.5, respectively. In BA, the frequency is set within the range [0, 1], with both loudness and pulse rate set to 0.5. For GA, the mutation rate and crossover rate are set to 0.1 and 0.8, respectively. LS represents the default method used in the MATLAB System Identification Toolbox. Similarly, the NN structure is optimized using MATLAB’s Regression Learner. ITAE is used as the fitness function for each algorithm.

During optimization, a step input is used to estimate the parameters. Table 4 presents a comparison of the performance of different methods in open-loop (OL) and closed-loop (CL) setups during optimization. For each method, the table shows the statistical analysis of ITAE values, the number of iterations required to achieve them and average CPU time. Based on the ITAE values obtained in this phase, it is evident that the closed-loop approach significantly reduces ITAE values across all algorithms compared to the open-loop case. The fixed PID controller in the CL configuration helps stabilize the system and improve identification accuracy. Among the metaheuristic algorithms, GWO achieves the lowest ITAE in the open-loop case (0.108766), while PSO shows the best performance in the closed-loop setting (0.003729). This suggests that GWO is more effective in open-loop identification, whereas PSO benefits more from the control feedback in the closed-loop system.

The standard deviations of ITAE indicate that the performance consistency varies among the algorithms. GWO exhibits the lowest standard deviation in both open-loop (0.081053) and closed-loop (0.000186), indicating stable optimization results. In contrast, NN and GA show higher variability, suggesting inconsistency across multiple runs.

With regard to iteration count, all algorithms complete identification within a similar range (approximately 43–50 iterations), with LS requiring the fewest iterations (38.5 in OL, 34 in CL). This highlights LS’s computational efficiency. The CPU time results further confirm this, as LS has the lowest computational cost (1219.264 s in OL, 1427.257 s in CL). On the other hand, NN is the most computationally expensive, requiring over 2000s in both settings.

To provide an unbiased evaluation of the algorithms’ performance, the response of the estimated DC motor model is tested using a variable step input and compared to the actual output in the validation phase. Table 5 tabulates the estimated perturbed DC motor parameters based on the best results (i.e., lowest ITAE) obtained in Table 4.

The results from the validation phase, recorded in Table 6, provide a deeper insight into the generalization capability of each system identification method. Among the tested approaches, PSO demonstrates the best overall performance in the CL case, achieving an ITAE of 1.055644. This suggests that PSO benefits from the presence of a PID controller, allowing it to accurately model the system dynamics under control. GWO, on the other hand, performs best in the OL validation, achieving an ITAE of 0.525352, which is the lowest among all methods. This aligns with its strong ITAE results in Table 4 and indicates that GWO is the most effective when no feedback control is present. Interestingly, while PSO outperforms GWO in the CL case, GWO excels overall, as it maintains the lowest ITAE across both OL and CL scenarios.

The LS method also shows strong CL performance (1.057431 ITAE), nearly matching PSO, reinforcing its effectiveness as a computationally efficient and reliable approach when a controller is available. However, its OL performance is notably weaker than GWO’s, suggesting that LS may not generalize as well in uncontrolled conditions. The metaheuristic algorithms BA and GA struggle in the OL validation phase, with significantly higher ITAE values (5.560765 and 4.151888, respectively). However, their performance improves drastically in the CL case, indicating that these methods rely heavily on the presence of a controller to enhance system identification accuracy. This highlights a key limitation of BA and GA-while they can achieve reasonable results in a controlled environment, their robustness is questionable when no feedback is available.

In contrast, NN performs the worst in both OL and CL validation, suggesting that it may require more extensive tuning, additional training data, or alternative architectures to perform optimally in this context. Its high ITAE values indicate a lack of generalization and possible overfitting to the training data.

Overall, the results suggest that while PSO is the best choice for CL system identification, GWO excels in OL scenarios and performs best overall due to its consistently low ITAE across both validation conditions. LS remains a competitive alternative due to its efficiency, but it may be less effective in OL settings. The findings emphasize that selecting the best system identification method depends on whether a controller is present, with PSO and GWO emerging as the most effective depending on the scenario.

Motorized systems play crucial roles across diverse applications but are challenged by noise, load fluctuations, and environmental changes. Advancing system identification methods is essential to tackle these issues effectively. This section presents potential research directions aimed at improving the robustness, efficiency, and adaptability of system identification techniques for perturbed motorized systems. Each direction explores innovative approaches to advance the field and meet evolving application needs. The potential research directions include:

i. Integration of Machine Learning Techniques Investigate the integration of advanced machine learning techniques, such as deep learning and reinforcement learning, into system identification for perturbed motorized systems. Explore their potential in capturing complex dynamics and improving model accuracy under diverse perturbation scenarios.

ii. Robust Model Predictive Control (MPC): Explore robust MPC strategies that incorporate detailed system models derived from advanced identification methods. Investigate how these models can enhance control performance by mitigating the effects of perturbations such as noise, load fluctuations, and parameter vairations.

iii. Multi-Objective Optimization: Investigate multi-objective optimization approaches for system identification that consider conflicting objectives, such as model accuracy vs computational efficiency, robustness against perturbations, and adaptability to varying operational conditions.

iv. Real-Time Implementation and Hardware-in-the-Loop Testing: Focus on real-time implementation challenges of identified models and their validation through HIL testing. Explore methodologies to ensure model accuracy and robustness in practical motorized systems subjected to dynamic perturbations.

v. Model Transferability and Generalization: Investigate techniques to improve the transferability and generalization of identified models across different motor types, operating conditions, and environmental settings. Address challenges related to model adaptation and calibration in diverse applications.

vi. Online Learning and Adaptive Control: Explore online learning algorithms and adaptive control strategies that continuously update system models based on real-time sensor data. Investigate their effectiveness in maintaining optimal performance amidst varying perturbation intensities.

vii. Fault Diagnosis and Condition Monitoring: Extend system identification methodologies to incorporate fault diagnosis and condition monitoring capabilities. Develop integrated frameworks that utilize identified models for early fault detection.

viii. Hybrid Learning-Based Methods: A promising research direction is the development of hybrid learning-based methods that integrate physics-based models with machine learning techniques to enhance system identification of perturbed motorized systems. By combining first-principle equations with data-driven approaches such as NNs or neuro-fuzzy systems, future studies can improve the adaptability and accuracy of motor models, particularly in handling nonlinear perturbations like load variations, friction, and hysteresis. Additionally, exploring the use of metaheuristic-assisted deep learning for optimizing neural network parameters could further enhance robustness and generalization, making hybrid modeling a powerful tool for real-time motor control applications.

ix. Adaptive Modeling Techniques: Since PSO and GWO demonstrate complementary strengths in system identification-PSO performing better in CL settings and GWO excelling in OL scenarios-a promising research direction is the development of an adaptive modeling technique that dynamically integrates both methods. This approach could involve real-time switching between PSO and GWO based on system conditions, such as error trends, stability requirements, or perturbation levels. Alternatively, a hybrid optimization strategy could be designed to combine PSO’s rapid convergence with GWO’s strong exploration ability, enhancing overall robustness across varying operating conditions. Future research can explore intelligent adaptation mechanisms, such as reinforcement learning or fuzzy logic-based decision frameworks, to autonomously determine the optimal optimization method under different system states.

By exploring these avenues, researchers can contribute to enhancing the reliability, efficiency, and adaptability of motorized systems in various operational environments.

In conclusion, this review has explored various system identification methods for perturbed motorized systems, highlighting their applications, challenges, and recent advancements. The findings emphasize the importance of accurate modeling in understanding and mitigating perturbations such as noise, load variations, and environmental disturbances. This study conducted a comprehensive identification and statistical analysis of perturbed motorized systems using several widely used optimization algorithms, providing valuable insights into their comparative performance.

Furthermore, future research should explore innovative approaches to enhance system identification for perturbed motorized systems. The integration of machine learning techniques, such as deep learning and reinforcement learning, offers promising potential for capturing complex system dynamics and improving model accuracy across varying perturbation scenarios. Dynamic parameter adaptation strategies should also be investigated to improve the metaheuristic algorithms. This could involve hybridization, such as combining GWO’s exploration strengths with PSO’s exploitation efficiency or integrating machine learning-based adaptations for improved responsiveness. Additionally, robust MPC frameworks leveraging advanced system identification techniques could be explored to enhance control performance under noise, load fluctuations, and parameter variations.

Other key research directions include multi-objective optimization, where system identification balances competing factors like accuracy, computational cost, and robustness against perturbations. Real-time implementation challenges and HIL testing should also be addressed to validate identified models in dynamic environments. Model transferability and generalization across different motor types and conditions remain crucial for broader applicability. Further, online learning and adaptive control strategies could be developed to continuously update models based on real-time data, improving resilience against varying perturbation intensities. Finally, expanding system identification methodologies to include fault diagnosis and condition monitoring could provide early fault detection capabilities. By pursuing these research directions, future studies can drive significant advancements in the accuracy, efficiency, and adaptability of system identification techniques for motorized systems.