Optimal and Stable Framework for Robust Field-Oriented Control of PMSM using Hybrid ALO-MRSO

This research demonstrates a control method that combines ALO and MRSO optimization methods to tackle the issues of dynamic stability and optimal performance in permanent magnet synchronous motor (PMSM) drives. The combination of ALO-MRSO facilitates the global optimization of controller parameters through the efficient exploration of high-dimensional search spaces to provide asymptotic stability by confining system trajectories within specified boundaries. This hybrid optimization paradigm provides an advantage over conventional methods by not only ensuring the convergence of the optimization process to feasible solutions but also by enforcing closed-loop stability conditions. The framework is integrated within a field-oriented control (FOC) structure, in which the ALO-MRSO algorithm systematically optimizes each proportional-integral (PI) regulator within the cascade control loop. The results verified the improvements of the combined ALO-MRSO method in both transient and steady-state performance over the ALO or MRSO method based on MATLAB Simulink. The results of the optimized system achieve a rise time of 0.68 seconds, a minimal overshoot of 0.03%, and a settling time of 0.63 seconds. These results highlight the ability to balance rapid dynamic response with exceptional stability. These results demonstrate an advancement in PMSM control, offering a systematic and theoretical basis for managing the exploration-exploitation balance in advanced electromechanical systems. Consequently, this methodology provides a scalable approach for a robust controller design and optimal performance for a nonlinear dynamic system.

Introduction

Permanent magnet synchronous motors (PMSMs) have recently attracted attention for various applications owing to their high efficiency, accurate dynamic response, and high power density [1], [2]. PMSMs control, on the other hand, demonstrates inherent challenges owing to uncertain parameters, sensitivity to external disturbances, and variations in system parameters during operation [3]–[5].

Researchers have addressed various algorithms for the precise control of PMSM drives, which are categorized into classical and non-classical methods. Classical techniques, including Proportional-Integral (PI) control, Field-Oriented Control (FOC), and Direct Torque Control (DTC), are frequently favored because of their ease and simplicity of implementation. They exhibit an acceptable performance under well-tuned operating conditions [1], [6], [7]. However, non-classical methods typically exhibit poor performance under parameter sensitivity and are not robust to nonlinearity, external disturbances, or parameter variations. Therefore, they may struggle to achieve optimum performance under various operating conditions [8].

Conversely, Non-classical control methods, which include model predictive control (MPC), Sliding Mode Control (SMC), fuzzy control, neural network control, and heuristic optimization techniques have emerged to overcome these limitations [4], [9]–[15]. However, most non-classical methods suffer from complexity, uncertainties, chattering effects, large amounts of training data, and intensive computation. However, Heuristic optimization techniques, such as the Ant Lion Optimization algorithm (ALO) and Modified Rat Swarm Optimization algorithm (MRSO), are highly efficient in optimizing the controller parameters for complex nonlinear systems, offering global search capabilities and flexibility [16]–[18]. Therefore, selecting an appropriate control strategy for PMSM drives requires a balance between complexity, robustness, computational intensity, and desirable performance outcomes.

Among the heuristic approaches, the Ant Lion Optimizer (ALO) and Modified Rat Swarm Optimization (MRSO) algorithms are proposed in this study to overcome the limitations of classical optimization techniques and address the challenges posed by non-classical optimization techniques [18], [19]. These are up-and-coming methods for resolving complex engineering problems and making them more effective tools for adjusting control parameters and improving system accuracy and robustness.

The remainder of this paper is organized as follows: Section II introduces the ALO and MRSO algorithms, along with their application to PMSM control. Section III explains the PMSM model and conventional control problems. Section IV provides the simulation and experimental results, and Section V concludes the paper with the primary findings and future research directions.

Proposed ALO-MRSO Heuristic Techniques in the PMSM Control Approach

The Ant Lion Optimizer (ALO) is inspired by the hunting behavior of antlions in nature. It is based on work that mimics the interaction between antlions and ants. The hunting search behavior of the antlion makes it particularly effective for global optimization, as it thoroughly scans the search space to avoid local optima. However, its exploitation is limited to improving precise solutions. Owing to slow convergence and the struggle with fine-tuning the solution, once a promising region is found, further refinement is challenging. Therefore, they can become stuck if the elite does not improve [16], [17], [19].

On the other hand, the modified rat swarm optimization (MRSO) algorithm is inspired by the social and behavioral characteristics of rats. This enhanced the crucial aspects of Exploration and Exploitation by striking a balance between them [18]. However, the aggressive behavior of rats may cause MRSO to converge too quickly.

Therefore, using the ALO algorithm as the leading global explorer and embedding MRSO’s mechanism as a powerful local exploiter to utilize the hybrid ALO-MRSO algorithm to avoid getting stuck in local optima and to find more accurate solutions, which improve local optimization and enable high-precision refinement near optimal points. The strengths of the new ALO-MRSO integrated algorithm are leveraged. Thus, ALO conducts common exploration to identify good regions in the solution space, and MRSO refines these solutions with localized precision. Therefore, the hybrid algorithm achieves a faster convergence speed and greater accuracy than single algorithms, which makes it particularly well-suited for dynamic PMSM control systems. The hybrid ALO-MRSO strength yields a properly balanced optimization structure to achieve a robust optimization platform for PMSM control systems, ensuring an optimal control performance across different operating regimes.

The ALO is also highly efficient in global optimization by controlling the balance between exploration and exploitation in the search space [16]. The MRSO further improves its local search accuracy through adaptive radial improvements, making it particularly suitable for dynamic systems [18], [20]. Tuning the PI gains becomes more accurate by combining the ALO and MRSO optimization algorithms. Therefore, the ALO-MRSO control strategy for PMSMs ensures a reduction in the current ripple, and enhances the transient response while maintaining stability in the system.

The ALO-MRSO hybrid method aims to maximize the control performance, enhance the robustness, and improve the computational efficiency. This is achieved through the automatic self-tuning of the PI gains, which then reduces the torque ripple and improves the current tracking accuracy. The hybrid approach ensured a stable performance under different working conditions and demonstrated the adaptability to parameter drifts and load changes. The hybrid approach reduces the computational burden over conventional optimization methods by leveraging the high convergence rate of the ALO and the accuracy of the MRSO.

The algorithm operates in three synergistic phases: Global Exploration Phase, the Transition Phase, and the Local Refinement Phase. A flow chart of these three phases is presented in Fig. 1. First, ALO is Dominant within the Global Exploration Phase. Adaptive Balancing was dominant during the transition phase. Finally, MRSO is Dominant within the local refinement phase.

Fig. 1. Flowchart of the ALO-MRSO optimization phases process.

ALO-MRSO hybridization demonstrates particular advantages for PMSM control applications from the viewpoint of:

• Simultaneous global parameter space search and local controller adaptation

• Dynamic adjustment to time-varying operating conditions using a dynamic balancing mechanism

• Maintenance of solution diversity to prevent premature convergence in multi-modal solution landscapes

• Effective allocation of computational resources between exploration and exploitation

Convergence is regulated by a hybrid stop criterion that measures both the stability of the solution and the optimization progress to ensure balanced performance in multiple operating regimes of PMSMs. This hybrid setup is specifically appropriate for challenging motor control optimization problems, particularly the simultaneous tuning of multiple controller parameters under dynamic load conditions.

The validity of the proposed method is confirmed by simulations, which demonstrate its superiority over the traditional PI-based and fixed-hysteresis control approaches.

Mathematical Representation of Permanent Magnet Synchronous Motor (PMSM)

The PMSM is characterized by its stator circuit, which can be effectively modelled based on the d-q reference frame. However, the modeling process depends mainly on the selection of the rotor reference frame, as it directly affects the generation of electromagnetic force, stator voltage, and the torque produced by the rotor [2], [21].

Certain complex factors are either reduced or eliminated to simplfy the modeling process and improve the control design [21], [22]. These factors include ignored phenomena such as magnetic saturation, rotor saliency, and system nonlinearities. Furthermore, the magnetic field distribution and back electromotive force are considered to be purely sinusoidal, while the harmonic components in the air gap and iron losses are neglected. These simplifications make the PMSM model more manageable, enabling faster computations and analysis, and helping with efficient control design. [14], [23].

The dynamic behavior of a PMSM can be mathematically modelled in the rotor reference frame (d-q axes) using the following fundamental equation [24]–[26].

1. Voltage and Electromagnetic Torque Equations is shown in (1):

(1) $$\begin{array}{l}{v}_q=R{i}_q+L_q{\displaystyle \frac{d}{dt}}{i}_q+{\omega }_r(L_d{i}_d+{\psi }_m)\\ v_d=R{i}_d+L_d{\displaystyle \frac{d}{dt}}{i}_d-{\omega }_r\left(L_q{i}_q\right)\\ T_e={\displaystyle \frac{3}{2}}\times {\displaystyle \frac{P}{2}}({\psi }_m{i}_q+(L_d-L_q)i_d{i}_q)\end{array}$$

In a surface-mounted PMSM, $L_d$ = $L_q$ = L, and the torque equation can be expressed as in (2):

(2) $$\begin{array}{r}{T}_e=\frac{3}{2}\times \frac{P}{2}\left({\psi }_m{i}_q\right)\end{array}$$

Assuming that magnetic flux linkage is constant, the torque is linearly dependent on the current along the q-axis. The Mechanical Motion equation for the electromagnetic torque can be expressed as in (3):

(3) $$\begin{array}{r}J\frac{d{\omega }_r}{dt}=T_e-T_L-B{\omega }_r\end{array}$$

The Rotor Position Relationship can be represented as (4):

(4) $$\begin{array}{r}{\theta }_r=\int {\omega }_{r}dt\end{array}$$

Field-Oriented Control (FOC) for PMSM Drives

Field-Oriented Control (FOC) or vector control represents a sophisticated control method that decouples the torque and flux components of a PMSM. This decoupling is achieved by transforming the motor variables into a rotating reference frame (d-q axes) that is referenced with respect to the rotor flux [6], [7], [27]. Within this framework, The id axis current governs the magnetic flux of the motor. By contrast, the iq-axis current independently regulates torque generation, thereby enabling independent and precise control over both critical parameters. Through this conversion, performed by Clarke and Park transformations, the nonlinear dynamics of the motor are efficiently linearized, enabling FOC to deliver high-performance behavior with low-ripple torque response and high efficiency across a wide speed range the block diagram of the field oriented control is shown in Fig. 2.

Fig. 2. Block diagram of the FOC.

Accurate rotor position feedback is essential for ensuring proper field orientation. The FOC is capable of providing maximum torque per ampere (MTPA) operation and enabling flux weakening for high-speed applications. Therefore, FOC is the preferred solution for the dynamic performance of many systems, such as electric vehicles, robotics, and industrial automation [6], [14], [27].

Optimization of a Multi-Objective Cost Function

The optimization cost function is designed as a mathematical criterion for evaluating and minimizing control errors to satisfy the system constraints in the context of PMSM control. The cost function usually incorporates several performance objectives within heuristic optimization algorithms; each component of this function focuses on a specific performance aspect, and they are combined to create a single metric (J). The performance objectives were as follows [28]–[31]:

• Tracking Error Minimization: This equation quantifies the primary control objective, ensuring that the actual output of the motor aligns with the desired reference value.

• Torque Ripple Reduction: This equation aims to minimize the torque ripple, thereby ensuring smooth and stable motor operation as shown in (5).

(5) $$\begin{array}{r}{J}_2=k_4\sum {\left(Te-T{e}^{\ast }\right)}^2\end{array}$$

• Energy Efficiency: This equation directly targets the energy efficiency of the motor as as shown in (6).

(6) $$\begin{array}{r}{J}_3=k_5\int (i{d}^2+i{q}^2)dt\end{array}$$

• Constraint Handling (Soft Constraints): This equation acts as a penalty function to handle operational constraints, particularly the voltage limits of the inverter as shown in (7).

(7) $$\begin{array}{rl}{J}_4& =k_6\sum max{(0,\mid vd\mid -vmax)}^2\\ & +k_7\sum max{(0,\mid vq\mid -vmax)}^2\end{array}$$

• Composite Cost Function: This final equation combines all the individual objectives into a single scalar value as shown in (8).

(8) $$\begin{array}{r}{J}_t=\sum _{i=1}^4{j}_i\end{array}$$

Weight Adaptation: Heuristic algorithms dynamically adjust weights ($k_1-k_7$) to prioritize objectives under different operating conditions (such as high torque versus high speed).

Multi-Objective Optimization: Balances competing goals (e.g., tracking vs. efficiency) using Pareto-front analysis.

Real-Time Tuning: Algorithms such as ALO-MRSO optimize controller gains (e.g., PI parameters, and hysteresis bands) by iteratively minimizing JJ.

This framework enables robust PMSM control with an optimized transient response, steady-state accuracy, and energy efficiency while adhering to hardware constraints.

Simulation Results

The simulation model in this study implements a comprehensive Field-Oriented Control (FOC) strategy within a MATLAB/Simulink environment. The architecture features comprise both outer and inner control loops to manage speed and current regulation, respectively. The outer speed control loop is responsible for generating a torque-producing current reference. Reference is achieved by measuring the error in the difference between the commanded speed and the actual motor speed through a proportional-integral (PI) controller. The inner control loops, on the other hand, consist of decoupled d-axis and q-axis current controllers, each regulated by separate PI controllers. The d-axis of the surface-mounted PMSMs is typically maintained at zero to maximize torque efficiency. The primary task of the model is to optimize the PI controller gain using three distinct metaheuristic optimization strategies: the Ant Lion Optimizer (ALO), Modified Rat Swarm Optimization (MRSO), and novel hybrid ALO-MRSO approach.

Enabling precise interaction between the controller and the three-phase motor system has been achieved using essential reference frame transformations, including abc-dq and dq-abc conversions. The control voltage signals are then converted into pulse-width modulated (PWM) waveforms by a two-level, three-phase voltage source inverter (VSI), powered by a 500 V DC link. The PMSM block models the electromagnetic and mechanical dynamics of the motor and calculates variables such as stator currents, electromagnetic torque, and rotor speed. The objective function of the system performance was quantitatively assessed using the weighted fitness of 40% error metrics, 30% overshoot, 20% rise time, and 10% ripple factor. Torque ripple measurements were used to evaluate the dynamic behavior of the system. The integration of metaheuristic optimization through MATLAB subroutines enables the systematic tuning of control parameters, resulting in significant improvements in speed tracking, torque ripple reduction, and overall dynamic response. The simulation provides a robust framework for investigating the effectiveness of advanced control strategies and optimization techniques in PMSM drive systems. The proportional (Kp) and integral (Ki) gains for both the speed loop and the inner d-axis and q-axis current control loops are listed in Table I.

Table I. The Optimal Proportional and Integral Gains for Different Methods

Method	Controller	Kp	Ki
ALO	Speed	7.2	17.3
	q-axis current	2.5	11.8
	d-axis current	1.4	8.6
MRSO	Speed	9.4	30.5
	q-axis current	3.6	13.7
	d-axis current	2.7	33.4
ALO-MRSO	Speed	1.5	55.8
	q-axis current	0.9	13.5
	d-axis current	0.9	11.2

The results illustrate the performance of different metaheuristic methods, specifically ALO, MRSO, and the hybrid ALO-MRSO. The figures depict the performance evaluated under various step changes in speed commands, offering insights into its capability to accurately track different reference speeds, as well as load torque, motor torque, and motor current. Fig. 3 illustrates the performance evaluated under various step changes of the ALO optimization method, whereas Figs. 4 and 5 show the performance of the MRSO and hybrid ALO-MRSO.

Fig. 3. Step response performance of the ALO method.

Fig. 4. Step response performance of the MRSO method.

Fig. 5. Step response performance of the ALO-MRSO method.

Fig. 3, representing the ALO-optimised controller, illustrates a relatively fast speed tracking response, characterized by a rise time of approximately 0.75 seconds, and a slight overshoot of around 0.08% during acceleration. While the system reaches a steady state quickly, minor oscillations in both torque and current are observed during the transient phase. Fig. 3 indicates that although ALO provides acceptable dynamic performance, its damping capabilities may exhibit slightly weaker under highly dynamic operation conditions.

In contrast, Fig. 4 details the response of the MRSO-based tuning, demonstrating notable improvements. This method exhibits enhanced torque smoothness, particularly during load transitions, and demonstrates a slightly faster rise time of approximately 0.7 seconds with reduced overshoot of around 0.05%. Furthermore, the current response exhibited better attenuation of high-frequency noise, indicating increased robustness and more effective current loop regulation. The MRSO approach appears to be more effective in minimizing the torque ripple and mitigating lower current spikes, reflecting its improved global search ability for optimizing PI gains.

Fig. 5 highlights the superior performance of the hybrid ALO-MRSO approach, which delivers the optimal performance among the three optimization methods. The speed curve exhibits negligible steady-state error, a minimal overshoot of less than 0.03%, and the fastest rise time among all methods at approximately 0.68 seconds. The torque waveform was remarkably smoother, and the current response exhibited the lowest ripple and noise across all operational phases. This confirms that the synergistic combination of exploration ability of the ALO with the exploitation strength of the MRSO results in superior optimized controller tuning. The ALO-MRSO method successfully combines a fast transient response with a high steady-state accuracy and robust current regulation. Overall, the ALO-MRSO strategy demonstrates the most balanced and reliable dynamic behavior, making it particularly suitable for real-world PMSM applications that require precision and stability under varying load and speed conditions.

Figs. 6–8 demonstrate the optimized performance of the system during the acceleration and deceleration phases across all four-quadrant operating conditions, which encompass positive speed/positive torque, positive speed/negative torque, negative speed/negative torque, and negative speed/positive torque. These figures correspond to the ALO, MRSO, and ALO-MRSO optimization approaches, respectively.

Fig. 6. Four-quadrant operation of the ALO method.

Fig. 7. Four-quadrant operation of the MRSO method.

Fig. 8. Four-quadrant operation of the ALO-MRSO method.

Fig. 6, illustrating the ALO optimized system, exhibits speed tracking with good responsiveness during both acceleration from (0 to around 1.5 seconds) and deceleration from (4 to around 4.5 seconds), with the actual speed closely aligning with the reference. However, during the transient phase, particularly around 1.5 and 4.5 seconds, noticeable oscillations and higher ripple are monitored in the torque and current responses. While the system effectively navigates the four quadrants, the dynamic stability, especially in terms of torque and current smoothness, could be further improved.

In contrast, Fig. 7, representing the system tuned with MRSO, illustrates a smoother dynamic response. The speed tracking performance remained excellent, comparable to the achievemrnts of the ALO-optimized system. Crucially, the torque and current waveforms reduce oscillations and ripples during both acceleration and deceleration phases. This indicates that the MRSO contributes to better damping characteristics and more robust current regulation, resulting in a more stable and efficient operation across all four quadrants. The observed improvements in torque and current smoothness suggest that the MRSO optimizes the PI controller gains more effectively than the ALO approach, particularly in mitigating transient disturbances.

Fig. 8, which illustrates the hybrid ALO-MRSO approach, demonstrates the superior dynamic performance. The achieved speed tracking is exact, exhibiting minimal deviation from the reference thought all acceleration and deceleration processes. The most substantial enhancement is observed in the torque and current responses, which are remarkably smooth with almost negligible ripple and spikes across the entire operating range, including the challenging four-quadrant transitions. This performance further confirms that the synergistic combination of ALO’s global exploration capabilities with MRSO’s refined local exploitation strength results an optimally tuned controller. Thus, the hybrid method effectively delivers a fast transient response coupled with high steady-state accuracy and robust current regulation, making it exceptionally suitable for demanding PMSM applications that require accurate precision and stability.

Fig. 9 highlights the unique strengths and weaknesses in managing the PMSM system performance of the three optimization methods: ALO, MRSO, and the hybrid ALO-MRSO. The ALO approach provides a reasonable dynamic response with acceptable rise times and overshoot andshows moderate torque oscillations and higher current ripples, indicating a limited ability to dampen sudden speed changes. Conversely, MRSO improves upon this by achieving better torque stability and reduced overshoot, owing to its superior local exploitation capabilities. However, the best results are consistently achieved by the hybrid ALO-MRSO approach, which effectively combines the global search efficiency inherent in ALO with the refined convergence characteristics of MRSO. This combination leads to faster response times, minimal overshoot, negligible steady-state errors, and lower torque and current fluctuations. The superior performance of the hybrid strategy underscores its robustness, making it suitable for high-precision, dynamic motor control applications. Table II presents the performance advantages of each method across key control parameters.

Fig. 9. A comparative analysis of the three optimization techniques.

Table II. Performance Comparison of Optimization Methods

Method	Rise time (s)	Overshoot (%)	Current ripple (A)
ALO	0.75	0.08	High
MRSO	0.7	0.05	Moderate
ALO_MRSO	0.68	0.03	Low

Conclusion

This study focuses on the control performance of a PMSM using a novel hybrid ALO-MRSO approach based on a field-oriented control framework. The effectiveness of the proposed multi-objective cost function is the objective of this study, which incorporates transient response, steady-state error, and stability constraints, particularly when synergistically combined with the ALO-MRSO method to guide the optimization process.

Simulation results highlighted that while both ALO and MRSO individually yielded notable enhancements in speed tracking and torque regulation, the hybrid ALO-MRSO approach consistently surpassed its performance across all evaluated metrics. Specifically, the hybrid approach achieved faster rise times, reduced overshoot, minimal steady-state errors, and enhanced robustness against parameter variations. Furthermore, the system demonstrated an excellent dynamic response across diverse input scenarios, thereby confirming the suitability of the hybrid strategy for high-precision motor-control applications.

Collectively, the contributions of this research establish a generalized and reliable framework for applying intelligent, heuristic-based optimization to the design of high-performance PMSM controllers. These contributions are summarized as follows:

• Development of multi-objective cost function: This paper introduces the ALO-MRSO-based multi-objective cost function, which integrates transient, steady-state, and stability performance metrics, enabling more robust controller tuning for PMSM speed regulation.

• Comparative evaluation of optimized techniques: This study conducts a comparative evaluation of ALO, MRSO, and a novel hybrid ALO-MRSO optimization technique within the context of PMSM control. The study demonstrates that the hybrid approach delivers superior convergence speed, minimal overshoot, and enhanced stability compared to the individual optimizers.

• Demonstration of enhanced robustness: this study validates the adaptability of the proposed controller under real-world operating uncertainties by demonstrating enhanced robustness to parameter variations.

• Simulation-based verification: The findings of the proposed approach were confirmed through extensive simulations using MATLAB/Simulink software. This provides a clear and comprehensive performance standard for the different operating conditions.

The ALO-MRSO hybrid heuristic optimization offers a strong and effective control solution for PMSM drives. This is particularly important in environments characterized by uncertain parameters and high demand. The flexibility of the proposed control framework makes it an excellent choice for any PMSM-driven systems that require optimal performance, adaptability, and efficiency.