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# Optimized voltage vector selection for dual-star induction motor: Robust predictive direct torque control-based hysteresis-free approach

Amel Kasri, Kamel Ouari, Youcef Belkhier, Djamel Ziane, Mohamed Fouad Benkhoris, Mohamed Benbouzid

► To cite this version:

Amel Kasri, Kamel Ouari, Youcef Belkhier, Djamel Ziane, Mohamed Fouad Benkhoris, et al. Optimized voltage vector selection for dual-star induction motor: Robust predictive direct torque control-based hysteresis-free approach. ISA Transactions, 2025, 10.1016/j.isatra.2025.09.015. hal-05339543

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https://hal.science/hal-05339543v1

Submitted on 30 Oct 2025

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Practice article

# Optimized voltage vector selection for dual-star induction motor: Robust predictive direct torque control-based hysteresis-free approach

Amel Kasriª,*, Kamel Ouariª, Youcef Belkhierb, Djamel Zianec, Mohamed Fouad Benkhorisc, Mohamed Benbouzidb,*,

## HIGHLIGHTS

* A novel, robust predictive Direct Torque Control (DTC) technique for dual-star induction motors using dual-voltage source inverters, replacing traditional hysteresis-based methods with cost function optimization.

* A reformulated model predictive control (MPC) cost function with integral action, effectively eliminating steady-state errors and improving disturbance rejection under model uncertainties.

* Validated through real-time simulations on an OPAL-RT platform, demonstrating superior performance over conventional approaches.

## ARTICLE INFO

**Keywords:**

Direct torque control

Dual star induction motor

Predictive direct torque control

Robust speed control

## ABSTRACT

The dual-star induction machine is a widely adopted multiphase machine in industrial applications, thanks to its superior reliability in managing power supply faults and enhanced robustness. This study presents a robust predictive DTC strategy for a dual-star induction motor supplied by dual-voltage source inverters. Relying on a cost function optimization, the suggested predictive DTC algorithm replaces traditional hysteresis regulators and switching tables typically employed in classical DTC, determining the optimal sequence of voltage space vectors. Regarding speed regulation, the proposed MPC framework incorporates integral action, eliminating steady-state inaccuracies and improving disturbance rejection under model uncertainties and external perturbations. Real-time validation on the OPAL-RT platform confirms the effectiveness of the proposed scheme. Compared with conventional DTC, torque and flux ripples are reduced by 83.95 % and 74.75 %, respectively, while current THD decreases by 77.89 %. In addition, the speed response time improves by 82.5 %, and the rejection time by 97 %. These results highlight the robustness and efficiency of the proposed RPDTC technique, making it a strong candidate for high-performance DSIM drive applications.

## 1. Introduction

Induction Machines (IMs) are among the most widely used electrical machines in the industrial sector worldwide. They are employed as motors or generators across various applications due to their numerous benefits compared to other machine types [1]. Their key advantages include cost-effective manufacturing and maintenance, and reliable operation under diverse environmental conditions [2]. However, the

operation of IMs at zero speed presents challenges. To prevent flux saturation in this region, it is essential to considerably reduce both the frequency and magnitude of the stator voltage [3]. Therefore, the impact of the stator resistance voltage loss, which changes with the operating conditions, is considered when estimating the flux. Furthermore, decreasing the stator current oscillation enhances the drift issue in the integration process for flux determination. Consequently, estimating flux at zero speed is susceptible to inaccuracies [4,5].

<footnote>ª Université de Bejaia, Faculté de Technologie, Laboratoire de Technologie Industrielle et de l'Information (LTII), 06000 Bejaia, Algeria</footnote>

<footnote>b University of Brest, UMR CNRS 6027 IRDL, 29238 Brest, France</footnote>

<footnote>c IREENA Laboratory, Nantes University, 44600 Saint Nazaire, France</footnote>

<footnote>* Corresponding author.
Email address: Mohamed.Benbouzid@univ-brest.fr (M. Benbouzid).</footnote>
Nomenclature

DSIM Dual star induction motor

MIM Multiphase induction machine

DTC Direct torque control

MPC Model predictive control

VSI Voltage source inverter

SVM Space vector modulation

FCS-MPC Finite control set model predictive control

CCS-MPC Continuous control set model predictive control

RPDTC Robust predictive direct torque control

$v_{sa1}, v_{sa2}, v_{sb1}, v_{sb2}, v_{sc1}, v_{sc2}$ Stator voltage components in the three-phase (a, b, c) system

$v_{sd1}, v_{sq1}, v_{sd2}, v_{sq2}$ dq stator voltages

$i_{sd1}, i_{sq1}, i_{sd2}, i_{sq2}$ dq stator currents

$\phi_{sd1}, \phi_{sq1}, \phi_{sd2}, \phi_{sq2}$ dq stator flux

$v_{rd}, v_{rq}$ dq rotor voltages

$i_{rd}, i_{rq}$ dq rotor currents

$\phi_{rd}, \phi_{rq}$ dq rotor flux

$R_{s1}, R_{s2}$ Stator resistance

$\omega_s$ Velocity of the synchronous reference frame

$L_{s1}, L_{s2}$ Stator inductance

$R_r$ Rotor resistance

$L_r$ Rotor inductance

$L_m$ Magnetizing inductance

$\omega_r$ Rotor's electrical angular velocity

$T_{em}$ Electromagnetic torque

$T_{em}^*$ Torque reference

$T_L$ Load torque

$L_A^l$ $l^{th}$ order Lie derivative

$y(t + \tau)$ Predicted system output at $\tau$ steps forward

$y_{ref}(t + \tau)$ Reference trajectory projected $\tau$ steps forward

$\sigma$ Output's relative degree

$\omega_{ref}(t)$ Trajectory of reference

$\Sigma_{\omega}(t + \tau)$ Prediction of the error integral

$T_p$ Predictive time

$\phi_{sd1,2}(k + 2), \phi_{sq1,2}(k + 2)$ Stator flux prediction

$T_{em}(k + 2)$ Predicted electromagnetic torque

This has led to the emergence of Multiphase Induction Machines (MIMs), which offer enhanced capabilities for these challenging industrial needs [6]. The MIMs are gaining significant traction in industrial applications because of their superior capabilities over traditional three-phase motor drives. They offer numerous advantages, including enhanced total power output, smoother torque generation, reduced rotor harmonic losses, power segmentation, and greater system reliability. These features make them ideal for high-power applications [7]. Among the various types of MIMs, the Dual-Star Induction Machines (DSIMs) emerge as one of the most prominent. Renowned for their simplicity and cost-effectiveness, they have become a focal point in the recent literature [8]. Their versatility has been highlighted in numerous studies that have emphasized their potential in a wide range of applications, including their role as generators in renewable energy systems and as motors in industrial sectors such as naval propulsion, large cargo vessels, electric aircraft, and rail traction [9–11]. The DSIM offers numerous benefits, such as power splitting, enhanced efficiency, and improved stability by reducing rotor harmonic currents. Furthermore, it is suitable for harsh operating conditions and maintains satisfactory performance even in phase failure, providing an advantage unmatched by three-phase machines [12].

## 1.1. Literature review

Several advanced control strategies have been proposed for DSIM, showcasing notable effectiveness and innovation. The Direct Torque Control (DTC) method is commonly recognized as one of the most efficient techniques for DSIM due to its ease of implementation and simplicity; it also provides a quick torque response and takes advantage of a clear distinction between torque and flux variables [8,13]. Nevertheless, the DTC strategy is not without limitations. Its reliance on hysteresis controllers makes controlling the variable frequency challenging while also generating substantial torque and magnetic flux ripples, which induce undesirable mechanical vibrations, affecting the output power quality [14,15]. Additionally, inaccuracies in the estimation of torque and electromagnetic flux arising from variations in stator resistance within DSIM control systems further exacerbate the instability of the DTC approach [16]. Furthermore, DTC encounters significant hurdles when combined with a PI controller. Throughout the operation, the motor must demonstrate a fast and accurate dynamic response, ensuring no steady-state error. However, substantial and frequent variations in load torque, along with typical parameter uncertainty in the model, lead to system instability [17]. These factors contribute to the generation of acoustic noise

and disrupt control processes, ultimately diminishing the effectiveness of the control algorithm [18,19].

Researchers have developed various types of control strategies to tackle these challenges and optimize dynamic effectiveness within the DTC framework. In [20], the authors present a second-order sliding mode control approach utilizing the Super Twisting Algorithm (STA) combined with Fuzzy Logic Control (FSOSMC) for the DTC of a DSIM drive. In this study, the FSOSMC parameters are optimized using a Genetic Algorithm (GA) to enhance control performance. This method effectively minimizes torque and flux ripples while suppressing chattering. Another intelligent strategy is proposed in [21], where an Ant Colony Optimization (ACO) algorithm is integrated into the DTC framework to optimally tune the PID controller gains. By employing a combined weighted cost function, this approach aims to achieve more precise and efficient torque and speed control. In [22], a refined DTC scheme utilizing PI controllers has been presented for cage induction motors. The approach focuses on enhancing dynamic performance by minimizing torque and flux oscillations and improving current waveform quality through THD reduction. This is achieved by feeding the machine with a diode-clamped five-level inverter configured with a phase opposition disposition modulation strategy. Additionally, integrating a stator flux-based MRAS observer enables sensorless operation, providing precise flux estimation and thereby enhancing the overall robustness and reliability of the drive system. The authors in [23] present a modeling approach for six-phase induction machines that incorporates the effects of space harmonics using Vector Space Decomposition (VSD). Multi-purpose optimization algorithms are employed to accurately predict the settings of multiphase machines, and the obtained results are validated against manufacturer-provided data. The studies in [24] introduce a control approach that integrates DTC with a Model Reference Adaptive System (MRAS)-based estimation technique. This strategy provides exceptional stability and effective disturbance rejection. Furthermore, an absolute error output is employed together with the reference and rotor velocity in the adaptive equation to improve the resilience of the controller. Nevertheless, the primary drawback of MRAS is its computational complexity and the lengthy processing time required. The authors in [8] introduce an improved DTC approach that integrates a PI Fuzzy Logic Controller (PIFLC) for a six-phase machine, optimized through Particle Swarm Optimization (PSO). In addition, the air-gap harmonics are significantly reduced by employing VSD and the enhanced vector pulse width modulation technique. These methods have proven highly effective in controlling and identifying nonlinear
systems, enabling precise regulation of switching frequency to achieve rapid flux and torque responses with minimal distortion. Regarding the issue of variable switching, the most common solution is the integration of DTC with the Space Vector Modulation (SVM) technique, often referred to as DTC-SVM [25]. This method replaces the traditional switching table with the SVM technique to determine the inverter's switching states and substitutes the conventional hysteresis band with standard controllers. Furthermore, when applied to the DSIM, this integrated approach significantly minimizes torque and flux ripples [26]. In [13], the authors propose a strategy to enhance the performance of DTC by integrating a fuzzy logic controller with DTC-SVM. This approach replaces the conventional PI controllers with a self-tuning fuzzy PI controller, which dynamically produces the voltage components in the direct and quadrature axes. In [26], the authors present a robust DTC-SVM strategy for a DSIM, which combines DTC-SVM with a Sliding Mode Flux Observer (SMFO) to enhance robustness against stator resistance variations. The use of SVM effectively reduces the torque and flux ripples inherent in conventional DTC, while the SMFO ensures precise estimation of mechanical speed and stator flux, thereby reinforcing the system's overall reliability. Although these advanced strategies have achieved notable improvements, such as reduced ripples, enhanced robustness, and superior dynamic performance, they still present several challenges. Many approaches involve complex algorithms that increase computational demands and are sensitive to parameter variations and tuning requirements. Moreover, the dependence on sophisticated observers or optimization techniques can hinder real-time implementation and compromise reliability in industrial settings. Consequently, there remains a strong need for control strategies that offer high performance and robustness while maintaining simplicity and practical feasibility for real-time applications.

Model Predictive Control (MPC) has gained prominence as a widely acclaimed approach valued for its straightforward execution, exceptional proficiency in managing system limitations and dealing with nonlinear behaviors, and its proven effectiveness in achieving superior system performance [27,28]. As a type of feedback control technique, MPC uses a predictive model to forecast upcoming outputs over a pre-defined period, which are used to compute an optimization problem based on a specific cost function to determine the optimal control inputs. The MPC framework comprises three key components: a predictive model, a recursive optimization process, and a feedback mechanism [29]. However, its performance is highly sensitive to the precision of the system parameters. Any discrepancies or uncertainties in these parameters during the control process can result in prediction errors, which can significantly impair the system's overall performance and stability [30]. In practical applications, uncertainty in model parameters is a common challenge [31]. Variations in the thermal conditions of the DSIM and the magnetic saturation in the stator winding during operation can lead to mismatches in parameters like stator resistance and inductance. Inaccurate system modeling or parameter discrepancies can significantly impact control precision, potentially causing system instability [32]. As a result, addressing model parameter mismatch has become a critical challenge, prompting increasing research efforts focused on the robust control of MPC to enhance system performance and stability. Moreover, the load torque experiences significant variations, and the drive speed servo system must deliver an effective response even when substantial external load disturbances occur. Therefore, innovative approaches are needed to manage the dynamic changes in load characteristics [30]. In [33], a Tube-Based Robust Model Predictive Control (RMPC) strategy is introduced, leveraging online optimization to effectively handle system constraints and uncertainties. However, its design is often overly conservative, as the tube is constructed based on worst-case disturbance scenarios. To overcome this limitation, the authors incorporate a learning-based Gaussian Process (GP) regression into the RMPC framework. The GP enables real-time prediction of both the mean and variance of the disturbances, allowing for an adaptive tube design that reflects the actual disturbance characteristics rather than conservative estimates.

This integration significantly reduces conservatism and enhances overall control performance. In [34], the paper's research offers an innovative and robust Predictive Current Control (PCC) for IMs, incorporating a discrete-time integral action. The authors design a discrete-time predictive model for the stator voltage that incorporates an integral term. This modification enhances the controller's performance, particularly when dealing with parameter variations. In this research, the cost function differs from standard MPC methods by using stator voltage vectors instead of torque, magnetic flux, and current. The findings show that the suggested control is more resilient than standard PCC, resulting in smaller oscillations in circumstances with considerable fluctuations in parameter values. In [35], the authors investigate speed control for the DSIM using a robust control approach. Three nonlinear controllers are utilized to manage the machine's operation: initially, an Active Disturbance Rejection Control (ADRC) is implemented, followed by an analysis of Sliding Mode Control (SMC) to meet performance objectives. The paper concludes with the presentation of a robust controller relying on the backstepping approach, which is founded on the stability principles of the Lyapunov function. While effective in reducing parameter sensitivity, this solution necessitates the creation of both observers, which adds complexity. The paper [36] introduces an innovative model predictive control approach grounded in Lyapunov theory for a multi-drive system utilizing a nine-switch inverter. The research addresses the stability issues commonly encountered in MPC by reinterpreting the cost function as a Lyapunov energy function, which enhances the stability of the closed-loop system. The method put forward identifies the control input that complies with the Lyapunov stability condition. In [37] the authors introduced an MPC approach incorporating an open-loop repetitive control structure to enhance the robustness of electrical motor drives. Furthermore, an output filter was implemented to mitigate frequency quantization issues, thereby improving frequency flexibility. Nevertheless, the study did not address stability analysis. A model-free predictive current control technique that integrates an ultra-local model for a doubly-fed induction generator is proposed in [38]. While this method enhances robustness against parameter variations, its complexity could result in higher computational demands.

## 1.2. Main contributions of the proposed approach

This study proposes a Robust Predictive Direct Torque Control (RPDTC) strategy specifically designed for DSIMs fed by two voltage inverters. The method employs a two-stage predictive control framework. In the first stage, a Finite Control Set Model Predictive Control (FCS-MPC) algorithm based on an optimized cost function replaces the traditional hysteresis regulators and switching tables commonly used in conventional DTC schemes. This enables the selection of the optimal voltage space vector sequence for the next sampling period, thereby minimizing torque and flux errors. Additionally, time delay issues are mitigated through the implementation of a two-step forward prediction technique. In the second stage, a Continuous Control Set MPC (CCS-MPC) strategy is employed for speed regulation. The robustness of the system is further enhanced by incorporating a newly formulated MPC law with integral action, which effectively eliminates steady-state errors and significantly improves disturbance rejection, even in the presence of model uncertainties and external perturbations. The main contributions of this study are summarized as follows:

* A robust two-stage predictive control strategy is proposed, specifically adapted for high-performance multiphase electric drives. The method combines an FCS-MPC algorithm for inner-loop torque control with a CCS-MPC strategy for outer-loop speed regulation.

* Unlike most predictive control strategies, which are primarily tailored for conventional three-phase machines, the proposed method is developed and experimentally validated on a dual-star induction motor driven by dual inverters. The integration of model predictive control in this context enriches the applicability of the approach to advanced multiphase configurations, naturally accounting for the
<figure><img src="image_4.png" /><figcaption>Fig. 1. DSIM phase winding representation.</figcaption></figure>

increased system dimensionality and inter-phase dynamics inherent to such topologies.

* In the outer speed loop, although based on the predictive control principle, a novel robust control law is introduced through the formulation of an innovative cost function. In contrast to conventional approaches that often depend on additional observers or disturbance estimators to improve robustness, thereby increasing computational complexity, the proposed method inherently achieves robustness against model uncertainties and external disturbances without requiring explicit disturbance modeling. Moreover, theoretical closed-loop stability is established using Lyapunov theory.

To provide a clear and logical flow for the reader, this paper is structured as follows: Section 2 presents the mathematical models of the DSIM. Section 3 describes the proposed robust predictive direct torque control strategy for DSIM. Section 4 discusses the simulation of both the conventional DTC and the suggested RPDTC, including an analysis of the results and robustness testing. Experimental findings acquired with the OP4510 real-time simulator from OPAL-RT are presented in Section 5. Section 6 offers a comparative study of various strategies applied to DSIM. Finally, the paper concludes with a summary of the study.

## 2. Dual-star induction motor model

Our study investigates a complex system comprising a DSIM driven by two voltage source inverters connected to its stator windings. The stator features a pair of independent three-phase windings, which are spatially shifted by 30°. The rotor maintains a conventional three-phase configuration, similar to that of a standard induction motor. Fig. 1 provides a schematic representation of the DSIM windings. To establish a mathematical model for the DSIM, specific assumptions are taken into account:

* Magnetic characteristics exhibit minimal saturation effects.

* Both stators share identical configurations.

* Motor windings are arranged in a sinusoidal distribution.

The DSIM model is expressed in the synchronously rotating dq reference frame, where the corresponding voltage equations are represented as [39]:

$$ 
\left\{
\begin{aligned}
v_{sd1} &= R_{s1}i_{sd1} + \frac{d}{dt}\phi_{sd1} - \omega_s\phi_{sq1} \\
v_{sq1} &= R_{s1}i_{sq1} + \frac{d}{dt}\phi_{sq1} + \omega_s\phi_{sd1} \\
v_{sd2} &= R_{s2}i_{sd2} + \frac{d}{dt}\phi_{sd2} - \omega_s\phi_{sq2} \\
v_{sq2} &= R_{s2}i_{sq2} + \frac{d}{dt}\phi_{sq2} + \omega_s\phi_{sd2} \\
v_{rd} &= R_{r}i_{rd} + \frac{d}{dt}\phi_{rd} - \omega_{sl}\phi_{rq} = 0 \\
v_{rq} &= R_{r}i_{rq} + \frac{d}{dt}\phi_{rq} - \omega_{sl}\phi_{rd} = 0
\end{aligned}
\right.
\quad (1) $$

The slip speed $\omega_{sl}$ is determined by $\omega_{sl} = \omega_s - \omega_r$. The components of the flux are represented by:

$$ 
\left\{
\begin{aligned}
\phi_{sd1} &= L_s i_{sd1} + L_m(i_{sd1} + i_{sd2} + i_{rd}) \\
\phi_{sq1} &= L_s i_{sq1} + L_m(i_{sq1} + i_{sq2} + i_{rq}) \\
\phi_{sd2} &= L_s i_{sd2} + L_m(i_{sd1} + i_{sd2} + i_{rd}) \\
\phi_{sq2} &= L_s i_{sq2} + L_m(i_{sq1} + i_{sq2} + i_{rq}) \\
\phi_{rd} &= L_r i_{rd} + L_m(i_{sd1} + i_{sd2} + i_{rd}) \\
\phi_{rq} &= L_r i_{rq} + L_m(i_{sq1} + i_{sq2} + i_{rq})
\end{aligned}
\right.
\quad (2) $$

The mechanical dynamics of the DSIM are governed by:

$$ J \frac{d\omega_r}{dt} = T_{em} - f_r \omega_r - T_L \quad (3) $$

The electromagnetic torque equation is presented as follows:

$$ T_{em} = \frac{3}{2} p (\phi_{sd1} i_{sq1} + \phi_{sd2} i_{sq2} - \phi_{sq1} i_{sd1} - \phi_{sq2} i_{sd2}) \quad (4) $$

## 3. Design of the suggested control system

This section introduces an innovative, robust predictive control strategy dedicated to a dual-star induction machine. The proposed approach enhances the core principles of conventional DTC, replacing hysteresis comparators and the switching table with a model-based predictive control driven by an optimization-based control algorithm. Additionally, the classical PI speed regulator is replaced by a robust predictive speed controller, designed using a novel control law to improve system robustness. The overall structure of the proposed RPDTC strategy is illustrated in Fig. 2, which highlights the coordination between the external speed control loop and the two internal predictive DTC blocks assigned to each stator winding set, Star 1 and Star 2. The operation of each predictive DTC block is illustrated in Fig. 3, which presents the flowchart of the model predictive DTC algorithm applied to Star 1. It describes the main steps: recording system variables, predicting future states of flux and torque, evaluating the cost function, and selecting the optimal voltage vector that minimizes this function. An identical structure is implemented for Star 2.

### 3.1. Predictive DTC algorithm for DSIM

The core concept of the predictive DTC method lies in forecasting the system's anticipated behavior over time based on the dynamic model of the DSIM, enabling accurate control of both flux and torque. The implementation of the predictive DTC algorithm involves two primary steps: first, predicting the controlled variables, and second, selecting the appropriate voltage vector to apply during the subsequent sampling period [18]. A cost function is developed to determine the optimal voltage vector for the upcoming interval, assessing potential vectors according to predefined objectives. The function minimizes errors while effectively reducing torque and flux ripples. The prediction of controlled variables is achieved through the Forward Euler approximation, ensuring an efficient estimation process. The prediction of the stator current is formulated through (5) and (6):

$$ 
\left\{
\begin{aligned}
i_{sd1}(k+1) &= \left(1 - \frac{T_s R_{s1}}{L_s}\right) i_{sd1}(k) + \frac{T_s}{L_s} v_{sd1}(k) + T_s \omega_s(k) i_{sq1}(k) \\
i_{sq1}(k+1) &= \left(1 - \frac{T_s R_{s1}}{L_s}\right) i_{sq1}(k) + \frac{T_s}{L_s} v_{sq1}(k) + T_s \omega_s(k) i_{sd1}(k)
\end{aligned}
\right.
\quad (5) $$
<figure><img src="image_5.png" /><figcaption>Fig. 2. Block diagram of the suggested RPDTC strategy.</figcaption></figure>

$$ \left\{ \begin{aligned} i_{sd2}(k+1) &= \left(1 - \frac{T_s R_{s2}}{L_s}\right) i_{sd2}(k) + \frac{T_s}{L_s} v_{sd2}(k) + T_s \omega_s(k) i_{sq2}(k) \\ i_{sq2}(k+1) &= \left(1 - \frac{T_s R_{s2}}{L_s}\right) i_{sq2}(k) + \frac{T_s}{L_s} v_{sq2}(k) + T_s \omega_s(k) i_{sd2}(k) \end{aligned} \right. \quad (6) $$

The stator flux prediction is represented as shown below:

$$ \left\{ \begin{aligned} \phi_{sd1}(k+1) &= L_s i_{sd1}(k+1) \\ \phi_{sq1}(k+1) &= L_s i_{sq1}(k+1) \end{aligned} \right. \quad (7) $$

$$ \left\{ \begin{aligned} \phi_{sd2}(k+1) &= L_s i_{sd2}(k+1) \\ \phi_{sq2}(k+1) &= L_s i_{sq2}(k+1) \end{aligned} \right. \quad (8) $$

$$ \phi_{s1}(k+1) = \sqrt{(\phi_{sd1}(k+1))^2 + (\phi_{sq1}(k+1))^2} \quad (9) $$

$$ \phi_{s2}(k+1) = \sqrt{(\phi_{sd2}(k+1))^2 + (\phi_{sq2}(k+1))^2} \quad (10) $$

The predicted electromagnetic torque is computed as described in (11):

$$ T_{em}(k+1) = \frac{3}{2}(\phi_{sd1}(k+1)i_{sq1}(k+1) + \phi_{sd2}(k+1)i_{sq2}(k+1) - \phi_{sq1}(k+1)i_{sd1}(k+1) - \phi_{sq2}(k+1)i_{sd2}(k+1)) \quad (11) $$

The cost function Y integrates the absolute deviations of torque and flux, intending to optimize the voltage vector selection process to achieve superior control of both. It is formulated as follows:

$$ \left\{ \begin{aligned} Y_1 &= |T_{em}^* - T_{em}(k+1)| + \Lambda|\phi_{s1}^* - \phi_{s1}(k+1)| \\ Y_2 &= |T_{em}^* - T_{em}(k+1)| + \Lambda|\phi_{s2}^* - \phi_{s2}(k+1)| \end{aligned} \right. \quad (12) $$

where $T_{em}^*$ and $T_{em}(k+1)$ correspond to the desired reference torque and the predicted torque at the next step, respectively. Similarly, $\phi_{s1}^*, \phi_{s2}^*$ and $\phi_{sq1}(k+1)$ and $\phi_{sq2}(k+1)$ denote the reference and predicted flux components. Additionally, a constant parameter, $\Lambda$, is introduced to adjust the relative importance of flux inaccuracies compared to torque inaccuracies, with its value set to 30 in this case. To provide a comprehensive view of the internal structure of the proposed predictive DTC algorithm, Fig. 4 presents a detailed schematic of the entire control process applied to the DSIM.

## 3.2. Time delay compensation

At time $(k+1)$, a specified voltage vector is applied with a single-step delay. Given the minimal sample duration, the delay may affect the operation of the system. Therefore, it is crucial to compensate for this delay to improve efficiency by reducing torque ripples [40]. Consequently, the

<figure><img src="image_6.png" /><figcaption>Fig. 3. Flowchart of the predictive DTC strategy applied to star 1 of the DSIM.</figcaption></figure>

prediction will be considered two steps ahead, at $(k+2)$. The following is an expression for the predicted stator currents at $(k+2)$:

$$ i_{sd1,2}(k+2) = \left(1 - \frac{T_s R_s}{L_s}\right) i_{sd1,2}(k+1) + \frac{T_s}{L_s} V_{ds1,2}(k+1) + T_s \omega_s(k+1) i_{sq1,2}(k+1) \quad (13) $$
<figure><img src="image_7.png" /><figcaption>Fig. 4. Internal structure of the proposed predictive DTC algorithm for the dual-star induction motor.</figcaption></figure>

$$i_{sq1,2}(k+2) = \left(1 - \frac{T_s R_s}{L_s}\right) i_{sq1,2}(k+1) + \frac{T_s}{L_s} V_{qs1,2}(k+1) + T_s \omega_s(k+1) i_{sq1,2}(k+1) \quad (14)$$

The predicted electromagnetic torque at (k + 2) can be expressed as:

$$T_{em}(k+2) = \frac{3p}{2}(\phi_{sd1}(k+2)i_{sq1}(k+2) + \phi_{sd2}(k+2)i_{sq2}(k+2) - \phi_{sq1}(k+2)i_{sd1}(k+2) - \phi_{sq2}(k+2)i_{sd2}(k+2)) \quad (15)$$

To mitigate overcurrent, the cost function is enhanced by incorporating a limitation on the current magnitude, which is denoted as:

$$\begin{aligned} \hat{h}(i_{sd1,2}(k+2), i_{sq1,2}(k+2)) &= \infty \quad \text{if } |i_{sd1,2}(k+2)| > i_{max} \quad \text{and} \quad 0 \quad \text{if } |i_{sd1,2}(k+2)| \le i_{max} \\ & \quad \text{or} \quad |i_{sq1,2}(k+2)| > i_{max} \quad \text{or} \quad |i_{sq1,2}(k+2)| \le i_{max} \end{aligned} \quad (16)$$

The cost function $\Upsilon_{1,2}$ incorporating compensation for computational time delays, is defined as:

$$\Upsilon_1 = |T_{em}^* - T_{em}(k+2)| + \Lambda|\phi_{s1}^* - \phi_{s1}(k+2)| + \hat{h}(i_{sd1}(k+2), i_{sq1}(k+2)) \quad (17)$$

$$\Upsilon_2 = |T_{em}^* - T_{em}(k+2)| + \Lambda|\phi_{s2}^* - \phi_{s2}(k+2)| + \hat{h}(i_{sd2}(k+2), i_{sq2}(k+2)) \quad (18)$$

### 3.3. Robust predictive speed control

The control strategy proposed in this study for the speed regulation loop is based on model predictive control, forecasting future outputs using the DSIM model during a specified prediction horizon. These predictions derive the optimal control action by addressing a specific optimization issue that adheres to a defined cost function. This approach strengthens the robustness of conventional MPC, which traditionally depends on Taylor series expansion via the Lie derivative through modifications to the cost function. The mechanical dynamics of the DSIM are represented in matrix form through the equation below:

$$\begin{cases} \dot{x}(t) = A(x) + B(x)u(t) \\ y(t) = h(x) \end{cases} \quad (19)$$

Where the vectors $x(t) = \omega_r$, $u(t) = T_{em}$, and $y(t) = \omega_r$ correspond to the system's states, inputs, and outputs, respectively. It is presumed

that the functions $A(x)$ and $h(x)$ exhibit continuous differentiability to an adequate degree, $B(x)$ presents a continuous function of $x$, which are specified as outlined:

$$\left\{ \begin{aligned} A(x) &= -\frac{f_r}{J_1} \omega_r \\ B(x) &= \frac{1}{J} \end{aligned} \right. \quad (20)$$

#### 3.3.1. Conventional cost function

In line with the generalized predictive control approach, the following provides the cost function [41]:

$$\xi = \frac{1}{2} \int_{0}^{T_p} [\omega_{ref}(t + \tau) - \omega_r(t + \tau)]^2 d\tau \quad (21)$$

where $T_p$ is the predictive time, $\omega_r(t + \tau)$ is the predicted output and $\omega_{ref}(t + \tau)$ is the anticipated reference. Regarding the traditional MPC, accurately determining the output's relative degree is essential for formulating the control law to refine the cost function. The relative degree is defined by the order of the output's derivative [42]. The expected output is represented by a Taylor series expansion of $\sigma^{th}$ order, as defined through the Lie derivative [43]:

$$\omega_r(t + \tau) = \omega_r(x) + \sum_{l=1}^{\sigma_j} \frac{\tau^l}{l!} L'_A h_j(x) + \frac{\tau^l}{\sigma_j!} L_B L'_A (\sigma_j - 1) h_j(x) u(t) \quad (22)$$

where $h(x)$ has the following Lie derivative to the vector field $A(x)$:

$$\left\{ \begin{aligned} L_A h_j(x) &= \frac{\delta h_j}{\delta x} A(x) \\ L'_A h_j(x) &= L_A (L^{-1} h_j(x)) \\ L_B L_A h_j(x) &= \frac{\delta L_A h_j}{\delta x} B_u(x) \end{aligned} \right. \quad (23)$$

where $L'_A h_j$ is the $l^{th}$ order Lie derivative of $h_j$. The command is considered optimal if the subsequent condition holds:

$$\frac{d\xi}{du} = 0 \quad (24)$$

At this stage, predictive control techniques based on Taylor series expansion are commonly used to manage system nonlinearities and ensure satisfactory tracking, considering that the motor settings are constant. However, this approach becomes unreliable in the presence of external disturbances or parameter variations [42]. To address these limitations, the following section introduces a robust MPC strategy with a newly modified cost function specifically designed to enhance resilience under such conditions.
<figure><img src="image_8.png" /><figcaption>Fig. 5. Structure of the suggested robust predictive speed control.</figcaption></figure>

### 3.3.2. Enhanced cost function

The cost function (21) is adjusted to incorporate an integral action within the control law to address the limitations of the earlier discussed MPC. This innovation significantly enhances the controller's performance while eliminating the necessity of directly identifying external disturbances or system parameters. The block diagram representing the proposed robust predictive speed controller is presented in Fig. 5. The integral action is effectively introduced through the revised cost function:

$$ \xi = \frac{1}{2} \int_{0}^{T_p} (\Sigma(t + \tau))^2 d\tau \quad (25) $$

with

$$ \Sigma(t) = \int_{0}^{t} (\omega_{ref}(t) - \omega_{r}(t)) d\tau \quad (26) $$

Given that the output has a relative degree of $\sigma = 1$, it implies that:

$$ \dot{\omega}(t) = L_A h(x) + \Gamma(x) u(t) \quad (27) $$

With $\Gamma(x) = \frac{f_r}{J^2}$. The anticipated term $\Sigma(t + \tau)$ is expressed as a Taylor series expansion of $(\sigma + 1)^{th}$ order:

$$ \Sigma(t + \tau) = \Sigma(t) + \tau[\omega_{ref}(\tau) - \omega_{r}(\tau)] + \frac{\tau^2}{2}[\dot{\omega}_{ref}(t) - L_A h(x) - \Gamma(x)u(t)] \quad (28) $$

In matrix form, (28) can be expressed as:

$$ \Sigma(t + \tau) = \Pi(\tau)[E(t) - \Psi(u)] \quad (29) $$

Where

$$ \Pi(\tau) = \begin{bmatrix} 1 & \tau & \frac{\tau^2}{2} \end{bmatrix}; E(t) = \begin{bmatrix} \int_0^t [\omega_{ref}(\tau) - \omega_r(\tau)] d\tau \\ \omega_{ref}(\tau) - \omega_r(\tau) \\ \dot{\omega}_{ref}(t) - L_A h(x) \end{bmatrix}; \Psi(u) = \begin{bmatrix} 0 \\ 0 \\ \Gamma(x) \end{bmatrix} u(t) \quad (30) $$

Replacing (29) into (25) yields the following cost function:

$$ \xi = \frac{1}{2} \left[ \begin{bmatrix} \Sigma(t) \\ \omega_{ref}(\tau) - \omega_r(\tau) \\ \dot{\omega}_{ref}(t) - L_A h(x) \end{bmatrix} - \begin{bmatrix} 0 \\ 0 \\ \Gamma(x) \end{bmatrix} u(t) \right]^T \Lambda(\tau) \left[ \begin{bmatrix} \Sigma(t) \\ \omega_{ref}(\tau) - \omega_r(\tau) \\ \dot{\omega}_{ref}(t) - L_A h(x) \end{bmatrix} - \begin{bmatrix} 0 \\ 0 \\ \Gamma(x) \end{bmatrix} u(t) \right] \quad (31) $$

where

$$ \Lambda(\tau) = \begin{bmatrix} 1 & \tau & \frac{\tau^2}{2} \end{bmatrix}^T \begin{bmatrix} 1 & \tau & \frac{\tau^2}{2} \end{bmatrix} \quad (32) $$

By satisfying the condition (24), the optimal control is determined as outlined below:

$$ u(t) = - \left( \frac{f_r}{J^2} \right)^{-1} \left[ \lambda_0 \int_0^t \epsilon(\tau) d\tau + \lambda_1 \epsilon(t) + \lambda_2 \dot{\epsilon}(t) \right] \quad (33) $$

With

$$ \lambda_0 = \frac{2}{T_p}; \quad \lambda_1 = \frac{2}{T_p}; \quad \lambda_2 = 1 \quad (34) $$

and

$$ \epsilon(t) = \omega_{ref}(t) - \omega_r(t) \quad (35) $$

To minimize the control effort, a second-order linear filter is applied to the reference speed signal, which is supplied by:

$$ F(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n + \omega_n^2} \quad (36) $$

With

$$ \begin{cases} \omega_n = 5 \\ \zeta = 0.7 \end{cases} \quad (37) $$

### 3.3.3. Analysis of system stability

Analyzing the error dynamics behavior at the starting point is instrumental in evaluating the stability of the entire closed-loop system. To achieve this, substitute (33) into (27) to produce the following feature formula:

$$ \lambda_0 \int_0^t \epsilon(\tau) d\tau + \lambda_1 \epsilon(t) + \lambda_2 \dot{\epsilon}(t) = 0 \quad (38) $$

Based on (38), the characteristic polynomial is expressed by the following:

$$ \lambda_2 s^2 + \lambda_1 s + \lambda_0 = 0 \quad (39) $$

The stability condition is derived from the solutions to (39), as follows:

$$ \left\{ \begin{array}{l} s_1 = -\frac{1}{T_p} + \frac{1}{T_p}i \\ s_2 = -\frac{1}{T_p} + \frac{1}{T_p}i \end{array} \right. \quad (40) $$

Given that all eigenvalues have negative real components, the closed-loop framework achieves asymptotic stability.

## 4. Simulation results

The control algorithm, previously detailed in theoretical discussions, has been implemented and tested through simulations using MATLAB/Simulink.

The simulations were conducted on a DSIM, with its specific parameters provided in Table 1, and a prediction horizon of $T_p = 1$ ms was employed. Various scenarios were simulated to assess the proposed control strategy's effectiveness and dynamic performance:

<table><caption>**Table 1**
Parameters of DSIM.</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Generator-rated power</td><td>P<sub>n</sub> = 4.5 Kw</td></tr><tr><td>Nominal frequency</td><td>f = 50 Hz</td></tr><tr><td>Number of pole pairs</td><td>P = 2</td></tr><tr><td>Stator resistances</td><td>R<sub>s1</sub> = R<sub>s2</sub> = 3.72 Ω</td></tr><tr><td>Rotor resistance</td><td>R<sub>r</sub> = 2.12 Ω</td></tr><tr><td>Stator inductance</td><td>L<sub>s</sub> = 0.0137 H</td></tr><tr><td>Rotor inductance</td><td>L<sub>r</sub> = 0.006 H</td></tr><tr><td>Mutual inductance</td><td>L<sub>m</sub> = 0.3672 H</td></tr><tr><td>Moment of inertia</td><td>J = 0.0625 kg.m<sup>2</sup></td></tr><tr><td>Friction coefficient</td><td>f<sub>r</sub> = 0.001 N.m.rad<sup>-1</sup>.s<sup>-1</sup></td></tr></tbody></table>
<figure><img src="image_9.png" /><figcaption>(a) Tracking speed</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(c) Electromagnetic torque</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(e) Stator flux with DTC</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(b) Tracking speed error</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(d) Stator currents</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(f) Stator flux with RPDTC</figcaption></figure>

<caption>Fig. 6. Simulated results under load torque disturbances.</caption>

## 4.1. Performance evaluation under load torque disturbances

The motor speed in this setup changes from 100 rad/s to –100 rad/s at $t = 2s$. A load torque of $T_L = 20Nm$ is applied during the time interval $t \in [1, 1.5]$ s and $T_L = 10Nm$ is introduced in the interval $t \in [3, 3.5]$ s.

Fig. 6(a) shows that while the traditional DTC successfully tracks the reference speed, it exhibits greater sensitivity to load torque disturbances. In contrast, the proposed RPDTC effectively eliminates the error, counteracting the effect of the load torque. The speed response consistently tracks the reference accurately, showcasing a notably fast and stable response time.

Fig. 6(b) shows the speed tracking error response for both controllers, highlighting the superior performance of the proposed RPDTC, even

under load torque fluctuations. This approach ensures quick responsiveness, eliminates steady-state errors, and minimizes overshoot. In contrast, the DTC exhibits increasing errors with each change in speed and load torque.

The results presented in Fig. 6(c) highlight several notable improvements in the electromagnetic torque response with the proposed RPDTC, demonstrating its superiority over the conventional DTC. The RPDTC provides a quick torque response with negligible overshoot and excellent responsiveness, with almost no delay. An enhancement of roughly 83.95 % is noted, illustrating the significant contribution of the RPDTC scheme in efficiently mitigating unwanted torque ripple in DTC.
<figure><img src="image_15.png" /><figcaption>Fig. 7. Frequency analysis of phase current with THD evaluation.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 8. Simulated results under load torque disturbances and variation of the stator resistance $R_s$.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 9. Simulated results under load torque disturbances and variation of inductance $L_s$.</figcaption></figure>

Fig. 6(d) depicts that the three-phase current of the first stator supplied to the DSIM follows a sinusoidal waveform, having its magnitude precisely proportional to the electromagnetic torque. The proposed RPDTC system achieves a remarkable reduction in the current ripple, effectively addressing one of the primary limitations of conventional DTC. This enhancement is particularly noteworthy, as excessive current ripple remains a critical challenge that the RPDTC successfully overcomes. As a result, the peak current is lowered, enhancing the overall power quality delivered to the motor.

Fig. 6(e) and (f) illustrate the trajectories of the flux amplitudes in the $(\alpha, \beta)$ plane. The fluxes are restricted within the defined boundaries of the variation bands. The RPDTC demonstrates a smoother trajectory circle with fewer oscillations than the conventional control method. This represents an improvement of 74.75% in flux ripple reduction, highlighting the efficiency of the suggested control approach in minimizing flux ripple.

Fig. 7 illustrates the results of the FFT applied to the phase current, along with the THD values obtained for both the proposed RPDTC
<figure><img src="image_18.png" /><figcaption>Fig. 10. Simulated results under varying inertia moments.</figcaption></figure>

and the standard DTC. The THD achieved using RPDTC is 2.72 %, significantly lower than the 12.31 % recorded with the conventional DTC approach, indicating a reduction of approximately 77.9 %. These simulation outcomes demonstrate the enhanced performance of the proposed RPDTC strategy over the traditional DTC method.

## 4.2. Robustness test under uncertainty in the electrical and mechanical parameters

First, a series of parameter adjustments was applied to the electrical machine to assess the effectiveness of the proposed control system under parameter mismatches. Specifically, the stator resistance was increased by 30 % and 50 %, while the inductance was reduced by 30 % and 50 % from their nominal values. In each test scenario, a load torque disturbance was introduced at $t = 1$ s. The comparative results between traditional DTC and proposed RPDTC, presented in Figs. 8 and 9, demonstrate that proposed RPDTC significantly reduces the effects of load torque disturbances and parameter variations. It precisely maintains the speed at its reference value with a fast response and achieves zero steady-state error. In contrast, the classical DTC shows notable deviations in steady-state speed tracking. Furthermore, the error curve of the proposed controller rapidly converges to zero and remains stable, highlighting its superior performance. In comparison, the classical DTC exhibits an error that progressively increases with load torque and parameter variations, underscoring its sensitivity to such variations.

The system is then evaluated under uncertainties of mechanical parameters, where the moment of inertia is increased by 10 % and 40 % of its nominal value. Fig. 10 presents the speed trajectory tracking and the corresponding tracking error. It is evident that when the proposed controller is applied, the steady-state error is eliminated and accurate speed tracking is successfully attained. The findings

showcase the exceptional efficiency of the proposed controller in speed-tracking performance. After each change in machine parameters, the speed error quickly converges to zero, highlighting the controller's robustness.

## 5. Real-time validation results

This part provides a real-time simulation of the suggested control using the OPAL-RT discrete real-time simulation platform. Fig. 11 illustrates the real-time simulation setup, which comprises the key components: (1) a host computer, (2) an OP4510 real-time digital simulator, (3) a data acquisition interface, and (4) a digital oscilloscope.

The initial step in the real-time simulation process involves model partitioning. The system is divided into computation and console subsystems. Components requiring computation, such as the RPDT controller and the DSI motor, are assigned to the computation subsystem, represented by the master (SM) block. Meanwhile, scopes and constants are allocated to the console (SC) subsystem. Once the compilation is complete, the generated code is transferred to the target through the TCP/IP interaction and performed using parallel processing. The digital oscilloscope's I/O channels are then used for tracking and visualizing the data collected from the display blocks. Following the decomposition and adaptation of the DSIM simulated system for implementation on the RT LAB platform, a real-time simulation was performed. To evaluate robustness under realistic operating conditions, different disturbance scenarios were introduced. In the first test, abrupt load torque disturbances and a speed reversal were applied to assess the controller's capability to reject external perturbations and maintain precise speed tracking, similar to the scenarios used in the simulation studies. Fig. 12–16 provide a detailed illustration of the experimental framework, comparing the suggested technique and traditional strategy under load torque disturbances.

<figure><img src="image_19.png" /><figcaption>Fig. 11. Configuration of the OPAL-RT platform for experimental testing.</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>Fig. 12. Experimental real-time evaluation under load torque disturbances.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 13. Real-time response of stator currents $i_{abc}$.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Fig. 14. Real-time response of stator currents $i_{a1}, i_{a2}$.</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>Fig. 15. Experimental real-time evaluation of speed tracking under electrical parameter variations and load torque disturbances.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Fig. 16. Experimental real-time evaluation of stator currents and electromagnetic torque under electrical parameter variations and load torque disturbances.</figcaption></figure>

The graphic in Fig. 12(a) highlights the exceptional accuracy achieved by the proposed controller, showcasing a near-perfect alignment between the actual speed and the reference trajectory. Unlike the classic DTC, the proposed method ensures superior performance by addressing the challenges of steady-state errors and sensitivity to disturbances. Even with fluctuations in load torque, the controller demonstrates its robustness by effectively eliminating steady-state errors and maintaining exact tracking performance.

Fig. 12(b) presents the evolution of the electromagnetic torque, while a detailed zoomed-in view is provided in Fig. 12(c) to allow closer examination, highlighting the controller's exceptional ability to precisely track the reference trajectory. Despite abrupt trajectory changes, the system effectively minimizes torque ripple, ensuring a smooth and accurate response.

The experimental results shown in Figs. 13 and 14 illustrate the stator currents, which closely match the simulation results, thus validating the precision and reliability of the proposed approach.

In the second test, significant parameter variations were introduced: a -30 % variation in stator inductance and a +50 % increase in stator resistance, with a load torque perturbation applied at $t = 1$ s. These variations were implemented to emulate modeling uncertainties and to validate the controller's robustness in the presence of parameter mismatches. The resulting data presented in Figs. 15 and 16 demonstrate the validity and effectiveness of the proposed control approach in the presence of parameter changes. Moreover, the close agreement between the simulation and the experimental results underscores the reliability of the proposed approach and its suitability for real-time implementation.
# 6. Discussion and comparison

## 6.1. Comparative performance analysis of RPDTC and DTC

A detailed quantitative comparison of the DTC and the proposed RPDTC is provided in Table 2, highlighting key dynamic and steady-state performance metrics. In terms of speed control, the proposed RPDTC achieves a significantly shorter response time of 0.021 s compared to 0.12 s for DTC, corresponding to an improvement of approximately 82.5 %. Furthermore, the overshoot is drastically reduced from 3 rad/s to only 0.15 rad/s, resulting in smoother acceleration and improved tracking precision. The rejection time is also considerably minimized from 0.34 s to 0.01 s, about 97 % improvement, enabling the system to promptly reject disturbances. Furthermore, the steady-state error is eliminated, reduced from 1.2 rad/s to 0, confirming the high precision and excellent setpoint tracking capability of RPDTC. Regarding torque control, the torque ripple is reduced by approximately 84 %, which significantly enhances mechanical robustness. Moreover, the torque response time is reduced to 0 s, implying an instantaneous torque dynamic behavior under the proposed strategy. Concerning current quality, the total harmonic distortion (THD) of the stator current $i_{sa}$ is notably decreased by approximately 78 %.

To further validate the robustness of the proposed RPDTC under parameter variations, a detailed analysis using the Integral of the Squared Error (ISE), Integral of the Absolute Error (IAE), and Integral of the Time-weighted Absolute Error (ITAE) was conducted, as summarized in Table 3. Under load torque ($T_L$) variations, the ISE is reduced by 98.7 %, IAE by 92.5 %, and ITAE by 93.3 %. Similar improvements are consistently observed for variations in stator resistance ($R_s$), inductance ($L_i$), and moment of inertia ($J$). These remarkable reductions in error indices indicate a superior disturbance rejection capability, robustness against parameter uncertainties, and excellent transient performance of the RPDTC. The comprehensive comparative analysis confirms that the proposed RPDTC strategy significantly outperforms the classical DTC in all key performance aspects, making it a strong candidate for high-performance and industrial-grade electric drive applications where high accuracy, robustness, and fast dynamic responses are critical.

<table><caption>**Table 2**

Comparative performance between proposed RPDTC and classical DTC.</caption><thead><tr><th>Performance metrics</th><th></th><th>DTC</th><th>RPDTC</th><th>Improvement (%)</th></tr></thead><tbody><tr><td rowspan="4">Speed</td><td>Response time (s)</td><td>0.12</td><td>0.021</td><td>82.5</td></tr><tr><td>Overshoot (rad/s)</td><td>3</td><td>0.15</td><td>95.0</td></tr><tr><td>Rejection time (s)</td><td>0.34</td><td>0.01</td><td>97.1</td></tr><tr><td>Steady-state error (rad/s)</td><td>1.2</td><td>0</td><td>100.0</td></tr><tr><td rowspan="2">Torque</td><td>Ripple (N·m)</td><td>3.8</td><td>0.61</td><td>84.0</td></tr><tr><td>Response time (s)</td><td>0.02</td><td>0</td><td>100.0</td></tr><tr><td>Flux</td><td>Stator flux ripple (Wb)</td><td>0.04</td><td>0.0101</td><td>74.7</td></tr><tr><td>Current</td><td>THD i<sub>sa</sub> (%)</td><td>12.31</td><td>2.72</td><td>77.9</td></tr></tbody></table>

<table><caption>**Table 3**

Comparative analysis of DTC and RPDTC using ISE, IAE, and ITAE metrics under parameter variations.</caption><thead><tr><th>Variation</th><th>Metric</th><th>DTC</th><th>RPDTC</th><th>Improvement (%)</th></tr></thead><tbody><tr><td rowspan="3">T<sub>L</sub></td><td>ISE</td><td>1.734</td><td>0.0219</td><td>98.7</td></tr><tr><td>IAE</td><td>1.469</td><td>0.1105</td><td>92.5</td></tr><tr><td>ITAE</td><td>2.640</td><td>0.1756</td><td>93.3</td></tr><tr><td rowspan="3">R<sub>s</sub></td><td>ISE</td><td>2.054</td><td>0.0943</td><td>95.4</td></tr><tr><td>IAE</td><td>1.295</td><td>0.0645</td><td>94.9</td></tr><tr><td>ITAE</td><td>1.584</td><td>0.0671</td><td>95.8</td></tr><tr><td rowspan="3">L</td><td>ISE</td><td>1.096</td><td>0.0050</td><td>99.5</td></tr><tr><td>IAE</td><td>0.881</td><td>0.0384</td><td>95.6</td></tr><tr><td>ITAE</td><td>0.723</td><td>0.0384</td><td>94.7</td></tr><tr><td rowspan="3">J</td><td>ISE</td><td>1.008</td><td>0.0049</td><td>99.5</td></tr><tr><td>IAE</td><td>0.821</td><td>0.0381</td><td>95.4</td></tr><tr><td>ITAE</td><td>0.701</td><td>0.0141</td><td>98.0</td></tr></tbody></table>

<table><caption>**Table 4**

Comparison with recent robust control approaches in the literature.</caption><thead><tr><th>Reference</th><th>Control strategy</th><th>Response time (s)</th><th>Overshoot (rad/s)</th><th>Rejection time (s)</th></tr></thead><tbody><tr><td>[44]</td><td>MFPSC-QRC</td><td>0.38</td><td>0.260</td><td>0.35</td></tr><tr><td>[44]</td><td>MBPSC</td><td>0.42</td><td>0.474</td><td>0.48</td></tr><tr><td>[45]</td><td>GA-DTC</td><td>0.0272</td><td>0</td><td>0.00182</td></tr><tr><td>[45]</td><td>ACO-DTC</td><td>0.0272</td><td>0</td><td>0.041</td></tr><tr><td>[46]</td><td>BC-SSMC-MRAS</td><td>0.37</td><td>0</td><td>3</td></tr><tr><td>[47]</td><td>ANN-DTC</td><td>0.0494</td><td>0</td><td>0.0152</td></tr><tr><td>[48]</td><td>P-MRAS-DTC</td><td>0.278</td><td>0</td><td>0.012</td></tr><tr><td>Proposed</td><td>RPDTC</td><td>0.018</td><td>0.15</td><td>0.021</td></tr></tbody></table>

<table><caption>**Table 5**

Comparative analysis of different DTC-based control strategies in the literature.</caption><thead><tr><th>Reference</th><th>Control strategy</th><th>Torque ripple (N·m)</th><th>Flux ripple (Wb)</th><th>THD (%)</th></tr></thead><tbody><tr><td>[45]</td><td>GA-DTC</td><td>2.1</td><td>0.04253</td><td>5.03</td></tr><tr><td>[45]</td><td>ACO-DTC</td><td>1.91</td><td>0.0401</td><td>4.25</td></tr><tr><td>[49]</td><td>NNC-DTC</td><td>1.5</td><td>0.054</td><td>2.1</td></tr><tr><td>[50]</td><td>Fuzzy-DTC</td><td>1.14</td><td>0.02</td><td>2.81</td></tr><tr><td>[51]</td><td>NN-PDTC</td><td>0.80</td><td>0.0032</td><td>-</td></tr><tr><td>[51]</td><td>CNN-PDTC</td><td>2.50</td><td>0.007</td><td>-</td></tr><tr><td>[46]</td><td>BC-SSMC-MRAS</td><td>0.01</td><td>0.009</td><td>2.61</td></tr><tr><td>[46]</td><td>BC</td><td>0.12</td><td>0.05</td><td>2.74</td></tr><tr><td>[47]</td><td>ANN-DTC</td><td>1.08</td><td>0.01827</td><td>3.26</td></tr><tr><td>[48]</td><td>P-MRAS-DTC</td><td>1.46</td><td>0.0148</td><td>2.22</td></tr><tr><td>[26]</td><td>RDTC-SVM</td><td>3.3</td><td>0.17</td><td>2.53</td></tr><tr><td>[18]</td><td>DTC</td><td>3.8</td><td>0.04</td><td>12.31</td></tr><tr><td>Proposed</td><td>RPDTC</td><td>0.61</td><td>0.0101</td><td>2.72</td></tr></tbody></table>

## 6.2. Comparison with recent robust control strategies

To further emphasize the robustness and novelty of the proposed RPDTC strategy, a comprehensive quantitative comparison with recently published robust control approaches is provided in Table 4. The selected references include advanced techniques such as Model-Free Predictive Speed Control with Quasi-Resonant Controllers (MFPSC-QRC) [44], Model-Based Predictive Speed Control (MBPSC) [44], Genetic Algorithm-based DTC (GA-DTC) and Ant Colony Optimization-based DTC (ACO-DTC) [45], Backstepping Synergetic-Sliding Mode Controller with MRAS (BC-SSMC-MRAS) [46], Artificial Neural Network-based DTC (ANN-DTC) [47], and Predictive MRAS-DTC (P-MRAS-DTC) [48].

The comparison demonstrates that the proposed RPDTC method outperforms these advanced controllers in terms of key dynamic performance indicators. Specifically, RPDTC achieves the shortest response time (0.018 s) and complete elimination of overshoot, while also exhibiting a significantly faster disturbance rejection time (0.021 s). These improvements are primarily attributed to the reformulated cost function with integral action, which enhances disturbance rejection capabilities and ensures zero steady-state error without requiring additional compensators or observers. In contrast, most existing robust strategies either rely on complex auxiliary observers or introduce additional computational overhead to mitigate steady-state errors and disturbances.

## 6.3. Comparative analysis with DTC-based control strategies

To clearly illustrate the advantages of the proposed RPDTC method over existing DTC-based strategies, a detailed comparative analysis is presented in Table 5. From the data, it is evident that the proposed RPDTC demonstrates a substantial improvement over conventional methods, which exhibit significantly higher torque ripple levels: 3.8 N·m for classical DTC, 2.1 N·m for GA-DTC, and 1.5 N·m for NNC-DTC. Furthermore, the flux ripple is minimized to 0.0101 Wb, outperforming the majority of existing approaches. In terms of THD, the RPDTC achieves an impressively low value, indicating superior current quality
<table><caption>**Table 6**
Performance comparison with other MPC-based control strategies for multiphase machines.</caption><thead><tr><td>Ref.</td><td>Machine type</td><td>Control strategy</td><td>Rejection time (s)</td><td>Overshoot (rad/s)</td><td>THD (%)</td><td>Torque ripple (Nm)</td></tr></thead><tbody><tr><td>[53]</td><td>Five-Phase IM</td><td>DMP VC</td><td>0.04</td><td>1.57</td><td>5.15</td><td>0.51</td></tr><tr><td>[52]</td><td>Five-phase PMSM</td><td>DB MPCC</td><td>0.05</td><td>2</td><td>2.17</td><td>0.21</td></tr><tr><td>[52]</td><td>Five-phase PMSM</td><td>Hysteresis MPTC</td><td>0.09</td><td>3.5</td><td>3.49</td><td>0.25</td></tr><tr><td>[54]</td><td>OW- PMSM</td><td>3D-MPCC</td><td>0.15</td><td>2.09</td><td>5.3</td><td>2.36</td></tr><tr><td>Proposed</td><td>Dual star IM</td><td>RPDTC</td><td>0.01</td><td>0.15</td><td>2.72</td><td>0.61</td></tr></tbody></table>

<table><caption>**Table 7**
Summary of the main advantages and limitations of the proposed RPDTC strategy.</caption><thead><tr><td>Aspect</td><td>Advantages</td><td>Limitations</td></tr></thead><tbody><tr><td>Optimization efficiency</td><td>Rapid convergence and effective avoidance of local minima thanks to the advanced predictive cost function.</td><td>-</td></tr><tr><td>Torque and flux ripple</td><td>Significant reduction of torque and flux ripples.</td><td>-</td></tr><tr><td>Dynamic response</td><td>Minimal overshoot and fast transient behavior enable precise torque and speed tracking.</td><td>-</td></tr><tr><td>Robustness</td><td>Strong robustness to external disturbances and parameter variations. Mitigates dependence on accurate parameter identification, thus enhancing reliability.</td><td>-</td></tr><tr><td>Computational complexity</td><td>Lower overall computational burden compared to conventional MPC approaches, enabling feasible real-time implementation.</td><td>Requires high sampling rates and powerful processors, which may limit use in cost-sensitive applications.</td></tr><tr><td>Cost function tuning</td><td>-</td><td>Weighting factors require careful tuning and experimental validation to achieve the desired trade-off between performance, robustness, and losses.</td></tr></tbody></table>

and smoother torque response. The results confirm that the RPDTC provides clear performance advantages, mainly due to the integration of the proposed optimization algorithm.

## 6.4. Comparison with other MPC-based control strategies for multiphase machines

Table 6 shows a detailed comparison between the proposed RPDTC strategy and several recent MPC-based control methods used for multiphase machines, including both induction and permanent magnet synchronous motors. As can be seen, the proposed RPDTC achieves the lowest rejection time (0.01 s), demonstrating its remarkable dynamic response and its ability to quickly reject disturbances. Additionally, it provides a minimal overshoot (0.15 rad/s), which is significantly smaller than the values reported for Deadbeat Model Predictive Voltage Control (DMPVC), Hysteresis MPTC, and 3D-Model Predictive Current Control (3D-MPCC) approaches.

Regarding steady-state performance, the RPDTC also offers a lower THD on the stator currents (2.72 %), comparable to or even better than the results achieved by other methods such as those in [52,53]. Moreover, the torque ripple obtained with the RPDTC (0.61 Nm) remains highly competitive and ensures a smoother electromagnetic torque profile. The proposed RPDTC strategy surpasses existing MPC-based control approaches for multiphase machines. These encouraging outcomes confirm its robustness and reliability, positioning it as a highly promising solution for advanced industrial applications requiring high-performance multiphase drives.

## 6.5. Discussion on advantages, limitations, and industrial feasibility

The proposed RPDTC strategy introduces several significant advantages over conventional DTC and other advanced control methods. It offers high optimization efficiency, rapid dynamic response, and strong robustness to disturbances and parameter variations. Moreover, it significantly reduces sensitivity to modeling errors and parameter mismatches. Unlike conventional approaches whose performance strongly depends on an accurate machine model and precise parameter identification, such as stator resistance, inductance, and complex coupling effects in multiphase machines, the proposed RPDTC strategy effectively mitigates these dependencies. It ensures robust control performance and high accuracy even in the presence of parameter mismatches, thus significantly improving system reliability and reducing sensitivity to

modeling uncertainties. However, despite its promising capabilities, some practical aspects remain to be addressed, which represent avenues for future research and optimization rather than critical limitations. Table 7 summarizes the main advantages and limitations of the proposed approach.

Regarding generalization, the proposed RPDTC strategy is not limited to dual-star induction machines. It can be extended to other multiphase machines, including five-phase and even permanent magnet synchronous machines, provided that the predictive model is appropriately adapted. The core control design and cost function formulation remain fundamentally applicable. Finally, extensive simulation results confirm that the proposed strategy exhibits excellent dynamic behavior under demanding operating scenarios, such as sudden load changes, reversals, and heavy-load startups. These features confirm its practical feasibility and strong potential for advanced multiphase drive applications.

## 7. Conclusion

This study introduces a novel control strategy for dual-star induction motors, grounded in a predictive direct torque control algorithm. By replacing conventional hysteresis regulators and switching tables with a prediction block driven by an optimization algorithm, the suggested controller enhances the efficiency of the traditional DTC method. To mitigate the effects of time delays, a two-step forward prediction strategy is integrated. Additionally, the speed regulation system is optimized through the introduction of a robust finite-horizon cost function, eliminating the requirement for directly measuring or observing external perturbations. A comparative analysis between classical strategies and the proposed RPDTC revealed significant improvements in system performance, emphasizing the potential of our approach for DSIM, where variations in system parameters and external disturbances are unavoidable. The proposed RPDTC accomplishes several key objectives: it ensures precise reference tracking, guarantees robustness against parametric uncertainties, and effectively mitigates external disturbances. Compared to conventional DTC, torque and flux ripples are reduced by 83.95 % and 74.75 %, respectively. The real-time simulation results, conducted on the OPAL-RT platform, validate the efficiency and robustness of the control scheme suggested. Future work will focus on optimizing the predictive control cost function by embedding intelligent algorithms
such as nature-inspired metaheuristic methods like ant colony optimization. The incorporation of ACO can be utilized to fine-tune control parameters and optimize the voltage vector selection process, leading to minimized losses and improved transient performance. The combination of predictive control with intelligent algorithms-based optimization strategies opens up new opportunities for self-adaptive and robust control frameworks. This approach can also be extended to the development of intelligent predictive direct control systems for multilevel inverters, which are increasingly employed in high-power applications utilizing multiphase induction machines.

### CRediT authorship contribution statement

**Amel Kasri:** Writing – original draft, Visualization, Validation, Methodology, Formal analysis, Conceptualization. **Kamel Ouari:** Writing – review & editing, Visualization, Validation, Methodology, Formal analysis. **Youcef Belkhier:** Writing – review & editing, Visualization, Validation, Methodology, Formal analysis. **Djamel Ziane:** Writing – review & editing, Visualization, Validation, Methodology, Formal analysis. **Mohamed Fouda Benkhoris:** Writing – review & editing, Visualization, Validation, Methodology, Formal analysis. **Mohamed Benbouzid:** Writing – review & editing, Visualization, Validation, Methodology, Formal analysis.

### Funding

This research received no external funding.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Data availability statement

The author declares that there are no data available for this research work.

### References

[1] Janisch G, Kugi A, Kemmetmüller W. A high-performance model predictive torque control concept for induction machines for electric vehicle applications. Control Eng Pract 2024;153:106128.

[2] Khoshhava MA, Zarchi HA, Markadeh GA. Optimal design of a dual stator winding induction motor with minimum rate reduction level. IEEE Trans Ind Electron 2020;68(2):1016–24.

[3] Nurettin A, Inanç N. Sensorless vector control for induction motor drive at very low and zero speeds based on an adaptive-gain super-twisting sliding mode observer. IEEE J Emerg Sel Top Power Electron 2023;11(4):4332–9.

[4] Paramasivam A, Kalaiyarasi D, Senthil Raja M, Pavaiyarkarasi R. An efficient coa approach-based open-end winding induction motor with direct torque control for minimize the power loss. Electr Eng 2024:1–15.

[5] Ayana T, Kolodziejek P, Morawiec M, Wogi L. Loss minimization based sensorless control of high-speed induction motor considering core loss. IEEE Access 2024;12:116341–53.

[6] Khoshhava MA, Zarchi HA, Markadeh GA, Mosaddegh H, Al-Haddad K. A novel highly efficient torque-sharing algorithm for dual stator winding induction machines for various speed regions. IEEE Trans Ind Electron 2023;71(6):5564–75.

[7] Mehedi F, Yahdou A, Djilali AB, Benbouhenni H. Direct torque fuzzy controlled drive for multi-phase ipsm based on svm technique. J Eur Syst Automat 2020;53(2):259–66.

[8] Belalia K, Mostefa A, Boulouira HM, Draou A, Denai M. Direct torque control of a dual star induction generator based on a modified space vector pwm under fault conditions. ISA Trans 2024;155:237–60.

[9] Milles A, Merabet E, Benbouhenni H, Debdouche N, Colak I. Robust control technique for wind turbine system with interval type-2 fuzzy strategy on a dual star induction generator. Energy Rep 2024;11:2715–36.

[10] Nesri M, Nounou K, Sifelislam G, Benkhoris MF, Azeddine H. Hybrid flatness-based control of dual star induction machine drive system for more electrical aircraft. Power Electron Drives 2024;9(1):50–62.

[11] Sowmiya M, Thilagar SH. Vector-controlled dual stator multiphase induction motor drive for energy-efficient operation of electric vehicles. IETE J Res 2023;69(6):3693–710.

[12] Abdelwanis MI. Optimizing the performance of six-phase induction motor-powered electric vehicles with fuzzy-pid and dtc. Neural Comput Appl 2025;37(16):9721–34.

[13] Reghioui H, Belhamdi S, Abdelkarim A, Lallouani H. Enhancement of space vector modulation based-direct torque control using fuzzy pi controller for doubly star induction motor. Adv Model Anal C 2019;74(2):27–34.

[14] Tarusan SAA, Jidin A, Jamil MLM. The optimization of torque ripple reduction by using dtc-multilevel inverter. ISA Trans 2022;121:365–79.

[15] Aissa O, Reffas A, Krama A, Benkercha R, Talhaoui H, Abu-Rub H. Advanced direct torque control based on neural tree controllers for induction motor drives. ISA Trans 2024;148:92–104.

[16] Fahassa C, Zahraoui Y, Akherraz M, Kharrich M, Elattar EE, Kamel S. Induction motor dtc performance improvement by inserting fuzzy logic controllers and twelve-sector neural network switching table. Mathematics 2022;10(9):1357.

[17] Mahfoud S, Derouich A, El Ouanjli N, Mossa MA, Motahhir S, El Mahfoud M, et al. Comparative study between cost functions of genetic algorithm used in direct torque control of a doubly fed induction motor. Appl Sci 2022;12(17): 8717.

[18] Kasri A, Ouari K, Belkhier Y, Bajaj M, Zaitsev I. Optimizing electric vehicle powertrains peak performance with robust predictive direct torque control of induction motors: a practical approach and experimental validation. Sci Rep 2024;14(1):14977.

[19] Mahfoud S, Derouich A, El Ouanjli N, Taoussi M, Mahfoud ME. Improved dtc of the pid controller by using genetic algorithm of a doubly fed induction motor. In: International conference on digital technologies and applications; Springer; 2021. p. 1687–98.

[20] Boukhalfa G, Belkacem S, Chikhi A, Bouhentala M. Fuzzy-second order sliding mode control optimized by genetic algorithm applied in direct torque control of dual star induction motor. J Cent South Univ 2022;29(12):3974–85.

[21] Mahfoud S, Derouich A, El Ouanjli N, Quynh NV, Mossa MA. A new hybrid ant colony optimization based pid of the direct torque control for a doubly fed induction motor. World Elect Veh J 2022;13(5):78.

[22] El Daoudi S., Lazrak L., El Ouanjli N., Ait Lafkih M., Improved dtc-spwm strategy of induction motor by using five-level pod-pwm inverter and mras sf estimator. Int J Dyn Control 2021;9:448–62.

[23] Abu-Seif MA, Ahmed M, Metwly MY, Abdel-Khalik AS, Hamad MS, Ahmed S, et al. Data-driven-based vector space decomposition modeling of multiphase induction machines. IEEE Trans Energy Convers 2023;38(2):2061–74.

[24] Holakooie MH, Ojaghi M, Taheri A. Modified dtc of a six-phase induction motor with a second-order sliding-mode mras-based speed estimator. IEEE Trans Power Electron 2018;34(1):600–11.

[25] Nahin NI, Hosain MK, Biswas SP, Biswas M, Mondal S, Ferdousi Z. An improved direct torque control scheme for pwm inverter based solar pv array fed induction motor drives. In: 2022 12th International Conference on Electrical and Computer Engineering (ICECE); IEEE; 2022. p. 304–7.

[26] Latrech FZ, Rhouma AB, Khedher A. Fpga implementation of a robust dtc-svm based sliding mode flux observer for a double star induction motor: hardware in the loop validation. Microelectron Reliab 2023;150:115118.

[27] Köhler J, Müller MA, Allgöwer F. Analysis and design of model predictive control frameworks for dynamic operation of an overview. Annu Rev Control 2024;57:100929.

[28] Kasri A, Ouari K. Robust intelligent nonlinear predictive control based on artificial neural network for optimizing pmsm drive performance. Period Polytech Electr Eng Comput Sci 2024;68(4):356–66.

[29] Yang Q, Karamanakos P, Liegmann E, Tian W, Geyer T, Kennel R, et al. A fixed switching frequency direct model predictive control for neutral-point-clamped three-level inverters with induction machines. IEEE Trans Power Electron 2023;38(11):13703–16.

[30] Shi H, Zuo L, Wang S, Yuan Y, Su C, Li P. Robust predictive fault-tolerant switching control for discrete linear systems with actuator random failures. Comput Chem Eng 2024;181:108554.

[31] Bighash EZ, Sadeghzadeh SM, Ebrahizadeh E, Blaabjerg F. Robust mpc-based current controller against grid impedance variations for single-phase grid-connected inverters. ISA Trans 2019;84:154–63.

[32] Köhler J, Kötting P, Soloperto R, Allgöwer F, Müller MA. A robust adaptive model predictive control framework for nonlinear uncertain systems. Int J Robust Nonlinear Control 2021;31(18):8725–49.

[33] Kiani S, Salmanpour A, Hamzeh M, Kebriaei H. Learning robust model predictive control for voltage control of islanded microgrid. IEEE Trans Autom Sci Eng 2024;22:3021–32.

[34] Oliani I, Figueiredo R, Lourenço LFN, Pelizari A, Sguarezi Filho AJ. Robust predictive current control using discrete-time integral action for induction motors. IEEE J Emerg Sel Top Power Electron 2023;11(6):5766–73.

[35] Oumar A, Chakib R, Labbadi M, Cherkaoui M. Robust nonlinear controller of the speed for double star induction machine in the presence of a sensor fault. Int J Intell Eng Syst 2020;13(3):124–33.

[36] Gulbudak O, Gokdag M, Komurcugil H. Lyapunov-based model predictive control of dual-induction motors fed by a nine-switch inverter to improve the closed-loop stability. Int J Electr Power Energy Syst 2023;146:108718.

[37] Liu Y, Cheng S, Ning B, Li Y. Robust model predictive control with simplified repetitive control for electrical machine drives. IEEE Trans Power Electron 2018;34(5):4524–35.

[38] Zhang Y, Jiang T, Jiao J. Model-free predictive current control of a dfig using an ultra-local model for grid synchronization and power regulation. IEEE Trans Energy Convers 2020;35(4):2269–80.

[39] Terfia E-S, Mendaci S, Rezgui SE, Gasmi H, Kantas W. Optimal third-order sliding mode controller for dual star induction motor based on grey wolf optimization algorithm. Heliyon 2024;10(12).
[40] Li G, Hu J, Li Y, Zhu J. An improved model predictive direct torque control strategy for reducing harmonic currents and torque ripples of five-phase permanent magnet synchronous motors. IEEE Trans Ind Electron 2018;66(8):5820-9.

[41] Ouari K, Rekioua T, Ouhrouche M. Real time simulation of nonlinear generalized predictive control for wind energy conversion system with nonlinear observer. ISA Trans 2014;53(1):76-84.

[42] Köhler J, Soloperto R, Müller MA, Allgöwer F. A computationally efficient robust model predictive control framework for uncertain nonlinear systems. IEEE Trans Autom Control 2020;66(2):794-801.

[43] Chen W-H, Ballance DJ, Gawthrop PJ. Optimal control of nonlinear systems: a predictive control approach. Automatica 2003;39(4):633-41.

[44] Sun Z, Deng Y, Wang J, Li H, Cao H. Improved cascaded model-free predictive speed control for pmsm speed ripple minimization based on ultra-local model. ISA Trans 2023;143:666-77.

[45] Mahfoud S, El Ouanjli N, Derouich A, El Idrissi A, Chetouani E, Loulijat A, et al. An advanced direct torque control for doubly fed induction motor using evolutionary computational techniques. Sci Rep 2025;15(1):22719.

[46] Zellouma D, Bekakra Y, Benbouhenni H. Robust synergetic-sliding mode-based backstepping control of induction motor with mras technique. Energy Rep 2023;10:3665-80.

[47] Mahfoud S, Derouich A, El Ouanjli N, el Mahfoud M. Enhancement of the direct torque control by using artificial neuron network for a doubly fed induction motor. Intell Syst Appl 2022;13:200060.

[48] El Ouanjli N, Mahfoud S, Al-Sumaiti AS, El Daoudi S, Derouich A, El Mahfoud M, et al. Improved twelve sectors dtc strategy of induction motor drive using backstepping speed controller and p-mras stator resistance identification-design and validation. Alex Eng J 2023;80:358-71.

[49] Gdaim S, Mtibaa A, Mimouni MF. Artificial neural network-based dtc of an induction machine with experimental implementation on fpga. Eng Appl Artif Intell 2023;121:105972.

[50] El Ouanjli N, Motahhir S, Derouich A, El Ghzizal A, Chebabhi A, Taoussi M. Improved dtc strategy of doubly fed induction motor using fuzzy logic controller. Energy Rep 2019;5:271-9.

[51] Sahu A, Mohanty KB, Mishra RN. Development and experimental realization of an adaptive neural-based discrete model predictive direct torque and flux controller for induction motor drive. Appl Soft Comput 2021;108:107418.

[52] Saeed MSR, Song W, Yu B, Wu X. Low-complexity deadbeat model predictive current control with duty ratio for five-phase pmsm drives. IEEE Trans Power Electron 2020;35(11):12085-99.

[53] Mossa MA, Vu Quynh N, Echeikh H, Do TD. Deadbeat-based model predictive voltage control for a sensorless five-phase induction motor drive. Math Probl Eng 2020;2020(1):4164526.

[54] Saeed MSR, Song W, Yu B, Xie Z, Feng X. Low-complexity deadbeat model predictive current control for open-winding pmsm drive with zero-sequence current suppression. IEEE Trans Transp Electr 2021;7(4):2671-82.