Enhanced Direct Torque Control Prediction for Torque Ripple Reduction in Switched Reluctance Motors

Abstract

In this study, a novel direct torque control (DTC) strategy is proposed to mitigate the torque ripple issue inherent in switched reluctance motors (SRMs), which is caused by the double salient pole configuration and the pulse power supply mode. The strategy is based on the prediction and optimization of a long-time-domain model. Central to this method is the development of a multi-step predictive optimization framework. By incorporating hysteresis control, the conventional approach of minimizing instantaneous error in predictive control is shifted towards minimizing tracking error over an extended time frame. A dual-objective evaluation function is also introduced, which simultaneously optimizes both torque smoothness and switching frequency, ensuring their collaborative enhancement. To validate the proposed method, a 6/4-pole SRM simulation model was implemented using MATLAB/Simulink 2024B, and comparisons were made with traditional methods. The results demonstrate that this strategy significantly reduces torque pulsation and lowers the system’s switching frequency, even under varying operational conditions such as different rotational speeds and sudden load variations. Consequently, this approach not only guarantees improved dynamic performance but also enhances the motor’s efficiency and stability.

1. Introduction

With the advancement of the “dual carbon” strategy, traditional fossil fuel consumption has imposed higher demands on energy transition and environmental protection. Electrification is accelerating in industrial and transportation sectors, with new energy vehicles emerging as a key strategic industry. As a core component, the efficiency, cost, and low-speed torque performance of drive motors directly impact the competitiveness of the entire vehicle. Compared to other motor types, the switch reluctance motor (SRM) offers distinct advantages. Its rotor consists solely of laminated silicon steel sheets, eliminating the need for windings or permanent magnets. This design results in a simpler structure, lower cost, and higher reliability. Additionally, the SRM delivers a high torque-to-current ratio, operates across a wide speed range, and features excellent fault tolerance and ease of maintenance. Consequently, it demonstrates strong competitiveness in new energy vehicles, flywheel energy storage, aerospace, and other fields [1,2]. Particularly in the context of electric vehicles, recent research has actively explored torque vector control strategies to enhance vehicle stability and energy efficiency, such as the energy-oriented torque vector control framework proposed in [3].

Significant research efforts have been devoted to addressing the challenge of high torque pulsation in switched reluctance motor (SRM) drive systems. Existing solutions can be broadly categorized into three main approaches: the first involves optimizing the structural design of the SRM itself [4,5]; the second focuses on rapidly constructing accurate nonlinear mathematical models for the motor [6,7,8]; and the third centers on improving the control strategies applied to the drive system.

Presently, torque control methodologies for motors are broadly categorized as either direct or indirect approaches. Within these categories, both the model predictive control (MPC) algorithm and the direct torque control (DTC) strategy have shown significant effectiveness in mitigating torque ripple. However, each method faces specific limitations: MPC typically involves substantial computational demands and difficulty in parameter tuning, whereas DTC may struggle with negative torque generation and relatively lower operational efficiency. Consequently, substantial research has been conducted to overcome these drawbacks. For instance, reference [9] provides a comparative analysis between direct instantaneous torque control (DITC) and direct torque and flux control (DTFC), highlighting that DTFC offers benefits such as reduced vibration and noise, along with simpler implementation for sensorless operations, while DITC demonstrates superior torque regulation and higher efficiency. Meanwhile, study [10] implements an online adaptive phase excitation predictive technique utilizing PWM to enable the high-dynamic four-quadrant control of SRMs. Another hybrid solution in the literature [11] integrates DTC with model predictive flux control (MPFC), yielding improved performance in minimizing high torque pulsation. For instance, ref. [12] applied neural network-based optimization to performance-based logistics systems, demonstrating the potential of intelligent algorithms in complex system optimization, which provides a useful reference for parameter optimization and state estimation in motor drive control. Through simulation and experiments, ref. [13] demonstrates that, while DTC provides a superior flux control, MPDTC (combined with TSF) offers better dynamic performance and torque ripple suppression.

In summary, the existing research has achieved certain results in improving dynamic performance and reducing torque ripple. However, common issues include high algorithm complexity, insufficient real-time capability, limited robustness, and significant discrepancies between experimental and simulation results. These shortcomings become increasingly apparent in practical applications, particularly in highly nonlinear systems such as SBRMs. Existing research indicates that SRM control methods are evolving from traditional SMC toward intelligent control and self-disturbance-tolerant control, and further toward the integration of model predictive control (MPC) and direct torque control (DTC). MPC possesses predictive and optimization capabilities, while DTC offers direct and rapid advantages. The combined model predictive direct torque control (MPDTC) strategy enhances dynamic performance by directly tracking torque and flux errors through predictive modeling. However, traditional MPDTC still exhibits limitations in error suppression. To address this, this paper introduces hysteresis control into MPDTC to further reduce torque ripple and improve stability. This approach constrains electromagnetic torque within the hysteresis loop through bandwidth selection, potentially maintaining identical switching vectors across multiple cycles. This achieves switching frequencies below the control frequency and improves steady-state speed performance. The method is validated via MATLAB/Simulink modeling to enhance the overall SRM control performance and promote its application in new energy vehicles and high-performance domains.

2. Mathematical Model of Switched Reluctance Motor

Based on the fundamentals of electromechanical energy conversion, a switched reluctance motor (SRM) can be modeled as a dual-port network, featuring electrical and mechanical terminal pairs interconnected by a coupled magnetic field. Key parameters of the mechanical domain encompass electromagnetic torque (Te), load torque (TL), and the viscous friction coefficient (D), as well as the combined inertia of the rotor and load (J). Correspondingly, the electrical domain for each phase winding k is characterized by its applied voltage (Uk), resistance (Rk), current (ik), and back electromotive force (ek). The magnetic flux linkage (ψk) of the k-th winding, which is dependent on the rotor’s angular position (θ), represents the key quantity within the coupled field. The relationship among these variables is defined by the law of electromagnetic induction:

$$e_k=-{\displaystyle \frac{d{\psi }_k}{dt}}$$

(1)

The governing equation for the k-th phase winding’s voltage balance is derived directly from fundamental circuit theory.

$$U_{\mathrm{k}}=R_k{i}_k-e_k=R_k{i}_k+{\displaystyle \frac{d{\psi }_k}{dt}}$$

(2)

The magnetic flux linkage in a phase winding is a function of both the phase current and the rotor displacement angle. This relationship can be represented as the product of the winding’s inductance and the current, namely

$${\psi }_k\left(\theta ,i_k\right)=L_k\left(\theta ,i_k\right)i_k$$

(3)

The phase inductance’s dependence on current arises from magnetic circuit saturation, while its variation with rotor position is a fundamental characteristic of SRMs and a prerequisite for torque generation. Substituting Equation (3) into (2) yields

$$\begin{array}{ccc}\hfill U_{\mathrm{k}}& =& R_k{i}_k+{\displaystyle \frac{d{\psi }_k}{d{i}_k}}{\displaystyle \frac{d{i}_k}{dt}}+{\displaystyle \frac{\partial {\psi }_k}{\partial \theta }}{\displaystyle \frac{d\theta }{dt}}\hfill \\ & =& R_k{i}_k+\left(L_k+i_k{\displaystyle \frac{\partial L_k}{\partial i_k}}\right){\displaystyle \frac{d{i}_k}{dt}}+i_k{\displaystyle \frac{\partial L_k}{\partial \theta }}{\displaystyle \frac{d\theta }{dt}}\hfill \end{array}$$

(4)

The physical meaning of each component on the equation’s right-hand side can be interpreted as follows: the initial term corresponds to the resistive voltage drop within the k-phase circuit. The subsequent term represents the electromotive force induced by variations in magnetic flux linkage due to changing current, which is termed the transformer EMF. The final term indicates the electromotive force induced by flux linkage alterations resulting from rotor movement, known as the motional EMF; this component is directly involved in electromechanical energy conversion.

Following mechanical principles, the rotational motion of the SRM’s rotor under the influence of both electromagnetic and load torques can be described by the subsequent equation:

$$T_e=J{\displaystyle \frac{d^2\theta }{dt}}+D{\displaystyle \frac{d\theta }{dt}}+T_L$$

(5)

Within switched reluctance drive systems, torque regulation constitutes the fundamental objective of speed control strategies. This emphasis is rooted in electromechanical energy conversion theory, where the SRM’s coupled magnetic field gives rise to two crucial interaction terms: an induced electromotive force at the electrical port and a corresponding electromagnetic torque at the mechanical port. These terms are critical for comprehending the motor’s operational behavior. Furthermore, as per the generalized motor theory, precise electromagnetic torque computation is essential not only for assessing dynamic performance but also for the integrated design optimization of the motor, its power converter, and the associated controller [14].

As shown in Figure 1, since the mutual inductance effect between phase windings is neglected, the electromagnetic torque analysis of SRM can be simplified to a single-phase independent study. The mathematical expression representing the mechanical energy produced by one phase winding over a complete operating cycle is as follows. The instantaneous electromagnetic torque value at any rotor position is derived based on the principle of virtual displacement, and the specific expression is

$$T_e={\displaystyle \frac{\partial W^{\prime }}{\partial \theta }}{|}_{i=const}=-{\displaystyle \frac{\partial W}{\partial \theta }}{|}_{\psi =const}$$

(6)

The average output torque of m-phase SRM can be obtained through an integral operation:

$$T={\displaystyle \frac{m{N}_r}{2\pi }}{\displaystyle {\int }_0^{2\pi /N_r}{T}_e}\left(\theta ,i\left(\theta \right)\right)d\theta$$

(7)

Here, m signifies the phase number; Nr indicates the number of teeth on the rotor.

3. Model Predictive Control of Switched Reluctance Motors

Owing to its extended magnetic path winding configuration, the mutual inductance among different phases in an SRM can be considered negligible. Figure 2 illustrates the relationship between the magnetic flux linkage and the current for a single phase under this condition.

The phase inductance is at its minimum. The slope of this linear characteristic is defined as the unsaturated inductance L_q. In the aligned position, consider magnetic saturation. Take the linear inductance before saturation L_d and saturation inductance L_dsat as function parameters. Therefore, the following equation describes the flux–current relationship for both the aligned and unaligned positions:

$$\left\{\begin{array}{l}{\psi }_p\left(i\right)=L_{dsat}i+A(1-e^{-Bi})\hfill \\ {\psi }_p\left(i\right)=L_{q}i\hfill \end{array}\right.$$

(8)

In formulas $A ={\Psi }_m-L_{dsat}{I}_m$ and $B={\displaystyle \frac{(L_d-L_{dsat})}{({\psi }_m-L_{dsat}{I}_m)}}$, ${\psi }_m,I_m$ are the magnetic chain and phase currents of the corresponding windings, respectively.

The angle correction expression is

$$f(\theta )=\left\{\begin{array}{l}{\displaystyle \frac{128{\theta }^3}{{\pi }^3}}-{\displaystyle \frac{48{\theta }^2}{{\pi }^2}}+1, \theta \in [0,{\displaystyle \frac{\pi }{4}}]\hfill \\ f({\displaystyle \frac{\pi }{2}}-\theta ) , \theta \in [{\displaystyle \frac{\pi }{4}},{\displaystyle \frac{\pi }{2}}]\hfill \end{array}\right.$$

(9)

The expression for the magnetization characteristics of the switched reluctance motor is given below:

$${\psi }_p=L_{q}i+[L_{\mathit{dsat}}i+A(1-{\mathrm{e}}^{-Bi})-L_{q}i]f(\theta )$$

(10)

As defined in Equation (10), the electromagnetic torque is derived from the partial derivative of the magnetic co-energy with respect to the rotor angle, leading to the following expression:

$$T_{e,p}=[{\displaystyle \frac{L_{\mathit{dsat}}-L_q}{2}}{i}_p^2+A{i}_p-{\displaystyle \frac{A}{B}}(1-{\mathrm{e}}^{-Bi})]f^{\prime }(\theta )$$

(11)

Euler’s method is employed to discretize the switched reluctance motor’s mathematical model across a single sampling interval:

$$\left\{\begin{array}{l}{i}_{k+1}=i_k+{\displaystyle \frac{\Delta T}{\partial \psi /\partial i_k}}[U_{k+1}-R{i}_k-{\displaystyle \frac{\partial \psi }{\partial {\theta }_k}}\omega ]\hfill \\ {\theta }_{k+1}={\theta }_k+\omega \Delta T\hfill \\ T_{k+1}=[{\displaystyle \frac{L_{\mathit{dsat}}-L_q}{2}}{i}_{k+1}^2+A{i}_{k+1}-{\displaystyle \frac{A}{B}}(1-{\mathrm{e}}^{-B{i}_{k+1}})]f^{\prime }({\theta }_{k+1})\hfill \end{array}\right.$$

(12)

According to Equation (12), it is possible to predict the motor state equation obtained from sampling at moment k and to see the input voltage corresponding to the switching vector selected at moment k + 1, utilizing the predicted current and rotor position for that same time step.

To establish the prediction model, the forward Euler method is employed to discretize the state equations of the motor. The choice of Euler discretization is primarily motivated by two considerations: first, although the electromagnetic dynamics of switched reluctance motors are relatively fast, within the selected control period (T_s; = XX μs), the Euler method approximates the state evolution with sufficient accuracy while maintaining low computational complexity, facilitating real-time implementation; second, the discretized prediction model yields a concise explicit form that facilitates the derivation of the recursive relationships between torque and flux predictions, providing a unified formulation for subsequent multi-step rolling optimization. Based on this discretization, the prediction horizon is defined as N steps. Its relationship with the “long-time-domain” concept introduced in this paper is as follows: conventional predictive torque control typically adopts a single-step or short prediction horizon (N ≤ 3), whereas the proposed approach integrates hysteresis rules with predictive control, enabling the controller to evaluate the cumulative trend of torque errors over N = 5–8 steps. By extending the prediction horizon, the evaluation function can anticipate the evolution of torque deviations under different switching actions, thereby adjusting switching signals proactively under the constraints of the hysteresis bounds. This avoids the myopic behavior commonly associated with short-horizon methods, which often leads to unnecessary switching actions. Within the model, the state prediction terms (flux linkage and current) are used to recursively compute the electromagnetic torque at future time steps. The torque prediction term directly participates in the evaluation function, while the output of the hysteresis comparator serves as the basis for dynamic weight adjustment, allowing the optimization objective to adaptively shift between torque tracking and switching loss reduction under different error intervals. The coordinated interaction of these components constitutes the complete long-time-domain predictive control logic, ensuring that the controller suppresses torque ripple while maintaining system efficiency.

4. Construct MPDTC Speed Controller Design Using Hysteresis Loop Rule

4.1. Power Converters and Sectorization

The power converter adopts a three-phase asymmetric half-bridge driving circuit topology, as shown in Figure 3, where each phase bridge arm consists of two IGBTs and two consecutive diodes. We define the upper and lower bridge arms conduct simultaneously as state “1”: at this point, the motor is in the forward excitation state. The motor operates in a freewheeling state when the upper switch is off and the lower switch is on, a condition denoted as state “0”. Conversely, a reverse demagnetization mode occurs when both the upper and lower switches are turned off simultaneously, which is defined as state “−1”. The switching combinations of the three bridge arms can be represented by a state vector.

As analyzed in the previous section, factors such as magnetic circuit saturation and the eddy current effect lead to the generation of torque pulsations in switched reluctance motors. In addition to this, the selection and division of sectors is also extremely critical for model predictive torque control, and appropriate sector division will reduce the torque pulsation as well as the computational amount of model predictive control. However, when the sector selection is inappropriate, such as the moment of entering a separate phase conduction zone from the phase change zone, the previous phase will provide a negative torque if it is still in the conduction state or the phase current fails to be reduced to less than 0.5 A to 1 A. Given that the total torque in a switched reluctance motor results from the summation of three-phase torques, improper sector definition augments torque ripple and adversely affects control precision. To mitigate this issue, a fuzzy logic-based adjustable sector strategy is introduced. This method dynamically modifies the sector width using current measurements at designated positions, thereby diminishing pulsations caused by negative torque and enhancing overall performance [15].

This version reorganizes the logic flow to first present the problem of computational complexity, then introduce the sector solution, and finally explain the torque control mechanism, using formal academic phrasing.

Figure 3 indicates that the power converter for each phase possesses three distinct switching states, leading to a total of 27 (3³) possible switching vector combinations for a three-phase SRM system. Performing predictions for all 27 vectors within one sampling interval would impose a significant computational burden.

To address this, a six-sector division scheme is implemented, segmenting one electrical cycle of the rotor into six distinct regions. The reference position (0°) is defined when the stator aligns with the center of the rotor notch for phase A, as detailed in

Table 1. Set of switch action vectors corresponding to different sectors.

Sector	Determination Criteria	Candidate Switch Vector
N1	${\theta }_{on}\le \theta <{\theta }_{on}+{\theta }_{ov}.$	[1,−1,1], [0,−1,0]
N2	${\theta }_{on}+{\theta }_{ov}\le \theta <{\theta }_{on}+{\theta }_{\alpha }$	[1,−1,−1], [0,−1,−1]
N3	${\theta }_{on}+{\theta }_{\alpha }\le \theta <{\theta }_{off}$	[1,1,−1], [0,0,−1]
N4	${\theta }_{off}\le \theta <{\theta }_{on}+2{\theta }_{\alpha }$	[−1,1,−1], [−1,0,−1]
N5	${\theta }_{on}+2{\theta }_{\alpha }\le \theta <{\theta }_{off}+{\theta }_{\alpha }$ ${\theta }_{off}+{\theta }_{\alpha }\le \theta <{\theta }_e$	[−1,1,1], [−1,0,0]
N6	$0\le \theta <{\theta }_{on}$	[−1,−1,1], [−1,−1,0]

. Within this framework, torque reduction is achieved via a “soft-chopping” technique, transitioning the bridge arm state from “−1” to “0”. Employing a “hard-chopping” approach by directly selecting state “−1” causes a rapid decay of current in the previously conducting phase. This leads to a sharp drop in torque contribution from that phase before the subsequent phase becomes active, creating a torque deficit. The resultant total torque insufficiency causes noticeable oscillations and pulsations.

Consequently, to minimize computational load, a unique subset of switching vectors is pre-selected for application within each specific sector, significantly reducing the number of candidates evaluated per cycle (see Table 1).

Traditional model predictive direct torque control (MPDTC) schemes primarily aim to minimize errors in torque and flux linkage tracking when selecting voltage vectors. However, this approach frequently results in elevated switching frequencies and associated power losses. To achieve a better balance between torque ripple suppression and switching loss reduction, the hysteresis band principle commonly found in direct torque control is incorporated into the predictive control framework. As illustrated in Figure 4, this integration facilitates more a flexible adjustment of the controller’s tracking performance versus switching frequency trade-off, thereby enhancing the system’s adaptability for various operational requirements.

4.2. Introduction of Hysteresis Loop Rules in MPDTC

Following the incorporation of hysteresis regulation, the bandwidth is defined as ±5% of a given torque reference. At sampling instant k, the phase current and rotor position are sampled and fed into the specified equation to compute the instantaneous torque T_e(k). This same model is then utilized to predict the state at the next interval, k + 1, providing estimates for the rotor position, phase current, and consequently the torque Te(k + 1).

The Switch Action Vector Set can be determined from the position of the rotor at the k + 1 moment, with different switching action vectors corresponding to different torque predictions. Take the torque T_e(k) at moment k and the torque T_e(k + 1) at moment k + 1 as a straight line at two points, and, based on the slopes, use linear extrapolation to calculate the length that extends over the range of alternatives when it intersects with the upper limit and the lower limit, as shown in Figure 5. Optimization Mechanisms Introducing Hysteresis Loop Rules.

The torque extension length within the hysteresis bandwidth is calculated as follows:

$$\left\{\begin{array}{l}{l}_{\mathrm{exten}}^{\mathrm{right}}={\displaystyle \frac{T_{\mathrm{e}}^{\mathrm{upp}}-T_{\mathrm{e}}(k+1)}{T_{\mathrm{e}}(k+1)-T_{\mathrm{e}}(k)}}\hfill \\ l_{\mathrm{exten}}^{\mathrm{left}}={\displaystyle \frac{T_{\mathrm{e}}^{\mathrm{low}}-T_{\mathrm{e}}(k+1)}{T_{\mathrm{e}}(k+1)-T_{\mathrm{e}}(k)}}\hfill \end{array}\right.$$

(13)

$$l_{exten}=\left\{\begin{array}{ll}{l}_{exten}^{right},\hfill & l_{exten}^{right}>l_{exten}^{left}\hfill \\ l_{exten}^{left},\hfill & l_{exten}^{right}<l_{exten}^{left}\hfill \end{array}\right.$$

(14)

$l_{exten}^{right}$ is the distance at which the line intersects the upper threshold of the hysteresis band, and $l_{exten}^{\mathrm{left}}$ is the distance at which the straight line intersects the lower threshold of the hysteresis band.

At this point, the evaluation function can be expressed as

$$J={\displaystyle \frac{1}{l_{exten}}}$$

(15)

The smaller the calculation result of Equation (15), the better the long-term tracking effect of the given torque, avoiding the traditional tracking control for the given torque tracking repeatedly between the forward excitation and reverse demagnetization state. Therefore, when the torque pulsation suppression effect is better, the switching rate of the power semiconductor is intrinsically lowered, and the switching loss is also smaller. Therefore, the selection of candidate vectors for switching action is performed based on the minimization criterion of the calculated value of the evaluation function.

To clearly illustrate the proposed control logic, Figure 5 presents the algorithmic flowchart. The core mechanism can be summarized as follows. First, based on the current flux linkage, torque, and switching state, the prediction model is used to recursively compute the torque evolution over N future control steps. Second, the hysteresis comparator outputs a status flag according to the current torque error, which dynamically adjusts the weights of the torque tracking term and the switching loss term in the evaluation function. Finally, by minimizing the extension length-based evaluation function (Equation (15)), the optimal switching sequence that maintains the torque error within the hysteresis band for an extended duration is selected. This mechanism embeds the hysteresis rules into the long-time-domain optimization framework, achieving a coordinated optimization of torque ripple suppression and switching loss reduction.

The proposed evaluation function based on torque extension length (Equations (13)–(15)) minimizes the reciprocal of the extension length. The rationale is that maximizing the extension length encourages the torque error to remain within the hysteresis bounds for longer durations, thereby reducing frequent boundary crossings. Minimizing the reciprocal inherently penalizes short extension lengths, which correspond to premature exits from the hysteresis band. In this way, the controller naturally balances torque ripple suppression and switching frequency reduction: longer extension lengths indicate sustained torque tracking with fewer switching actions, while short extension lengths incur a large penalty, implicitly discouraging unnecessary switching. This formulation thus enables the controller to achieve coordinated optimization over an extended time horizon.

Figure 5 illustrates the complete control system block diagram. The scheme employs a cascaded control structure with speed and torque loops. The speed controller’s input is the velocity error, and its output generates the torque setpoint for the inner loop. Based on the predicted torque, the appropriate switching signal for the next cycle is chosen and designated for the converter.

The control flowchart is shown in Figure 6:

5. Simulation Analysis

5.1. Simulation Parameters

A simulation model of a 6/4-pole switched reluctance motor was developed within the MATLAB/Simulink environment to assess the performance of the proposed model predictive control strategy incorporating hysteresis regulation. The key motor and simulation parameters are provided in

Table 2. Parameters of switched reluctance motors.

Algorithm	Value	Algorithm	Value
Number of stator poles	6	Rotor pole arc	45 mm
Number of rotor poles	4	Stator pole arc	32 mm
Air gap	0.2 mm	DC voltage	240 V
Outer diameter of stator	142 mm	Coefficient of static friction	0.01
Moment of inertia	0.0862	Number of phases in the winding	3

and

Table 3. Simulation control parameters.

Parameters	Value
Given speed	1000 r/min 200 r/min
Load torque	4→8→6 N·m
Hysteresis loop width selection for MPDTC	5% of the given torque
Torque hysteresis loop width of DITC	±0.01 N·m
Chain hysteresis loop width for DITC	±0.01 Wb
DITC’s given magnetic chain	0.3 Wb
Current hysteresis loop width for CCC	±0.1 A
CCC conduction angle and turn-off angle	42°, 77°

, respectively.

This paper compares the traditional SRM control strategy, Chopper Current Control (CCC), with the more advanced direct instantaneous torque control (DITC). The simulation control parameters for the three algorithms are shown in the table below (Table 3).

5.2. Simulation Results and Analysis at High Rotational Speeds

Figure 7 presents the simulated torque and current waveforms under CCC, DITC, and the proposed method, under a constant speed of 1000 rpm and a load torque that transitions sequentially from 4 N·m to 8 N·m, and finally to 6 N·m.

The standard deviation of the torque and torque pulsation factor enables the quantitative evaluation of torque simulation results [16]:

$$T_{std}=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\left(T_e-T_{av}\right)}^2}}{n}}}$$

(16)

where n is the sample size (statistics), T_e represents the electromagnetic torque value at the sampling instant, and T_av represents the mean value of the electromagnetic torque across a set of n sampling instants.

$$T_{RC}={\displaystyle \frac{T_{\max }-T_{\min }}{T_{av}}}$$

(17)

T_max, T_min, and T_av are the maximum, minimum, and average values of torque pulsations, respectively.

The standard deviations of the torque and torque pulsation coefficient for a load torque of 6 N⋅m are shown in

Table 4. Comparison of torque standard deviations.

Algorithm	Tstd
CCC	1.643
DITC	1.554
MPDTC	0.976

and

Table 5. Comparison of torque pulsation coefficients.

Algorithm	TRC
CCC	0.757
DITC	0.734
MPDTC	0.619

.

The data in Table 4 and Table 5 show that the standard deviation of the torque of the algorithm is reduced by 0.676 compared to CCC and 0.587 compared to DITC. The torque pulsation coefficient is reduced by 0.138 compared to CCC and 0.115 compared to DITC, which indicates that the algorithm has a better suppression of torque pulsation compared to the conventional algorithm and has a better suppression of torque pulsation and better torque steady state performance at high rotational speeds.

The simulated performance at a fixed speed of 1000 rpm is compared in Figure 8. Regarding response rapidity, the CCC method demonstrates quicker velocity tracking than both DITC and the proposed approach. Nonetheless, the new algorithm achieves superior precision in steady-state velocity regulation and exhibits minimized fluctuation during abrupt load torque variations, indicating enhanced dynamic stability.

5.3. Simulation Results and Analysis at Low Rotational Speeds

Figure 9 compares the simulated torque and current waveforms for the CCC, DITC, and proposed methods under a low speed of 200 rpm and a load torque that undergoes step changes from 4 N·m to 8 N·m, and finally to 6 N·m.

The standard deviation of the torque and the torque pulsation coefficient for a load torque of 6 N⋅m are shown in

Table 6. Comparison of torque standard deviation factors.

Algorithm	Tstd
CCC	1.55
DITC	1.378
MPDTC	1.005

and

Table 7. Comparison of torque pulsation coefficients.

Algorithm	TRC
CCC	0.872
DITC	1.201
MPDTC	0.182

.

The data presented in Table 6 and Table 7 indicate that the proposed algorithm achieves reductions in torque standard deviation of 0.545 and 0.0373 relative to CCC and DITC, respectively. The corresponding decreases in the torque pulsation coefficient are 0.69 and 1.091, confirming its superior torque steady-state performance and enhanced pulsation suppression capability at low speeds. Figure 10 displays the speed response at 200 rpm. CCC and the proposed method exhibit comparable response rapidity, both surpassing DITC. A key advantage of the proposed strategy is the absence of overshoot in the speed trajectory, ensuring stability without compromising responsiveness. Although a marginally larger speed error occurs during low-speed torque transients, the overall difference is negligible. Consequently, the proposed algorithm demonstrates distinct advantages in terms of speed dynamic performance.

6. Conclusions

To enhance the robustness and disturbance rejection capability of switched reluctance motor systems, an improved model predictive direct torque control strategy is developed in this work. The following conclusions are supported by theoretical analysis, simulation results, and experimental tests.

(1)

At high speeds, the proposed algorithm demonstrates a higher accuracy in speed error compared to conventional methods, exhibits smaller speed fluctuations during load torque transients, and shows a superior dynamic stability. Simultaneously, the torque ripple coefficient and torque standard deviation are significantly reduced, resulting in enhanced steady-state performance. At low speeds, the proposed algorithm and CCC exhibit comparable response speeds, both markedly outperforming DITC. However, the proposed algorithm eliminates overshoot in the speed curve, ensuring rapid response while maintaining superior stability. It effectively suppresses torque ripple and torque standard deviation, demonstrating more pronounced dynamic performance advantages under low-speed conditions.

(2)

Compared to traditional CCC and DITC algorithms, the proposed algorithm exhibits lower torque ripple across varying speeds and loads, delivering robust performance under diverse operating conditions. From the perspective of the control mechanism, the performance improvements of the proposed method over CCC and DITC can be attributed to the following design features. First, by extending the prediction horizon over multiple steps, the controller anticipates the cumulative trend of torque errors and proactively adjusts switching actions before the torque approaches the hysteresis boundaries, thereby suppressing additional ripple caused by frequent boundary crossings. Second, the dynamic weighting evaluation function enables the optimization objective to adaptively shift between torque tracking and switching loss reduction, avoiding the switching frequency fluctuations commonly associated with fixed hysteresis bands in CCC. Third, the extension length-based penalty mechanism encourages the controller to prioritize switching sequences that maintain the torque within the hysteresis band for longer durations, thereby reducing switching losses without compromising torque quality. These mechanisms collectively explain why the proposed method achieves superior performance over CCC and DITC in terms of both torque ripple suppression and switching frequency reduction under both 1000 r/min and 200 r/min operating conditions.