A High-Performance Indirect Torque Control Strategy for Switched Reluctance Motor Drives

This paper proposes a high-performance indirect control scheme for torque ripple minimization in the switched reluctance motor (SRM) drive system. Firstly, based on the nonlinear torque-angle characteristic of SRM, a novel torque sharing function is developed to obtain the optimal current profiles such that the torque ripple is minimized with reduced copper losses. Secondly, in order to track current accurately and indirectly achieve high-performance torque control, a robust current controller is derived through the Lyapunov stability theory. The proposed robust current controller not only considers the motor parameter modeling errors but also realizes the fixed frequency current control by introducing the pulse width modulation method. Further, a disturbance-observer-based speed controller is derived to regulate the motor speed accurately, and the load torque is considered an unknown disturbance. The simulations and experiments on a 1.5 kW SRM prototype are carried out to demonstrate the effectiveness of the proposed high-performance indirect torque control strategy. Results verify the superiority of the proposed strategy with respect to the torque ripple suppression, system efficiency, and antidisturbance.


Introduction
Switched reluctance motor (SRM) has recently attracted much attention from the industrial and academic communities due to its own advantages, such as simple and strong structures, high reliability, wide speed range, no need for rare Earth permanent magnetic materials, and low manufacturing cost [1][2][3][4]. However, relatively higher torque ripples are viewed as a significant disadvantage in comparison with other types of motors [5,6]. erefore, it is of great significance to study the torque ripple suppression of SRM for its further popularization and application. e literature review reveals that the torque ripple of SRM can be effectively reduced by means of an appropriate control algorithm [7]. At present, the torque control methods of SRM mainly include direct torque control (DTC) and indirect torque control (ITC). In DTC, the switching signal of the power converter is directly generated by combining the error between the command torque and the instantaneous torque of SRM with the appropriate commutation logic [8][9][10]. e instantaneous torque of the SRM is usually calculated from the measured rotor position and phase current [11]. DTC has the advantage of simple structure and easy implementation because it has no current loop [12]. However, the shortcomings of DTC strategy, such as high sampling rate, no over-current protection, and variable switching frequency, greatly limit its popularization and application.
Alternatively, ITC methods for SRM can partly overcome these disadvantages. In ITC, torque sharing function (TSF) is adopted to distribute the electromagnetic torque command to each phase of the motor, and then the optimized phase current reference is obtained by the nonlinear mapping of current-torque-angle i(T, θ). Torque control can be indirectly realized by regulating the current to track the optimized current reference [13][14][15]. erefore, the key to determining the performance of the ITC algorithm is the TSF and the inner-loop current controller. In [16], the effects of different TSFs such as linear, cosine, quadratic, and exponential to reduce the torque ripple of SRMs are compared and evaluated. However, these TSFs will produce a larger peak current during commutation, which will increase the torque ripple and reduce the driving efficiency of the SRM. An offline optimization method of TSF is used in [17] to minimize torque ripple, and its objective function combines both the phase current and the rate of change of flux-linkage. In [18], the objective function of offline TSF is further improved and simplified by using a single weight parameter. In [19], an online compensation of TSF method is developed to smooth the torque output by applying positive and negative compensation to the outgoing phase and incoming phase, respectively. In [20], a proportional-integral (PI) controller is adopted to compensate for the tracking error of the torque in real time. However, in [19,20], a large memory is needed to store additional torque characteristics T(i, θ) for the on-line estimation of torque.
For ITC, the performance of the current controller will also directly affect the torque ripples of the motor. Hysteresis controllers are widely used in current loops because of their simple structure, model independence, and fast dynamic response [21,22]. However, when hysteresis current controller is used in the SRM drive system, the switching frequency of power converter is uncontrollable, which will produce some unpredictable acoustic noise. To solve this problem, a digital pulse-width-modulation (PWM) current controller for SRM is studied with constant switching frequency in [23]. However, the digital PWM current controller cannot guarantee that the tracking error tends to zero when the current reference is a time-varying signal. A model-based predictive current controller is investigated in [24]. e predictive controller achieves the fixed switching frequency. However, the performance of the predictive controller needs a large gain, which may degrade the performance of the controller. Additionally, the robustness of the predictive current controller was not investigated.
In order to simultaneously solve the aforementioned problems, a high-performance indirect torque control (HPITC) strategy for SRM drive is proposed in this paper. To the best of our knowledge, it is the first time in the literature that TSF optimization, accurate current tracking, and disturbance rejection are simultaneously dealt with in the SRM drive system. A novel TSF is developed to suppress the torque ripple while reducing the peak current during commutation.
en, based on the Lyapunov stability theory, a robust current controller is derived to achieve accurate current tracking. Moreover, a novel speed controller associated with a load torque observer has been developed to achieve accurate speed regulation and improve the antidisturbance performance of the SRM drive system. Due to the comprehensive improvement of the HPITC strategy, the proposed method offers the feasibility of effectively reducing the torque ripple, improving system efficiency, and enhancing antidisturbance ability. e effectiveness of the HPITC method is demonstrated by simulations and experiments.

SRM Model
Neglecting the effect of mutual inductance, the phase torque of SRM can be calculated as where W c and W s denote the magnetic coenergy and the magnetic energy storage, which can be calculated as where T p , i p , and ψ p are the p-th phase torque, the p-th phase current, and the p-th phase flux-linkage, respectively. θ is the rotor position. e total electromagnetic torque T e of the studied threephase SRM is described as follows: In order to improve the energy conversion, the SRM always works in the magnetic saturation region. In this region, the electromagnetic characteristics of the SRM exhibit high nonlinearity. As a result, it is difficult to establish an accurate nonlinear model of SRM based on conventional electromagnetic and physical characteristics deduction. At present, most of the nonlinear models of SRMs are identified based on the sample data of electromagnetic characteristics obtained by finite element analysis (FEA) or experiment [25]. An analytical model [26] that is embedded in the SimPowerSystems toolbox of Matlab/ Simulink software is adopted in this paper. As analyzed in [26], the flux-linkage profile of SRM is represented analytically as where L q denotes the inductance at unaligned rotor position (q-axis) and L dsat denotes the saturated inductance at aligned rotor position (d-axis). A, B, and f(θ p ) can be further represented as follows: where L d denotes the nonsaturated inductance in d-axis and I m denotes the rated maximum current with corresponding flux-linkage ψ m . Substituting (4) into (1), the stator phase torque T p can be calculated as 2 Mathematical Problems in Engineering e experimental measurement of the studied SRM was conducted using the method in [25]. e measured fluxlinkage and torque profiles of the motor are shown in Figures 1(a) and 1(b), respectively. Based on the measured electromagnetic characteristics, as shown in Figure 1, the model coefficients ψ m � 0.9, L q � 0.0226, L dsat � 0.0185, L d � 0.3152, and I m � 10 are obtained by the least-squares fitting. us, the nonlinear model of SRM is established.

High-Performance Indirect Torque Control
e schematic diagram of the HPITC system is shown in Figure 2, involving speed controller, TSF, torque-to-current module, current controller, 12/8 SRM, position and current sensors, and power converter.
As shown in Figure 2, the total torque command T * e is derived by the designed disturbance-observer-based speed controller. e inputs of disturbance observer are speed ω and reference torque T * e . In the torque control scheme, the torque reference T * e is distributed for every single phase by using a novel torque sharing function. en the phase current command is generated from the current-torqueposition i(T, θ) characteristics. Finally, the proposed robust current controller is used to generate the required phase voltage u p so that the phase current i p can track the current command i * p and then the electromagnetic torque T e can track the reference torque T * e .

Torque Sharing Function.
At present, the most commonly used TSFs mainly include linear, cubic, exponential, and sinusoidal functions [16]. A typical torque sharing curve of the sinusoidal TSF is shown in Figure 3(a). e phase torque command T * p can be obtained by a predefined TSF as follows: where f rise (θ) and f fall (θ) denote the rising TSF and decreasing TSF, respectively. θ on , θ off , θ ov , and θ p represent turn-on angle, turn-off angle, overlap angle, and rotor period, respectively. In the overlap region, the total electromagnetic torque command is equal to the sum of torque commands for incoming phase and outgoing phase. Hence, the relationship between f rise and f fall can be described as

From Figures 1(b) and 3(a)
, it is observed that the torque curves in the rising and falling stages are quite different from conventional TSF curves. If the conventional TSF is used, the incoming phase will be allocated a large reference torque in the initial stage of excitation, which will increase the peak current of stator winding during commutation. As a result, the copper losses of the SRM will increase and the drive efficiency of the motor system will decrease.
In order to further reduce copper losses while minimizing torque ripples, a novel nonlinear TSF is presented as shown in Figure 3 e proposed TSF uses a power function to distribute the total torque reference in the overlap region of incoming and outgoing phases. e functions f rise and f fall of the proposed TSF can be described as where α ≥ 2 is a positive constant. e larger the value of α, the smaller the torque reference allocated to the incoming phase at the initial stage. e TSF parameters θ on , θ off , and θ ov can be linked as where ε � 360/(mN r ).

Robust Current Controller Design.
For SRM, phase voltage u p is described as follows: where R denotes the resistance of each phase winding of the motor. e phase flux-linkage ψ p (i p , θ p ) is a function of the phase current i p and rotor position θ p . us, the differential of flux-linkage is derived as Substituting (12) into (11), the derivative of phase current i p of SRM can be represented as where L inc � (zψ p /zi p ) denotes the increment inductance and E bmf � (zψ p /zθ p ) is the back electromotive force constant. Further, according to the flux-linkage expression (4), the constituent terms in (13) like incremental inductance L inc and back electromotive force constant E bmf can be calculated as follows: Mathematical Problems in Engineering 3 Define the phase current tracking error and its differential as where i * p is the current reference. e control objective of this subsection is to design a robust current controller such that   phase current i p can accurately track its reference i * p in the presence of model uncertainty.
Without considering the modeling errors, the ideal control law can be derived as where c > 0 is the controller parameter. Lyapunov stability theory is adopted to design the control law u(t) for current tracking control of SRM. Select the following Lyapunov function candidate: e time derivative of (17) is By substituting (13), (15), and (16) into (18), (18) can be rewritten as It should be noted that _ V < 0 for all the conditions except e � 0. Equation (19) indicates that the phase current i p can track the current reference i * p asymptotically. However, in the real application, the terms L inc and E bmf cannot be calculated accurately because of the modeling errors in equation (4). Accordingly, the real values of the increment inductance L inc and back electromotive force constant E bmf can be divided into the estimated values and their modeling errors. us, equation (14) is rewritten as A robust control design method is used to overcome the above model uncertainty.
e robust current controller allows the use of a simplified nonlinear model of SRM in a control system design, which is readily available for magnetization curves at two extreme positions. Using the estimated values L inc and E bmf instead of the actual values L inc and E bmf , the control law (16) can be reformulated as where According to the Lyapunov stability condition, equation (22) needs to satisfy the following condition: Let the absolute value of current tracking error converge to error bound e b , which is very close to zero, and inequality (24) can be rewritten as It can be seen from equation (25) that the controller parameter c can be adjusted adaptively according to the value of g(·). By substituting (20) in (23), equation (23) can be rewritten as Substituting (20) and (26) into (25), the condition in (25) can be further simplified as follows: In the practical applications, excessive feedback gain will result in oscillatory response near unaligned rotor position. erefore, a feedback gain c varying according to (27) is adopted to achieve better current control performance. For a specific SRM, the values of ΔL inc and ΔE bmf are usually fixed and can be obtained by calculating the error between the real and simplified model parameters [27]. In this paper, ΔL inc and ΔE bmf are taken to be constant as 50% of the nominal value over the entire range of operation.
To achieve the fixed switching frequency control, PWM control is embedded in the proposed HPITC algorithm. e duty cycle of the PWM control is described as where λ p is the duty cycle of the pth phase.

Speed Controller Design.
In this subsection, the objective is to explore a disturbance-observer-based speed controller to provide the total electromagnetic torque reference for the middle loop (torque loop). Assuming that the torque loop bandwidth is sufficiently large, the equation of motion is given by where A 1 � k ω /J, A 2 � 1/J, k ω is the friction coefficient, J is the moment of inertia, and the load torque T L is taken as an unknown external disturbance. Let the idea control law of speed loop be

Mathematical Problems in Engineering
where e � ω * − ω is speed error and k 1 is a positive parameter.
Combining with (29) and (30), we get Equation (31) shows that the real speed ω can track the speed command ω * asymptotically. However, the desired control law (30) cannot be implemented because the load disturbance T L of the SRM drive system is unknown. Since the external load disturbance of the motor system is constantly changing and has finite energy, the external load acting on the motor can be seen as the unknown, timevarying yet bounded signals with the finite change rates [28]. erefore, the following assumption can be made. e load disturbance T L is unknown and time-varying yet bounded and there exists an unknown positive constant σ such that | _ T L | ≤ σ.

Simulation Results
In order to demonstrate the effectiveness of the HPITC method, the SRM drive system is simulated under different operating conditions using the Matlab/Simulink software. In this paper, a 1.5 kW three-phase 12/8-pole SRM is selected as an example. e measured flux-linkage ψ(i, θ) and torque T(i, θ) characteristics are shown in Figures 1(a) and 1(b), respectively. e sampling time in all simulations is 10 μs. e specification and parameters of the SRM used in the simulation and experimental studies are presented in Table 1.
To quantitatively evaluate the performance of the proposed HPITC method, its performance indexes are defined as follows: where T rf represents the torque ripple factor and T e max , T e min , and T e av denote the maximum torque, the minimum torque, and the average torque, respectively. I peak is the peak value of phase current. P loss denotes the copper losses of SRM. I RMSE represents the root mean square error of phase current tracking. N is the number of samples.
In the first simulation, the comparisons between the proposed power TSF and traditional sinusoidal TSF are carried out to verify their steady-state performance. e proposed robust controller is applied to the current control loop. e turn-on angle θ on and turn-off angle θ off of the proposed TSF and traditional TSF are set to 22.5°and 37.5°, respectively. Figures 4(a) and 4(b) exhibit the simulation results of the proposed TSF and traditional linear TSF at low speed of 100 r/min, respectively. Figures 5(a) and 5(b) present the results of the proposed TSF and traditional sinusoidal TSF at high speed of 1500 r/min, respectively. e waveforms of the motor speed, total electromagnetic torque, phase torque, and stator phase current are presented in each subfigure. From Figures 4 and 5, we can see that both proposed TSF and conventional sinusoidal TSF can control the torque ripple within a certain range and have excellent torque ripple suppression ability. But the peak current of the proposed power TSF is much smaller than that of the traditional sinusoidal TSF. e current waveform of the proposed TSF is smoother than that of the conventional TSF. Moreover, Figures 6 and 7 show the comparison results of phase torque waveform between conventional and proposed TSFs at low speed of 100 r/min and high speed of 1500 r/min, respectively. As shown in Figures 6 and 7, the reference phase torque of the proposed TSF is much lower than that of the traditional sinusoidal TSF in the initial stage of inductance rise, which makes the peak current of the proposed TSF smaller than that of the traditional TSF. Correspondingly, in the second half of the inductance rise, the reference phase torque of the proposed TSF is higher than that of the conventional TSF to keep the total torque constant, but the phase current required to generate the same torque is smaller than that in the initial stage of inductance rise. As a result, the copper losses of the proposed TSF will be lower than those of the conventional TSF.
Further, the comparison of torque ripple factors T rf , peak current I peak , and copper losses P loss between the proposed TSF and the traditional sinusoidal TSF is summarized in Table 2. From Table 2, it is seen that the torque ripple factors T rf of the proposed TSF and the conventional sinusoidal TSF are very similar in a wide speed range. However, the peak current I peak is reduced from 8.51 A to 7.48 A in conventional TSF to 5.42 A and 5.12 A at the speeds of 100 r/min and 1500 r/min, respectively. As a result, the conventional TSF has higher copper losses P loss of 4.9887 and 5.5328 at speed commands of 100 and 1500 r/min, respectively. On the contrary, the copper losses P loss of the proposed TSF are reduced to 4.2083 and 5.1164. us, the copper losses P loss for the proposed TSF are lower by 15.6% and 10.1% compared with the traditional sinusoidal TSF. e simulation result demonstrates that the efficiency of the proposed TSF is higher than that of the traditional TSF.
In the next simulation, the comparative study of the proposed robust current controller and traditional hysteresis current controller is carried out at low speed of 100 r/min and high speed of 1500 r/min with load torque T L � 5 Nm to test their current tracking capability. e switching frequency of the proposed robust current controller is set to 50 kHz. e current band of the hysteresis controller is set to ±0.1 A, and its switching frequency is between 1 kHz and 100 kHz depending on the speed and current. e waveforms of the robust current controller and hysteresis controller at low speed of 100 r/min are shown in Figures 8 and  9. Figures 10 and 11 exhibit the current and electromagnetic torque waveforms of the two controllers at a high speed of 1500 r/min, respectively. As shown in Figures 8 and 9, the proposed robust controller has dynamic response and current tracking accuracy similar to those of the traditional hysteresis current controller. However, the robust current controller has a lower current ripple than the conventional hysteresis controller. It can be seen from Figures 9 and 11 that the reduced current ripple by the proposed controller also results in reduced torque ripple. Table 3 gives a quantitative comparison of current tracking performance between the proposed controller and the hysteresis controller. According to Table 3, it can be seen that the RMSE of the traditional hysteresis controller is 0.1554 and 0.8076, while I RMSE of the proposed controller is 0.1410 and 0.7745. As shown in Table 3, moreover, the torque ripple is reduced from 0.3762 to 0.5484 in hysteresis controller to 0.2503 and 0.3995 at the speed of 100 r/min and 1500 r/min, respectively.
ese results indicate that the performance of the proposed current controller is superior to that of the traditional hysteresis controller.
In the final simulation, the tracking performance and robustness of the proposed speed controller are investigated by controlling the SRM to work in the states of starting, acceleration, deceleration, and constant speed. Figure 12 shows the speed command changing from 0 r/min to 1500 r/ min to 1000 r/min with load torque of 5 Nm. It can be seen that the motor speed can track the reference speed very well during starting, accelerating, and decelerating. In the whole tracking process, the maximum dynamic tracking error of motor speed is less than 3 r/min, and the steady-state error is less than 1 r/min. Further, in order to test the robustness of the proposed controller, a step load torque is added to the SRM drive system, which is taken into account as a disturbance. In the simulations, the load torque T L is stepchanged from 1 Nm to 6 Nm at a constant speed of 1000 r/ min. Moreover, the proposed disturbance-observer-based  [30]. e parameters k p and k i are 2.25 and 6.25, respectively. e responses to external load disturbance are illustrated in Figures 13(a) and 13(b) for the proposed disturbance-observer-based speed control and conventional PI methods, respectively. As shown in Figures 13(a) and 13(b), although the speed drops are         roughly similar in both cases, the proposed controller can be returned to the original speed reference more quickly than the conventional PI controller. It is also observed that the motor torque increases rapidly to maintain the load torque. In Figure 13, the maximum speed dips are 24 and 26 r/min with respect to the proposed disturbance-observer-based speed controller and conventional PI control. e speed error of the proposed method backs to the range of ±1 r/min only through 0.055 s, and the speed error of the traditional PI method recovers to the range of ±1 r/min through 0.35 s. e proposed disturbance-observer-based speed controller exhibits stronger antidisturbance ability than the traditional PI algorithm.
is is due to the fact that the external load disturbance is compensated accurately by the disturbance observer in the proposed control scheme.
In general, the HPITC algorithm has an excellent performance in efficiency, minimization of torque ripple, and robustness for the SRM drive system.

Experimental Results
In this section, the availability of the HPITC algorithm is further verified by experiments. A three-phase 12/8-pole 1.5 kW SRM is selected as the experimental prototype. e hardware schematic diagram of the SRM drive system is depicted in Figure 14(a). e photograph of the experimental platform is shown in Figure 14(b). e experimental platform consists of two parts: the mechanical system and electrical system. e mechanical system comprises the SRM prototype, torque and speed sensor, and magnetic particle brake, allowing adjusting the load torque. e electrical system mainly includes control unit (dSPACE DS1103), measurement device (digital oscilloscope), and asymmetric halfbridge power electronic converter. e sampling periods of speed and current control in the experimental study are set as 1 ms and 100 us, respectively. e following experiments were carried out to test the performance of the HPITC method.

Current Tracking Performance.
e HPITC method was implemented at different speeds (200 r/min, 1000 r/min, and 1500 r/min) with load torque T L � 5 Nm to confirm the current tracking performances. Figures 15(a)-15(c) show the experimental results of the speeds of 200 r/min, 1000 r/ min, and 1500 r/min, respectively. e stator phase current i p and its reference i * p are observed, respectively, through channel 1 (CH1) and channel 2 (CH2) of the oscilloscope. It is observed that the stator phase current can quickly track the phase current reference at different speeds. In Figure 15(a), the maximum value of the stator phase current is 7.8 A, and the copper loss P loss is 5.43. It can be calculated from Figure 15(b) that the peak current is 7.5 A and the copper loss P loss is 5.92. In Figure 15(c), the maximum value of the stator phase current is 7.9 A, and the copper loss P loss is 6.37. Figure 15 shows that the proposed controller has good current tracking performance.

Torque Ripple Suppression Performance.
A speed command changing from 200 to 600 r/min with external load torque T L � 1 Nm is implemented to test the torque ripple suppression capability of the HPITC method. Figure 16 presents the ability of the proposed HPITC strategy to minimize the torque ripple of SRM. As shown in Figure 16, the total torque (CH2) can always maintain a low torque ripple in both steady and dynamic states. According to equation (42), the torque ripple factor T rf can be calculated to be 0.5614 in this case. It can be noted that lower torque ripple ensures that the motor speed (CH1) can track command speed well. From these results, it can be concluded that the HPITC algorithm has an excellent capability for torque ripple suppression.

Robustness against External Load Disturbance.
In order to test the antidisturbance ability of the HPITC algorithm, the experiments are carried out with a step change of external load torque during the steady operation. Figure 17 shows the experimental results. In this test, the motor speed is 400 r/min and the external load torque is changed from 0 Nm to 6 Nm. In Figure 17, the maximum speed drop of the SRM is 39 r/min, and it only takes 60 ms for the motor to recover to the initial speed reference. It is seen that under a load torque step variation the total electromagnetic torque follows the load torque value closely and always sustains low torque ripples. ese results show that the HPITC method has an excellent antidisturbance ability.
In sum, the experimental results verify the effectiveness of the proposed control scheme.

Conclusion
In this paper, a high-performance control technology was developed to reduce the torque ripple of the SRM drive system. Based on the nonlinear electromagnetic      characteristics of SRM, a more efficient TSF is used to ensure that the copper loss is reduced, while the torque ripple is minimized. en, a robust current controller with variable feedback gain is designed to accurately track the current reference. e stability of the robust current controller has been proved by the Lyapunov stability analysis. Moreover, a novel speed controller is designed to improve the speed tracking accuracy and antidisturbance performance of the SRM drive system. e simulation and experimental results demonstrated that the HPITC method has the advantages of suppressing the torque ripple, improving system efficiency, and enhancing antidisturbance ability.
Data Availability e data that support the findings of this study are included within the article.

Conflicts of Interest
e authors declare no conflicts of interest regarding the publication of this paper.