Stator Current VectorOrientation-Based Backstepping Predictive Control of Torque Ripple Minimization for PMSM with consideration of Electrical Loss

Existing research studies on torque ripple suppression mostly ignore the electrical loss of PMSM. However, the electrical loss will not only decrease the operating efficiency but also adversely influence the suppression of torque ripple. )is paper attempts to construct a unified framework to suppress torque ripple with consideration of electrical loss. Firstly, a dynamic mathematical model of PMSM under current vector orientation is established with a combination of electrical loss. )e constraints that can achieve the control of both torque ripple and electrical loss for PMSM are derived. )en, on the basis of the backstepping control principle, a closed-loop I/f integrative control method under stator current vector orientation is proposed. Meanwhile, this paper also proposes a speed estimation algorithm of PMSM based on the least-squares method to realize wide-range speed identification and an online prediction algorithm for control parameters of backstepping control to enhance the stability of the motor in operation. Both simulations and experiments have been performed to verify the effectiveness of the proposed control method, and the results indicate that torque ripple is suppressed effectively, operating efficiency is significantly improved, and all variables are regulated to track their reference signals correctly and rapidly.


Introduction
Permanent magnet synchronous motor (PMSM) is broadly used in various industrial fields for its advantages of simple structure, good dynamic response, and high-power factor [1]. However, for reasons like asymmetric winding structure and flux harmonics produced by the permanent magnet, torque ripple is a crucial problem of PMSM that would lead to speed fluctuation and weaken dynamic performance during operation [2]. Meanwhile, the operating efficiency of PMSM is another issue that deserves more and more attention to energy-saving [3,4]. If the damping windings are ignored, the loss of PMSM is comprised of electrical and mechanical losses, where electrical loss includes iron loss and copper loss as well. e control of the motor's loss will improve its operating efficiency. erefore, simultaneous completions of suppressing torque ripple and reducing operating efficiency have dual beneficial effects on overall drive performance of PMSM. For the suppression of torque ripple in PMSM, the approaches can be mostly classified into two categories: structure of motor ontology optimized to produce air-gap field more closely to sinusoidal and decrease cogging torque [5][6][7] and appropriate control strategies devised to smooth electromagnetic torque [8][9][10][11][12][13]. e second kind of method is more suitable for motors in service. e iterative learning control strategy is adopted in [8,9]; the system regulates the current components continuously to achieve the suppression of torque ripple based on previous information. However, this control method is difficult to provide a suitable error compensation signal under the condition of variable speed. Meanwhile, there are multiple PI controllers in [9] that need to be adjusted, which requires a relatively large calculation amount. A current compensation method is proposed in [10], and a rotor flux observer based on Kalman filter is designed to suppress torque ripple to some extent. But the performance of the Kalman filter depends greatly on the parameters of the motor, which makes the proposed method to be less robust. Other compensation methods are investigated in [11][12][13], but these methods have compensation errors more or less, and their applications are sometimes limited.
Above all, existing research on torque ripple suppression often disregards electrical loss. In reality, these two problems are interacting with each other, and ignoring electrical loss will not only decrease the operating efficiency of PMSM but also bring some unfavorable influence on the overall control scenario. As for the operating efficiency of a motor, it is determined by the control of its loss. When a PMSM runs stably at a certain speed, its mechanical loss could hardly be reduced since it is mostly restricted by the speed. Moreover, the mechanical loss is only a small proportion of a total loss. erefore, the key point of the loss control is to focus on reducing electrical loss, which means lower iron and copper losses. Control strategies used to realize a minimum electrical loss in recent years are commonly divided into two types of online search-based minimum power control [14,15] and loss-based model control [16][17][18]. e former is used for steady-state operation. But this method has a high requirement of the accuracy in detecting input power and normally takes a long time to converge. Now, it is gradually taken place by the latter method. e latter establishes an accurate loss model of the motor and achieves optimal operation by analyzing the relationship of various losses to exciting current and speed. is method has a fast response speed and is suitable for variable speed operation. e variation of the inductance during operation is considered in [16], and the approximately linear optimal current equation is derived based on Kuhn-Tucker theory. In terms of loss equation, the optimal output voltage by using the Lyapunov stability theory is obtained in [17]. However, the whole schemes of the above two control methods are complicated. e approximate approach is used in [18] to find the optimal solution, which reduces the calculation amount but also weakens the calculation accuracy.
Hence, this paper attempts to construct a unified control method to realize torque ripple suppression with consideration of electrical loss. Nowadays, I/f control is used in the startup for sensor-less PMSM drives [19]. However, from the viewpoint of the control structure, conventional I/f control is an open-loop control strategy that exhibits the problem of stability and will consume more energy with fixed current amplitude. Hence, this paper will make a significant contribution to the existing I/ f control method and introduce an improved method as the closed-loop control framework.
rough the integration of torque ripple suppression with electrical loss reduction, an overall dynamic mathematical model of PMSM under closed-loop I/f control framework is constructed and various constraints are derived to inhibit torque ripple and pursuit optimal operating efficiency simultaneously. A backstepping control-based closed-loop I/f integrative controller is proposed to suppress torque ripple and reduce electrical loss instantaneously with a combination of the least-square-based speed estimation algorithm. Meanwhile, a nonlinear predictive control algorithm is introduced to adjust multiple control parameters online, which improves the control precision and enhances the stability of the proposed control scheme. Simulations and hardware experiments have demonstrated the feasibility and practicability of the proposed controller. e results display that both steady-state and dynamic performances are improved, and torque ripple in PMSM is suppressed effectively while operating efficiency is also enhanced. e following context is arranged as follows: an overall mathematical model of the system is described in Section 2.
e control algorithm of torque ripple suppression with consideration of the electrical loss for PMSM is proposed in Section 3. e method of speed identification is designed in Section 4. e online prediction algorithm for control parameters is designed in Section 5. e simulation and experimental analysis are conducted in Section 6. e last section summarizes the research results.

Constraint Condition under Minimum Loss Operation for PMSM.
e equivalent circuits of PMSM in d-q axis including iron loss resistance are plotted in Figure 1 and corresponding dynamic equations can be expressed in the following equation: where i wd and i wq are the active current components in d and q axes, i cd and i cq are the iron loss current components in d and q axes, L d and L q are the inductors in d and q axes, u sd and u sq are the voltages in d and q axes, R c is the equivalent iron loss resistance, n p is the number of pole pairs of rotor, λ r is the linkage of permanent magnets, and ω r is the mechanical angular velocity of the rotor. e effective electromagnetic torque T e is written as e total electrical loss P Loss including copper loss P Cu and iron loss P Fe of PMSM are described as follows: In terms of electromagnetic torque T e in equation (2), i wq can be acquired. By a partial derivative of P Loss in equation 2 Mathematical Problems in Engineering (3) with respect to i wd , optimal stator current to achieve minimum total loss is given by Combined with equation (1), the constraint condition of d-axis current in PMSM under minimum totalloss is

Stator Current Vector Orientation-Based Overall Model of PMSM.
e originally rotating coordinate frame of PMSM is supposed to be dqo. Another rotating coordinate frame d * q * o shown in Figure 2 is established to express stator current vector i s , where d * and q * axes indicate the new real and imaginary axes individually, and θ L is used to represent the angle between d axis and q * axis.
For PMSM, its dynamic equations of stator winding between voltage and current in d * q * o are described as [20] where u s d * and u sq * are the voltages in d * and q * axes, i s d * and i sq * are the currents in d * and q * axes, λ s d * and λ sq * are the flux linkages in d * and q * axes, and ω i is the rotating speed in d * q * o frame. In Figure 2, the stator current i s is oriented along q * axis, which makes i sq * to be equal to i s and i s d * to be limited to zero. Furthermore, equations (6) and (7) can be rearranged as where L d * is the inductor in d * axis and L q * is the inductor in q * axis. Additionally, the relation of θ L among ω r and ω i can be described as follows: Ultimately, the motion equation of the rotor is shown as where T L is the load torque, B is the viscous coefficient, and J is the moment of inertia. e overall dynamical model of PMSM in d * q * o coordinate frame is built from equations (8) to (11). It is worth mentioning that θ L appears to be restricted to the range of (0°, 180°) and a suitable acceleration of the speed should be required in conventional I/f control exhibited to run the motor properly and stably.

Constraint Condition for Torque Ripple Suppression of PMSM.
From the perspective of magnetic coenergy, T e can be expressed as [21] where p indicates d(·)/dθ in the article, K p � 3n p /2, ψ d and ψ q are the flux linkages in d and q axes, and T cog indicates the cogging torque. And because of where k is the harmonic order, I sk and ϕ sk are the kth harmonic current and its phase angle, ψ 0 is the average magnetic linkage in d axis, ψ dk and ψ qk are the kth harmonic components of permanent magnet flux in d and q axes, ϕ ψk is the kth flux harmonic's phase angle, and T ck and ϕ ck are the kth harmonic's magnitude and phase angle in cogging torque.
By substituting equation (13) into equation (12), a more specific expression of T e can be obtained. In reality, ψ dk , ψ qk , and I sk are only small proportions in their respective normal values. e product terms among three variables of ψ dk , ψ qk , and I sk will be smaller which can be ignored. Hence, the expression of T e can be simplified as where T 0 is the active component of T e and ϕ k and φ k are two introduced angles. Considering L d � L q in surface-mounted PMSM, If all harmonics in T e can be inhibited, we can get Furthermore, for the kth harmonic current in equation (14), it also should be suppressed: Because θ L is restricted to the range of (0°, 180°), D k is supposed to be larger than zero. e optimal angle ϕ opt sk for the kth harmonic current will be determined. us, In the combination of equations (15) and (18), the optimal harmonic current I sk can be solved as erefore, equations (5) and (19) constitute constraint conditions for torque ripple minimization with consideration of electrical loss.

Control of Torque Ripple with consideration of Electrical Loss
In this section, a backstepping controller with optimal current conditions is designed. e θ , e θk , e ω , e i , and e ik indicate the tracking errors of angles θ L ,θ Lk , ω r , i s0 , and i sk , where i s0 and i sk are the amplitudes of fundamental current and harmonic current, respectively, and θ Lk is the angle between d axis and harmonic current; , and i * sk are the expected reference signals of θ L , θ Lk , ω r , i s0 , and i sk , respectively. i * sk equals to I opt sk , derived in equation (19). e first virtual control variable θ * L will be devised to regulate the speed and also achieve optimal loss: In accordance with the backstepping control principle and because of e θ � θ L − θ * L , the derivative of e θ can be written as en, the second virtual control variable ω * r can be acquired as follows: where k θ is a pending positive number. By substituting equation (22) into equation (21), equation (21) will be rewritten as

Mathematical Problems in Engineering
In terms of Figure 2 and equation (2), T e can be rewritten as Due to e ω � ω r − ω * r , the derivative of e ω could be expressed as erefore, the third virtual control variable i * s0 can be given by where k ω is a pending positive number. en, equation (20) can be more specific by substituting equation (5) and equation (26) into it. In addition, substituting equation (26) into equation (25), we can get Because of e i � i s0 − i * s0 , the derivative of e i can be expressed as Control variable u sq0 * can be designed as follows: (29) where k i is a pending positive number. By substituting equation (29) into equation (28), equation (28) can be rewritten as Due to and e θk � θ Lk − θ * Lk , the derivative of e θ could be expressed as Control variable u s dk * can be chosen as where k θk is a pending positive number. By substituting equation (33) into equation (32), equation (32) will be rewritten as Because i * sk equals to I opt sk and e ik � i sk − i * sk , the derivative of e ik can be calculated as Control variable u s dk * can be chosen as where k ik is a pending positive number. By substituting equation (36) into equation (35), equation (35) will be rewritten as Based on the above derivation, a closed-loop I/f controller used for suppressing torque ripple and meanwhile guaranteeing minimum electrical loss is proposed. e final voltage control equations of the proposed controller are e schematic of the control scenario is plotted in Figure 3, where speed identification and control parameters prediction will be investigated in the subsequent sections.

Speed Identification of PMSM Based on the Least-Squares Method
e control performance of the proposed method relies on precise speed information. A sensor-less speed identification algorithm is presented in this section. In general, the speed identification algorithm of PMSM could be divided into two types of low speed and high speed. For medium-low speed, the injection of various high-frequency signals [22,23] is widely used to acquire speed information that will inevitably give rise to unnecessary torque ripple. For medium-high speed, applications of stator flux linkage versus position feature [24,25] and various observers such as sliding mode [26,27] and extended Kalman filter (EKF) [28,29] to detect rotor position are normally used. To obtain the stator flux linkage, it is necessary to integrate back electromotive force. But the integrator exhibits zero drift and phase shift during operation, which will limit the accuracy of estimation. e controller of sliding mode is simple and easy to implement, but it may cause the system to chatter. EFK has a strong antiinterference ability. However, it relies heavily on the Mathematical Problems in Engineering parameters of the motor and the algorithm is relatively complex which requires a larger computation. e least-squares method with a forgetting factor (LSFF) has been verified that it has an excellent performance to track a changing signal. e principle of LSFF is described as follows [30]: where B indicates the identified parameters that could be a multidimensional vector; L, P, and φ are the vectors of gain, covariance, and information; y produces the outputs; n represents the sampling point; and ξ is the forgetting factor, 0 <ξ ≤1.
In allusion to the insufficiency in present algorithms of speed identification, LSFF in the study is used to estimate the rotor position of PMSM and the identified structure is built as follows: −n p λ r cos θ L (n) ω r (n) � u s d * (n) + n p ω r (n)L q * i s (n) where T s is the sampling time. Compared with equation (40), the variables required in speed identification from equation By substituting equation (42) into equation (40), the identification law of rotor speed for PMSM is completed.

Online Prediction of Control Parameters
Backstepping control has obvious potential advantages in controlling uncertain nonlinear systems [31]. Merely, the control gains in backstepping control lack the ability to adjust to the operating conditions adaptively. Model predictive control (MPC) provides an effective method to deal with uncertain problems by optimizing a given objective function online [32][33][34]. MPC is introduced in this section to propose a nonlinear predictive algorithm that online optimizes control parameters in backstepping control and improves the robustness of the controller. In order to satisfy the needs of online computation, the model of the system is discretized through the differential method and shown in equation (43).
θ Lk (n + 1) � θ Lk (n) + T s −u sd * k (n) − kn p λ rk ω r (n)cos θ Lk (n) − kn p ω i (n)L q * i sk (n) Mathematical Problems in Engineering en, virtual control variables θ * L , ω r * , and i * s0 are discretized as equation (44), and the voltage control equations are e objective function F is selected as follows: where e � [e θ , e ω , e i , e θk , e ik ] T , M is the predictive time domain, and E is a positive definite matrix. e control parameters K � [k θ , k ω , k i , k θk , k ik ] that make the objective function to be optimal can be obtained by solving constrained linear programming online:

Simulation Certification.
e simulations performed in MATLAB are designed to testify the control method proposed in this section. e study has revealed that only a finite number of harmonics occupy the dominant position in PMSM. eir frequencies are the numbers of six times in dqo coordinate frame, such as the 6th, 12th, and 18th [35]. e investigation in the study demonstrates that the sixth harmonic component occupies a dominant place in total harmonic components, which is selected to be suppressed. e parameters of PMSM are described in Table 1.
e result of the first case is given in Figure 4 to validate the proposed least-squares speed estimation algorithm. Two sets of reference steady-state speed are determined as 20 r/ min and 150 r/min to represent low, medium, and high speeds, respectively. It is clear that the estimation algorithm can track a certain range of reference speeds rapidly and accurately, and the effectiveness of the algorithm is proved. e second case compares closed-loop I/f control with existing I/f control without considering torque ripple and electrical loss. e load torque T L is 15 N·m and PMSM starts up from standstill to 60 r/min. Comparative control performances are given in Figure 5. Compared to the conventionally open-loop I/f control with a larger amplitude of stator current and speed ripple, significant superiority of closed-loop I/f control is less energy consumption and more stable speed. e controller of torque ripple suppression without electrical loss is tested in the third case. e load torque T L is 15 N·m, and PMSM starts up from standstill to 60 r/min. Backstepping controls with and without torque ripple suppression are taken into effect at 15 s, respectively, and the results are plotted in Figures 6(a)-6(d). With the aid of the Mathematical Problems in Engineering      torque ripple suppression algorithm, operating parameters are more stable, and the peak-to-peak value of torque ripple reduces from more than 5 N·m to less than 0.1 N·m. In practice, the electrical loss that is unavoidable, while taking electrical loss into account, the torque ripple δT, and total electrical loss P Loss in the simulation are plotted in Figures 7(a)-7(b). Compared with Figure 6(d), the electrical loss will adversely affect the control performance of torque ripple suppression and P Loss turns out to be quite large which is hard to be ignored. e fourth case is designed to testify the effectiveness of minimum electrical loss control. e load torque T L is 15 N·m, and PMSM starts up from standstill to 60 r/min. e results of backstepping control with and without considering the constraint condition of optimal electrical loss are presented in Figures 8(a)-8(d). Apparently, consideration of optimal loss constraint condition enables to decrease energy consumption and stabilizes operation variables of PMSM effectively.
e fifth case will investigate the closed-loop I/f control proposed in the context that integrates torque ripple suppression with optimal electrical loss simultaneously. e load torque T L is 15 N·m and PMSM starts up from standstill to 60 r/min. Figures 9(a)-9(c) and Figures 10(a)-10(b) display the operating parameters, torque ripple δT, and total electrical loss P Loss . In this way, the peak-to-peak value of torque ripple is suppressed to less than 0.1 N·m. And in comparison with Figure 7(b), P Loss is reduced from 48.7 W to 40.5 W. It can be seen that with the proposed closed-loop I/f controller, all variables are regulated to be more stable torque ripple is suppressed effectively, and electrical loss is reduced to a certain extent. e stability and operating efficiency of PMSM are significantly enhanced.

Test motor
Load motor

Hardware Setup and Implementation.
In order to further verify the algorithm proposed in this paper, a hardware platform is established and exhibited in Figure 11. In this section, the dynamic performance of the proposed closedloop I/f control method is tested where the parameters of PMSM are consistent with the simulation. e first experiment is to test the control performance of PMSM with the proposed control when load torque changes suddenly. PMSM starts up from standstill to 60 r/ min. e load torque T L is firstly set to be 10 N·m and increases to 15 N·m at 30 s. en, it decreases to be 10 N·m at 50 s and lasts to the end. Figures 12(a)-12(c) indicate the operating variables. Figure 13(a) gives the waveform of torque ripple δT. Figure 13(b) compares the results of total electrical loss P Loss with and without optimal electrical loss condition. When load torque changes suddenly, all the operating parameters will only suffer a slight oscillation and will restore to its reference very quickly. e proposed closed-loop I/f control can suppress torque ripple and improve operating efficiency. Meanwhile, it shows a good dynamic performance with a sudden variation of load torque. In this experiment, control parameters are firstly chosen by an online prediction algorithm as K � [94, 80, 283, 121, 177]. While load torque changes suddenly, the prediction algorithm swiftly adjusts control parameters to K � [140,163,254,192,287].  e objective function F is plotted in Figure 12(d), which indicates that control parameters can be quickly adjusted if load torque is abruptly changed. e second experiment is to verify the control performance of PMSM with the proposed control when the speed reference changes suddenly. PMSM starts up from standstill to steady state and load torque T L is 15 N·m. e reference speed is firstly set to be 60 r/min and increases to 100 r/min at 30 s; then, it reduces to 60 r/min at 50 s. e waveforms of various variables, torque ripple δT, and total electrical loss P Loss are, respectively, shown in Figures 14(a)-14(c) and 15(a)-15(b). When the speed reference changes suddenly, the variables will still only suffer a slight oscillation and will recover to its reference rapidly, which demonstrates a good dynamic performance with a sudden change of reference speed. In the experiment, control parameters are regulated from K � [94, 80, 283, 121, 177] to K � [207,188,314,268,192] when the speed reference has been changed suddenly at 15 s. e objective function F is presented in Figure 14(d) under the control of the proposed algorithm, which indicates that the proposed prediction algorithm is proved to be effective and robust.

Conclusion
is paper proposed a closed-loop I/f control to combine torque ripple suppression with electrical loss. Meanwhile, this paper also makes some improvements on open-loop I/f control and nonadjustable control parameters of backstepping control. Feasibility and effectiveness of the proposed approach has been validated by both simulations and experiments. e conclusions are as follows: (1) rough the proposed control scheme, the torque ripple of PMSM is suppressed effectively. Electrical loss of PMSM is significantly reduced, and operating efficiency is successfully improved. (2) Every variable of PMSM operates stably with the proposed method; the ripple and oscillation of speed and current barely happen. (3) e dynamic performance of the proposed method is remarkable. Even the reference signal suffers a sudden change; the whole system will remain stable.
Only a slight oscillation will occur at the moment of sudden change and all variables are able to recover to steady-state in a very short time. (4) Online MPC algorithm for control parameters in backstepping control and speed estimation algorithm based on the least-squares method proposed in this paper significantly make up for the short-board of the existing algorithm and greatly improve the stability of the control scheme. However, the parameter optimization criterion used in the prediction algorithm is the sum of squared errors in each prediction period, which is not directly aimed at overshoot. Hence, we will aim at modifying the criterion and increasing the constraint on overshoot in the later research.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest. Mathematical Problems in Engineering 15