Harmonic Injection Control of Permanent Magnet Synchronous Motor Based on Fading Memory Kalman Filtering

Abstract

In order to reduce the harm of harmonic disturbances of the permanent magnet synchronous motor (PMSM) control system to the drive and improve the accuracy of the control system, a harmonic injection control method based on asymptotic fading memory Kalman filtering is proposed. Compared with the traditional harmonic injection method, this method reduces the torque pulsation of the PMSM, converges faster, and realizes fast stabilization of the control system. In order to improve the control accuracy of the system, extended Kalman filtering is used to estimate the mechanical angular velocity and optimize the harmonic extraction process to reduce the interference of noise signals in the extraction process. At the same time, a fading memory factor is introduced to replace the fixed gain in the Kalman filter, which can correct the system error and effectively prevent the filtering dispersion, thus enhancing the system’s stability. Finally, the system is simulated and experimentally analyzed, and the results show that the method can improve the dynamic response speed, stability, and control accuracy of the system compared with the traditional harmonic injection method.

1. Introduction

The PMSM system has advantages of light transmission weight and strong overload resistance. It has been rapidly applied and developed in the metallurgical industry and the new energy vehicle industry. Due to the motor cogging effect, the winding design, the dead zone effect of the inverter, the measurement error of the current, and the angular velocity sensors, there are a series of harmonic interferences in the current of the PMSM, which significantly affects the stability of the control system and increases the drive loss. Among them, the fifth and seventh harmonic components will bring the sixth pulsation to the PMSM electromagnetic torque, which has a particular impact on the control stability, noise, and vibration of the motor [1,2].

To reduce torque pulsations and losses in the drive circuit, the harmonic currents must be effectively suppressed. On the one hand, harmonic-induced PMSM torque pulsations can be suppressed and improved by a structural optimization design. In terms of motor body design, the primary research to suppress harmonic currents is to improve the motor body’s cogging and magnetic field distribution to avoid the resonance mode phenomenon in specific cases [3,4,5]. Literature [6] proposed a permanent magnet synchronous motor with an asymmetric turn-self structure, which can effectively reduce the distortion rate of rotor leakage and reverse electromotive force and realize the suppression of harmonic current. Prof. Xu Xiaofeng established a model of cogging torque harmonics and the connection between motor cogging effects and cogging torque harmonics [7].

On the other hand, harmonic suppression through the control strategy is also a more effective method. The commonly used PI controller has a simple structure and is easy to implement, but it cannot effectively track the harmonic signals and has a poor control effect [8]. Feed-forward control has a specific control effect for harmonic suppression, but it relies on high-precision parameters and accurate mathematical modeling [9]. Many scholars have achieved harmonic suppression by improving controllers, such as sliding mode control and self-adaptation control. These controllers are simple in structure and easy to implement, but they need to be optimized by algorithms in practical applications [10,11]. In [12], Prof. Ying Qu proposed a self-resistant control strategy with the linear–nonlinear switching state error feedback control law, which can improve the system control accuracy but also has the problem of amplifying the noise signal. In [13], the stability of the system is improved by introducing fuzzy control and the extended state observer to increase the order of the self-immobilizing controller, thereby improving the suppression of harmonic currents. In [14], the combination of a self-adaptation controller and a Fal function filter effectively avoids the problem of amplification of unwanted noise signals, but the process of determining the parameters in the Fal function is complicated. In addition to the suppression of harmonic currents through controllers, harmonic injection control methods have better harmonic suppression capabilities and are widely used in permanent magnet synchronous motors [15,16,17,18]. Harmonic injection algorithms rely on filtering and extracting harmonic currents, so effectively extracting harmonic signals is crucial for the control effect. In literature [19], the adaptive control algorithm improves the filter to optimize the harmonic current extraction process. Still, this method is suitable for processing smooth signals and has the disadvantages of a complex training algorithm and a long time. To suppress the error caused by harmonic injection signals, a position observer with a memory factor can be constructed for realistic error tracking and compensation through an adaptive iterative learning control strategy [20,21].

Considering the effectiveness of the harmonic injection method for PMSM harmonic suppression as well as the robustness and reliability of sensorless, this paper proposes an asymptotic fading memory Kalman-filtered harmonic injection (FMKF-HI) method for torque pulsation suppression, which optimizes the extraction process of the traditional harmonic injection method, obtains a high-precision electrical signal, and improves the robustness and dynamic response of the whole system. It has good torque ripple suppression and phase current harmonic suppression. Simulations and experiments verify the effective suppression effect of harmonic current and ripple torque. The control algorithm innovations proposed in this paper are as follows:

(1) Improvement of system nonlinearity. In the state estimation of the control system, the formula derivation process under the new coordinate system is rewritten to reduce the system’s nonlinearity, shorten the sampling period, save the calculation time, and improve the control precision.

(2) Improve the accuracy. To achieve higher control precision and reduce system interference from observation noise, the FMKF-HI method is used to suppress harmonics, and the fading memory factor is introduced in the realization process, which can dynamically adjust the state estimation and prediction process, prevent the system divergence, and improve the robustness of the system.

(3) System verification. Compared with PI control and existing suppression control algorithms, the designed FMKF-HI control method has better control accuracy and robustness, which verifies the superiority of the designed system.

2. Harmonic Voltage Compensation Model

PMSM in the operation process, because of the motor body structure design, control, and other reasons, will cause the current distortion so that it contains 5, 7, 11, and other harmonic components. In the drive circuit, because the dead zone presents a 6 k ± 1 voltage harmonic, the tube voltage drop will also cause a 6 k ± 1 harmonic. Whether the motor back electromotive force harmonic or the inverter output voltage harmonic, the harmonic amplitude is attenuated with an increase in harmonic frequency, so the 5^th and 7^th harmonics are mainly considered. In this paper, the fifth and seventh harmonics that cause the sixth ripple of electromagnetic torque are taken as the research object, and the relevant mathematical model and harmonic suppression model are established.

In order to facilitate the establishment of the mathematical model and the design of the controller, the synchronous rotating coordinate system $d$—$q$ is selected to establish the parametric equation. The three-phase PMSM stator voltage equation is [22]:

$$\left\{\begin{array}{l}{u}_d=L_d\frac{d{i}_d}{dt}-\omega L_q{i}_q+R_s{i}_d\\ u_q=L_q\frac{d{i}_q}{dt}+\omega L_d{i}_d+R_s{i}_q+\omega {\phi }_f\end{array}\right.$$

(1)

The steady-state voltage equation of surface-mounted three-phase PMSM is rewritten in the $d$—$q$ axis as follows:

$$\left\{\begin{array}{l}{u}_d=-\omega L_q{i}_q+R_s{i}_d\\ u_q-\omega {\phi }_f=\omega L_d{i}_d+R_s{i}_q\end{array}\right.$$

(2)

where $u_d$ and $u_q$ are PMSM $d$ and $q$ axis voltage, $i_d$ and $i_q$ are $d$ and $q$ axis current, $L_d$ and $L_q$ are $d$ and $q$ axis inductance, ${\phi }_f$ is PMSM permanent magnet flux parameter, $R_s$ is stator resistance, and $\omega$ is the angular velocity of fundamental wave voltage.

In an ideal case, without considering harmonics, the above formulas are all DC components. However, in practice, due to the structural design and control of the motor body, it will contain a series of harmonic components, and the suppression and attenuation of harmonic current components cannot be realized by relevant coordinate transformation. The existence of a large number of harmonic components also does not take advantage of subsequent controller design and control accuracy improvement. In the actual control process, the fifth and seventh harmonics account for a large proportion; so this paper will analyze and study the fifth and seventh harmonics.

In the static three-phase coordinate system, the rotation speed of the fifth harmonic voltage component is 5$\omega$, and the rotation direction of the harmonic voltage component is opposite to the rotation direction of the fundamental wave voltage vector. The rotation speed of the seventh harmonic voltage component is 7$\omega$, and the rotation direction of the harmonic voltage component is the same as the rotation direction of the fundamental wave voltage vector. Based on the above analysis, the three-phase PMSM current equation can be rewritten as:

$$\left\{\begin{array}{ll}{u}_a=& u_1\sin (\omega t+{\theta }_{u1})+u_5\sin (-5\omega t+{\theta }_{u5})\\ & +u_7\sin (7\omega t+{\theta }_{u7})+\cdot \cdot \cdot \\ u_b=& u_1\sin (\omega t+{\theta }_{u1}-\frac{2}{3}\pi )+u_5\sin (-5\omega t+{\theta }_{u5}-\frac{2}{3}\pi )\\ & +u_7\sin (7\omega t+{\theta }_{u7}-\frac{2}{3}\pi )+\cdot \cdot \cdot \\ u_c=& u_1\sin (\omega t+{\theta }_{u1}+\frac{2}{3}\pi )+u_5\sin (-5\omega t+{\theta }_{u5}+\frac{2}{3}\pi )\\ & +u_7\sin (7\omega t+{\theta }_{u7}+\frac{2}{3}\pi )+\cdot \cdot \cdot \end{array}\right.$$

(3)

where $u_1$, $u_5$, and $u_7$ are the amplitudes of fundamental wave, 5^th harmonic voltage, and 7^th harmonic voltage, respectively; ${\theta }_{u1}$, ${\theta }_{u5}$, and ${\theta }_{u7}$ are the initial phase angles of the fundamental, 5^th, and 7^th harmonic voltages, respectively.

According to the theoretical analysis and verification above, to simplify the mathematical model, the voltage equation in PMSM was changed into an equation containing fundamental voltage and harmonic component voltage in the synchronous rotating coordinate system, which can be expressed as:

$$\left\{\begin{array}{l}{u}_d=u_{d1}+u_5\cos (-6\omega t+{\theta }_{u5})+u_7\cos (6\omega t+{\theta }_{u7})\\ u_q=u_{q1}+u_5\sin (-6\omega t+{\theta }_{u5})+u_7\sin (6\omega t+{\theta }_{u7})\end{array}\right.$$

(4)

where $u_{d1}$ and $u_{q1}$ represent the direct axis and quadrature axis voltage components of the fundamental voltage on the synchronous rotating coordinate axis.

In the process of coordinate transformation, in the case of trigonometric function operation, the voltage harmonic components mentioned above will appear as 6th-order components in the $d$—$q$ synchronous rotation coordinate axis.

When PMSM is in a motorized state, factors such as motor heating, motor structure, and magnetic circuit saturation effects can cause flux distortion. In order to simplify the model formula, the fundamental magnetic flux pole pair of three-phase PMSM is set to P. According to theoretical derivation and motor model analysis, it can be obtained that the 5^th harmonic magnetic flux pole pair exhibits 5P, the 5^th harmonic magnetic flux rotation direction is different from the fundamental ware, the 7^th harmonic magnetic flux pole pair exhibits 7P, and the 7^th harmonic magnetic flux rotation direction is the same as that of the fundamental ware. PMSM permanent magnet flux ${\phi }_f$ will generate corresponding electromotive force $u_{\phi }$ in the rotation process, and the electromotive force $u_{\phi }$ generated by magnetic flux rotation can be transformed into synchronous rotation coordinates by coordinate transformation.

PMSM permanent magnet flux ${\phi }_f$ generates corresponding electromotive force $u_{\phi }$ during the rotation process, which can be obtained by transforming the electromotive force $u_{\phi }$ generated by the rotating flux into synchronous rotation coordinates through coordinate transformation.

$$u_{\phi }=\omega {\phi }_{f1}-5\omega {\phi }_{f5}\sin (-6\omega t+{\theta }_{\phi 5})+7\omega {\phi }_{f7}\sin (6\omega t+{\theta }_{\phi 7})$$

(5)

In the formula, ${\phi }_{f1}$, ${\phi }_{f5}$, and ${\phi }_{f7}$ are the fundamental wave, 5^th harmonic flux, and 7^th harmonic flux amplitude of permanent magnet, respectively. ${\theta }_{\phi 5}$ and ${\theta }_{\phi 7}$ are the initial phase angles of the 5^th and 7^th harmonic motion potentials.

Hypothesis

$$\left\{\begin{array}{l}{u}_d^{*}=u_d\\ u_q^{*}=u_q-u_{\phi }\end{array}\right.$$

(6)

Joining Equation (2) with Equation (6), we get:

$$\left\{\begin{array}{l}{u}_d^{*}=-\omega L_q{i}_q+R_s{i}_d\\ u_q^{*}=\omega L_d{i}_d+R_s{i}_q\end{array}\right.$$

(7)

The harmonic voltage problem caused by the motor cogging effect and inverter control circuit can also cause the generation of harmonic currents, which contain similar high-order harmonic components. The rotation and rotation direction of each order of harmonic currents and the corresponding order of harmonic voltages are the same.

Similar to harmonic voltage analysis, the current equation in PMSM is changed into an equation containing fundamental wave current and harmonic component current, and the current equation is rewritten as:

$$\left\{\begin{array}{l}{i}_d=i_{d1}+i_5\cos (-6\omega t+{\theta }_{i5})+i_7\cos (6\omega t+{\theta }_{i7})+\cdot \cdot \cdot \\ i_q=i_{q1}+i_5\sin (-6\omega t+{\theta }_{i5})+i_7\sin (6\omega t+{\theta }_{i7})+\cdot \cdot \cdot \end{array}\right.$$

(8)

where $i_{d1}$ and $i_{q1}$ are the direct axis and quadrature axis current components of the fundamental current under the synchronous rotation axis. ${\theta }_{i5}$ and ${\theta }_{i7}$ are the initial phase angles of the 5^th and 7^th harmonic currents.

According to the principle of coordinate transformation and the correspondence between higher-order harmonic components and coordinate systems, components with the same rotation direction and speed as synchronous rotation axis systems exhibit DC characteristics. Therefore, 5^th and 7^th harmonic components exhibit DC components in their corresponding 5^th and 7^th rotation axis systems, while AC components exhibit AC components in the rotation axis of fundamental and other frequencies. The schematic diagram for selecting a coordinate system is shown in Figure 1.

By combining the above analysis and substituting Equation (8) into Equation (7), the following is obtained:

$$\left\{\begin{array}{l}{u}_d^{*}=-\omega L_q{i}_{q1}\sin (6\omega t+{\phi }_1)+R_s{i}_{d1}\cos (6\omega t+{\phi }_1)\\ +5\omega L_q{i}_{q5th}+R_s{i}_{d5th}-7\omega L_q{i}_7\sin (12\omega t+{\phi }_7)\\ +R_s{i}_7\cos (12\omega t+{\phi }_7)+\cdot \cdot \cdot \\ u_q^{*}=\omega L_d{i}_{d1}\cos (6\omega t+{\phi }_1)+R_s{i}_{q1}\sin (6\omega t+{\phi }_1)\\ -5\omega L_d{i}_{d5th}+R_s{i}_{q5th}+7\omega L_d{i}_7\cos (12\omega t+{\phi }_7)\\ +R_s{i}_7\sin (12\omega t+{\phi }_7)+\cdot \cdot \cdot \end{array}\right.$$

(9)

where $i_{d5th}$ and $i_{q5th}$ are the 5^th harmonic currents of the direct axis and the quadrature axis in the synchronous rotating coordinate system, respectively.

Due to the fact that only specific higher-order harmonics are represented as DC components in the corresponding harmonic coordinate system, while they are represented as AC components in the rotating coordinate system corresponding to other orders, in order to obtain the specific harmonic components, the AC components are dropped, and the corresponding state control equation can be obtained by analyzing the 5^th harmonic as the object:

$$\left\{\begin{array}{l}{u}_{d5th}=5\omega L_q{i}_{q5th}+R_s{i}_{d5th}\\ u_{q5th}=-5\omega L_d{i}_{d5th}+R_s{i}_{q5th}\end{array}\right.$$

(10)

Similar to the above analysis, the state control equation for the 7^th harmonic component is rewritten as:

$$\left\{\begin{array}{l}{u}_{d7th}=-7\omega L_q{i}_{q7th}+R_s{i}_{d7th}\\ u_{q7th}=7\omega L_d{i}_{d5th}+R_s{i}_{q7th}\end{array}\right.$$

(11)

where $i_{d7th}$ and $i_{q7th}$ are the 7^th harmonic currents of the $d$-axis and the $q$-axis in the synchronous rotating coordinate system, respectively.

During the operation of PMSM, in order to reduce the proportion of harmonic components and improve the accuracy of the control system, adding corresponding voltage compensation components in the SVPWM modulation process can effectively weaken specific order harmonic components and thereby reduce the adverse effects caused by disturbances.

On the basis of the goals and requirements of the control system, the control objectives are $i_{d5th}^{*}=0$,$i_{q5th}^{*}=0$, $i_{d7th}^{*}=0,$and $i_{q7th}^{*}=0$. The higher-order harmonic component presents the DC component in the coordinate system with corresponding rotation frequency and becomes the AC component in the coordinate system with non-corresponding rotation frequency. Therefore, the corresponding harmonic current can be extracted by designing a suitable filter, which are $i_{d5th}$, $i_{q5th}$, $i_{d7th}$, and $i_{q7th}$, respectively. Take the seventh current harmonic as an example. The difference between the given values $i_{d7th}^{*}=0$ and $i_{q7th}^{*}=0$ and the actual feedback values $∆i_{d7th}$ and $∆i_{q7th}$ of the seventh current harmonics $i_{d7th}$ and $i_{q7th}$ and the PI link with the cross product term is designed according to the steady-state equation obtained by the above analysis. The precise harmonic voltage can be obtained for PMSM with different parameters under different speed conditions. In order to make the system have fast dynamic response and better control accuracy, the feedforward item is constructed in the controller design process to achieve the related control performance. The corresponding harmonic voltage components are obtained by a harmonic suppression algorithm. In the three-phase PMSM control system, the control part is divided into speed outer loop control, current inner loop control, and SVPWM modulation. The compensations $u_{\alpha }^{com}$ and $u_{\beta }^{com}$ obtained from the harmonic current loop constructed by the above analysis and the corresponding sums of $u_{\alpha }^{pi}$ and $u_{\beta }^{pi}$ obtained from the double closed-loop control can form the pulse train control three-phase PMSM after SVPWM modulation.

3. Harmonic Injection Method of Fading Memory Kalman Filter

The calculation amount of the Kalman filter is small, and the state can be used to obtain the optimal estimation of the current state at the present moment. The integration of motor control and combined navigation data has been widely used; however, it is known that it is difficult to do in reality [23,24,25]. Therefore, the problem of filtering and divergence often occurs. In response to this problem, the fading memory factor can effectively suppress the phenomenon of filtering and divergence.

In this paper, a harmonic injection method is proposed to accurately obtain the mechanical angular velocity of PMSM by fading memory Kalman filtering. The mechanical angular velocity is predicted and updated according to the outputs $u_{\alpha }$ and $u_{\beta }$ of the controller and the three-phase current$i_{\alpha }$ and $i_{\beta }$ in the static coordinate system, and the angular velocity information with high accuracy is obtained. Aiming at the divergence problem of the traditional Kalman filter due to the unknown system model and the statistical characteristics of noise, a weighted factor automatic selection algorithm is introduced, which can constantly update the use efficiency of historical data so that the whole filter is not easy to diverge and the control is more stable. On the basis of the traditional harmonic injection, the system adds a mechanical angular velocity update link. This link aims to obtain more accurate angular velocity information and better convergence and then compensates the voltage harmonic component, which is injected into the SVPWM modulation link to offset the voltage harmonic component, which can improve the electromagnetic torque ripple problem of PMSM.

Kalman filtering is used to obtain the optimal estimation of the current state of PMSM. A more accurate state estimation depends on choosing a suitable coordinate system and modeling the motor accurately. In the process of model establishment, the static coordinate system and the rotating coordinate system are common alternative reference systems. In the process of establishing a model using a synchronous rotating coordinate system, the measured electrical parameters of the motor need to be converted into state components in the corresponding coordinate system using a coordinate transformation matrix. In the coordinate transformation process, the coupling term of the trigonometric function will be attached to the flux. On the one hand, the nonlinearity of the system and controller will be greatly increased. On the other hand, the recursive time to obtain the optimal state estimation increases. Therefore, in order to avoid the above problems, the fading memory Kalman filter model can be built through the static coordinate system.

Based on the harmonic injection method of fading memory Kalman filter proposed for linear systems, and in order to reduce the nonlinear and recursive calculation time of system model establishment and improve control accuracy, the steady-state current equation of surface-mount PMSM was rewritten in the static coordinate system as follows [26]:

$$\left\{\begin{array}{l}\frac{d{i}_{\alpha }}{dt}=-\frac{R}{L_s}{i}_{\alpha }+{\omega }_e\frac{{\phi }_f}{L_s}\sin {\theta }_e+\frac{u_{\alpha }}{L_s}\\ \frac{d{i}_{\beta }}{dt}=-\frac{R}{L_s}{i}_{\beta }-{\omega }_e\frac{{\phi }_f}{L_s}\cos {\theta }_e+\frac{u_{\beta }}{L_s}\end{array}\right.$$

(12)

where $i_{\alpha }$ and $i_{\beta }$ are currents in two phase orthogonal coordinate systems, ${\omega }_e$ is the electric angular velocity.

When the motor is in a steady state, there are the following relations:

$$\left\{\begin{array}{l}\frac{d{\omega }_e}{dt}=0\\ \frac{d{\theta }_e}{dt}={\omega }_e\end{array}\right.$$

(13)

According to the establishment of the PMSM system model, appropriate state variables are selected to establish the state equation of the motor control model:

$$\left\{\begin{array}{l}\frac{d}{dt}x=Ax+Bu\\ y=cx\end{array}\right.$$

(14)

$$x=\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\\ {\omega }_e\\ {\theta }_e\end{array}\right] u=\left[\begin{array}{c}{u}_{\alpha }\\ u_{\beta }\end{array}\right] y=\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]$$

(15)

$$A(x)=\left[\begin{array}{c}-\frac{R}{L_s}{i}_{\alpha }+{\omega }_e\frac{{\phi }_f}{L_s}\sin {\theta }_e\\ -\frac{R}{L_s}{i}_{\beta }-{\omega }_e\frac{{\phi }_f}{L_s}\cos {\theta }_e\\ 0\\ {\theta }_e\end{array}\right]$$

(16)

$$B=\left[\begin{array}{c}\begin{array}{cc}\frac{1}{L_s}& 0\\ 0& \frac{1}{L_s}\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}\end{array}\right] C=\left[\begin{array}{cc}\begin{array}{cc}1& 0\\ 0& 1\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\end{array}\right]$$

(17)

The control system is discretized with

$$\begin{array}{c}x(k+1)=A[x(k)]+B(k)u(k)+V(k)\\ y(k)=C(k)x(k)+W(k)\end{array}$$

(18)

where $V\left(k\right)$ and $W\left(k\right)$ are system noise and measurement noise, respectively.

Define the covariance matrix Q and R.

$$\begin{array}{l}\mathrm{cov}(V)=E\left\{V{V}^T\right\}=Q\\ \mathrm{cov}(W)=E\left\{W{W}^T\right\}=R\end{array}$$

(19)

Use the optimal estimate $\widehat{x}\left(k\right)$ and system input $u\left(k\right)$ to obtain the predictive value of the next moment $\tilde{x}\left(k+1\right)$.

$$\tilde{x}(k+1)=\widehat{x}(k)+T_s[A(\widehat{x}(k))+B(k)u(k)]$$

(20)

where $T_s$ is the sampling period of the control system.

Calculation predicted the corresponding control output $\tilde{y}\left(k+1\right)$.

$$\tilde{y}(k+1)=C\tilde{x}(k+1)$$

(21)

Calculated corresponding to the state of the error covariance matrix $\tilde{p}\left(k+1\right)$, in order to make timely use of the new status value to estimate line correction and suppress filter, here, the introduction of fading factor ${\rho }_k$.

$$\tilde{p}(k+1)=\widehat{p}(k)+T_s{\rho }_k[\underset{¯}{A}(k)\widehat{p}(k)+\widehat{p}(k){\underset{¯}{A}}^T(k)]+Q$$

(22)

Among them

$$\underset{¯}{A}(k)=\frac{\partial A(x)}{\partial x}\left|{}_{x=\widehat{x}(k)}\right.$$

(23)

In the process of normal data iteration, ${\rho }_k=1$, if the problem of filtering divergence occurs, it will make the predicted residual increase and lead to an increase in the covariance matrix, so as to increase the fading factor. By introducing the fading factor, the mean square error of prediction can be influenced to make better use of the current system measurement.

The determination method of ${\rho }_k$ is:

$$\begin{array}{l}{\rho }_k=max\left\{1,\frac{1}{n}abs\frac{Tr(N_k)}{Tr(M_k)}\right\}\\ \left\{\begin{array}{l}{N}_k=P_{vk}=CQ{C}^T\\ M_k=CA\tilde{p}(k)A^T{C}^T\end{array}\right.\end{array}$$

(24)

$P_{vk}$ is the covariance matrix for predicting residuals.

$v_k$ is the residual vector of $\widehat{x}\left(k\right)$, expressed as:

$$P_{vk}=E(v_k{v}_k^T)={\displaystyle \sum _i^k{v}_k{v}_k{}^T}$$

(25)

$$v_k=y(k)-C\widehat{x}(k)$$

(26)

${Tr(N}_k)$ and ${Tr(M}_k)$ are traces of the matrices $N_k$ and $M_k$.

Calculate the gain matrix $K(k+1)$:

$$K(k+1)=\tilde{p}(k+1)C^T{[C\tilde{p}(k+1)C^T+R]}^{-1}$$

(27)

The prediction state $\tilde{x}\left(k+1\right)$ obtained by a recursive operation is iteratively optimized, and then the optimal state estimation $\widehat{x}\left(k+1\right)$ at the current moment is obtained. Finally, the compensation state estimation ${\omega }_e^{k+1}$ is added to the harmonic current loop to improve the compensation accuracy. The optimal state estimate of the system is determined by the following equation.

$$\widehat{x}(k+1)=\tilde{x}(k+1)+K(k+1)[y(k+1)-\tilde{y}(k+1)]$$

(28)

where $\tilde{x}\left(k+1\right)$ is the predicted system state, $\widehat{x}\left(k+1\right)$ is the optimal state of the system, and ${\omega }_e^{k+1}$ is the optimal electrical angular velocity of the system.

For the next state estimation prediction, the error covariance matrix needs to be updated, which can be obtained as follows:

$$\widehat{p}(k+1)=K(k+1)C\tilde{p}(k+1)$$

(29)

In the above model construction process, so as to achieve effective suppression of ripple torque and improve the system control accuracy, harmonic suppression is introduced, harmonic components are separated, and harmonic voltage compensation is constructed. In order to improve the accuracy of state estimation and the stability of the system, the fading memory Kalman filter is introduced to improve the anti-interference ability of the system and the control accuracy of the system. The fading factor ${\rho }_k$ is introduced to prevent the divergence of the filter and make the prediction more accurate.

To sum up, the principle of the fading memory Kalman filter harmonic injection torque ripple suppression method is as follows: according to the sampling current of three-phase PMSM, the corresponding 5^th and 7^th harmonic currents are extracted, and the control target is constructed to construct the harmonic current loop. The angular velocity of PMSM is predicted and compensated by the fading memory Kalman filter. With the effect of the fading memory factor, the errors of the system can be corrected in real-time to improve the efficiency and accuracy of the utilization of system observation data.

4. Validation Analysis

4.1. Simulation Experiment

The control model of PMSM was built by Simulink2020b, and the control algorithms in double closed-loop PI, traditional HI control, and FMKF-HI control were compared and verified. The electromagnetic torque waveform, direct axis current waveform, and three-phase current were obtained, and the harmonic analysis was carried out by fast Fourier transform. The parameters of the motor applied for simulation verification are shown in

Table 1. Main parameters of the motor.

Parameter	Value
Rated speed	1000 r/min
Rated power	5.5 kw
Phase resistance	0.62 Ω
Phase inductance	4 mH
Polar number	4
Flux linkage	0.35 Wb

below:

4.1.1. Simulation Results of Torque Ripper

The circuit of Figure 2, showing a PMSM system structure block diagram, is established by Simulink 2020b.

Based on the above analysis, the simulation model parameters are set as follows: Kp = 10, Ki = 3, DC side voltage $u_{dc}$ = 311 v, PWM switching frequency $f_{pwm}$ = 10 kHz, initial load ${\mathrm{T}}_L$ = 0, load torque ${\mathrm{T}}_L$ = 20 N·m at t = 0.2 s, reference rotating speed $N_{nef}$ = 1000 r/min.

The electromagnetic torque of the three control methods is compared in Figure 3. The electromagnetic torque fluctuation curves of the system using PI, HI, and FMKF-HI are shown in the figure. In general, the three control schemes can maintain the stability of electromagnetic torque, but there is a big difference in the convergence rate of electromagnetic torque pulsation and harmonic injection.

Compared with the double-closed-loop method, the periodic torque ripple of the traditional HI and FMK-HI methods is significantly reduced. It can be seen from Figure 3 that the fluctuation amplitude of the double closed-loop method is 6 N·m, that of the traditional HI control method is 3.5 N·m, and that of the FMK-HI method is 1.5 N·m. It can be seen that compared with the traditional PI control and the traditional HI control, the amplitude is reduced by 19.6% and 2.5%, respectively. Compared with the conventional HI control method, the FMK-HI method has a better attenuation and convergence effect in the early stage of harmonic injection, which can satisfy the control accuracy of the system faster.

4.1.2. Current Simulation Results

The given speed of the motor was set to 1000 r/min, and a torque load of 20 N was added during operation. Due to the use of the fading memory Kalman filter harmonic suppression algorithm, it can extract the harmonic components and compensate them. The waveform of the compensated voltage is shown in Figure 4. The harmonic components can be effectively suppressed due to the injection of a compensating voltage in the inverter circuit.

The FMK-HI algorithm improves motor currents and reduces motor drive losses. To show that this method has little influence on the output of the inverter, the bridge arm terminal voltage of the inverter under the action of the three methods is simulated, respectively, at different speeds, and the simulation results are shown in Figure 5.

To more vividly compare the output voltage of the inverter under different control methods and speeds, the average output voltage of the bridge arm on the inverter is introduced as the analysis index, and the simulation results are shown in Figure 6. As the motor speed changes, the average value of the output voltage of the upper bridge arm of the inverter fluctuates up and down at 155.5 V, and the amplitude of the fluctuation is 0.15 V. After the above analysis, the proposed FMK-HI algorithm has less effect on the inverter output voltage.

The currents of the three methods are analyzed in the current simulation verification. The motor runs at a speed of 1000 r/min. The current distortion degree in Figure 7a is relatively high, and the current waveform has significant distortion. Figure 7b shows that the ordinary harmonic injection method has improved the current waveform. However, there is still a certain distortion at the peak of the three-phase current. Figure 7c shows the three-phase current waveform with Kalman harmonic suppression introduced by fading memory. The PMSM phase current has a better sinusoidal wave and better control accuracy.

Figure 8 shows the simulation results of direct-axis current fluctuation under different controls. Figure 8a shows that under PI control, the direct axis current fluctuates up and down with a value of 1.2 A, which is a significant fluctuation. Figure 8b shows that the direct axis current fluctuates up and down under HI control by 0.7 A, and there is a significant sudden change. Figure 8c indicates that the FMK-HI control current fluctuates at 0.55 A, which is more stable. Therefore, compared with the speed and current dual PI control and harmonic injection methods, the straight shaft current fluctuation of FMK-HI control is minor, and the control system is more stable.

The PI control cannot effectively suppress the 5^th harmonic current in the direct and quadrature axes, and the 5^th harmonic current in the quadrature axis has a magnitude of 0.7 A. The simulation results are shown in Figure 9a. The HI control has a certain suppression effect on the 5^th harmonic current. Still, the 5^th harmonic current in the quadrature axis has a magnitude of 0.3 A, and there is a large fluctuation. The simulation results are shown in Figure 9b. Figure 9c shows that the 5^th harmonic current in the straight axis is effectively suppressed under FMK-HI control, and the amount of harmonic current fluctuation is only 0.16 A.

The three-phase current harmonics are analyzed using the fast Fourier transform, and the results are shown in Figure 10. From Figure 10a–c, it can be seen that the corresponding harmonic components decrease from 19.56% to 5.29% and 2.04% and from 2.74% to 1.99% and 1.07%, respectively. The total harmonic distortion of the PI algorithm is 7.95%, and the HI algorithm has a total harmonic distortion of 3.76%. In comparison, the FMK-HI calculation in this paper has a total harmonic distortion of 2.83%. Compared to other algorithms, the FMK-HI control has a better effect on harmonic suppression.

To further verify the superiority of the proposed algorithm, the algorithm in this paper is compared with the adaptive filtering harmonic current suppression algorithm (AF-HCS), and the simulation results are shown in Figure 11. Since AF-HCS is suitable for smooth signals, a long iteration time exists, so the convergence is poor. After stabilizing the system, the electromagnetic torque fluctuation is 2.5 N·m under AF-HCS control and 1.5 N·m under FMK-HI control. The three-phase current waveforms under both control algorithms are improved. However, the current waveform of the AF-HCS control algorithm is still distorted to a certain extent, and the total harmonic component is higher than that of this paper’s algorithm. Therefore, this paper’s proposed control algorithm has better torque pulsation suppression, and the torque pulsation suppression effect is better.

The electromagnetic torque ripple, three-phase current distortion degree, and THD data of the motor are compared, and the comparison data are shown in

Table 2. Comparison of control algorithms.

Control Algorithm	Electromagnetic Torque Ripple Amplitude/N·m	Three Phase Current Distortion Degree	THD
PI	6	Severe distortion	7.95%
HI	3.5	Moderate distortion	3.76%
FMK-HI	1.5	Less distortion	2.83%
AF-HCS	2.5	Less distortion	3.34%

. As shown in Table 2, compared with PI, the torque ripple of HI and AF-HCS is reduced by 2.5 N·m and 3.5 N·m, respectively, but there is still a certain distortion of the three-phase current crest and trough. The FMK-HI algorithm proposed in this paper reduces the electromagnetic torque pulsations by 4.5 N·m and THD from 7.95% to 2.83%. Compared with AF-HCS, FMK-HI electromagnetic torque pulsation is improved by 5%, and the three-phase current waveform is closer to a sinusoidal waveform.

4.2. Experimental Analysis

For the purpose of further verifying the harmonic disturbance compensation algorithm proposed in this paper, experimental verification is carried out through the motor drive experimental platform. The experimental platform comprises dSPACE, PMSM, a control drive system, and the load. The given speed adopted in the experiment is 1000 r/min, the same as in the simulation verification process. The main control CPU is TMS320F28335 of TI Company, the DC sensor model is TBC100BS, the AC sensor model is LAH 25-NP, and the power device is Infineon’s three-phase full-bridge module model FS100R12KT4G. The PWM frequency is 10 kHz. The model is downloaded through a Matlab file to the rapid control prototype (RCP) during the experiment. The experiment includes double-closed-loop control, traditional HI, and FMK-HI methods. Electromagnetic torque information and FFT analysis are obtained through the upper computer and an electrical signal. The experimental platform built into this paper is shown in Figure 12.

Figure 13 shows the phase current waveform obtained using three control methods. Figure 13a–c shows a group of comparison waveforms before and after using the harmonic voltage compensation algorithm when the fundamental frequency is 66.6 Hz. As can be seen from the experimental results in Figure 13, after the addition of the harmonic current suppression algorithm, the current waveform is significantly improved, and the proportion of high-order harmonics in single-phase current in PMSM decreases from 5.92% and 2.68% to 0.1% and 0.04%, respectively.

Three sets of experimental comparisons verify that the FMK-HI algorithm proposed in this paper significantly affects harmonic suppression and reduction of the three-phase current distortion rate. The electromagnetic torque experiments are compared at a given speed of 1000 r/min and a load torque of 20 N·m, and the torque waveform is shown in Figure 14.

As illustrated in Figure 14, compared with PI control, the electromagnetic torque fluctuation of the harmonic injection control method using the fading memory Kalman filter decreases from 11 N·m to 3 N·m; compared with the ordinary harmonic injection control method, the torque fluctuation decreases from 5 N·m to 3 N·m.

5. Conclusions

To solve the problems of low accuracy and poor stability of the harmonic injection control algorithm in the PMSM control system, this paper researches and designs a permanent magnet synchronous motor harmonic injection control method based on attenuated memory Kalman filter. Firstly, considering the influence of system noise and external interference on the harmonic extraction process, an extended Kalman filter is designed to optimize electrical parameters in the harmonic extraction process, effectively reducing the dependence on mechanical sensors. To improve the stability of the control system, the gradual elimination factor is introduced, which can better utilize the measured values of the system and correct the system error according to the predicted data of the system, effectively preventing the problem of filter dispersion and improving the stability of the system. Finally, MATLAB/Simulink simulation and experimental verification were carried out to verify the effectiveness and reliability of the study’s design. Future work will consider the harmonic injection control strategy by system identification under the variation of motor parameters.