## Abstract

A discrete mathematical model of a permanent magnet synchronous motor (PMSM) is established, then the fifth-order cubature Kalman filter (CKF) algorithm is introduced. A Gauss–Newton iterative method is introduced into the iterative process of the fifth-order CKF algorithm to generate the innovation variance and covariance. Therefore, an iterative fifth-order CKF algorithm is proposed as the basis of a PMSM sensorless control is implemented. Then, a PMSM sensorless control based on the iterative fifth-order CKF algorithm is applied to an electric power steering (EPS) system, whose control system is constructed by adopting the typical assist and return control strategy. Finally, to verify the performance of the proposed PMSM sensorless control method, the EPS system model of the PMSM sensorless control is built by using the common phase-locked loop (PLL), the CKF algorithm, the fifth-order CKF algorithm, and the proposed iterative fifth-order CKF algorithm. The simulation analyses and the experimental tests show that the proposed iterative fifth-order CKF algorithm can estimate the PMSM speed with good accuracy and has a strong resistance to disturbances in the load and speed. The assist and return performances of the EPS system are also improved.

## 1 Introduction

Compared with the traditional hydraulic power steering system, the electric power steering (EPS) has many advantages, such as a high efficiency, energy-saving properties, a small size, and easy installation. Hence, it has been increasingly used in various types of automobiles. There are three kinds of electric motors for the EPS system, including the brushless direct current motor, induction motor, and the permanent magnet synchronous motor (PMSM). Among them, the PMSM has several advantages; for instance, a small volume, high power density, high torque-inertia ratio, a small torque fluctuation, and a high reliability. This motor has been increasingly used in EPS systems, which has made the PMSM become the development trend of the automobile EPS system [1,2].

The performance of the EPS system mainly depends on the control of the power-assisted motor; for the PMSM, the rotor position, and speed signal are essential for good control. Currently, the acquisition of the rotor position signal has mainly been achieved by installing a photoelectric encoder or rotary transformer. However, this will not only increase the cost but also reduce the reliability of the system. To solve this problem, many scholars have studied the PMSM sensorless technology in three main aspects. First, the back electromotive force (EMF) integral method has been used [3,4], which has less dependence on the motor parameters and can estimate the rotor position more accurately when the speed is higher and the back EMF is larger. However, when the speed of the motor is lower, the estimation error will become larger when the back EMF is too small. Second, the high-frequency injection method has been mainly applied to salient pole type PMSMs [5,6]. The estimation error of the surface mounted PMSM is large, and accurate motor parameters are needed. Moreover, the high-frequency signal generation requires additional devices, which will reduce the reliability of the system. Third, the model-based adaptive observing method has been adopted, the most common of which is the sliding mode observer method [79], which is insensitive to the parameter perturbation of the motor and has good robustness. However, the chattering phenomenon will affect the accuracy of the rotor position and speed observation. Other model-based adaptive observing methods include the extended Kalman filter method [10,11] and the unscented Kalman filter method [12,13]. The extended Kalman filter method uses a Taylor expansion of the nonlinear system, truncation of the high-order terms and linearization of the nonlinear system. Therefore, a truncation error will occur, and the Jacobian matrix needs to be computed, which requires a large calculation. While the unscented Kalman filter method does not require calculation of the Jacobian matrix, in the process of the numerical calculation, the rounding errors and other issues will cause many defects, such as an asymmetric covariance or a nonpositive definite covariance, which can easily diverge.

To overcome the above shortcomings and achieve an accurate observation of the state of the nonlinear systems, some scholars have proposed a new state estimation method, named that cubature Kalman filter (CKF) [14,15], which has been applied to estimate the rotor position and speed of PMSMs [16]. The CKF is a new method for the state estimation of strongly nonlinear systems, which uses a third-order phase-diameter volume numerical integration rule to calculate the mean and covariance of the random variables after the nonlinear transformation. Compared with previous methods, the CKF has a better nonlinear approximation performance, numerical accuracy, and filter stability, realizes simplicity, and can achieve third-order accuracy.

Based on the above analysis, the discrete mathematical model of the PMSM is established by using the Euler method. The fifth-order spherical-phase radius volume rule and the fifth-order CKF algorithm with accuracy up to the fifth-order are introduced. The innovation variance and covariance generated by the iteration process are improved using a Gauss–Newton iteration method. An iterative fifth-order CKF method is proposed. The algorithm is used to estimate the rotor speed of the PMSM installed in the EPS system, and the PMSM achieves sensorless control. The EPS control system is built using PMSM sensorless control based on the fifth-order CKF algorithm. Finally, the simulation results show that the proposed iterative fifth-order CKF method can effectively reduce the speed estimation error. The PMSM sensorless control based on the iterative fifth-order CKF algorithm can improve the control performance of the EPS system, reduce torque fluctuation, and improve steering comfort.

The rest of this paper is organized as follows: The discrete mathematical model of the PMSM is first established in Sec. 2. Next, the iterative fifth-order cubature Kalman filter algorithm is introduced in Sec. 3. Furthermore, the EPS system design based on the PMSM sensorless control is introduced in Sec. 4. The simulation and experimental tests results are, respectively, given in Secs. 5 and 6. Finally, the conclusion section ends this paper in Sec. 7.

## 2 Permanent Magnet Synchronous Motor Model

In this paper, an EPS system installed in the surface mounted PMSM is studied. The voltage equation of the surface mounted PMSM in the αβ coordinate system is as follows:
${iα′=−RsLsiα−eαLs+uαLsiβ′=−RsLsiβ−eβLs+uβLs$
(1)

In Eq. (1), $iα$ and $iβ$ are the currents of the α-axis and β-axis, respectively, and $Rs$ and $Ls$ are the stator resistance and inductance, respectively.

The back EMF can be expressed as follows:
${eα=−ψfωe sin θeeβ=ψfωe cos θe$
(2)

In Eq. (2), $θe$ is the electric degree of the PMSM rotor, $ωe$ is the electric angular velocity of the rotor, and $ψf$ is the rotor flux.

The equation of the electromagnetic torque can be expressed as follows:
${ω′e=1J(Te−TL−Bωe)θ′e=ωe$
(3)
The state variables can be set as $x=[iα,iβ,ωe,θe]T$, the system input is $u=[uα,uβ]T$, and the system output is $y=[iα,iβ]T$. To facilitate the study, it is assumed that the motor runs steadily in the process of speed estimation, that is to say, the electromagnetic torque $Te$ is equal to the load torque $TL$. Therefore, the PMSM model can be shown as follows:
${x′=f(x(t))+Bu(t)+w(t)y=h(x(t))+v(t)$
(4)
In Eq. (4), $w(t)$ is the process noise, which mainly includes the system disturbance, parameter perturbation and other system noise, and $v(t)$ is the measurement noise. It is assumed that $w(t)$ and $v(t)$ are mutually independent where
$wheref(x(t))=[−RsLsiα(t)+ψfωe(t)sin θe(t)Ls−RsLsiβ(t)−ψfωe(t)cos θe(t)Ls−BJωeωe],B=[1Ls001Ls0000],h(x(t))=[iαiβ]̊$
The Euler method can be used to discretize Eq. (4), so the PMSM discrete model can be described as follows:
${xk+1=f′(xk)+B′uk+wkyk+1=h′(xk)+vk$
(5)

In Eq. (5), $xk$ is the state variable of time $k$, $uk$ is the input variable of time $k$, and $wk$ and $vk$ are the process noise and measurement noise of time $k$, respectively, which satisfy $wk∼(0,Qk)$ and $vk∼(0,Rk)$.

Assume that $T$ is the sample period, then,
$f′(xk)=[(1−TRsLs)iα,k+Tψfωe,k sin θe,kLs(1−TRsLs)iβ,k−Tψfωe,k cos θe,kLs(1−TBJ)ωe,kθe,k+Tωe,k],B′=[TLs00TLs0000],h′(xk)=[iα,kiβ,k].$

## 3 The Iterative Fifth-Order Cubature Kalman Filter Algorithm

Consider the following nonlinear discrete system
${xk=f(xk−1)+wk−1yk=h(xk)+vk$
(6)

In Eq. (6), $xk$ is the state vector, $yk$ is the measurement vector, and $xx∈Rn,yk∈Rm$; $wk−1$ and $vk$ are the Gauss white noise and measurement Gauss noise of the zero mean value system, respectively. They are mutually independent, and their variances are $Qk−1$ and $Rk$, respectively.

The nonlinear Gauss filtering method is often used to filter the nonlinear system as shown in Eq. (6), but in many cases, the solution to the Gauss integral of the nonlinear Gauss filtering algorithm is not easy to solve. Therefore, many methods are used in application to approximately solve the Gauss integral, in which the Kalman filtering method that utilized the third-order spherical-radial cubature rule to approximately solve the Gauss integral is called the CKF method.

In recent years, the fifth-order spherical-radial cubature rule has been applied to approximately solve the Gauss integral with higher a filtering accuracy, which is called the fifth-order CKF method. In this paper, the iterative fifth-order CKF is adapted to achieve the PMSM sensorless control of the EPS system. The fifth-order spherical-radial cubature rule is described in Sec. 3.1.

### 3.1 The Fifth-Order Spherical-Radial Cubature Rule.

The Gauss integral can be transformed into Eq. (7)
$I(f)=∫f(x)exp(xTx)dx$
(7)

where $f(x)$ is the function of the vector or matrix.

If set $x=ry,yTy=1$, then $xTx=r2$, $r∈[0,∞)$. Therefore, Eq. (7) can be transformed into a spherical integral and a radial integral, expressed as follows:
$S(r)=∫Unf(ry)dσ(y)$
(8)

$R=∫0∞S(r)rn−1e−r2dr$
(9)

where $Un$ is the n-dimensional unit sphere, and $σ(⋅)$ are the elements on $Un$.

By using the Gauss–Hermite criterion and the spherical criterion, Eq. (7) can be transformed into
$I(f)=∫0∞rn−1e−r2∫Unf(ry)dσ(y)dr≈∑i=1Nr∑j=1Nswr,iwy,jf(riyj)$
(10)

where $ri$ and $wr,i$ are the points and weights of the phase path integral, respectively, $yj$ and $wy,j$ are the points and weights of the spherical integral, respectively, and $Nr$ and $Ns$ are the points of the phase path integral and spherical integral, respectively.

Equation (10) can be solved by utilizing a different cubature rule, and the different integral results can be obtained. Although the estimation accuracy of the arbitrary-order CKF algorithm based on the arbitrary-order spherical-radial cubature rule will be higher, the operation time of the higher-order CKF algorithm will increase, which will affect the real-time performance of the algorithm. Considering the real-time performance and the system dimension is less than 7 [17], the fifth-order CKF can meet the estimation accuracy requirements. Therefore, in this paper, the fifth-order spherical-radial cubature rule is used to solve Eq. (10).

The fifth-order spherical-radial cubature rule can be described as follows [18]:
$∫Un,5f(y)dσ(s)=wy,1∑j=1n(n−1)/2(f(yj+)+f(−yj+)+f(yj−)+f(−yj−))+wy,2∑j=1n(f(ej)+f(−ej))$
(11)
where the weights $wy,1$ and $wy,2$ can be expressed as
$wy,1=Ann(n+2)$
(12)

$wy,2=(4−n)An2n(n+2)$
(13)

In Eqs. (12) and (13), $An=2πn/Γ(n/2)$ is the superficial area of the unit sphere, and the function $Γ(z)=∫0∞e−λλz−1dλ$.

The point set $yj+$ and $yj−$ can be expressed as
${{yj+}={12(ek+el):k
(14)

where $ej$ is the unit vector of the space $Rn$, meaning the $jth$ element is 1.

According to the moment matching method, when $Nr=2$, the integral points and weights of the fifth-order spherical-radial cubature rule need to meet the following conditions [19]:
${wr,1r10+wr,2r20=12Γ(12n)wr,1r12+wr,2r22=14Γ(12n)wr,1r14+wr,2r24=12(12n+1)(12n)Γ(12n)$
(15)
If $r1$ is regarded as a free variable and its value is 0, then the points and the weights of the fifth-order spherical-radial cubature rule can be obtained according to Eq. (15)
${r1=0r2=12n+1$
(16)

${wr,1=(1(n+2))Γ(n2)wr,2=(n2(n+2))Γ(n2)$
(17)
According to Eqs. (10), (11), (16), and (17), the fifth-order spherical-radial cubature rule $(Nr=2,Ns=2n2)$ meets the standard Gauss distribution $x∼N(x;0,I)$ and can be deduced, expressed as follows:
$∫Un,5f(x)N(x;0,I)dx≈1πn/2∑i=1Nr∑j=1Nswr,1wy,jf(2riyj)=2n+2f(0)+1(n+2)2∑j=1n(n−1)/2[f(n+2yj+)+f(−n+2yj+)]+1(n+2)2∑j=1n(n−1)/2[f(n+2yj−)+f(−n+2yj−)]+4−n2(n+2)2∑j=1n[f(n+2ej)+f(−n+2ej)]$
(18)

From Eq. (18) above, there are $2n2+1$ cubature points applied in the fifth-order spherical-radial cubature rule. When $j=1,2,…,2n(n−1)$, then $wy,j=wy,1$; however, when $j=2n(n−1)+1,2n(n−1)+2,…,2n2$, then $wy,j=wy,2$.

Similarly, for the general Gauss distribution $x∼N(x;x̂,P)$, the fifth-order spherical-radial cubature rule can be expressed as follows:
$∫Un,5f(x)N(x;x̂,P)dx=∫Un,5f(Sx+x̂)N(x;0,I)dx=2n+2f(x̂)+1(n+2)2∑j=1n(n−1)/2[f(n+2S⋅yj++x̂)+f(−n+2S⋅yj++x̂)]+1(n+2)2∑j=1n(n−1)/2[f(n+2S⋅yj−+x̂)+f(−n+2S⋅yj−+x̂)]+4−n2(n+2)2∑j=1n[f(n+2S⋅ej+x̂)+f(−n+2S⋅ej+x̂)]$
(19)

### 3.2 The Iterative Algorithm.

To further improve the accuracy of the fifth-order CKF algorithm, in this paper the Gauss–Newton iteration method is used to improve the variance and covariance of the new information produced by its iteration process, which can reduce the estimation error of the target initial state and the transmission error introduced by the linearized measurement equation.

The following likelihood function is established by taking the one-step state prediction results $x̂k+1/k$ and $P̂k+1/k$ in the fifth-order CKF algorithm as the initial values [20]
$L(xk+1/k,yk+1/k)=exp {12[(xk+1/k−x̂k+1/k)TP̂k+1/k−1(xk+1/k−x̂k+1/k)+(yk+1/k−h(xk+1/k))TRk+1/k−1(yk+1/k−h(xk+1/k))]}$
(20)
The cost function can be defined as follows:
$J(xk+1/k)=(xk+1/k−x̂k+1/k)TP̂k+1/k−1(xk+1/k−x̂k+1/k)+(yk+1/k−h(xk+1/k))TRk+1/k−1(yk+1/k−h(xk+1/k))$
(21)
The maximum likelihood estimation of the likelihood function expressed by Eq. (20) is equivalent to solving for the minimum value of the cost function. The minimum iterative formula of the cost function defined by Eq. (21) can be obtained by using the Gauss–Newton nonlinear iterative method and the linearized measurement equation, which is shown as follows:
$x¯k+1/k(i+1)=x̂k+1/k+P̂k+1/k(Hk+1(i))T(Hk+1(i) P̂k+1/k(Hk+1(i))T+Rk+1/k)−1 ⋅[yk+1/k−h(x¯k+1/k(i))−Hk+1(i)(x̂k+1/k−x¯k+1/k(i))]$
(22)

where $Hk+1/k(i)=∂h(xk+1/k)/∂xk+1/k|xk+1/k=x¯k+1/k(i)$.

According to Eq. (22), the approximate variance and covariance after linearizing the measurement equation can be described as follows:
$Pyy,k+1(i)=Hk+1(i)P̂k+1/k(Hk+1(i))T+Rk+1/k$
(23)

$Pxy,k+1(i)=P̂k+1/k(Hk+1(i))T$
(24)

When calculating Eqs. (23) and (24), the Jacobian matrix $Hk+1/k(i)$ is usually obtained by utilizing the first-order Taylor series of the measurement equation. Therefore, larger truncation errors will appear in the linearization process.

Consequently, in order to decrease the errors, the following method can be used to calculate the Jacobian matrix
${Yk+1/k,j(i)=h(Xk+1/k,j(i))y¯k+1/k(i)=∑j=1mwjYk+1/k,j(i)$
(25)

$Pyy,k+1(i)=∑j=1mwjYk+1/k,j(i)(Yk+1/k(i))T−y¯k+1/k(i)(yk+1/k(i))T+Rk+1/k$
(26)

$Pxy,k+1(i)=∑j=1mwjXk+1/k,j(i)(Yk+1/k,j(i))T−x¯k+1/k(i)(y¯k+1/k(i))T$
(27)

In Eq. (25), $Xk+1/k,j(i)$ is the cubature point of the $ith$ iterative calculation. From Eqs. (25), (26), and (27) above, the cubature points of the measurement equation are applied to calculate the variance $Pyy,k+1(i)$ and covariance $Pxy,k+1(i)$ of the new information, which has no need for linearization. Therefore, the higher accuracy variance $Pyy,k+1(i)$ and covariance $Pxy,k+1(i)$ of the new information can be obtained.

### 3.3 Iterative Fifth-Order Cubature Kalman Filter Algorithm Steps.

For the nonlinear system described in Eq. (6), assume that the state vector $x$ of time $k$ meets $xk∼N(xk;x̂k,Pk)$, then the iterative fifth-order CKF algorithm steps containing the time update and the iterative measurement update are as follows.

1. Time update

• Calculate the cubature points $xk,i$$(i=0,1,…,2n2)$
$xk,i=x̂k+Skγi$
(28)
where $Pk=SkSkT$, and the values of the vector $γi$ can be expressed as follows:
$γi={[00⋯0]T,i=0n+2yi+,i=1,2,…,n(n−1)/2−n+2yi−n(n−1)/2+,i=n(n−1)/2+1,n(n−1)/2+2,…,n(n−1)n+2yi−n(n−1)−,i=n(n−1)+1,n(n−1)+2,…,3n(n−1)/2−n+2yi−3n(n−1)/2−,i=3n(n−1)/2+1,3n(n−1)/2+2,…,2n(n−1)n+2ei−2n(n−1),i=2n(n−1)+1,2n(n−1)+2,…,n(2n−1)−n+2ei−n(2n−1),i=n(2n−1)+1,n(2n−1)+2,…,2n2$
(29)
In Eq. (29), $yj+$ and $yj−$ can be shown by Eq. (14).
• The cubature points generated by Eq. (28) are transferred according to the state transition equation, which can obtain a new cubature point $χk+1/k,i$
$χk+1/k,i=f(xk,i)$
(30)
• The state prediction value at time $k+1$ is $x̂k+1/k$, which can be shown as follows:
$x̂k+1/k,i=∑i=02n2wiχk+1/k,i$
(31)
where the weight value of the $wi$ can be obtained by Eq. (19), shown as follows:
$wi={2/(n+2),i=01/(n+2)2,i=1,2,…,2n(n−1)(4−n)/(n+2)2,i=2n(n−1)+1,2n(n−1)+2,…,2n2$
(32)
• The state error covariance matrix at time $k+1$ is $Pk+1/k$, which can be expressed as follows:
$Pk+1/k=∑i=02n2wiχk+1/k,i(χk+1/k,i)T−x̂k+1/k(x̂k+1/k)T+Qk$
(33)
2. Iterative measurement update

The iterative measurement update is implemented by using $x̂k+1/k$ and $Pk+1/k$ as the initial values. The estimated value of the jth iteration is set as $x¯k+1/kj$, the variance is set as $P̂k+1/kj$($j=0,1,2,…,Nmax$), and $Nmax$ is the maximum number of iterations.

• The new cubature points generated by the jth iteration can be expressed as follows:
$xk+1/k,ij=S¯k+1/kjγi+xk+1/kj$
(34)
where $P̂k+1/kj=S¯k+1/kj(S¯k+1/kj)T$.
• The state and variance estimation generated by the jth iteration are shown as follows:
$Kk+1/kj=Pxy,k+1/kj(Pyy,k+1/kj)−1$
(35)

$xk+1j+1=x̂k+1/k+Kk+1/kj[yk+1/k−h(x¯k+1/kj)−(Pxy,k+1/kj)TPk+1i/k−1(x̂k+1/k−x¯k+1/kj)]$
(36)

$Pk+1j+1=Pk+1/k−Kk+1/kj(Pyy,k+1/kj)(Kk+1/kj)T$
(37)
where $Pxy,k+1/kj$ and $Pyy,k+1/kj$ can be solved by using Eqs. (26) and (27).
• The condition of the iterative termination is described by the following equation:
$‖xk+1j+1−x¯k+1/kj‖≤ε or i=Nmax$
(38)
where $ε$ and $Nmax$ are the preset threshold and the maximum iterations, respectively.
If the number of the iterative termination is $N$, then the estimated state and the covariance at time $k+1$ can be expressed as follows:
$x̂k+1=x̂k+1N$
(39)

$Pk+1=Pk+1N$
(40)

If given the initial values $x̂0|0$ and $P0|0$, the iterative fifth-order CKF algorithm can be realized according to the calculation steps mentioned earlier.

According to the iterative fifth-order CKF algorithm steps mentioned above, the flowchart of the iterative fifth-order CKF algorithm can be shown as in Fig. 1.

## 4 The Electric Power Steering System Design Based on the Permanent Magnet Synchronous Motor Sensorless Control

The sensorless control of the PMSM studied in this paper is achieved by utilizing the iterative fifth-order CKF algorithm, and the PMSM is applied into an automobile EPS system. Therefore, when determining the output torque of the PMSM, the torque needs of the EPS system should be considered. It is necessary, to consider the EPS system torque control strategy when designing the EPS system based on the PMSM sensorless control.

The typical control strategy of the EPS torque control mainly consists of two control modes, namely, assisted control and the return control. Assisted control is mainly used into the operating condition where the vehicle steering resistance is large in pivot steering or at a low vehicle speed to provide an assisted moment for the driver to steer lightly and comfortably. The return control is mainly applied under the two operating conditions: the overshoot of the returnability and the lack of the returnability, which are disadvantageous to the steering handling and stability.

The main function of the EPS system torque control strategy is to determine the EPS control mode according to the vehicle steering operating conditions, then the motor torque $T∗$ required for the EPS system is obtained based on the control mode determined previously.

The main aim of this paper is to research the PMSM sensorless control, so the EPS control mode is not overly discussed. The study of the multimode control of the EPS system can be seen in Ref. [21].

According to the above analysis, the EPS system design based on PMSM sensorless control can be shown as in Fig. 2, which mainly consists of three parts. The first part is the EPS system torque control, whose main function is to determine the two control modes of the EPS system, such as the assisted control and return control. According to the vehicle steering operating conditions, the motor torque $T∗$ required by the EPS system is obtained on the basis of the determined control mode. The second is the determination of the resistance moment of the EPS system, which mainly considers the friction resistance moment of the EPS system itself, the return moment caused by gravity and the self-return moment. It is introduced in detail in Ref. [22], which is not repeated here. The third is the PMSM control system itself. The PMSM direct torque control based on the sensorless control, is implemented by using the iterative fifth-order CKF algorithm introduced in Sec. 2.

## 5 Simulation and Analysis

### 5.1 The Permanent Magnet Synchronous Motor Speed Estimation Simulation Based on the Iterative Fifth-Order Cubature Kalman Filter Algorithm.

To verify the effectiveness of the proposed iterative fifth-order CKF filtering algorithm for the PMSM speed estimation, a PMSM sensorless control system for the EPS system shown in Fig. 2 is established in matlab/simulink. The PMSM speed estimation is implemented by using the PLL method, the CKF algorithm, the fifth-order CKF algorithm, and iterative fifth-order CKF algorithm under different operating conditions, such as starting in no-load, load change, and speed change.

The main PMSM parameters used in the simulation are as follows: the stator resistance is 2.875 Ω, the AC and DC inductors are 8.5 mH, the flux of the permanent magnet is 0.175 Wb, the pole of the rotor has four pairs, the moment of the inertia is 0.0008 kg m2, and the damping coefficient is 0.001 S−1. The given speed of the rotor is 600 rpm, the relationship between the number of iterations of the iterative fifth-order CKF algorithm and the speed estimation error is shown in Fig. 3. It can be seen from the Fig. 3 that when the number of iterations N is greater than or equal to 20, the speed estimation error is less than 0.1 rpm. Therefore, in order to improve the real-time performance of the algorithm, the number of iterations N of the iterative fifth-order CKF algorithm is set at 20 times in the simulation. The speed estimation error and average computation time of one run for these estimation methods are shown in Table 1. Because of the large estimation error of PLL, the computation time is not counted. The given speed of the rotor is 600 rpm. The speed estimation and the estimated error of starting with no load are shown in Figs. 3 and 4, respectively. The speed estimation and the estimated error when the PMSM load torque changes from 3 N·m to 0 N·m are shown in Figs. 5 and 6, respectively. The speed estimation and the estimated error are, respectively, shown in Figs. 7 and 8, when the PMSM speed changes from 600 rpm to 500 rpm.

Figures 4 and 5 are the curves of the speed estimation and estimated error, respectively, when starting with no load to the steady speed of 600 rpm. From the figures, we can see that it takes a long time for the PLL speed estimation method to achieve a stable state. While the final small fluctuation of the PLL estimated values is around the steady speed of 600 rpm, the speed estimated values fluctuate greatly in the start-up process, which is not conducive to the stable start-up of the PMSM. However, the estimated values of the CKF algorithm, the fifth-order CKF algorithm and iterative fifth-order CKF algorithm can quickly achieve a stable state, but in the start-up stage the speed estimation of the CKF algorithm also has a certain fluctuation compared with the fifth-order CKF algorithm and the iterative fifth-order CKF algorithm. And in the stable stage, the estimation error of the iterative fifth-order CKF algorithm is about 0.03 rpm, which is smaller than that of the fifth-order CKF algorithm. In the entire process, from the start-up to the steady speed of 600 rpm, the estimated values of the iterative fifth-order CKF algorithm are consistent with the actual speed of the PMSM, which proves that the iterative fifth-order CKF algorithm has both good accuracy from the start-up to the steady speed of 600 rpm.

The curves of the PMSM speed estimation and the estimated error are shown in Figs. 6 and 7, respectively, when the load torque changes at 0.25 s from 3 N·m to 0 N·m. It can be seen from the figures that when the load changes, the estimation error of the PLL speed estimation algorithm reaches more than 50 rpm, while that of the iterative fifth-order CKF algorithm is the smallest, and the estimation error of the CKF algorithm, the fifth-order CKF algorithm, and the iterative fifth-order CKF algorithm are all below 1 rpm, and the estimated values of the iterative fifth-order CKF algorithm achieve the steady speed of 600 rpm more quickly than the CKF algorithm and the fifth-order CKF algorithm. But the estimation error of the iterative fifth-order CKF algorithm is approximately 0.3 rpm, which is smaller than that of the fifth-order CKF algorithm. Therefore, the simulation results show that when the load torque changes, the estimated accuracy of the iterative fifth-order CKF algorithm is higher, and the time to reach the given speed is the shortest. That is, the PMSM sensorless control based on the iterative fifth-order CKF algorithm has strong antiload disturbance characteristics.

To verify the estimated accuracy of each estimation algorithm when the PMSM speed changes, the given speed of the PMSM is reduced from 600 rpm to 500 rpm at 0.25 s, with the simulation results shown in Figs. 8 and 9. As seen from the figures, the estimation error of the PLL estimation method is very large, reaching nearly 200 rpm. The estimation error of the CKF algorithm is approximately 3 rpm and more accurate than the PLL estimation algorithm, but it takes a long time to reach the steady speed of 500 rpm. The estimation error of the fifth-order CKF algorithm is approximately 1.8 rpm, and the time to reach 500 rpm is shorter than that of the CKF algorithm, but it is still longer than that of the iterative fifth-order CKF algorithm. The estimated values of the iterative fifth-order CKF algorithm also have fluctuations when the PMSM speed changes. However, from the error curve, it is known that its error is approximately zero and it can reach the steady speed of 500 rpm more quickly. The simulation results show that when the speed changes, the iterative fifth-order CKF algorithm has the highest estimated accuracy and the strongest antispeed disturbance ability.

By comparing the simulation analysis, we can see that the proposed iterative fifth-order CKF algorithm has the highest estimated accuracy of the PMSM speed, and has the best ability to antiload disturbance and the antispeed disturbance.

### 5.2 The Electric Power Steering System Simulation of the Permanent Magnet Synchronous Motor Based on the Iterative Fifth-Order Cubature Kalman Filter Algorithm.

To verify the application effect of the PMSM sensorless control proposed in this paper, in the EPS system, the system model shown in Fig. 2 was built in matlab/simulink. The estimation of the PMSM speed signal is implemented using four methods, namely, the PLL method, CKF algorithm, fifth-order CKF algorithm, and the iterative fifth-order CKF algorithm. By comparing the simulation results of the EPS system based on these four PMSM sensorless control systems, the superiority of the iterative fifth-order CKF algorithm is verified. The simulation of the assisted control and return control was conducted for the EPS system based on the PMSM, and the results are shown in Figs. 10 and 11.

Figure 10 shows the simulation of the assistance performance of the EPS system based on the four methods of the PMSM sensorless control described above. As seen from the figure, all four PMSM sensorless control methods mentioned above can be applied to the EPS system, and the assisted control of the EPS system can also be better completed. However, when the steering wheel begins to turn, the torque fluctuation of the PLL estimation method is the largest, the CKF algorithm is the second, the fluctuation of the fifth-order CKF algorithm is smaller than that of CKF, and the iterative fifth-order CKF algorithm is the smallest, which is also consistent with the estimation accuracy of the PMSM speed simulation. Therefore, the EPS system of the PMSM sensorless control based on the iterative fifth-order CKF algorithm has the best steering comfort performance and the least torque fluctuation during the steering manipulation.

Figure 11 shows the simulation of the return ability of the EPS system based on the above four methods of the PMSM sensorless control. The steering wheel is turned in a circle to one side, held for a period of time, then released. It can be seen from the figure that the EPS system of the PMSM sensorless control based on the above four methods can make the steering wheel return to the center. However, in the process of returning to the center, the steering wheel angle fluctuation of the EPS system based on the PLL estimation method is the largest, the CKF algorithm is the second, the fifth-order CKF algorithm fluctuates less than that of CKF, and the fluctuation of the EPS system based on the iterative fifth-order CKF algorithm is the smallest. Therefore, the EPS system utilizing the PMSM sensorless control based on the iterative fifth-order CKF algorithm has the best steering comfort performance and a smaller steering wheel angle fluctuation during the steering handling.

## 6 Experimental Tests

### 6.1 The Permanent Magnet Synchronous Motor Speed Estimation Experiment Based on the Iterative Fifth-Order Cubature Kalman Filter Algorithm.

To further verify the practical effect of the iterative fifth-order CKF algorithm proposed in this paper for the PMSM speed estimation, the motor test platform, as shown in Fig. 12, is built. The PLL method, CKF algorithm, fifth-order CKF algorithm, and iterative fifth-order CKF algorithm are used to estimate the PMSM speed under three operating conditions, namely, starting with no load, load change ,and speed change. The motor test platform includes the motor driven test box, the upper monitor, the PMSM and loading motor. The control circuit and driving circuit are integrated in the motor driving experiment box. The main control chip TMS320F28335 DSP (Texas Instruments, Dallas, TX) provides the SVPWM signal, and the driving circuit drives the PMSM to realize the driving control of the PMSM. In the test, the actual speed measured by the photoelectric encoder on the test platform is compared with the estimated value, and the test results are shown in Figs. 1318.

Figures 13 and 14 are the curves of the speed estimation and estimated error, respectively, when starting with no load to the steady speed of 600 rpm. It can be seen from Fig. 13 that under the PMSM starting with no load, the PLL algorithm has the maximum speed overshoot, the speed overshoot of the CKF algorithm and fifth-order CKF algorithm is between the PLL algorithm and iterative fifth-order CKF algorithm, and the iterative fifth-order CKF algorithm has the minimum speed overshoot. After the PMSM runs smoothly, the estimation value of the iterative fifth-order CKF algorithm is the closest to the actual speed of the motor. From the speed estimation error curve in Fig. 14, it can be seen that the speed estimation accuracy of the iterative fifth order CKF is highest and the error is minimum. According to the test data, the root-mean-square error of the PLL algorithm estimation error is greater than 80 rpm; the root-mean-square error of the CKF algorithm estimation error is approximately 24.51 rpm, with an average value of 7.44 rpm; the root-mean-square error of the fifth-order CKF algorithm estimation error is approximately 16.62 rpm, with an average value of 4.81 rpm; and the root-mean-square error of the iterative fifth-order CKF algorithm estimation error is approximately 12.27 rpm, with an average value of less than 1 rpm. Therefore, in the entire process, from startup to the steady speed of 600 rpm, the estimated values of the iterative fifth-order CKF algorithm are consistent with the actual speed of the PMSM, which indicates that the iterative fifth-order CKF algorithm has good accuracy from startup to the steady speed of 600 rpm.

To verify the estimation accuracy of each algorithm when the PMSM load changes, the load change test is carried out. During the test, when the load of the PMSM changes from 3 N·m to zero at 0.25 s, the speed estimation and error curve of PMSM are as shown in Figs. 15 and 16. It can be seen from Fig. 15 that the overshoot of the PLL algorithm speed estimation is the largest, and the speed estimation accuracies of the CKF algorithm, fifth-order CKF algorithm, and iterative fifth-order CKF algorithm are good. When t = 0.25 s, the load changes. Other than the PLL algorithm, the algorithms have small speed fluctuations. From Fig. 16, we can see that the PMSM speed estimation error of the iterative fifth-order CKF algorithm is the smallest, and the error overshoot is less than 10 rpm. The speed estimation error overshoot of the CKF algorithm is approximately 18 rpm, and that of the fifth-order CKF algorithm is approximately 26 rpm. Therefore, the experimental results show that when the load torque changes, the estimated accuracy of the iterative fifth-order CKF algorithm is higher, and the time to reach the given speed is the shortest. That is, the PMSM sensorless control based on the iterative fifth-order CKF algorithm has strong antiload disturbance characteristics.

To verify the estimated accuracy of each estimation algorithm when the PMSM speed changes, the given speed of the PMSM is reduced from 600 rpm to 500 rpm at 0.25 s, with the experimental results shown in Figs. 17 and 18. From the speed estimation experiment curve in Fig. 17, we can see that in the process of a speed change, the four estimation algorithms can realize the PMSM speed estimation, but the PMSM speed estimation accuracy of the iterative fifth-order CKF algorithm is the highest, the speed fluctuation is the smallest, and the PLL algorithm is the worst. It can be seen from the speed estimation error curve in Fig. 18 that the PLL algorithm has the largest speed estimation error and takes a long time to reach stability, with a poor dynamic performance. After the speed changes, the speed overshoot is the largest, and the antispeed disturbance performance of the PLL algorithm is the worst. Compared with the PLL algorithm, the speed estimation errors of the CKF algorithm and fifth-order CKF algorithm are obviously improved, and the error fluctuation is approximately 35 rpm. The PMSM speed estimation error of the iterative fifth-order CKF algorithm is the smallest and can estimate the rotor speed well even when the speed changes; the time to reach the stability is the shortest, and the error fluctuation is within 10 rpm in the steady-state. The experimental results show that when the speed changes, the iterative fifth-order CKF algorithm exhibits the highest estimated accuracy and the strongest antispeed disturbance ability.

### 6.2 The Electric Power Steering System Experiment of the Permanent Magnet Synchronous Motor Based on the Iterative Fifth-Order Cubature Kalman Filter Algorithm.

To verify the EPS system performance of the PMSM sensorless control based on the iterative fifth-order CKF algorithm, the hardware-in-the-loop (HIL) test platform of the EPS system equipped with the PMSM is built as shown in Fig. 19.

The HIL test system consists of a personal computer monitoring system, a real-time simulation system, a hardware system, and a high-speed interface system. The main function of the personal computer monitoring system is as a human–computer interaction platform, through which the parameters of the running model can be easily modified in the real-time simulation system, and the running state of the simulation model under the control system can also be monitored. The real-time simulation system is the core of the whole HIL test system (Anhui Cusp Intelligent Technology Co., Ltd., Hefei, China). In this paper, the PXI host of NI company is used as the real-time simulation system, and the PMSM sensorless EPS controllers based on PLL, CKF algorithm, fifth-order CKF algorithm and iterative fifth-order CKF algorithm are constructed by LabVIEW software according to Fig. 2 and run on the PXI host in real-time. The hardware system mainly includes the EPS system equipped with PMSM, the various kinds of sensors, the electronic control unit circuit board and steering resistance simulation system. The high-speed interface system connects its control output signal with the PMSM of the EPS system to control the PMSM and feeds back the signals collected by sensors after control to the controller running on PXI in real-time to realize closed-loop control.

In this paper, the HIL test platform of the EPS system is used to carry out the assisted control and return control test, and the PMSM sensorless control performances of the PLL, the CKF algorithm, fifth-order CKF algorithm, and iterative fifth-order CKF algorithm are compared. The test results are shown in Figs. 20 and 21.

The results of the assisted control HIL test curve shown in Fig. 20 are consistent with the simulation results. When the steering wheel begins to turn, the torque fluctuation of the PMSM sensorless EPS control system based on the PLL estimation method is large, while that of the CKF algorithm and fifth-order CKF algorithm is smaller than that of the PLL algorithm. The torque fluctuation of the PMSM sensorless EPS control system based on the iterative fifth-order CKF algorithm is the smallest, and the HIL test results also show that the speed estimation accuracy of the PLL method is the lowest, so the performance of the PMSM sensorless EPS control system based on the PLL method is the worst, the speed estimation accuracy of the iterative fifth-order CKF algorithm is the highest, the control performance is the best, and the torque fluctuation is the smallest and is relatively gentle. Therefore, the EPS system of the PMSM sensorless control based on the iterative fifth-order CKF algorithm has the best steering comfort performance.

Figure 21 shows the HIL test curves of the return ability of the EPS system based on the above four methods of the PMSM sensorless control. The steering wheel is turned in a circle to one side, held for a period of time, and then released. The EPS system based on PMSM sensorless control with four methods can make the steering wheel return to the center, and the steering residual angle fluctuates near zero. However, in the process of returning to the center, the steering wheel angle fluctuation of the EPS system based on the PLL estimation method is the largest, and the fluctuation of the EPS system based on the iterative fifth-order CKF algorithm is the smallest, which also shows that the return performance of the PMSM sensorless EPS control system based on the iterative fifth-order CKF algorithm is better than that of the other three methods and has the best steering comfort performance.

Through the HIL comparison test of the PMSM sensorless EPS control system based on the four methods, it can be seen that the performance of the PMSM sensorless EPS control system based on the iterative fifth-order CKF algorithm with its higher estimation accuracy, is better than that of the other three PMSM sensorless control methods. The experiment also shows that the performance of the EPS system equipped with the PMSM can be improved by improving the PMSM control performance.

## 7 Conclusion

Aimed at the PMSM sensorless control problems of the EPS system in this paper, a PMSM discrete mathematical model is established using the Euler method, and an iterative fifth-order CKF algorithm is applied to the PMSM speed estimation. The iterative fifth-order CKF algorithm is compared with the common PLL method, the CKF algorithm, and the fifth-order CKF algorithm speed estimation in the simulation and experimental tests. The results show that the iterative fifth-order CKF algorithm has the highest speed estimation accuracy for the PMSM. Compared with the other three methods, under the conditions of starting in no-load, load change, and speed change, the PMSM sensorless control based on the iterative fifth-order CKF algorithm is more accurate and has a better antiload disturbance and the antispeed disturbance. Then, the PMSM sensorless control based on the iterative fifth-order CKF algorithm is applied to the EPS system. The steering assist and return performances of EPS system based on the PMSM sensorless control proposed in this paper are verified by simulation and experimental tests. The results show that the EPS system using PMSM sensorless control based on the iterative fifth-order CKF algorithm has the best assist and return control performances, which also shows that EPS system performance can be improved by enhancing the PMSM sensorless control performance. The iterative fifth-order CKF algorithm has the best estimation accuracy in the simulations presented in this paper, but its real-time performance is hampered by the large amount of computation demanded by the iterative process. Therefore, in order to promote its practical application, two ways of improving its real-time performance can be investigated: First, the iterative fifth-order CKF algorithm can be improved by updating the cubature points directly by iteration and avoiding generating the cubature points by using the Gauss approximation and the root-mean-square method. This approach could reduce the calculation time and improve the algorithm's real-time performance. Second, the real-time performance of the iterative fifth-order CKF algorithm could be optimized by programming in a more efficient computer language. Thus, improving the real-time performance will be the next important work.

## Acknowledgment

Authors are grateful for helpful comments from referees to improve this paper. And thanks are due to Anhui Cusp Intelligent Technology Co., Ltd., for assistance with the experiments.

## Funding Data

• National Natural Science Foundation of China (Grant Nos. 51605003 and 51575001; Funder ID: 10.13039/501100001809).

• Anhui Science and Technology Project (Grant No. 1604a0902158; Funder ID: 10.13039/501100010816).

• Introduction of Talents Science Research Foundation of Anhui Polytechnic University (Grant No. 2016YQQ002).

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