Sensorless Position Estimation for Permanent Magnet Synchronous Linear Motor with Variable Coupling Length

Affected by the structure size of a permanent magnet synchronous linear motor, the coupling length of the mover and stator will change when the relative motion occurs. This paper analyzes the influence of the linear motor coupling length change on the position estimation accuracy of the position sensorless algorithm and proposes a Gauss-Newton iterative method to estimate the position. Compared with the existing high-frequency injection method, the proposed method is unaffected by the change of the coupling length, has strong robustness to the change of inductance parameters, and can obtain high position accuracy in the entire motion stroke. In this paper, the finite element method is used to analyze the variation law of the inductance parameter caused by the change of the coupling length, and the influence of the change of the inductance parameter on the position estimation accuracy of the position sensorless algorithm. The high-frequency differential current is obtained by the stationary reference frame high-frequency square-waveform injection method and difference processing, Moreover, the fast Fourier transform in the position domain is carried out to obtain the high-frequency differential current model function. The Gauss-Newton iterative method is used to obtain the estimated position of the linear motor in the entire motion stroke by estimating the position of each position sampling point. Lastly, the position accuracy of the proposed method and the traditional phase-locked loop method is compared and verified through experiments, and the corresponding conclusions are drawn.


I. INTRODUCTION
Permanent magnet synchronous linear motor (PMSLM) is mainly composed of the stator and mover. The stator mainly includes the winding coil and shell, and the mover includes the permanent magnet and mover iron. PMSLM has the advantages of simple structure, high efficiency, good dynamic performance, and direct driving load. It is widely used in the field of linear servo drive such as medical treatment, numerical control (NC) machine tools, and automatic machining [1]- [5]. The position feedback of the linear motor mainly includes the Hall sensor, grating sensor, and so on. These sensors can provide accurate position information to the linear motor, but they have the disadvantages of large space occupation and high cost. Therefore, sensorless methods have received increasing attention and studied in recent years to improve reliability, reduce cost, and reduce system volume.
Position sensorless technology mainly includes two methods: the fundamental frequency back electromotive force (EMF) method and the high-frequency injection method. The fundamental frequency back EMF method [6], [7] uses the fundamental frequency electromotive force of the motor [8], [9] and establishes an observer [10], [11] to obtain the position information of the rotor. This method is suitable for the motor in the high-speed motion region, it is difficult to estimate the rotor position at low speed, and it is not available at zero speed. Another method is the highfrequency injection method [12]- [14], this method is mainly used in medium and low-speed motion regions. The principle of which is to use the saliency characteristics of the motor.
The high-frequency voltage is injected into the fundamental frequency voltage of the stator coil. After a coordinate transformation, rotor position information is included in the high-frequency current response, and the position of the rotor can be extracted by appropriate position demodulation.
For PMSLM, given that the coupling length changes with the movement of the mover, inductance parameters change, thereby making the sensorless position estimation accuracy change substantially. This is a problem that needs to be studied in depth.
At present, many scholars have carried out research on the sensorless algorithm for PMSLM. The inductance unbalance characteristics of the three-phase linear motor and the relationship between the inductance and the thrust are quantitatively analyzed, and a three-stage primary modular structure is proposed to improve the inductance unbalance and suppress the thrust fluctuation in [2]. Inductance characteristics of the two-phase cylindrical permanent magnet synchronous linear motor were analyzed using the finite element method, and the actual inductance parameters of the full coupling stage were introduced into the stationary reference frame high-frequency square-waveform injection algorithm. The extended Kalman filter (EKF) extracts the rotor position and realizes the position detection of the mover in [15]. In [16], flux linkage and inductance characteristics of the TPMSLM in the entire range of motion were analyzed. Inductance and flux linkage expressions of the entire coupling region were established using analytical methods and were verified by finite element and experimental tests. The preceding studies have carried out a series of investigations on the inductance parameters and position detection accuracy of the PMSLM. How to improve the position observation accuracy of the linear motor in the entire coupling stage still needs to be deeply studied.
In this paper, given the large position estimation error of the traditional high-frequency injection sensorless algorithm caused by the change in the linear motor coupling process, a Gauss-Newton iterative position estimation algorithm is proposed to perform high-frequency differential current modeling and position estimation to improve the linear motor. This method is not sensitive to the change of the inductance parameters and can still obtain high position estimation accuracy when the coupling length of the linear motor changes. The remainder of this paper is structured as follows. In the second section, the high-frequency square-waveform injection method of the stationary reference frame is analyzed, and the Gauss-Newton iterative method is proposed to demodulate the position of the linear motor mover. In Section III, inductance characteristics of the linear motor are simulated and analyzed, the high-frequency differential current curve modeling is carried out by fast Fourier transform (FFT) and the simulation analysis of position extraction is carried out by Gauss-Newton iteration. Section Ⅳ compares the proposed method with traditional sensorless position estimation methods through experiments. Lastly, Section V provides the corresponding conclusions.

A. STATIONARY REFERENCE FRAME HIGH-FREQUENCY SQUARE-WAVEFORM INJECTION METHOD
Permanent magnet synchronous motor (PMSM) in the synchronous reference frame can be expressed by (1): where q u and d u are the stator voltage components, d i and q i are the stator current components, respectively, R is the stator winding resistance, d L and q L are the stator inductance components, e  is the mover electrical angular velocity, f  is the permanent magnet flux linkage, and P is the differential operator.
The sensorless algorithm of high-frequency squarewaveform injection aims to increase the frequency of the injected voltage to the order of PWM switching frequency. At this time, the linear motor can be regarded as a purely inductive circuit, so the partial pressure caused by highfrequency impedance can be disregarded [17]. The highfrequency voltage equation in the synchronous reference frame is as follows: According to (4), the high-frequency current equation of the stationary reference frame can be represented as follows: is the inverse matrix of inductance matrix, which is expressed as The high-frequency square-waveform injection method of the stationary reference frame is to inject a high-frequency voltage signal into the - axis. Amplitude of the highfrequency square-waveform voltage in two adjacent PWM cycles is equal and the sign is opposite. The equation is as follows: where inj V is the high-frequency voltage, and h V is the amplitude of the injected high-frequency voltage.
Thereafter, (8) is brought into (5), and the discretized highfrequency current differential equation can be derived as follows: From the (9), there is a DC offset p L in the highfrequency differential current equation of the - axis, and the - and - axes contain the second harmonic related to position. In addition, the amplitudes of the expression related to the position  are equal in the high-frequency differential current equations of the - and - axes.

B. HIGH-FREQUENCY DIFFERENTIAL CURRENT
From the derivation process of the preceding static axis highfrequency square-waveform sensorless method, the highfrequency differential current contains the mover position information. Moreover, the high-frequency differential current is affected by the injected high-frequency voltage amplitude and inductance parameters. The object of this study is the tubular permanent magnet synchronous linear motor (TPMSLM), and its related characteristics are relatively different from the permanent magnet synchronous rotating motor [8]. The TPMSLM is mainly composed of the stator and the mover. In particular, the stator includes the two-phase windings and the stator iron and belongs to a motor with a slotless structure. The mover includes the permanent magnet and the mover iron, and the magnetization direction of the permanent magnet is axial magnetization. The simplified structure of TPMSLM is shown in Fig. 1.

FIGURE 1. Schematic of TPMLSM.
To extract the high-frequency differential current of PMSLM, the high-frequency voltage equation in the stationary reference frame is first expressed as follows: is expressed as the actual inductance matrix [18] of the stationary reference frame as follows: Bringing the actual inductance matrix (11) into the differential current equation in (5), and the derivation results are as follows:   two adjacent PWM cycles, In addition, the differential value of high-frequency current of the - and - axes are obtained through current sampling. The differential expression of the acquired high-frequency current in two adjacent PWM cycles can be represented as follows.
The high-frequency current differential term of the stationary reference frame in the first PWM cycle is expressed as follows: The high-frequency current differential term of the stationary reference frame in the second PWM cycle is expressed as follows: The difference operation is performed on the highfrequency differential current obtained from two adjacent PWM cycles, and the results are as follows: Equation (17) includes the DC offset and second harmonic components. After removing the DC offset part, it can be deduced as follows: After removing the DC offset, the high-frequency differential current equation only contains the second harmonic related to the position. However, the coefficient of the equation is composed of the inductance parameters of TPMSLM, and the preceding inductance parameters also change with the position. Therefore, the position error proposed by the traditional arctangent method or phaselocked loop (PLL) method will have a large position estimation error.

C. DEMODULATE THE MOVER POSITION OF LINEAR
In the process of mover position demodulation, to improve the accuracy of position estimation, this paper proposes the Gauss-Newton iterative method for position estimation and compares it with the traditional PLL method.

1) PLL METHOD
A traditional method for extracting the mover position is the PLL method [19], which consists of a proportionalintegral (PI) controller and integrator, which can convert input error signal into a position signal. In this process, harmonic interference can be eliminated, and the speed and position information can be output. The high-frequency differential currents and the second harmonic of the output position are calculated by difference, and the result is used as the input of PLL. Thereafter the position estimate can be obtained. The block diagram of the mover position demodulation of the PLL method is shown in Fig. 3.

2) GAUSS-NEWTON ITERATIVE METHOD
The method of arctangent or PLL can obtain the position of the mover. However, the length of the mover for TPMSLM is higher than the stator length. In the actual moving process, if the running displacement is large, then the mover and stator will be partially decoupled stage. Therefore, the coupling relationship between the mover and stator during the linear motor motion is divided into two forms, namely, full and partial couplings, in which partial coupling includes left and right partial couplings. edge of the Hall sensors, the coupling length is reduced from 9 pole pitches to 5 pole pitches. Compared with the traditional rotating PMSM, the change of electrical parameters will be considerably complicated. In the partial coupling process, with the decrease of the coupling length of the mover and the stator, self and mutual inductance in the inductor will experience DC offset and attenuation of the second harmonic amplitude, which makes the position extraction markedly difficult [20]. The high-frequency differential current coefficient is composed of the motor's self and mutual inductance parameters. When the inductance parameter changes substantially, it will directly affect the accuracy of the position estimation. The high-frequency differential current is no longer a pure sinusoidal variation law in the entire motion stroke, but a polynomial form containing numerous harmonic components. In this case, the PLL method is no longer applicable. Therefore, this paper proposes an iterative method to estimate the position of the linear motor.
First, the actual spectral distribution of the high-frequency differential current is obtained through FFT, and then the polynomial function of the high-frequency current curve is established as follows: where () Thereafter, the nonlinear least squares method is used to minimize the sum of squares of the residual values of the two phases. This function is used as the objective function of the iterative method, which is expressed as follows: where m () A fx and m () B fx are the measured data of highfrequency differential currents.
The Jacobian matrix () AB x J is derived as follows: The measured value function matrix () m x f of the highfrequency differential currents is shown as follows: The iteration step k h of the Gauss-Newton method can be calculated as follows: Lastly, the iterative estimation of mover position is carried out, which is expressed as follows: For a TPMSLM, the initial value is 0 0 x  , Thereafter, the stator position is obtained using one-step Gauss-Newton iteration method. This position is used as the initial value of the next iteration, and the position iteration of the next sampling point is performed. The specific iterative process is shown in Fig. 5.

A. TPMSLM INDUCTANCE PARAMETER SIMULATION
Given that the characteristics of self and mutual inductance directly determine the amplitude of the high-frequency differential current, this chapter firstly simulates and analyzes the inductance characteristics of the linear motor. The PMSLM structural simulation model shown in Fig. 1 is established by ANSYS Maxwell software, and the material properties of each part are set according to Table 1. Inductance parameter simulation under different coupling lengths is carried out according to the relative positions of the stator and mover shown in Fig. 2. This paper takes the zerophase point of the mover near the middle position of the stator as the center position, and the distance of 30 mm before and after is used as the full range of motion. That is, 6 VOLUME XX, 2017 the motion displacement is 60 mm. Self-inductance and mutual inductance parameters after simulation are shown in Fig. 6.

FIGURE 6. Simulation results of variation of inductor parameters: selfinductance (a), mutual inductance (b).
The simulation results in Fig. 6 show that the self and mutual inductance of the AB phase are sinusoidal curves, and there are DC offsets and second harmonics related to the position. This variation law is markedly consistent with Equation (11). In all the intermediate coupling stages, as shown in Fig. 4(c), the self-induced DC offset and the amplitude of the second harmonic remain unchanged. Meanwhile, in some coupling stages, as shown in Fig. 4(a) and (b), the amplitude of the DC offset and the second harmonic will attenuate to varying degrees. If the mover position is obtained by using the position demodulation process in the traditional high-frequency square-waveform injection algorithm, then a large position estimation error will occur in some coupling stages.

B. OBTAIN HIGH-FREQUENCY CURRENT CURVE PARAMETERS
The simulation results in the previous chapters show that because the inductance parameters of PMSLM directly constitute the amplitude of high-frequency differential currents and its value changes with the position of the mover, the complex inductance parameters will directly affect the accuracy of sensorless position estimation.
The high-frequency differential current mainly includes fundamental frequency and second harmonic components. Therefore, based on the inductance simulation results and measured high-frequency differential current, a preliminary simulation model can be established to preliminarily verify the effectiveness of the Gauss-Newton position iteration algorithm.
Specific data in the simulation process include the fundamental frequency and second harmonic parameters of the two-phase high-frequency differential current model of linear motor: To verify the effectiveness of the adopted Gauss-Newton iterative position estimation algorithm, random noise random  is added based on the simulation model to simulate the measured value of high-frequency differential current, as shown in (27) Fig. 7 shows the phase AB model curve of high-frequency differential current and simulated data. Fig. 8 shows the theoretical and iterative estimates of position estimates. Fig.  9 shows the Gauss-Newton iterative position estimation error. Fig. 7, Fig. 8, and Fig. 9 are simulation curves, abscissa coordinate is the position, which represents the displacement of the motor movement, and its maximum value is 60 mm. This process is consistent with the displacement of the linear motor in the subsequent experiments.

IV. EXPERIMENTAL VERIFICATION
The sensorless position estimation method proposed in this paper is verified by a PMSLM single-axis experimental platform, as shown in Fig. 10. The control system of the experimental platform is built based on TI's TMS320F28335 digital processing chip. The linear motor adopts a two-phase cylindrical PMSLM. The bus voltage of the linear motor is 48 V, the current sampling frequency is 20 kHz, the highfrequency injection voltage is 16V and the injection frequency is 10 kHz. The inductance parameters are measured by the LCR bridge, and the grating sensor is used as the sensor for position calibration. The structure of the linear motor is shown in Fig. 11, and the detailed dimensions are shown in Table 2.

A. MEASURED INDUCTANCE VALUE
For the cylindrical PMSLM in this paper, the self-inductance and mutual inductance parameters can be measured through the LCR bridge. The self-inductance coefficient can be measured directly through the bridge, and the mutual inductance value can be measured indirectly: (1) connect the AB phases of the motor in forward and reverse series to measure the value, and (2) process the difference between the two sets of data, and 1/4 of the difference is the mutual inductance value we want to obtain [21]. Fig. 12(a) and Fig. 12(b) are the self-inductance and mutual inductance values measured by this method, respectively. As shown in Fig. 12, inductance value measured by the LCR bridge has a deviation of about 5% compared with the simulation results. This result may be caused by machining and assembly errors of PMSLM. However, the results of the finite element simulation and measured value have the same variation law with the mover position. 8 VOLUME XX, 2017

B. ESTABLISH HIGH-FREQUENCY DIFFERENTIAL CURRENT CURVE MODEL FUNCTION
The preceding analysis indicates that the high-frequency differential current obtained by the high-frequency squarewaveform injection method includes DC offset and positionrelated second harmonic. Moreover, and the amplitude of the second harmonic is mainly affected by the self-inductance and mutual inductance parameters of the linear motor itself.   Position error experiment in this paper verifies the proposed position estimation method under two different maximum speeds. Fig. 13 and Fig. 14 show the highfrequency differential current curves at the maximum speeds of 20 mm/s and 200 mm/s, respectively. The experimental procedure is as follows: a. Single-axis experimental platform shown in Fig. 10 was used in this experiment. Take the 0 electrical angle point VOLUME XX, 2017 9 shown in Fig. 11 as the middle point of the movement, and take the point back 30 mm as the starting point of the movement. b. In the current loop of the servo system, we injected high-frequency voltage signals with equal amplitudes but opposite symbols into the αaxis in the stationary reference frame. The injection sequence is shown in Fig. 2. The amplitude of the injected signal is 16 V, and the injection period is half of the PWM switching frequency, which is 10 kHz.
c. Position motion command is applied to the position loop of the servo system, the total displacement is 60 mm, and the type of velocity planning is T-type. Maximum speeds of 20 mm/s and 200 mm/s during the experiment are the maximum values during the acceleration and deceleration processes, respefctively.
d. We sampled the currents of the actual phase AB and differentiated them. The sampling period is the same as the PWM period, (i.e., 20 kHz). e. We performed a difference operation on the highfrequency differential currents obtained in two adjacent PWM cycles in step (4). Operation results are the high differential current signals, as shown in Fig. 13 and Fig. 14.
As shown in Fig. 13 and Fig. 14, with the change of the position of the mover, the currents of the two-phase A and B show a sine wave law, and both have a DC offset. Given that the high-frequency differential current is not a sine wave with a single frequency, it contains numerous harmonic components. In this paper, the FFT is used to analyze the spectral characteristics of the obtained high-frequency differential current, which is used to establish the highfrequency differential current. The current curve model lays the foundation. The specific spectrum analysis results are shown in Fig. 15 and Fig. 16.
The spectrum analysis in Fig. 15 and Fig. 16 show that the high-frequency differential current is mainly composed of fundamental frequency current. Moreover, and the harmonic component is mainly second harmonic, which is consistent with the theoretical analysis result of the Equation (19). At the same time, the high-frequency differential current also contains more low-frequency harmonics, including some fractional harmonics. The sampling resolution is 1/3 Hz because the sampling period of linear motor current in the position domain is 20 mm and sampling length is 60 mm. To better estimate the position, this paper takes the frequency of the first 5 Hz as the frequency distribution range of the model function (i.e., 1/3 Hz to 5 Hz), which is specifically divided into fractional and integer frequencies, as shown in Table 3.

C. HIGH-FREQUENCY CURRENT CURVE MODEL
According to the preceding high-frequency differential current spectrum distribution diagram of the first 5 Hz, the curve modeling is carried out according to Equation (20). To model the high-frequency differential current obtained at different speeds, so as to realize the Gauss-Newton position iteration method proposed earlier in this study, we convert the high-frequency differential current data from the timedomain coordinate to the position domain coordinate. In the subsequent experimental processing, the methods proposed are carried out in the position domain coordinate. Evidently, the modeling curve can fit the measured value well. Fig. 17 and Fig. 18 are curve modeling at maximum speeds of 20 mm/s and 200 mm/s, respectively.

D. POSITION ESTIMATION ACCURACY
In the position estimation accuracy experiment, the PLL method in the high-frequency square-waveform injection method and the Gauss-Newton iterative method used in this paper are compared and verified. The experimental process was verified at two maximum speeds of 20 mm/s and 200 mm/s. The velocity planning is T-type acceleration and deceleration. The entire motion process includes five stages: rest, acceleration, uniform speed, deceleration, and stop. The experimental results are shown in Fig. 19 to Fig. 22.  As shown in Fig. 20 and Fig. 22, the position estimation accuracies of the PLL method are 0.671 mm and 1.07 mm, respectively. Moreover, the maximum errors at both speeds are near the starting and ending points of the mover motion position. The reason is that in the foregoing analysis, the mover motion starting and ending points are positions with less coupling between the mover and stator (i.e., Fig. 4 represents the partial coupling stage). The fundamental reason for the large error in these two areas is that the amplitude of the high-frequency differential current is affected by the inductance parameters of the motor. In addition, the inductance parameters will be attenuated to different degrees with the reduction of the coupling length in the partial coupling stage. Therefore, if the existing PLL method is used, then a large position error occurs.
As shown in Fig. 20 and Fig. 22, by comparing errors at two maximum speeds, the position error is shown to be larger when the maximum speed is 200 mm/s. This result is due to the difference in processing the sampled current in the process of extracting the high-frequency differential current. The premise of this processing method is that the fundamental frequency current will not change substantially in two PWM cycles. This method can easily obtain the highfrequency differential current. However, with an increase in movement speed, the fundamental frequency current of PMSLM will change violently. At this time, there will be large errors in the different processing methods, resulting in worse position estimation accuracy when the movement speed is high.   Fig. 26 show the position estimation and error analysis, respectively, using the Gauss-Newton iterative method at two maximum speed. In this part of the experiment, experimental conditions are consistent with the PLL method. that is, the velocity planning is T-type acceleration and deceleration, and maximum speeds are 20 mm/s and 200 mm/s. The motion process at two speeds consists of five stages: rest, acceleration, constant speed, deceleration, and stop. Fig. 24 and Fig. 26 show that the maximum position error is 0.22 mm at a maximum speed of 20 m/s and 0.637 mm at a maximum speed of 200 mm/s, which are 32.7% and 59.5%, respectively, of the PLL method. The position estimation accuracy is substantially improved. With an increase in speed, the position error increases. Given that this method is also based on high-frequency differential current obtained by the difference method, and the position iterative estimation is carried out on this basis, this method will inevitably be affected by the change of fundamental frequency current. Table 4 shows the RMS, maximum, and peak-to-peak error results of the two methods at different maximum speeds. Note that the Gauss-Newton iteration method has a greater improvement in position estimation accuracy than the PLL method.

V. CONCLUSION
In this work, a new high-frequency injection sensorless position estimation method using the Gauss-Newton iteration estimation was investigated. This method is unaffected by the change of coupling length, has strong robustness to the change of the inductance parameters, and can obtain higher position accuracy in the entire stroke. Affected by the change in the coupling length of the mover and stator of the linear motor, the inductance parameters change differently in different coupling stages. In the full coupling stage, it is the second harmonic of the position, and in the partial coupling stage, the amplitude and DC offset attenuation phenomenon occurs. These changes lead to the failure of traditional position sensorless algorithms. The finite element method is used to analyze the variation of the inductance parameters in the entire motion stroke and to verify by the measured values.

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In the position estimation process, the high-frequency differential current is obtained by high-frequency squarewaveform injection and interpolation processing, and the FFT is performed to obtain the curve model function. The motor has high positional accuracy over the entire travel range. Using this method, experimental verification shows that estimation accuracy is improved by 67.3% and 40.5% compared with the traditional PLL method when the maximum speed of the T-shaped acceleration and deceleration are 20 mm/s and 200 mm/s, respectively. Evidently, the proposed method also causes the drive system to generate acoustic noise. How to reduce the noise requires further research. The proposed method can also be used for the sensorless position estimation of various linear motors.