Speed Estimation Method of Linear Motor Extended Kalman Filter Based on Attenuation Memory

Abstract

In allusion to the phenomenon that the extended Kalman filter is easy to diverge in the mover position estimation of permanent magnet synchronous linear motor, a linear motor extended Kalman filter speed estimation method based on attenuation memory is designed. By setting the attenuation factor, α, the extended Kalman filter is introduced to increase the weight of the latest speed data and restrain the divergence of the filter, so as to achieve a better speed tracking effect. In the simulation experiment of the sensorless control of a linear motor, the AMEKF algorithm can significantly improve the speed estimation accuracy of standard EKF, and the speed estimation error is reduced by 0.75%. At the same time, it still maintains a good speed tracking effect and good dynamic performance under variable speed and different load conditions.

1. Introduction

In the field of modern motor control, permanent magnet synchronous linear motor (PMSLM) is widely used in high-end manufacturing industries such as CNC machine tools, high-performance laser cutting machines, and semiconductor manufacturing equipment because of its high thrust density, high dynamic response, long stroke, and direct drive without intermediate links [1,2,3]. However, the use of a speed sensor destroys the characteristics of a simple structure, limits its application range, and reduces the robustness of the whole system. Therefore, speed sensorless control has become an important research direction of the modern AC drive [4,5,6].

The Extended Kalman filter (EKF) algorithm is the extended application of the Kalman filter algorithm for linear system state estimation in a nonlinear system [7]. Because the filter gain can adapt to the environment and adjust automatically, it provides an iterative nonlinear estimation algorithm and avoids differential operation. Therefore, EKF itself is an adaptive system that can estimate the system state online and realize the real-time control of the system [8,9,10,11].

In reference [12], Zhu Jun et al. studied the sensorless control of PMLSM based on an improved volumetric Kalman filter, and proposed an improved algorithm of a volumetric Kalman filter for the problem of filter divergence. Although its speed estimation curve can keep up with the change trend of actual speed, the error is small. In reference [13], Li Xingyu et al. proposed a low-speed sensorless control strategy of a permanent magnet linear motor based on the pulse vibration high-frequency injection method, which has a good speed dynamic tracking effect.

The research shows that the EKF linearizes the estimated state vector at each sampling time. As K gets bigger, the ratio of historical data in the filtered value is too large, while the proportion of new data is too small [14]. When there are model errors and calculation errors in the system, the effect of new observation data on state estimation is too small to effectively suppress the influence of errors on state estimation, resulting in distortion of the covariance matrix and a divergence of filtering [15,16,17].

To solve the above problems, an attenuated memory-extended Kalman filter (AMEKF) speed estimation method is proposed. The attenuation factor is introduced to increase the weight of the latest speed data and restrain the filter divergence, so as to achieve a better speed tracking effect. This method has low requirements for the accuracy of the system model and parameters, high system stability, and strong robustness.

2. PMLSM Mathematical Model

The motor selected in this paper is the basic model, as illustrated in Figure 1.

PMLSM is a strong coupling and multivariable nonlinear system. In this paper, the mathematical model of a synchronous rotating coordinate axis is selected for controller analysis. It is assumed that in the motor air gap, the excitation magnetic field and armature reaction magnetic field have a sinusoidal distribution, ignoring the magnetic saturation effect, motor eddy current, and hysteresis loss, and ignoring nonlinear friction and other factors [18,19,20]. The voltage equation of PMLSM can be obtained as:

$$\{\begin{array}{l}{u}_d(t)=R_s{i}_d+L_d\frac{d{i}_d}{dt}-\frac{\pi }{\tau }\nu {\psi }_q\\ u_q(t)=R_s{i}_q+L_q\frac{d{i}_q}{dt}+\frac{\pi }{\tau }\nu {\psi }_d\end{array}$$

(1)

In the above formula, R_S is the stator resistance; $\tau$ is the PMLSM polar distance; v is the running speed of the linear motor; u_d and u_q is the voltage of the shaft; i_d, i_q is the current of the shaft; L_d, L_q is the inductance component of the shaft; ${\psi }_f$ is the permanent magnet flux linkage. The equation of the stator flux is:

$$\{\begin{array}{c}{\psi }_d=L_d{i}_d+{\psi }_f\\ {\psi }_q=L_q{i}_q\end{array}$$

(2)

From Formula (1), the current state equation is:

$$\{\begin{array}{l}\frac{d{i}_d}{dt}=-\frac{R_s}{L_d}{i}_d+\frac{\pi L_q}{\tau L_d}\nu i_q+\frac{u_d}{L_d}\\ \frac{d{i}_q}{dt}=-\frac{R_s}{L_q}{i}_q-\frac{\pi L_d}{\tau L_q}\nu i_d+\frac{u_q}{L_q}-\frac{\pi {\psi }_f}{\tau L_q}\nu \end{array}$$

(3)

The electromagnetic thrust equation of PMLSM is:

$$F_{em}=p_n\frac{3\pi }{2\tau }[{\psi }_f{i}_q+(L_d-L_q)i_d{i}_q]$$

(4)

In the above formula, F_em is the thrust of linear motor and p_n is the pole logarithm of the linear motor. Because the PMLSM is used, the thrust equation can be simplified to:

$$F_{em}=p_n\frac{3\pi }{2\tau }{\psi }_f{i}_q$$

(5)

The equation of motion of PMLSM is:

$$m\frac{d\nu }{dt}=F_{em}-f-B\nu$$

(6)

In the above formula, m is the mover mass, B is the viscous friction coefficient, and f is the system disturbance. Simultaneous Formulas (5) and (6) can be obtained:

$$\frac{d\nu }{dt}=a_M{i}_q+b_M\nu +c_{M}f$$

(7)

In the above formula:

$$a_m=\frac{1}{m}{p}_n\frac{3\pi }{2\tau }{\psi }_f$$

$$b_m=-\frac{B}{m}$$

$$c_m=-\frac{1}{m}$$

3. Extended Kalman Filter

Because Kalman is based on a linear system, it is only suitable for the state estimation of linear systems. In order to effectively estimate nonlinear systems, an extended Kalman filter is proposed. EKF simply linearizes the nonlinear system model by first-order or second-order Taylor extension of the nonlinear system function near the best estimation point, ignoring the high-order components. The state estimation of EKF is roughly split into two stages: the first part is the prediction stage, followed by the correction stage. Its advantage is that the system state can still be estimated when the system is noisy [21,22,23]. The relationship is shown by the following formula:

$$\{\begin{array}{l}\frac{d{w}_e}{dt}=0\hfill \\ \frac{d{q}_e}{dt}=w_e\hfill \end{array}$$

(8)

The general form of the EKF equation is:

$$\{\begin{array}{l}\frac{d\widehat{x}}{dt}=A(\widehat{x})\widehat{x}+Bu+K(y-\widehat{y})\hfill \\ \widehat{y}=C\widehat{x}\hfill \end{array}$$

(9)

Among:

$$\widehat{x}=\left[\begin{array}{c}{i}_a\\ i_b\\ w_e\\ q_e\end{array}\right],u=\left[\begin{array}{l}{u}_a\hfill \\ u_b\hfill \end{array}\right],\widehat{y}=\left[\begin{array}{c}{i}_a\\ i_b\end{array}\right]$$

(10)

$$A(\tilde{x})=\left[\begin{array}{c}-\frac{R}{L_s}{i}_a+w_e\frac{y_l}{L_s}sin{q}_e\\ -\frac{R}{L_s}{i}_b-w_e\frac{y_e}{L_s}cos{q}_e\\ 0\\ q_e\end{array}\right]$$

(11)

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

(12)

where $\stackrel{⌢}{x}$ is a state variable; $\stackrel{⌢}{y}$ is the observational measurement; Y is the real measurement; A is the state matrix; Input matrix for B; C is the observation matrix; U is the input variable; K is the gain matrix.

Discretize Equation (9) to obtain

$$\{\begin{array}{l}{\widehat{x}}_{k+1}=A^{\prime }{\widehat{x}}_k+B^{\prime }u+K_{k+1}(y-\widehat{y})\hfill \\ \widehat{y}=C^{\prime }\widehat{x}\hfill \end{array}$$

(13)

Figure 2 shows its implementation flow chart:

The specific steps are as follows: predict the state vector, that is, estimate x from the system input U(k) and the state of the last sampling time to realize the estimation of the state vector at the next sampling time:

$${\tilde{x}}_{k+1}=A^{\prime }{\widehat{x}}_k+B^{\prime }{u}_k$$

(14)

Calculate error covariance matrix:

$${\tilde{P}}_k=F_k{P}_{k-1}{F}_k^T+Q_k$$

(15)

Calculate Kalman filter gain matrix:

$$K_k={\tilde{P}}_k{H}_k^T{\left(H_k{\tilde{P}}_k{H}_k^T+R\right)}^{-1}$$

(16)

State predictive value correction:

$${\widehat{x}}_k={\tilde{x}}_k+K_k\left(y_k-C_k{\tilde{x}}_k\right)$$

(17)

Error covariance matrix correction:

$${\widehat{P}}_{k+1}={\tilde{P}}_{k+1}-K_{k+1}{H}_{k+1}{\tilde{P}}_{k+1}$$

(18)

where A’ is the discrete state matrix, B’ is the discrete input matrix, and C’ is the discrete observation matrix; “∼” indicates the predicted value; “^” indicates the check value. The gradient matrix is $H_k$, and $K_k$ is the filter gain of the extended Kalman filter, expressed as the degree of deviation between the filter estimation and the real state.

4. Attenuated Memory Extended Kalman Filter Algorithm

4.1. Filtering Algorithm

As a filter of wireless growth memory, the optimal estimation data at time K comes from all observation data before this time. Moreover, with the increase of K, the proportion of historical data will also be large, and the ratio of new data is too small. Especially in the motor start-up stage, the data changes violently, which has a great impact on the overall data. When there is calculation or system error, the correction effect of new observation data on state estimation is too small to suppress the impact of error on state estimation, resulting in filter divergence [24,25,26]. Therefore, in order to avoid the strong proportion of start-up data in filtering, which leads to the increase of tracking error, for the filtering divergence caused by model error, we should try to increase the role of new observation data and relatively reduce the influence of historical data on filtering value. Enter the AMEKF algorithm to reduce the influence of historical data information on filtering and the proportion of corresponding new data will increase within a certain range, so as to improve the tracking of motor speed. The attenuation memory algorithm is to introduce an attenuation factor into the observation noise covariance matrix, α, order:

$$P(k+1)=aP(k)$$

(19)

where attenuation factor, α, is an empirical value and α ≥ 1. In theory, when α > At 1:00, the weight of the past observation data will be weakened, and the weight of the current latest observation data will be increased. Meanwhile, if the value of α is too large, the filter will vibrate.

Under the attenuated memory extended Kalman filter algorithm, the error covariance matrix, gain matrix, and state prediction value correction equation can be expressed as:

$${\tilde{P}}_k^{\ast }=F_{k}a{P}_{k-1}{F}_k^T+Q_k$$

(20)

$$K_k^{\ast }={\tilde{P}}_k^{\ast }{H}_k^T{\left(H_k{\tilde{P}}_k{H}_k^T+R\right)}^{-1}$$

(21)

$${\widehat{x}}_k^{\ast }={\tilde{x}}_k+K_k^{\ast }\left(y_k-C_k{\tilde{x}}_k\right)$$

(22)

where “*” represents the extended Kalman equation after the memory attenuation algorithm.

4.2. AMEKF Control Structure

The PMLSM model of the AMEKF sensorless control scheme is illustrated in Figure 3. where “*” indicates the current input to the current loop controller.

The system adopts three PI regulators to form a double closed-loop speed regulation system with outer speed loop and inner current loop.

As shown in Figure 4, the discrete PI regulator is used to estimate the deviation of output speed according to the given speed of the system and AMEKF, where it is possible to adjust the given value of torque current through the set K_P and K_i, and output it.

As shown in Figure 5, The discrete PI regulator is adopted. The torque current and excitation current PI regulator adjusts the corresponding stator voltage setting values of the d and q axes according to the deviation between the detection current and the current setting value.

The input of the AMEKF speed estimation module is the stator current and the stator voltage in a static coordinate system, and the output is the estimated speed.

5. Simulation

The PMSLM sensorless control model based on AMEKF is established in Matlab/Simulink. The model adopts the double closed-loop speed regulation system of outer speed loop and inner current loop, and the simulation model is established according to the motor driving parameters, as shown in

Table 1. The main parameters of motor.

Parameter	Value
Stator resistance Rs/Ω	2.875
d-q axis inductance Ldq/mH	8.5
Mover mass m/kg	1.425
Viscous friction coefficient B/N/m⋅s	20
Continuous thrust F/N	32
Polar distance τ/m	0.018

.

As shown in Figure 6a,b, the comparison results of the measured and estimated speed values using EKF and AMEKF are shown, respectively. In the simulation, the motor runs without load after starting. The graph shows that the speed estimation errors of the two algorithms have certain errors in the rising stage of speed, but the speed estimation error decreases gradually with the rise of speed and stable operation.

Comparing Figure 7a,b shows that when the motor speed reaches the reference speed, EKF has a large estimation error, and the maximum fluctuation amplitude is 0.04. If the AMEKF estimation is used, the fluctuation is significantly reduced, and the maximum steady-state fluctuation amplitude is 0.02.

The reference speed of the control system is set as the sinusoidal waveform and fluctuates at the median speed of 2m/s, as shown in Figure 8a,b, which is the actual speed waveform and speed estimation waveform of EKF and AMEKF. Figure 8c,d show the speed estimation errors of EKF and AMEKF. It can be seen from the figure that AMEKF can effectively suppress divergence, even under long-time operation conditions. Compared with EKF, AMEKF has less actual speed estimation error and a better tracking effect.

As shown in

Table 2. Velocity fluctuation error.

	Fluctuation Range at 2 m/s Speed (m/s)	Error Percentage	Velocity Fluctuation Range under Sinusoidal Velocity	Error Percentage
EKF	−0.015–0.03	2.25%	±0.04	4%
AMEKF	−0.01–0.02	1.5%	±0.025	2.5%

. The above verification shows that AMEKF has better performance than EKF in speed estimation. In order to verify its dynamic performance, the motor starts without load and adds different loads at 2 S. As shown in Figure 9, the AMEKF algorithm is used to add 100 N, 200 N, 300 N, and 400 N loads for simulation. The results indicate that the control system has the better dynamic response under different torque loads and different loads, and the control system has good robustness.

The curves of torque versus time of a linear motor with 100 N, 200 N, 300 N, and 400 N loads added at 2 S are shown in Figure 10.

The motor has no stall in the process of sudden change of load torque and long-time operation. The speed estimation and mover position error show that AMEKF can well estimate the motor mover position under sudden change of load torque and constant working conditions. The verification results indicate that the proposed method has strong robustness and verify the reliability of this method.

6. Experiment

The experimental platform built in this paper is shown in Figure 11. The platform uses a TMS320F28335 chip to control PMLSM. The mechanical sensors of the controlled motor are used for speed comparison and observer verification. The motor parameters in the experiment are consistent with those in the simulation.

The given speed of the control system is 0.5 m/s. At 0.3 s, the given speed increases to 1 m/s, and at 0.8 s, the given speed of the system decelerates to 0.7 s/m. Figure 12 shows the actual speed and estimated speed waveform of the EKF control system, and Figure 13 shows the actual speed and estimated speed waveform of the AMEKF control system. It can be seen that, in the experiment, the AMEKF algorithm designed in this paper has better speed estimation performance than EKF.

In Figure 14a,b, the speed errors of the motor under EKF and AMEKF are shown, respectively. It can be seen from the figure that, in the case of sudden speed change, the two-speed estimation algorithms have certain estimation error fluctuations, but when the speed is stable and at the given speed, the AMEKF algorithm has little fluctuation and estimation errors.

7. Conclusions

In this paper, speed estimation based on the Extended Kalman Filter method of a linear motor based on attenuation memory is proposed to estimate the motor speed. The purpose is to estimate the speed of a linear motor. This method can estimate the speed of a linear motor well under no-load and sudden torque change. The motor runs smoothly without bad phenomena such as stall. The control system has strong robustness to motor parameters and model errors. This method does not require a high accuracy of motor parameters and model, but has certain requirements for the accuracy of measurement data. During the actual debugging, the attenuation factor can be adjusted according to the system conditions to balance the impact of model inaccuracy and parameter measurement error on the system and realize the optimization of the system. Compared with the EKF algorithm, the AMEKF algorithm has better estimation performance in speed estimation, but it has a large amount of computation in practical application. Therefore, simplification of the AMEKF algorithm still needs to be further studied. The speed estimation algorithm designed in this paper has a certain reference value for the speed sensorless control system of a linear motor.