A Hybrid Algorithm for PMLSM Force Ripple Suppression Based on Mechanism Model and Data Model

Abstract

The force ripple of a permanent magnet synchronous linear motor (PMSLM) caused by multi-source disturbances in practical applications seriously restricts its high-precision motion control performance. The traditional single-mechanism model has difficulty fully characterizing the nonlinear disturbance factors, while the data-driven method has real-time limitations. Therefore, this paper proposes a hybrid modeling framework that integrates the physical mechanism and measured data and realizes the dynamic compensation of the force ripple by constructing a collaborative suppression algorithm. At the mechanistic level, based on electromagnetic field theory and the virtual displacement principle, an analytical model of the core disturbance terms such as the cogging effect and the end effect is established. At the data level, the acceleration sensor is used to collect the dynamic response signal in real time, and the data-driven ripple residual model is constructed by combining frequency domain analysis and parameter fitting. In order to verify the effectiveness of the algorithm, a hardware and software experimental platform including a multi-core processor, high-precision current loop controller, real-time data acquisition module, and motion control unit is built to realize the online calculation and closed-loop injection of the hybrid compensation current. Experiments show that the hybrid framework effectively compensates the unmodeled disturbance through the data model while maintaining the physical interpretability of the mechanistic model, which provides a new idea for motor performance optimization under complex working conditions.

1. Introduction

As the core driving component of high-end equipment, the stability of the force output of a permanent magnet synchronous linear motor (PMSLM) directly determines the control accuracy of the precision motion system [1,2]. The generation of a force ripple is due to the complex multi-physical field coupling effect inside the motor [3]. On the one hand, the inherent electromagnetic disturbances such as the cogging effect and end effect can be modeled by magnetic energy analysis [4,5]. On the other hand, time-varying factors such as temperature drift and mechanical deformation lead to model parameter mismatch, while unmodeled dynamics such as friction nonlinearity further aggravate the force ripple. Although the traditional suppression method based on the mechanistic model is physically interpretable, it is difficult to accurately characterize the nonlinear characteristics of the system under actual working conditions [6,7,8]. Although the data-driven method can approximate the measured data through black box modeling, it relies on a large number of training samples and has a heavy real-time computing burden, which restricts its deployment capability on embedded platforms [9,10].

The traditional force ripple suppression method starts from two aspects, namely, suppressing the cogging magnetic resistance and the end magnetic resistance by modifying the body design parameters and using the control strategy to compensate the force ripple in real time. In [11], the mechanistic model of the cogging force is established, and the thrust ripple of the DSPMLSM is effectively suppressed by changing the phase of the cogging force by using the tooth cutting method. In [12], a trapezoidal Halbach magnetic pole was designed. By optimizing the shape of the magnetic pole, the output thrust of the PMLSM can be improved and the thrust ripple can be reduced.

In the use of control strategies to suppress the force ripple, ref. [13] used ST-SMC and DOB-SMV to improve the response speed and control accuracy of the system. However, it can be seen that the current control strategy regards the thrust fluctuation and load disturbance as the external disturbance of the system. In fact, for a designed PMLSM, the thrust ripple can be effectively predicted and feedforward compensated.

In view of the above problems, the hybrid modeling framework combining the physical mechanism and a data-driven approach has become a breakthrough direction. The modeling framework takes into account both model accuracy and computational efficiency through a hierarchical collaborative mechanism: the basic mechanism layer parses the modeled disturbance components [14], and the data layer captures the residual ripple characteristics through sensor feedback [15]. This structure not only retains the causal correlation of the mechanistic model, but also compensates the unmodeled dynamics by using the adaptive ability of the data model, which provides methodological support for complex disturbance suppression [16].

Therefore, aiming at the force ripple problem of permanent magnet linear motor, this paper constructs non-model parameters such as nonlinear friction and mechanical deformation and modellable parameters such as cogging force and end force, and proposes an adaptive hybrid force ripple suppression algorithm. The software and hardware platforms are built to collect multi-dimensional signals such as current, position, and acceleration in real time. The mechanistic model is used to calculate the ideal force ripple of the motor and consider the actual force ripple error, so as to realize the efficient suppression of the force ripple under the influence of multi-dimensional factors. Based on this idea, this paper explores a new path of force ripple suppression through algorithm–hardware co-design.

2. The Force Ripple Modeling of PMLSMs

2.1. The Structure and Force Ripple Composition of PMLSM

The typical structure of the PMLSM is composed of a mover, a stator, and an end plate. The mover includes a workbench and a core, and the stator includes a permanent magnet and a bottom plate. The structural diagram is shown in Figure 1.

The electromagnetic part of the PMLSM includes iron core and permanent magnet, which can be divided into two parts: a middle tooth and an end tooth according to the magnetic field area. The cogging force is composed of the tangential component of the excitation magnetic field generated by the permanent magnet and the force between the iron core and the cogging, corresponding to the intermediate tooth and the intermediate magnetic field region in Figure 2. The mechanism of the end effect is that the iron core of the PMLSM is linear, and the disconnected state at both ends will cause distortion of the magnetic field at both ends of the iron core, corresponding to the end tooth and the end magnetic field area in Figure 2. Due to the inconsistent magnetic flux entering the core at both ends, the force at both ends of the primary core will be unbalanced, resulting in an end force ripple.

In addition, the back electromotive force generated by the permanent magnet magnetic field in the winding during PMLSM operation contains harmonic components, which interact with the driving current to generate a force ripple. In practical applications, the output current is obtained through the inverter, resulting in the current also containing some higher harmonics, which will also produce a force ripple.

The mechanical parts of the PMLSM include the guideway, slider, and end plate. In the mechanical part, the force ripple of the PMLSM mainly includes friction disturbance and load disturbance. In PMLSMs, friction may be one of the main causes of instantaneous oscillation and system steady-state error when the speed is zero. The friction force in the system is usually composed of linear and nonlinear parts. Among them, the linear part mainly refers to the viscous friction which is proportional to the speed. The nonlinear part includes static friction force and sliding friction force. Because friction is related to many factors, it is difficult to be considered in traditional modeling analysis.

From the analysis, the force ripple sources of PMLSM can be classified, including the electromagnetic force ripple generated by the electromagnetic components and the mechanical force ripple generated by the mechanical components. It is not difficult to see that the source of the electromagnetic force ripple is clear, and the source of mechanical force ripple is complex, and it is often not regular and cannot be modeled. Therefore, it is difficult to effectively suppress the current force ripple.

2.2. Modeling of Electromagnetic Force Ripple

The analytical model of the force ripple of the PMLSM can accurately predict the magnitude of force ripple according to structural parameters and clarify the contribution of various factors to the force ripple, so as to achieve the purpose of minimizing the force ripple.

The cogging magnetic resistance is the tangential force caused by the relative motion between the permanent magnet and the primary iron core, which is essentially the result of the interaction between the permanent magnet magnetomotive force and the air gap permeance. The size of the electromagnetic part of the PMLSM is shown in Figure 3. In the figure, L_fe is the length of the core, h is the height of the core, δ is the length of the air gap, h_pm is the length of the magnetization direction of the permanent magnet, w_pm is the width of the permanent magnet, τ is the polar distance, b_t is the width of the tooth top, b_d is the width of the end tooth, and h_d is the height of the end tooth. The red, green, and blue parts in the iron core represent the A-phase, C-phase, and B-phase windings, respectively. The points inside the graph represent the current outflow plane, and the crosses represent the current inflow plane.

According to the Kirchhoff’s law of magnetic circuit, the air-gap magnetic flux density B_δ of PMLSM can be expressed as

$$B_{\mathsf{\delta }}={\displaystyle \frac{B_r(x)h_{\mathrm{pm}}}{{\mu }_0}}\left({\lambda }_0+{\displaystyle \sum _{k=1}^{\infty }{\lambda }_{\mathrm{k}}\cos \left[\frac{2n\pi }{\tau }\left(x+x_0\right)\right]}\right)$$

(1)

where λ₀ is the DC component of permeability. λ_k is the harmonic component of permeability.

The magnetic field energy stored in the PMLSM can be considered as the magnetic field energy in the permanent magnet and the air gap, which can be expressed as follows [17]:

$$W={\displaystyle {\int }_V\frac{B_{\mathsf{\delta }}^2}{2{\mu }_0}dV}={\displaystyle \frac{h_{\mathrm{pm}}^2{L}_{\mathrm{fe}}p\tau \delta }{2{{\mu }_0}^3}}\cdot \left[{\lambda }_0{B}_{\mathrm{r}0}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{kz=np}{B}_{\mathrm{rn}}{\lambda }_{\mathrm{k}}\cos \left(n\frac{2\pi }{\tau }x\right)}\right]$$

(2)

where p is the pole pair number.

The magnetic resistance of the groove can be expressed as the derivative of the magnetic energy relative to the displacement, and the mechanistic model can be obtained as follows [17]:

$$F_{\mathrm{c}}={\displaystyle \frac{\pi z{h}_{\mathrm{pm}}^2{L}_{\mathrm{fe}}\delta }{2{{\mu }_0}^3}}{\displaystyle \sum _{kz=np}k{B}_{\mathrm{rn}}{\lambda }_{\mathrm{k}}sin\left(n\frac{2\pi }{\tau }x\right)}$$

(3)

where z is the slot number.

The end teeth will cause distortion of the air gap magnetic field and generate additional force ripples. As shown in Figure 4, when the core is in different positions relative to the permanent magnet, the degree of distortion is different.

According to the magnetic circuit Ohm’s law, the maximum magnetic flux φ_m passing through the longitudinal end edge of the mover core can be expressed as

$${\phi }_{\mathrm{m}}=2H_{\mathrm{pm}}{h}_{\mathrm{pm}}{\displaystyle \frac{{\mu }_{0}A}{L_{\mathrm{eq}}}}$$

(4)

where H_pm is the magnetic field strength of permanent magnet. A is the iron core longitudinal end area. L_eq is the equivalent magnetic path length. A and L_eq can be further defined as

$$\left\{\begin{array}{c}A=L_{\mathrm{ef}}\left({\displaystyle \frac{\tau }{2}}-\delta -{\displaystyle \frac{h_{\mathrm{pm}}}{2}}\right)\\ L_{\mathrm{eq}}={\displaystyle \frac{\pi \tau }{8}}+{\displaystyle \frac{3h_{\mathrm{pm}}}{2}}+\delta \end{array}\right.$$

(5)

where L_ef is the iron core lamination thickness.

The variation in magnetic field energy storage can be expressed as

$$\Delta W={\displaystyle \frac{K_{\mathrm{c}}\delta {\phi }_{\mathrm{m}}^2}{2{\mu }_0{k}_1\tau L_{\mathrm{ef}}}}\left[{\displaystyle \frac{1}{3}}+{\displaystyle \frac{4}{{\pi }^2}}{\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2}\cos n\frac{2\pi }{\tau }}x\right]$$

(6)

where k_c is the air gap coefficient. k₁ is the flux coefficient.

According to the principle of magnetic energy-virtual displacement, the force ripple of the end effect can be defined as follows [18]:

$$F_{\mathrm{d}}={\displaystyle \frac{2K_{\mathrm{c}}\delta {\phi }_{\mathrm{m}}^2}{{\mu }_0{k}_1{\tau }^2{L}_{\mathrm{ef}}}}\left[\pm 1+{\displaystyle \frac{2}{\pi }}{\displaystyle \sum _{n=1}^{\infty }\frac{1}{n}\sin n\frac{2\pi }{\tau }\left(x+\lambda \right)}\right]$$

(7)

The PMLSM is usually designed with fractional slot concentrated winding, so the harmonics in the back electromotive force will be effectively suppressed. The force ripple introduced by the back-EMF harmonics is ignored in the analysis of this paper.

Therefore, the force ripple caused by the electromagnetic part can be expressed as the sum of the cogging magnetic resistance F_c and the end magnetic resistance F_d.

2.3. Modeling of Mechanical Force Ripple

Friction exists widely in servo systems and is the main factor affecting the accuracy of the system. The nonlinear characteristics of friction force will affect the steady-state error of the control system, which makes it difficult to control the system accurately. The PMLSM mainly runs at low speed, so the classical Stribeck model is used to model and analyze the friction force [19].

The Stribeck friction model can be expressed as

$$F_{\mathrm{m}}=\left[f_{\mathrm{c}}+\left(f_{\mathrm{m}}-f_{\mathrm{c}}\right)e^{-{\left(v/x_{\mathrm{s}}\right)}^2}\right]\mathrm{sgn}\left[v\right]+k_{\mathrm{v}}v$$

(8)

where F_m is the frictional force. f_c is the Coulomb friction. f_m is the static frictional force. x_s is the Stribeck critical speed. sgn[v] is the symbolic function. k_v is the friction coefficient. v is the moving speed of the PMLSM.

The static friction force is related to the pressure on the mover. For a PMLSM, the normal pressure includes gravity and normal electromagnetic force. The latter is generated by the magnetic field between the core and the permanent magnet, which can be expressed as

$$F_{\mathrm{ri}}\left(t\right)={\displaystyle \sum _Z{\displaystyle {\int }_{i{L}_{\mathrm{ef}}/Z-b_{\mathrm{t}}/2}^{i{L}_{\mathrm{ef}}/Z+b_{\mathrm{t}}/2}{p}_{\mathrm{r},\mathrm{mn}}}\left(\theta ,t\right)b_{\mathrm{t}}{l}_{\mathrm{ef}}d\theta }$$

(9)

where F_ri(t) is the normal electromagnetic force. p_r,mn is the m-frequency n-order electromagnetic force density.

The static frictional force f_m is the product of the friction coefficient μ and the total normal pressure F_N, which can be expressed as

$$f_{\mathrm{m}}=\mu \cdot F_{\mathrm{N}}=\mu \cdot \left(F_{\mathrm{g}}+F_{\mathrm{ri}}\right)$$

(10)

where F_g is the total gravity of moving parts in the PMLSM.

It can be seen from the above theory that there are many factors affecting the friction force, and the modeling often depends on the accurate friction coefficient, which leads to the inaccuracy of the friction force modeling. In addition, the load disturbance that causes a mechanical force ripple is difficult to characterize by theoretical modeling, which affects that the mechanical force ripple cannot be feedforward suppressed by the mechanistic model like the electromagnetic force ripple.

Therefore, this paper proposes a PMLSM force ripple suppression method based on a mechanism-data model. The acceleration sensor is used to collect the force ripple data of the linear motor. When the mass of the mover is known, the acceleration sensor is used to measure the acceleration of the mover in real time to obtain the force ripple data. The acceleration sensor offers advantages such as compact size and suitability for measuring force ripples in narrow working conditions. Additionally, it enables measurement under dynamic conditions compared to uniform speed methods that only allow measurements under static conditions. The measuring acceleration is depicted in Figure 5.

Since linear motor force ripple is directly proportional to motor acceleration, an empirical model for linear motor force ripples can be established using sub-acceleration values, and its equation is as follows.

$$F_{\mathrm{s}}=ma$$

(11)

where F_s is the force ripple of the empirical model, m is the moving mass, and a is the moving acceleration.

In order to convert the vibration acceleration signal into a force ripple signal, it is necessary to calculate the running frequency of the PMLSM and the mass of the moving parts. The PMSM parameters in this paper are shown in

Table 1. The parameters of PMLSM.

Symbol	Parameters	Value
Z/P	Slot/Pole	14/12
δ	Air gap length	1 mm
τ	Pole pitch	14 mm
hpm	Permanent magnet height	4.5 mm
wpm	Permanent magnet width	10.2 mm
bt	Tooth width	11.5 mm
Lfe	Core length	168 mm
h	Core height	54 mm
Ce	Back-emf coefficient	105.6 V/ms−1
Ct	Force coefficient	64.7 N/Arms
m	Total mass of moving parts	3.2 kg

.

The rated running speed of the linear motor in this paper is 0.12 m/s, and its fundamental current frequency is the ratio of speed to pole distance, which is calculated to be 5 Hz. The PMLSM is tested repeatedly along the guideway at the rated speed, and the mover acceleration data of the PMLSM is measured. The mover acceleration data when the PMLSM is running smoothly is selected to avoid the influence of the excessive acceleration signal on the accuracy of the force ripple data of the linear motor when the PMLSM starts and stops.

Further, the obtained acceleration signal can be filtered to obtain low-frequency acceleration. Finally, the measured force ripple curve including the tooth groove force, end force, and ripple force can be calculated. The mechanical force ripple of the PMLSM can be obtained by subtracting the force ripple curve from the force ripple function established by the mechanistic model. The acceleration curve is fitted into a mechanical force ripple model considering mechanical friction and load disturbance using the superposition method of multiple sine functions.

3. Establishment of PMLSM Control Model

3.1. Control Model of PMLSM

The previous section analyzes the interference factors that cause the force ripple of the linear motor and establishes a mechanistic model that can be expressed by these models according to its generation mechanism. In this section, the control model of the PMLSM is built, and the mechanistic model and data model of the force ripple are coupled into the control system. The voltage equation of the PMLSM can be expressed as

$$\left[\begin{array}{c}{u}_{\mathrm{d}}\\ u_{\mathrm{q}}\end{array}\right]=R_s\left[\begin{array}{c}{i}_{\mathrm{d}}\\ i_{\mathrm{q}}\end{array}\right]+p\left[\begin{array}{c}{\psi }_{\mathrm{d}}\\ {\psi }_{\mathrm{q}}\end{array}\right]+{\displaystyle \frac{\pi v}{\tau }}\left[\begin{array}{c}-{\psi }_{\mathrm{q}}\\ {\psi }_{\mathrm{d}}\end{array}\right]$$

(12)

where u_d and u_q are the d-axis and q-axis voltages of armature winding. R_s is the armature resistance. i_d and i_q are the d-axis and q-axis current armature winding. p is the differential operator, v is the PMLSM running speed, and τ is the linear motor pole pitch. ψ_d and ψ_q are the armature winding d-axis and q-axis flux linkage.

The flux linkage in (12) can be expressed as

$$\left[\begin{array}{c}{\psi }_{\mathrm{d}}\\ {\psi }_{\mathrm{q}}\end{array}\right]=\left[\begin{array}{cc}{L}_{\mathrm{d}}& 0\\ 0& L_{\mathrm{q}}\end{array}\right]\left[\begin{array}{c}{i}_{\mathrm{d}}\\ i_{\mathrm{q}}\end{array}\right]+\left[\begin{array}{c}{\psi }_{\mathrm{PM}}\\ 0\end{array}\right]$$

(13)

where L_d and L_q are the d-axis and q-axis inductance of armature winding. ψ_PM is the flux linkage generated by the permanent magnet.

The electromagnetic force equation and mechanical motion equation of the PMLSM can be expressed as

$$F_{\mathrm{em}}={\displaystyle \frac{3\pi P}{2\tau }}\left[\left(L_{\mathrm{d}}-L_{\mathrm{q}}\right)i_{\mathrm{d}}{i}_{\mathrm{q}}+{\psi }_{\mathrm{PM}}{i}_{\mathrm{q}}\right]$$

(14)

$$\left(m+m_0\right){\displaystyle \frac{dv}{dt}}=F_{\mathrm{em}}-F_{\mathrm{l}}-Dv$$

(15)

where F_em is the electromagnetic force. m₀ is the total mass of mover and iron core. m is the additional mass of the load. F_l is the load resistance. D_v is the viscous friction coefficient.

In this paper, the vector control method of i_d = 0 is adopted. It can be seen from (14) that when i_d = 0, the electromagnetic force is only related to the q-axis current i_q, so the force of the PMLSM can be adjusted according to the q-axis current. Because the mathematical model of the PMLSM does not consider influence factors such as the end effect and cogging effect, this paper adds the mechanistic model of disturbance factors such as the end effect and cogging effect to the electromagnetic force, The control block diagram and flowchart of the proposed control method is shown in Figure 6 and Figure 7. In Figure 6, “*” represents the given values of each controlled object. For example, i_q^* represents the expected value of q-axis current passing through the control system.

3.2. The Force Ripple Acquisition of PMLSM

The force ripple generated by the core and the permanent magnet of the PMLSM can be calculated according to (1)–(7), while the friction and load disturbance need to be fitted and modeled by the measured data. Therefore, in this paper, the vibration acceleration test bench of the PMLSM is built, and the force ripple data considering mechanical friction and disturbance are measured and the model is established. The experimental platform of the PMLSM is shown in Figure 8.

The experimental platform includes the tested PMLSM, a vibration acceleration sensor from the DEWE company placed in the moving direction of the mover and a supporting data acquisition card from the NI company, Shenzhen, China, a building block power electronic power module of the Runit Company, Hangzhou, China, a programmable real-time controller of Runit Company, DC power supply, and a computer for collecting data.

According to the experimental design described above, the target linear motor was tested multiple times by moving in both positive and negative directions at a low uniform speed. Since the measured data from the acceleration sensor includes not only the acceleration signal of force ripple but also other factors, it is necessary to process this data to minimize the influence of external disturbances. Considering that the low-frequency force ripple is predominant in linear motors based on their mechanism, a low-pass filter was applied to process the acceleration signal. The acceleration ripple curve and the processed acceleration ripple curve, obtained from an acceleration sensor while the linear motor was running forward at a speed of 0.12 m/s, are presented in Figure 9.

Based on the vibration acceleration data of the traveling direction obtained from the experiment, the force ripple can be calculated according to (11). Because the low-frequency force ripple has a great influence on the operation of the PMLSM, the high-frequency component of the measured data can be filtered out. The measured and calculated force ripple and the force ripple calculated by the mechanistic model are shown in Figure 10.

It can be analyzed that due to certain components of force ripple that cannot be accurately modeled by mechanisms—for example, the generation mechanism of friction force is too complicated to be accurately described by the mechanistic model, and the friction force model needs to be obtained by experimental measurement, which leads to the lack of friction in the force ripple mechanistic model, resulting in differences between the force ripple mechanistic model and the actual force ripple—as well as the potential influence of process and parameters, the mechanistic model falls short in fully describing the actual force ripple.

The traditional PMLSM force control strategy has difficulty fully considering the friction and other load disturbances. From the above analysis, it can be seen that the force ripple includes the cogging magnetic resistance, the end magnetic resistance, the ripple force, the friction force, and other disturbing forces that can be effectively modeled.

The force ripple calculated based on the measured vibration acceleration data includes the above two sources of force ripple. The mechanical force ripple part of the PMLSM can be obtained by removing the mechanistic model part. After multi-sinusoidal function fitting, the mechanical force ripple which is difficult to model can be modeled and added to the control strategy as a part of feedforward compensation.

4. Evaluation and Analysis of Force Ripple Suppression Effect

To verify the effectiveness of the proposed algorithm, a 14-pole 12-slot PMLSM was used for experimental validation. The motor parameters are listed in Table 1, and the test platform is shown in Figure 6.

4.1. Suppression Effect of Force Ripple at Rated Speed

The experiment is first conducted at 0.12 m/s, at which the fundamental frequency is 5 Hz. The experiment results of the proposed vibration suppression algorithm are shown in Figure 11, Figure 12 and Figure 13.

By comparing the speed response curves under three compensation strategies, the system verifies the suppression effect of the proposed hybrid algorithm on the PMLSM force ripple. The experimental results show that at the rated speed (0.12 m/s), when the compensation strategy is not adopted, the motor speed exhibits significant high-frequency oscillation and steady-state error, and the ripple range exceeds ±10%, indicating that the cogging effect, end effect, and nonlinear friction have a serious negative impact on the stability of the system. When only the mechanistic model is used for compensation, the velocity ripple amplitude is reduced to about ±5%, and the residual periodic oscillation reveals the limitation of the pure mechanistic model to the unmodeled mechanical disturbance. After using the hybrid compensation algorithm of mechanism and data fusion, the ripple range is suppressed to below ±1%.

It can be seen from Figure 11 and

Table 2. The result at rated speed.

	The Control Method Without Compensation	The Control Method with Mechanistic Model Compensation	The Control Method with the Proposed Hybrid Algorithm
Speed Ripple	0.014 m/s	0.007 m/s	0.003 m/s
Torque Ripple	52 N	46 N	34 N

that the force ripple amplitude is significant without compensation, which reflects the characteristics of high-frequency oscillation. After introducing the mechanistic model compensation, the ripple range is reduced but there is still periodic residual oscillation, which indicates the limitation of the mechanistic model to the unmodeled mechanical disturbance. After using the hybrid algorithm (mechanism and data fusion), the force ripple is effectively suppressed. Compared with the uncompensated case, the thrust ripple is suppressed by more than 50%.

4.2. Suppression Effect of Force Ripple at Half Speed

The effect of the hybrid force ripple suppression algorithm proposed in this paper is further analyzed at different motion speeds. In the experiment, the low-speed operation state of half rated speed is selected. In this state, the speed ripple, output thrust, and dq-axis current of PMLSM are shown in Figure 14 and Figure 15.

Although

Table 3. The result at half rated speed.

	The Control Method Without Compensation	The Control Method with Mechanistic Model Compensation	The Control Method with the Proposed Hybrid Algorithm
Speed Ripple	0.008 m/s	0.006 m/s	0.002 m/s
Torque Ripple	38 N	32 N	31 N

shows that the proposed hybrid algorithm does not significantly reduce the motor force ripple compared to the mechanistic model method, it can be seen from Figure 15 that the proposed hybrid algorithm effectively reduces the low-frequency force ripple, which is the reason why the proposed hybrid algorithm can effectively reduce the motor speed fluctuation.

It can be seen from Figure 14 and Figure 15 that the thrust ripple of each source will still affect the running speed of the motor at low speed. The hybrid algorithm based on the mechanistic model and data model proposed in this paper can also effectively suppress the force ripple and speed ripple at lower speeds and has strong applicability.

4.3. Discussion

Through theoretical analysis and experimental verification, it can be seen that the hybrid algorithm proposed in this paper can effectively suppress the force ripple of the PMLSM and can achieve better results at different speeds. However, due to the current experimental conditions, the load experiment cannot be further carried out; that is, the effect of the current hybrid force ripple suppression algorithm cannot consider the influence of back electromotive force harmonics and current harmonics, which will be further improved in the future.

In addition, it can also be seen from the method of this paper that the suppression of the force ripple depends on the accurate mechanistic model and the measured vibration acceleration data. This means that on the one hand, it is necessary to accurately obtain the back electromotive force coefficient and size parameters of the PMLSM. On the other hand, it is also necessary to accurately measure the vibration acceleration in the feed direction at a certain speed in order to calculate the force ripple. This means that independent measurements and control strategies need to be developed for different PMLSM systems. For different speeds, different vibration acceleration values need to be measured to fit the data model. Therefore, the algorithm proposed in this paper can only be applied to steady-state conditions. In the future, this algorithm will be extended to transient conditions. The data model is collected and analyzed online and the deadbeat control logic is used to suppress the force ripple.

5. Conclusions

The permanent magnet synchronous linear motor has a significant force ripple problem under multi-source disturbances. The traditional force ripple suppression method is limited because it cannot take into account the physical characterization of electromagnetic disturbances and the dynamic characteristics of mechanical disturbances. In this paper, a hybrid algorithm of fusion mechanism and data is proposed. In the mechanism layer, the model of cogging force and end force is established based on electromagnetic field theory. In the data layer, the acceleration sensor is used to capture the dynamic response signal in real time, and the residual ripple model is constructed by combining frequency domain analysis and parameter fitting, which can effectively compensate for the disturbance that cannot be modeled in traditional methods such as friction nonlinearity and mechanical deformation. A PMLSM experimental platform was built on the basis of the hybrid algorithm. The experimental results show that the scheme suppresses the speed ripple amplitude from more than 10% to less than 3% under the rated speed condition, and the thrust ripple is reduced by more than 50%. The half-speed test further confirms its strong robustness, which is significantly better than the compensation effect of the single-mechanism model.