Adaptive Feedforward Control Based on Estimated Electromagnetic Parameters upon a Novel Two-Degrees-of-Freedom Actuator with a Unibody Magnetic Circuit

Abstract

This paper focuses on the study of the tracking performance of a novel two-degrees-of-freedom (two-DOF) voice coil motor (VCM). Intelligent control algorithms estimate the motor control model by the relationship between the input and output values, and it is hard to achieve excellent tracking performance due to the wider variational range of motor electromagnetic parameters (MEPs). This paper proposes estimated methods for MEPs and an adaptive feedforward control algorithm based on estimated MEPs (ACBE). ACBE can alter its control parameters from inside out in time with the change in MEPs. Experiments are carried out to verify that the proposed ACBE realized excellent tracking performance upon the novel two-DOF VCM. This study indicates that an adaptive control algorithm with high-precision control parameters has better tracking performance upon a motor control model with large variation.

1. Introduction

This paper focuses on the study of the tracking performance of a novel two-DOF VCM, which is applied as an end-effector of an optical fiber alignment system. The end-effectors of high-performance multi-dimensional motion platforms directly act on their terminal properties [1,2,3]. In order to improve the precision and reduce equipment size, multi-degrees-of-freedom motors are potential force sources for these end-effectors [4,5]. The novel two-DOF VCM has the advantage of a higher power–mass ratio and lower thrust (torque) ripple [6]. The structure of this novel two-DOF VCM is shown in Figure 1. Its linear mover and rotary rotor share one stator; this stator employs one permanent magnet (PM) to excite the main magnetic field, and the PM is magnetized in the direction of y. A Lorentz force will act on the linear coil of this two-DOF VCM when current flows through it; then, a linear reciprocating motion along direction x is produced. A Lorentz force also will be imposed on the rotary coil when current flows through it, but a rotary motion around the axis of the precision bearing is generated due to the special mechanism of the rotary coil and its bracket. The peculiarities of this novel two-DOF VCM are as follows:

(1)

The linear coil and swing coil share one iron yoke; their main magnetic circuits are the same. When the current in one coil changes, a related electromotive force (EMF) will be induced by the other coil.

(2)

Due to the rotation of the rotor, the relative position of the linear coil and the rotary coil will change; this leads to variations in the mutual inductance between the two coils and the leakage inductance of the rotary coil.

(3)

The swing coil also has back EMF when the linear motion is active separately.

The novel two-DOF VCM will carry optical fibers and scan for a light source. Excellent position tracking control performance is helpful for improving the quality and efficiency of fiber alignment. Many control strategies are used to improve the accuracy and tracking performance of motion systems. In [7], fuzzy control was adopted to achieve 15% speed tracking error upon a butterfly-shaped linear ultrasonic motor. To deal with the hysteresis of piezoelectric actuators, a fuzzy adaptive internal model controller was designed to improve the tracking performance, and 0.32% tracking error was realized [8]. Further, Long Cheng provided an adaptive fuzzy model-based predictive controller for piezoelectric actuators and reduced the tracking error to 0.2% [9]. Fuzzy fractional-order sliding-mode control effectively attenuated the chattering upon a permanent magnet linear synchronous motor and realized 0.26% tracking error [10].

Pan Huihui proposed disturbance observer-based adaptive tracking control for a class of nonlinear systems with parametric uncertainty, bounded external disturbance, and actuator saturation. The modeling inaccuracy and disturbance were integrated by a disturbance observer, which was estimated and compensated [11]. Liu Xiufeng took position and velocity as variates to establish a linear extended state observer for an iron-less permanent magnet linear synchronous motor. The precision control could compensate the slow-varying disturbance due to the observer, and 1.5 µm tracking performance was obtained [12]. Further, a periodic adaptive disturbance observer was provided for a permanent magnet linear synchronous motor, which estimated parametric errors, friction force, and force ripple. Depending on the observer, a 2.7 µm tracking performance was realized [13]. Zhao Ling designed observer-based integral sliding-mode tracking control for a pneumatic cylinder with varying loads. The generalized nonlinear extended state observer estimated the total disturbance and its derivative for the system, which reduced the tracking error to 0.25 mm [14]. An internal model control PID method based on a model linear extended state observer was used to improve the estimation accuracy for the system states and total unknown uncertainties, which was applied to a permanent magnet linear synchronous motor servo system, and 0.18% tracking error was accomplished [15]. An improved sliding-mode observer was also presented, which was based on the design idea of “identification–adjustment–adaptation”. It could adaptively estimate the load torque during variable-speed and variable-load operation and improved the response to load change successfully [16].

Active disturbance rejection control (ADRC) is an effective method for precision motion. In [17], a particle swarm optimization method was suggested to tune the parameters of ADRC for a magnetic levitation system, which enhanced the dynamic performance of the control system. The control strategy had better disturbance rejection capability. Predictive control is another control strategy for precision motion systems. Model predictive control was provided for a three-phase permanent magnet synchronous motor to deal with parameter variation and improve control precision, which was based on an indirect reference vector with an extended control set, and 8.9% current total harmonic distortion was realized [18]. Further, recurrent-neural-network-based model predictive control was presented for a nano-positioning system. The nonlinear dynamics of the nano-positioning system were accurately captured, which effectively guarantees the track performance of the system [19]. Xu Qingsong provided digital integral terminal sliding-mode predictive control for a piezoelectric-driven motion system. The control strategy was output-based and did not require a hysteresis model and a state observer and achieved 0.16% tracking performance [20].

The traditional identification methods of motor parameters include the Kalman filter [21,22], least square method [23,24], etc. The Kalman filter algorithm involves a lot of matrix and vector operations and needs to pre-process the motor mathematical model, so the process is complicated. The least square method needs to calculate the derivative of the objective function with respect to the motor parameters, which is susceptible to interference from measurement noise and motor speed fluctuations, which means unsteadiness. Compared with traditional methods, the intelligent optimization algorithm has the advantages of high efficiency, strong robustness, and low requirements for the objective function, such as the neural network algorithm [25,26]. However, it still has some problems, such as weak global search ability, poor anti-interference ability, and slow convergence speed.

Adaptive fuzzy control does not require an exact control model and has the advantages of stronger robustness and better tracking performance, but the fuzzy rule is key in precision motion and hard to obtain. The observer establishes equations of state from inputs and outputs and identifies the changes in the control model. However, the performance of the observer in one control system depends on precise state equations, which are non-universal and need be gradually optimized by lots of test data. The effect of ADRC and predictive control essentially depends on the accuracy of their control parameters, which are given by another intelligent algorithm. Nevertheless, the applicable intelligent algorithm also needs to be optimized by lots of data or be deduced by complex mathematical models.

Some of the above-discussed control algorithms require a lot of data, such as fuzzy control and neural networks; some are limited by the compensation accuracy of the inverter, such as the state observer; and some require a lot of computational time, such as the least squares method. At the same time, the control algorithms discussed above do not consider the characteristics of the control object, which is not conducive to the further optimization of the control performance. Based on the structure of the novel two-DOF VCM, this paper proposes a real-time estimated method for MEPs to deal with their changefulness. The estimated MEPs include the electrical time constant, torque/thrust constant, and back-EMF constant. The estimated method of the electrical time constant is based on the estimation of armature inductance, which is elaborated in Section 2. The estimated method of the torque/thrust constant and back-EMF constant is based on the estimation of air-gap flux density, which is elaborated in Section 3. This paper also proposes an adaptive feedforward control algorithm from inside out based on estimated parameters (ACBE) in Section 4. ACBE does not require measured data for optimization or complex control rules, and the control performance of ACBE depends mainly on the accuracy of the estimated electromagnetic parameters. Section 5 provides the experimental verification and analysis. The conclusion is given in Section 6.

2. Estimated Method of Electrical Time Constant

The estimated method of the electrical time constant in this paper takes core-loss resistance and magnetizing inductance as constant and time-varying parameters, respectively, in one current cycle. When the swing motion is active, the linear coil is treated as the tested coil, and the equivalent circuit of the swing motion is shown in Figure 2. R₁ and L_1σ(θ) are the resistance and leakage inductance of the swing coil, respectively, M_12σ(θ) is the mutual inductance between the swing and linear coil, u_eω(t) and θ are the back EMF and angle of swing motion, respectively, u₁(t) and i₁(t) are the voltage and current of the swing coil, respectively, N₁ and N₂ are the turns of the swing/linear coil, respectively, R_Fe1 and i_Fe1(t) are core-loss resistance and current for swing motion, respectively, and L_μ1(t), i_μ1(t), and e₁(t) are the magnetizing inductance, current, and voltage for the swing motion, respectively. L_1σ(θ) and M_12σ(θ) are changed with θ.

As i₁(t) is variational, the magnetic flow in the iron core φ_i(t) is generated only by i_μ1(t), and the open-circuit voltage e₂(t) of the linear coil is induced by L_μ1(t) and M_12σ(θ). The phase difference between i₁(t) and i_μ1(t) is γ. R_m1(t) and L_m1(t) are the excitation resistance and inductance for the swing motion, respectively, which are accorded with

$$R_{\mathrm{m}1}\left(t\right)=4{\pi }^2{f}^2{R}_{\mathrm{Fe}1}{L}_{\mathsf{\mu }1}^2\left(t\right)/\left[R_{\mathrm{Fe}1}^2+4{\pi }^2{f}^2{L}_{\mathsf{\mu }1}^2\left(t\right)\right],L_{\mathrm{m}1}\left(t\right)=R_{\mathrm{Fe}1}^2{L}_{\mathsf{\mu }1}\left(t\right)/\left[R_{\mathrm{Fe}1}^2+4{\pi }^2{f}^2{L}_{\mathsf{\mu }1}^2\left(t\right)\right],$$

where f is the frequency of i₁(t). The electrical time constant of the swing motion τ_a1(t) is equal to [L_1σ(θ) + L_m1(t)]. Then, τ_a1(t) is related to R_Fe1, L_μ1(t), and L_1σ(θ). R_Fe1 is estimated by the analysis of iron loss. The iron yoke in the 2-DOF VCM is not laminated; its iron loss P_Fe consists of hysteresis loss P_h and eddy current loss P_ec. Considering the effect of B_m and f on P_h, P_h can be given by

$$P_{\mathrm{h}}=C_{\mathrm{h}}\left(B_{\mathrm{m}},f\right)f^{\alpha \left(B_{\mathrm{m}},f\right)}V{B}_{\mathrm{m}}^{\beta \left(B_{\mathrm{m}},f\right)},$$

(1)

where V is the volume of iron, and C_h(B_m, f), α(B_m, f), and β(B_m, f) are the coefficients of hysteresis loss, frequency, and flux density, respectively. The coefficients can be obtained by the analysis of the hysteresis curve under different flux density amplitudes. B_m is the maximum flux density of the alternating magnetic field which is excited by i_μ1(t). For Equation (1), the accuracy of P_h depends on B_m greatly, and B_m is derived as follows. e₂(t) is excited by φ_i(t), and N₁e₂(t)/N₂ = N₁dφ_i(t)/dt + M_12σ(θ)di₁(t)/dt. Supposing the sample period is t₀, which is tiny, e₂(t) can be described as

$$N_1{e}_2\left(t\right)/N_2=N_1\left[{\phi }_i\left(t\right)-{\phi }_i\left(t-t_0\right)\right]/t_0+M_{12\sigma }\left[i_1\left(t\right)-i_1\left(t-t_0\right)\right]/t_0,$$

(2)

Then, φ_i(t) can be expressed as

$${\phi }_i\left(t\right)=\left\{N_1{e}_2\left(t\right)t_0-N_2{M}_{12\sigma }\left[i_1\left(t\right)-i_1\left(t-t_0\right)\right]\right\}/\left(N_1{N}_2\right)+{\phi }_i\left(t-t_0\right),$$

(3)

where φ_i(0) = φ_PM(x_l); φ_PM(x_l) is the initial magnetic flow caused by the PM only, and its value is related with the location of linear coil x_l. φ_imax and φ_−imax are the maximum positive and negative values of φ_i(t) in one current cycle, respectively, which are obtained by Equation (3). Considering the bias of the PM, B_m can be given by B_m = (φ_imax − φ_−imax)/2S, where S is the cross-area of iron. Then, P_h is obtained in one period.

Based on the structure of the 2-DOF VCM, the initial magnetic flux caused by the PM to the different parts of the iron is different. This paper equally divides the iron into 2n parts along the main magnetic path, which is shown in Figure 3a. The width of each part is l₀; l₀ is tiny, and l₀ = l/2n; l is the length of the main magnetic path. There are many eddy current loops within the cross-area of jth divided iron, and a supposed eddy current loop of the area of 4rz² is shown in Figure 3b, which is z away from the center line. φ_i(z) is the magnetic flux of 4rz² that is caused by i_μ1(t), r = l_cy/l_m, and l_cy and l_m are the lengths of the iron core in directions y and z, respectively; l is the length of the iron core.

The alternating magnetic density B_i(y,z) of the iron core is excited by i_μ1(t) within the cross-area, which is uneven. Considering the skin effect, its effective value B_rms(y,z) can be approximately expressed as

$$B_{\mathrm{rms}}\left(y,z\right)=B_{0\mathrm{rmsj}}\left[\cosh \left(k_{\mathrm{jy}}\right)+\cosh \left(k_{\mathrm{jz}}\right)\right],$$

(4)

where B_0rmsj is the effective value at the central point of jth divided iron, and

$$B_{0\mathrm{rmsj}}={\displaystyle {\int }_{-l_{\mathrm{m}}/2}^{l_{\mathrm{m}}/2}{\displaystyle {\int }_{-l_{\mathrm{cy}}/2}^{l_{\mathrm{cy}}/2}\frac{B_{\mathrm{rms}}\left(y,z\right)}{\cosh \left(k_{\mathrm{jy}}\right)+\cosh \left(k_{\mathrm{jz}}\right)}}}\mathrm{d}y\mathrm{d}z=\frac{k_{\mathrm{j}}{\phi }_{\mathrm{irms}}}{2\left[l_{\mathrm{cy}}\sinh \left(k_{\mathrm{j}}{l}_{\mathrm{m}}/2\right)+l_{\mathrm{m}}\sinh \left(k_{\mathrm{j}}{l}_{\mathrm{cy}}/2\right)\right]},$$

φ_irms is the effective value of φ_i(t), k_j is the skin coefficient of jth divided iron, and k_j = $\sqrt{2\mathsf{\pi }f{\mu }_{\mathrm{rmsj}}\sigma \mathrm{Fe}}$, where σ_Fe is the electrical conductivity of iron, and μ_rmsj is the magnetic conductivity of jth divided iron in one current cycle. Μ_rms can be given by μ_rms = μ_PMj + f (φ_irms/S). μ = f (B) is the normal magnetization curve of iron, and μ_PMj is the initial magnetic resistance of jth divided iron that is caused by the PM. Supposing E_rmsj(z) and φ_irmsj(z) are the effective values of the induced voltage and magnetic flow of 4rz², respectively, E_irms and φ_irms are the effective values of the induced voltage and magnetic flow of S, respectively. E₁ is the effective value of e₁(t); then,

$$\frac{E_{\mathrm{rmsj}}\left(z\right)}{E_{\mathrm{irms}}}=\frac{E_{\mathrm{rmsj}}\left(z\right)}{E_1/N_1}=\frac{{\phi }_{\mathrm{irmsj}}\left(z\right)}{{\phi }_{\mathrm{irms}}}=B_{0\mathrm{rmsj}}\frac{4rz\sinh \left(k_{\mathrm{j}}z\right)+4z\sinh \left(k_{\mathrm{j}}z\right)}{k_{\mathrm{j}}{\phi }_{\mathrm{irms}}}$$

is founded, which deduces

$$E_{\mathrm{rmsj}}\left(z\right)=2E_1\left[rz\sinh \left(k_{\mathrm{j}}z\right)+z\sinh \left(r{k}_{\mathrm{j}}z\right)\right]/\left\{N_1\left[l_{\mathrm{cy}}\sinh \left(k_{\mathrm{j}}{l}_{\mathrm{m}}/2\right)+l_{\mathrm{m}}\sinh \left(k_{\mathrm{j}}{l}_{\mathrm{cy}}/2\right)\right]\right\}.$$

(5)

Referring to Figure 3b, the conductance dG corresponding to E_rmsj(z) can be expressed as

$$\mathrm{d}G=k_{re}{\sigma }_{\mathrm{Fe}}r{l}_0\mathrm{d}z/\left[4\left(1+r^2\right)z\right],$$

(6)

k_re is the path correction coefficient, which depends on the shape and size of the cross-area of iron. Referring to Equations (5) and (6), the eddy current loss of jth divided iron P_ecj can be written as

$$P_{\mathrm{ecj}}={\displaystyle {\int }_{-l_{\mathrm{m}}/2}^{l_{\mathrm{m}}/2}{E}_{\mathrm{rmsj}}^2\left(z\right)\mathrm{d}G}=\frac{k_{re}{\sigma }_{\mathrm{Fe}}r{l}_0{E}_1^2{\displaystyle {\int }_{-l_{\mathrm{m}}/2}^{l_{\mathrm{m}}/2}z{\left[rz\sinh \left(k_{\mathrm{j}}z\right)+z\sinh \left(r{k}_{\mathrm{j}}z\right)\right]}^2\mathrm{d}z}}{N_1^2\left(1+r^2\right){\left[l_{\mathrm{cy}}\sinh \left(k_{\mathrm{j}}{l}_{\mathrm{m}}/2\right)+l_{\mathrm{m}}\sinh \left(k_{\mathrm{j}}{l}_{\mathrm{cy}}/2\right)\right]}^2},$$

Then,

$$R_{\mathrm{Fe}1}=E_1^2/\left[C_{\mathrm{h}}\left(B_{\mathrm{m}},f\right)f^{\alpha \left(B_{\mathrm{m}},f\right)}V{B}_{\mathrm{m}}^{\beta \left(B_{\mathrm{m}},f\right)}+{\displaystyle {\sum }_{\mathrm{j}=1,2,\dots .}^{2n}{P}_{\mathrm{ecj}}}\right].$$

(7)

The estimated method for R_Fe1 has fully considered the distribution of magnetic density; moreover, the tested E₁ is helpful in weakening the influence of non-sinusoidal current on core loss, which makes the estimated R_Fe1 more accurate. Referring to Figure 2,

$$\begin{array}{cc}{e}_1\left(t\right)=L_{\mathsf{\mu }1}\left(t\right){i_{\mathsf{\mu }1}}^{\prime }\left(t\right)=N_1{e}_2\left(t\right)/N_2-M_{12\sigma }{i_{\mathsf{\mu }1}}^{\prime }\left(t\right),& i_{\mathsf{\mu }1}\left(t\right)=i_1\left(t\right)-i_{\mathrm{Fe}}\left(t\right)=i_1\left(t\right)-e_1\left(t\right)/R_{\mathrm{Fe}1}\end{array}$$

are founded, which deduce that

$$L_{\mathsf{\mu }1}\left(t\right)=\left\{\begin{array}{cc}{L}_{\mathsf{\mu }1}\left(t-t_0\right)\hfill & f\left(\theta ,t\right)=0\hfill \\ R_{\mathrm{Fe}1}\left[N_1{e}_2\left(t\right)/N_2-M_{12\sigma }\mathrm{d}{i}_{\mathsf{\mu }1}\left(t\right)/\mathrm{d}t\right]/f\left(\theta ,t\right)\hfill & f\left(\theta ,t\right)\ne 0\hfill \end{array}\right.$$

(8)

where

$$f\left(\theta ,t\right)=R_{\mathrm{Fe}1}{i_1}^{\prime }\left(t\right)+{M_{12\sigma }}^{\prime }\left(\theta \right){\theta }^{\prime }\left(t\right){i_1}^{\prime }\left(t\right)+M_{12\sigma }\left(\theta \right){i_1}^{″}\left(t\right)-N_1{e_2}^{\prime }\left(t\right)/N_2$$

and t₀ is the sampling interval. Equation (8) indicates that L_μ1(t) is an instantaneous value.

3. Estimated Method of Torque Constant

To the novel 2-DOF VCM, the torque constant C_f is the same as the back-EMF constant C_e. Due to different x_s(t) and armature reaction, the gap flux density is not immobile, which leads to C_f and C_e that are variational. X_s(t) is the equivalent position of the swing coil. Swing motion is integrated in linear motion, x_s(t) = x_l + d + (d_b + b_st + l_sz/2)sin[θ(t)], which is shown in Figure 4b. x_l is the position of the linear coil, and d is the distance between the linear and swing coils as θ = 0. Supposing the equivalent gap flux density of the swing motion is B_δe[x_s(t), φ_i], and the flux through the forced edge of the swing coil is φ_δ[x_s(t), φ_i], φ_δ[x_s(t), φ_i] = B_δ[x_s(t), φ_i]A_s, and A_s is the cross-area of the swing coil on the plane xoz. The real-time torque constant C_f [x_s(t), φ_i] can be written as

C_f[x_s(t), φ_i] = C_e[x_s(t), φ_i] = B_δe[x_s(t), φ_i]C_e[x_s(t), 0]/B_δe[x_s(t), 0],

(9)

where C_e[x_s(t), 0] and B_δe[x_s(t), 0] are the back-EMF constant and equivalent gap flux density at x_s(t) when φ_i = 0, respectively. C_e[x_s(t), 0] can be obtained by the test of back EMF. B_δe[x_s(t), 0] can be obtained by the calculation or simulation of the static magnetic field.

When φ_i = 0, φ_δ[x_s(t), 0] is excited only by F(b_s), F(b_s) is the magnetomotive force of the PM under swing coil, and the center line of the swing coil passes through pth and qth divided iron, which are shown in Figure 4a. φ_δ[x_s(t), 0] is divided into φ_mn and φ_mp, and its equivalent magnetic circuit can be treated as Figure 4c. R_δ1 is the magnetic resistance of gap δ₁; its value is changeless and can be written as δ₁/[μ₀(l_m + δ₁)(l_cy + δ₁)].

R_mz[x_s(t), 0] and R_my[x_s(t), 0] are the magnetic resistance of iron located to the left and right of the swing coil, respectively, when t = 0, and

$$R_{\mathrm{mz}}\left[x_s\left(t\right),0\right]={\displaystyle {\sum }_{h=q,\left(q+1\right)\dots }^p{l}_0/\left[g\left(B_{\mathrm{PM}h}\right)S\right]},$$

(10)

$$R_{\mathrm{my}}\left[x_s\left(t\right),0\right]={\displaystyle {\sum }_{i=1,2\dots }^{\left(q-1\right)}{l}_0/\left[g\left(B_{\mathrm{PM}i}\right)S\right]}+{\displaystyle {\sum }_{j=\left(p+1\right),\left(p+2\right)\dots }^{2n}{l}_0/\left[g\left(B_{\mathrm{PM}j}\right)S\right]}.$$

(11)

where μ = g(B) is the initial magnetization curve, B_PMk is the initial flux density of kth divided iron, and k belongs to [1 2n]. R_δ0[x_s(t)] is the magnetic resistance of gap δ₀ for the swing coil at x_s(t), which is hard to calculate directly. To avoid the uncertain path of magnetic field lines within gap δ₀, this paper treats F(b_s) as a constant along the x-axis, but φ_δ[x_s(t), 0] is a variate. Referring to Figure 4c,

$$R_{\mathsf{\delta }0}\left[x_s\left(t\right)\right]=\frac{F\left(b_{\mathrm{s}}\right)}{{\phi }_{\mathsf{\delta }}\left[x_s\left(t\right),0\right]}-\frac{\left\{R_{\mathsf{\delta }1}+R_{\mathrm{my}}\left[x_s\left(t\right),0\right]\right\}\left\{R_{\mathsf{\delta }1}+R_{\mathrm{mz}}\left[x_s\left(t\right),0\right]\right\}}{2R_{\mathsf{\delta }1}+R_{\mathrm{my}}\left[x_s\left(t\right),0\right]+R_{\mathrm{mz}}\left[x_s\left(t\right),0\right]}.$$

When i₁(t) is active, its equivalent magnetic circuit is as shown in Figure 4d. The added flux φ_i(t) in iron will alter the original flux densities of divided irons, which lead to R_mz[x_s(t), φ_i] and R_my[x_s(t), φ_i] being changed. The variational R_mz[x_s(t), φ_i] and R_my[x_s(t), φ_i] cause the change in B_δ[x_s(t), φ_i]A_s, so accurate R_mz[x_s(t), φ_i] and R_my[x_s(t), φ_i] are the key for B_δ[x_s(t), φ_i]. Supposing R_mzr[x_s(t), φ_i] and R_myr[x_s(t), φ_i] are the magnetic resistance of iron located to the left and right of the swing coil regardless of the magnetic saturation, respectively, they can be expressed as

$$R_{\mathrm{mzr}}\left[x_s\left(t\right),{\phi }_i\right]=\left\{{\displaystyle {\sum }_{h=q,\left(q+1\right)\dots }^p{l}_0/\left\{g\left(B_{\mathrm{PM}h}\right)+f\left[{\phi }_i\left(t\right)/S\right]\right\}}\right\}/S,$$

$$R_{\mathrm{myr}}\left[x_s\left(t\right),{\phi }_i\right]={\displaystyle {\sum }_{i=1,2\dots }^{\left(q-1\right)}\frac{l_0/S}{g\left(B_{\mathrm{PM}i}\right)+f\left[{\phi }_i\left(t\right)/S\right]}+{\displaystyle {\sum }_{j=\left(p+1\right),\left(p+2\right)\dots }^{2n}\frac{l_0/S}{g\left(B_{\mathrm{PM}j}\right)+f\left[{\phi }_i\left(t\right)/S\right]}}}.$$

In the initial magnetic path of the PM, the fluxes in 1th~nth divided irons have opposite directions to those in (n + 1)th~2nth divided irons, which make φ_i(t) magnetized in one part and demagnetized in the other part. So, only one of R_mzr[x_s(t), φ_i] and R_myr[x_s(t), φ_i] is affected by magnetic saturation. If φ_i(t) has the same direction as φ_mn, R_my[x_s(t), φ_i] = R_myr[x_s(t), φ_i]; else, R_mz[x_s(t), φ_i] = R_mzr[x_s(t), φ_i]. Referring to Figure 4d, R_my[x_s(t), φ_i] + R_mz[x_s(t), φ_i] = $N_1^2$/L_μ1(t) − 2R_δ1 is founded. Then, R_mz[x_s(t), φ_i] and R_my[x_s(t), φ_i] can, respectively, be expressed as

$$R_{\mathrm{mz}}\left[x_s\left(t\right),{\phi }_i\right]=\left\{\begin{array}{cc}{R}_{\mathrm{mzr}}\left[x_s\left(t\right),{\phi }_i\right]\hfill & sign\left[{\phi }_i\left(t\right)\right]=sign\left({\phi }_{\mathrm{mn}}\right)\hfill \\ N_1^2/L_{\mathsf{\mu }1}\left(t\right)-2R_{\mathsf{\delta }1}-R_{\mathrm{myr}}\left[x_s\left(t\right),{\phi }_i\right]\hfill & sign\left[{\phi }_i\left(t\right)\right]\ne sign\left({\phi }_{\mathrm{mn}}\right)\hfill \end{array}\right.$$

$$R_{\mathrm{my}}\left[x_s\left(t\right),{\phi }_i\right]=\left\{\begin{array}{cc}{R}_{\mathrm{myr}}\left[x_s\left(t\right),{\phi }_i\right]\hfill & sign\left[{\phi }_i\left(t\right)\right]=sign\left({\phi }_{\mathrm{mn}}\right)\hfill \\ N_1^2/L_{\mathsf{\mu }1}\left(t\right)-2R_{\mathsf{\delta }1}-R_{\mathrm{mzr}}\left[x_s\left(t\right),{\phi }_i\right]\hfill & sign\left[{\phi }_i\left(t\right)\right]\ne sign\left({\phi }_{\mathrm{mn}}\right)\hfill \end{array}\right.$$

Then, B_δe[x_s(t), φ_i] can be deduced by

$$B_{\delta e}\left[x_s\left(t\right),{\phi }_i\right]=\frac{N_1^2\mathrm{F}\left(b_{\mathrm{s}}\right)}{A_{\mathrm{s}}\left\{L_{\mathsf{\mu }1}\left(t\right)\left\{R_{\mathsf{\delta }1}+R_{\mathrm{mz}}\left[x_s\left(t\right),{\phi }_i\right]\right\}\left\{R_{\mathsf{\delta }1}+R_{\mathrm{my}}\left[x_s\left(t\right),{\phi }_i\right]\right\}+N_1^2{R}_{\mathsf{\delta }0}\left[x_s\left(t\right)\right]\right\}}.$$

Then, C_f [x_s(t), φ_i] and C_e [x_s(t), φ_i] are presented.

4. Adaptive Feedforward Control Based on Estimated Electromagnetic Parameters

The proposed ACBE is a typical three-loop control system; ACR1 is the PI controller for the current loop, ASR1 is the PI controller for the speed loop, and APR1 is the P controller for the position loop, as shown in Figure 5. In particular, the parameters of ACR1, ASR1, and APR1 are changed as estimated electromagnetic parameters, which always ensure the expected performance. Further, the position feedforward controller (PFC) and drag feedforward controller (DFC) are added to the ACBE. The PFC and DFC are beneficial to tracking performance, especially as they update parameters in real time.

In Figure 5, K and T₀ are the voltage amplification factor and equivalent time constant of the PWM converter, respectively. T_fi1 and T_fn1 are the filtering time of current feedback and position feedback, respectively. k₁₁ and k₁₂ are the current and speed feedback coefficients, respectively. J is the rotational inertia of the swing motion. T_e and T_f are the electromagnetic and drag torque, respectively. w_ACR1(s), w_ASR1(s), and w_APR1(s) are the transfer function of ACR1, ASR1, and APR1, respectively.

w_ACR1(s) = K_c1(t)[1 + τ_i1(t)s]/[τ_i1(t)s], w_ASR1(s) = K_s1(t)[1 + τ_n1(t)s]/[τ_n1(t)s], w_APR1(s) = K_p1(t).

where K_c1(t) and τ_i1(t) are the amplification coefficient and integration time constant of ACR1, respectively, K_s1(t) and τ_n1(t) are the amplification coefficient and integration time constant of ASR1, respectively, and K_p1(t) is the amplification coefficient of APR1. Ignoring the influence of speed on the bearing resistance, these variates are changed as τ_a1(t), L_m1(t), L_1σ(θ), and C_e [x_s(t), φ_i]. Taking the current loop as a second-order system, based on the traditional performance optimization tuning method,

K_c1(t) = [L_1σ(θ) + L_m1(t)]/[2K(T₀ + T_fi1)k₁₁], τ_i1(t) = τ_a1(t)= [L_1σ(θ) + L_m1(t)]/[R₁ + R_m1(t)].

Which indicates that the parameters of w_ACR1(s) are changed as τ_a1(t), L_m1(t), and L_1σ(θ) and always keep the performance of the current loop. Considering back EMF, the open-loop transfer function of current-loop W_oc(s) can be written as

$$W_{oc}\left(s\right)=K_{\mathrm{c}1}\left(t\right)k_{11}\left[1+{\tau }_{\mathrm{i}1}\left(t\right)\right]/\left\{{\tau }_{\mathrm{i}1}\left(t\right)\left(T_{0}s+1\right)\left(T_{\mathrm{fi}1}s+1\right)\left\{L_s\left(t\right)s^2+R_s\left(t\right)s+C_e^2\left[x_s\left(t\right),{\phi }_i\right]/J\right\}\right\}.$$

(12)

In order to reduce the impact of bearing resistance on the tracking performance, the DFC is added at the input of the current loop. The transfer function of the DFC is F_n(s), and

$$F_n\left(s\right)=k_{11}\left[1+W_{oc}\left(s\right)\right]/\left\{W_{oc}\left(s\right)\left(T_{\mathrm{fi}1}s+1\right)C_f\left[x_s\left(t\right),{\phi }_i\right]\right\},$$

which indicates that the parameters of F_n(s) are changed as τ_i1(t), K_c1(t), and C_e [x_s(t), φ_i] and effectively keep the accuracy of the DFC. A filter link is added after ASR1, and the current loop is treated as an inertia link; then, the open-loop transfer function of the speed-loop W_os(s) can be written as

$$W_{os}\left(s\right)=C_f\left[x_s\left(t\right),{\phi }_i\right]K_{\mathrm{s}1}\left(t\right)k_{12}\left[1+{\tau }_{\mathrm{n}1}\left(t\right)s\right]/\left\{J{k}_{11}{\tau }_{\mathrm{n}1}\left(t\right)s^2\left[T_e\left(t\right)s+1\right]\right\},$$

(13)

where T_e(t) is the equivalent time constant of the current loop. Referring to Equation (12) and omitting the higher term of the small time constant,

$$T_e\left(t\right)=L_s\left(t\right)/\left[R_s\left(t\right)+K_{\mathrm{c}1}\left(t\right)k_{11}\right]+{\tau }_{\mathrm{i}1}\left(t\right)\left[R_s\left(t\right)+K_{\mathrm{c}1}\left(t\right)k_{11}\right]/\left[K_{\mathrm{c}1}\left(t\right)k_{11}\right]+T_{\mathrm{fn}1}.$$

Equation (13) indicates that W_os(s) is a typical type II system. Supposing h is the intermediate frequency width of the speed loop, K_s1(t) and τ_n1(t) can be deduced as

$${\tau }_{\mathrm{n}1}\left(t\right)=h{T}_e\left(t\right),K_{\mathrm{s}1}\left(t\right)=J{k}_{11}/\left\{\sqrt{h}{T}_e\left(t\right)C_f\left[x_s\left(t\right),{\phi }_i\right]k_{12}\right\},$$

which indicates that the parameters of w_ASR1(s) are changed as L_m1(t), L_1σ(θ), τ_i1(t), K_c1(t), h, and C_f [x_s(t), φ_i] and always keep the performance of the speed loop. In order to further improve the track performance of the speed loop, the PFC is added at the input of the speed loop. The transfer function of the PFC is F_r(s), and

$$F_r\left(s\right)=k_{12}\left[W_{oc}\left(s\right)+1\right]/\left[W_{oc}\left(s\right)\left(s{T}_{\mathrm{fn}1}+1\right)\right].$$

Supposing the gain of the speed loop is K_os(t),

$$K_{os}\left(t\right)=K_{\mathrm{s}1}\left(t\right)k_{12}{C}_f\left[x_s\left(t\right),{\phi }_i\right]/\left[Jh{T}_e\left(t\right)k_{11}\right].$$

Referring to Figure 5, the close-loop transfer function of position-loop W_cp(s) can be written as

$$W_{cp}\left(s\right)=K_{\mathrm{p}1}\left(t\right)K_{os}\left(t\right)/\left\{k_{12}\left[T_e\left(t\right)s^4+s^3+K_{os}\left(t\right){\tau }_{\mathrm{n}1}\left(t\right)s^2+K_{os}\left(t\right)s\right]+K_{\mathrm{p}1}\left(t\right)K_{os}\left(t\right)\right\}$$

$$\approx K_{\mathrm{p}1}\left(t\right)/\left\{k_{12}\left\{{\tau }_{\mathrm{n}1}\left(t\right)+T_e\left(t\right)+1/\left[K_{os}\left(t\right){\tau }_{\mathrm{n}1}\left(t\right)\right]\right\}+k_{12}s+K_{\mathrm{p}1}\left(t\right)\right\},$$

which is treated as a second-order system. Supposing the damping ratio is ζ, and the undamped natural frequency is ω_n, then

$$2\zeta {\omega }_{\mathrm{n}}=1/\left\{{\tau }_{\mathrm{n}1}\left(t\right)+T_e\left(t\right)+1/\left[K_{os}\left(t\right){\tau }_{\mathrm{n}1}\left(t\right)\right]\right\},{\omega }_{\mathrm{n}}^2=K_{\mathrm{p}1}\left(t\right)/\left\{k_{12}\left\{{\tau }_{\mathrm{n}1}\left(t\right)+T_e\left(t\right)+1/\left[K_{os}\left(t\right){\tau }_{\mathrm{n}1}\left(t\right)\right]\right\}\right\}$$

is founded, which deduces

$$K_{\mathrm{p}1}\left(t\right)=k_{12}/\left\{4{\zeta }^2\left\{{\tau }_{\mathrm{n}1}\left(t\right)+T_e\left(t\right)+1/\left[K_{os}\left(t\right){\tau }_{\mathrm{n}1}\left(t\right)\right]\right\}\right\}.$$

The above expositions show that the control parameters of ACBE are changed as estimated electromagnetic parameters from the inside out; once the estimated electromagnetic parameters are accurate, the expected tracking performance can be realized.

5. Experimental Test and Discussion

Based on the proposed control model, an experimental platform was conducted, as shown in Figure 6.

The permeability of the core is always changed by the variational current. Figure 7a is the input current of the swing coil, and the tested curve of e₂(t) is shown in Figure 7b. The phase between i₁(t) and e₂(t) is not 90 degrees due to R_m1(t). Based on the tested e₂(t) and Equation (7), R_Fe1 = 0.166 Ω is obtained in this cycle. Then, the estimated τ_a1(t) is deduced by the proposed method, which is shown in Figure 7c. The estimated τ_a1(t) indicates that the electrical time constant of the swing coil has changed greatly within a cycle of i₁(t) = 1.4sin(2πt), and it is significant to estimate the electrical time constant and update the related control parameters. Due to the hysteresis characteristic of the core, τ_a1(t) lags behind i₁(t) obviously. When the absolute value of i₁(t) is greater than 1A, τ_a1(t) has less change due to the core being saturated.

The accuracy of the estimated τ_a1(t) is important for the performance of the proposed ACBE, but both the amplitude and frequency of i₁(t) can affect the estimated accuracy. Figure 8 shows the effect of different current amplitudes on the accuracy of the estimated τ_a1(t). Due to the increasing B_m and core reluctance, the estimated R_Fe1 increases slightly as the current amplitude increases, but the estimated L_μ1e decreases, which is shown in Figure 8a,b. L_μ1e is the effective value of L_μ1(t). The tested R_s and L_s in Figure 8c,d are obtained by the digital electric bridge, and the estimated R_s and L_s are deduced by the estimated R_Fe1 and L_μ1e. In comparison with the tested R_s and L_s, the estimated values are smaller due to the accuracy of the estimated R_Fe1 being related to the tested error of e₂(t) and the estimated error of the core loss. The effective value of noise of e₂(t) has a greater proportion as the value of E₂ is smaller, which makes the error of the estimated R_Fe1 larger at a lower current, and the accuracy of the estimated R_s and L_s increases as I changes from 0.1 A to 1 A. When I exceeds 1 A, the core becomes saturated, and the estimated core loss based on the proposed method is relatively bigger than that when it is not saturated, which leads to a slightly smaller estimated R_Fe1. The value of the estimated L_μ1(t) is related to the estimated R_Fe1; thus, the variation tendency of L_μ1(t) is similar to that of R_Fe1. So, the estimated R_s and L_s have the highest accuracy around I = 1 A. Based on Figure 8c,d, Figure 8e shows that the highest accuracy of the estimated electrical time constants under the proposed method is around 1.31%, which appears at the saturation point of the core.

Figure 9 shows the effect of different current frequencies on the accuracy of the estimated τ_a1(t). Due to the increasing f, the estimated R_Fe1 increases obviously with the increase in current frequency, but the estimated L_μ1e also decreases, which is shown in Figure 9a,b. The accuracy of the estimated core loss is depressed with the rise in current frequency, which leads to the accuracy of the estimated R_Fe1 decreasing gradually. The variation tendency of L_μ1(t) is also similar to that of R_Fe1. So, the estimated R_s and L_s are smaller than the tested R_s and L_s, and the estimation accuracy decreases with increasing frequency, which is shown in Figure 9c and Figure 9d, respectively. Figure 9e shows that the estimation accuracy of the electrical time constants also decreases with increasing frequency, which is from 1.31% to 2.02% as f goes from 1 Hz to 10 Hz.

Figure 8 and Figure 9 illustrate that the electrical time constant estimation based on the proposed method has high accuracy around the saturation point of the core and decreases with increasing frequency.

Figure 10a,b show the palstance ω and armature voltage E_bEMF of the back-EMF test, respectively. The tested static C_f is obtained by E_bEMF/ω, which is shown in Figure 10c. Due to the effective length of the swing coil being increased as cos (θ) becomes smaller, the static C_f has minimum and maximum values of around θ = 0° and θ = ±8°, respectively, which are 28.356 mN/A and 28.905 mN/A. The armature current can enhance or reduce the magnetic density of the air gap. When I = 1 A, the tested dynamic C_f for size 0.5° within [−8° 8°] is shown in Figure 10d. Compared to the static C_f, the dynamic C_f under I = 1 A is larger and also has minimum and maximum values of around θ = 0° and θ = ±8°, respectively. Without the proposed method, the error of C_f under I = 1 A is around 4.6%, which is shown in Figure 10e. Figure 10d shows that the estimated dynamic C_f is smaller than the tested value because the estimated L_u1(t) is smaller than the actual value. The estimated accuracy is reduced as cos (θ) becomes smaller because the difference between R_mz and R_my becomes bigger as cos (θ) becomes smaller. Figure 10e shows that the error of C_f under I = 1 A with the proposed method is around 1.5%.

In order to verify the track performance of ACBE, this paper takes fuzzy control (FZC) as a comparative object. The given angle and palstance input of reciprocating motion are shown in Figure 11a and Figure 11b, respectively; the angle is from −8° to +8°, f = 1 Hz, and palstance is 0.698 rad/s at uniform motion. Under the given input, Figure 11c shows the current of the swing coil with ACBE and FZC; I_u is the current at uniform motion, which is around 1 A. The magnitude and fluctuation of the armature current can affect the accuracy of the control parameters of ACBE and vice versa. The control parameters of ACBE have higher accuracy when I_u = 1 A and f = 1 Hz, which make its current fluctuations lower than those of FZC. The varying torque (acceleration) will increase the amplitude of the armature current, which is not beneficial to the tracking performance of ACBE.

Figure 11d shows that the track error of palstance e_ω(t) under ACBE is larger than that of FZC as the torque changes (acceleration) but lower in the region of uniform motion. Under the given input of palstance, the region of uniform motion of ACBE and FZC means the steady-state region of the response curve of palstance, which is also called effective travel. The effective travel of ACBE is around 11.687°, which is lesser than that of FZC (12°). Within the region of uniform motion, the palstance track performance of ACBE is better than that of FZC, which is shown in Figure 11e,f. The palstance track error at uniform motion of ACBE is less than 4 × 10⁻³ rad/s, and that of FZC is 7.5 × 10⁻³ rad/s. The root-mean-square (RMS) of the palstance track error can partly reflect the palstance track performance of the control method at uniform motion, where ACBE is around 1.3 × 10⁻³ rad/s and FZC is around 2.5 × 10⁻³ rad/s. Similar to the palstance track error, Figure 11g shows that the track error of angle e_θ(t) under ACBE is more than that of FZC at the beginning of motion and lesser at uniform motion. Within the region of uniform motion, the e_θ(t) of ACBE is changed from 18.8 × 10⁻³ degrees to 20.5 × 10⁻³ degrees, and its RMS value is 19.7 × 10⁻³ degrees, which is shown in Figure 11h. Meanwhile, the e_θ(t) of FZC is changed from 26.9 × 10⁻³ degrees to 29.3 × 10⁻³ degrees, and its RMS value is 28.3 × 10⁻³ degrees, which is shown in Figure 11i. Figure 11 reveals that ACBE with higher accuracy control parameters has better track performance than FZC at the region of uniform motion but lesser effective travel. Both the armature current and reciprocating frequency can affect the accuracy of the control parameters of ACBE.

I_u is the armature current at uniform motion, which can be increased by adjusting the friction torque. Different I_u changes the accuracy of the control parameters of ACBE, which determine the tracking performance of ACBE. θ_e is effective travel, and e_ωRMS and e_θRMS are the RMS values of e_ω(t) and e_θ(t) at uniform motion, respectively, which are evaluating indicators of tracking performance. Figure 12 shows the impact of different I_u on the θ_e, e_ωRMS, and e_θRMS of ACBE and FZC. When I_u is changed from 0.4 A to 1.2 A (step size 0.1 A), the θ_e of FZC is always a constant, the e_ωRMS of FZC hardly changes, and the e_θRMS of FZC is changed from 25.7 × 10⁻³ degrees to 28.9 × 10⁻³ degrees as the I_u increases. For ACBE, the θ_e, e_ωRMS, and e_θRMS have a similar change trend as the increase in I_u; their best points are all around I_u = 1 A, which is the critical saturation point of the iron core. When ACBE works at the critical saturation point of the iron core, its parameters have the least fluctuation. The values of I_u not only determine the accuracy of the ACBE parameters at the uniform motion but also at the acceleration motion. When the armature current is closer to 1 A, the accuracy of ACBE parameters is higher. Compared to FZC, ACBE is more affected by the change in armature current, but within I_u = [0.8 A 1 A], the tracking performance of ACBE is better than that of FZC, especially when I_u is around 1 A.

Except for the armature current, the reciprocating frequency f also can affect the accuracy of the control parameters of ACBE. The angle range of the reciprocating motion is kept at −8° to +8°, and I_u is around 1 A. Figure 13 shows the impact of different f on the θ_e, e_ωRMS, and e_θRMS of ACBE and FZC. As f increases, the acceleration and amplitude of the given palstance curve (trapezoidal wave) become bigger, which increases the armature current at acceleration motion and the displacement at the sampling interval. So, the tracking performance of ACBE and FZC declines as f increases. An increased armature current seriously degrades the accuracy of the control parameters of ACBE at acceleration motion; the θ_e of ACBE decreases more than that of FZC as f increases, which is shown in Figure 13a. At uniform motion, increased f also reduces the accuracy of the control parameters of ACBE. The e_θRMS and e_ωRMS of ACBE are more than those of FZC when f exceed 3 Hz and 4 Hz, respectively, which is shown in Figure 13b,c. ACBE is more affected as f increases, but within f = 3 Hz, the tracking performance of ACBE is still better than that of FZC.

The above experiments show that the proposed methods for MEPs have higher accuracy at low frequency and around the critical saturation point of the iron core. The tracking performance of ACBE depends on the accuracy of the estimated electromagnetic parameters.

6. Conclusions

In this paper, through the analysis of the armature current and the flux of the main magnetic circuit, the real-time armature inductance, the reluctance of the main magnetic circuit, and the air-gap magnetic density are deduced. Further, the electromagnetic parameters are estimated. According to the estimated electromagnetic parameters, ACBE adjusts the control parameters of the current loop, speed loop, position loop, and feedforward controller from inside out. A higher accuracy of the estimated electromagnetic parameters and better tracking performance of ACBE are observed. With the proposed method, the palstance tracking error of the swing motion is within 4 × 10⁻³ rad/s, and the angle tracking error is around 19.7 × 10⁻³ degrees. This study indicates that adaptive algorithms based on a real-time control model have better tracking performance at uniform motion in a motor system. In addition, ACBE can not only be adopted in the two-DOF VCM but can also be used in other linear VCMs which need just one detective coil, which means that the proposed ACBE can be applied more widely.