Hysteresis nonlinearity modeling and position control for a precision positioning stage based on a giant magnetostrictive actuator

Abstract

A precision positioning stage based on giant magnetostrictive actuator (PPS-GMA) shows nonlinear displacement when it is used in the field of precision positioning control. To improve the defect, the Jiles–Atherton hysteresis model and the dynamic recurrent neural network (DRNN) feed forward-fuzzy PID feedback control strategy were adopted. An accurate hysteresis nonlinearity model of PPS-GMA was established with the Jiles–Atherton model and its parameters were identified using the particle swarm optimization (PSO) algorithm. A dynamics inverse model of the PPS-GMA was established with the DRNN learning method to compensate the hysteresis nonlinearity characteristic. A fuzzy PID feedback control was used to compensate for the mapping error of DRNN. Using these control methods, the positioning accuracy of the precision positioning stage was improved. The simulation and experimental results show that the Jiles–Atherton hysteresis model can describe the hysteresis nonlinear characteristic of the precision positioning stage, the PSO algorithm has high precision for parameter identification, the DRNN feed forward-fuzzy PID feedback control strategy can effectively eliminate the nonlinear characteristics of the PPS-GMA, which has practical significance for improving the positioning accuracy of the PPS-GMA.

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Introduction

The precision positioning stage based on a giant magnetostrictive actuator (PPS-GMA) is a precision positioning device using a giant magnetostrictive actuator (GMA) as the driving system. Compared with the traditional positioning stage based on a piezoelectric ceramic actuator, the PPS-GMA has some excellent performances such as fast responses, high control precision and large loads.^1–3 However, a GMA based on the Joule effect of a giant magnetostrictive material (GMM) has hysteresis nonlinearity between the output magnetization and input magnetic field which leads to a hysteresis error of output displacement of about 20%.⁴ In order to meet the demand of precision positioning, a novel hysteresis nonlinearity model of a GMA is urgently required. Then a reasonable control mode can be adopted to improve the control precision of GMAs and to broaden the applied range of GMMs in the field of precision and ultra precision positioning.

At present, the modeling method of hysteresis nonlinearity for GMMs is mainly divided into two categories at home and abroad, one category is a physical model from the material internal mechanism and its representative is the Jiles–Atherton model, the other is a mathematical model from a pure mathematics method and its representative is the Preisach model.⁵

The Preisach model is complex and has more parameters that need to be identified, while the Jiles–Atherton model has been widely used because of it reflecting the characteristics of a magnetic field.⁶ Li Xinxin developed a control system of hysteresis compensation based on the Jiles–Atherton model, which effectively eliminated the hysteresis effect and improved the positioning precision, but its anti-jamming capability is poor.⁷ Li Guokang used a model of an adaptive PID controller, which can effectively improve the dynamic and steady-state performances of a GMA under the circumstances of less interference and only the parameter changes slowly.⁸ Tuning the Jiles–Atherton model and quadratic domain magnetostriction model, Dapino established a hysteresis model description between the input current and output displacement.⁹ Meng Aihua put forward a cerebellar model arithmetic computer (CMAC) inverse neural network feed forward compensation and fuzzy PID control strategy, which effectively eliminated the influence of hysteresis nonlinearity. But the control process takes up a lot of controller memory and reduces the execution efficiency.¹⁰

Therefore, in this paper, a hysteresis nonlinear model of a PPS-GMA was established based on the Jiles–Atherton model, and a learning controller was built by using a DRNN feed forward-fuzzy PID feedback control strategy, which improved the dynamic performance of the system and increased the positioning accuracy of the PPS-GMA.

The structure and working principle of the PPS-GMA

The structure of the PPS-GMA

The PPS-GMA has three parts: the GMA, a flexible hinge mechanism and the workbench. The GMA included the base, a preload bolt, a cooling device, a coil, a GMM rod, an output shaft, a preloading disc spring, an end cover and an outer wall socket, and its structure is shown in Fig. 1(a). The structure of the flexible hinge and the workbench is shown in Fig. 1(b).

Fig. 1 Schematic diagram of the 2D precision positioning stage (a) schematic diagram of the GMA (b) schematic diagram of the flexure hinge and workbench.

The working principle of the PPS-GMA

When the drive coil is passed into a current, a magnetic field is generated inside the GMA. Under the action of the magnetic field, the GMM rod produces a certain output force, which loads to the flexible hinge mechanism to make the positioning stage deformation. The different sizes of control current have different sizes of output displacement, so as to realize the positioning function. According to the working principle of the PPS-GMA, it can be divided into the GMA and flexible hinge two parts, and the GMA includes the coil and GMM rod. The work flow of the positioning stage is shown in Fig. 2.

Fig. 2 Work flow of the positioning stage.

Mathematical model of the PPS-GMA

According to the working principle of the PPS-GMA, it can be divided into four parts, respectively the model of the magnetic field, the hysteresis model, the magnetostrictive model and the model of the flexible hinge.

The model of the magnetic field

The magnetic field of the GMA is generated by the double coil, which is equal to the sum of the magnetic field of the inner and outer coils.¹¹

H(t) = H_q + H_p = f_qI(t) + f_pI_p (1)

where: H(t) refers to the magnetic field of the GMA. H_q refers to the driving magnetic field by the inner coil. H_p refers to the bias magnetic field by the outer coil. f_q refers to the coefficient of the magnetic field for the inner coil. f_q refers to the coefficient of the magnetic field for the outer coil. I(t) refers to the current for the drive coil. I_p refers to the current for the bias coil.

Hysteresis model and its parameters identification

Jiles–Atherton is a hysteresis model built by two physicists Jiles and Atherton based on the theory of the domain wall of ferromagnetic materials. In this model, there is a need to determine the relationship between the additional magnetic field H and the magnetization M from five aspects.^12,13where: H_e refers to the effective magnetic field of magnetic materials. M_an refers to no hysteresis magnetization. M_irr refers to a non reversible magnetization. M_rev refers to a reversible magnetization. M refers to the total magnetization. H refers to an external magnetic field. H_σ refers to the magnetic field produced by σ₀. α′ = α + 9λ_sσ₀/(2u₀M_s²). When dH/dt > 0, δ = 1; while when dH/dt < 0, δ = −1.

So the relationship between magnetization M and drive magnetic field H is as follows:

where: c refers to the reversible weight coefficient. α refers to the interaction coefficient of the domain wall. a refers to the shape factor of no hysteresis magnetization. k refers to the irreversible loss coefficient. M_s refers to the saturation magnetization. The five parameters can obtain through the experiment curve λ–H using parameters identification.¹⁴

In this paper, an experimental system of the GMA is built and is shown in Fig. 3. It mainly includes a homemade GMA, a programmable power supply with three channels for IT6332B, a dual-frequency laser interferometer for XL-80 and other devices of auxiliary clamping. The core part of the GMA is a TbDyFe alloy which is Φ 10 × 100 mm in size. The function of the XL-80 is a real-time measure of the output displacement of the GMA and the accuracy level of the measurement is nanometer. The temperature sensor of the PT100 real-time displays the temperature of the GMM rod through the temperature display.

Fig. 3 The experimental system of the GMA.

The experimental data was measured in the experimental system as sample data, and the parameters of this hysteresis model are identified using the standard particle swarm optimization (PSO) algorithm, and the detailed operation steps are as follows:

(1) (Initialization) the population size (N) is determined and N individuals are randomly generated within the range of the variable values. The acceleration parameters (c₁, c₂, c₃) and the inertia weight (w) of the PSO are set.

(2) (Calculation) the positions and speeds of the particles are initialized and the fitness values of the particles are calculated looking for the extremums of individuals and groups.

(3) (Updating) the positions and speeds of the particles are updated and the fitness values of the particles are also calculated. The extremums of individuals and groups are also updated.

(4) (Termination conditions) the steps (2)–(3) are repeated until the iteration number (MAXGEN) reaches the maximum.

(5) (End algorithm) the optimal solution (pg) is acquired, namely the value of the bulletin board.

The flow chart of this algorithm is shown in Fig. 4.

Fig. 4 The flow chart of PSO algorithm.

The population size N is set to 40, for the acceleration parameters of the PSO c₁ is set to 2, c₂ is set to 1 and c₃ is set to 1, the inertia weight w is set to 0.2, the largest iterations number MAXGEN is set to 30.

We respectively used the artificial fish algorithm (AF), the genetic algorithm (GA), and the standard particle swarm optimization (PSO) to identify the parameters of the model to enable an intuitive comparison using the measured experimental data as the sample data. The evolution curve between the fitness function value and the iteration number is shown in Fig. 5 and the final optimal parameters are shown in Table 1, wherein E_min signifies the minimum of the fitness function values.

Fig. 5 Evolution curves of the parameters identification.

Table 1 Results of the parameters identification

Parameters	AF	GA	PSO
Ms	2.70 × 10^5	2.92 × 10^5	6.5 × 10^5
α	0	0.0001	0.0104
k	2197.2	1918.1	2748.1
a	8904.8	9923.0	13 723
c	0.0500	0.0246	0.0340
Emin	1.5220	0.9067	0.6834

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The characteristics of the three parameters identification algorithms can be intuitively shown in Fig. 5. The PSO algorithm has a fast convergence rate and a high precision of parameters identification, the minimum fitness function value (E_min) of PSO is 0.6834 from Table 1 when the number of iteration is 30, which shows the best optimal in the three kinds of optimization algorithms.

Magnetostrictive model

According to the secondary domain model, the relationship between magnetostrictive strain λ and magnetization M for the isotropic materials can be show as:¹⁵where: M_s refers to saturation magnetization and λ_s refers to the saturation magnetostriction coefficient.

According to the knowledge of material mechanics, the magnetostrictive force F shows as:

F = E^HA_rλ (5)

where: E^H refers to the elasticity modulus of the GMM rod and A_r refers to the cross-sectional area of the GMM rod.

Establishment and analysis of the flexible hinge model

The schematic diagram of the flexure hinge is shown in Fig. 6 and it is cut by a two symmetry cylinder perpendicular to end plane.

Fig. 6 Schematic diagram of the flexure hinge.

Where: b refers to the width of the flexible hinge. t refers to the minimum thickness of the flexible hinge. R refers to the cutting radius. h refers to the height of the flexible hinge. θ_r refers to the central angle. M_f refers to the torque of the hinge. θ refers to the angular distortion of the hinge.

When t ≪ h, the simplified calculation formula of the rotational stiffness of the flexible hinge offered by Paros is shown as:¹⁶

where: E refers to the elasticity modulus of the flexible hinge material.

According to the principle of mechanism movement of the flexible hinge, when the force F is acting on the positioning stage, the simplified schematic diagram of flexible hinge deformation is shown in Fig. 7.

Fig. 7 Simplified schematic diagram of flexible hinge deformation.

According to the knowledge of mechanical principle and materials mechanics, the results are shown:

So the displacement X is shown as follows:

According to Fig. 1(b), the flexible hinge positioning stage is completely symmetric in structure. In order to verify this theory, the structure of the flexible hinge positioning stage is analyzed using the finite element method in an ANSYS Workbench. The displacement nephogram of the work platform is shown in Fig. 8 when applied force (F = 1300 N) in the X and Z directions and four screw holes are completely constraint, wherein the displacement nephogram of the work platform in the X direction is shown in Fig. 8(a), the displacement nephogram of the work platform in the Y direction is shown in Fig. 8(b) and the total displacement nephogram of the work platform is shown in Fig. 8(c).

Fig. 8 Displacement nephograms of the work platform (a) in the X direction (b) in the Y direction (c) total displacement.

According to Fig. 8, the displacement of the work platform in the X direction is almost equal to the displacement of the work platform in the Y direction, which shows the flexible hinge model is common in both the X and Y directions.

The controller of the DRNN feed forward and fuzzy PID feedback

The principle of the controller of the DRNN feed-forward and fuzzy

PID feedback. DRNN is one kind of neural network having a fixed weight, external input and internal state. And it can be seen as about the behavior of internal state dynamics based on weight and external input parameters. According to the hysteresis nonlinearity characteristic of the GMA, a three-layer DRNN is constructed, with its structure having W input nodes, N hidden layer nodes and one output node as shown in Fig. 9.

Fig. 9 The structure of the DRNN.

The output of the input layer nodes of the DRNN are shown:

where: O⁽¹⁾_M(k) refers to the threshold of the input layer and I⁽¹⁾_i(k) = [r(k), y(k − 1), …, y(k − n), I(k − 1), …, I(k − W)]^T.

The input and output of the hidden layer of the DRNN are shown respectively:

where: ω⁽²⁾_ij refers to the weight coefficient connecting to the input layer nodes and hidden layer nodes. ω^D_j refers to the feedback weight coefficient from the hidden layer nodes. f[·] refers to the activation function of the hidden layer.

The input and output of the output layer of the DRNN are shown respectively:

where: ω⁽³⁾_jg refers to the weight coefficient connecting to the hidden layer and output layer nodes and f[·] refers to the activation function of the output layer.

The principle of the DRNN feed forward and fuzzy PID controller of the PPS-GMA is shown in Fig. 10.

Fig. 10 Principle of the DRNN feed forward and fuzzy PID controller.

The DRNN controller realizes the feed forward control and learns the approximate inverse dynamic model of the PPG-GMA. The fuzzy PID controller realizes the closed loop feedback control and improves the performance of the output displacement of this system.

Where: r(k) refers to the input reference signal. y(k) refers to the output displacement of the PPS-GMA. I(k) refers to the total control variable of the PPS-GMA and is equal to the sum of the feedback control signal I_b(k) and the feed forward control signal I_f(k). That is:

I(k) = I_b(k) + I_f(k) (13)

Simulation and analysis of the models. According to the above models, the simulation model of the DRNN feed forward inverse compensation and fuzzy PID control of the PPS-GMA is established in the MATLAB/SIMULINK platform and the simulation block diagram is shown in Fig. 11. The simulation block diagram of the PPS-GMA is shown in Fig. 12. And the simulation block diagram of the Jiles–Atherton model is shown in Fig. 13. Because the iterative relations complex, the inverse model of the PPS-GMA in Fig. 11 and the irreversible magnetization M_irr in Fig. 13 are established using the S-function module of SIMULINK.

Fig. 11 Simulation block diagram of DRNN feed forward inverse compensation and fuzzy PID control.

Fig. 12 Simulation block diagram of the PPS-GMA model.

Fig. 13 Simulation block diagram of the Jiles–Atherton model.

The current signal is passed into the drive coil of the GMA and linear changed in accordance with the 0 A–2 A–0 A. According to Fig. 10 and 11, the conclusion of the simulation model are shown as: the relationship curve between the output magnetization intensity M of the Jiles–Atherton model and the input current I as shown in Fig. 14 and the relationship curve between output displacement y of the PPS-GMA and input current I as shown in Fig. 15.

Fig. 14 The curve between magnetization M and current I.

Fig. 15 The curve between displacement y and current I.

The hysteresis model based on the Jiles–Atherton model can effectively describe the hysteresis nonlinearity characteristic of the PPS-GMA from Fig. 14. The PPS-GMA has a hysteresis error and it is necessary to design an error compensator to compensate for the error from Fig. 15.

In order to eliminate the hysteresis nonlinearity and reduce the hysteresis error, an inverse model of the PPS-GMA is online training by a way of a combination of a DRNN structure and a learning algorithm of error back propagation fixed network weights. The contrast curve between the output signal of the inverse model and input current signal of the inverse model is shown in Fig. 16. According to the model of the DRNN feed forward and fuzzy PID feedback control shown in Fig. 10, the contrast curve between output displacement and input reference signal is shown in Fig. 17.

Fig. 16 The contrast curve between the output signal of the inverse model and input current signal.

Fig. 17 The contrast curve between output displacement y and reference input signal.

In order to test the stability of the system, a simulation block diagram of the DRNN feed forward inverse compensation and fuzzy PID control with input perturbation signal is established and shown in Fig. 18, wherein the source is a step signal. Through the numerical simulation analysis, the step response curve of this system is acquired and is shown in Fig. 19.

Fig. 18 Simulation block diagram of the DRNN feed forward inverse compensation and fuzzy PID control with input perturbation signal.

Fig. 19 Step response curve of this system.

According to Fig. 16, the output signal of the inverse model is consistent with the input current signal in the changing trend aspect, which verifies that the inverse model of the PPS-GMA trained by a combination of the DRNN structure and a learned algorithm of error back propagation fixed network weights is effective.

According to Fig. 17, the output displacement of the PPS-GMA follows the input reference signal changing and its tracking error is small, which shows that the inverse model can effectively eliminate the nonlinear characteristic and improve the control precision of the output displacement of the PPS-GMA.

According to Fig. 19, the step response curve of this system is stabilized quickly, which denotes this system has good dynamics and its stability is satisfied.

Experimental

In order to validate the theory of the DRNN feed forward-fuzzy PID control strategy, an experimental platform is built and shown in Fig. 20.

Fig. 20 The experimental platform of the PPS-GMA.

Through experimental testing, the contrast curve between the actual output displacement and ideal output displacement is shown in Fig. 21 and the error curve between the actual output displacement and ideal output displacement is shown in Fig. 22.

Fig. 21 The contrast curve between the actual output displacement and ideal output displacement.

Fig. 22 The error curve between the actual output displacement and ideal output displacement.

According to the Fig. 21, the changing trend of the actual output displacement of the PPS-GMA is consistent with the ideal output displacement and according to Fig. 22, the maximum error between the actual output displacement and the ideal output displacement is equal to 17 μm, namely 4.25%, which shows that it is effective to eliminate the nonlinear characteristic and improve the control precision of the output displacement of the PPS-GMA by using the DRNN feed forward-fuzzy PID control strategy.

Conclusion

In this paper, a mathematical model of a PPS-GMA was built based on the classic Jiles–Atherton model and an inverse compensation model of the PPS-GMA was obtained using online learning by a control strategy of a dynamic recurrent neural network (DRNN) feed forward and fuzzy PID feedback. The research results show that the Jiles–Atherton hysteresis model can describe the hysteresis nonlinear characteristic of the PPS-GMA, that the PSO algorithm has high precision for parameters identification, and that the DRNN feed forward and fuzzy PID feedback control strategy can eliminate the nonlinear characteristics of the PPS-GMA, which has practical significance for improving the positioning accuracy of the PPS-GMA.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Project No. 51075001, 51575002) and the Chinese Key Technologies Program of Anhui Province (No. 1301022074).

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