Hysteresis modeling with frequency-separation-based Gaussian process and its application to sinusoidal scanning for fast imaging of atomic force microscope

Hysteresis modeling with frequency-separation-based Gaussian process and its application to sinusoidal scanning for fast imaging of atomic force microscope

Sensors and Actuators A 311 (2020) 112070 Contents lists available at ScienceDirect Sensors and Actuators A: Physical journal homepage: www.elsevier...

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Sensors and Actuators A 311 (2020) 112070

Contents lists available at ScienceDirect

Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna

Hysteresis modeling with frequency-separation-based Gaussian process and its application to sinusoidal scanning for fast imaging of atomic force microscope Yi-Dan Tao a,b , Han-Xiong Li b , Li-Min Zhu a,c,∗ a

State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, China Department of Systems Engineering and Engineering Management, City University of Hong Kong, Hong Kong, China c Shanghai Key Lab of Advanced Manufacturing Environment, Shanghai 200240, People’s Republic of China b

a r t i c l e

i n f o

Article history: Received 13 January 2020 Received in revised form 29 April 2020 Accepted 12 May 2020 Available online 27 May 2020 Keywords: Atomic force microscope Hysteresis Sinusoidal scanning Gaussian process Tracking control

a b s t r a c t Rate-dependent hysteresis of the piezoelectric tube scanner (PTS) used in the atomic force microscope (AFM) deteriorates the tracking performance of the PTS and thus causes image distortion of the AFM, especially in high-speed operations. Additionally, the traditional raster pattern scanning technique also limits fast imaging of the AFM. In this work, the frequency-separation-based Gaussian Process (FSGP) is proposed to model the hysteresis of the PTS for sinusoidal scanning. In order to properly describe the rate-dependency of the hysteresis, the training dataset of the model is obtained by exciting the PTS using a sinusoidal chirp signal. So, it contains a large number of datapoints which brings a heavy computational burden. Different from the conventional Gaussian Process (GP) which utilizes the whole training dataset at test-stage, the FSGP separates the training dataset according to the target frequency of the testing reference. Only the optimal subset of the training dataset is selected for making predictions. By this way, the computational efficiency as well as the model accuracy are improved significantly. Without the inversion calculation, an inverse hysteresis compensator (IHC) is directly constructed by using the FSGP. Based on the IHC, open-loop and closed-loop controllers are designed and tested. Experiments are carried out on a commercial AFM. The tracking and imaging results demonstrate the effectiveness and superiority of the FSGP-based modeling and compensation method. © 2020 Elsevier B.V. All rights reserved.

1. Introduction Atomic force microscope (AFM) is a versatile investigative tool in nano-measurement and nano-manipulation [1,2]. By moving the cantilever probe and measuring the interaction force between the probe and the sample, the AFM can show the three-dimensional topography of the sample surface with a high spatial resolution down to nano-scale [3]. The piezoelectric tube scanner (PTS) is adopted as the nanopositioner in commercial AFM, due to its high performance in high-resolution, fast response time and low cost [4]. In the past few years, special attention has been paid to improving the accuracy and the scanning speed of AFM, driven by the ever-growing demand of atomic-scale characterizations in nanotechnology and biological researches. However, the PTS has its limitations in high speed operations due to the lightly-damped resonant modes, and inherent nonlinearities especially the ratedependent hysteresis [5,6]. These features may deteriorate the

∗ Corresponding author at: State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, China. https://doi.org/10.1016/j.sna.2020.112070 0924-4247/© 2020 Elsevier B.V. All rights reserved.

positioning accuracy of the PTS and thus lead to image distortion of the AFM. One cause for the limitations of PTS is the lightly-damped resonant modes. The most common scanning patterns used in the AFM is the raster scanning pattern, which is constructed from a triangular signal tracked by the lateral fast axis of the PTS and a staircase or a ramp signal tracked by the lateral slow axis. But unfortunately, the high-frequency components of the triangular signal would excite the lightly-damped resonant modes and thus lead to the vibration of the PTS. Consequently, in practice, the highest scanning rate of AFM under raster scanning is limited to about 1%∼10 % of the PTS’s resonance frequency [7]. In order to overcome this limitation and to further improve the scanning rate, the sinusoidal scanning pattern is adopted for AFM imaging in this work, which is obtained by replacing the triangular signal used in the raster scanning pattern by a sinusoid. Therefore, the scanning rate can be improved without inducting vibration of the lateral fast axis. Another major cause is the rate-dependent hysteresis. With the increase of the input voltage frequency, the hysteresis loop becomes larger and rounder [8]. This would cause large track-

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Y.-D. Tao, H.-X. Li and L.-M. Zhu / Sensors and Actuators A 311 (2020) 112070

ing error of the lateral axes of the PTS, especially in high speed applications. Over the past few decades, many approaches have been suggested to suppress the rate-dependent hysteresis and to improve the tracking performance of the PTS [9]. They can be roughly classified into two categories. One is hardware upgrading, which works well for eliminating the hysteresis. For example, the error caused by hysteresis was reduced to less than 1% of the scan range by using charge drives [10]. However, it is difficult to implement the additional modifications into the current AFM framework. The other is based on the control strategies with feedforward and/or feedback controls [8,11–15]. Among these control strategies, the feedforward control with an inverse hysteresis compensator (IHC) has been demonstrated to be simple but effective to cancel the hysteresis [8]. The key of the IHC is to find a suitable model that can precisely describe the hysteresis. Various models have been proposed for describing the rate-independent hysteresis effect, including Jiles-Atherton model [16], Duhem model [17], Preisach model [18] and Prandtl-Ishlinskii (P-I) model [19]. Recently proposed works show that making some improvements on these conventional rate-independent models can endow the models with the ability to describe the rate-dependent behavior of the hysteresis. In [20], the least-squared  -density function optimization method along with the fast Fourier transform was adopted to endow the classic Preisach model with the ability to form a rate-dependent compensator. Instead of identifying the frequency components and then linearly forming the density function, directly modifying the density function with the changing rate of the input signal also worked well for the Preisach model [21], and this method was later introduced to the P-I model [22]. Additionally, replacing the fixed weights or fixed thresholds of the play operators with dynamic weights [23] or dynamic thresholds [24] in the P-I model were reported to successfully describe the ratedependent hysteresis. Although the aforementioned models have the advantage of interpretability, they may lack expressive power for some intricate applications, such as the cases with complex driven signals in wide frequency range [12,25]. Moreover, most of them bring a large number of parameters to be identified, which complicates the modeling process [26]. Apart from the models above, there is a current interest in intelligent models based on the machine learning methods since they have powerful capability of modeling nonlinear systems. In [27], for instance, a novel Takagi-Sugeno fuzzy-system-based hysteresis model was proposed, and the precise tracking was realized though fuzzy internal model controller. Applying the neural networks to modeling and compensation of rate-dependent hysteresis also showed good results [28]. It should be noted that, for fuzzy system, the key components are the rule base and the membership function, which should be selected properly. For neural networks, the model structure as well as the active function greatly influence the model accuracy. So, both fuzzy-system-based methods and neural-networks-based methods may be blocked in applications due to the lack of a principled framework to determine the key components in the model. As an alternative, kernel-based methods, including support vector machine (SVM) [29] and Gaussian Process (GP) [12], could provide flexible and accurate models which are practical to work with. As a non-parametric Bayesian approach, GP can describe the nonlinear system with no need to determine the structure and parameters of the model. It can not only easily accommodate prior knowledge in the form of covariance functions which make it very flexible at test-stage, but also return the estimates of the predictive confidence. In our previous work [12], GP is introduced for the first time to model and compensate the rate-dependent hysteresis of a piezoelectric actuator. However, GP typically requires computing time in O(N 3 ) with N being the number of data points, which makes it computationally expensive [30]. Furthermore, the model

accuracy would decrease with the increase of the data quantity and data variance. In order to avoid the high computational burden and the accuracy loss caused by large data quantity, this paper proposes a frequency-separation-based Gaussian Process (FSGP) method. According to the target frequency of the reference, the observations (training dataset) are separated. Only the observations relevant to the reference are selected for constructing the covariance function, which improves the computational efficiency. Additionally, due to the fact that only the most relevant information is selected for making the prediction, the modeling accuracy of FSGP is improved compared to the conventional GP. Then, an inverse FSGP hysteresis compensator is constructed, which bases the designs of the open-loop and closed-loop controllers. The controllers are used for enhancing the tracking performance of the PTS, and thus improving the imaging quality of AFM. The paper is outlined as follows. In Section 2, the FSGPbased rate-dependent hysteresis model is presented, and then validated with comparison to the conventional GP. The control approaches, together with the tracking and imaging results of AFM, are described in Section 3. Finally, conclusions are drawn in Section 4. 2. Rate-dependent hysteresis modeling of PTS In the sinusoidal scanning operations of AFM, the fast axis (Xaxis) of the PTS is configured to track a sinusoid, while the slow axis (Y-axis) of the PTS tracks a staircase or a ramp signal. The X-axis of the PTS is affected by the rate-dependent hysteresis more seriously than the Y-axis. Hence, the rate-dependent hysteresis of the X-axis is modeled and compensated in this work. The training dataset for modeling the rate-dependent hysteresis is obtained as D = {X, y} = {xi , yi }N , where X and y denote the i=1 inputs and outputs of the model, respectively, N is the number of date points, yi ∈ R is the displacement of the PTS, and xi = (v, v˙ ) ∈ R2 is the model input which is composed of the voltage value v and its changing rate v˙ . Given this training dataset, the underlying function f (x) between the inputs and the outputs can be assumed to make predictions for new inputs X ∗ that are not in the training dataset. 2.1. GP for rate-dependent hysteresis modeling Formally, a GP is a collection of random variables, in which any finite number of variables follow a joint Gaussian distribution. It can be completely specified by a mean function m(x) and a covariance function k(xp , xq ), m(x) = E[f (x)]

(1)

k(xp , xq ) = E[(f (xp ) − m(xp ))(f (xq ) − m(xq )]

(2)

where x = (vx , v˙ x ) , xp = (vxp , v˙ xp ) and xp = (vxq , v˙ xq ) are the input vectors. In practical uses, the mean function is often initialized as m(x) = 0. The covariance function k(xp , xq ), also referred to as the kernel, describes the similarity between xp and xq . One of the most powerful and popular kernel is the squared exponential (SE) function, expressed as k(xp , xq ) = f2 exp(−

xp − xq 2 2l2

)

(3)

where f2 is the signal variance and l > 0 is the length scale. Generally, these two indexes are called as the hyperparameters of the 2 GP, and they can be denoted as  = ( f , l). Then the GP can be represented as f (x)∼GP(m(x), k(xp , xq ))

(4)

Y.-D. Tao, H.-X. Li and L.-M. Zhu / Sensors and Actuators A 311 (2020) 112070

In realistic modelling situations, noises always exist in the observations. Therefore, the observations can be assumed as y = f (x) + ε , where ε∼N(0, n 2 ) is the noise term with variance n 2 . Therefore, the joint distribution of the observed y and the predictions y ∗ at the testing inputs X ∗ can be written as



y y*





∼N



0,

K (X, X) + n2 I

K (X, X * )

K (X * , X)

K (X * , X * )



(5)

where K (X, X) denotes the covariance matrix which collects the covariances between all inputs in X and has the following expression:



k(x1 , x1 )

⎢ ⎢ k(x2 , x1 ) K (X, X) = ⎢ ⎢ .. ⎣. k(xN , x1 )

k(x1 , x2 )

· · · k(x1 , xN )

k(x2 , x2 )

···

k(x2 , xN )

.. . . . . .. . k(xN , x2 )

···

k(xN , xN )

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(6)

y *  E[y * |X, y, X * ] = K (X * , X) [K (X, X) + n2 I]



−1

cov (y * ) = K (X * , X * ) − K (X * , X) K (X, X) + n2 I

(7)

y

−1

input-output pairs under several sinusoidal inputs. Fig. 1 shows the modeling scheme. For a specific model testing case (the voltage signal is of one single frequency f∗ ), the training dataset is separated and only a subset which contains the data points relevant to the testing inputs is chosen: Df* = {X f* , y f* } = {(xi , yi )|(xi , yi ) ∈ D and i = k −

K (X, X * )

(8)

where y ∗ is the vector of posterior means which serves as the model predictions for the displacement of PTS; cov(y ∗ ) is the posterior covariance matrix, which can be used to describe the uncertainties of the predictions. The hyperparameter  in the covariance function k(xp , xq ) can be optimized by maximizing the logarithmic marginal likelihood, which is defined as

  N   1 1 logp yX,  = − y T Ky−1 y − log K y  − log(2) 2 2 2

(9)

2.2. FSGP method In order to properly train a rate-dependent hysteresis model, the training dataset is often obtained by exciting the PTS using a sinusoidal chirp signal. The range of the frequency is of interest in applications [12,31,32]. Therefore, the training dataset often contains a large number of data points. Given a training dataset with N data points and testing input X ∗ with M data points (usually  M), the model predictions can be obtained by using the close form expression in Eq. (7). However, calculating the inverse of the N × N covariance matrix typically requires the computation time scaling in O(N 3 ) [30], and this computational complexity limits the applications of the GP especially in the cases with large training dataset. Additionally, from Eq. (7), it is found that the prediction for a new input is made by using the information from the whole training dataset. In practice, however, not all data is useful for making the prediction. Rather, some data may induce negative effects on the predictive results. Less but more relevant data seems to be more suitable for making the prediction. Thus, a more flexible and computationally inexpensive model as well as the accompanying method for selecting relevant observations are required for the cases with large training dataset. Here, a FSGP method is proposed which separates the training dataset into subsets and fits the specific testing inputs to the optimal subset to get a submodel for making prediction. Computing the submodel on the specific subset with n data points requires O(n3 ) operations. So, substantial saving in computation would be realized by setting n  N. In this work, the training dataset is obtained by exciting the PTS with a chirp signal whose frequency increases linearly from f0 to fend . The testing dataset is composed of the

n n , ..., k + } 2 2

(10)

where k is the separation index, which can be calculated by roundf −f ing N × f ∗ −f0 ; n is an even number which is chosen as the number end

0

of data points in the optimal subset. Therefore, the predictive equation under testing inputs X ∗ for the FSGP-based submodel can be expressed as p(y ∗ |X f∗ , y f∗ , X ∗ )∼N(y ∗ , cov(y ∗ )), with









 



y *  E[y * |X f* , y f* , X * ] = K X * , X f* [K X f* , X f* + n2 I]



The entries K (X, X ∗ ), K (X ∗ , X) and K (X ∗ , X ∗ ) have similar definitions to that of K (X, X). According to [25], we arrive at the key predictive equation for GP regression of the rate-dependent hysteresis, p(y ∗ |X, y, X ∗ )∼N(y ∗ , cov(y ∗ )), with

3

cov (y * ) = K (X * , X * ) − K X * , X f*



K X f* , X *



K X f* , X f* + n2 I

−1

y f*

(11)

−1 (12)

It should be noted that the hyperparameter  in the covariance function k(xp , xq ) for this submodel is calculated by Eq. (9), which uses the whole training dataset for optimizing the hyperparameter. Therefore, the training processes for the FSGP and the conventional GP are the same. 2.3. Experimental setup The experiments are carried out on a commercial AFM (NTEGRA Prima) designed by NT-MDT company. The PTS of the AFM is utilized to drive the probe to fulfill the scanning and imaging operations under sinusoidal scanning. The scanning range of the PTS along X, Y and Z axes are 100, 100, and 10 m, respectively. The three axes of the PTS are configured with capacitive sensors to capture the real-time displacement with high precision. Due to the low resonance of the Z-axis of the PTS, the AFM is operated at a constant height mode. A contact probe (MSSET/200, NT-MDT) is selected to measure the topography of the sample (an optical grating TGZ1 from NT-MDT, with a period of 3 m). The proposed modeling and control algorithms are programmed in the Simulink/MATLAB environment. A dSPACE-DS1103 system equipped with 16-bit ADCs and 16-bit DACs interfaces is configured to implement the control approaches. A voltage amplifier (produced by Harbin Core Tomorrow Science and Technology Co., Ltd) with a constant amplification factor of 30 is used to provide the actuation voltage for the PTS. A signal access module is provided to deliver the actuation voltage to the PTS and the incorporated controller transform the position signal from the sensors to the dSPACE. Fig. 2 shows the experimental setup. 2.4. Model testing results Since the bandwidth of the X-axis with proportional-integral (PI) control is about 180 Hz, the frequency range of the chirp signal, which is used for exiting the X-axis to obtain the training dataset, is determined as [f 0, fend ] = [0.01, 150]. In this work, the number of the data points in the optimal subset (n) of the FSGP model is determined by making the frequency range of the subset equals to [f∗ − 5, f∗ + 5], and f∗ is the frequency of the reference. To validate the FSGP-based rate-dependent hysteresis model, three indexes are chosen: (1) the normalized root-mean-squared error (enrms ); (2) the relative maximum error (erm ); (3) the runtime t. The run-time of the testing tasks with different methods

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Fig. 1. Modeling scheme of the FSGP. The training dataset is obtained by exciting the PTS with a chirp signal. At test-stage, an optimal subset of the training dataset is selected according to the frequency of the testing signal for making predictions.

Table 1 Model Testing Results. FSGP

GP

f/Hz

10 30 50 120

enrms /%

erm /%

t/s

enrms /%

erm /%

t/s

1.39 0.75 0.62 0.95

2.80 2.07 1.08 1.81

36.1 37.8 36.3 35.2

0.77 0.24 0.17 0.17

1.49 0.52 0.33 0.40

0.63 0.61 0.63 0.65

than those of the GP-based model. For other sinusoidal signals, the predictive errors of the GP-based model are all reduced by 70 % by using the FSGP-based model. Moreover, the run-time of the FSGP model is much shorter than that of the GP model. In conclusion, the FSGP method, which separates the training dataset and uses the corresponding submodel for making predictions, shows better predictive accuracy and faster computation than the GP method. 3. Controller design

are profiled using the same machine (a dual-core CPU clocked at 3.1 GHz and 8 GB memory). The enrms and erm are defined as

  M   2 1 (yi − yˆ i ) M

i=1

enrms =

maxyi − minyi

× 100%

(13)

3.1. Hysteresis compensator

(14)

Owing to the fact that the inversion of hysteresis effect is by nature hysteresis loops, the direct inverse hysteresis compensation concept is widely used in literature to help eliminate the hysteresis effect [33]. The feedforward compensation approach (shown in Fig. 4) can be expressed as

i

i

max|yi − yˆ i | erm =

i

maxyi − minyi i

In this section, an IHC is designed with the inverse FSGP model. Based on the IHC, open-loop and closed-loop controllers are designed and validated to realize the high-precision tracking control.

× 100%

i

where M is the number of the testing data, yi is the true output of the PTS, and yˆ i denotes the model prediction. For comparison, a conventional GP-based rate-dependent model is constructed and identified using the same training dataset. Sinusoidal signals with different frequencies are chosen as the input driven signals to generate the testing dataset. Testing results of the two models are shown in Table 1 and Fig. 3. The model predictive errors of the FSGP-based model, whether enrms or erm , for the 10 Hz sinusoidal signal are almost 50 % less

yout = H[H −1 [yr ]](t)

(15)

where yr and yout are the input and output, respectively. H −1 [ · ] is the inverse hysteresis model, which is construct directly by interchanging the voltage and displacement as the input and output of the FSGP-based model described in Section 2. For comparison, a GP-based inverse hysteresis compensator is established using the same approach.

Y.-D. Tao, H.-X. Li and L.-M. Zhu / Sensors and Actuators A 311 (2020) 112070

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Fig. 2. The experimental setup: (a) the experimental platform; (b) the block diagram. The scan range of the PTS of AFM is 100m × 100m × 10m. Table 2 Tracking results of the IHC.

3.2. Hybrid controller The existence of the modeling error and the perturbations that the control system might experience would cause tracking errors. Therefore, a PI controller is introduced in the feedback loop to compensate for the model imperfection and other disturbances. The block diagram of the hybrid controller is shown in Fig. 5. Fig. 5 Block diagram of the hybrid controller. The control signal is composed of two parts: the output of the IHC vff (t) = H −1 [yr (t)],

t

and the output of the PI controller vfb (t) = Kp e(t) + Ki 0 e()d. e(t) is the tracking error. Kp and Ki are the proportional and integral gains of the PI controller, respectively.

f (Hz) GP-based IHC FSGP-based IHC

Sinusoids with frequencies of 10 Hz, 30 Hz, 50 Hz and 120 Hz are selected as the references. The tracking results of the openloop IHCs are listed in Table 2. It is found that the FSGP-based IHC performs better than the GP-based IHC. The FSGP-based hybrid controller, composed of a FSGP-based IHC and a PI feedback controller, is validated using the same testing signals as in the IHC testing tasks. As comparison, the performance of the pure PI feedback controller and the GP-based hybrid controller are also tested. The proportional gain Kp and the integral gain Ki are optimized as 0.1 and 300 for all the PI controller through

30

50

120

1.68 3.09 1.21 2.38

0.84 1.71 0.26 0.70

0.98 2.21 0.26 0.66

1.39 3.96 0.66 1.59

Table 3 Tracking results of the pure PI controller and the hybrid controllers. f (Hz) Pure PI feedback

3.3. Tracking results

enrms ,% erm ,% enrms ,% erm ,%

10

GP-based hybrid controller FSGP-based hybrid controller

enrms ,% erm ,% enrms ,% erm ,% enrms ,% erm ,%

10

30

50

120

14.33 20.68 0.82 2.28 0.64 1.16

45.57 65.13 0.68 1.69 0.22 0.61

81.63 116.03 0.97 2.34 0.27 0.54

234.23 331.58 1.40 3.34 0.62 1.32

the trial-and-error method. Tracking results of the pure PI feedback controller and the designed hybrid controllers are listed in Table 3. In order to better illustrate the performance of the proposed FSGP-based hybrid controller, time-domain tracking results of the sinusoid references are shown in Fig. 6. Table 3 and Fig. 6 show that as the frequency of the reference increases, the tracking error of the pure PI controller increases

6

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Fig. 3. Model testing results of the GP-based rate-dependent hysteresis model and the FSGP-based rate-dependent hysteresis model. Sinusoidal testing signals at the frequencies of (a) 10 Hz; (b) 30 Hz; (c) 50 Hz; (d) 120 Hz.

Fig. 4. Block diagram of the inverse hysteresis compensator. H[ · ] denotes the hysteresis effect of the PTS. H −1 [ · ] is the inverse hysteresis model.

Fig. 5. Block diagram of the hybrid controller. The control signal is composed of two parts: the output of the IHC vff (t) = H −1 [yr (t)], and the output of the PI controller

vfb (t) = Kp e (t) + K i

t 0

e () d. e (t) is the tracking error. Kp and Ki are the proportional and integral gains of the PI controller, respectively.

Y.-D. Tao, H.-X. Li and L.-M. Zhu / Sensors and Actuators A 311 (2020) 112070

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Fig. 6. Tracking results of the pure PI controller, the GP-based hybrid controller and the FSGP-based hybrid controller: (a) 10 Hz; (b) 30 Hz; (c) 50 Hz; (d) 120 Hz.

rapidly. It means that the rate-dependent hysteresis cannot be suppressed with only PI feedback, especially in high-frequency conditions. The methods which contain the inverse hysteresis compensators display better tracking performance than the pure PI controller, since the enrms s and the erm s of them are all below 1.5 % and 3.5 %, respectively. Comparing the results of the IHCs and those of the hybrid controllers, it is found that with the help of PI controller, the tracking accuracy is improved. So, it is better to use the hybrid controllers to accomplish the tracking tasks. Furthermore, with the FSGP-based hybrid controller, the enrms s of the tracking experiments under sinusoids with four different frequencies are 22 %, 67.6 %, 72.2 % and 55.7 % less than those of the tracking experiments with the GP-based hybrid controller, respectively. Compared with the GP-based hybrid controller, the erm s of the tracking experiments with the FSGP-based hybrid controller are all reduced by at least 50 %. In conclusion, the FSGP-based hybrid controller is an effective method for eliminating the rate-dependent hysteresis of the PTS.

trol approaches are shown in Fig. 7. Due to the limitation of the Z-axis of the AFM used in this work, only 10 Hz and 30 Hz scanning operations are carried out. The scanning time of the 10 Hz scanning operation is 10 s, and that of the 30 Hz scanning operation is 3.3 s. The pure PI control results for sinusoidal scanning are shown in Fig.7 (a) and (d). Due to the poor tracking capability of the pure PI controller, the scanning image is seriously distorted. In addition, the distortion becomes more and more pronounced with the increase of the scanning frequency. This kind of image distortion deviates from the sample’s topography which makes the imaging result of the AFM unreliable. Compared to the images generated by the GP-based method and the FSGP-based method, it is found that this image distortion would be alleviated with the help of introducing the inverse hysteresis compensator. The periodicities of the patterns shown in Fig.7 (b), (c) (e) and (f) are more consistent with the sample’s topography. Furthermore, the imaging results of the FSGP-based method are of better quality than those of the GP-based method, which demonstrates the effectiveness of the proposed FSGP-based method.

3.4. Imaging results A standard strip grating with a period of 3 m is selected as the sample for imaging. To illustrate the performance of the proposed FSGP-based hybrid controller on AFM imaging works, the X-axis of the PTS is configured to track the sinusoid signal with the FSGP-based hybrid controller, while the Y-axis is designed to track a ramp signal with pure PI controller. For comparison, the pure PI controller and the GP-based hybrid controller are also used for compensating the rate-dependent hysteresis effect of the X-axis during AFM imaging. The images with different scanning frequencies (the frequency of the sinusoid signal for the X-axis) under different con-

4. Conclusion GP is flexible enough to describe the rate-dependent hysteresis with no need to determine the structure and parameters of the model. However, when the data quantity of the training dataset is large, GP would face the problems of the heavy computational burden and the accuracy loss. To avoid these problems, this work proposed a FSGP-based hysteresis modeling and compensation method. Instead of using the entire training dataset at test-stage, the FSGP separates the training dataset and then fits the testing

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Fig. 7. Imaging results of a standard strip grating with a period of 3 m under different control schemes. Pure PI method: (a) 10 Hz and (d) 30 Hz; GP-based method: (b) 10 Hz and (e) 30 Hz; FSGP-based method: (c) 10 Hz and (f) 30 Hz.

inputs to the optimal subset in order to form a submodel for making predictions. Compared to the conventional GP, this procedure not only reduces the computational burden, but also improves the model accuracy. An FSGP-based IHC is constructed, by which the open-loop and closed-loop controllers are designed for tracking the desired trajectories with high accuracy. Furthermore, the FSGPbased hybrid controller is also applied to AFM imaging. The images obtained by the FSGP-based hybrid controller are of much better quality than those generated by the pure PI controller or the GP-based hybrid controller. CRediT authorship contribution statement Yi-Dan Tao: Conceptualization, Methodology, Software, Data curation, Writing - original draft, Visualization, Investigation. HanXiong Li: Formal analysis, Validation, Writing - review & editing. Li-Min Zhu: Methodology, Supervision, Writing - review & editing, Funding acquisition. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work was partially supported by the National Natural Science Foundation of China under grant No. 51975375.

References [1] I.A. Mahmood, S.O. Moheimani, Fast spiral-scan atomic force microscopy, Nanotechnology 20 (36) (2009) 365503, http://dx.doi.org/10.1088/09574484/20/36/365503, September 9. [2] H. Xiaodong, et al., Effect of surfactants in aqueous solutions on oil-resisting performance of membrane surfaces with charges by atomic force microscopy, Nanotechnol. Prec. Eng. 1 (1) (2018) 9–15, http://dx.doi.org/10.13494/j.npe. 20170022, 2018/03/01/. [3] Y. Wu, Y. Fang, X. Ren, A high-efficiency Kalman filtering imaging mode for an atomic force microscopy with hysteresis modeling and compensation, Mechatronics 50 (2018) 69–77, http://dx.doi.org/10.1016/j.mechatronics. 2018.01.010. [4] M.S. Rana, H.R. Pota, I.R. Petersen, A survey of methods used to control piezoelectric tube scanners in high-speed AFM imaging, Asian J. Control 20 (4) (2018) 1379–1399, http://dx.doi.org/10.1002/asjc.1728. [5] M.S. Rana, H.R. Pota, I.R. Petersen, Improvement in the imaging performance of atomic force microscopy: a survey, IEEE Trans. Autom. Sci. Eng. 14 (2) (2017) 1265–1285, http://dx.doi.org/10.1109/tase.2016.2538319. [6] Y. Liu, J. Shan, U. Gabbert, Feedback/feedforward control of hysteresis-compensated piezoelectric actuators for high-speed scanning applications, Smart Mater. Struct. 24 (1) (2015), http://dx.doi.org/10.1088/ 0964-1726/24/1/015012. [7] A. Bazaei, Y.K. Yong, S.O.R. Moheimani, Combining spiral scanning and internal model control for sequential AFM imaging at video rate, IEEE/ASME Trans. Mechatron. 22 (1) (2017) 371–380, http://dx.doi.org/10.1109/TMECH. 2016.2574892. [8] M.-J. Yang, C.-X. Li, G.-Y. Gu, L.-M. Zhu, Modeling and compensating the dynamic hysteresis of piezoelectric actuators via a modified rate-dependent Prandtl–Ishlinskii model, Smart Mater. Struct. 24 (12) (2015) 125006, http:// dx.doi.org/10.1088/0964-1726/24/12/125006. [9] Q. Xu, K.K. Tan, Advanced Control of Piezoelectric Micro-/nano-positioning Systems, Springer, 2016. [10] A.J. Fleming, K.K. Leang, Charge drives for scanning probe microscope positioning stages, Ultramicroscopy 108 (12) (2008) 1551–1557, http://dx. doi.org/10.1016/j.ultramic.2008.05.004, 2008/11/01/. [11] X. Mao, Y. Wang, X. Liu, Y. Guo, A hybrid feedforward-feedback hysteresis compensator in piezoelectric actuators based on least-squares support vector

Y.-D. Tao, H.-X. Li and L.-M. Zhu / Sensors and Actuators A 311 (2020) 112070

[12]

[13]

[14]

[15]

[16] [17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25] [26]

[27]

[28]

machine, IEEE Trans. Ind. Electron. 65 (7) (2018) 5704–5711, http://dx.doi. org/10.1109/tie.2017.2777398. Y.-D. Tao, H.-X. Li, L.-M. Zhu, Rate-dependent hysteresis modeling and compensation of piezoelectric actuators using Gaussian process, Sens. Actuators A Phys. 295 (2019) 357–365, http://dx.doi.org/10.1016/j.sna.2019. 05.046, 2019/08/15/. Q. Xu, Precision motion control of piezoelectric nanopositioning stage with chattering-free adaptive sliding mode control, IEEE Trans. Autom. Sci. Eng. 14 (1) (2017) 238–248, http://dx.doi.org/10.1109/tase.2016.2575845. W. Zhu, X.-T. Rui, Hysteresis modeling and displacement control of piezoelectric actuators with the frequency-dependent behavior using a generalized Bouc–Wen model, Precis. Eng. 43 (2016) 299–307, http://dx.doi. org/10.1016/j.precisioneng.2015.08.010, 2016/01/01/. W. Zhu, L.X. Bian, L. Cheng, X.T. Rui, Non-linear compensation and displacement control of the bias-rate-dependent hysteresis of a magnetostrictive actuator, Precis. Eng. 50 (2017) 107–113, http://dx.doi.org/ 10.1016/j.precisioneng.2017.04.018, 2017/10/01/. D.C. Jiles, Theory of ferromagnetic hysteresis, D. L. J. J. o. m. Atherton, and m. Mater. 61 (1-2) (1986) 48–60. C.-J. Lin, P.-T.J.C. Lin, Tracking control of a biaxial piezo-actuated positioning stage using generalized Duhem model, Comput. Math. With Appl. 64 (5) (2012) 766–787. P.-B. Nguyen, S.-B. Choi, B.-K. Song, A new approach to hysteresis modelling for a piezoelectric actuator using Preisach model and recursive method with an application to open-loop position tracking control, Sens. Actuators A Phys. 270 (2018) 136–152, http://dx.doi.org/10.1016/j.sna.2017.12.034. M. Rakotondrabe, Classical Prandtl-Ishlinskii modeling and inverse multiplicative structure to compensate hysteresis in piezoactuators, in: in 2012 American Control Conference (ACC), IEEE, 2012, pp. 1646–1651. X. Shunli, L. Yangmin, Modeling and high dynamic compensating the rate-dependent hysteresis of piezoelectric actuators via a novel modified inverse preisach model, IEEE Trans. Control. Syst. Technol. 21 (5) (2013) 1549–1557, http://dx.doi.org/10.1109/tcst.2012.2206029. R.B. Mrad, H. Hu, A model for voltage-to-displacement dynamics in piezoceramic actuators subject to dynamic-voltage excitations, IEEE/ASME Trans. Mechatron. 7 (4) (2002) 479–489, http://dx.doi.org/10.1109/TMECH. 2002.802724. U.X. Tan, T.L. Win, W.T. Ang, Modeling piezoelectric actuator hysteresis with singularity free prandtl-ishlinskii model, in: In 2006 IEEE International Conference on Robotics and Biomimetics, 17-20 December 2006, 2006, pp. 251–256, http://dx.doi.org/10.1109/ROBIO.2006.340162. Y. Qin, B. Shirinzadeh, Y. Tian, D. Zhang, Design issues in a decoupled XY stage: static and dynamics modeling, hysteresis compensation, and tracking control, Sens. Actuators A Phys. 194 (2013) 95–105, http://dx.doi.org/10. 1016/j.sna.2013.02.003, 2013/05/01/. M. Al Janaideh, P. Krejci, Inverse rate-dependent prandtl–Ishlinskii model for feedforward compensation of hysteresis in a piezomicropositioning actuator, IEEE/ASME Trans. Mechatron. 18 (5) (2013) 1498–1507, http://dx.doi.org/10. 1109/tmech.2012.2205265. C.K. Williams, C.E. Rasmussen, Gaussian Processes for Machine Learning (no. 3), MIT Press, Cambridge, MA, 2006. P.-K. Wong, Q. Xu, C.-M. Vong, H.-C. Wong, Rate-dependent hysteresis modeling and control of a piezostage using online support vector machine and relevance vector machine, IEEE Trans. Ind. Electron. 59 (4) (2012) 1988–2001, http://dx.doi.org/10.1109/tie.2011.2166235. P. Li, P. Li, Y. Sui, Adaptive fuzzy hysteresis internal model tracking control of piezoelectric actuators with nanoscale application, IEEE Trans. Fuzzy Syst. 24 (5) (2016) 1246–1254, http://dx.doi.org/10.1109/tfuzz.2015.2502282. L. Deng, R.J. Seethaler, Y. Chen, P. Yang, Q. Cheng, Modified Elman neural network based neural adaptive inverse control of rate-dependent hysteresis, in: In 2016 International Joint Conference on Neural Networks (IJCNN), 24-29 July 2016, 2016, pp. 2366–2373, http://dx.doi.org/10.1109/IJCNN.2016. 7727493 [Online]. Available: https://ieeexplore.ieee.org/ielx7/7593175/ 7726591/07727493.pdf?tp=&arnumber=7727493&isnumber=7726591.

9

[29] Q. Xu, P.-K. Wong, Hysteresis modeling and compensation of a piezostage using least squares support vector machines, Mechatronics 21 (7) (2011) 1239–1251, http://dx.doi.org/10.1016/j.mechatronics.2011.08.006. [30] R.B. Gramacy, H.K.H. Lee, Bayesian treed gaussian process models with an application to computer modeling, J. Am. Stat. Assoc. 103 (483) (2012) 1119–1130, http://dx.doi.org/10.1198/016214508000000689. [31] Q. Xu, Identification and compensation of piezoelectric hysteresis without modeling hysteresis inverse, IEEE Trans. Ind. Electron. 60 (9) (2013) 3927–3937, http://dx.doi.org/10.1109/TIE.2012.2206339. [32] G. Wang, G. Chen, Identification of piezoelectric hysteresis by a novel Duhem model based neural network, Sens. Actuators A Phys. 264 (2017) 282–288, http://dx.doi.org/10.1016/j.sna.2017.07.058. [33] Z. Sun, et al., Asymmetric hysteresis modeling and compensation approach for nanomanipulation system motion control considering working-range effect, IEEE Trans. Ind. Electron. 64 (7) (2017) 5513–5523, http://dx.doi.org/ 10.1109/tie.2017.2677300.

Biographies

Yi-Dan Tao received the B.E. degree (with honors) in Mechanical Engineering and Automation from Xi’an Jiaotong University in 2015, and she is currently working toward the Doctor’s degree in Shanghai Jiao Tong University and City University of Hong Kong. Her research interests include modeling and controlling of the piezoelectric actuators, tracking control of the AFM in high speed and AFM image reconstruction.

Han-Xiong Li received the B.E. degree in aerospace engineering from the National University of Defense Technology in 1982, the M.E. degree in electrical engineering from Delft University of Technology, The Netherlands, in 1991, and the Ph.D. degree in electrical engineering from the University of Auckland, New Zealand, in 1997. He is currently a Professor with the Department of Systems Engineering and Engineering Management, City University of Hong Kong. He serves as an Associate Editor for the IEEE TRANSACTIONS ON SMC: SYSTEM, and was an Associate Editor for the IEEE TRANSACTIONS ON CYBERNETICS (2002–2016) and the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS (2009–2015) Li-Min Zhu received the B.E. degree and the Ph.D. degree in mechanical engineering from Southeast University in 1994 and 1999, respectively. He is currently the “Cheung Kong” Chair Professor, head of the Department of Mechanical Engineering and vice director of the State Key Laboratory of Mechanical System and Vibration in Shanghai Jiao Tong University. He was the recipient of the National Science Fund for Distinguished Young Scholars in 2013 and selected into the National High-level Personnel of Special Support Program in 2016. He is an Associate Editor for the IEEE Transactions on Automation Science and Engineering and a Technical Editor for the IEEE/ASME Transactions on Mechatronics.