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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 4, APRIL 2012

Rate-Dependent Hysteresis Modeling and Control of a Piezostage Using Online Support Vector Machine and Relevance Vector Machine Pak-Kin Wong, Qingsong Xu, Member, IEEE, Chi-Man Vong, Member, IEEE, and Hang-Cheong Wong

Abstract—Hysteresis nonlinearity degrades the positioning accuracy of a piezostage and requires a suppression for precision micro-/nanopositioning applications. This paper proposes two new approaches to modeling and compensating the rate-dependent hysteresis of a piezostage driven by piezoelectric stack actuators. By formulating the hysteresis modeling as an online nonlinear regression problem, online least squares support vector machine (SVM) (LS-SVM) and online relevance vector machine (RVM) models are proposed to capture the hysteretic behavior. Both hysteresis models are capable of updating continually with subsequent samples. After a comparative study on modeling performances, an inverse model-based feedforward combined with proportional–integral–derivative feedback control is presented to alleviate the hysteresis effect. Experimental results show that the LS-SVM model-based control scheme is over 86% more accurate than the RVM model-based one in the motion tracking task, whereas the latter is 14 times faster than the former in terms of updating time. Moreover, both LS-SVM and RVM model-based control schemes can suppress the rate-dependent hysteresis to a negligible level, which validates the feasibility and effectiveness of the proposed approaches. Index Terms—Hysteresis, least squares support vector machines (SVMs) (LS-SVMs), motion control, piezoelectric actuator, relevance vector machine (RVM).

I. I NTRODUCTION

A

PIEZOSTAGE refers to a micro-/nanopositioning stage actuated by piezoelectric stack actuators (PSAs). In comparison with other types of actuators [1], PSA is capable of positioning with subnanometer resolution, rapid response, and large blocking force. Hence, it is widely adopted in diverse micro-/nanopositioning applications such as scanning probe microscopy [2], [3] and biological manipulation [4], [5]. Nevertheless, PSA introduces nonlinearity into the system due to Manuscript received March 14, 2011; revised June 2, 2011; accepted August 14, 2011. Date of publication August 30, 2011; date of current version November 1, 2011. This work was supported by the Research Committee of the University of Macau under Grants MYRG149(Y1-L2)-FST11-WPK and SRG006-FST11-XQS in part by the Macao Science and Technology Development Fund under Grant 024/2011/A. P.-K. Wong, Q. Xu, and H.-C. Wong are with the Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Macao, China (e-mail: [email protected]; [email protected]; hcwong@ umac.mo). C.-M. Vong is with the Department of Computer and Information Science, Faculty of Science and Technology, University of Macau, Macao, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2011.2166235

the piezoelectric hysteresis and creep effects [6]. The hysteresis is embodied as a nonlinear relationship between the input voltage and output displacement and induces a severe openloop positioning error as high as 10%–15% of the motion range of the positioning system. Therefore, the hysteresis has to be suppressed in high-precision application scenarios [7]–[9]. Although the hysteresis can be greatly alleviated by using a charge-driven approach or a capacitor insertion method [10], it is at the cost of stroke reduction. Thus, voltage actuation is still widely adopted in practice. Generally, the hysteresis is modeled using the Preisach model [11], [12], Prandtl–Ishlinskii model [13], Bouc–Wen model [14], etc. Recently, some novel mathematical models have also been proposed [15]. Then, an inverse hysteresis model is constructed and utilized as an input shaper to cancel the hysteresis effect [11]. However, the hysteresis is dependent not only on the amplitude but also on the frequency of input signals. It is a major challenge to capture the complicated rate-dependent hysteretic behavior precisely. Typically, the existing techniques for the ratedependent hysteresis modeling can be classified into three categories. The first type is based on phenomenological models. Recent implementations include the modified Preisach model [16], modified Prandtl–Ishlinskii model [17], and generalized Prandtl–Ishlinskii model [18]. However, these models have a lot of parameters to be determined, which complicate the hysteresis modeling process and may block their applications in real-time control. The second category contains mathematical models. For instance, an approximate model consisting of a variable gain and a variable time delay is proposed in [19] to describe the rate-dependent hysteresis of a PSA, where the variable gain and time delay are assumed to be functions of the magnitude and frequency of the input command, respectively. A mathematical model based on a family of ellipses, which relate the frequencies and amplitude of the input signal to the major and minor axes and orientation of the ellipse, has been reported in [20] and [21]. Nevertheless, a great number of experiments are required to establish proper relationships between the model parameters and hysteresis characteristics. In contrast, it appears that the third type of approach, which is based on artificial neural networks (ANNs), provides an efficient way to model the nonlinear hysteresis [22]–[24]. However, there is no universal method to determine an optimal ANN structure in terms of the number of hidden layers and number of neurons in each layer. Moreover, ANN has the shortcomings of overfitting and sinking into local op-

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WONG et al.: HYSTERESIS MODELING AND CONTROL OF A PIEZOSTAGE USING ONLINE SVM AND RVM

tima. Alternatively, support vector machines (SVMs) unfold a promising means to estimate nonlinear system models accurately [25]. Based on statistical learning theory and structural risk minimization principle, SVM is capable of modeling nonlinear systems by transforming the regression problem into a convex quadratic programming (QP) problem and then solving it with a QP solver. Compared to ANN, SVM exhibits the major advantages of global optimization and higher generalization capability [26]. Furthermore, least squares SVMs (LS-SVMs) utilizes equality constraints instead of the inequality constraints as in the ordinary SVM. Thus, it simplifies the regression to a problem that can be easily solved from a set of linear equations [27]. The potentials of LS-SVM in modeling the rate-dependent hysteretic behavior have been demonstrated in a few of previous works [28], [29]. However, LS-SVM just represents an offline algorithm which cannot continually update and correct the hysteresis models with subsequent samples. Although there is a pruning-based online version of LS-SVM, namely, decremental LS-SVMs (DLSSVMs) [30], it is commonly believed that this algorithm still suffers from two drawbacks. The first one lies in that there are two important hyperparameters in DLSSVM which substantially affect the function estimation accuracy and generalization. Optimizing these two hyperparameters is time consuming, and the optimality of the hyperparameter values is not guaranteed after the decremental update process. The second drawback is that DLSSVM makes unnecessarily liberal use of the basis functions since the number of support vectors (SVs) in DLSSVM typically grows linearly with respect to the size of the training data set. In order to overcome the deficiencies of ANN and LS-SVM, an advanced function estimation technique, i.e., relevance vector machine (RVM), has been recently proposed in [31] and [32]. RVM is an inherent online machine learning technique which utilizes a more flexible and sparser function without additional regularization parameters. In the current research, both online LS-SVM and RVM are adopted and tailored to model and suppress the rate-dependent hysteretic behavior of a piezostage system, and a comparative study is carried out. In the literature, various strategies have been presented to compensate for the hysteresis effect, e.g., the H∞ control [33], [34], sliding mode control [35]–[39], fuzzy control [40]–[42], etc. In the current research, to demonstrate the effectiveness of the proposed approaches, the nonlinear hysteresis is suppressed by using inverse hysteresis model-based feedforward (FF) control combined with feedback (FB) control. The aim of this paper is to investigate the feasibility of the two new approaches in predicting and mitigating the rate-dependent hysteresis behavior and to perform a comparison between the established models via experimental studies. To the best knowledge of the authors, this research is the first attempt at extending online LS-SVM and RVM to the domain of hysteresis modeling and control. In the rest of this paper, the test bed employed in the research is presented, and the problem is formulated in Section II. Then, the modeling of hysteresis based on the online LS-SVM and RVM models is described in Sections III and IV, respectively. The derived models are validated and compared with a classical online model based on recurrent radial basis

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Fig. 1. Experimental setup of a micro-/nanopositioning piezostage system.

function (RBF) neural network (RBFNN) (RRBFNN) [24] by experimental studies performed in Section V. Afterward, the controller design and verification are carried out in Section VI. The concluding remarks and future works are provided in Section VII. II. S YSTEM D ESCRIPTION AND P ROBLEM F ORMULATION The test bed employed in the current research and its hysteresis properties are described in this section. A. Experimental Setup The experimental setup is depicted in Fig. 1. The piezostage is composed of a flexure parallel mechanism driven by three 30-µm stroke PSAs from Physik Instrumente Company, Ltd. The stage output platform delivers amplified translational motions by making use of elastic deformations of the lever displacement amplifiers, and the output positions are measured by three capacitive sensors from the Physik Instrumente. A dSPACE DS1005 rapid prototyping system equipped with 16-b analog-to-digital (A/D) and digital-to-analog (D/A) boards is employed to implement the controller. The D/A device produces an analogy voltage which is then amplified by a threeaxis voltage amplifier to provide a voltage ranging from −20 to 100 V for driving the PSA. In addition, the sensor output voltage signals are passed through a signal conditioner and then simultaneously acquired by the A/D board. Control algorithms are developed with Matlab/Simulink software and downloaded to the DS1005 board by resorting to the ControlDesk interface to realize the real-time control. B. Hysteresis Characterization By applying a triangular-wave voltage with the constant feed rate (8.0 V/s) and variable amplitudes to PSA #1, the openloop position responses of the piezostage in the three working axes are shown in Fig. 2(a). It is seen that the stage has a spatial workspace around 250 × 250 × 250 µm3 with a maximal crosstalk of 1.24% among the axes, which confirms that the output motion is approximately decoupled. For brevity, only the treatment of the x-axis motion is presented in this paper. The x-axis hysteresis loops in Fig. 2(a) reveal that the shape of the hysteresis loop is dependent on the amplitude of the input. The larger the amplitude of the input signal, the wider the output hysteresis loop. In addition, by applying sine waves with the same amplitude and different frequencies, the x-axis position output is depicted in Fig. 2(b). It is observed that the hysteresis shape also relies on the input rate. The higher the frequency of the input signal, the greater the output hysteresis. It is noticed that the position errors of the piezostage result-

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valued mapping into a one-to-one mapping, an input sequence matrix was formulated in [28] by investigating the hysteresis loop directions. However, it is not feasible, particularly when the input signals are arbitrary and unknown beforehand. The current input and input variation rate were introduced in [22] to generate a single-valued mapping. Nevertheless, in the case that the input data are accompanied with noises, the variation rate is not smooth and will induce modeling error. Moreover, it is still not clear how many orders of the variation rate are sufficient to establish the mapping. In [43], an expanded input space was developed by resorting to a hysteresis operator to construct a one-to-one mapping, and nonlinear moving average model with exogenous inputs and autoregressive moving average model with exogenous inputs were implemented to identify the ratedependent hysteresis. However, it appears a complicated procedure to implement such an expanded input space. In this paper, the multivalued mapping is transformed into a single-valued one by employing the current and previous inputs and previous outputs as exogenous inputs [29], and a nonlinear regression model is formulated to predict the current output yˆk = f (xk )

(1)

xk = [xk , xk−1 , . . . , xk−n , yk−1 , . . . , yk−m ]

(2)

with

where xk−1 and yk−1 denote the input voltage and output position of the system at time instant k − 1 and yˆk is the estimated model output at the current time instant k. In addition, n ≥ 0 and m ≥ 1 define the order of the system. The regression model is established based on LS-SVM and RVM techniques to predict the displacement output in the following sections. III. H YSTERESIS M ODELING W ITH LS-SVM Fig. 2. (a) Hysteresis loops of the three-axis motions with PSA-1 driven by triangular-wave voltage input with constant feed rate. (b) x-axis hysteresis loops obtained by applying sinusoidal voltage inputs with different frequencies.

ing from different frequencies of the driving signal are much smaller than those for a stand-alone PSA. This is caused by high mechanical stiffness of the stage, which reduces the free stroke of the PSA. The aforementioned open-loop test results indicate that the complicated hysteresis effects are dependent not only on the amplitude but also on the frequency of input signals. The modeling and compensation of the hysteretic nonlinearity are carried out in the current research. C. Regression Model of Hysteresis Owing to the hysteresis effects, an input voltage corresponds to multiple position outputs. Typically, LS-SVM and RVM only treat the problem of single-valued mapping between the input and output. Thus, it is nontrivial to extend the LS-SVM and RVM to the hysteresis modeling problem. One of the challenges lies in how to convert the multivalued mapping problem into a single-valued one. Various approaches have been adopted in previous works. For instance, in order to transform the multi-

In this section, the LS-SVM is employed to model the piezoelectric hysteresis for the piezostage system. A. LS-SVM Modeling LS-SVM maps the input data into a high dimensional feature space and constructs a linear regression function therein. The unknown hysteresis function is approximated by y(x) = wT ϕ(x) + b

(3)

with the given training data set {xk , yk }N k=1 , where N represents the number of training data sets, xk ∈ Rn+m+1 is an input vector as shown in (2), and yk ∈ R are the output data. Additionally, w is a weight vector, ϕ(·) denotes a nonlinear mapping from the input space to a higher dimensional feature space, and b is the bias. The LS-SVM approach formulates the regression as an optimization problem in the primal weight space. Then, the conditions for optimality are obtained by solving a series of partial derivatives, which are used to construct the dual formulation, i.e.,

WONG et al.: HYSTERESIS MODELING AND CONTROL OF A PIEZOSTAGE USING ONLINE SVM AND RVM



1TN −1

0 1N

Ω+γ

 IN



  b 0 = α y

(4)

where α = [α1 , α2 , . . . , αN ]T is called the SV. The support values are αk = γek with γ ∈ R denoting the regularization factor. In addition, 1N = [1, 1, . . . , 1]T , y = [y1 , y2 , . . . , yN ]T , and IN is an identity matrix. Moreover, the kernel trick is employed to derive that Ωkj = ϕ(xk )T ϕ(xj ) = K(xk , xj ),

k, j = 1, 2, . . . , N (5)

where K is a predefined kernel function. The purpose of introducing the kernel function is to avoid the explicit computation of the map ϕ(·) in dealing with the high-dimensional feature space. It is observed that the LS-SVM approach utilizes the equality constraints instead of the inequality constraints as in the ordinary SVM. Thus, it simplifies the regression to a problem that can be easily solved from a set of linear equations. After calculating b and α from (4), one can obtain the solution for the regression problem y(x) =

N 

αk K(x, xk ) + b

(6)

k=1

where K is the kernel function satisfying Mercer’s condition, xk is the training data, and x denotes the new input data. By adopting the RBF as a kernel function   x − xk 2 K(x, xk ) = exp − (7) σ2 with σ > 0 denoting the width parameter (which specifies the kernel sample variance σ 2 ) and  ·  representing the Euclidean distance, the LS-SVM model for the hysteresis model estimation becomes   N  x − xk 2 αk exp − + b. (8) y(x) = σ2 k=1

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is used to update the trained LS-SVM whenever a new sample is added to the training set, and a decremental pruning algorithm is adopted to delete old samples based on a sorted support value-based criterion. The incremental updating and decremental pruning algorithms are briefly described hereinafter. First, (4) is rewritten into the form AN αN = YN

(9)

where  AN =

0

1TN

1N

Ω + γ −1 IN



 ,

αN

 b = , α

YN

  0 = . y (10)

Whenever a new sample (xN +1 , yN +1 ) is added to the training set, the model can be updated by an incremental relation using the (N + 1) pairs of data AN +1 αN +1 = YN +1

(11)

where the notations of AN +1 , αN +1 , and YN +1 can be found in [30]. The online incremental training algorithm aims to update A−1 N +1 efficiently without the explicit computation of the matrix inverse. The aforementioned online incremental algorithm has an infinite memory length. Hence, the sparseness property in LS-SVM is lost due to the condition αk = γek for the optimality. To solve this issue, a decremental algorithm based on a pruning procedure is adopted to remove the most unimportant support value when a pair of its corresponding training data is −1 removed [30]. Specifically, A−1 N is updated from AN +1 without explicitly computing the matrix inverse. Then, the coefficients of the LS-SVM are updated using (9). With the said incremental and decremental algorithms, the LS-SVM is capable of online updating. Moreover, a sparse LS-SVM is obtained by the gradual decremental pruning algorithm. Instead of using the fixed-size LS-SVM [30], the pruning procedure is carried out by omitting the training samples that have smaller absolute support values [45]. Specifically, in each pruning step, the training data points whose absolute support values are smaller than a threshold are removed [46].

With the assigned regularization parameter γ and kernel parameter σ, the purpose of the training process is to determine the support values αk and the bias b. The high generalization ability of the LS-SVM model relies on the appropriate tuning of the two hyperparameters (γ and σ). In the current research, the leave-one-out cross-validation approach is adopted to infer the values of the hyperparameters.

In this section, the RVM technique [31] is employed to establish a regression model of the piezostage system to predict the position output. The hysteresis modeling procedure is presented as follows.

B. Online LS-SVM

A. RVM Modeling

The classical LS-SVM is an offline algorithm, i.e., the established hysteresis model cannot be updated continually with the subsequent samples for model correction unless retrained from scratch. However, this process is time consuming and unsuitable for the real-time control purpose. To solve this issue, an online LS-SVM technique, i.e., DLSSVM [30], [44], is employed. In the online LS-SVM, an incremental algorithm

Given a training data set D of N input vectors xk , k = 1 to N , along with N corresponding scalar-valued output yk , the input vector xk ∈ Rn+m+1 in (2) contains the previous measured hysteresis time series at a specific time instant. The corresponding target output value at that time is defined as the output yk ∈ R which is assumed to contain zero-mean Gaussian noise with variance σ 2 . Hence, the probability of prediction

IV. RVM-BASED H YSTERESIS M ODELING

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error εk for yk is a Gaussian distribution of zero mean and variance σ 2 , i.e., p(εk |σ 2 ) = N (0, σ 2 ), with yk = g(xk , ω) + εk p(yk |xk , ω, σ 2 ) = N (ˆ yk , σ 2 )

(12)

N 

ωk K(x, xk ) = ωφ(x)

(14)

k=1

where K(x, xk ) is a basis function and φ(x) = [1, K(x, x1 ), . . . , K(x, xN )]T . In the current research, RBF is chosen as the basis function K. Under the aforementioned formulation, the likelihood function of the output vector y = yk , k = 1 to N , is constructed by a multivariate Gaussian   ˆ 2 y − y 2 −N/2 −N p(y|ω, σ ) = (2π) σ exp − (15) 2σ 2 ˆ = Φω and Φ represents an N × (N + 1) design mawhere y trix with the notation Φ = [φ(x1 ), . . . , φ(xN )]T , and φ(xk ) = [1, K(xk , x1 ), . . . , K(xk , xN )]T , k = 1 to N . The likelihood function (15) is complemented by a prior over the weights, ω = {ωp }, p = 0 to N , to control the complexity of the predicted hysteresis model function and avoid overfitting. The prior is a zero-mean Gaussian distribution and is defined over every weight ωk as follows:

N  αp ωp2 −N/2 1/2 . (16) p(ω|α) = (2π) αp exp − 2 p=0 T

The hyperparameter vector α = [α0 , . . . , αN ]T controls how far each weight ωp is allowed to deviate from zero. Given α, using Bayes’ rule, the posterior over ω is expressed as p(ω|y, α, σ 2 ) =

p(y|ω, σ 2 )p(ω|α) = N (Ω|µ, Σ). p(y|α, σ 2 )

(17)

The posterior mean µ and covariance Σ for ω are given as follows [32]: Σ = (A + σ −2 ΦT Φ) µ =ω = σ

−2

T

ΣΦ y

−1

L(α) = log p(y|α, σ ) = log

(18a) (18b)

where A is defined as diag(α0 , . . . , αN ). The posterior mean µ is an estimation of the weight vector ω for prediction. The only unknowns in (18) are the vector α which can be estimated via a type-II maximum likelihood procedure [32]. It is called sparse Bayesian learning which is formulated as the local maximization with respect to α of the marginal likelihood or, equivalently, its logarithm L(α)

p(y|ω, σ 2 )p(ω|α) dω

−∞

(13)

where yˆk = g(xk , ω) is the output prediction of the true value yk and ω = [ω0 , . . . , ωN ] is the weight vector for the RVM model. Generally, the prediction can be represented by yˆ = g(x, ω) = ω0 +

∞ 2

=−

1 N log 2π + log |C| + yT C−1 y 2

(19)

where C = σ 2 I + ΦAΦT . By maximizing (19) over α, the most probable values αM P can be generated. Then, A can be obtained by substituting αM P into (18). Afterward, the covariance matrix Σ and posterior mean µ = [µ0 , µ1 , . . . , µN ]T can be estimated. Therefore, the hysteresis model function g can be found by setting ω = µ yˆ = g(x, Ω) = g(x, µ) = µ0 +

N 

µk K(x, xk )

k=1

= µ0 +

N  k=1

  x − xk  µk exp − σ2

(20)

where yˆ is the prediction of the position output y with the unseen input data x containing the recent measured data series. One crucial observation is that the optimal values of many hyperparameters αp are typically infinite. Considering that ωp = µp ∝ (1/αp ), this leads to a parameter posterior infinitely peaked at zero for many weights ωp . In consequence, the posterior mean µ consists of very few nonzero elements.

B. Online RVM Modeling In order to train and update the hysteresis model function continually, an online algorithm for RVM is employed and briefly described hereinafter. More details about the online RVM algorithm can be found in the literature [32]. The algorithm starts with an empty set of basis function for the hysteresis model and sequentially adds basis functions φk = [1, K(xk , x1 ), . . . , K(xk , xN )]T to increase the marginal likelihood and modify their corresponding weights ωk . Within the same principal framework, the likelihood can also be updated by deleting those basis functions which subsequently become redundant. Removing a basis function φk implies that the corresponding xk is no longer important and can be excluded from the design matrix Φ in constructing the hysteresis model g, which can be simply done by setting the corresponding hyperparameter αp equal to infinity (practically a very large value). In this way, the corresponding weight value ωp = µp ∝ (1/αp ) becomes zero. Since ω0 is a bias corresponding to no training data, only the weights ω1 , . . . , ωN are considered. The data xk with nonzero weights ωk , k = 1 to N , are referred to as relevant vectors (RVs). Since the basis functions are sequentially added to or deleted from the predicted hysteresis model by RVM, the likelihood can be continually updated. Hence, this mechanism makes online update of the hysteresis model feasible.

WONG et al.: HYSTERESIS MODELING AND CONTROL OF A PIEZOSTAGE USING ONLINE SVM AND RVM

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Fig. 3. Hysteresis modeling results with the first data set. (a) Whole data sets. (b) Training results of RRBFNN, LS-SVM, and RVM models. (c) Testing results of RRBFNN, LS-SVM, and online LS-SVM models. (d) Testing results of RRBFNN, RVM, and online RVM models.

V. E XPERIMENTAL S TUDIES ON H YSTERESIS M ODELING In this section, the hysteresis model for a piezostage is established using the aforementioned online LS-SVM and online RVM models via experimental studies. The purpose of hysteresis modeling is to get an understanding and prediction of the hysteresis behavior of the system. First, the effects of the order (n and m) selection on the modeling results are investigated. It is found that the larger the model order, the lower the training error. However, the testing error is not monotonic with respect to the order selection, which is coincident with the result reported in [29]. In order to make a compromise between the training and testing errors, n = 3 and m = 3 are selected for an illustration in the current research.

A. Data Sampling To evaluate the performance of the established LS-SVM and RVM models, two experimental time series are collected using different voltage input signals

 ur (t) =



5e−0.1t sin(6πte−0.346t − 1.5) + 1.0 , r = 1 5e−0.13t cos(3πte−0.09t − 3.15) + 1.0 , r = 2. (21)

The two data sets acquired with a sampling time of 0.01 s are depicted in Figs. 3(a) and 4(a), respectively. For the rth (r = 1, 2) experimental data pairs (N = 1000), the first 200 ones are used as training data set Br to build the LS-SVM and RVM hysteresis models, and the remaining 800 ones are adopted as the testing data set Tr . For the online LS-SVM and online RVM models, the testing data sets Tr are also utilized as the updating data set Ur for real-time updating by every D steps. Both LS-SVM and RVM models are implemented in Matlab environment. Based on the displacement error e = ymodel − y (model represents LS-SVM or RVM) of the model output, the mean absolute error (MAE) and root-mean-square (rms) error (rmse) of each model are defined as follows:

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Fig. 4. Hysteresis modeling results with the second data set. (a) Whole data sets. (b) Training results of RRBFNN, LS-SVM, and RVM models. (c) Testing results of RRBFNN, LS-SVM, and online LS-SVM models. (d) Testing results of RRBFNN, RVM, and online RVM models.

B. RRBFNN Model Training and Testing Nt 1  MAE = |ei |, Nt i=1

  Nt  1  rmse =  e2 Nt i=1 i

(22)

where Nt = 800 is the number of testing data set. In addition, the pruning rate is directly related to the sparseness and updating time of the model, which are important issues for real-time control and update. In the current research, the pruning rate for each online model is defined as follows:  ρmodel =

1−

NV NT

 × 100%

(23)

where the subscript “model” represents online LS-SVM or online RVM, NT is the size of the overall training data set, and NV denotes the nonzero SV of the LS-SVM model or the nonzero RV of the RVM model, respectively. A large pruning rate means a small number of nonzero SV or RV values.

It is well known that RBFNN owns universal approximation capabilities. The structure of RBFNN allows it to be trained with less computation than others, e.g., multilayer perceptron neural networks. For comparison purposes, a recurrent neural network, i.e., RRBFNN [24], is trained to estimate the regression model (1) of the hysteresis. Like the treatment in [24], a spread parameter of 40 is chosen for the RBFs for both of the two training data sets (B1 and B2 ). The training and testing results of the two data sets are depicted in Figs. 3 and 4. It is observed that the rms training errors for B1 and B2 are 0.273 and 0.288 µm (i.e., 0.11% and 0.12% of the motion range), respectively. In contrast, the rms testing errors with T1 and T2 are 0.583 and 0.849 µm (i.e., 0.43% and 0.45% of the motion range), respectively. C. LS-SVM Model Training and Testing Given the training data set Br (r = 1, 2), the hyperparameters of the LS-SVM model are obtained by the leaveone-out cross-validation method (γ = 1.4253 × 108 and σ =

WONG et al.: HYSTERESIS MODELING AND CONTROL OF A PIEZOSTAGE USING ONLINE SVM AND RVM

1.6945 for B1 ; γ = 6.0696 × 105 and σ = 0.6521 for B2 ). They are used to train the LS-SVM to generate the values of αi and b. Once the training process is completed, the LS-SVM model produces the outputs as illustrated in Figs. 3(b) and 4(b) for the two training sets. Concerning the training errors, it is observed that the rmses for the two training sets are 0.014 and 0.015 µm (i.e., 0.01% and 0.01% of the motion range), respectively, which are about 95% lower than the RRBFNN model result. Using the update data set Ur , the online LS-SVM model (DLSSVM) is implemented, which is updated every D = 50 steps. In the pruning algorithm, the training data set with the absolute values of the SVs less than the threshold sr is eliminated in each step. The smaller the threshold, the lower the pruning rate. By trial-and-error method, s1 = 4000 and s2 = 40 are set to obtain better performance. The testing results of the LS-SVM and online LS-SVM models are depicted in Figs. 3(c) and 4(c), respectively. For the data set T1 , the LS-SVM and online LS-SVM produce the rms testing errors of 1.277 and 0.196 µm (i.e., 0.94% and 0.14% of the motion range), respectively. It appears that the online LS-SVM reduces the rmse by 84.65% as compared with the offline LS-SVM model and alleviates the error by 66.38% over the RRBFNN model. Nevertheless, with the data set T2 , the LS-SVM and online LS-SVM produce the maximum rms testing errors of 0.617 and 1.015 µm (i.e., 0.33% and 0.54% of the motion range), respectively. That is, the online LS-SVM model result is 39.21% worse than the offline LS-SVM model output and 16.35% worse than the RRBFNN model result. Even so, it is observed that the offline LS-SVM model result is 27.33% better than that of the RRBFNN model. In order to get a more clear understanding of the online LS-SVM model, the model results are investigated using different updating step sizes D. The mean (MAE) and standard derivation (i.e., rmse here) error bars of the two models are compared in Fig. 5(a), where the upper and lower graphs correspond to the testing data sets T1 and T2 , respectively. For the data set T1 , the MAE increases slowly as the step size rises, whereas the MAE decreases greatly as the step size increases (up to D = 50) in data set T2 . The nonmonotonic variation tendency of MAE with respect to the step size in the two data sets mainly comes from the fact that the generalization ability of LS-SVM is very dependent on the two hyperparameters γ and σ. For both data sets, the hyperparameters are tuned by using the training data set Br , whereas they remain constant in the following updating process over Ur . It implies that different hyperparameters should be inferred for different data sets. This is the major drawback of the LS-SVM-based hysteresis model. The final pruning rate and cumulative updating time versus the updating step size are plotted in Fig. 5(b) and (c), respectively. It is found that the two performances remain almost constant as the variation of the step size. Further trials show that the generalization capability of the online LS-SVM model is influenced by the threshold sr used in the decremental algorithm. The smaller the threshold, the lower the pruning rate

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and the better the generalization ability, whereas the higher the model complexity and the longer the updating time at the same time. The problem of determining a suitable threshold value is not an easy work and is time consuming if the searching method is based on random trials. This issue will be brought to a future study.

D. RVM Model Training and Testing The RVM model is established by the procedure presented in Section IV. Instead of inferring an optimal parameter σ as in the LS-SVM, only an initial value σ of the basis width is required by RVM since σ is updated by the RVM algorithm automatically. A large σ means a stronger smoothing of the kernel function. For both B1 and B2 , σ = 32 is set by the trialand-error approach to yield better results. After the training process with the two data sets Br , the RVM model produces the results as shown in Figs. 3(b) and 4(b). For the two data sets, the rmses are 0.386 and 0.284 µm (i.e., 0.16% and 0.12% of the motion range), respectively. In comparison with the online LS-SVM model, the online RVM has a larger training error which is about 20 times worse than the former. However, the training error tells nothing about the generalization ability. Using the updating data set Ur , the online RVM model is realized, which is also updated every D = 50 steps. The testing results of the RVM and online RVM models are described in Figs. 3(d) and 4(d). Concerning the data set T1 , the RVM and online RVM produce the rms testing errors of 0.636 and 0.288 µm (i.e., 0.47% and 0.21% of the motion range), respectively. It means that the online RVM reduces the displacement error by 54.72% as compared with the offline RVM model and by 50.60% over the RRBFNN model. With the data set T2 , the RVM and online RVM give the rms testing errors of 1.138 and 0.347 µm (i.e., 0.61% and 0.19% of the overall motion range), respectively. That is, the model accuracy of the online RVM model has been improved by 69.51% with respect to the offline RVM model and by 59.13% as compared to the RRBFNN model due to a good generalization ability. Unlike the online LS-SVM model which has different generalization capabilities for different data sets, the online RVM always exhibits an improved generalization ability over the offline RVM model. Further trial experiments show that the training and testing errors are dependent on the selection of the initial basis width value σ. The smaller the basis width, the smaller the training error, whereas the larger the testing error. Hence, a tradeoff between the training error and generalization capability is required to choose the initial value of σ. In order to discover the influence of the updating step size on the online RVM model accuracy, the model outputs are generated by varying the step size D. The error bars for the two data sets Tr are depicted in the upper and lower plots in Fig. 5(a). Compared with the online LS-SVM model, the online RVM model is insensitive to the step size since the MAE remains almost constant as the variation of the step size. This arises from the fact that the model parameter σ is updated in each step by the online RVM.

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Fig. 5. Performance comparisons of the online LS-SVM and online RVM for hysteresis modeling with the training data-set sizes of (a)–(c) 200 and (d)–(f) 1000. (a) and (d) Error bars of the absolute displacement errors versus updating step size. (b) and (e) Pruning rate versus step size. (c) and (f) Cumulative updating time versus step size. In each subfigure, the upper and lower plots correspond to the first and second data sets, respectively.

For the two updating data sets Ur , the pruning rates versus the step size are shown in Fig. 5(b). It is found that the online RVM model has a greater pruning rate (over 80% for U1 and 83% for U2 ) in comparison with the online LS-SVM model (above 25% for U1 and 54% for U2 ), which indicates that less data pairs are utilized in the online RVM model. Further trials display that the pruning rate is dependent on the selection of the parameter σ. On the contrary, the cumulative updating time as compared in Fig. 5(c) exhibits that the online RVM model requires a much longer time for updating than the online LS-SVM model. E. Influences of Data-Set Size on Modeling Accuracy The comparative study performed earlier shows that both LS-SVM and RVM models are superior to the RRBFNN model in generalization ability. In order to discover more about the modeling accuracy, the training and updating processes of the two online models are carried out using larger data sets. Specifically, a smaller sampling time interval of 0.002 s is used to acquire the input and output data pairs. Thus, the sizes of data sets are increased by four times, i.e., the sizes of training and updating data sets (Br and Ur ) become 1000 and 4000, respectively. As the updating step size rises, the results of the two models are illustrated in Fig. 5(d)–(f). It is observed that the online RVM model has a significant pruning rate (over 97%) which results in a shorter updating time in comparison with the online LS-SVM model. The latter owns a much lower pruning rate, i.e., less than 12% and 26% for the updating data sets U1 and U2 , respectively. As the updating

step size varies, the updating time of the online LS-SVM almost keeps unchanged, whereas the updating time of the online RVM reduces gradually as the step size increases. The modeling errors as compared in Fig. 5(d) indicate that, as the training data set increases, the online LS-SVM model outperforms the online RVM model in terms of generalization ability, although the accuracy of the former is greatly influenced by the step size in the data set T1 . Thus, it appears that, with a small training data set (Br = 200), the online RVM is slightly better than the online LS-SVM in terms of generalization errors, whereas the former needs more updating time than the latter. On the other hand, in the case of a large training data set (Br = 1000), the online LS-SVM produces much smaller generalization errors than the online RVM model at the cost of longer updating time. VI. C ONTROLLER D ESIGN AND V ERIFICATION Based on the hysteresis modeling accuracy confirmed in the previous section, controllers are designed based on the two online models and verified for the hysteresis compensation in this section. A. Controller Design In order to suppress the hysteresis nonlinearity, an FF control based on the inverse hysteresis model can be constructed. To establish a regression model for capturing the inverse hysteresis, the same principle as shown in Section II-C is employed by selecting the voltage (x) as the output variable. The exogenous input vector is formed by the current and

WONG et al.: HYSTERESIS MODELING AND CONTROL OF A PIEZOSTAGE USING ONLINE SVM AND RVM

Fig. 6. Block diagram of FF plus FB hybrid controller for a piezostage system using an online LS-SVM or RVM inverse hysteresis model.

previous displacement and previous voltage data for the online LS-SVM and online RVM inverse model training. Once trained offline using the scheme as shown in the upper graph in Fig. 6, the LS-SVM or RVM inverse model provides the FF control (model means LS-SVM or RVM). Owing to the signal umodel FF existence of modeling error, the hysteresis cannot be eliminated completely by the stand-alone inverse model-based FF compensator. Therefore, an FB control signal uF B is adopted to create a hybrid control as described in the lower graph in Fig. 6. Specifically, the proportional–integral–derivative (PID) control algorithm is implemented as an FB control due to its robustness and popularity. By adopting an incremental PID algorithm, the overall control input can be derived in the discretized form u(kT ) = umodel F F (kT ) + uF B (kT − T ) + Kp [e(kT ) − e(kT − T )] + Ki e(kT ) + Kd [e(kT ) − 2e(kT − T ) + e(kT − 2T )]

(24)

where e represents the displacement tracking error, uF B (kT − T ) is the FB control command in the previous time step, and the FF term umodel F F (kT ) is given by the online LS-SVM or RVM inverse hysteresis model. Additionally, Kp , Ki , and Kd are the proportional, integral, and derivative gains, respectively, which can be finely tuned by resorting to the Ziegler–Nichols method. It is noticeable that, unlike the inversion-based control where the system dynamics model is required to be inverted [3], the FF control constructed in the current research is based on the inverse hysteresis function only, which is established using the online LS-SVM and RVM, respectively. Thus, the FF control does not influence the stability of the control system. As far as FB control is used, the system stability is guaranteed by the fine-tuned PID controller. B. Controller Verification and Discussions In the control experiments, the sampling time interval is assigned as T = 0.001 s. The two inverse hysteresis models based on online LS-SVM and RVM are trained using the first

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2000 data pairs and then updated by a step size of 500 using the remaining 8000 data pairs. The PID gains are tuned as Kp = 0.0142, Ki = 2.8476, and Kd = 1.7798 × 10−5 . For a reference displacement as shown in Fig. 7(a), The LS-SVM inverse hysteresis model is trained with the hyperparameters γ = 1.4153 × 107 and σ = 0.8916, which are inferred by the leave-one-out cross-validation method. The online LS-SVM model-based FF and online LS-SVM model-based FF+FB tracking results and errors are shown in Fig. 7(a) and (b). Moreover, the online RVM inverse hysteresis model is trained with the initial hyperparameter σ = 50, which is then updated online in each updating step. Using the same reference input as shown in Fig. 7(d), the tracking results and errors of the online RVM model-based FF and online RVM model-based FF+FB control are shown in Fig. 7(d) and (e). Fig. 7(b) reveals that the FF control using the LS-SVM model produces the MAE and rmse of 0.04% and 0.05% of the motion range. By contrast, the RVM model-based FF approach [see Fig. 7(e)] gives the MAE and rmse of 0.61% and 0.76%, respectively, which are 15 times worse than the LS-SVM model results. Furthermore, the tracking results of the hybrid control as shown in Fig. 7(b) indicate that the LS-SVM model-based FF+FB control leads to the MAE of 0.03% and rmse of 0.04%. While the RVM model-based hybrid control [see Fig. 7(e)] substantially suppresses the MAE and rmse to 0.22% and 0.33%, respectively. Although the RVM-based hybrid controller has reduced the MAE and rmse by 63.93% and 56.58% in comparison with the FF control alone, the results of the hybrid control are about seven times worse than those obtained by the LS-SVM inverse model-based hybrid controller. The better tracking results of LS-SVM is the consequence of the its better model generalization ability with a larger training data set. Moreover, to disclose the efficiency of the hybrid control based on each of the inverse hysteresis models, the tracking results of the stand-alone FF and FB and the combined FF+FB control based on the LS-SVM and RVM models are illustrated in Fig. 7(a)–(f), respectively. Concerning the LS-SVM inverse model, Fig. 7(b) exhibits that the online LS-SVM-based FF+FB produces the MAE and rmse of 0.03% and 0.04%, respectively, which have been only improved by 28.61% and 34.53% in comparison with the FF tracking results. This means that the LS-SVM model-based FF tracking results are already satisfactory due to a fine generalization accuracy. With the hybrid controller, the MAE and rmse have been substantially enhanced by 97.95% and 151.23% as compared with the stand-alone FB outputs, respectively. As a result, the hysteresis effects are suppressed by the LS-SVM-based FF+FB approach to a negligible level (0.2%) as indicated in Fig. 7(c). Although the PID FB tracking results can be improved by tuning the control gains to be larger, it is at the risk of oscillations in the motion tracking as shown in the close-up view in Fig. 7(a). Regarding the RVM-based inverse model [see Fig. 7(d) and (e)], the FF+FB hybrid control gives the MAE and rmse of 0.22% and 0.33%, which indicate an improvement of tracking accuracy by 64.44% and 69.82%, respectively, in comparison with the stand-alone FF outputs. Moreover, the two types of tracking errors have been reduced by 84.19% and 130.10% with

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Fig. 7. Tracking results of (a)–(c) online LS-SVM and (d)–(f) online RVM inverse model-based stand-alone and hybrid control schemes. (a) and (d) FF, FB, and FF+FB results. (b) and (e) Tracking errors of the three control methods. (c) and (f) Displacement–displacement hysteresis loops obtained by the three approaches.

respect to the PID FB tracking results. Thus, the hysteresis has also been suppressed to a low level (1.8%) by the RVM modelbased hybrid control as depicted in Fig. 7(f).

For an explicit comparison, the motion tracking results by employing the LS-SVM and RVM models are summarized in Table I, where the MAE and rmse are calculated as percentage

WONG et al.: HYSTERESIS MODELING AND CONTROL OF A PIEZOSTAGE USING ONLINE SVM AND RVM

TABLE I T RACKING P ERFORMANCES OF O NLINE LS-SVM AND RVM I NVERSE M ODEL -BASED C ONTROLLERS

Fig. 8. Comparisons of updating performances of online LS-SVM and RVM inverse models in each iteration. (a) Pruning rate versus iteration number. (b) Updating time versus iteration number. TABLE II R ECOMMENDED C ONTROL S YSTEMS BASED ON O NLINE LS-SVM OR RVM M ODEL FOR D IFFERENT R EQUIREMENTS

values with respect to the overall motion range. It is evident that the LS-SVM model is superior to the RVM model in terms of tracking errors due to an improvement of more than 86% for both FF and hybrid FF+FB methods. Comparisons of the pruning rates and updating time of the LS-SVM and RVM inverse models are displayed in Fig. 8. It is observed that, owing to the lower pruning rate, the updating time of LS-SVM increases as the iteration rolls over. Ultimately, the updating time of the online LS-SVM is 15 times that of the online RVM model. According to the performance requirements, suitable control systems for different cases are recommended in Table II. This conclusion is analyzed hereinafter. First, while the pruning rate of online LS-SVM can be arbitrarily set by choosing a proper threshold in the decremental algorithm, the pruning rate of the online RVM cannot be tuned at random by selecting the basis width σ value since σ also influences the training and testing errors at the same time. Therefore, the online LS-SVM can be tuned to have a lower pruning rate and, hence, a higher tracking

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accuracy. Second, the lower pruning rate of online LS-SVM means a higher model complexity, which significantly increases the computation time (particularly for real-time applications) along with a longer updating time as evident from Fig. 8. On the other hand, the online RVM is more preferable in terms of fast updating time. Third, from the viewpoint of tuning, the online LS-SVM is more flexible since its performance has more degrees of freedom to adjust with more parameters. On the contrary, the RVM has fewer parameters to be adjusted and needs less efforts in parameter tuning, which is more attractive for setting up a system in a limited working period. In this research, the online LS-SVM and RVM are utilized to model the piezoelectric hysteresis which is rate dependent in nature. Assuming that the hysteresis is rate independent, i.e., the rate dependence is not taken into account, much smaller modeling errors are expected for these models, but the rate-independent models are insufficient for many practical applications. The experimental results show that both online models are superior to the offline ones and the typical RRBFNN models and give promising results. As compared to traditional hysteresis models (e.g., the Prandtl–Ishlinskii model) which are directly identified from preacquired data sets, the intelligent models are capable of updating continually with new data sets. However, their limitations lie in that some crucial parameters require to be finely tuned and a training process is needed before they can give satisfactory results. In addition, the current research focuses on the hysteresis modeling and compensation. The creep effect is not explicitly treated here as it is suppressed by the FB control. In the future, the creep nonlinearity will be modeled as well. VII. C ONCLUSION The investigation presented in this paper shows that the ratedependent hysteresis of a piezostage can be accurately modeled by the online LS-SVM and online RVM regression models which are new attempts at the domain of hysteresis modeling. Comparative studies of the online model-based hysteresis modeling and compensation reveal the following. 1) The performances of the online LS-SVM model heavily rely on the two hyperparameters (γ and σ), whereas the online RVM is relatively insensitive to its sole initial parameter (σ) selection. 2) The generalization capability of online LS-SVM can be enhanced by choosing smaller threshold value in pruning procedures while at the cost of a high model complexity and longer updating time. The generalization ability of online RVM is limited since the tuning of σ value also influences the training accuracy. 3) With a small training data set, the performance of the online RVM model is slightly better than the online LS-SVM model in terms of testing error. In the case of a large training data set, the online LS-SVM outperforms the online RVM model instead, whereas the latter requires a much shorter updating time. Moreover, this paper has also proposed a new hybrid control algorithm using the online LS-SVM or RVM inverse modelbased FF control combined with a PID FB control for hysteresis

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compensation. The most important feature of the algorithm is its online model update ability which can compensate the hysteresis model error when there are any changes in the real hysteresis behavior. Experimental results demonstrate that the hybrid control algorithm is capable of suppressing the hysteresis nonlinearity effectively. In the current research, the comparative study is carried out for the two online models with the popular RBF kernel function. More comparisons with other types of kernel functions will be performed in the future. The proposed modeling and control approaches will be extendable to other types of hysteretic systems as well. Moreover, an attempt will be conducted to improve the generalization ability while maintaining a shorter updating time for the online RVM model. R EFERENCES [1] M.-Y. Chen, H.-H. Huang, and S.-K. Hung, “A new design of a submicropositioner utilizing electromagnetic actuators and flexure mechanism,” IEEE Trans. Ind. Electron., vol. 57, no. 1, pp. 96–106, Jan. 2010. [2] N. Bonnail, D. Tonneau, F. Jandard, G.-A. Capolino, and H. Dallaporta, “Variable structure control of a piezoelectric actuator for a scanning tunneling microscope,” IEEE Trans. Ind. Electron., vol. 51, no. 2, pp. 354–363, Apr. 2004. [3] K. K. Leang, Q. Zou, and S. Devasia, “Feedforward control of piezoactuators in atomic force microscope systems: Inversion-based compensation for dynamics and hysteresis,” IEEE Control Syst. Mag., vol. 29, no. 1, pp. 70–82, Feb. 2009. [4] Y. Zhang, K. K. Tan, and S. Huang, “Vision-servo system for automated cell injection,” IEEE Trans. Ind. Electron., vol. 56, no. 1, pp. 231–238, Jan. 2009. [5] A. A. Ramadan, T. Takubo, Y. Mae, K. Oohara, and T. Arai, “Developmental process of a chopstick-like hybrid-structure two-fingered micromanipulator hand for 3-D manipulation of microscopic objects,” IEEE Trans. Ind. Electron., vol. 56, no. 4, pp. 1121–1135, Apr. 2009. [6] H.-J. Shieh, F.-J. Lin, P.-K. Huang, and L.-T. Teng, “Adaptive displacement control with hysteresis modeling for piezoactuated positioning mechanism,” IEEE Trans. Ind. Electron., vol. 53, no. 3, pp. 905–914, Jun. 2006. [7] Y. K. Yong, S. Aphale, and S. O. R. Moheimani, “Design, identification and control of a flexure-based XY stage for fast nanoscale positioning,” IEEE Trans. Nanotechnol., vol. 8, no. 1, pp. 46–54, Jan. 2009. [8] H. C. Liaw and B. Shirinzadeh, “Robust adaptive constrained motion tracking control of piezo-actuated flexure-based mechanisms for micro/nano manipulation,” IEEE Trans. Ind. Electron., vol. 58, no. 4, pp. 1406–1415, Apr. 2011. [9] Y. Li and Q. Xu, “A novel piezoactuated XY stage with parallel, decoupled and stacked flexure structure for micro/nano positioning,” IEEE Trans. Ind. Electron., vol. 58, no. 8, pp. 3601–3615, Aug. 2011. [10] J. Minase, T.-F. Lu, B. Cazzolato, and S. Grainger, “A review, supported by experimental results, of voltage, charge and capacitor insertion method for driving piezoelectric actuators,” Precis. Eng., vol. 34, no. 4, pp. 692– 700, Oct. 2010. [11] X. Tan and J. S. Baras, “Adaptive identification and control of hysteresis in smart materials,” IEEE Trans. Autom. Control, vol. 50, no. 6, pp. 827– 839, Jun. 2005. [12] M. Ruderman, F. Hoffmann, and T. Bertram, “Modeling and identification of elastic robot joints with hysteresis and backlash,” IEEE Trans. Ind. Electron., vol. 56, no. 10, pp. 3840–3847, Oct. 2009. [13] K. Kuhnen, “Modeling, identification and compensation of complex hysteretic nonlinearities: A modified Prandtl–Ishlinskii approach,” Eur. J. Control, vol. 9, no. 4, pp. 407–421, 2003. [14] Y. Li and Q. Xu, “Adaptive sliding mode control with perturbation estimation and PID sliding surface for motion tracking of a piezo-driven micromanipulator,” IEEE Trans. Control Syst. Technol., vol. 18, no. 4, pp. 798–810, Jul. 2010. [15] C. Ru, Y. Zhang, Y. Sun, Y. Zhong, X. Sun, D. Hoyle, and I. Cotton, “Automated four-point probe measurement of nanowires inside a scanning electron microscope,” IEEE Trans. Nanotechnol., vol. 10, no. 4, pp. 674– 681, Jul. 2011. [16] Y. Yu, Z. Xiao, N. G. Naganathan, and R. V. Dukkipati, “Dynamic Preisach modelling of hysteresis for the piezoceramic actuator system,” Mech. Mach. Theory, vol. 37, no. 1, pp. 75–89, Jan. 2002.

[17] W. T. Ang, P. K. Khosla, and C. N. Riviere, “Feedforward controller with inverse rate-dependent model for piezoelectric actuators in trajectorytracking applications,” IEEE/ASME Trans. Mechatronics, vol. 12, no. 2, pp. 134–142, Apr. 2007. [18] M. A. Janaideh, S. Rakheja, and C.-Y. Su, “Experimental characterization and modeling of rate-dependent hysteresis of a piezoceramic actuator,” Mechatronics, vol. 19, no. 5, pp. 656–670, Aug. 2009. [19] M.-S. Tsai and J.-S. Chen, “Robust tracking control of a piezoactuator using a new approximate hysteresis model,” Trans. ASME, J. Dyn. Syst. Meas. Control, vol. 125, no. 1, pp. 96–102, Mar. 2003. [20] G. Gu and L. Zhu, “High-speed tracking control of piezoelectric actuators using an ellipse-based hysteresis model,” Rev. Sci. Instrum., vol. 81, no. 8, pp. 085104-1–085104-9, Aug. 2010. [21] P. Tian, G. Li, and R. W. Jones, “Ellipse-based modelling of hysteresis in a smart actuator,” in Proc. IEEE/ASME Int. Conf. Adv. Intell. Mechatronics, Montreal, QC, Canada, 2010, pp. 872–877. [22] S. Yu, G. Alici, B. Shirinzadeh, and J. Smith, “Sliding mode control of a piezoelectric actuator with neural network compensating rate-dependent hysteresis,” in Proc. IEEE Int. Conf. Robot. Autom., Barcelona, Spain, 2005, pp. 3641–3645. [23] R. Dong, Y. Tan, H. Chen, and Y. Xie, “A neural networks based model for rate-dependent hysteresis for piezoelectric actuators,” Sens. Actuators A, Phys., vol. 143, no. 2, pp. 370–376, 2008. [24] J. P. Lien, A. York, T. Fang, and G. D. Buckner, “Modeling piezoelectric actuators with hysteretic recurrent neural networks,” Sens. Actuators A, Phys., vol. 163, no. 2, pp. 516–525, Oct. 2010. [25] J. A. K. Suykens, “Support vector machines: A nonlinear modeling and control perspective,” Eur. J. Control, vol. 7, no. 2/3, pp. 311–327, Aug. 2001. [26] P. K. Wong, L. M. Tam, K. Li, and C. M. Vong, “Engine idle-speed system modelling and control optimization using artificial intelligence,” Proc. Inst. Mech. Eng. Part D-J. Automobile Eng., vol. 224, no. 1, pp. 55–72, 2010. [27] P.-K. Wong, C.-M. Vong, L.-M. Tam, and K. Li, “Data preprocessing and modelling of electronically-controlled automotive engine power performance using kernel principal components analysis and least-square support vector machines,” Int. J. Vehicle Syst. Model. Test., vol. 3, no. 4, pp. 312–330, 2008. [28] X.-F. Yang, W. Li, Y.-Q. Wang, and X.-P. Su, “A multi-loop hysteresis model of piezo actuator based on LS-SVM,” in Proc. Asian Simul. Conf. and 7th Int. Conf. Syst. Simul. Sci. Comput., Beijing, China, 2008, pp. 1451–1454. [29] W. Lei, J. Mao, and Y. Ma, “A new modeling method for nonlinear ratedependent hysteresis system based on LS-SVM,” in Proc. 10th Int. Conf. Control, Autom., Robot. Vis., Hanoi, Vietnam, 2008, pp. 1442–1446. [30] H. Tang, S. Xue, R. Chen, and T. Sato, “Online weighted LS-SVM for hysteretic structural system identification,” Eng. Struct., vol. 28, no. 12, pp. 1728–1735, Oct. 2006. [31] M. E. Tipping, “The relevance vector machine,” in Advances in Neural Information Processing Systems, vol. 12, S. A. Solla, T. K. Leen, and K.-R. Muller, Eds. Cambridge, MA: MIT Press, 2000, pp. 652–658. [32] M. E. Tipping and A. C. Faul, “Fast marginal likelihood maximization for sparse Bayesian models,” in Proc. 9th Int. Workshop Artif. Intell. Statist., C. M. Bishop and B. J. Frey, Eds., Key West, FL, 2003. [33] B. E. Helfrich, C. Lee, D. A. Bristow, X. H. Xiao, J. Dong, A. G. Alleyne, S. M. Salapaka, and P. M. Ferreira, “Combined H∞ -feedback control and iterative learning control design with application to nanopositioning systems,” IEEE Trans. Control Syst. Technol., vol. 18, no. 2, pp. 336–351, Mar. 2010. [34] R. Wang, G.-P. Liu, W. Wang, D. Rees, and Y.-B. Zhao, “H∞ control for networked predictive control systems based on the switched Lyapunov function method,” IEEE Trans. Ind. Electron., vol. 57, no. 10, pp. 3565– 3571, Oct. 2010. [35] J.-X. Xu and K. Abidi, “Discrete-time output integral sliding-mode control for a piezomotor-driven linear motion stage,” IEEE Trans. Ind. Electron., vol. 55, no. 11, pp. 3917–3926, Nov. 2008. [36] X. Yu and O. Kaynak, “Sliding mode control with soft computing: A survey,” IEEE Trans. Ind. Electron., vol. 56, no. 9, pp. 3275–3285, Sep. 2009. [37] S. Khan, A. Sabanovic, and A. O. Nergiz, “Scaled bilateral teleoperation using discrete-time sliding-mode controller,” IEEE Trans. Ind. Electron., vol. 56, no. 9, pp. 3609–3618, Sep. 2009. [38] B. Veselic, B. Perunicic-Drazenovic, and C. Milosavljevic, “Improved discrete-time sliding-mode position control using Euler velocity estimation,” IEEE Trans. Ind. Electron., vol. 57, no. 11, pp. 3840–3847, Nov. 2010. [39] A. J. Mehta, B. Bandyopadhyay, and A. Inoue, “Reduced-order observer design for servo system using duality to discrete-time sliding-surface

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Pak-Kin Wong received the M.S. degree in manufacturing systems engineering from the University of Warwick, Coventry, U.K., in 1992, and the Ph.D. degree in mechanical engineering from The Hong Kong Polytechnic University, Hong Kong, China, in 1997. He is currently an Associate Professor and the Head of the Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Macao, China. His research interests include automotive engineering, fluid transmission and control, engineering applications of artificial intelligence, and engineering vibration. He has published over 100 scientific papers in refereed journals, book chapters, and conference proceedings.

Qingsong Xu (M’09) received the B.S. degree in mechatronics engineering (with honors) from Beijing Institute of Technology, Beijing, China, in 2002, and the M.S. and Ph.D. degrees in electromechanical engineering from the University of Macau, Macao, China, in 2004 and 2008, respectively. He was a Visiting Scholar at the Swiss Federal Institute of Technology (ETH), Zurich, Switzerland. He is currently an Assistant Professor of electromechanical engineering with the University of Macau. His current research interests include parallel manipulators, microelectromechanical systems (MEMS) devices, micro-/nanorobotics, micro-/nanomanipulation, smart materials and structures, and computational intelligence. Dr. Xu is a member of the American Society of Mechanical Engineers.

2001

Chi-Man Vong (M’10) received the M.S. and Ph.D. degrees in software engineering from the University of Macau, Macao, China, in 2000 and 2005, respectively. He is currently an Assistant Professor with the Department of Computer and Information Science, Faculty of Science and Technology, University of Macau. His research interests include machine learning methods and intelligent systems.

Hang-Cheong Wong received the B.S. and M.S. degrees in electromechanical engineering from the University of Macau, Macao, China, in 2008 and 2010, respectively, where he is currently working toward the Ph.D. degree. His research interests include automotive powertrain control, vehicle vibration control, and engineering applications of artificial intelligence.

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