Nonlinear Modeling and Decoupling Control of XY ... - IEEE Xplore

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 18, NO. 3, JUNE 2013

821

Nonlinear Modeling and Decoupling Control of XY Micropositioning Stages With Piezoelectric Actuators Yangqiu Xie, Yonghong Tan, and Ruili Dong

Abstract—In this paper, a modeling method of XY micropositioning stage with piezoelectric actuators is proposed. In the modeling scheme, a sandwich model consists of both input and output linear submodels, and an embedded neural-network-based hysteresis submodel is used to describe the motion behavior of each axis of the stage. Moreover, a neural-network-based submodel is constructed to describe the nonlinear interactive dynamics caused by the movement of another axis. Then, a tracking control scheme combined with a nonlinear decoupling control is proposed to compensate for the effect of the interactions between axes and track the reference trajectory. Then, the robust design method for the tracking and decoupling control is discussed. Finally, the experimental results on an XY micropositioning stage are presented. Index Terms—Decoupling control, expanded input space, hysteresis, micropositioning stage, neural network, sandwich system.

I. INTRODUCTION SUALLY, XY micropositioning stages are used in ultraprecision manufacturing systems [1]–[8], which are driven by piezoeloectric actuators (PEAs) [1]–[5], magnetical actuators [6], or walking piezoactuators [7], [8]. Among those driven devices, PEAs have been widely utilized in such stages in order to obtain nanoscale displacement due to PEAs’ high stiffness, nanometer displacement resolution, large bandwidth, and fast frequency response [9]. However, the performance of XY micropositioning stages is often degraded by the hysteresis inherent in those driven devices. As hysteresis is a nonsmooth nonlinearity with multivalued mapping, it often leads to oscillation and unexpected residuals [11]. Therefore, the compensation

U

Manuscript received March 8, 2011; revised June 27, 2011 and November 26, 2011; accepted December 21, 2011. Date of publication March 8, 2012; date of current version January 18, 2013. Recommended by Technical Editor Y. Sun. This work was supported in part by the Leading Academic Discipline Project of Shanghai Normal University under Grant DZL811, in part by the Innovation Program of Shanghai Municipal Education Commission under Grant 09ZZ141 and Grant 11YZ92, in part by the Advanced Research Grant of Shanghai Normal University under Grant DYL201005, Grant DYL201006, and Grant DRL904, in part by the National Science Foundation of China under Grant 60971004 and Grant 61171088, in part by the Natural Science Foundation of Shanghai under Grant 09ZR1423400 and Grant 10ZR1422400, and in part by the Science and Technology Commission of Shanghai Municipality under Grant 09220503000 and Grant 10JC1412200. Y. Xie is with the the College of Mechanical and Electronic Engineering, Shanghai Normal University, Shanghai 200234, China, and also with the School of Electronic Engineering, Xidian University, Xi’an 710071, China (e-mail: [email protected]). Y. Tan and R. Dong are with the College of Mechanical and Electronic Engineering, Shanghai Normal University, Shanghai 200234, China (e-mail: [email protected]; [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/TMECH.2012.2187794

of the oscillation and undesired system error caused by hysteresis in piezoelectric actuators is one of the important issues in nanoscale positioning systems. In [10], an adaptive inverse control for hysteresis was implemented, while in [11], the authors considered the hysteresis as a model uncertainty and developed a robust adaptive control strategy to handle it. In [12], a Presach inverse model is constructed to compensate for the effect of hysteresis for piezoelectric actuators. In addition, an adaptive sliding control is proposed to control a system with piezoelectric actuator [13]. Moreover, Li and Xu [14] proposed a modified Prandtl–Ishlinskii hysteresis model-based control method to suppress the effect of hysteresis in the PEAs of a micropositioning stage. Besides, the generalized Prandtl–Ishlinskii inverse model [15], Dahl inverse model [5], and an inverse model based on neural network cooperated with a linear autoregressive submodel with exogenous input [34] are, respectively, employed as the feedforward compensators to eliminate the effect of hysteresis, and then, the corresponding tracking control is performed by a PID controller. On the other hand, it is noted that the performance of stages is also affected by the interactive disturbances between axes, which are, in fact, a nonlinear coupling behavior. Therefore, in order to obtain satisfactory performance, it is necessary to compensate for the effects of both hysteresis in the actuators and interactive disturbances between axes. Recently, there have been some schemes to handle the compensation of coupling effects existing in micropositioning stages, such as the special mechanical structures are designed to reduce the coupling effects between axes [1], [16]–[18]. Some decoupling control strategies based on linear coupling models are developed [19]–[22]. Among those decoupling control schemes, an H2 /H∞ -based multiple-input multiple-output (MIMO) robust linear control scheme is proposed in [19], while a linear dynamic model-based feedforward decoupling control method is developed in [20]. Moreover, a MIMO controller with lateral feedback is proposed in [21] to suppress the coupling effects, and in [22], the authors implemented an iterative control approach to eliminate the displacement errors caused by coupling motions. Actually, the coupling effects between axes are nonlinear and dynamic behaviors. A nonlinear and dynamic decoupling method will be more applicable to this situation. Note that the design of a control strategy to eliminate the effects of both hysteresis and nonlinear interactions existing in the micropositioning stage relies on the models to describe the dynamics, hysteresis phenomena, and coupling behaviors between axes of the stage. Therefore, constructing the proper and accurate models of the stage and developing the corresponding model-based compensators to eliminate the effects of hysteresis

1083-4435/$31.00 © 2012 IEEE

822


Fig. 2.

Fig. 1.

Experimental setup.

and interactions in terms of the derived models are the main tasks of our research. In this paper, the architecture of an XY micropositioning stage is described first. Then, the identification of the nonlinear dynamic behavior of each axis that is involved with hysteresis in the PEA is presented. To describe the behavior of the stage, the so-called sandwich models with hysteresis are constructed. Each sandwich model consists of an input linear submodel, an output linear submodel, and a hysteresis submodel embedded between the two linear submodels. On the other hand, in order to describe the nonlinear interaction caused by another axis, a neural-network-based coupling submodel is developed. After that, an inverse sandwich compensator is built to compensate the hysteretic effect, and a neural-network-based nonlinear decoupling controller is also developed to cancel the effect of the interactive disturbance. Considering the existence of model uncertainty and decoupling compensation error, the robust design scheme is proposed, so that the control scheme can guarantee the system to achieve the desired performance in order to track the reference trajectory. Finally, the experimental results on an XY micropositioning stage are presented. II. CONFIGURATION OF THE XY MICROPOSITIONING STAGE The corresponding experimental setup of the micropositioning stage with piezoelectric actuators and the corresponding control system is shown in Fig. 1. In the stage, the X-axis is perpendicular to the Y -axis. For each axis, the piezoceramic actuator driven by an amplifier–filter circuit is used to drive a flexure device. The displacement of the flexure hinge device of each axis is measured by a noncontact capacitive-type sensor. Then, the measured signal is sampled by an A/D converter and the obtained digital signal is sent to a PC. Then, the control signal from the PC is sent to the amplifier through a D/A converter. III. MODEL OF A SINGLE AXIS OF THE STAGE In some proposed schemes, a Hammerstein model with hysteresis was used to describe the behavior of the positioning stages [15], [26]. In this case, the dynamic effect of the

Sandwich model with hysteresis of a single axis.

amplifier–filter circuit for piezoelectric actuator has been ignored. In order to derive a fast and accurate positioning control performance, the dynamic effect of the amplifier should be considered in the model architecture. Thus, the so-called sandwich model with hysteresis should be adopted. In this system, the effect of the hysteresis existing in piezoelectric actuators cannot be ignored, especially when nanoscale positioning is implemented. On the other hand, the flexure device is usually considered as a spring–damping system, which is approximated by a secondorder low-pass linear system [23]. Furthermore, the behavior of the amplifier–filter circuit in this stage can be described by a first-order linear system. Hence, for each axis of the stage, the structure of the model shown in Fig. 2 can be used to describe the motion behavior of a single axis by considering the effect of hysteresis existing in the piezoelectric actuator. In Fig. 2, L1 (·), H(·), and L2 (·) denote the amplifier–filter circuit, piezoceramic actuator, and flexure device, respectively. In this model, both L1 (·) and L2 (·) are linear submodels, while H(·) represents hysteresis. This model is named as the sandwich model with embedded hysteresis. Note that w1 (k) and w2 (k) are internal variables that represent the output of the amplifier and the actuator displacement, respectively. These are assumed to be unmeasurable directly. Moreover, u(k) and y(k) are the input voltage of the amplifier and the displacement of the flexure hinge device, individually. A. Description of the Sandwich Model With Hysteresis Based on Fig. 2, the mapping between the output y(k) and input u(k) of the sandwich model can be described by y(k) = L2 {H [L1 (u(k))]}.

(1)

Thus, the corresponding sandwich model can be represented as w1 (k) = −

n1a

a1i w1 (k − i) +

i=1

n1b

b1j u(k − q1 − j). (2)

j =1

w2 (k) = H[w1 (k)]

(3)

and y(k) = −

n2a i=1

a2i y(k − i) +

n2b

b2j w2 (k − q2 − j)

(4)

j =1

where n1a , n1b and n2a , n2b are the orders; a1i , b1j and a2i , b2j are the coefficients; and q1 and q2 are the time delays, respectively. Here, we compose the following assumption. Assumption 1: Both input and output linear submodels of the sandwich model are subject to the following conditions: A1: stable and minimum phase; A2: gains are positive constants; A3: both L1 (·) and L2 (·) are coprime, so is their product.

XIE et al.: NONLINEAR MODELING AND DECOUPLING CONTROL OF XY MICROPOSITIONING STAGES WITH PIEZOELECTRIC ACTUATORS

Remark 1: In the assumption, if the linear subsystem satisfies condition A1, it implies that the inverse model of the linear subsystem exists; if the linear submodel obeys condition A2, it means that the sign of the linear subsystem in steady state will only be affected by the sign of the input; moreover, condition A3 in the assumption implies that there are no cancellation between the zeros and poles in the linear submodels so as to guarantee the unique representation of the linear submodels. It is known that most of the systems, in practice, can satisfy the conditions given in Assumption 1. During the recent decade, identification of the dynamic systems with nonsmooth nonlinearities has become one of the interesting topics in mechatronics and control engineering. There have been several identification methods for Wiener or Hammerstein models with nonsmooth nonlinearities [25], [26]. The modeling methods for nonsmooth sandwich systems with dead zone or backlash are also studied in [27], [29], [33], and [36]. Some methods of modeling of rate-dependent hysteresis can be found in [34], [35], and [37]. However, in those methods, both input and output of the hysteresis can be measured directly. In the XY micropositioning stage, the input and output of the hysteresis embedded in the stage may not be measurable in practice. Hence, in the following, a two-step identification method for the sandwich model with hysteresis will be presented. B. Degeneration of the Hysteresis Subsystem As hysteresis is a nonsmooth and nonlinear function with multivalued mapping, the traditional identification techniques are usually unavailable to model such a system. Hence, a socalled degeneration input signal is designed to degenerate hysteresis to a static polynomial with one-to-one mapping within a certain operation region. In this case, the sandwich model can be simplified to a model linearized in coefficients. Definition 1: For a sandwich system with hysteresis, an input signal is called the degeneration input if it satisfies the following conditions. 1) The nonsmooth nonlinear hysteresis embedded in the sandwich system is constrained in a certain operation region but not allowed to enter the other operation regions. 2) The input is a persistent excitation signal. Note that the operation region of the hysteresis will be changed if the direction of its input signal is changed. Hence, if a monotonic input is implemented, the response of the hysteresis will also present a monotonic feature, so that H(·) will maintain in the specified operation region and not enter the other operation regions. Moreover, in order to estimate the parameters of L1 (·) and L2 (·), an independently incremental signal that satisfies the persistent excitation condition is designed. Thus, a degeneration input is selected as u1 (k) = u1 (k − 1) + Δu1 (k)

(5)

and Δu1 (k) = n(k) + nm in + ap

(6)

823

Fig. 3. Degenerated process of hysteresis H(·). (a) Hysteresis block H (·). (b) Degenerated block H (·).

where Δu1 (k) is the incremental signal of u1 (k), while n(k) is an independent identically distributed (i.i.d.) white noise sequence with zero mean value and variance σ, nm in = min |n(k)| and ap (ap > 0) is an offset, respectively. In (6), nm in is adopted to ensure that Δu1 (k) is always positive in the specified operation region, so that u1 (k) increases monotonically, while ap is used to counteract the dynamic effect of L1 (·). Note that ap can be determined by Theorem 1. Theorem 1: Based on conditions A1 and A2 of Assumption 1, an independently incremental signal u1 (k) described by (5) and (6) is a degeneration input for a sandwich system with hysteresis if Ψ1 − 1 · |nm in | (7) ap ≥ k1 is held, where k1 is the static gain of L1 (·) and Ψ1 is the supremum of the expanded amplitude scale of L1 (·), respectively. Proof: Refer to Appendix A. Remark 2: The purpose of Theorem 1 implies that if formula (7) is satisfied, the input of the hysteresis, i.e., w1 (k), is a monotonically increased signal with persistent exciting property so as to guarantee the hysteresis to be degenerated to a polynomial with one-to-one mapping within the specified operation region. In this case, the hysteresis H(·) is degenerated to a static function with one-to-one mapping shown in Fig. 3, where H (·) is the function degenerated from the hysteresis that can be approximated by a polynomial. Furthermore, in order to estimate the parameters of L1 (·) and L2 (·), the equations with incremental variables should be considered to remove the monotonically increasing tendency of the data. Thus, we have L1 (·): Δw1 (k) =

n1b j =1

b1j Δu1 (k−q1 −j)−

n1a

a1i Δw1 (k − i)

i=1

(8) H (·): Δw2 (k) = Δw1 (k) + c1 Δw12 (k) + · · · + cp−1 Δw1p (k) (9) and L2 (·): Δy(k) =

n2b j =1

b2j Δw2 (k − q2 − j) −

n2a

a2i Δy(k − i)

i=1

(10) where ci (i = 1, . . . , p − 1) are the coefficients of the polynomial. In order to obtain a model that is linear in parameters but the variables are nonlinear, the key term separation principle [25]

824


is applied. Moreover, let b21 = 1 in order to obtain the unique expression. If the first term of L2 (·) is selected to be the key term, half-substituting [25] (9) into (10) results in Δy(k) = −

n2a

a2i Δy(k − i) +

i=1

n2b

b2j · Δw2 (k − q2 − j)

j =2

+ cp−1 Δw1p (k − q2 − 1).

(11)

Then, half-substituting (8) into (11) leads to n2a

a2i Δy(k − i) +

i=1

n2b

ˆ1 (k − q2 ) + cˆ1 (k − 1) · Δw ˆ12 (k − q2 ) Δw ˆ2 (k − q2 ) = Δw + · · · + cˆp−1 (k − 1) · Δw ˆ1p (k − q2 )

+ Δw1 (k − q2 − 1) + c1 Δw12 (k − q2 − 1) + · · ·

Δy(k) = −

and

b2j · Δw2 (k − q2 − j)

j =2

where Δw ˆ1 , Δw ˆ2 , a îj , ˆbij , and cˆl denote the estimate of Δw1 , Δw2 , aij , bij , and cl , respectively, and q = q1 + q2 .

+

ˆ θ(k) = arg min θ

−

N

ˆ − 1))θ(k ˆ − 1)]2 . ˆ T (k|θ(k [Δyp (k) − h

k =1

(19) Therefore, the parameter vector θ can be estimated by

b1j Δu1 (k − q2 − q1 − j − 1)

j =1 n1a

(18)

Then, the Recursive General Identification Algorithm (RGIA) [27] is utilized to estimate the coefficients of the model. Define the cost function as

+ c1 Δw12 (k−q2 −1) + · · · + cp−1 Δw1p (k − q2 − 1) n1b

(17)

a1i Δw1 (k − q2 − i − 1).

(20)

ˆ ˆ − 1) + K(k)e(k) θ(k) = θ(k

(21)

(12)

i=1

C. Estimation of the Linear Subsystems Consider that the internal variables of the system cannot be measured directly. The estimation of Δw1 and Δw2 should be implemented based on models (8) and (9) with the coefficients estimated at previous sampling period. Thus, based on (12), we use ˆ − 1))θ(k ˆ − 1) ˆ − 1)) = h ˆ T (k|θ(k Δˆ y (k|θ(k

ˆ − 1) ˆ T (k)θ(k e(k) = Δyp (k) − h

(13)

to represent the estimation relying on the parameters derived at previous sampling step. In (13), ˆ − 1)) = [Δu1 (k−q−2), . . . , Δu1 (k − q − n1b − 1), ˆ θ(k h(k|

K(k) =

ˆ P(k − 1)h(k) ˆ T (k)P(k − 1)h(k) ˆ h + μ(k)γ(k)

P(k) =

1 ˆ T (k)] · P(k − 1) · [I − K(k) · h μ(k)

(22)

ˆ T (k)]T + K(k) · γ(k) · KT (k) (23) · [I − K(k) · h γ(k) = γ(k − 1) + ρ(k)[e2 (k) − γ(k − 1)]

(24)

where e(k), K(k), P (k), and γ(k) are the modeling error, gain vector, covariance matrix, and estimation value of the correlation of model error, and ρ(k) ∈ (0, 1) and μ(k) ∈ (0, 1) are the convergence and forgetting factors, respectively. Based on [27], the convergence and forgetting factors can be determined by ρ(k) = (k + 1)−5/8 and μ(k) =

ρ(k − 1) [1 − ρ(k)] . (25) ρ(k)

If the input is in the monotonically decreased direction, we obtain the similar results.

− Δw ˆ1 (k − q2 − 2), . . . , − Δw ˆ1 (k − q2 − n1a − 1), Δw ˆ2 (k − q2 − 2), . . . , Δw ˆ2 (k − q2 − n2b ),

D. Identification of the Embedded Hysteresis Model

− Δy(k − 1), . . . , −Δy(k − n2a ),

When the coefficients of L1 (·) and L2 (·) are determined, the internal variables w1 (k) and w2 (k) can also be estimated based on (8) and (9) with the previously obtained estimated coefficients. Thus, a neural network is employed to model the hysteresis exists in the piezoelectric actuator using the expanded input space method [29]. In order to excite the mode of hysteresis, input u2 (k) can be designed as

Δw ˆ12 (k − q2 − 1), . . . , Δw ˆ1p (k − q2 − 1)]T (14) ˆ θ(k) = [ˆb11 (k), . . . , ˆb1n 1 b (k), a ˆ11 (k), . . . , a ˆ1n 1 a (k), ˆb22 (k), . . . , ˆb2n (k), a ˆ21 (k), . . . , a ˆ2n 2 a (k), 2b cˆ1 (k), . . . , cˆp−1 (k)]T Δw ˆ1 (k − q2 ) =

n1b

(15)

ˆb1j (k − 1) · Δu1 (k − q − j)

j =1

−

n1a i=1

a ˆ1i (k − 1) · Δw ˆ1 (k − q2 − i)

(16)

u2 (k) = e−α t A[sin(2πf e−δ t k + ϕ) + 1] + ζ

(26)

where α and δ are the amplitude and frequency attenuating coefficients, and ζ, A, f , and ϕ are the offset, amplitude, frequency, and phase of u2 (k), respectively. As the input with time-varying frequency is implemented, the hysteresis in the piezoelectric stage presents the rate-dependent behavior. Then, based on the previously estimated parameters of the linear submodels and


825

the measured system output data, we can reconstruct the corresponding input and output of the hysteresis, i.e., w ˆ1 (k) = −

n1a

a ˆ1i w ˆ1 (k − i) +

i=1

n1b

ˆb1j u2 (k − q1 − j) (27)

j =1

and w ˆ2 (k) = −

n2b

ˆb2j w ˆ2 (k − j + 1) + 1 +

j =2

q 2 +1

a ˆ2i

i=1

+

n2a

a ˆ2i y(k + q2 + 1 − i). (28)

i=q 2 +2

Based on the expanded input space method [29], a neural network model is used to model the behavior of hysteresis with rate-dependent characteristic. The corresponding neural hysteresis model is described by o(k) = PT3 · σ1 PT1 · Σ + PT4 · σ2 PT2 · O (29) where Σ = {w ˆ1 (k − i), Ψ(k − i)|i = 1, . . . , m} and O = {w ˆ2 (k − j)|j = 1, . . . , n}; I m ×1 , Ψm ×1 , and O n ×1 are the input, hysteretic operator output, and network feedback vectors; m and n are the time lags; moreover, o(k) is the output of the neural model. In this model, the hidden layer consists of two parts, i.e., the feedforward part connected to the expanded input space [I m ×1 , Ψm ×1 ] to mimic the hysteretic performance, and the feedback part connected to the previous model output O n ×1 in order to describe the dynamics or rate-dependent behavior of the hysteresis; P i (i = 1, 2, 3, 4) are the weight matrices. In this paper, the logarithmic sigmoid σ 1 (·) and linear function σ 2 (·) are, respectively, used as the activation functions for the hidden nodes. In this case, the hysteretic operator Ψ(·) with the wiping-out property is generated by the superposition of backlash operators, so that the expanded input space is expanded to [I m ×1 , Ψm ×1 , O n ×1 ]T . Then, the Levenberg–Marquarqt algorithm [30] is implemented to train the neural model. Remark 3: In the sandwich system with hysteresis, both input and output of the hysteresis cannot be measured directly. In order to estimate the coefficients of the linear submodels ahead and preceded the hysteresis, respectively, the so-called degenerated input is designed to decompose the hysteresis to a polynomial within a specified region. Then, the coefficients of the linear submodels can be estimated. Note that the polynomial only partly represents the property of hysteresis. Therefore, the whole characteristic of the hysteresis within the whole operation region should be modeled by a neural network on an expanded input space. However, both input and output are not measured directly in the sandwich system. In this case, the input and output of the hysteresis are reconstructed by the obtained linear submodels. With the reconstructed input and output, neural network can be trained to approximate the hysteresis. IV. MODELING OF INTERACTIONS BETWEEN AXES In this section, the interactive effect of the coupling movement of the micropositioning stage is presented through an experiment. In the experiment, let uX (k), i.e., the input of the

Fig. 4. Coupling motion of x-axis. (a) Time-domain output. (b) Frequencydomain output.

X-axis, be zero, while uY (k), i.e., the input of the Y -axis, be a sequence of white noise. In this case, the displacement of the X-axis is not equal to zero due to the coupling effect caused by the movement of the Y -axis, i.e., yX (k) = yY X (k).

(30)

Fig. 4 illustrates the measured coupling displacement of the Xaxis caused by the movement of the Y -axis. Fig. 4(a) shows the coupling response in time domain, while Fig. 4(b) illustrates the coupling response in frequency domain. It is known that the peak-to-peak amplitude of the coupling displacement is 0.2564 μm. Moreover, the frequency band is between 0.8 and 2.1 kHz. Obviously, the effect caused by the interaction is unacceptable for a nanometer-scale micropositioning system. According to what mentioned previously, a sandwich model with hysteresis and interactions is illustrated in Fig. 5 to describe the complex behavior of the XY micropositioning stage. In Fig. 5, uX , yX , LX 1 (·), HX (·), and LX 2 (·) are the input, output, input linear submodel, hysteresis sumodel, and output linear submodel of the X-axis, respectively; while uY , yY , LY 1 (·), HY (·), and LY 2 (·) are the variables and blocks of the Y -axis, which are the same as those defined in the X-axis,

826


this coupling submodel, the logarithmic sigmoid function υ1 (·) and the linear function υ2 (·) are, respectively, selected as the activation functions for the hidden neurons and the output neuron. Similarly, the Levenberg–Marquarqt algorithm is also adopted to train this neural-network-based coupling submodel. V. TRACKING AND DECOUPLING CONTROL SCHEME

Fig. 5.

Model structure of the XY micropositioning stage.

respectively; wY i and wX i (i = 1,2) are internal variables; yY X is the coupling output of CX (uY ), which is the coupling mapping between the X-axis and Y -axis, while yX Y is the coupling output of CY (uX ), which is affected by the movement of the X-axis. Hence, we obtain yX (k) = yXX (k) + yY X (k),

for X-axis

(31)

yY (k) = yY Y (k) + yX Y (k),

for Y -axis.

(32)

and

Without loss of generality, only the case of X-axis is analyzed in this paper. In order to identify the coupling submodel between Y -axis and X-axis of the stage, just let the input of the X-axis be zero. In this case, the response of the feedforward path of the axis becomes zero, i.e., yXX (k) = 0. At the same time, feed the excitation signal to the Y -axis while measure the corresponding displacement of the X-axis, i.e., yX (k) = yY X (k). Then, the coupling submodel between X-axis and Y -axis, i.e., CX (·) can be estimated based on uY (k) and yX (k). As the micropositioning stage involved with nonlinear devices, the coupling submodel CX (·) of the X-axis is not just a linear function. Therefore, a neural-network-based nonlinear submodel is built to describe the characteristic of the coupling motion of the X-axis. According to the analyses in [31], the main factors affecting the coupling motions are the input voltage and velocity output of another axis in the micropositioning stage. Thus, a neural network to describe the coupling motion of the X-axis can be constructed. Therefore, we have yˆY X (k) = υ2 {(Q2q ×1 )T · υ1 [(Q1α ×q )T · Ωα ×1 ]}

(33)

where the input space Ω = [U Y , ΔY Y ]T , which consists of the input voltage {uY (k − i1 ) | i1 = 1, . . . , α1 } and the difference of the displacement, i.e., {ΔyY (k − i2 ) = yY (k − i2 ) − yY (k − i2 − 1) | i2 = 1, . . . , α2 }, of the Y -axis, where both α1 and α2 are the time lags. Moreover, yˆY X (k) is the output of the neural network and the neuron number of the input space, hidden layer, and output space are α (α = α1 + α2 ), q, and 1, respectively. Moreover, Q1α ×q is the weight matrix connecting the input space Ωα ×1 and the hidden layer, while Q2q ×1 is the weight matrix connecting the hidden layer and the output yˆY X (k). In

In this paper, a tracking control scheme combined with coupling compensation strategy is proposed to handle the control of the micropositioning stage. To simplify the description, only the case of X-axis is discussed. In the decoupling scheme, a nonlinear feedforward decoupling compensator is constructed to compensate the interaction caused by the Y -axis. Simultaneously, the tracking control strategy is applied in order to suppress the effect of hysteresis effect and coupling compensation residual as well as to force the output of the stage to track the reference trajectory. A. Inverse Sandwich Model With Hysteresis In order to design a tracking controller, an inverse sandwich model with hysteresis is constructed. Similar to the modeling methods shown in Section III, for the sandwich model of the X-axis, the corresponding inverse model should satisfy the following relation: ˆ −1 {H ˆ −1 [L−1 (yX (k))]} uX (k) = L X X2 X1

(34)

ˆ −1 (·), L ˆ −1 (·), and L ˆ −1 (·) are the inverse submodels where H X X1 X2 of the hysteresis and two linear subsystems, respectively. ˆ −1 (·) can be directly obtained by ˆ −1 (·) and L In this case, L X1 X2 LX 1 (·) and LX 2 (·), respectively, i.e., ˆ −1 (·) : uX (k) = − L X1

n1b

ˆb1j uX (k − j)

j =1

+

n1a

a ˆ1i vX 1 (k + q1 − i)

(35)

i=1

and ˆ −1 (·) : vX 2 (k) = − L X2

n2b

ˆb2j vX 2 (k − j)

j =1

+

n2a

a ˆ2i yX (k + q2 − i).

(36)

i=1

ˆ −1 (·) has the same Moreover, the inverse neural model H X structure as (29) but the input vector is ¯ I = {ˆ vX 2 (k − i)|i = ¯ = {ˆ 1, . . . , m} and output vector is O vX 1 (k)}. Similarly, the expanded input space of the inverse hysteresis model consists of an inverse hysteretic operator generated by the superposition of backlash operators. ¯ m ×1 ], and In this case, the expanded space is Σ = [I¯ m ×1 , Ψ we have ¯ TX 3 · σ1 (P ¯ TX 1 · Σ) + P ¯ TX 4 · σ2 (P ¯ T2 · OX ). (37) vX 1 (k) = P It is noted that this neural model can also be trained by the Levenberg–Marquarqt algorithm.


827

Theorem 2: Suppose uX f =

GX (z −1 ) eX FX (z −1 )

(43)

where eX = rX − yX is the system error of the X-axis, X gX i = 0, and FX (z −1 ) = fX (z −1 )Δ, where GX (1) = ni=0 −1 Δ = 1 − z is the incremental operator and all the roots of fX (z −1 ) are within the unit circle. If we choose ˆ −1 (H ˆ −1 (L ˆ −1 (.))) RX (.) = L X1 X X2

Fig. 6.

Parameter-estimation results of the linear subsystems.

B. Control Strategy Based on the architecture shown in Fig. 5, we have yX = PX (uX ) + CX (uY ), for X-axis yY = PY (uY ) + CY (uX ),

for Y -axis

(38)

where Pl (ul ) = Ll2 (Hl (Ll1 (.))), {l = X, Y } represents the relationship between the input ul and output yl . For the case of X-axis, in order to design a control strategy to handle the compensation of coupling effect and guarantee the tracking of the reference trajectory, we compose the following assumptions. Assumption 2: For the model residual βX =

ˆ X 2 (H ˆ X (L ˆ X 1 (.))) LX 2 (HX (LX 2 (.))) − L ˆ X 2 (H ˆ X (L ˆ X 1 (.))) L

(39)

there exists a bound βx > 0, such that |βX | ≤ βx . Assumption 3: For the coupling model error αC X = CX (uY ) − CˆX (uY )

(40)

a bound αX > 0exists, such that |αC X | ≤ αX . Remark 4: Both Assumption 2 and Assumption 3 are composed for the design of robust control strategy. They are the up bounds of the modeling errors of sandwich model and the coupling submodel, respectively. In engineering practice, the up bounds of the modeling residuals can be estimated based on ratios between the maximum values of the model validation errors and the steady-state gains of the obtained models. Assumption 4: There exists a continuous transformation RX (.), such thatPX RX = 1 and the control strategy uX (k)) uX (k) = RX (ˆ

(41)

u ˆX = rX − CˆX (uY ) + uX f

(42)

where

where CˆX (uY ) is the model to describe the coupling effect caused by the movement of Y -axis, while uX f is the tracking control output. Then, we compose the following theorem for the design of the corresponding tracking control strategy.

(44)

then the system output yX of the X-axis can track the reference trajectory rX accurately. Proof: The corresponding proof of the theorem is shown in Appendix B. Consider the effect of model mismatch. The corresponding robust design of the control strategy will be shown in Theorem 3. Theorem 3: If the system illustrated by (38) is controlled by the strategy described by (41)–(44) and the model errors are subject to Assumptions 2 and 3, the robust stability of the system can be achieved by choosing a proper function, i.e., GX (z −1 )/FX (z −1 ), to satisfy ∗ ∗ GX (ω ) ≤ 1, ω ∗ ∈ (−π, π) |MX (ω )| (45) FX (ω∗) where MX = 1 + βx , ω∗ = e−2π /T and T is the sampling period. Proof: The proof of Theorem 3 is shown in Appendix C. Based on Theorems 2 and 3, the strategy for both decoupling and tracking control can be designed even though there exist both bounded model uncertainty and coupling model residual. Furthermore, we can design the corresponding decoupling and tracking control strategy for Y -axis based on the methods similar to the case of X-axis. VI. EXPERIMENT The proposed identification and control scheme is applied to an XY micropositioning stage with piezoelectric actuators (MPT2MRL005) with the setup shown in Fig. 1. In this system, the nominal displacement is within 0–50 μm and the input voltage is located in the range of 0–10 V, while the digital controller consists of a PCI-1716 L card (Advantech Co.) for 16 bits A/D converter and a PCI-1723 card (Advantech Co.) for 16 bits D/A converter. For the high-speed positioning stage, the response frequency of the piezoceramic actuator is very fast, which can be up to several thousands hertz [8], [24]. In order to derive more precise performance, the high-speed sampling should be implemented. Thus, the corresponding control strategy is programmed by Borland C. In this case, the hardware addresses and registers can be directly operated. Based on the requirement of the system, the sampling frequency is selected as 30 kHz. A. Identification of the Sandwich Systems With Hysteresis It is known that, in this system, the output of each axis includes two parts, i.e., the output of the feedforward path and the output

828


of the coupling path that represents the interactive action caused by the movement of another axis. In order to identify the model of the feedforward path that can be described by the sandwich model with hysteresis, just let the input of another axis be zero. In this case, it can get rid of the interactive effect caused by the movement of another axis. For the structures of the linear submodels of the sandwich model with hysteresis, it is known that the piezoelectric actuator is driven by an amplifier–filter circuit that can be described by a first-order linear submodel. Moreover, the flexure hinge device that is, in fact, a spring– damping system that can be described by a second-order linear submodel. Hence, we can identify the model of the X-axis PX (.) free of interaction, first. Suppose the behavior of the feedforward path of the X-axis is represented by L1 (·) : w1 (k) = a11 w1 (k − 1) + b11 u(k − 1)

Fig. 7.

Model validation result of hysteresis on X -axis.

(46)

H(·) : w2 (k) = N N [w1 (k − i1 ), Ψ(w1 (k − i1 )), w2 (k − i2 )] (47) L2 (·) : y(k) = a21 y(k − 1) + a22 y(k − 2) + w2 (k − 1) (48) where i1 = 0, 1 and i2 = 1, 2. It implies that L1 (·) and L2 (·) are described by the first-order and second-order linear subsystems, respectively, while the hysteresis block H(·) is modeled by a neural network based on the expanded input space, which consists of four input neurons (m = n = 2), 12 hidden neurons (9 neurons for the hysteretic part, and three neurons for the dynamic part, respectively), and one output neuron. Then, a monotonic and independently incremental signal u1 (k) is designed to excite the system. Choose n(k) as a white noise signal with zero mean value and variance σ = 0.02 and the offset ap = 0.5 × |nm in |. Then, the degenerated sandwich model is reconstructed by (8)–(10), so that the RGIA is applied to the estimate of the linear submodels L1 (·) and L2 (·) with the initialization: P (0) = 106 · I, γ(0) = 0.001, and θ(0) = [0.001, . . . , 0.001]T . The corresponding stop criterion of estimation is

e2 (k − 1) + e2 (k) < ε

(49) −6

where e(k) is the modeling error and ε is chosen as 10 . The identification results are shown in Fig. 6. After that, we applied 2π800e−20T k T k − π −10T k u2 (k) = 4.5e sin + 1.0 + 0.3 2 (50) to the excitation of the stage to identify the inherent hysteresis ˆ2 (k) are rein the PEA. Then, the internal signal w ˆ1 (k) and w ˆ 1 (·) constructed by the previously identified linear submodels L ˆ 2 (·). Thus, the neural-network-based hysteresis submodel and L ˆ2 }, while the shown in (29) is trained by the data pairs {w ˆ1 , w hysteretic operator Ψ(·) is generated by the superposition of ten backlash operators. After 32 epochs, the training of neural network is finished and MSE = 1.4236e-006. To test the obtained neural hysteresis

submodel, an input for model validation is designed as uval (k) = 3.0e−15T k [sin(2π300e−10T k T k+π/2)+1.0]+1.0. (51) The corresponding model validation result is shown in Fig. 7. It is obvious that the obtained hysteresis submodel can accurately model the behavior of the hysteresis existing in the PEA of the X-axis. Similarly, we can also derive the model of Y -axis PY (.) which is free of coupling effect caused by the motion of X-axis. B. Identification of the Coupling Submodel Let input uX (k) on the X-axis be zero but input uY (k) on the Y -axis be a white noise signal with the amplitude ranged between 0.3 and 9.7 V. Measure the corresponding output of the X-axis, i.e., yX (k). Based on the measured dataset {uY (k), yY (k), yX (k)}, the corresponding neural-network-based submodel described by (33) is trained to mimic the coupling behavior happened on the X-axis. Here, the input space of the neural coupling model is Ω = [UY , ΔYY ]T . There are six input neurons (α1 = α2 = 3), 14 hidden neurons, and one output neuron in the architecture of this submodel. The Levenberg– Marquarqt algorithm is implemented to train the neural network model. The MSE of the training procedure is 1.6339e-007. On the other hand, the prediction of the obtained coupling submodel CX (·) and the corresponding prediction error are illustrated in Fig. 8, while the input used for model validation is a pseudorandom binary sequence with the amplitude ranged between 3 and 7 V. Noted that the MSE of the coupling model prediction is 8.9393e-006. Therefore, the obtained neural network model well matches the coupling behavior caused by the motion of the Y -axis. In terms of the same way, we can also obtain the coupling submodel for Y -axis. C. Decoupling and Tracking Control of the Stage In this section, the proposed decoupling and tracking control scheme applying to the control of the XY micropositioning stage will be illustrated. The reference trajectory is a rectangular func-


Fig. 8. data].

829

Validation of the coupling submodel [(- - -) real data, (—) prediction

tion. In the control system, the controller output is constrained between 0 and 10 V. Based on the obtained sandwich model with hysteresis, the corresponding inverse sandwich models of both X-axis and Y -axis are established based on (35)–(37). In each sampling period, the predictive outputs of the coupling submodels, i.e., yˆY X (k) and yˆX Y (k) are implemented. According to (33) and (41)–(44), we obtain the corresponding decoupling compensators CX (·) and CY (·), as well as the corresponding tracking controllers uX f and uY f , respectively. For the design of the tracking controllers, based on Theorems 2 and 3, the corresponding controllers are chosen as

Fig. 9. Tracking control response on XY plane. (a) Proposed control scheme [(—) setpoint, (- - -) displacement of the stage]. (b) Tracking control scheme without decoupling compensation [(—) setpoint, (- - -) displacement of the stage].

GX (z −1 ) = 3.5443 − 5.1833z −1 + 0.1634z −2 , fX (z −1 ) = 1, GY (z fY (z

−1

−1

) = 3.5317 − 6.9404z

D. Comparison of Hammerstein System With Hysteresis −1

+ 0.1539z

−2

,

and

) = 1,

respectively. In this system, the reference signals of both X-axis and Y -axis are the 20 μm × 30 μm square trajectories. The corresponding control result is illustrated in Fig. 9(a) and the maximum tracking error is 0.04 μm. On the other hand, Fig. 9(b) presents the control result with the tracking control strategy without decoupling compensation. It is noted that the maximum tracking error of this case is 0.11 μm. Obviously, the proposed scheme has obtained much better control performance and smaller control error by comparing with the method without decoupling consideration. For the micropositioning stage with piezoelectric actuators, the hysteresis inherent in the piezoelectric devices may deteriorate the performance of the stage. Therefore, the compensation of the hysteretic effect existing in the piezoelectric actuators is an important issue to derive satisfactory positioning performance.

Furthermore, in order to compare the proposed method with the tracking and decoupling control based on the Hammerstein model with hysteresis, the corresponding experimental results on the X-axis are shown in Fig. 10. In this case, the dynamic effect of the amplifier–filter circuit is ignored. Hence, it is simplified as a proportional gain. Thus, the stage is modeled by a Hammerstein model with hysteresis. In this experiment, the reference signal is r(k) = 6.0 × [sin(2π × 100 × k) + 1] + 2.0

(μm).

(52)

The control scheme based on the Hammerstein model are designed as GX (z −1 ) = 6.3263 − 8.107z −1 + 3.795z −2 and fX (z −1 ) = 1. From Fig. 10, it can be seen that the corresponding maximum tracking error is 0.3 μm for the control method based on the Hammerstein model and 0.1 μm for the strategy based on the sandwich model. Based on the results of this experiment, it is also known that the dynamic effect of the amplifier–filter circuit cannot be

830


As both nm in and ap are positive constant offsets, in negative direction, the maximum amplitude of{Δu1 (k)} is determined by {n(k)}, i.e., amp− {Δu1 (k)} = amp− {n(k)} = nm in > 0.

(A.3)

Hence, the minimum value of Δu1 (k) should be min{Δu1 (k)} = E{Δu1 (k)} − amp− {Δu1 (k)} = ap > 0. (A.4) It is clear that Δu1 (k) is a positive sequence that makes u1 (k) a monotonically increased signal. Note that L1 (·) usually has the characteristic of a low-pass filter. Then, in terms of conditions A1 and A2 of Assumption 1, we have Fig. 10. Control error of X -axis. The tracking error [(—) Hammerstein model, (- - -) Sandwich model].

neglected in the case of high precise tracking control. Otherwise, it will lead to larger tracking error. Therefore, the sandwich model with hysteresis will be more suitable to the design of tracking control for the high precise and fast positioning stage. VII. CONCLUSION In this paper, a sandwich model with hysteresis is constructed to describe the characteristic of each axis of the micropositioning stage with piezoelectric actuators. In this sandwich model, the amplifier–filter circuit and work platform are modeled by linear dynamic submodels, respectively, and a neural network is used to describe the hysteresis behavior in piezoelectric actuator. Moreover, we have also constructed a neural submodel to describe the coupling motion caused by another axis. Finally, a tracking control strategy combined with the decoupling scheme is proposed to compensate the effects of both hysteresis and interactive disturbances. The advantage of the proposed decoupling control scheme is that it does not rely on the inverse coupling model. Therefore, it is easier to be implemented. Then, the robust design is discussed to handle the influence of model mismatch and the coupling compensation error. The experiments on an XY micropositioning stage have demonstrated that the proposed method is much better in performance than the tracking control without coupling compensation. APPENDIX A Proof: First, it is proved that u1 (k) satisfies the first condition of Definition 1. From (6), it is known that n(k) is an i.i.d. white noise sequence with zero mean value. Thus, it leads to E{Δu1 (k)} = nm in + ap > 0

(A.1)

where E{.} denotes the expectation operation. Define the maximum amplitude of {n(k)} in negative direction as amp− {n(k)} = nm in > 0.

(A.2)

E{Δw1 (k)} = k1 E{Δu1 (k)} = k1 (|min{n(k)}| + ap ) > 0. (A.5) Then, amp_{Δw1 (k)} of Δw1 (k), the maximum amplitude of {Δw1 (k)} in negative direction will also be determined by {n(k)}, i.e., amp− {Δw1 (k)} = ko1 amp− {Δu1 (k)} ≤ Ko1 |min{n(k)}| (A.6) where ko 1 and Ko 1 are defined as the expanded amplitude scale of L1 (·) and its corresponding maximum value, respectively [28]. Then, we have min{Δw1 (k)} = E{Δw1 (k)} − amp− {Δw1 (k)} ≥ k1 (|min{n(k)}| + ap ) − Ko1 |min{n(k)}| = (k1 − Ko1 )|min{n(k)}| + k1 ap .

(A.7)

If L1 (·) is an overdamped system, we have k1 − Ko1 > 0. Then, it leads to min{Δw1 (k)} > 0. On the other hand, if L1 (·) is an underdamping system, Ko 1 approximates the maximum peak of the response. In this case, ko 1 will be larger than k1 . Thus, min{Δw1 (k)} is probably a negative value. In order to ensure min{Δw1 (k)} to be larger than zero, the following inequality should be satisfied: Ko1 − 1 · |min{n(k)}|. (A.8) ap ≥ k1 If the variance σ of Δu1 (k) is properly selected, then we have min {Δw1 (k)} > 0. It implies that Δw1 (k) is positive and w1 (k) is also monotonic. Hence, the multivalued mapping of the hysteresis embedded in the system can be degenerated into a polynomial shown in Fig. 3(b). Then, we prove that u1 (k) will also satisfy the second condition of definition 1. Suppose L1 (·) is excited by the signal shown in (5) and (6). Based on the results obtained previously, it is known that w1 (k) will be a monotonic function and Δw1 (k) will be an independent sequence, respectively. On the other hand, L1 (·) can be considered a low-pass filter. Assume that the cutoff frequency of the filter is ω c 1 and the bandwidth of Δw1 (k) is 0–ω c 1 . Note that H(·) has been degenerated into a polynomial which is a static and one-to-one mapping when w1 (k) passes through


it. Then, w2 (k) is also a monotonic response and Δw2 (k) is an independent sequence, respectively. Nevertheless, due to the impact of high-order harmonics, the bandwidth of Δw2 (k) is much greater than that of Δw1 (k), so that Δw2 (k) has a wide enough spectrum to excite the output linear subsystem L2 (·). Therefore, u1 (k) designed based on (5) and (6) also satisfies the second condition of Definition 1. Therefore, if the variance σ of Δu1 (k) is properly selected and the offset ap satisfies formula (7), then u1 (k) designed according to (5) and (6) is a degenerating input for the sandwich systems with hysteresis. APPENDIX B The proof of Theorem 2 is as follows. Based on Assumptions 3 and 4, combining (41), (42), and (44) results in ˆ −1 (H ˆ −1 (L ˆ −1 (rX − CˆX (uY ) + uX f ))). uX = L X1 X X2

(B.1)

Substituting (B.1) in (38) and considering (40) and (43) leads to yX (k) = rX + αC X +

GX (z −1 ) eX (k). fX (z −1 )Δ

(B.2)

Rearranging (B-2) results in 0 = αC X fX (z −1 )Δ + [fX (z −1 )Δ + GX (z −1 )]eX (k). (B.3) Let z −1 → 1. It yields GX (1)eX (1) = 0.

(B.4)

As GX (1) = 0, we have z −1 →1

yX −→ rX . APPENDIX C Suppose the model error of the X-axis exists while the coupling compensation error cannot be neglected. Moreover, the model error is subject to Assumption 2 and the coupling compensation error is subject to Assumption 3. Considering (38) and (41)–(44) leads to βX (z −1 )CˆX (uY ) − βX (z −1 )rX (k) − αX (z −1 ) . 1 + (1 + βX (z −1 ))(GX (z −1 )/FX (z −1 )) (C.1) The equivalent characteristic equation is eX (k) =

1 + (1 + βX (z −1 ))

GX (z −1 ) = 0. FX (z −1 )

(C.2)

According to the small-gain theorem [32], a proper GX (z −1 )/FX (z −1 ) can be selected to satisfy ∗ ∗ GX (ω ) |1 + βX (ω )| FX (ω ∗ ) GX (ω ∗ ) < 1, ω ∗ ∈ (−π, π) (C.3) ≤ |1 + β| FX (ω ∗ ) where ω ∗ = e−π /T .

831

REFERENCES [1] Y. Li and Q. Xu, “Development and assessment of a novel decoupled XY parallel micropositioning platform,” IEEE/ASME Trans. Mechatronics, vol. 15, no. 1, pp. 125–135, Feb. 2010. [2] Q. S. Xu, Y. M. Li, and N. Xi, “Design, fabrication, and visual servo control of an XY parallel micromanipulator with piezo-actuation,” IEEE Trans. Autom. Sci. Eng., vol. 6, no. 4, pp. 710–719, Oct. 2009. [3] Y. K. Yong, S. 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. [4] A. J. Fleming and K. K. Leang, “Integrated strain and force feedback for high-performance control of piezoelectric actuators,” Sens. Actuators A, Phys., vol. 161, pp. 256–265, 2010. [5] Q. S. Xu and Y. M. Li, “Dahl model-based hysteresis compensation and precise positioning control of an XY parallel micromanipulator with piezoelectric actuation,” J. Dyn. Syst., Meas., Control, vol. 132, no. 4, pp. 425– 437, 2010. [6] G. R. Jayanth and C. H. Menq, “Modeling and design of a magnetically actuated two-axis compliant micromanipulator for nanomanipulation,” IEEE/ASME Trans. Mechatronics, vol. 15, no. 3, pp. 360–370, Jun. 2010. [7] S. P. Salisbury, R. Ben Mrad, D. Waechter, and S. E. Prasad, “Design, modeling and closed loop control of a complementary clamp piezoworm stage,” IEEE/ASME Trans. Mechatronics, vol. 14, no. 6, pp. 724–732, Dec. 2009. [8] R. J. E. Merry, N. C. T. de Kleijn, M. J. G. van de Molengraft, and M. Steinbuch, “Using a walking piezo actuator to drive and control a high-precision stage,” IEEE/ASME Trans. Mechatronics, vol. 14, no. 1, pp. 21–31, Feb. 2009. [9] B. Bhikkaji and S. O. Reza Moheimani, “Integral resonant control of a piezoelectric tube actuator for fast nanoscale positioning,” IEEE/ASME Trans. Mechatronics, vol. 13, no. 5, pp. 530–537, Oct. 2008. [10] G. Tao and P. V. Kolotovic, “Adaptive control of plants with unknown hysteresis,” IEEE Trans. Autom. Control, vol. 40, no. 2, pp. 200–212, Feb. 1995. [11] X. Chen, C.-Y. Su, and T. Fukuda, “Adaptive control for systems preceded by hysteresis,” IEEE Trans. Autom. Control., vol. 53, no. 4, pp. 1019– 1025, May 2008. [12] P. Ge, “Modelling and control of hysteresis in piezoceramic actuator,” Ph.D. dissertation, Univ. Rhode Island, Kingston, RI, 1996. [13] H. Y. Chen and J. W. Liang, “Adaptive sliding control with self-tuning fuzzy compensation for a piezoelectrically actuated X–Y table,” Control Theory Appl., vol. 4, no. 11, pp. 2516–2526, 2010. [14] Y. Li and Q. Xu, “Precise tracking control of a piezoactuated micropositioning stage based on modified Prandtl-Ishlinskii hysteresis model,” in Proc. 6th IEEE Int. Conf. Autom. Sci. Eng., Aug. 2010, pp. 692–697. [15] Al Janaideh, M. Rakheja, and C.-Y. Su, “An analytical generalized Prandtl–Ishlinskii model inversion for hysteresis compensation in micropositioning control,” IEEE/ASME Trans. Mechatronics, vol. 16, no. 4, pp. 734–734, Aug. 2011. [16] Q. Yao, J. Dong, and P. Ferreira, “Design, analysis, fabrication and testing of a parallel-kinematic micropositioning XY stage,” Int. J. Mach. Tools Manuf., vol. 47, no. 6, pp. 946–961, 2007. [17] S. Awtar and A. H. Slocum, “Constraint-based design of parallel kinematic XY flexure mechanisms,” ASME J. Mech. Des., vol. 129, no. 8, pp. 816– 830, 2007. [18] S. Polit and J. Y Dong, “Development of a high-bandwidth XY nanopositioning stage for high-rate micro-/nanomanufacturing,” IEEE/ASME Trans. Mechatronics, vol. 16, no. 4, pp. 724–733, Aug. 2011. [19] X. Chen, S. Zhang, X. Bao, and H. Zhao, “Master and slave control of a dual-stage for precision positioning,” in Proc. 3rd IEEE Int. Conf. Nano/Micro Eng. Molecular Syst., Jan. 6–9, 2008, Sanya, China, pp. 583– 587. [20] J. Zheng, G. Guo, and Y. Wang, “Feedforward decoupling control design for dual-actuator system in hard disk drives,” IEEE Trans. Magn., vol. 4, no. 4, pp. 2080–2082, Jul. 2004. [21] A. Daniele, S. Salapaka, M. V. Salapaks, and M. Dahleh, “Piezoelectric scanners for atomic force microscope: Design of lateral sensors, identification and control,” in Proc. Amer. Control Conf., San Diego, CA, 1999, pp. 253–257. [22] S. Tien, Q. Zou, and S. Devasia, “Iterative control of dynamics-couplingcaused errors in piezoscanners during high-speed AFM operation,” IEEE Trans. Control Syst. Technol., vol. 13, no. 6, pp. 921–931, Nov. 2005. [23] A. J. Fleming, “Nanopositioning system with force feedback for highperformance tracking and vibration control,” IEEE/ASME Trans. Mechatronics, vol. 15, no. 3, pp. 433–447, Jun. 2010.

832

[24] U. X. Tan, W. T. Latt, C. Y. Shee, and W. T. Ang, “A low-cost flexurebased handheld mechanism for micromanipulation,” IEEE/ASME Trans. Mechatronics, vol. 16, no. 4, pp. 773–777, Aug. 2011. [25] J. Vörös, “Parameter identification of wiener systems with multisegment piecewise-linear nonlinearities,” Syst. Control Lett., vol. 56, no. 2, pp. 99– 105, 2007. [26] F. Giri, Y. Rochdi, F. Z. Chaoui, and A. Brouri, “Identification of Hammerstein systems in presence of hysteresis-backlash and hysteresis-relay nonlinearities,” Automatica, vol. 44, no. 3, pp. 767–775, 2008. [27] Y. Tan and R. Dong, “Recursive identification of sandwich systems with dead-zone and application,” IEEE Trans. Control Syst. Technol., vol. 17, no. 4, pp. 945–951, Jul. 2009. [28] Y. Xie, Y. Tan, and R. Dong, “Identification of sandwich systems with hysteresis based on two-stage method,” in Proc. Int. Conf. Modelling, Identification Control, 2010, Okayama, Japan, pp. 370–375. [29] X. Zhao and Y. Tan, “Modeling hysteresis and its inverse model using neural networks based on expanded input space method,” IEEE Trans. Control Syst. Technol., vol. 16, no. 3, pp. 484–490, May 2008. [30] M. T. Hagan and M. B. Menhaj, “Training feedforward networks with the Marquardt algorithm,” IEEE Trans. Neural Netw., vol. 5, no. 6, pp. 989– 993, Nov. 1994. [31] M. Tahani and M. Mirzababaee, “Higher-order coupled and uncoupled analyses of free edge effect in piezoelectric laminates under mechanical loadings,” Mater. Design, vol. 30, no. 12, pp. 2473–2482, 2009. [32] John Dorsey, Continuous and Discrete Control Systems. New York: McGraw-Hill, 2001. [33] R. Dong and Y. Tan, “On-line identification algorithm and convergence analysis for sandwich systems with backlash,” Int. J. Control Autom. Syst., vol. 9, no. 3, pp. 588–594, 2011. [34] X. Zhang, “Neural networks based identification and compensation of rate-dependent hysteresis in piezoelectric actuators,” Physica B: Phys. Condensed Matter, vol. 405, no. 12, pp. 2687–2693, 2010. [35] X. Zhang and Y. Tan, “A hybrid model for rate-dependent hysteresis in piezoelectric actuators,” Sens. Actuators A, Phys., vol. 157, no. 1, pp. 54– 60, 2010. [36] R. Dong and Y. Tan, “On-line modeling of sandwich systems with dead zone,” in Proc. Int. Conf. Control Autom. Syst., Seoul, Korea, 2007, pp. 1529–1533. [37] R. Dong, Y. Tan, and H. Chen, “A neural networks based model for rate-dependent hysteresis for piezoceramic actuators,” Sens. Actuators A, Phys., vol. 143, no. 2, pp. 370–376, 2008.


Yangqiu Xie received the M.S. degree from Guilin University of Electronic Technology, Guilin, China. He is currently working toward the Ph.D. degree jointly in the College of Information, Mechanical and Electrical Engineering, Shanghai Normal University, Shanghai, China, and the School of Electronic Engineering, Xidian University, Xi’an, China. His research interests include mechatronics, modeling, and control of nonlinear systems.

Yonghong Tan received the Ph.D. degree in electrical engineering from the University of Ghent, Ghent, Belgium. He was a Postdoctoral Fellow at Simon Fraser University, Vancouver, BC, Canada. He was a Visiting Professor at Colorado State University, Fort Collins, from January to July 2001, and at Concordia University, Montreal, QC, Canada, from November 2001 to January 2002. He held professorships at Guilin University of Electronic Technology and the University of Electronic Science and Technology of China. He is currently a Full Professor in the College of Information, Mechanical and Electrical Engineering, Shanghai Normal University, Shanghai, China. His research interests include modeling and control of nonlinear systems, mechatronics, and intelligent control.

Ruili Dong received the Ph.D. degree from Shanghai Jiaotong University, Shanghai, China. She is currently an Associate Professor in the College of Information, Mechanical and Electrical Engineering, Shanghai Normal University, Shanghai. Her research interests include mechatronics and identification and control of nonlinear systems.