Adaptive inverse control for the pickup head flying height of near-field optical disk drives

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Adaptive inverse control for the pickup head flying height of near-field optical disk drives

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2006 Smart Mater. Struct. 15 1632

(http://iopscience.iop.org/0964-1726/15/6/015)

Smart Mater. Struct. 15 (2006) 1632–1640 doi:10.1088/0964-1726/15/6/015

Adaptive inverse control for the pickup

head flying height of near-field optical disk

drives

C C Hsiao, T S Liu

and S H Chien

Department of Mechanical Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan, Republic of China

E-mail:[email protected]

Received 10 July 2006, in final form 24 August 2006

Published 3 October 2006

Online at

stacks.iop.org/SMS/15/1632

Abstract

Near-field optical disk drives represent a promising technique for optical data

recording to achieve an even larger storage density and capacity than

DVD-ROM and Blu-ray disk drives. To realize near-field optics, unlike

conventional optical disk drives, a flying pickup head is required. In this

study, the pickup head consists of a suspension arm, a slider and a bimorph

piezoelectric bender between the suspension arm and the slider. The dynamic

model of the pickup head is identified using measurement, but the whole

dynamics including both pickup head and interface dynamics between the

disk and slider is unmodeled. Adaptive inverse control of robustness is

developed to track the vibratory deformation of rotating optical disks, so that

the flying height of a pickup head can remain stable in the presence of

modeling error. Experimental results demonstrate that using the proposed

method the pickup head can not only track disk deformation but also

maintain the flying height.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

The storage capacity of optical disk drives has been increasing in the past decade. Although nowadays high density optical storage media like DVD disks and Blu-ray are available, people are still attempting to find a revolutionary way to record a dramatically increased number of data. Near-field optical disk drives (NFODD) represent one potential device [1, 2], for which a capable servo system must be constructed to facilitate reading and writing. Yatsui et al [3] used a pyramidal silicon probe on a slider to perform near-field recording and reading. Kurita et al [4] adjusted the flying height for a magnetic head slider using a piezoelectric micro-actuator. To achieve high precision in focusing for near-field data recording, it is essential to develop servo systems that can maintain a constant height at which the pickup head flies above a rotating disk. Hence in this work we attach a bimorph piezoelectric (PZT) bender to the suspension of a slider to compensate for the 1 _{Author to whom any correspondence should be addressed.}

surface deformation of rotating optical disks. To control the flying height, the PZT bender bends under an applied voltage to maintain a constant height between a rotating disk and the pickup head. However, the PZT bender has inherent hysteresis nonlinearity which will cause imprecision when using conventional linear controllers.

Ge and Jouaneh [5] presented a tracking control approach for a piezoelectric actuator, incorporating a feedforward loop with a feedback controller. Zhou et al [6] dealt with a piezohydraulic actuator with a hybrid model consisting of a dynamic part and a static nonlinear part. Takaishi et al [7] developed a dual-stage controller composed of a voice coil motor and a PZT for precise position tracking.

Owing to the time-varying and nonlinear properties inherent in piezoelectric materials, a robust controller is needed to achieve high accuracy and a fast operating speed. Besides, the dynamics between the pickup head and the rotating disk is hard to obtain in practice. It is impossible for most controllers to achieve high accuracy since the overall system dynamics is unmodeled. Thus, in this paper adaptive inverse control (AIC)

Adaptive inverse control for the pickup head flying height of near-field optical disk drives Suspension arm Pivot PZT VCM Slider

Figure 1. Pickup head including the PZT bender.

is developed to achieve accurate motion control in NFODD. Plett [8] presented AIC for unmodeled linear systems. Kaelin and von Grunigen [9] proposed an approach to decreasing computational complexity and accelerating convergence rate by splitting AIC controllers into a long fixed and a short adaptive filter. Widrow and Plett [10] proposed feedforward control for both linear and nonlinear plant.

In this paper, section2introduces the PZT characteristics and identifies the mathematical model of a pickup head. In section3, an adaptive inverse controller is developed to control the flying height of the pickup head. Experimental results and conclusions are presented in sections4and5, respectively.

2. Pickup head identification

PZT elements have been widely applied as actuators in addition to sensors, and their performance is qualified to generate fast and precise movement. Among various PZT actuators, PZT benders are popular in many structural applications, such as individual blade control in rotorcraft, vibration dampening and positional control. Since a large deformation of the disk surface during disk rotation may well degrade focusing performance in near-field disk drives, in the present pickup head a PZT bender attached to a suspension arm, as shown in figure1, serves as an actuator to maintain the flying height in the presence of vibratory disk deformation.

In this study, a swept-sine method is employed to obtain the dynamic characteristics of the pickup head, and hence its Bode plot, from which a transfer function is determined. As a result, the model of the pickup head is identified as

Gp(s) = {1.779×108s2+2.055×1010s +1.886×1015}{s4+1203s3+1.047×108

+2.657×1010+4.235×1014}−1. (1) At sampling timeTs=0.0001 s, equation (1) can be written in discrete form Gp(z) = 0.7935z3_−0.7306z2_{−0.656z+0.7557} z4−₂._965z3+₃._877z2−₂._762z+₀.₈₈₆₇. (2) 102 ₁₀3 ₁₀4 Frequency (rad/sec) Magnitude (dB) Phase (deg) -30 -20 -10 0 10 20 30 -270 -225 -180 -135 -90 -45 0

Figure 2. Bode plot of the pickup head.

100 101 102 103 104 Frequency (rad/sec) -60 -40 -20 0 20 40 60 Magnitude (dB) Phase (deg) -270 -225 -180 -135 -90 -45 0

Figure 3. Bode plot of the pickup head after compensation.

Its Bode plot is depicted in figure2, where resonances at 2000 and 10 000 rad s−1exist. Hence, a compensator is designed to filter out the resonances. Cascading the compensator with the pickup head and utilizing feedback network results in a system with better dynamics. The compensator is designed as

Cp(z) = {0.05(z −0.9233)(z2−1.027z+0.9057) × (z2_{−1.938z+0.9794)}{(z −0.9)}

× (z2_{−1.883z+0.9878)(z}2_{−1.97z+0.9702)}}−1_. (3) It consists of proportional-integral (PI) control and two notch filters, which aim to suppress both resonances. The Bode plot of the compensated system is depicted in figure3, where both resonances have been suppressed. Figure2depicts the dynamics of the pickup head, but it does not represent the whole system dynamics. The whole system dynamics should include the dynamics of the pickup head and the interface dynamics between the disk and slider; i.e. the air bearing effect. Therefore, the whole system dynamics is unmodeled. Although the pickup head modelGp(z)does not include the air bearing effect, in this study the air bearing effect is treated as a disturbance and will be taken into account in the process of modeling AIC.

Σ Σ Σ ) (k x ) (k d Disturbance n(k) ) ( ˆ k y

Modeling error e(k) ) (k y Σ ) ( Copy z Pˆ ) ( ˆ Copy z C ) ( ˆ Controller z C ) ( Reference Model z M ) ( Adaptive Model z Pˆ

System error es(k) ) (k x (z) GP Pickup head (z) CP Compensator Command input Output

Figure 4. Adaptive inverse control.

3. Adaptive inverse control

AIC is robust and adjusts itself to optimize the overall dynamic response. It is implemented by using an adaptive filter containing a tapped delay line with adjustable parameters. Parameters are adjusted to minimize a least mean square (LMS) error function. Figure 4 depicts the block diagram of AIC [11], which consists of two main parts: one is adaptive modeling, i.e. nonlinear identification, and the other is inverse control. With good adaptive modeling, an LMS based algorithm can be used to train the AIC controller. Based on the LMS algorithm in minimizing the mean square error, adaptive modeling aims to estimate the model of the real plant. 3.1. Adaptive modeling

As depicted in figure4, the transfer function of a plant isP(z)

whose output is y(k). A discrete-time additive disturbance

n(k)appears at the output where the discrete time index isk. The overall measured output isd(k), given by

d(k) = y(k) + n(k). (4)

The impulse response of the adaptive model ˆP(z)in the vector form is written as

ˆP(z) = [ ˆp1 ˆp2· · ·]T. (5) The adaptive model ˆP(z)yields an output responseˆy(k), being the convolution of its input signal with its impulse response:

ˆy(k) = x(k) ∗ ˆp(k). (6)

The adaptive model ˆP(z)is an estimation of the plant P(z)

aiming to yield a response ˆy(k)as close as possible tod(k). The plant P(z) consists of pickup head model Gp(z), its compensator Cp(z) and disturbance n(k). Taking the z -transform for equation (6) gives

ˆY(z) = X(z) ˆP(z). (7)

Parameters in the adaptive model ˆP(z)are adjusted by adaptive algorithms to minimize the modeling errore(k)in the sense of the mean square error. Defining a performance function called

Jms, the mean square error is written as

Jms= E 

e2(k) (8)

where E{·}denotes an expectation operator. Minimizing the mean square error leads to

Jms∼= Jmin. (9) The minimization procedure [12] using the mean square error will be described in the next subsection.

3.1.1. Least mean square. The objective of the adaptive model ˆP(z)is to find an optimal weightingW0_{that minimizes} the performance functionJ, the expected value of the square error. Defining the modeling error as

e(k) = d(k) − ˆy(k) (10)

the performance function can be written as

J = Ee2(k)= Ed(k) − ˆy(k)2 = Ed2(k)−2Ed(k) · ˆy(k)+ Eˆy2(k). (11) Since ˆy(k) = N−1  l=0 x(k − l)wl= XT(k)W, (12) whereN is the number of weights and

X(k) = [x(k)x(k −1)x(k −2) · · · x(k − N +1)]T (13)

W= [w0w1· · · wN−1]T. (14) Equation (11) can be rewritten as

J = Ed2(k)−2WTE{d(k)X (k)}

+ WT_E_X(k)XT_(k)_W∗_{. (15)}

For convenience, we define the expected power ofd(k)as

D ∼= Ed2(k). (16) The ensemble autocorrelation matrix ofx(k)is expressed by

R ∼= EX(k)XT(k). (17) The ensemble average cross-correlation vector is written as

P ∼= E {d(k)X (k)} . (18) To makeD,PandRtime invariant,d(k)andx(k)are assumed stationary, in a wide sense. Accordingly,

Adaptive inverse control for the pickup head flying height of near-field optical disk drives

J = D −2WT_{P+ W}T_{RW. (19)} An optimal weighting is found when

∇wJ = −2P+2RW ∼=0. (20) This leads to

W0= R−1P. (21) In such an optimal condition, the performance function in equation (19) is a minimum; i.e.

Jmin= D −2W0

T

P+ W0TRW0= D − W0TP. (22) LetWbe the actual weighting, related to the optimal weighting

W0_by

W= W0+ V. (23) Substituting equations (22) and (23) into equation (19) yields

J = Jmin+ VTRV (24) i.e. if V = W − W0 ∼= 0, the performance function will be minimized. With recursive training, this can indeed be done. Nevertheless, some stability conditions must be satisfied to guarantee convergence.

3.1.2. FIR filter based on the gradient descent approach. A gradient descent approach to finding the optimal weighting is expressed by

W(k +1) = W (k) − μ∇wJ|W=W(k) (25)

where μ represents the updating step size. Different from equation (20), differentiation of equation (24) yields another form of gradient

∇wJ =2RV. (26)

We define

R= QQ−1 (27)

 =diag [λ1 λ2 · · · λN] (28)

wheredenotes an eigenvalue matrix ofRin a diagonal form, and Qis aN-by-N matrix, formed by eigenvectors ofR. We define a transformed version ofVas

V = QV. (29) Equally, the transformation equation (29) can be applied to the weight vector

W= QW. (30) Accordingly, equation (24) becomes

J = Jmin+ VTV. (31) Using equations (23), (26), (27) and (31), equation (25) becomes

V(k +1) = (I −2μ) V(k). (32)

Solving the difference equation (32) gives

V(k) = (I −2μ)kV(0) (33)

V(0) = W(0) − W0(0) (34) whereV(0)is an initial condition. To ensure convergence in equation (33), the step sizeμis chosen such that

|1−2μλi| <1, for 1 i  N. (35)

Therefore, the condition for stability of the weight vector is 0< μ < 1

λmax

(36) whereλmaxis the largest eigenvalue ofR.

3.2. Adaptive inverse controller

Figure4depicts an AIC strategy [11], where the training signal of the controllerC is the system error signales(k), i.e. the reference model output minus the actual system outputd(k), not from the modeling error signal e(k). Even if P(z) is truncated and is not a perfect match to P(z), the adaptive algorithm will tend to minimize the overall system error by optimizing the choice of ˆC(z). It is noted that adaptation is still active when the plant P(z) is identified by the adaptive modeling process. Hence, this ensures that AIC can adjust itself and further control the whole system when the system is affected by the air bearing and nonlinear properties inherent in piezoelectric materials.

To explain how the forward controller copy ˆC(z)serves as an ‘inverse control’ to achieve AIC, we focus on the lower part in figure4. Cascading the model copyP(z)and controller ˆC(z)can be treated as an open-loop transfer function. The output ¯x(k)of both copy P(z)and system errores(k)act as updating signals to optimize the parameters of the controller ˆC(z) and further eliminate system errores(k)based on the LMS algorithm. Once the system errores(k)is eliminated, the open-loop transfer function equals the reference modelM(z), which can be expressed as

ˆP(z) · ˆC(z) = M(z). (37) Since the adaptation only updates parameters in controller

ˆC(z), we rewrite equation (37) as ˆC(z) = M(z) · 1

ˆP(z) (38)

where controller ˆC(z)is expressed as a product of the reference modelM(z)and the inverse of the adaptive model ˆP(z).

The forward path at the top of figure4cascades the copy ˆC(z)and plantP(z)(after adaptive modelling ˆP(z)substitutes for P(z)), and the open-loop transfer function relating the command inputx(k)and system output ˆy(k)is written as

ˆY (z)

X(z)= ˆC(z) · ˆP(z) = M(z). (39)

The reference modelM(z)is chosen to have the same dynamic response that the designer wants for the system. Hence, adaptive modeling aims to estimate the real plant based on the LMS algorithm to eliminate modeling error e(k). Once the adaptive model ˆP(z) can be estimated by the adaptive modeling process, an inverse controller ˆC(z)can be obtained from equation (38) and used to eliminate system errores(k). After convergence, cascading the copy ˆC(z) and the plant

P(z) results in the same dynamic response as the reference model does.

4. Experimental results

The experimental setup consists of a PC, two laser Doppler vibrometers that respectively generate a single laser beam for displacement sensing, two A/D–D/A cards that receive two displacement signals sensed by two laser Doppler vibrometers, the pickup head and PZT amplifier, and near-field optical disks, 1635

Figure 5. Experimental setup using dual-laser beams to sense the pickup head and disk displacements.

(a) (b)

Figure 6. Photograph of the slider: (a) top, (b) bottom.

-50 -40 -30 -20 -10 0 10 20 30 40 50

without AIC compensation

with AIC compensation

Output displacement ( m) μ Command input ( m)μ -60 -40 -20 0 20 40 60

Figure 7. Hysteresis loops for 50μm with and without

compensation.

as depicted in figure 5. The control method is mastered by a PC that receives both the disk displacement signal and the pickup head displacement signal via two A/D–D/A cards. The PC generates a driving signal calculated by AIC via a D/A converter and a signal amplifier to control the pickup head

-5 -4 -3 -2 -1 0 1 2 3 4 5

without AIC compensation

with AIC compensation

Output displacement ( m) μ Command input ( m)μ -6 -4 -2 0 2 4 6

Figure 8. Hysteresis loops for 5μm with and without compensation.

displacement. In experiments the sampling rate is 10 kHz and the disk rotation speed is 5400 rpm. Figures6(a) and (b) show the top and bottom photographs of the slider, respectively.

AIC can be separated into two main procedures in experiments. Adaptation for the plant P(z) should be done ahead of adaptive inverse control. It is desired to obtain ˆP(z)as close as possible toP(z). The adaptive model ˆP(z)chosen as FIR form with a weight vectorWof 100 elements in adaptation for the plant model P(z). The adaptive algorithm employs the LMS method, but is modified as a normalized LMS of the form [13]

W(k +1) = W (k) + μe(k)X (k) (40)

μ(k) = α

γ + XT(k)X (k) (41)

whereαandγare set as 0.004 and 1, respectively. Rather than constant step sizes, the step size functionμ(k)in normalized LMS is time varying and is updated by the input signal

X(k). The step sizeμ(k) automatically shrinks or enlarges according to the product of XT_{(k) · X (k). Parameters in} the adaptation and adaptive algorithm of controller ˆC(z)are the same as in the adaptive modeling process. A pure delay model usually results in good performance control. Here, the reference model M(z) is chosen to be a delay of 20 samples. AIC in experiments employs the LMS algorithm

Adaptive inverse control for the pickup head flying height of near-field optical disk drives 1.5 1.7 1.9 Time (sec) 2.1 2.3 2.5 Displacement (nm) Deformation (nm) Tracking error (nm) (a) 1.5 1.7 1.9 Time (sec) 2.1 2.3 2.5 (b) 1.5 1.7 1.9 Time (sec) 2.1 2.3 2.5 (c) -1000 0 1000 2000 3000 -1000 0 1000 2000 3000 -1000 0 1000 2000 3000

Figure 9. Pickup head tracks 2μm disk deformation: (a) pickup head displacement, (b) disk deformation, (c) tracking error.

2.45 2.46 2.47 _{Time (sec)} 2.48 2.49 2.5 (a) 2.45 2.46 2.47 Time (sec) 2.48 2.49 2.5 (b) 2.45 2.46 2.47 Time (sec) 2.48 2.49 2.5 (c) Displacement (nm) Deformation (nm) Tracking error (nm) 0 500 1000 1500 2000 0 500 1000 1500 2000 -100 -50 0 50 100

Figure 10. Enlarged view in figure9from 2.45 to 2.5 s.

to save computational time. Adaptive modeling is performed before 2 s, while AIC is carried out after 2 s. From 0 and 2 s, AIC only drives the PZT bender without control by using a training signal so as to capture plant dynamicsP(z)and further obtain an adaptive model ˆP(z). After 2 s, AIC indeed controls the pickup head to track the command input.

4.1. Hysteresis compensation

The hysteresis property of the PZT bender causes imprecision in tracking, for which hysteresis compensation is undertaken by using AIC. The hysteresis properties of the PZT bender cause 15% imprecision. Hence, hysteresis compensation is

necessary for AIC to control the displacement of the pickup head. Figures 7 and 8, respectively, compare experimental hysteresis loops for 50 and 5μm with and without AIC. In figure 7, the hysteresis is reduced by 90% by using AIC. In figure8, using AIC the hysteresis loop shrinks to become a straight line.

4.2. Tracking disk deformation

To demonstrate the control method in compensating for disk deformation, the pickup head tracks a real disk deformation of 2 μm. Figures 9and 10depict results of an experiment in which the pickup head tracks 2 μm disk deformation. 1637

Figure 11. Experimental setup using a dual-beam laser to sense the flying height. Flying Height (nm) 0 500 1000 1500 2000 2500 Time (sec) 0.5 1 1.5 2 2.5 0 3

Figure 12. Free-flying height without AIC control.

Figures 9(a) and (b) respectively depict the pickup head displacement and disk deformation, while figure9(c) depicts the corresponding tracking error. Figure10enlarges figure9 from 2.45 and 2.5 s. The tracking error lies between±50 nm, which is 5% of the disk deformation. Experimental results verify that with AIC the pickup head can efficiently track disk deformation.

4.3. Measurement of flying height

Another experimental setup depicted in figure 11, different from figure5, is used to measure the flying height of the pickup head. The present experiment uses only one laser Doppler vibrometer that generates both laser beams 1 and 2 focusing on the slider and the disk surface, respectively.

Figure 12 depicts the flying height of the pickup head without AIC control. Figures 13(a) and (b) enlarge figure 12 from 0.25 to 0.35 and 2 to 2.1 s, respectively. The slider takes off from disk surface at 0.262 s and flies stably after 2 s between 900 and 2200 nm. According to the measurement results, the flying height can be measured by

using the experimental setup depicted in figure11, which will be employed in subsequent experiments on control of flying height.

4.4. Flying height control

Two experimental cases are presented for control of flying height.

Case 1: Desired average flying height 1650 nm. Figure14(a) depicts histories of control of flying height by the AIC controller when the desired average flying height is set at 1650 nm. The AIC controller is turned on and starts at 6 s and error converges at 9 s. Figures14(b) and (c) are enlarged views from 5 to 5.1 and 9 to 9.1 s, respectively. In figure14(b) the flying height varies between 3000 and 1500 nm, and the average flying height is 2250 nm. In figure14(c), the average flying height is(1900+1400)/2=1650 nm and the variation in the flying height reduces to 1900−1400=500 nm. The AIC controller lowers the average flying height by 2250−1650= 600 nm and reduces the measured variation of the flying height by 67%.

Adaptive inverse control for the pickup head flying height of near-field optical disk drives 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 Time (sec) (a) Time (sec) (b) 2 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.1 Flying Height (nm) Flying Height (nm) 0 500 1000 1500 500 1000 1500 2000 2500

Figure 13. Enlarged view of figure12from (a) 0.25 to 0.35 s, (b) 2 to 2.1 s.

0 2 4 6 8 10 Time (sec) (a) Time (sec) (b) Time (sec) (c) Flying Height (nm) Flying Height (nm) Flying Height (nm) 0 500 1000 1500 2000 2500 3000 3500 0 500 1000 1500 2000 2500 3000 3500 0 500 1000 1500 2000 2500 3000 3500 9 9.02 9.04 9.06 9.08 9.1 5 5.02 5.04 5.06 5.08 5.1

Figure 14. (a) Control history for a desired average flying height of 1650 nm. (b) Enlarged view from 5 to 5.1 s. (c) Enlarged view from

9 to 9.1 s.

Case 2: Desired average flying height 800 nm. Figure 15(a) depicts histories of control of flying height using AIC when the desired flying height is set at 800 nm. Figures15(b) and (c) are enlarged views from 5 to 5.1 and 9 to 9.1 s, respectively. In figure15(b) the flying height varies between 2800 and 1100 nm and the average flying height is 1950 nm. In figure 15(c), the average flying height is

(1100+500)/2 = 800 nm and the variation in the flying height reduces to 1100−500=600 nm. The AIC controller

lowers the average flying height by 1950−800 = 1150 nm and reduces the variation in measured flying height by 65%.

5. Conclusion

To maintain a stable flying height of the pickup head, a novel pickup head is presented to meet the demands of near-field optical disk drives. A PZT bender serves as an actuator to track the vibratory deformation of the disk surface. However, unmodeled system dynamics results in 1639

0 2 4 _{Time (sec)} 6 8 10 (a) Time (sec) (b) Time (sec) (c) Flying Height (nm) Flying Height (nm) Flying Height (nm) 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 9 9.02 9.04 9.06 9.08 9.1 5 5.02 5.04 5.06 5.08 5.1

Figure 15. (a) Control history for a desired average flying height of 800 nm. (b) Enlarged view from 5 to 5.1 s. (c) Enlarged view from 9 to

9.1 s.

difficulty in controlling the flying height. AIC is very suitable for controlling the unmodeled system dynamics due to its robustness. This study has carried out experiments using AIC to effectively compensate for hysteresis and track disk deformation. Moreover, this study has sensed and controlled the flying height of the pickup head.

Acknowledgment

This work was supported by National Science Council in Taiwan, ROC under Grant No NSC93-2752-E009-009-PAE.

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