Sliding mode based learning control for track-following in hard disk drives

(1)

Sliding mode based learning control for

track-following in hard disk drives

W.C. Wu, T.S. Liu

*

Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 30010, Taiwan, ROC

Accepted 21 May 2004

Abstract

This study aims to develop a sliding mode based learning controller for track-following in hard disk drives. The proposed controller incorporates characteristics of sliding mode control into learning control. The reason for using sliding mode control is attributed to its robust properties dealing with model uncertainty and disturbances. The learning algorithm utilizes shape functions to approximate inﬂuence functions in integral transforms and estimate the control input to reduce repetitive error. Mathematical derivation of the control law and sta-bility analysis are presented. To validate the proposed method, this work conducts track-following experiments.

Keywords: Sliding mode control; Learning control; Track-following control; Repetitive error

1. Introduction

The development of hard disk drive techniques has come to maturity. A pivoted voice coil motor (VCM) has been the common rotary actuator to perform track seeking and following in hard disk drives. While the hard disk drive data storage capacity and track density rapidly increase, repetitive disturbance degrades the track-following performance even more than before. Hence the control performance has to be elevated since the data track width and track pitch become smaller.

Disturbances in hard disk drive track-following motion can be classiﬁed into repetitive and nonrepetitive components. Nonrepetitive disturbances generally come

*_{Corresponding author. Fax: +886-3-5720634.} E-mail address:tsliu@mail.nctu.edu.tw(T.S. Liu).

(2)

from the mechanical resonance, external impact, windage-induced disk flutter, etc. [1]. In addition, the repetitive disturbances mostly caused by the geometric bias of a disk, the relative position bias of the data track and rotary center, and the spindle motor defects [2]. In general, repetitive disturbances hurt track-following precision more than the nonrepetitive one [1,2]. In the conventional track-following control design, which only considers nominal plant models and nonrepetitive factors, such as PID, phase lead–lag, or notch filter can provide adequate gain and phase margin and reduce the resonance but can not perform well dealing with periodic disturbances [2]. Recently, many repetitive control methods have been proposed such as internal model type controllers [2], external model controller with a basis function algorithm [3] or with a learning algorithm [4]. These repetitive controllers can be categorized as internal mode based and external model based ones [5]. Controllers of internal model based are linear and constructed with a periodic signal generator inside a control loop. Conversely, external model based ones generate a cancellation signal outside the loop to eliminate the repetitive error. Comparison [5] of these methods indicates tradeoffs of error convergence speed and disturbance rejection ability. The internal model based approach with advantageous properties of linearity and easy analysis can converge rapidly but changes loop gains and influences sensitivity to distur-bances. The robustness to unmodelled dynamics and noises is thus reduced. The external model controller is more complex in implementation. However, it applies a disturbance model outside the basic feedback loop, which can be adaptively adjusted to match the actual disturbance. Hence, the control compensation is more like a feedforward one that affects the nominal open loop gain less than the former.

Integrating the adaptive control [6] and learning control, the adaptive learning control [7] does not require exact knowledge of the plant model and can eﬀectively eliminate the repetitive error after periods of learning. In addition, the nonlinear properties of plant and disturbance can be compensated by learning estimation [8]. It can be treated as a learning feedforward control, which is not designed completely prior to operation but keeps learning during control. Repetitive errors are com-pensated by a learning component that is operated in a feedforward path after training [9]. In addition, sliding mode control [10] has been developed and examined in various systems including nonlinear systems, discrete time systems, large scale systems, stochastic systems, multi-input, multi-output systems, etc. [11].

Dealing with model uncertainty and repetitive error motivates this work to integrate sliding mode control and learning control. As long as the error signal period is known in advance, without exact plant models the proposed sliding mode based learning controller can reduce repetitive error. The learning algorithm utilizes shape functions to approximate inﬂuence functions in integral transforms and esti-mate the control input to reduce repetitive error. Once the learning error converges, the sliding mode reaching condition [10,12] is obtained and the position error will converge to zero on a prescribed sliding surface.

This paper is organized as follows. Section 2 presents the sliding mode based learning control devoted to the repetitive tracking control. Some properties of the proposed controller are investigated. Section 3 describes a VCM model and simu-lation results. Experimental results are shown in Section 4.

(3)

2. Sliding mode based learning control

In this section, a sliding mode based learning controller is developed and its properties are presented.

2.1. Dynamic model and prescribed conditions

The equation of motion for an n-dimensional system can be expressed as MðqÞ€qþ Cðq; _qÞ _q þ GðqÞ ¼ u ð1Þ where qðtÞ, _qðtÞ, and €qðtÞ are respectively n 1 position, velocity, and acceleration vectors, u denotes a n 1 actuator input vector, MðqÞ is a symmetric positive deﬁnite inertia matrix, Cðq; _qÞ _q results from Coriolis and centripetal forces, and GðqÞ is a gravitational force vector.

Deﬁne Kerðt; sÞ as a periodic Hibert–Schmit kernel function [8] that satisﬁes Z T

0

Kerðt; sÞ2ds¼ ker < 1; Kerðt; sÞ ¼ kerðt þ T ; sÞ P 0 Condition 1. There exists an inﬂuence function aðsÞ such that

MðqÞ_t þ Cðq; _qÞt þ GðqÞ ¼ Z T

0

Kerðt; sÞaðsÞ ds ð2Þ where tðtÞ 2 Rn_{is a vector function.}

Condition 2. Using a proper deﬁnition of matrix Cðq; _qÞ, both MðqÞ and Cðq; _qÞ in Eq. (1) satisfy

qTð _M 2CÞq ¼ 0 8q 2 Rn _ð3Þ

That is,ð _M 2CÞ is a skew-symmetric matrix. In particular, elements in Cðq; _qÞ can be deﬁned as Cij¼ 1 2 _q ToMij oq " þX n k¼1 oMik oqj oMjk oqi _qk # : 2.2. Controller design

The present track-following control is to position a read/write head precisely at the center of desired tracks in the presence of disturbances. In order to deal with the periodic position error caused by repetitive disturbance, deﬁne periodic motion with a known period T as

qdðt þ T Þ ¼ qdðtÞ; _qdðt þ T Þ ¼ _qdðtÞ; €qdðt þ T Þ ¼ €qdðtÞ

where qdðtÞ, _qdðtÞ, and €qdðtÞ are desired position, velocity, and acceleration,

respec-tively. For position output qðtÞ, the resultant position error is written as

(4)

Based on Condition 1, periodic motion corresponding to the reference velocity _qr can be written as MðqÞ€qrþ Cðq; _qÞ _qrþ GðqÞ ¼ w ¼ Z T 0 Kerðt; sÞaðsÞ ds ð5Þ

The reference velocity _qris deﬁned based on the adaptive control [13] as

_qrðtÞ ¼ _qdðtÞ KeðtÞ C

Z

eðtÞ dt ð6Þ

and the error between the output velocity and reference velocity is

_erðtÞ ¼ _qðtÞ _qrðtÞ ð7Þ

Substituting Eq. (6) into (7) and employing Eq. (4) lead to _er¼ _q _qd Ke C Z edt ¼ _e þ Ke þ C Z edt

which implies that once the reference velocity error _er equals zero, the system

tra-jectory under control will satisfy

_er¼ _e þ Ke þ C

Z

edt¼ 0

Concerning sliding mode control, to have the steady state position error stay on a sliding surface and consequently eliminate the position error [10], a sliding vector can be deﬁned by letting s¼ _erwhich leads to

sðtÞ ¼ _eðtÞ þ KeðtÞ þ C Z

eðtÞ dt ð8Þ

where K¼ kiIdand C¼ ciIdare both positive deﬁnite matrices, and Idrepresents the

identity matrix. Hence, this study deﬁnes a control input uSMLC that integrates an

estimated learning compensation used to approximate w in Eq. (5), a proportional, integral, and derivative (PID) feedback control term, and a discontinuous control term; i.e. uSMLC¼ ~w KI Z edt KPe KD_e Q sgnð_erÞ ¼ ~w KD _e þ K1 D KPeþ KD1KI Z edt Q sgnðsÞ ð9Þ Denoting K¼ K1

D KP and C¼ KD1KIand substituting Eq. (8) into (9) yield

uSMLC¼ ~w Ks Q sgnðsÞ ð10Þ

where Q and K ¼ KD¼ kId are both positive deﬁnite matrices. The control block

diagram is shown in Fig. 1.

Diﬀerent from conventional controllers designed based on nominal plant models, the proposed controller Eq. (10) does not require exact knowledge of plant and

(5)

disturbance models. The three terms in Eq. (10) are correlated based on Eqs. (5), (6) and (8). It will be proved in Section 2.3 that the proposed controller after learning will constitute an equivalent sliding mode controller that satisﬁes the sliding mode reaching condition i.e. making the tracking error trajectory approach and stay on a prescribed sliding surface. Accordingly, the proposed controller with properties of both learning control and sliding mode control can be carried out doing without complicated procedure and any plant model. To verify stability of the proposed controller, a mathematical proof is given in the following section. Moreover, the reaching condition for the sliding mode control is also veriﬁed.

Based on the learning control [7], the desired input ~wcan be estimated as

~ wðtÞ ¼

Z T

0

Kerðt; sÞeIðt; sÞ ds ð11Þ

where eIðt; sÞ is the unknown estimated inﬂuence function.

Deﬁnition: Let CkðT Þ denote a subset of CðT Þ (which is the space of continuous T

-period functions IðÞ : Rþ! Rn) such that every IðÞ is piecewise continuously

dif-ferentiable, and sup t2½0;T d dtIðÞ 6k

Given a collection for shape functionsfUig and / > 0, there exist a ﬁnite number of

shape functionsfU0;U1;U2; . . . ;UNg that uniformly approximate members of CkðT Þ

within / > 0, i.e. for every I2 CkðT Þ, there exist constant vectors C0; C1;

C2; . . . ; Cn2 Rn such that sup t2½0;T IðtÞ XN i¼0 CiUi </

To estimate the desired inﬂuence function IðtÞ, it can be approximated by a linear combination of shape functions Ui. Hence,

IðtÞ ﬃX N i¼0 CiUiðtÞ K Q Plant d q d dte e s sgn(s) w u q edt dt d Learning Law ∫ ∫ ∼ Λ − − − + + + + + Γ

(6)

where Ci2 Rnrepresent unknown coeﬃcient vectors for each shape function Uiat an

instant, and Nþ 1 denotes the total number of shape functions. Hence, the estimated term is generated by determining coeﬃcients eCi, i.e.,

eIðt; sÞ ¼XN

i¼0

e

Ciðt; sÞUiðsÞ ð12Þ

where eCiðt; sÞ are estimated corresponding coeﬃcients. In addition, the sliding vector

is introduced into an adaptation law as o

otCeiðt; sÞ ¼ KLKerðt; sÞUiðsÞs ð13Þ where the learning gain KL¼ klId is a symmetric positive deﬁnite matrix.

This study employs a set of piecewise linear functions, as depicted in Fig. 2. Accordingly, in each interval of½iT =N ; ði þ 1ÞT =N , only two linear shape functions,

1 0 0 N T N T 2 N T 3 T 1 0 1 N T N T 2 N T 3 T 1 2 N T N T 2 N T 3 T 1 N N T N T 2 N T 3 T 0 0 Φ Φ Φ Φ

(7)

Ui and Uiþ1, are required; i.e. there are only two corresponding coeﬃcients, ci and

ciþ1, to be updated at any instant. The linear shape functions are a B-spline set of

second order. A B-spline of order n consists of piecewise polynomial functions of order n 1. The time span of each shape function is deﬁned as its support. If there are Nþ 1 equally spaced shape functions, the period can be expressed by

T ¼ Nd=2 ð14Þ

where d is the shape function support. It in general can be regarded as a ﬁlter as shown in Fig. 3 with N ¼ 5, for example. The accuracy of the ﬁltering or called approximating process depends on the support of the shape functions [14]. For ease of computing kernel functions, piecewise linear functions shown in Fig. 4 are used as kernel functions for integral transforms, where the span ‘sp’ denotes a subinterval length.

2.3. Stability analysis

Applying the control input given by Eq. (10) into the system in Eq. (1), system dynamics subject to the present controller is written as

MðqÞ€qþ Cðq; _qÞ _q þ GðqÞ ¼ ~w Ks Q sgnðsÞ

The approximation error w¼ ~w w of control inputs between Eqs. (5) and (11) can be written as w¼ ~w w ¼ Z T 0 Kerðt; sÞeIðt; sÞ ds Z T 0 Kerðt; sÞaðsÞ ds ¼ ½MðqÞ€qþ Cðq; _qÞ _q þ GðqÞ þ Ks þ Q sgnðsÞ ½MðqÞ€qrþ Cðq; _qÞ _qrþ GðqÞ ¼ MðqÞ_s þ Cðq; _qÞs þ Ks þ Q sgnðsÞ ð15Þ T 0 t 1 0 T 5 T 0 5 2 3 1 4 t support I 0 C C1 C2 C₃ C4 C5 Φ Φ Φ Φ Φ Φ ∼ ∼ ∼ ∼ ∼ ∼

Fig. 3. The inﬂuence function approximating process by shape functions with the shape functions number of N¼ 5.

(8)

Taking the time derivative of Eq. (15) leads to _

w¼ _~w _w

It follows from Eqs. (11)–(13) that _ w¼ o ot Z T 0 Kerðt; sÞeIðt; sÞ ds Z T 0 Kerðt; sÞaðsÞ ds ¼ KLZðKer; UÞs ð16Þ

where ZðKer; UÞ is a function of the kernel function and the shape function. In order to verify stability of the proposed control method, i.e. s¼ _er

asymptotically converges to zero, prescribe a Lyapunov function candidate of the form

V ¼1

2ðs

T_Ms_þ_wT_K

LwÞ P 0 ð17Þ

Taking the time derivative of V leads to _ V ¼1 2s T_{M s}_{_} _{þ s}T_{M _s}_þ_wT_K Lw_ ð18Þ From Eq. (15), MðqÞ_s ¼ w Cðq; _qÞs Ks Q sgnðsÞ (a) (b) (c)

τ

Fig. 4. Piecewise linear kernel functions. (a) 0 < t1< sp 2, (b) sp 2< t2< T sp 2, (c) T sp 2< t3< T.

(9)

Hence, Eq. (18) can be rewritten as _ V ¼1 2s T_{M s}_{_} _{þ s}T_½_w_{Cðq; _qÞs Ks Q sgnðsÞ þ}_wT_K Lw_ ¼1 2s T_{ð _}_M_{2CÞs þ s}T_½_w_{Ks Q sgnðsÞ þ}_wT_K Lw_

Applying Condition 2, it becomes _

V ¼ sT_½_w_{Ks Q sgnðsÞ þ}_wT_K

Lw_

Substituting Eq. (16) into the above equation leads to _ V ¼ sT_½_w_{Ks Q sgnðsÞ}_wT_K2 LZðKer; UÞs ¼ sT_½I d KL2ZðKer; UÞw s T_Ks_sT_Q_sgnðsÞ _ð19Þ

Assuming that both wand ZðKer; UÞ are bounded, i.e. kwk 6 W and ZðKer; UÞk k 6 f, where W and f are positive, one has

k½Id KL2ZðKer; UÞwk 6 kId KL2ZðKer; UÞk w

6_½kI_d_{k þ kK}2 LZðKer; UÞkkwk 6 ð1 þ k 2 lfÞW ð20Þ

Eq. (19) can be rewritten as _

V ¼ sT_f½I

d KL2ZðKer; UÞw Q sgnðsÞg s

T_Ks

Hence, letting Q½ð1 þ k2

lfÞW þ dId and applying Eq. (20) yields

_

V 6 dksk kksk2 ð21Þ

where d > 0. Since V > 0 from Eq. (17) and according to Eq. (21), the negative deﬁniteness of _V implies the convergence of s. In addition, a ﬁnite reaching time to the sliding surface s¼ 0 is ensured by designing d P g. Hence, the reaching condition [12,15]

_

V 6 gksk; g >0

for the sliding mode control is ensured. As a consequence, the position error will converge to zero on the sliding surface s¼ _e þ Ke þ CRedt¼ 0.

3. Plant model and simulation results

Since hard disk drives employ a pivoted VCM as the track seeking and follow-ing actuator, in this study the control performance is investigated with a VCM plant model whose parameters were identified and listed in Table 1 [16]. The iden-tified plant model is adopted to perform simulation. The nominal plant model is defined as QðsÞ UðsÞ¼ PVCMðsÞ ¼ k 1 s2þ X2 i¼1 c_i s2_{þ 2f} ixisþ x2i ! eTdS _ð22Þ

where qðtÞ denotes the position output (lm) of the pickup head, uðtÞ is the input current (A), k is a constant gain, fi, ci, and xiare ith mode damping factor, residue

(10)

and resonant frequency, respectively, and Td is a time delay. The time delay includes

the eﬀect of delay caused by zero order hold (ZOH), which is half of the sampling period. The term of the time delay is approximated by a third-order Pade approx-imation [17].

To demonstrate the periodic disturbance rejection of the track-following control with the proposed controller, a input periodic disturbances is applied as a composite sinusoidal signal consisting of multiple frequencies of x¼ 60 Hz. Thus, the control input disturbance dIis expressed as

dIðtÞ ¼ 0:01 ½0:1 þ sinð2pxtÞ þ 0:5 sinð4pxtÞ þ 0:25 sinð8pxtÞðAÞ ð23Þ

In order to evidence the periodic error convergence capability, a PD controller is used to compare with the proposed controller. It can be found from Fig. 1 that the proposed controller without the learning control and the discontinuous switching terms is equivalent to a PD controller based on the sliding variable deﬁnition

sðeÞ ¼ _e þ Ke

Hence, the equivalent PD controller can be rewritten as uPD¼ Ks ¼ Kð_e þ KeÞ ¼ KD_e KPe

where KD¼ K and KP¼ KK are corresponding PD control gains. The corresponding

controller gains in both controllers will be the same in the discrete time simulation at 100 kHz sampling rate for impartial comparison. Assuming the repetitive tracking error is caused by a periodic input disturbance, Figs. 5 and 6 compare repetitive errors in amplitudes and power spectrums between both control methods, respec-tively, where the PD controller cannot cope with the disturbance. By contrast, using the proposed controller, the repetitive tracking error caused by the input disturbance is eliminated after ﬁve learning periods. The power spectrum in Fig. 6 shows that error component due to fundamental frequency x¼ 60 Hz has been removed.

Furthermore, this study superimposes a white noise of zero mean and variance 5 mA on a periodic input disturbance and another white noise of zero mean and variance 2 lm in the position measurement to examine the learning control robustness to nonrepetitive disturbances. Fig. 7 shows that the proposed controller is robust in the presence of white noise and the repetitive error convergence capability is not degrade at all. At a lower sampling frequency of 1 kHz, Fig. 8 indicates the

Table 1

VCM model parameters

Fs Sampling frequency 11 kHz

x1 First resonant frequency 5200 Hz

x2 Second resonant frequency 6100 Hz

f1 First resonant damping factor 0.02

f2 Second resonant damping factor 0.01

c1 Residue of ﬁrst resonant mode )1.2

c2 Residue of second resonant mode )0.2

K Constant gain 1.3· 108

(11)

repetitive error convergence conditions subject to diﬀerent disturbance frequencies, where the lower the disturbance frequency is, the faster the tracking error converges.

4. Experiment and results

As shown in Fig. 9, this experimental setup contains a VCM that drives a sus-pension arm in a 3.5’’ hard disk drive. For the position sensing purpose, this study

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Tracking Error ( µ m) Time (sec)

Fig. 5. Tracking error amplitudes of sliding mode based learning control (solid line) and PD control (dotted line) under periodic input disturbances.

1 10 100 1000 10000 10-6 10-4 10-2 100 102 104 106 1 10 100 1000 10000 10-10 10-8 10-6 10-4 10-2 100 102 Frequency (Hz) Tracking Error (dB) Tracking Error (dB)

Fig. 6. Tracking error power spectrums of sliding mode based learning control (solid line) and PD control (dotted line) under periodic input disturbances.

(12)

0.00 0.05 0.10 0.15 0.20 0.25 0.30 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Tracking Error ( µ m) Time (sec)

Fig. 7. Tracking errors of sliding mode based learning control (solid line) and PD control (dotted line) subject to white noise in both control input and measurement signal.

0 1 2 3 4 5 6 -12 -8 -4 0 4 8 12 16 Tracking Error ( µ m) Time (sec) 1 Hz 5 Hz 10 Hz

Fig. 8. Tracking error of sliding mode based learning control subject to various frequencies of periodic

(13)

uses a Renishaw RGH22S digital optical readhead and a fan-shaped component with a reﬂective tape scale attached at the tip of the suspension arm, as shown in Fig. 10. The reﬂective tape scale is scanned by a digital optical readhead. The readhead generates a digital square wave signal, which is encoded in a NI PCI-7344 Flex-Motion control card as position feedback signals. Accordingly, the positioning resolution can achieve 0.1 lm.

Fig. 11 depicts this experimental setup. It consists of a Pentium II PC to perform control algorithm calculation at 1 kHz sampling rate that is the same as simulation, a motion control card to encode the digital square wave signal and convert digital control signal to analog output, an ampliﬁer to drive the VCM, and a modiﬁed VCM with the suspension arm as the control plant.

To validate the controller proposed in this work, experimental results of both equivalent PD controller and sliding mode based learning controller are compared. In practice, the repetitive position error is caused by spindle motor bias, disk runout and deformation, etc. As long as the period of the disturbance is known, the present method can eliminate the repetitive error. Further, the periodic disturbance pre-scribed in Eq. (23) is included on purpose in the control input. Repetitive errors of PD control and sliding mode based learning control are compared in Fig. 12, where in the presence of 1 Hz disturbance the proposed method outperforms the PD control. Tracking results compared with those of 5 and 10 Hz disturbances are presented in Fig. 13. The learning performance with a ﬁxed learning gain degrades with increasing disturbance frequency due to sampling rate limitation of the con-troller hardware.

D/A

Encoder Motion Control Card

PCI Bus PC

Amplifier

Plant

Readhead signal

Fig. 11. Experimental setup.

Optical readhead Optical tape Fan-shaped component VCM Encoder

(14)

Modern disk drives have much higher spin speed, say 12,000 rpm that causes a 200 Hz high frequency disturbance. To deal with a higher frequency disturbance, using hardware of higher sampling rate to implement the proposed controller will be an eﬀective solution. In tracking sinusoidal signals, the total number N of shape functions in a learning period T determines error magnitudes in steady state. Since the number N of shape functions is prescribed as 100 in experiments that result in Fig. 13, to obtain comparable performance in dealing with a 200 Hz disturbance of

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -25 -20 -15 -10 -5 0 5 10 15 20 25 30

Tracking Error (Track)

Time (sec)

Fig. 12. Experimental results in sliding mode based learning control (solid line) and PD control (dotted line). 0 1 2 3 4 -15 -10 -5 0 5 10 15 20 25

Tracking Error (Track)

Time (sec)

1 Hz 5 Hz 10 Hz

Fig. 13. Experimental results of sliding mode based learning control among diﬀerent frequencies of dis-turbances.

(15)

period T ¼ 5 ms, it is also prescribed that N ¼ 100. Substituting T ¼ 5 ms and N ¼ 100 into Eq. (14) gives d ¼ 0:1 ms. However, according to the last paragraph in Section 2.2, the shape function support d of a piecewise linear shape function as shown in Fig. 2 has to be at least twice the sampling period Dt to implement the proposed controller in discretized form. Hence, Dt¼ 0:05 ms; i.e. the sampling rate must be at least 20 kHz. An alternative is to use other shape functions such as Fourier series bases [4] that may better approximate the desired input even at a low sampling rate, which however will increases computation time.

5. Conclusion

A sliding mode based learning control method has been proposed for a VCM in hard disk drives to perform disturbance rejection in track-following control. Mathematical derivation of the control law and stability proof have been carried out. According to experimental results, without an exact plant model the proposed control method achieves rejection of periodic disturbances. Additionally, the sliding mode based learning control not only exhibits error convergence faster than the PD control but also can eliminate the repetitive error. However, in experiments due to time delay caused by I/O, the sampling rate is conﬁned to 1 kHz at most. The tracking error can be reduced more eﬀectively with lower frequency disturbance signals or using higher sampling rate if available. The latter can be achieved by using more advanced control hardware.

Acknowledgements

This work was supported by ‘Photonics science and technology for tera era’, Center of Excellence, Ministry of Education, Taiwan under Contract 89-E-FA06-1-4.

References

[1] Atsumi T, Arisaka T, Shimizu T, Yamaguchi T. Vibration servo control design for mechanical resonant modes of a hard disk drive actuator. JSME Int J C: Mech Syst Mach Elements Manuf 2003;46(3):819–27.

[2] Chew K, Tomizuka M. Digital control of repetitive errors in disk-drive systems. IEEE Control Syst Mag 1990;10:16–20.

[3] Tomizuka M, Kempf C. Design of discrete time repetitive controllers with applications to mechanical system. In: Proceeding of 11th International Federation of Automatic Control World Congress, Tallinn, Estonia, USSR, 1990. p. 5.

[4] Cao WJ, Xu JX. Fourier series-based repetitive learning variable structure control of hard disk drive servos. IEEE Trans Magn 2000;36(5):2251–4.

[5] Kempf C, Messner W, Tomizuka M, Horowitz R. Comparison of four discrete-time repetitive control algorithm. IEEE Control Syst Mag 1993;13(6):48–54.

[6] Sadegh N, Horowitz R. Stability and robustness analysis of a class of adaptive controllers for robotic manipulators. Int J Robotics Res 1990;9(3):74–92.

(16)

[7] Sadegh N, Guglielmo K. A new repetitive controller for mechanical manipulators. J Robotic Syst 1990;8(4):507–29.

[8] Messner W, Horowitz R, Kao W, Boals M. A new adaptive learning rule. IEEE Trans Automatic Control 1991;36(2):188–97.

[9] Velthuis WJR, de Vries TJA, Schaak P, Gaal EE. Stability analysis of learning feedforward control. Automatica 2000;36(12):1889–95.

[10] Utkin VI. Sliding modes in control optimization. New York: Springer-Verlag; 1992.

[11] Hung JY, Gao WB, Hung JC. Variable structure control: a survey. Special Issue on Sliding Mode Control. IEEE Trans Ind Electron 1993;40(1):2–22.

[12] Bailey E, Arapostathis A. Simple sliding mode control scheme applied to robot manipulators. Int J Control 1987;45(4):1197–209.

[13] Slotine JJE, Li W. On the adaptive control of robot manipulators. Int J Robotics Res 1987;6(3):49– 59.

[14] Velthuis WJR, de Vries TJA, Gaal EE. Experimental veriﬁcation of the stability analysis of learning feedforward control. In: Proceedings of the 37th IEEE International Conference on Decision and Control, Florida, USA, December 16–18, 1998. p. 1225–9.

[15] Slotine JJE, Coetsee JA. Adaptive sliding controller synthesis for non-linear systems. Int J Control 1986;43(6):1631–51.

[16] Ohno K, Abe Y, Maruyama T. Robust following control design for hard disk drives. In: Proceedings of the IEEE International Conference on Control Applications, Mexico City, Mexico, September 5–7, 2001. p. 930–5.

[17] Golub GH, Van Loan CF. Matrix computations. Baltimore: Johns Hopkins University Press; 1989. p. 557–8.