Frequency Adaptive Technique of Compact Disk Drives for Rejecting Periodic Runout

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Control Engineering Practice 12 (2004) 31–40

Frequency adaptive control technique for rejecting periodic runout

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Jieng-Jang Liu, Yee-Pien Yang*

Department of Mechanical Engineering, National Taiwan University, No. 1, Roosevelt Road, Sec. 4, Taipei, Taiwan, 106, ROC Received 8 April 2002; accepted 11 November 2002

Abstract

This paper proposes a novel adaptive controller for rejecting the periodic runout of a track-following system in the compact disk drive (CDD) with dual actuators. The control objective is to attenuate adaptively the speciﬁc frequency contents of runout disturbances without amplifying its rest harmonics. This controller can be implemented in a plug-in manner to an existing feedback control system without changing the original control setup. It is applicable to both the spindle modes of constant linear and constant angular velocity for various operation speeds. The experimental results show that the novel control strategy leads to a satisfactory performance in terms of the reduction of tracking error of CDDs.

Keywords: Compact disk drive; Runout rejection; Adaptive control; Track-following system

1. Introduction

Periodic disturbance occurs in various control en-gineering applications of the rotational mechanical systems. For compact disk drives (CDD), the spirally shaped tracks are usually not perfectly circular and the assembly of the disk and spindle motor is unavoidably eccentric. The resulting periodic disturbance is, there-fore, synchronous with the disk rotation, and becomes particularly noticeable for the track-following servo system. These disturbances cause a large steady-state error at the frequencies of integer multiples of disk rotation. The conventional feedback regulation strategy usually did not consider these periodic disturbances; therefore no better asymptotical track-following can be achieved. For that reason, the runout components pose the perpetual tracking errors and degrade system performances.

According to the compact disk operation, there are various playing speeds of spindle modes of constant angular velocity (CAV) and constant linear velocity (CLV). Depending on the media types of stored data, the system processor selects a suitable running speed and

spindle type for correct media playing. Between the two rotating types, the disk rotation is ﬁxed at high speed for the CAV operation, while the speed is descent linearly for the CLV operation. Once the disk is running, the runout disturbances are present for both spindle modes, and the disturbance cancellation is necessary for better tracking performances.

The methods for eliminating periodic disturbances with fixed fundamental frequency were classified as four different repetitive control algorithms (Kempf, Messner, Tomizuka, & Horowitz, 1993). Among them, the adaptive repetitive controller (ARC) (D.otsch & Smak-man, 1995) and adaptive feedforward cancellation (AFC) (Weerasooriya, Zhang, & Low, 1996; Onuki, Ishioka, & Yada, 1998; Lee, 1997) schemes were most widely used for disk drives; some analysis and synthesis methods were also reported (Bodson, Sacks, & Khosla, 1994; Sacks, Bodson & Messner, 1995; Messner & Bodson, 1994). These methods can be classified as the internal model principle (IMP) based controller design (Bodson et al., 1994). For the typical AFC method, the coefficients were continuously estimated and adapted for the on-line description of the sinusoidal disturbance. Experimental results reported in Weera-sooriya et al. (1996),Lee (1997)and Sacks, Bodson and Messner (1995) verified that the modified AFC method was efficiently implemented on the disk drive, while the improved works dedicated by Sacks, Bodson and Messner (1995) andMessner and Bodson (1994)showed

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This research is accomplished by the aid of ﬁnancial supports from the National Science Council of Taiwan, Republic of China, under Contract No. NSC90-2213-E002-083.

*Corresponding author. Fax: +886-2-23631755. E-mail address:[email protected] (Y.-P. Yang).

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various schemes to improve the robustness and perfor-mance of the AFC. In addition, depending on the available adaptation method used in AFC design, the work (Bayard, 1997) formulated the necessary and sufﬁcient conditions for the AFC controller design. However, it is shown by Sacks, Bodson and Messner (1995) that, an AFC algorithm designed to cancel a speciﬁc harmonic could attenuate or amplify the rest ones as well. Furthermore, as shown by Kempf et al. (1993), the necessary requirement of the plug-in repetitive controllers was that the sampling time had to be adjusted so that an integer number of samples, N; matched the period of the fundamental frequency. In this regard, the implementation of AFC is limited to the disk operation at a CAV. For optical disk drives, however, variable rotating speeds with two spindle types make such implementation of their proposed repetitive controllers impossible.

In order to deal with the repeatable runout cancella-tion for both the CAV and CLV spindle modes in the optical disk drive, a novel frequency adaptive control technique (FACT) is proposed. The new frequency adaptive controller design depends only on the disk position, thereby independent of the spindle type and playing speed of the disk. Indeed, the new development of the on-line calculation method involves the creation of a frequency sampling filter (FSF) bank, by which the individual components of the runout disturbance can be extracted. This figure of merits leads to the result that the rejection of specific components of runout harmo-nics will not influence the rest ones. Therefore, the proposed method becomes suitable for both the spindle modes of CAV and CLV without changing the control setup. Implemented with the original feedback control-ler, the novel frequency adaptive controller is realized by a fixed-point FPGA to validate its application for a CDD. In the proceeding sections, a brief description of the CDD is depicted in Section 2. The frequency adaptive control mechanism is introduced in Section 3, where the frequency sampling filtering and the adaptation algo-rithm are elaborated. Next, the proposed frequency adaptive controller is implemented with a CD-ROM drive, and the experiments are performed in Section 4 . Experimental results show that the proposed controller has successfully rejected the undesired harmonics of the runout disturbance to reach a satisfactory tracking performance. Final section summarizes the conclusions.

2. The conventional servo system of CDD

The schematic diagram of a typical optical disk system is shown inFig. 1. The mechanism is composed of a spindle motor for the rotation of the disk, an optical pick-up head for focusing and track following, and a coarse actuator to move the pick-up assembly in radial

direction. During the operation of track-following, the laser spot follows the track by two-dimensional actua-tors: a DC sled motor for coarse motion, and a voice coil motor for ﬁne motion adjustment. These two actuators possess different dynamic characteristics. The ﬁne actuator has faster response with a narrow displacement range, while the coarse actuator moves across the entire disk radius with lower bandwidth. The position xf of the objective lens and xc of the pick-up

head determine the laser spot position, thereby forming a multi-input single-output servo system as shown in

Fig. 2. The transfer functions of ﬁne and coarse actuators are denoted as GfðsÞ and GcðsÞ; while the

variables ufðtÞ and ucðtÞ are the control inputs to GfðsÞ

and GcðsÞ; respectively. The feedback controller CfðsÞ

and CcðsÞ are designed so that the closed-loop system is

stable, rejecting the external disturbance dðtÞ and reaching a tolerable level of tracking error. Without knowing the relative displacement between the pick-up and the ﬁne actuator during the track-following phase, only measured is the tracking error signal eðtÞ; which is commonly abbreviated to TES.

3. Algorithm of FACT

The frequency adaptive control mechanism is realized in a common used plug-in conﬁguration as shown in

Fig. 3(Ohmori, Narita, & Sano, 1998;Sutton & Elliott, 1995), in which the proposed algorithm of FACT is illustrated in Fig. 4. The plant GfðsÞ effected by the

pi

f

i k p

pick-up

Fig. 1. Optical disk mechanism.

uc (t) uf (t) f Gf G TES d (t)

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periodic disturbance dðtÞ is controlled by the controller CfðsÞ as well as the compensation vðtÞ through the

plug-in FACT. The FACT algorithm consists of a frequency sampling filter and a gradient descent algorithm. The former calculates the magnitude and phase of individual frequency component from the error signal eðtÞ which is measured with a fixed number of samples at each period, while the later adapts the magnitude and phase of the control output to compensate the error signal. To achieve accurate track-following, the TES is first filtered by a low pass filter, and sampled at N equally spaced points for each complete rotation of the disk. Then, the complete frequency contents of eðtÞ caused by the disturbance dðtÞ are extracted through the FACT. During the disk operation, the FACT only requires the last N samples of the error signal for the on-line calculation of the magnitude and phase of its individual component. Theoretically, the higher the harmonic components to be rejected, the more are points to be sampled for each rotation of the disk. Elaborated in the following section are the bank of frequency sampling filters and the adaptation mechanism that constitute the FACT.

3.1. Frequency sampling filter

The frequency sampling filter (FSF) is a finite-impulse-response filter and is similar to the discrete Fourier transform used in the digital signal processing (Bitmead & Anderson, 1981). The impulse response of FSF can be completely described by the N points equally spaced around the unit circle; the corresponding frequency contents are extracted through a bank of

transfer functions as follows: HnðzÞ ¼ 1 zN 1 Wn Nz1 ¼ 1 þ Wn Nz1þ WN2nz2þ ? þ W nðN1Þ N zðN1Þ; ð1Þ where Wn

N¼ expðjð2pn=NÞÞ and n ¼ 0; 1; y; N 1: The

input and the output of the nth elemental ﬁlter of the FSF bank are denoted by eðkÞ and xnðkÞ; respectively,

and a batch of N samples of eðtÞ per period is needed for each ﬁltering process, as shown in Fig. 5. Since the elemental ﬁlter HnðzÞ has N 1 zeros at points z ¼ WNi

for ian on the unit circle, only the magnitude and phase of the nth harmonics of the speciﬁc periodic signal are identiﬁed. Therefore, this bank of HnðzÞ produces a

collection of spectral components, which approximate the frequency contents of the input periodic signal. Theorem 1. Let the periodic signal eðtÞ be given by the sinusoidal functions with fundamental frequency o1; i.e.,

eðtÞ ¼X

M

n¼1

ðancos ont þ bnsin ontÞ; ð2Þ

where M is the highest harmonic order, on¼ no1; and an

and bnare unknown variables indicating the in-phase and

quadrature of individual harmonic component. Then, (I) the output of each elementary FSF for input eðtÞ; which is sampled at N equally spaced points, is described as xnðkÞ;

xnðkÞ ¼ HnðzÞeðkÞ ¼ N 2 ðanþ jbnÞ; ð3Þ where an¼ ancos ont þ bnsin ont; b_n¼ ansin ont bncos ont; ( ð4Þ n ¼ 1; 2; y; M; and the least value of N is

N ¼ 2M þ 2; ð5Þ

(II) the parameters an and bn in (2) are obtained by the

coordinate transformation an¼ ancos ont þ bnsin ont; bn¼ ansin ont bncos ont: ( ð6Þ d (t) e (t) v (t) uf (t) f Gf G

Fig. 3. The plug-in conﬁguration of frequency adapting control.

a_n

b_n

e (t) X (i+1) = X (i)−

Y (i+1) = Y (i)−_∂Y∂∂J X ∂J FSF Low Pass + x_n y_n cos ω_nt sin ω_nt v (t) X X

Fig. 4. The block representation of the algorithm of FACT.

z-1 _z-1 _z-1 n N W W e (k) N N W2n (N-1)n n (k)

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Proof. The proof is delineated in the appendix.

From Theorem 1, the parameters anand bndescribing

the measured periodic signal eðtÞ in (2) are calculated on-line by the bank of FSF defined in (1). It is also worth noticing that not a full bank of FSF filters but only the first M elementary filters is needed for FSF algorithm to be implemented.

3.2. Adaptation formulation

The runout disturbance dðtÞ is periodic and causes the measured tracking error eðtÞ with the same frequency. As shown inFig. 3, the transfer function from vðtÞ to the tracking error eðtÞ of the plant is expressed as

e v¼

Gf

1 þ GfCf þ GcCcCf

: ð7Þ

Since the bandwidth of the coarse actuator is much smaller than the runout frequencies, only the ﬁne actuator is implemented with the FACT algorithm to reject the runout effects. This leads to the transfer of reduced order as e v¼ Gf 1 þ GfCf ¼ W ðsÞ: ð8Þ

The addition of FACT output vðtÞ to the input ufðtÞ

results in a compensated control input, TRO, to the ﬁne actuator GfðsÞ: The control objective aims at canceling

the runout effect asymptotically by the compensation of vðtÞ: In terms of the ﬁrst M harmonics of eðtÞ; the adaptation signal vðtÞ is deﬁned as

vðtÞ ¼X

M

n¼1

ðxncos ont þ ynsin ontÞ; ð9Þ

where, the on-line adaptation of variables xnand ynare

to be formulated. The following adapting mechanism is modiﬁed from, and more efﬁcient than, the work by

Sutton and Elliott (1995).

Let vectors A; B; X and Y be A ¼ ½a1a2?a_MT; B ¼ ½b1b2?b_MT; X ¼ ½x₁x₂?x_MT; Y ¼ ½y₁y₂?y_MT; where, an and bn are given by (6) and xn and yn are

variables described in (9). The penalty function for the adaptation is deﬁned as the total energy of the measured error at each harmonic frequencies on such that

J ¼1 2ðA

T_{A þ B}T_BÞ: _ð10Þ

The gradient descent algorithms are described in terms of xn and yn as Xði þ 1Þ ¼ XðiÞ mqJ qX; Yði þ 1Þ ¼ YðiÞ mqJ qY; 8 > < > : ð11Þ

where, the adaptation gain m is positive such that the descendent of J is guaranteed. The gradients of the penalty function J with respect to the vectors X and Y

are written as qJ

qX¼ FaxA þ FbxB; qJ

qY¼ FayA þ FbyB: ð12Þ By the deﬁnition, the Jacobean matrices Fij in (12) are

diagonal with constant coefﬁcients, and represent the rate of change of the elements of A and B with respect to X and Y; respectively.

Since the vectors A and B are updated on-line by the FSF described in Theorem 1, if the Jacobean matrices in (12) are known, this adaptation algorithm can be used to adjust the value of vðtÞ such that the measured error eðtÞ is minimized. It is clear that the response of the linear system W ðsÞ at a speciﬁc frequency depends only on the excitation of the same frequency. Therefore, the frequency response of W ðsÞ excited by the sinusoidal input of frequency onis denoted by ½WrðonÞ þ jWiðonÞ;

and the input–output relationship becomes

anþ jbn¼ fWrðonÞ þ jWiðonÞgðxnþ jynÞ; ð13Þ

where, WrðonÞ and WiðonÞ are, respectively, the real and

imaginary parts of the frequency response of W ðsÞ at on:

Differentiating (13) with respect to xn and yn yields

qan qxn ¼ WrðonÞ; qan qyn ¼ WiðonÞ; qbn qxn ¼ WiðonÞ; qbn qyn ¼ WrðonÞ: ð14Þ

Hence, the combination of (11)–(14) constitutes an adaptation law as follows:

xnði þ 1Þ ¼ xnðiÞ m½WrðonÞanþ WiðonÞbn;

ynði þ 1Þ ¼ ynðiÞ m½WiðonÞanþ WrðonÞbn

(

ð15Þ with n ¼ 1; 2; y; M: It is worth noting that only the frequency responses at on of system W ðsÞ are required

to formulate the proposed adaptation law.

4. Implementation of FACT

4.1. Experimental setup

The compact disk used in the experiment is a commercial high-speed CD-ROM drive. The drive functions properly with the original control structure, including spindle, focus, track following and track seeking servos. In order to validate the FACT, only the track following servo is rebuilt in the experimental setup as shown inFig. 6. The TES is sampled at 175 kHz by an 8-bit A/D converter; frequency generator (FG) accounting for the disk position decoder signal has 6 logic outputs of square form in one revolution; tracking output (TRO) represents the ﬁne actuator output driven

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by an 8-bit D/A converter; and feed motor output (FMO) is the output of the coarse actuator driven by a PWM power module. A single ALTERA EPF10KA FPGA device (ALTERA, 2002) is used to implement the track following controllers CfðsÞ and CcðsÞ as well as the

proposed FACT algorithms. The FPGA is a program-mable logic device. The ALTERA provides powerful logic design tools to fulﬁll the user’s requirements. The implementation of FACT function as well as the controllers can be realized through the design environ-ment of MAX-Plus II provided by ALTERA.

4.2. Experimental results

In this section, experimental results for the track-following mechanism implemented with the proposed FACT are described under the disk operating at variable high-speed CAV mode and the slow CLV mode. The proposed controller successfully rejects the runout disturbance and promotes better track following quality. 4.2.1. Constant angular velocity mode at 75 Hz

In the ﬁrst experiment of the track-following control, the disk operates at the CAV mode of 75 Hz ð4500 rpmÞ: The corresponding frequency response W ðsÞ at frequen-cies between 10 and 1 kHz is obtained as shown in

Fig. 7. The real and imaginary parts of the first four harmonic frequencies at 75, 150, 225 and 300 Hz are listed in Table 1. The real time TES under the conventional feedback control is shown inFig. 8, where ch1 presents the time series of TES over three disk revolutions and ch2 presents its power spectrum through fast Fourier transformation (FFT), in which one voltage time response stands for 0:02 mm runout. The power spectrum of the TES shows that the runout error occurs at the integer multiples of the fundamental frequency, i.e., 75, 150, 225, 300, 375 Hz; etc. It is clear that the first four harmonics are dominant. In this case, the FACT is designed to cancel the first four harmonics, i.e. M ¼ 4; and the least number of equally spaced sampling points of one revolution of the disk is N ¼ 2 4 þ 2 ¼ 10 as formulated in (5). When the fourth-order FACT is added to the system by means of the plug-in manner, the power spectrums of the TES are shown as illustrated in

Fig. 9 under various conditions of runout rejection.

These four lines illuminate the corresponding FFTs of the compensated TES for the ﬁrst through fourth bank of FACT, respectively. Compared to the ch2 inFig. 8, it is evident that the FACT rejects the target frequencies without attenuating or amplifying other harmonics. Furthermore, when these four banks of FACT are added to the system simultaneously, as shown inFig. 10, the ﬁrst four harmonics of the tracking error are cancelled completely.

In view of the stability, it is important to exam-ine whether the stability margins would be affected by the compensation of the FACT. Regardless of the runout disturbance dðtÞ; the sensitivity function

CD D

Fig. 6. Experimental setup.

4 3 2 1 0 101 102 ₁₀3 0 -1 -2 -3 -4 101 102 ₁₀3 Frequency (Hz) Image Reil

Fig. 7. Frequency response of W ðsÞ for disk running at CAV 75 Hz:

Table 1

The frequency response of W ðonÞ

CAV mode at 75 Hz CAV mode at Hz

ea mae ea mae o o o o o o 75 5 5 5 5 5 5 5 57

Fig. 8. The TES (ch1) and its FFT waveforms (ch2) for disk running at 75 Hz when the FACT turned off.

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SðsÞ and the complementary sensitivity function T ðsÞ can be deﬁned as S ¼ *e r¼ 1 1 þ GfCf ; ð16Þ T ¼e r¼ GfCf 1 þ GfCf : ð17Þ

They are identiﬁed by the synchronized sinusoidal excitation rðtÞ shown in the measuring setup Fig. 11. Since the runout disturbances dðtÞ exist during the identiﬁcation process, the relationship among *eðtÞ; rðtÞ and dðtÞ is described by

*e ¼ 1 1 þ GfCf

ðr dÞ; ð18Þ

where the measurement of *eðtÞ is always contaminated by the runout disturbance dðtÞ with significant contents at the frequencies of 75, 150, 225, 300 Hz; etc. This implies that the identification of (16) becomes impos-sible. Alternatively, defining the approximate sensitivity

function as

Sa¼*e=r ð19Þ

then SaðsÞ is the same as SðsÞ for all frequencies except at

those harmonics and is extracted experimentally by measuring the gain and phase between output *eðtÞ and sinusoidal input rðtÞ sweeping from 40 to 4 kHz:

Figs. 12 and 13 show the frequency responses of SaðsÞ

before and after the FACT function were added on, respectively. Without the FACT in Fig. 12, there are signiﬁcant peak values at the ﬁrst four harmonic frequencies, which arise from the disturbance dðtÞ at

Fig. 9. The individual component cancellation of the FACT. The uncompensated components will neither be enlarged nor reduced. Plots (a)–(d) are the 1st, 2nd, 3rd and 4th harmonics been cancelled, respectively.

Fig. 10. The TES and its FFT waveforms after four banks of FACT turned on.

ρ

f G_f

Fig. 11. The identiﬁcation measuring setup for track-following system.

10 0 -10 -20 -30 -40 -50 -60 Mag (dB) 102 ₁₀3 Frequency (Hz)

Fig. 12. The magnitude of SaðsÞ for disk running at CAV 75 Hz without the FACT.

10 0 -10 -20 -30 -40 -50 -60 Mag (dB) Frequency (Hz) 102 103

Fig. 13. The magnitude of SaðsÞ after the four banks of FACT turned on.

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their harmonic frequencies as depicted in (18). This leads to the result that the system has poor steady-state performance at these frequencies. It is clear that, as shown in Fig. 13, the FACT provides a mechanism causing the magnitude of SaðsÞ small at low frequencies,

which is desired for good tracking and disturbance rejection. Hence, the stability margins at the frequencies of the ﬁrst four harmonics have been extended, thereby achieving better system stability. Similarly, the approx-imate complementary sensitivity function Ta¼ e=r is

also contaminated by the runout disturbance dðtÞ; and can be identiﬁed without and with the FACT as shown, respectively, in Figs. 14 and 15. The FACT makes the magnitude of TaðsÞ close to unity at the ﬁrst

four harmonic frequencies, while the roll off behavior at high frequencies is reserved for high-frequency noise rejection.

4.2.2. Variable play speed verification

When the disk is speed up to 113 Hz ð6780 rpmÞ; the ﬁrst four harmonic components become 113, 226, 339

and 452 Hz; and the corresponding frequency responses of W ðsÞ are also listed inTable 1.Figs. 16 and 17show the time series and power spectrum of TES, before and after, respectively, the four banks of FACT are turned on. It is clear that the FACT works properly through a new set of coefﬁcients of W ðonÞ at an alternative disk

speed.

More experiments are done on the disk running at slower CLV modes. As used on the CAV mode, the FACT can be also implemented without changing the original control structure. Since the disk speed varies for different tacking ranges at the CLV mode, the values of WrðonÞ and WiðonÞ in (15) must change

accordingly. Fortunately, these two variables have little change in low-frequency range as indicated in

Fig. 7. That leads to the choice of a set of mean values of these two variables over a speciﬁc disk speed range at CLV mode to fulﬁll the FACT satisfactorily.Figs. 18 and 19 show the experimental results for the CLV case. The former presents the uncompensated TES signal and its power spectrum, while the latter reveals

8 6 4 2 0 -2 -4 -6 -8 102 103 Frequency (Hz) Mag (dB)

Fig. 14. The magnitude of TaðsÞ for disk running at CAV 75 Hz without the FACT.

8 6 4 2 0 -2 -4 -6 -8 Mag (dB) Frequency (Hz) 102 ₁₀3

Fig. 15. The magnitude of TaðsÞ when the FACT were turned on.

Fig. 16. The TES and its FFT waveforms for disk running at 113 Hz when FACT turned off.

Fig. 17. The TES and its FFT waveforms for disk running at 113 Hz after FACT turned on.

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satisfactory runout rejection by the FACT. It is apparent that the dominant fundamental harmonic has been cancelled completely.

4.3. The settling properties of FACT

Indeed, the CD-ROM drives operate between the track-following and track-seeking one after another. At the end of seeking, the track-following proceeds immediately to read data from the target track. There-fore, the settling ability at the beginning of track-following becomes another requirement for runout rejection. The proposed FACT algorithm provides not only an efﬁcient mechanism to cancel the runout effect during the track-following operation, but also a simple switching setup to fulﬁll the connection to the track-seeking phase. According to the block diagram of the FACT in Fig. 4, the adapting function in (15) can be turned off temporarily during the seeking operation. The last adapted values of xn and yn generate the

compensated output vðtÞ during the track-seeking, resulting in the lens moving in phase of the disk runout. For instance, a short outward seeking is examined for 10 tracks at a CAV mode of 113 Hz to show the suitable settling performance of the FACT. InFig. 20, there is a steady and signiﬁcant tracking error before and after the seeking operation for the feedback track-following without FACT. An acceptable settling behavior at the

transient period of the track-following may be obtained through well designed feedback controllers and the velocity profile used in the seeking operation. Having the FACT added on, the tracking error can be reduced significantly before and after the seeking period as shown inFig. 21. It is clear that the fast settling process during the track pull-in phase is reserved and the runout rejection is fulfilled immediately as soon as the controller switched from the seeking operation.

By contrast with the method raised bySri-Jayantha, Dang, Yoneda, Kitazaki, and Yamnmoto (2001), a settle-out penalty was used at the transient period of the track-following for suppressing the disk runout. How-ever, the proposed FACT has the merit of compensating the disk runout all the way through seeking and track-following operations with continuously adapted control parameters. Once the compensation parameters xn and

yn are initially tuned and converged during the ﬁrst

track-following mode, they are remained during the following track seeking operation; and the ﬁne tuning proceeds at each start of the track-following, while no additional penalty is needed for the reduction of

Fig. 18. The TES and its FFT waveforms for disk running at CLV when FACT turned off.

Fig. 19. The TES and its FFT waveforms for disk running at CLV after FACT turned on.

Fig. 20. The settling property between track-following and seeking operation in the original system without the FACT.

Fig. 21. The settling property with the FACT added on. The fast settling process could be reserved without the settle-out penalty.

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tracking error. That way, the convergence rate is not critical and the control performance will not be changed between the switching of the track-following and seeking operation.

5. Conclusion

A novel frequency adaptive control technique is proposed and examined on a CDD. Both the theoretical and experimental results show that the periodic runout harmonics can be rejected simultaneously and efﬁciently for both CAV and CLV spindle modes under variable playing speed. The rejection devoted to some candidate harmonic components will not inﬂuence the uncompen-sated ones. In view of application requirement, the proposed FACT can be extended for the runout rejection to various compact disk drives, such as CD-ROM, CD-RW or DVD-ROM drives.

Appendix A. Proof of the Theorem 1

Let

W_Nn ¼ exp j2p Nn

¼ cos ynþ j sin yn;

where yn¼ 2pn=N and n ¼ 0; 1; y; N 1: Since the

FSF has N 1 zeros at z ¼ W_Ni for ian; the output of the nth elementary FSF for input eðtÞ given by (2) is the same as the output driven by the nth input component enðtÞ ¼ ancos ont þ bnsin ont: ð20Þ

Once the input eðtÞ has been sampled, its nth component can be denoted by

enðkÞ ¼ ancos kynþ bnsin kyn: ð21Þ

Thus,

xnðkÞ ¼ HnðkÞeðkÞ

¼ eðkÞ þ W_Nneðk 1Þ þ W_N2neðk 2Þ þ ? þ W_NðN1Þneðk N þ 1Þ ¼X

N1

i¼0

ðcos iynþ j sin iynÞðancosðk iÞyn

þ bnsinðk iÞynÞ

¼ 2 X

N=21

i¼0

ðcos iynþ j sin iynÞðancosðk iÞyn

þ bnsinðk iÞynÞ; ð22Þ

where the following properties of trigonometric func-tions applies:

cosðN þ iÞyn¼ cos iyn; sinðN þ iÞyn¼ sin iyn;

cos N 2 þ i yn ¼ ð1Þncos iyn; sin N 2 þ i yn¼ ð1Þnsin iyn:

For any kXN; summing up all the real and imaginary parts of (22) yields

xnðkÞ ¼

N

2 ½ðancos kynþ bnsin kynÞ

þ jðansin kyn bncos kynÞ: ð23Þ

Hence, Eqs. (3) and (4) follow. Furthermore, elementary trigonometric function properties

cos ynþN=2¼ cos yn; sin ynþN=2¼ sin yn

implies

xnþN=2¼ xn ð24Þ

which reveals that only N=2 outputs for the bank of FSF are independent. It follows that the least value of N is N ¼ 2 M þ 2 for the maximum order M: Finally, the transformation from (4) to (6) is straightforward by the transforming matrix

D ¼ cos ot sin ot sin ot cos ot

" #

:

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