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ADAPTIVE SUPPRESSION OF HIGH-FREQUENCY WIDE-SPECTRUM VIBRATIONS

WITH APPLICATION TO DISK DRIVE SYSTEMS

Liting Sun

Department of Precision Machinary

and Precision Instrumentation

University of Science and Technology

of China

Hefei, Anhui, China, 230027

Email: litingsun@me.berkeley.edu

Xu Chen

Department of Mechanical Engineering

University of California at Berkeley

Berkeley, California, 94720

Email: maxchen@me.berkeley.edu

Masayoshi Tomizuka

Department of Mechanical Engineering

University of California at Berkeley

Berkeley, California, 94720

Email: tomizuka@me.berkeley.edu

ABSTRACT

In the big-data era, requirements for storage capacity and

access speed in modern Hard Disk Drive (HDD) systems are be-

coming more and more stringent. As the track density of HDDs

increases higher and higher, vibration suppression of the record-

ing arm in HDDs has become more and more challenging. Vi-

brations in modern HDDs are environment/product-dependent.

Furthermore, they can occur at very high frequencies with wide

spectral peaks. This paper presents an adaptive algorithm to

identify and suppress these high-frequency wide-spectrum vibra-

tions. We design a vibration-compensation controller based on

an adaptive disturbance observer (DOB), and devise parame-

ter adaptation for not only the vibration frequency but also the

width of the vibration spectral peak. The peak-width parame-

ters are adaptively tuned online to minimize the error ampliﬁca-

tions resulted from the waterbed effect of the sensitivity function.

The proposed algorithm is veriﬁed by simulations of HDDs for

the problem of suppressing high-frequency wide-spectrum vibra-

tions.

1 INTRODUCTION

According to International Data Corporation (IDC) study

[1], the total amount of data is now explosively increasing and

will reach 8ZB (270 bytes) in 2015. The majority of the data

needs to be stored in hard disk driver systems (HDDs), e.g., for

cloud computing, analysis and management. This has created

demanding and stringent requirements of promoting the storage

capacity and data access speed in modern HDDs. However, as

the track density becomes higher and higher, the control of the

read/write head becomes more and more challenging. One of

the greatest challenges comes from vibrations, which can oc-

cur in modern HDDs with energy highly concentrated at sev-

eral frequency bands, i.e., wide spectral peaks. Both the center

frequencies and the peak widths can change in different opera-

tion environments or within different HDD products. Further-

more, the spectral peaks can sometimes lie beyond the open loop

servo bandwidth [2, 3], which makes these high-frequency wide-

spectrum vibrations difﬁcult to suppress by traditional feedback

controllers.

Various control algorithms have been developed for vibra-

tion suppression. Among them, disturbance observers (DOBs)

[4] has attracted a great amount of attention due to its simplicity,

light computational burden and good performance. For exam-

ple, White et. al [5] augmented a typical feedback loop of a disk

drive servo system with a DOB and realized 61%−96% reduc-

tion of the vibrations at frequencies below 100Hz. Zheng and

Tomizuka [6] suggested an adaptive disturbance observer which

estimated the frequency of the disturbance and then canceled it.

Jia [7] also incorporated the adaptive frequency estimation into

the traditional disturbance observer and achieved vibration rejec-

tion without losing the phase margin of the feedback system. Xu

et. al [8–11] introduced a minimum-parameter adaptive Q ﬁlter

in DOB and extended it to multiple-band cases, where distur-

bances with multiple spectral peaks can be accurately estimated

and effectively rejected.

Most of previous studies have been focusing on the adap-

tation of the center frequencies with ﬁxed width parameters

[6,8–11, 13, 14]. The adaptation of the peak width, which also

greatly inﬂuences the control performance, has seldom been dis-

cussed. Levin and Ioannou [15] proposed an controller with

adaptive bandwidth using multirate adaptive notch ﬁlter (ANF),

but still adaptation only happens to the estimate of the system

modal frequency and the tuning of the bandwidth is realized by

multiple pre-designed controllers. This paper proposes a con-

troller based on adaptive DOB to suppress the above mentioned

high-frequency wide-spectrum vibrations. Different from pre-

vious studies, in the proposed algorithm, we devise parameter

adaptation for not only the vibration frequency but also the spec-

tral peak width of the vibration. Another difference is that we

incorporates a new lattice-form IIR (Inﬁnite Impulse Response)

notch ﬁlter [16] into the Q-ﬁlter design in DOB, which offers

us more convenience and ﬂexibility to adaptively tune the width

parameter.

The remainder of the paper is organized as follows. Section

2 describes the hard disk drive system and formulates the prob-

lem. The proposed DOB structure and Q-ﬁlter design based on

a new lattice-formed IIR notch ﬁlter are presented in Section 3.

Section 4 gives the adaptation algorithm for the width of the vi-

bration spectral peak. Section 5 veriﬁes the effectiveness of the

proposed algorithm through simulation results. Conclusions are

summarized in Section 6.

2 HARD DISK DRIVE SYSTEM

Notations throughout this paper are as follows:

P(z−1)– HDD full-order plant

Pn(z−1)– HDD nominal model without delay

z−m–m-step delay in the HDD plant

C(z−1)– Baseline controller in HDD feedback loop

Q(z−1)– Q ﬁlter in DOB

y(k)– Output signal

e(k)– Position error signal (PES)

u(k)– Input control signal

d(k)– Disturbance signal

n(k)– Measurement noise

ˆ

d(k)– Estimated disturbance signal

S0(z−1)– Baseline sensitivity function

S(z−1)– Sensitivity function of proposed control scheme

Figure 1 shows the frequency responses of the full-order

HDD plant P(z−1)and its low-order nominal model z−mPn(z−1)

provided by the well formulated open-source HDD simulation

benchmark [17]. The sampling frequency Fsis 26400Hz. Sev-

eral pre-designed notch ﬁlters1have been added into P(z−1)for

high-frequency-resonance cancellation. From the frequency re-

sponse, we can see that the nominal model accurately matches

the actual dynamics of the plant up to about 2kHz.

The baseline sensitivity function S0(z−1)obtained by a PID

controller C(z−1)is shown in Fig. 2. Note that above the

1These Notch ﬁlters are designed to cancel the physical resonances of the

plant. Four second-order notch ﬁlters are incorporated, with notches centered at

4.1kHz, 5.0kHz, 8.2kHz and 12.3kHz. Note that the center frequencies of the

notches introduced here differ fromthe one addressed in Section 3.

−20

0

20

40

60

80

100

Magnitude (dB)

101102103104

−180

−90

0

90

180

Phase (deg)

Frequency (Hz)

HDD full−order plant

HDD nominal model

Figure 1. Frequency responses of the full-order plant and its nominal

model

servo bandwidth, the system has limited disturbance rejection

property and has a wide error ampliﬁcation region due to the

”waterbed” effect. If HDDs encounter high-frequency wide-

spectrum excitations, the ampliﬁed vibrations will greatly de-

grade the track-following performance. Thus, more considera-

tions must be given at those frequencies to ”locally” shape the

sensitivity function for enhanced vibration suppression [18]. In

this paper, we assume that all excitations enter the HDDs as a

lumped disturbance d(k).

101102103104

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−40

−30

−20

−10

0

10

Magnitude (dB)

Frequency (Hz)

Figure 2. Frequency responses of the baseline sensitivity function

3 DOB STRUCTURE WITH NEW LATTICE-FORM

NOTCH FILTER IN Q-FILTER DESIGN

3.1 DOB Structure for Vibration Suppression

The proposed DOB structure for vibration suppression is

shown in Fig. 3, where the disturbance estimate ˆ

d(k)is con-

structed as follows:

ˆ

d(k) = P(q−1)(d(k) + u(k)) + n(k)P−1

n(q−1)−q−mu(k)

=P(q−1)P−1

n(q−1)d(k) + P(q−1)P−1

n(q−1)−q−mu(k)

+P−1

n(q−1)n(k),(1)

where q−1denotes the one-step delay operation in time domain.

The transfer function from d(k)to e(k)is

Gd2e(z−1) = −(1−z−mQ(z−1))P(z−1)

1+P(z−1)C(z−1) + Q(z−1)P(z−1)P−1

n(z−1)−z−m

(2)

e(k)

C(z¡1

)

u(k

)

y(k

)

P(z

¡1

)

z

¡m

Q(z¡1)

P

¡1

n

(z

¡1

)

r(k)=0

^

d(k

)

Adaptation Algorithm based

on Lattice Notch Filter

n(k

)

d(k)

F(z¡1

)

Figure 3. The proposed DOB structure for vibration suppression

Notice that an accurate model is available at frequencies

lower than 2kHz (see Fig. 1), i.e., z−mPn(z−1)≈P(z−1)below

2kHz. Apply this approximation to simplify Eqs. (1) and (2) as:

ˆ

d(k)≈q−md(k) + P−1

n(q−1)n(k)(3)

Gd2e(z−1)≈ − P(z−1)(1−z−mQ(z−1))

1+P(z−1)C(z−1)(4)

Note that the factor 1/[1+P(z−1)C(z−1)] in Eq. (4) is the base-

line sensitivity function S0(z−1). Thus, we can derive the closed-

loop sensitivity function S(z−1)as:

S(z−1)≈1−z−mQ(z−1)S0(z−1)(5)

i.e., S(z−1)is approximately the product between S0(z−1)and

1−z−mQ(z−1). This approximation transforms the shaping of

the sensitivity function into the design of a proper Q ﬁlter, thus

strong design ﬂexibility is introduced.

When model uncertainty exists such that:

P(z−1) = z−mPn(z−1)(1+△(z−1)) (6)

where △(z−1)represents the multiplicative uncertainty term.

Then Gd2e(z−1)in Eq.(2) becomes

Gd2e(z−1) = P(z−1)(z−mQ(z−1)−1)

1+P(z−1)C(z−1) + Q(z−1)z−m△(z−1)(7)

A sufﬁcient condition to guarantee the robust stability of

the closed-loop system is: |Q(z−1)z−m△(z−1)|<|1+

P(z−1)C(z−1)|, i.e.,

|Q(e−jω)|<

1

S0(e−jω)△(e−jω)

,∀ω(8)

3.2 Q-ﬁlter Design based on Lattice-form IIR Notch

Filters

Equation (5) indicates that the design of Q ﬁlter in this DOB

structure is of great importance for vibration suppression. Recall

from Eq. (3), that the estimated disturbance ˆ

d(k)is a contam-

inated m-step delayed disturbance signal. Therefore, in order

to effectively compensate and cancel the wide-band disturbance,

the Q ﬁlter should be a band-pass ﬁlter whose passband is lo-

cated at the wide spectral peak of the disturbance [8]. With a

well tuned passband in Q, main frequency components of the

disturbance will be ﬁltered out and fed back into the control sig-

nal for cancellation. The steady state PES e(k)will be given by

(within frequency ranges where z−mPn(z−1)≈P(z−1)is satisﬁed

and the measurement noise n(k)is small compared to the distur-

bance contribution):

e(k) = −1−q−mQ(q−1)S0(q−1)P(q−1)d(k)

=−S0(q−1)AQ(q−1)−q−mBQ(q−1)

AQ(q−1)P(q−1)d(k)(9)

where BQ(q−1)and AQ(q−1)are the numerator and denominator

of the Q ﬁlter, respectively.

Assume that d(k)contains only one spectral peak centered

at f0(in Hz), then according to Internal Mode Principle (IMP),

we have (1−2cosω0z−1+z−2)sin(ω0k+φ) = 0 , where ω0=

2πf0Ts. Therefore, to obtain a small steay state e(k)in the pres-

ence of d(k), 1−z−mQ(z−1)in Eq. (9) should contain an IMP-

based notch ﬁlter N(z−1), whose notch width (NW, difference

between the upper and lower frequencies where the notch ﬁlter

gains are -3dB) can be optimally devised to cover the spectral

peak of d(k). Here, we choose the denominator of Q(z−1)to be

equal to that of N(z−1), i.e., AQ(z−1) = AN(z−1), as shown in

Eqs. (10) - (12). J(z−1)in Eq. (10) is a FIR (Finite Impulse

Response) polynomial of z−1.

1−z−mQ(z−1) = N(z−1)J(z−1)(10)

AQ(z−1)−z−mBQ(z−1)

AN(z−1)=BN(z−1)

AN(z−1)J(z−1)(11)

1−2cosω0z−1+z−2=BN(z−1)(12)

With this choice, Eq. (11) can be written as:

AN(z−1) = z−mBQ(z−1) + BN(z−1)J(z−1)(13)

Equation (13) is a Diophantine Equation (DE) where we can

solve for BQ(z−1)and J(z−1)once a proper notch ﬁlter N(z−1)

is designed.

Now we havetransformed the shaping of S(z−1)into design-

ing a proper notch ﬁlter with desired notch width at the center fre-

quency of the vibration spectral peak, f0. As for the structure of

notch ﬁlters, previous researches on ANF or adaptive DOB have

studied the usage of direct-form notch ﬁlters. Transfer function

of the direct-form notch ﬁlter is given in Eq. (14), where ω0de-

termines the center frequency of the notch and αdetermines the

notch width, as deﬁned in Eq. (15). The smaller the parameter α

is, the wider is the notch width (NW).

N(z−1) = 1−2cosω0z−1+z−2

1−2αcosω0z−1+α2z−2=A(z−1)

A(αz−1)(14)

NW ≈π(1−α)(15)

As mentioned earlier, the wide-band vibrations are always

environment- or product-dependent. Both the center frequency

of the spectral peak and the peak width will change in dif-

ferent situations. Adaptation to both the frequency parameter

cosω0and width parameter αis required for better vibration-

suppression performance. However, the nonlinearity of the

direct-form notch ﬁlter in Eq. (14) with respect to αmakes the

adaptation of width quite difﬁcult to implement. Also, the fre-

quency response of N(z−1)in Eq. (14) is not symmetric in low

and high frequencies, as shown in Fig. 4. Although N(z−1)

provides enhanced low gain near the notch center and at low

frequencies, the high frequency gain should be carefully han-

dled [18] to avoid undesired noise/disturbance ampliﬁcations.

The wider the notch, the more pronounced the increase of the

high-frequency gains, which may become problematic.

To solve this problem, a new lattice-form notch ﬁlter

[16], denoted by NL(z−1), is introduced in the proposed wide-

spectrum vibration suppression algorithm. The transfer func-

tion of NL(z−1)and its frequency responses with different notch

101102103104

−160

−140

−120

−100

−80

−60

−40

−20

0

20

40

Magnitude (dB)

Frequency (Hz)

Direct−form notch filter with α=0.5

Direct−form notch filter with α=0.95

Figure 4. Frequency responses of direct-form notch ﬁlters with different

notch widths

widths are shown in Eq. (16) and Fig. 5, respectively.

NL(z−1) = 1−2cosω0z−1+z−2

1−(1+αL)cosω0z−1+αLz−2=BNL(z−1)

ANL(z−1)(16)

NW =2arctan 1−αL

1+αL(17)

Compared to Eq. (14), we can see that NL(z−1)is bilinear with

respect to both the frequency parameter cosω0and the width pa-

rameter αL, and that it is quite amenable to adaptive algorithms.

Also Fig. 5 shows that lattice notch ﬁlters have nice symmetric

gains at low and high frequencies, regardless of the notch loca-

tion and width.

101102103104

−160

−140

−120

−100

−80

−60

−40

−20

0

20

40

Magnitude (dB)

Frequency (Hz)

Lattice−form notch filter with αL = 0.5

Lattice−form notch filter with αL = 0.95

Figure 5. Frequency responses of lattice-form notch ﬁlters with different

notch widths

Combining Eqs. (11), (13) and (16), we can solve for the Q

ﬁlter once NL(z−1)is designed and the passband of the Q ﬁlter is

also determined by αL. For example, if the system has one-step

delay, i.e., m=1, then the corresponding Q ﬁlter is given by

Q(z−1) = (1−αL)cosω0+ (αL−1)z−1

1−(1+αL)cosω0z−1+αLz−2.(18)

Figure 6 shows the frequency response of Eq. (18), where the

band-pass property is evident.

−40

−30

−20

−10

0

Magnitude (dB)

101102103104

−90

0

90

180

Phase (deg)

Frequency (Hz)

Q filter designed based on a lattice−form notch filter

Figure 6. Frequency response of the Q ﬁlter designed based on a lattice-

form notch ﬁlter (m=1)

4 PEAK-WIDTH ADAPTATION ALGORITHM

Recalling Eqs. (9), (10) and (11), we aim at designing an

adaptive Q ﬁlter to minimize the PES signal e(k)in the presence

of wide-spectrum vibrating excitations d(k). The design of Q ﬁl-

ter requires knowledge of the frequency information β=cosω0

and the width parameter αLwhich are not available in advance.

As a number of previous studies have addressed the frequency

identiﬁcation methods explicitly, such as [10, 11, 19], we assume

in this section that βis known and focus on the adaptation of

αL. As mentioned above, to suppress the wide-band vibrations,

NL(z−1)should have desired notch width which is wide enough

to remove the spectral peak, but not too wide to become a non-

notch ﬁlter 2and bring intolerant ampliﬁcations at other frequen-

cies due to the ”waterbed” effect. A nice balance between the

positive effect (effective removal of the peak) and the negative

effect (ampliﬁcations at other frequencies) should be found adap-

tively.

Equation (9) suggests that minimizing e(k)without chang-

ing S0(z−1)is equivalent to minimizing (1−q−mQ(q−1))eb(k),

where eb(k) = P(q−1)S0(q−1)d(k)reﬂects the baseline PES.

2As the notch becomes wider, the depth of the notch becomes shallower.

ˆ

d(k), as a m-step delayed contaminated estimate of d(k)(Eq.

(3)), has similar spectral characteristics with d(k)(when n(k)is

small). In Fig. 3, let F(z−1) = Pn(z−1)/(1+Pn(z−1)C(z−1)),

i.e., F(z−1)is a nominal version of P(z−1)S(z−1)and pass ˆ

d(k)

through F(z−1). The output is denoted by c(k) = F(q−1)ˆ

d(k),

which will thus preserve the spectral characteristics of eb(k).

Therefore, the best notch width αo

Lis obtained by minimizing

the cost function as follows:

Vk=

k

∑

j=1

1

2λk−j(k)[e0

L(j)]2(19)

where λ(k)is a time-varying forgetting factor satisfying:

λ(k) = λend −(λend −λ(k))λrate ∈(0,1),

and e0

L(j)is the priori estimation error, obtained by passing c(k)

through an adaptive lattice-form IIR notch ﬁlter NL(z−1), i.e.,

e0

L(k) = NL(q−1)c(k) = 1−2βq−1+q−2c(k)

1−(1+ˆ

αL(k−1))βq−1+ˆ

αL(k−1)q−2,

which is equivalent to

e0

L(k) = β(1+ˆ

αL(k−1))e0

L(k−1)−ˆ

αL(k−1)e0

L(k−2)

+c(k)−2βc(k−1) + c(k−2).(20)

For the related adaptation, we apply the Recursive

Prediction-Error Method (RPEM) (chapter 11 in [20]) which

guarantees unbiased local convergence during adaptation. The

recursive adaptation algorithm is expressed as follows:

F(k) = 1

λ(k)F(k−1)−F(k−1)φ(k−1)φT(k−1)F(k−1)

λ(k) + φT(k−1)F(k−1)φ(k−1)

(21)

ˆ

αL(k) = ˆ

αL(k−1) + F(k−1)φ(k−1)e0

L(k)(22)

where φ(k−1)is the gradient of e0

L(k)with respect to the

latest estimated parameter ˆ

αL(k−1), deﬁned by φ(k−1) =

−∂e0

L(k)/∂ˆ

αL(k−1).

To obtain φ(k−1), we notice the following relationship,

φ(k−1) = ∂e0

L(k)

∂ˆ

αL(k−1)=∂NL(ˆ

αL(k−1),q−1)

∂ˆ

αL(k−1)c(k)(23)

=∂NL(ˆ

αL(k−1),q−1)

∂ˆ

αL(k−1)N−1

L(ˆ

αL(k−1),q−1)e0

L(k)

=−q−1(q−1−β)

1−(1+ˆ

αL(k−1))βq−1+ˆ

αL(k−1)q−2e0

L(k)

So φ(k−1)can be expressed as follows:

φ(k−1) = β[1+ˆ

αL(k−1)]φ(k−2)−ˆ

αL(k−1)φ(k−3)

+βe0

L(k−1)−e0

L(k−2)(24)

To improve the estimation precision and increase the con-

vergence rate, the posteriori error ¯eL(k)is introduced to update

Eq. (20) and Eq. (24) [21]. The adaptation algorithm can then

be summarized as follows:

Initialization:

ˆ

α(0) = 0.5; F(0) = 100/E[e0

L(0)]2;φ(0) = φ(−1) = 0; ¯eL(−1) =

¯eL(−2) = 0.

Main loop: for k=1,2,...

e0

L(k) = β(1+ˆ

αL(k−1)) ¯eL(k−1)−ˆ

αL(k−1)¯eL(k−2)

+c(k)−2βc(k−1) + c(k−2)(25)

ˆ

αL(k) = ˆ

αL(k−1) + F(k−1)φ(k−1)e0

L(k)(26)

F(k) = 1

λ(k)F(k−1)−F(k−1)φ(k−1)φT(k−1)F(k−1)

λ(k) + φT(k−1)F(k−1)φ(k−1)

(27)

¯eL(k) = β(1+ˆ

αL(k)) ¯eL(k−1)−ˆ

αL(k)¯eL(k−2)

+c(k)−2βc(k−1) + c(k−2)(28)

φ(k) = β[1+ˆ

αL(k)]φ(k−1)−ˆ

αL(k)φ(k−2)

+β¯eL(k)−¯eL(k−1)(29)

REMARK 1: if no priori knowledge of the center frequency

parameter βis available, adaptation will be divided into two

stages because of the product term of βand αLin NL(z−1). In

the ﬁrst stage, βis estimated with a ﬁxed notch width using ANF

techniques. At this stage, αLcan be set to be close to 1 for more

accurate frequency estimation. Then with the estimated β, adap-

tation will be switched to αLusing the proposed algorithm.

REMARK 2: Due to the local-minima convergence of

RPEM, initial values of αLwill inﬂuence the performance. For

stability enhancing, lower and higher bounds for αLshould be

introduced during the adaptation.

5 CASE STUDY AND SIMULATION RESULTS

The proposed high-frequency wide-spectrum vibration sup-

pression algorithm based on adaptive DOB was implemented in

the HDD simulation benchmark [17]. As mentioned in Section

2, several notch ﬁlters have been constructed and added into the

HDD system for high-frequency-resonance cancellation. The de-

lay in the augmented plant P(z−1)is m=3, with a sampling time

of Ts=3.7878×10−5sec. A set of vibration excitations gener-

ated from actual HDD measurements with wide high-frequency

spectral peaks are used for algorithm veriﬁcation.

Figure 7 and Fig. 8 show the spectrum of the vibration sig-

nal and the corresponding baseline PES signal. Recall the base-

line controller in Fig. 3 and the baseline sensitivity function in

Fig. 2, we can see clearly that the ﬁrst spectral peak centered at

around 100Hz has been signiﬁcantly attenuated by the baseline

controller, but the second peak at 1172Hz is greatly ampliﬁed,

resulting in a large PES in HDDs.

100101102103104

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Frequency (Hz)

Spectrum of a vibration signal with wide spectral peaks

1172Hz

Figure 7. Spectrum of a vibration signal with wide spectral peaks

100101102103104

0

0.005

0.01

0.015

0.02

Frequency (Hz)

Spectrum of the baseline PES 1172Hz

Figure 8. Spectrum of the baseline PES signal in the presence of vibra-

tions in Figure 7

Figure 9 illustrates the relationship between the peak width

parameter αLand the vibration-suppression performance of the

DOB structure with the center frequency at 1172Hz. It indicates

that (1) with a DOB compensator as shown in Fig. 3 (within

the dashed box), vibrations will be greatly suppressed; (2) there

exists an optimal value for the width parameter αL(αopt

L=0.83

in this case study) for minimization of the 3σvalue of the PES

signal.

The proposed passband-adaptive DOB is implemented for

vibration suppression. The adaptation of the passband parameter

αLin Q ﬁlter is shown in Fig. 10, where the converged value

is αopt

L=0.86, very close to the manually tuned optimal value

in Fig. 9. This is due to the local convergence of the RPEM

0.4 0.5 0.6 0.7 0.8 0.9 1

0.132

0.138

0.144

0.150

Width parameter in designing Q filter (αL)

3σ of PES w/o DOB compensation

3σ of PES w/ DOB compensation

Figure 9. Relationship between the peak width parameter αLand the

performance of the DOB compensator

algorithm. However, as shown in Fig. 9, the converged value of

the proposed algorithm is within the suboptimal set which will

provide sufﬁcient suppression to the vibrations.

0 0.5 1 1.5 2 2.5 3

0.2

0.4

0.6

0.8

1

Time(s)

Width parameter αL

Figure 10. PES time trace with and without passband-adaptive DOB

Time trace and spectrum of the PES obtained by the pro-

posed adaptive algorithm are shown in Fig. 11 and Fig. 12,

respectively. It can be seen that the 3σvalue of the PES has been

reduced from 18.57% to 13.25%. The spectral peak at 1172Hz

also has been effectively removed without causing large ampliﬁ-

cations at other frequencies. The resulted new sensitivity func-

tion in Fig. 13 shows a signiﬁcant performance enhancement at

around 1172Hz.

6 CONCLUSIONS

In this paper, an adaptive DOB control algorithm is proposed

to suppress high-frequency wide-spectrum vibrations in HDD

systems. To handle the time-varying characteristics of the center

frequency and spectral peak width, a new lattice-form notch ﬁlter

is introduced to the design of Q ﬁlter such that the passband of

0 1 2 3 4 5 6 7

−30

−20

−10

0

10

20

30

Time(s)

PES(%TP)

T0

3 σ of PES = 13.25%TP

w/ compensation

3 σ of PES =18.57%TP

w/o compensation

Figure 11. PES time trace with and without passband-adaptive DOB

0 500 1000 1500 2000 2500 3000 3500 4000

0

0.01

0.02

0 500 1000 1500 2000 2500 3000 3500 4000

0

0.01

0.02

Frequency(Hz)

Spectrum of PES with the adapted passband in Q

Spectrum of baseline PES

Figure 12. Relationship between the peak width parameter αLand the

performance of the DOB compensator

Frequency (Hz)

101102103104

−100

−80

−60

−40

−20

0

20

Magnitude (dB)

New sensitivity function with

adapted width parameter α_L

Baseline sensitivity function

Figure 13. The baseline sensitivity function and the new sensitivity func-

tion with passband-width adaptive DOB

the Q ﬁlter can be adaptively tuned in the presence of different

vibrations. Simulations on a open-source HDD benchmark show

that the proposed algorithm can effectively ﬁnd a proper width

parameter which effectively removes the main spectral peak in

the error signal without degrading the performance at other fre-

quencies.

ACKNOWLEDGMENT

This work was supported by the Chinese Scholarship Coun-

cil (CSC) of China.

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