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Adaptive Suppression of High-Frequency Wide-Spectrum Vibrations With Application to Disk Drive Systems

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In the big-data era, requirements for storage capacity and access speed in modern Hard Disk Drive (HDD) systems are becoming more and more stringent. As the track density of HDDs increases, vibration suppression of the recording arm in HDDs is becoming more challenging. Vibrations in modern HDDs are environment/product-dependent with different frequency characteristics. 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 widespectrum vibrations. We design a vibration-compensation controller based on an adaptive disturbance observer (DOB), and devise parameter adaptation algorithms not only for the vibration frequencies but also for the spectral peak widths of the vibration. The peak-width parameters are adaptively tuned online to maximally attenuate the vibration with minimal error amplifications at other frequencies. The proposed algorithm is verified by simulations of HDDs for the problem of suppressing highfrequency wide-spectrum vibrations.
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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(z1)– HDD full-order plant
Pn(z1)– HDD nominal model without delay
zmm-step delay in the HDD plant
C(z1)– Baseline controller in HDD feedback loop
Q(z1)– 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(z1)– Baseline sensitivity function
S(z1)– Sensitivity function of proposed control scheme
Figure 1 shows the frequency responses of the full-order
HDD plant P(z1)and its low-order nominal model zmPn(z1)
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(z1)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(z1)obtained by a PID
controller C(z1)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)
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−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).
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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(q1)(d(k) + u(k)) + n(k)P1
n(q1)qmu(k)
=P(q1)P1
n(q1)d(k) + P(q1)P1
n(q1)qmu(k)
+P1
n(q1)n(k),(1)
where q1denotes the one-step delay operation in time domain.
The transfer function from d(k)to e(k)is
Gd2e(z1) = (1zmQ(z1))P(z1)
1+P(z1)C(z1) + Q(z1)P(z1)P1
n(z1)zm
(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., zmPn(z1)P(z1)below
2kHz. Apply this approximation to simplify Eqs. (1) and (2) as:
ˆ
d(k)qmd(k) + P1
n(q1)n(k)(3)
Gd2e(z1) P(z1)(1zmQ(z1))
1+P(z1)C(z1)(4)
Note that the factor 1/[1+P(z1)C(z1)] in Eq. (4) is the base-
line sensitivity function S0(z1). Thus, we can derive the closed-
loop sensitivity function S(z1)as:
S(z1)1zmQ(z1)S0(z1)(5)
i.e., S(z1)is approximately the product between S0(z1)and
1zmQ(z1). 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(z1) = zmPn(z1)(1+(z1)) (6)
where (z1)represents the multiplicative uncertainty term.
Then Gd2e(z1)in Eq.(2) becomes
Gd2e(z1) = P(z1)(zmQ(z1)1)
1+P(z1)C(z1) + Q(z1)zm(z1)(7)
A sufficient condition to guarantee the robust stability of
the closed-loop system is: |Q(z1)zm(z1)|<|1+
P(z1)C(z1)|, i.e.,
|Q(ejω)|<
1
S0(ejω)(ejω)
,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 zmPn(z1)P(z1)is satisfied
and the measurement noise n(k)is small compared to the distur-
bance contribution):
e(k) = 1qmQ(q1)S0(q1)P(q1)d(k)
=S0(q1)AQ(q1)qmBQ(q1)
AQ(q1)P(q1)d(k)(9)
where BQ(q1)and AQ(q1)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 (12cosω0z1+z2)sin(ω0k+φ) = 0 , where ω0=
2πf0Ts. Therefore, to obtain a small steay state e(k)in the pres-
ence of d(k), 1zmQ(z1)in Eq. (9) should contain an IMP-
based notch filter N(z1), 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(z1)to be
equal to that of N(z1), i.e., AQ(z1) = AN(z1), as shown in
Eqs. (10) - (12). J(z1)in Eq. (10) is a FIR (Finite Impulse
Response) polynomial of z1.
1zmQ(z1) = N(z1)J(z1)(10)
AQ(z1)zmBQ(z1)
AN(z1)=BN(z1)
AN(z1)J(z1)(11)
12cosω0z1+z2=BN(z1)(12)
With this choice, Eq. (11) can be written as:
AN(z1) = zmBQ(z1) + BN(z1)J(z1)(13)
Equation (13) is a Diophantine Equation (DE) where we can
solve for BQ(z1)and J(z1)once a proper notch filter N(z1)
is designed.
Now we havetransformed the shaping of S(z1)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(z1) = 12cosω0z1+z2
12αcosω0z1+α2z2=A(z1)
A(αz1)(14)
NW π(1α)(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(z1)in Eq. (14) is not symmetric in low
and high frequencies, as shown in Fig. 4. Although N(z1)
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(z1), is introduced in the proposed wide-
spectrum vibration suppression algorithm. The transfer func-
tion of NL(z1)and its frequency responses with different notch
101102103104
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−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(z1) = 12cosω0z1+z2
1(1+αL)cosω0z1+αLz2=BNL(z1)
ANL(z1)(16)
NW =2arctan 1αL
1+αL(17)
Compared to Eq. (14), we can see that NL(z1)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
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−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(z1)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(z1) = (1αL)cosω0+ (αL1)z1
1(1+αL)cosω0z1+αLz2.(18)
Figure 6 shows the frequency response of Eq. (18), where the
band-pass property is evident.
−40
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0
Magnitude (dB)
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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(z1)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(z1)is equivalent to minimizing (1qmQ(q1))eb(k),
where eb(k) = P(q1)S0(q1)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(z1) = Pn(z1)/(1+Pn(z1)C(z1)),
i.e., F(z1)is a nominal version of P(z1)S(z1)and pass ˆ
d(k)
through F(z1). The output is denoted by c(k) = F(q1)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
2λkj(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(z1), i.e.,
e0
L(k) = NL(q1)c(k) = 12βq1+q2c(k)
1(1+ˆ
αL(k1))βq1+ˆ
αL(k1)q2,
which is equivalent to
e0
L(k) = β(1+ˆ
αL(k1))e0
L(k1)ˆ
αL(k1)e0
L(k2)
+c(k)2βc(k1) + c(k2).(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(k1)F(k1)φ(k1)φT(k1)F(k1)
λ(k) + φT(k1)F(k1)φ(k1)
(21)
ˆ
αL(k) = ˆ
αL(k1) + F(k1)φ(k1)e0
L(k)(22)
where φ(k1)is the gradient of e0
L(k)with respect to the
latest estimated parameter ˆ
αL(k1), defined by φ(k1) =
e0
L(k)/ˆ
αL(k1).
To obtain φ(k1), we notice the following relationship,
φ(k1) = e0
L(k)
ˆ
αL(k1)=NL(ˆ
αL(k1),q1)
ˆ
αL(k1)c(k)(23)
=NL(ˆ
αL(k1),q1)
ˆ
αL(k1)N1
L(ˆ
αL(k1),q1)e0
L(k)
=q1(q1β)
1(1+ˆ
αL(k1))βq1+ˆ
αL(k1)q2e0
L(k)
So φ(k1)can be expressed as follows:
φ(k1) = β[1+ˆ
αL(k1)]φ(k2)ˆ
αL(k1)φ(k3)
+βe0
L(k1)e0
L(k2)(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(k1)) ¯eL(k1)ˆ
αL(k1)¯eL(k2)
+c(k)2βc(k1) + c(k2)(25)
ˆ
αL(k) = ˆ
αL(k1) + F(k1)φ(k1)e0
L(k)(26)
F(k) = 1
λ(k)F(k1)F(k1)φ(k1)φT(k1)F(k1)
λ(k) + φT(k1)F(k1)φ(k1)
(27)
¯eL(k) = β(1+ˆ
αL(k)) ¯eL(k1)ˆ
αL(k)¯eL(k2)
+c(k)2βc(k1) + c(k2)(28)
φ(k) = β[1+ˆ
αL(k)]φ(k1)ˆ
αL(k)φ(k2)
+β¯eL(k)¯eL(k1)(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(z1). 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(z1)is m=3, with a sampling time
of Ts=3.7878×105sec. 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
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−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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... Sun et. al [17] developed an adaptive DOB with a lattice-form IIR (Infinite Impulse Response) notch filter, which can automatically tune the width of the notch filter online for an optimal overall performance. ...
... γ= {1, α}. This special notch filter N (z −1 ) provides 1) a notch at frequency f 0 and 2) symmetric gains w.r.t f 0 [17]. If there are multiple spectral peaks in d(k), for example, n peaks at ...
... The servo system is subjected to both repeatable (periodic) and non-repeatable (random) disturbances/noises that are due to the imperfection in fabrication and assembly processes, internal and external vibrations Sun et al. (2014Sun et al. ( , 2013; Zheng et al. (2014a,b), and electronic interferences. Fig. 1.2 (left) can be adopted to abstract the block diagram of a single stage HDD servo system in track-following mode. ...
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