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


Abstract and Figures

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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Liting Sun
Department of Precision Machinary
and Precision Instrumentation
University of Science and Technology
of China
Hefei, Anhui, China, 230027
Xu Chen
Department of Mechanical Engineering
University of California at Berkeley
Berkeley, California, 94720
Masayoshi Tomizuka
Department of Mechanical Engineering
University of California at Berkeley
Berkeley, California, 94720
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-
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
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
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.
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
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.
Magnitude (dB)
Phase (deg)
Frequency (Hz)
HDD full−order plant
HDD nominal model
Figure 1. Frequency responses of the full-order plant and its nominal
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).
Magnitude (dB)
Frequency (Hz)
Figure 2. Frequency responses of the baseline sensitivity function
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:
d(k) = P(q1)(d(k) + u(k)) + n(k)P1
n(q1)d(k) + P(q1)P1
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
Adaptation Algorithm based
on Lattice Notch Filter
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:
d(k)qmd(k) + P1
Gd2e(z1) P(z1)(1zmQ(z1))
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:
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.,
3.2 Q-filter Design based on Lattice-form IIR Notch
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)
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)
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
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
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
NW =2arctan 1αL
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.
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
Figure 6 shows the frequency response of Eq. (18), where the
band-pass property is evident.
Magnitude (dB)
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)
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-
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.
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
through F(z1). The output is denoted by c(k) = F(q1)ˆ
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:
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.,
L(k) = NL(q1)c(k) = 12βq1+q2c(k)
which is equivalent to
L(k) = β(1+ˆ
+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) + φT(k1)F(k1)φ(k1)
αL(k) = ˆ
αL(k1) + F(k1)φ(k1)e0
where φ(k1)is the gradient of e0
L(k)with respect to the
latest estimated parameter ˆ
αL(k1), defined by φ(k1) =
To obtain φ(k1), we notice the following relationship,
φ(k1) = e0
So φ(k1)can be expressed as follows:
φ(k1) = β[1+ˆ
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:
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,...
L(k) = β(1+ˆ
αL(k1)) ¯eL(k1)ˆ
+c(k)2βc(k1) + c(k2)(25)
αL(k) = ˆ
αL(k1) + F(k1)φ(k1)e0
F(k) = 1
λ(k) + φT(k1)F(k1)φ(k1)
¯eL(k) = β(1+ˆ
αL(k)) ¯eL(k1)ˆ
+c(k)2βc(k1) + c(k2)(28)
φ(k) = β[1+ˆ
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.
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.
Frequency (Hz)
Spectrum of a vibration signal with wide spectral peaks
Figure 7. Spectrum of a vibration signal with wide spectral peaks
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
in this case study) for minimization of the 3σvalue of the PES
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
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
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.
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
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
0 500 1000 1500 2000 2500 3000 3500 4000
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)
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-
This work was supported by the Chinese Scholarship Coun-
cil (CSC) of China.
[1] Vesset, D., etal, 2012. ”Worldwide Big Data Technology and
Services 2012 2015 Forecast”. 1, Framingham, March, MA,
[2] Guo, L., and Chen, Y., 2001. Disk flutter and its impact on
hdd servo performance. IEEE Transactions on Magnetics,
37(2), pp. 866-870.
[3] Ehrlich, R., and Curran, D., 1999.Major HDD TMR sources
and projected scaling with TPI. IEEE Transactions on Mag-
netics, 35(2), pp. 885891.
[4] Ohnishi, K., 1987. A new servo method in mechatronics.
Transactions of Japanese Society of Electrical Engineering,
107(D), pp. 8386.
[5] White, M. T., Tomizuka, M., and Smith C., 2000. ”Improved
Track Following in Magnetic Disk Drives Using a Distur-
bance Observer”. IEEE/ASME Transactions on Mechatron-
ics, 5(1), pp. 3-11.
[6] Zheng, Q., and Tomizuka, M., 2008. A disturbance ob-
server approach to detecting and rejecting narrow-band dis-
turbances in hard disk drives. In Proceedings of 2008 IEEE
International Workshop on Advanced Motion Control, pp.
[7] Jia, Q. W., 2009. Disturbance rejection through disturbance
observer with adaptive frequency estimation. IEEE Transac-
tions on Magnetics, 45(6), pp. 26752678.
[8] Chen, X. and Tomizuka, M., 2010. ”Unknown Multiple
Narrow-Band Disturbance Rejection in Hard Disk Drives -
an Adaptive Notch Filter and Perfect Disturbance Observer
Approach”. In proceedings of 2010 ASME Dynamic Sys-
tems and Control Conference, September 13-15, Cambridge,
Massachusetts, USA.
[9] Chen, X. and Tomizuka, M., 2010. ”An Indirect Adap-
tive Approach to Reject Multiple Narrow-Band Disturbances
in Hard Disk Drives”. In proceedings of 2010 IFAC Sym-
posium on Mechatronic Systems, September 13-15, Cam-
bridge, Massachusetts, USA.
[10] Chen, X. and Tomizuka, M., 2012. ”A Minimum Parame-
ter Adaptive Approach for Rejecting Multiple Narrow-Band
Disturbances with Application to Hard Disk Drives”. IEEE
Transactions on Control Systems Technology, 20(2), pp.
[11] Chen, X. and Tomizuka, M., 2013. ”Selective model in-
version and adaptive disturbance observer for time-varying
vibration rejection on an active-suspension benchmark”. Eu-
ropean Journal of Control, 19(4), pp. 300-312.
[12] Nehorai, A., 1985. ”A Minimal Parameter Adaptive Notch
Filter With Constrained Poles and Zeros”. IEEE Transac-
tions on Acoustics, Speech, and Signal Processing, 33(4),
pp. 983-996.
[13] Landau, I. D., Aurelian, C., and Rey, D., 2005. ”Adaptive
narrow band disturbance rejection applied to an active sus-
pensionan internal model principle approach”. Automatica,
41(4), pp. 563-574.
[14] Abraham, C. S., Landau, I. D., and Airimioaie, T. B., 2013.
”Direct adaptive rejection of unknown time-varying narrow
band disturbances applied to a benchmark problem”. Euro-
pean Journal of Control, 19(4), pp. 326-336.
[15] Levin, J., and Ioannou, P., 2008. ”Multirate Adaptive
Notch Filter with an Adaptive Bandwidth Controller for Disk
Drives”. In proceedings of 2008 American Control Confer-
ence, June, pp. 4407-4412.
[16] Regalia, P. A., 1991. ”An improved lattice-based adaptive
IIR notch filter”. IEEE Transactions on Signal Processing,
39(9), pp. 2124-2128.
[17] Hirata, M., 2007. ”NSS benchmark problem of hard disk
drive system”.
[18] Chen, X., Tomizuka, M., 2013. ”nverse-Based Local Loop
Shaping and IIR-Filter Design for Precision Motion Con-
trol”. In proceedings of 6th IFAC Symposium on Mecha-
tronic Systems, pp. 490-497.
[19] Li, G., 1997. ”A stable and efficient adaptive notch filter for
direct frequency estimation”. IEEE Transactions on Signal
Processing, 45(8), pp. 2001-2009.
[20] Ljung, L., 1999. System Identification: Theory for the
User, 2ed. Prentice Hall PTR.
[21] Nehorai, A., 1985. ”A Minimal Parameter Adaptive Notch
Filter With Constrained Poles and Zeros”. IEEE Transac-
tions on Acoustics, Speech, and Signal processing, 33(4) pp.
... 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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Control methodologies for deterministic disturbance rejection and trajectory tracking have been of great interest to researchers in the elds of controls, mechatronics, robotics and signal processing in the past two decades. The applications of these methods span a wide range from satellite attitude control requiring an accuracy of a few meters, to positioning of the read/write head in hard disk drives with an accuracy of less than one nanometer. This dissertation addresses the problem of trajectory tracking and deterministic disturbance rejection in discrete time systems when the trajectory/disturbance is unknown, but can be realized as an a ne combination of known basis functions. Despite the prior work on this problem that assumes known and time invariant plant dynamics, we consider multi input single output systems with unknown dynamics. Moreover, we investigate the cases where the disturbance or system dynamics varies slowly or abruptly but infrequently. Within the broad class of disturbances/trajectories that satisfy the stated criteria, an elaborate study is conducted on periodic and superposition of multiple sinusoidal sequences. We propose two novel adaptive control methods for the aforementioned problem. The rst scheme can be classi ed as an indirect adaptive algorithm since it consists of two parts, namely a system identi cation mechanism that provides a dynamic model of the closed loop system, and the adaptive compensator which deploys the aforementioned dynamic model to synthesize the control law. The second proposed method is a direct adaptive controller, meaning that the control law is generated directly and the stated separation is not possible. Besides providing theoretical guarantees, we experimentally evaluate our algorithms on a challenging control task for nano scale positioning of the read write head in a dual stage hard disk drive (HDD). Even with the advent of NAND ash based data storage devices, the HDD continues to thrive as the most cost e ective, reliable solution for rewritable, very high density data storage. It remains a key technology particularly with the tremendously growing popularity of server based cloud computing and novel hybrid enterprise storage solutions. We described that the control methodologies that can address the problem under our study are crucial for Bit Patterned Media Recording which is one of the two breakthroughs in magnetic recording that have been immensely investigated in the past few years. Extensive computer simulations and implementation on a digital signal processor unit are performed to validate the e ectiveness of our proposed algorithms in full spectrum compensation of the repeatable runout in dual stage HDDs. This application introduces unique control challenges since it requires estimating a very large number of parameters that is order(s) of magnitude greater than prior work and frequency contents span from 120Hz to extremely large values (above 20KHz) where the plant dynamic uncertainties are large.
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Vibration rejection is a key technology of practical engineering, especially in optical telescopes with a stable accuracy of urad level. The closed-loop performance of optical telescopes is largely determined by the control bandwidth, while it is severely limited by the low sampling rate and large time delay of the image sensor, so it is difficult to mitigate structural vibrations in optical telescopes, especially wideband vibrations, because they exist universally and greatly influence the stability of the system. This paper develops an improved error-based disturbance observer (EDOB) based on the Youla parameterization approach to mitigate wideband vibrations in optical telescopes. This novel method can greatly improve the vibration rejection ability of the system by designing a proper Q-filter to accommodate wideband vibrations when their frequencies can be acquired. Because wideband vibrations in optical telescopes can be considered as multiple narrow-band vibrations with similar central frequencies, a novel Q-filter instead of a single wideband notch filter is proposed to mitigate wideband vibrations when considering the stability and closed-loop performance of the system. Moreover, this method only relies on a low frequency model, leading to a reduction in model dependence. Both the simulations and experimental results show that the error-based disturbance observer based on Youla parameterization can greatly improve the closed-loop performance of the system compared with the traditional feedback control loop.
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For safe and efficient planning and control in autonomous driving, we need a driving policy which can achieve desirable driving quality in long-term horizon with guaranteed safety and feasibility. Optimization-based approaches, such as Model Predictive Control (MPC), can provide such optimal policies, but their computational complexity is generally unacceptable for real-time implementation. To address this problem, we propose a fast integrated planning and control framework that combines learning- and optimization-based approaches in a two-layer hierarchical structure. The first layer, defined as the "policy layer", is established by a neural network which learns the long-term optimal driving policy generated by MPC. The second layer, called the "execution layer", is a short-term optimization-based controller that tracks the reference trajecotries given by the "policy layer" with guaranteed short-term safety and feasibility. Moreover, with efficient and highly-representative features, a small-size neural network is sufficient in the "policy layer" to handle many complicated driving scenarios. This renders online imitation learning with Dataset Aggregation (DAgger) so that the performance of the "policy layer" can be improved rapidly and continuously online. Several exampled driving scenarios are demonstrated to verify the effectiveness and efficiency of the proposed framework.
Conference Paper
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Iterative learning control (ILC) is an effective control technique for servo improvement in systems that repetitively execute the same tasks. In the learning process, the measured tracking error from the current iteration is incorporated to generate a new feedforward compensation signal to improve the system performance in the next iteration. Due to its discrete-time implementation , conventional ILC only considers errors at the sampled output points without inter-sample learning ability. Therefore, its achievable performance is limited by the output sampling rate. In this paper, a multirate ILC (MRILC) approach is proposed. Based on multirate Kalman Smoother and multirate feedforward control, the ILC update law in a multirate two-degree-of-freedom (2-DOF) control system with a fast feedforward ILC input but a slow output sampling rate is derived. The bandwidth of the learning loop may then be extended beyond that of the feedback loop for enhanced inter-sample learning. The effectiveness of the proposed MRILC is verified by experiments on a wafer scanner system.
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Modern hard disk drive (HDD) systems are subjected to various external disturbances. One particular category, defined as wide-band disturbances, can generate vibrations with their energy highly concentrated at several frequency bands. Such vibrations are commonly time-varying and strongly environment/product-dependent; and the wide spectral peaks can occur at frequencies above the servo bandwidth. This paper considers the attenuation of such challenging vibrations in the track-following problem of HDDs. Due to the fundamental limitation imposed by the Bode’s Integral Theorem, the attenuation of such wide-band disturbances may cause unacceptable amplifications at other frequencies. To achieve a good performance and an optimal tradeoff, an add-on adaptive vibration-compensation scheme is proposed in this paper. Through parameter adaptation algorithms that online identify both the center frequencies and the widths of the spectral peaks, the proposed control scheme automatically allocates the control efforts with respect to (w.r.t) the real-time disturbance characteristics. The effect is that the position error signal (PES) in HDDs can be minimized with effective vibration cancellation. Evaluation of the proposed algorithm is performed by experiments on a Voice-Coil-Driven Flexible Positioner (VCFP) system.
Conference Paper
Enhanced anti-windup (AW) schemes are designed for the dual stage hard disk drive (HDD) system to solve the amplitude saturation problem of the secondary actuator. The sensitivity decoupling approach is used for designing the nominal linear controllers for the dual stage HDD systems. The AW compensators for the system are synthesized by solving a linear matrix inequality optimization. The AW scheme is further enhanced with a filter, which can be designed by the robust control methodologies. The proposed schemes and an existing AW scheme are evaluated and compared by simulation on a dual stage HDD benchmark control problem with white noise vibration disturbance.
Conference Paper
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This paper presents an indirect adaptive control scheme that rejects unknown multiple narrow-band disturbances in hard disk drive systems. The proposed algorithm first finds the model of the disturbance (the internal model) and then adaptively estimates its parameters. The design of a band-pass filter with multiple narrow pass-bands is then presented and used to construct a disturbance observer (DOB) for disturbance rejection. The proposed algorithm estimates the minimal amount of parameters, and is computationally simple. Evaluation of the proposed algorithm is performed on a benchmark problem for HDD track following.
Conference Paper
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In this paper, an adaptive control scheme is developed to reject unknown multiple narrow-band disturbances in a hard disk drive. An adaptive notch filter is developed to efficiently estimate the frequencies of the disturbance. Based on the correctly estimated parameters, a disturbance observer with a newly designed multiple band-pass filter is constructed to achieve asymptotic perfect rejection of the disturbance. Evaluation of the control scheme is performed on a benchmark problem for HDD track following.
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This paper presents an adaptive control scheme for identifying and rejecting unknown and/or time-varying narrow-band vibrations. We discuss an idea of safely and adaptively inverting a (possibly non-minimum phase) plant dynamics at selected frequency regions, so that structured disturbances (especially vibrations) can be estimated and canceled from the control perspective. By taking advantage of the disturbance model in the design of special infinite-impulse-response (IIR) filters, we can reduce the adaptation to identify the minimum amount of parameters, achieve accurate parameter estimation under noisy environments, and flexibly reject the narrow-band disturbances with clear tuning intuitions. Evaluation of the algorithm is performed via simulation and experiments on an active-suspension benchmark.
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In this paper, a new adaptive disturbance rejection scheme is introduced. In the proposed scheme, adaptive frequency estimation technique is incorporated into the traditional disturbance observer method. The frequency of the dominant disturbance is estimated online by a fast stable adaptive algorithm, and the estimated frequency is used to adaptively tune the bandwidth of the disturbance observer. Application to a HDD system shows that the proposed method exhibits good capability of disturbance rejection. Comparing with the traditional disturbance observer method, the new method incurs much less drop in phase margin.
Conference Paper
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This paper presents the simulation results performed of a multirate adaptive notch filter with adaptive bandwidth controller for disk drives. The resonant modes of a disk drive may be uncertain and vary between units, and can also lie near or beyond the nyquist frequency. Suppressing these modes can be difficult. However, with an adaptive notch filter that is able to accurately track the resonant frequencies, the effects of such modes can be suppressed. By adding a multirate scheme to the adaptive notch filter, it can suppress modes at higher frequencies. As the multirate adaptive notch filter tracks the plant mode frequencies, the adaptive bandwidth controller ensures that stability and performance requirements are satisfied. The simulation results that are included show the benefit of the adaptive control scheme.
The paper presents a direct adaptive algorithm for the rejection of unknown time-varying narrow band disturbances, applied to an adaptive regulation benchmark. The objective is to minimize the residual force by applying an appropriate control signal on the inertial actuator in the presence of multiple and/or unknown time-varying disturbances. The direct adaptive control algorithm is based on the internal model principle (IMP) and uses the Youla–Kučera (YK) parametrization. A direct feedback adaptive regulation is proposed and evaluated both in simulation and real-time. The robustness is improved by shaping the sensitivity functions of the system through band stop filters (BSF).
This paper presents a methodology for feedback adaptive control of active vibration systems in the presence of time varying unknown narrow band disturbances. A direct adaptive control scheme based on the internal model principle and the use of the Youla–Kucera parametrization is proposed. This approach is comparatively evaluated with respect to an indirect adaptive control scheme based on the estimation of the disturbance model. The comparative evaluation is done in real time on an active suspension system.