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Adaptive Loop Shaping for Wide-Band Disturbances Attenuation in Precision Information Storage Systems

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 5, MAY 2017 3301813
Adaptive Loop Shaping for Wideband Disturbances Attenuation
in Precision Information Storage Systems
Liting Sun1, Tianyu Jiang2,andXuChen
2
1Department of Mechanical Engineering, University of California at Berkeley, Berkeley, CA 94720 USA
2Department of Mechanical Engineering, University of Connecticut, Storrs, CT 06269 USA
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 the real-time disturbance characteristics. The effect is that the position error signal in HDDs can be minimized with
effective vibration cancelation. Evaluation of the proposed algorithm is performed by experiments on a voice-coil-driven flexible
positioner system.
Index Terms—Adaptive loop shaping, hard disk drives (HDDs), parameter adaptation algorithm (PAA), vibration compensation,
wideband disturbance.
I. INTRODUCTION
THE production of data is expanding at an astonishing
pace. As the main cost-effective data storage device,
modern hard disk drive (HDD) systems are facing ever
stringent requirements to further increase the storage capac-
ity (track density) and the data access speed. However, as
the track density increases, the regulation of the read/write
head becomes more difficult, particularly in vibration-intensive
environments. For instance, in all-in-one compact comput-
ers, the HDDs are subjected to significant audio vibrations
generated by the audio systems [1]. Similar problems occur
in smart TVs, data centers, and cloud storage. In the latter
cases, a large number of HDDs are stacked and mounted in
compact enclosures; it is thus crucial to attenuate the vibration
disturbances that get excited and transmitted among the drives
and the cooling fans.
Extensive research efforts in the fields of design, control,
and signal processing have been made to attenuate external
disturbances in HDDs in the past decades. From the hard-
ware viewpoint, Hirono et al. [2] installed a spoiler between
the disks and studied the effect of the spoiler in reducing
the flow-induced vibrations. Choi and Rhim [3] applied a
herringbone groove pattern to the plane disk damper and
effectively suppressed the axial vibrations. Mou et al. [1]
analyzed the structurally transmitted vibrations in HDDs
induced by speakers, fans, and CD/DVD drives in notebook
PCs and concluded that soft chassis and mounting could
be helpful to suppress such vibrations at frequencies above
1000 Hz. Certainly, the hardware solutions are potentially
Manuscript received September 2, 2016; revised November 28, 2016;
accepted January 12, 2017. Date of publication January 17, 2017; date of
current version April 17, 2017. Corresponding author: X. Chen (e-mail:
xchen@uconn.edu).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMAG.2017.2654200
time-consuming and cost-expensive. They are also subject to
performance loss due to limited adaptiveness in the presence
of time-varying or environment/product-dependent vibrations.
On the other hand, control designs (software solution), par-
ticularly those with adaptive vibration compensation algo-
rithms, offer the opportunity of more flexibilities and good
performance with lower cost. For instance, through adap-
tive noise cancelation (ANC) technique [4], the unknown
transmission paths from external disturbances to the HDD
systems can be adaptively identified such that effective can-
celation signals can be generated based on the measured
disturbance signals [5], [6]. As an adaptive feedforward
control approach, ANC requires additional sensors. Adap-
tive feedback control schemes have also been developed
to deal with time-varying disturbances in HDDs. Examples
include: 1) adaptive repetitive control, which cancels repeti-
tive disturbances based on internal model principle [7], [8];
2) loop shaping via Youla parameterization with an adaptive
Qfilter [9]–[11] or special state-space formulations [12], [13];
3) adaptive peak filters [14], [15]; and 4) adaptive disturbance
observer (DOB) [16]–[21].
Although we have successfully dealt with many con-
ventional problems (such as basic noise cancelation and
rejection of internal vibrations) and achieved relatively
good servo performance of HDDs through the above-listed
approaches, one particular category, defined as wide-band
disturbance hereinafter, remains difficult to effectively com-
pensate. Such wide-band disturbances include the audio
and chassis vibrations described at the beginning of this
section. They can induce widely spanned spectral peaks
in contrast to the conventional single-frequency excitations
(an example is provided in Fig. 1). Moreover, the spectral
peaks are time-varying, product-dependent, and can appear
at frequencies above the bandwidth of the baseline servo
loop [22], [23].
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3301813 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 5, MAY 2017
Fig. 1. Examples of the disturbance spectrums.
One of the greatest challenges in compensating these wide-
band disturbances stems from the fundamental “waterbed”
limitation imposed by the Bode’s integral theorem. It states
that in linear feedback control systems, it is impossible to
attenuate disturbances at all frequencies unless for some
special cases that are rarely feasible in motion control applica-
tions. The attenuation of disturbances at one frequency range
will inevitably cause amplifications at other spectral regions.
Although approaches 3) and 4) can theoretically attenuate dis-
turbances at any customer-specified frequencies (even above
the bandwidth of the servo loop), they focus on the rejection
of narrow-band disturbances [see Fig. 1(a)] or single-peak
band-limited disturbances where the “waterbed” amplification
is relatively small. Hence, in these approaches, the adaptation
focuses mainly on finding the optimal attenuation frequen-
cies. Yet for multiple wide-band disturbance attenuation, such
undesired amplifications are much more dangerous, and should
be systematically considered in controller design to avoid
deterioration of the overall performance.
In view of the needs and challenges, an adaptive loop
shaping scheme is proposed in this paper for multiple wide-
band disturbance attenuation. By considering adaptations with
respect to not only the attenuation frequencies but also the
attenuation widths, this scheme automatically tunes for the
optimal controller parameters that offer better overall per-
formance. Such a two-degree-of-freedom (2-DOF) adaptation
allows an optimal allocation of control efforts over all fre-
quencies. It balances the preferred disturbance attenuation and
undesired amplification with minimum position errors. As an
extension to our previous work [18]–[20], [24]–[26], this paper
contributes in three aspects: 1) the adaptive controller design
covers both single and dual-stage HDDs; 2) wideband distur-
bances with the important extension to multiple spectral peaks
are addressed; and 3) experimental verification on a voice-coil-
driven flexible positioner (VCFP) system is performed.
The remainder of this paper is organized as follows.
Section II presents the proposed adaptive control structure for
both single and dual-stage HDDs. Section III discusses the
performance-oriented Qfilter design with tunable passband
widths. In Section IV, the 2-DOF parameter adaptation algo-
rithms (PAAs) are formulated. Stability analysis is given in
Section V, followed by the experimental results on a VCFP
Fig. 2. Structure of proposed control scheme for single-stage HDDs.
Fig. 3. Typical magnitude response of the baseline sensitivity function.
system in Section VI. Finally, Section VII concludes this
paper.
II. PROPOSED ADAPTATION STRUCTURE WITH MODEL
INVERSION
A. General Structure
Throughout this paper, we focus on the track-following
problem of HDD systems in the presence of the above
discussed wideband disturbances. Fig. 2 shows the proposed
adaptive control structure in single-stage HDDs. Without the
add-on compensator, it reduces to a basic feedback loop
where the system P(z1)is stabilized by a controller C(z1),
which achieves a baseline sensitivity function whose magni-
tude response is similar to the one shown in Fig. 3. Such a
baseline servo design is typical in practice, and can commonly
achieve a bandwidth of around 1000 Hz for single-stage
HDDs and around 2000 Hz for dual-stage HDDs. Above the
bandwidth, disturbances are amplified due to the “waterbed”
effect. Therefore, new customized compensator is desired for
enhanced disturbance attenuation at high frequencies.
Within the add-on compensator in Fig. 2, zmseparates1the
delay out of system P(z1), such that P(z1)zmPm(z1)
and Pn(z1)represents the nominal model of Pm(z1),i.e.,
Pn(z1)Pm(z1)at least at frequencies where distur-
bance attenuation is desired. The objective of the add-on
compensator is to generate a cancelation signal c(k),which
1For discrete-time systems, we always have m1 and the separation of
zmassures a realizable inverse of Pm(z1)and its nominal model described
afterward.
SUN et al.: ADAPTIVE LOOP SHAPING FOR WIDEBAND DISTURBANCES ATTENUATION 3301813
can effectively cancel out the influence of d(k). Assume that
the noise signal n(k)is relatively small compared with the
dominating vibration signal d(k). Then, one can notice that
ˆ
d(k)=P1
n(q1)[P(q1)(d(k)+u(k)) +n(k)]−qmu(k)
P1
n(q1)Pm(q1)qm(d(k)+u(k)) qmu(k)(1)
and
c(k)=Q(q1)ˆ
d(k)
Q(q1)P1
n(q1)Pm(q1)qmd(k)
+Q(q1)(P1
n(q1)Pm(q1)1)u(km)(2)
where q1is the one-step delay operator in time domain and
P1
n(z1)represents the stable inverse2of Pn(z1). Thus, for
perfect cancelation, i.e., c(k)approximately equals to d(k),
the filter Q(z1)should satisfy
Q(ejω)P1
n(ejω)Pm(ejω)ejmω1(3)
Q(ejω)P1
n(ejω)Pm(ejω)10(4)
at all target frequencies.
Compared with the regular DOB structure where the Qfilter
is typically a low-pass filter [28], the formulation in (3) and
(4) offers several flexibilities and benefits:
1) Enhanced Performance at High Frequencies: First, with
Qbeing a low-pass filter, a regular DOB mainly focuses
on the attenuation of disturbances at low frequencies
(within the bandwidth of the Qfilter, to be more
specific). While through (3) and (4), the yielded Qcan
additionally have band-pass and/or high-pass properties,
depending on the available model information and the
spectral distribution of the disturbances. Second, a reg-
ular DOB structure focuses more on the magnitude and
does not explicitly consider the phase compensation that
can significantly deteriorate the achievable performance
at high frequencies. On the contrary, (3) and (4) sys-
tematically include the phase influence for enhanced
performance.
2) Relaxed Requirements for Plant Inversion: In a regular
DOB, it is required that P1
n(ejω)Pm(ejω)1at
all frequencies within the bandwidth of a low-pass Q
filter. As the bandwidth of the Qfilter increases, this
becomes difficult to satisfy. While in (3) and (4), the
Qfilter (possibly of band-pass property) serves as a
selective factor, such that the model only needs to be
stably invertible “locally” at frequencies where enhanced
disturbance attenuation is desired. At these frequencies,
(1) and (2) reduce to
ˆ
d(k)d(km)(5)
c(k)=Q(q1)ˆ
d(k)Q(q1)d(km). (6)
3) “Model-Free” Adaptability: At interested frequencies,
P1
n(ejω)Pm(ejω)1 is satisfied. Then, the sensi-
2When Pn(z1)is nonminimum phase, stable inversion strategies such
as zero-phase error tracking [27] or model-matching techniques via H
formulation could be used.
Fig. 4. Structure of proposed control scheme for dual-stage HDDs.
tivity function of the structure in Fig. 2 becomes
S(z1)
=(1zmQ(z1))
×1
1+P(z1)C(z1)+Q(z1)P1
n(z1)P(z1)zm
(1zmQ(z1))S0(z1)(7)
where S0(z1)1/[1+P(z1)C(z1)]is the baseline
sensitivity function without the add-on compensator.
Hence, such formulation offers an approximately model-
free shaping factor 1 zmQ(z1)that allows us to
shape the sensitivity function without influencing the
nice properties achieved by S0(z1).3Moreover, in the
presence of unknown time-varying disturbances, such a
“model-free” Qfilter in Fig. 2 is easily adaptable, not
only to automatically identify the center frequencies of
the spectral peaks, but also to optimally allocate the
attenuation strength over frequencies to minimize the
position errors.
Remark: Note that the control structure in Fig. 2 treats
d(k)as a lumped equivalent input disturbance. The effects
of different error sources on the system output, for instance,
input disturbance, output disturbance, and model nonlineari-
ties/uncertainties [29], are lumped into d(k), such that they can
be compensated by c(k)from the input side of the plant. For
the general case with y(k)=P(z1)u(k)+Md(z1)d(k),the
output disturbance becomes Md(z1)d(k), and an equivalent
input disturbance compensation signal c(k)is generated to
provide the compensation.
B. Extension to Dual-Stage HDDs
The control scheme in Fig. 2 can be easily extended to
dual-stage HDD systems using the decoupled DOB technique
as in [19]. Fig. 4 shows one example design for the dual-stage
case, where Pi(z1)and Ci(z1)(i=1,2) represent the two
actuators (voice coil motor for the first stage and piezoelec-
tric actuator for the second) and the corresponding baseline
feedback controllers, respectively. Similar to Fig. 2, Pi,n(z1)
3More detailed analysis about this can be found in our previous work [18].
3301813 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 5, MAY 2017
denotes the delay-free nominal model of Pi(z1),andm2is
the delay steps in P2(z1), i.e., P2(z1)=zm2P2m(z1).
Then, the corresponding requirements for Qfilter in (3)
and (4) become
Q(ejω)P1
2n(ejω)P2m(ejω)ejm2ω1(8)
Q(ejω)P1
2n(ejω)P2m(ejω)10(9)
and the sensitivity function satisfies [19]
S(z1)(1zm2Q(z1))S
0(z1)
1zm2Q(z1)
1+P1(z1)C1(z1)+P2(z1)C2(z1).(10)
One can notice that the Qfilter design in both single and dual-
stage HDDs are quite similar. Therefore, the structure in Fig. 2
will be used as an example for analysis and implementation
hereinafter.
III. QFILTER DESIGN
A. Performance-Oriented Formulation
This section discusses the Qfilter design for loop shaping.
Recalling Fig. 2 and (7), the relationship between the steady-
state position error e(k)and the disturbance signal d(k)can
be expressed as
e(k)=−(1qmQ(q1))S0(q1)P(q1)d(k). (11)
Hence, to attenuate wideband disturbances with, for example,
nspectral peaks centered at fi(i=1,2,...,n,inHz),itis
necessary to generate nnotches/stopbands centered at these
frequencies with proper widths. Namely, the add-on shaping
factor 1zmQ(z1)should generate a structure N(z1)with
nwide notches/stopbands such that N(q1)d(k)0.
Define such a stable structure as N(z1)=
BN(z1)/AN(z1)and let Q(z1)=BQ(z1)/ AQ(z1).
Then, for effective cancelations, we should have
1zmQ(z1)=N(z1)J(z1)(12)
where J(z1)is additional polynomial for the existence
of solutions. To further simplify the solution, we let
AQ(z1)=AN(z1), and then, (12) reduces to
zmBQ(q1)+BN(z1)J(z1)=AN(z1)(13)
which is a Diophantine equation with unknowns BQ(z1)and
J(z1). Compared with directly assigning the parameters of
the Qfilter (e.g., by assuming some bandpass structures based
on the theory and practice of digital filters), the Diophantine-
based indirect approach benefits from: 1) more intuitive
performance-oriented designs, i.e., it converts the problem into
finding an optimal N(z1)that cancels d(k)based on its
spectral characteristics; 2) the bandpass property and phase
compensation in (3) are systematically included in the solution
of (13); and 3) stability of the Qfilter is guaranteed with
astableN(z1)and hence the closed-loop stability to be
discussed later in Section V.
Fig. 5. Example magnitude responses of direct-form notch filters.
B. N(z1)With Tunable Notch/Stopband Widths
To construct N(z1), recall from digital filter design, that
there are two main types of notch filters (a.k.a. band-stop
filters): 1) direct-form notch filter (denoted with superscript d)
and 2) lattice-form notch filter (denoted with superscript l).
The structures of the two are given by (with single notch)
Nd(z1)=12cosω0z1+z2
12αdcos ω0z1+d)2z2(14)
Nl(z1)=12cosω0z1+z2
1(1+αl)cos ω0z1+αlz2(15)
where ω0=2πf0Ts(Tsis the sampling time) controls the
location of the notch and αd,l(0,1)controls the notch width
(denoted as NW, which is the difference between the upper
and lower frequencies where the magnitudes are 3dB)by
the following relationships:
NWdπ(1αd)(16)
NWl2arctan1αl
1+αl.(17)
Figs. 5 and 6 give some examples of the two types of notch
filters with different center frequencies and notch widths. It can
be seen that consistent with (16) and (17), a smaller αd,lgives
a wider notch width. Meanwhile, as the notch width grows, the
distortions (i.e., nonunity gains) at other frequencies become
more significant.
Most of the previous studies on adaptive notch filter or
adaptive DOB have adopted the direct form for online identi-
fication of the unknown center frequency (ω0)ofthespectral
peak in disturbances, treating αdas a fixed predesigned
parameter. For wideband disturbance attenuation, we need to
perform loop shaping based on not only ω0but also αd,l.
The latter controls the strength of desired attenuation and
undesired “waterbed” amplifications. Therefore, control para-
meters should be adaptively updated with respect to both ω0
and αd,l. The nonlinear term d)2in (14), however, makes
the direct form difficult to be adaptable in such scenarios.
SUN et al.: ADAPTIVE LOOP SHAPING FOR WIDEBAND DISTURBANCES ATTENUATION 3301813
Fig. 6. Example magnitude responses of lattice-form notch filters.
Moreover, as shown in Fig. 5, the magnitude response of
the direct-form notch filter is not symmetric at low and high
frequencies, i.e., it intrinsically brings more amplifications
at high frequencies. On the contrary, the lattice-form notch
filter only contains linear terms with respect to parameter αl
[see (15)] and its magnitude response (as shown in Fig. 6) is
symmetric at low and high frequencies. Mathematically, we
have |Nl(z1)|z=1,1=2/(1+αl), and namely, the filter has
the same gain at dc and Nyquist frequencies.
Therefore, for the adaptation of notch width, the lattice-form
notch filter in (15) is adopted. To generate multiple notches at
nfrequencies fi,i=1,...,n,N(z1)in (12) is given by
N(z1)=
n
i=1
Nlωi
l
i,z1
=
n
i=1
12cosωiz1+z2
11+αl
icos ωiz1+αl
iz2(18)
with ωi=2πfiTs.
Recalling (12) and (13), one can notice that the Qfil-
ter is an implicit function of the parameter vector θ
[cos ω1,...,cos ωn
l
1,...,α
l
n]T,thatis
Q(z1)=) =cos ω1,...,cos ωn
l
1,...,α
l
n
(19)
where cos ωiand αl
i(i=1,...,n), respectively, control the
center frequencies and widths of the passbands. For example,
if m=1 and only a single notch is needed, namely, N(z1)=
Nl(z1)as in (15), then solving (13) yields
Q(z1)=(1αl)cos ω0+l1)z1
1(1+αl)cos ω0z1+αlz2(20)
which is a bandpass filter whose center frequency and width
of passband depend on cos ω0and αl, as shown in Fig. 7.
Fig. 7. Frequency responses of the bandpass Q filters.
IV. ADAPTIVE ATTENUATION ALGORITHMS
In this section, the adaptation algorithm for Qfilter is
presented. Recalling the structure in Fig. 2 and (11), and
by letting F(z1)=Pn(z1)/(1+zmPn(z1)C(z1)) and
dF(k)=F(q1)ˆ
d(k)F(q1)d(km), one can approxi-
mately represent the steady-state position error signal (PES)
as
e(k)=−(1qmQ(q1))S0(q1)P(q1)d(k)
≈−(1qmQ(q1))S0(q1)Pn(q1)d(km)
≈−(1qmQ(q1))F(q1)d(km)
≈−(1qmQ(q1))dF(k). (21)
Suppose that d(k)generates vibrations with nunknown
wide spectral peaks, then based on (19), (21) suggests
that e(k)is also a function of parameters in θ=
[cos ω1,...,cos ωn
l
1,...,α
l
n]T. Our control goal is to adap-
tively tune these parameters, such that |e(k)|, i.e., |(1
q1Q(q1))dF(k)|is minimized.
The cost function, however, may not be convex with respect
to the parameter vector θ. As shown in (18)–(20), if we
separate θinto θf[cos ω1,...,cos ωn]Tand θW
[αl
1,...,α
l
n]Tthat, respectively, controls the center frequen-
cies and passband widths of the Qfilter, then the cross-product
term of θfand θWmakes it difficult to simultaneously update
both parameter vectors through linear adaptation algorithms.
To cope with the challenge, we separate the PAA into two
stages.
1) Stage I: Adaptation with respect to θfwith preas-
signed θW.
2) Stage II: Adaptation with respect to θWwith converged
θfin Stage I.
A. Adaptation of the Center Frequencies
In this stage, the width parameter vector θWis fixed.
Moreover, to help the center frequency parameters in θf
3301813 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 5, MAY 2017
converge to the dominant spectral peaks, narrow notches are
desired, which makes it necessary and practically efficient
to set all αl
i(i=1,...,n)close to 1 and identical, e.g.,
αl
i=αl∈[0.9,0.995], given the relationship in (17). Then,
the adaptation problem with respect to θfreduces to one that
is similar to adaptive narrow-band disturbances attenuation
in [18].
Combining (12) and (18), one can obtain (1
qmQ(q1)) =J(q1)(BN(q1)/ AN(q1)) where
BN(q1)=
n
i=1
(12cosωiq1+q2)
=1+
n1
i=0
ai(qi+q2n+i)+anqn+q2n
(22)
ANl,q1)=
n
i=1
(1(1+αl)cos ωiq1+αlq2)
=1+
n1
i=1
bi(qi+l)niq2n+i)+bnqn
+l)nq2n.(23)
In (22) and (23), we have used the fact that coefficients of
1andq2are the same in 1 2cosωiq1+q2and differ
by a scalar gain of αlin 1 (1+αl)cos ωiq1+αlq2.
Moreover, due to the affine property, we can establish affine
relationships between the coefficients biand ai,i=1,...,n.
For example, after some algebra, it has the following.
1) n=1
a1=−2cosω1
b1=(1+αl)a1/2.
2) n=2
a1=−2(cos ω1+cos ω2)
a2=(2+4cosω1cos ω2)
b1=(1+αl)a1/2
b2=(a22)(1+αl)2/4+2αl.
3) n=3
a1=−2(cos ω1+cos ω2+cos ω3)
a2=3+4(cos ω1cos ω2+cos ω1cos ω3
+cos ω3cos ω2)
a3=−4(cos ω1+cos ω2+cos ω3)
8cosω1cos ω2cos ω3
b1=(1+αl)a1/2
b2=(1+αl)2
4a2+3αl3(1+αl)2
4
b3=(1+αl)3
8a3+αl(1+αl)(1+αl)3
4a1.
Substituting such affine relationships into (23), it then
becomes
ANl,q1)=1+
n
i=1
aigil,q1,...,q2n+1)
+l)nq2n(24)
where gil,q1,...,q2n+1), i=1,...,nare polynomials
whose coefficients are nonlinear functions of predesigned
parameter αl. Additionally, the specified conditioning on αl
i
allows us to drop J(q1)in the adaptation, since its influence
to narrow notches is small. Therefore, our adaptation model
becomes y(k)=−e(k)=(BN(q1)/ AN(q1))dF(k),which
is equivalent to
AN(q1)y(k)=BN(q1)dF(k). (25)
With (22)–(24), (25) yields the following difference
equation:
y(k)=ψ(k1)Tθa+dF(k)+dF(k2n)
l)ny(k2n)(26)
where θa[a1,...,an]Tis the unknown parameter vector
and ψ(k1)[ψ1(k1),...,ψ
n(k1)]Tis the regressor
vector satisfying
ψi(k1)=dF(ki)+dF(k2n+i)gil,y(k1),
...,y(k2n+1)), i=1,...,n1 (27)
ψn(k1)=dF(kn)gnl,y(k1),
...,y(k2n+1)). (28)
Here, gil,y(k1),...,y(k2n+1)),i=1,...,n,
is obtained by replacing q1,...,q2n+1with
y(k1),...,y(k2n+1)in gil,q1,...,q2n+1), i=
1,...,n.
Notice that in regulation problems in HDDs, the desired
output is zero, which means that the output y(k)also serves
as the estimation error (k). Replacing θain (26) with its
adaptive version, the difference equation for the adaptive
system becomes
(k)=ψe(k1)Tˆ
θa(k)+dF(k)+dF(k2n)
l)n(k2n)(29)
where ψe(k1)is defined similar to ψ(k1), with y(k)
replaced by (k).
Hence, we can see that (k)and ˆ
θa(k)in (29) are actually
the a posteriori estimation error and the a posteriori parameter
estimate. The aprioriestimation error is similarly defined as
0(k)=ψe(k1)Tˆ
θa(k1)+dF(k)+dF(k2n)
l)n(k2n). (30)
Based on (26)–(30), an output-error-based PAA with a fixed
compensator can be adopted for its simplicity and good
performance in noisy environments [18]. The implementation
of the algorithm is summarized as follows.
SUN et al.: ADAPTIVE LOOP SHAPING FOR WIDEBAND DISTURBANCES ATTENUATION 3301813
1) Select a fixed compensator4C(q1)1+l)nq2n+
n
i=1cigil,q1,...,q2n+1)that guarantees the
strict positive realness of (C(q1)/AN(q1)) (1/2).
2) Define the augmented a posteriori and aprioriestima-
tion errors
v(k)=(k)+l)n(k2n)
+
n
i=1
cigil,(k1),...,(k2n+1))
(31)
v0(k)=0(k)+l)n(k2n)
+
n
i=1
cigil,(k1),...,(k2n+1)).
(32)
3) Update the parameter estimation by
ˆ
θa(k)
=ˆ
θa(k1)F(k1)ψe(k1)v0(k)
1+ψe(k1)TF(k1)ψe(k1)
(33)
F(k)
=1
λ(k)F(k1)
F(k1)ψe(k1)ψT
e(k1)F(k1)
λ(k)+ψT
e(k1)F(k1)ψe(k1)
(34)
where λ(k)=10.99(1λ(k1)) with λ(0)=0.97
is a slowly growing forgetting factor in the region of
(0,1), which helps to increase the convergence speed.
Once the parameter vector θa=[a1,...,an]Tconverges,
we can then get bi,i=1,...,nin (23), and the actual
center frequency parameter vector θfand real frequencies
ωi(or fi), i=1,...,ncan be calculated by solving nfunc-
tions. For instance, when n=2, from (22), ω1and ω2are
found by solving
a1=−2(cos ω1+cos ω2)
a2=(2+4cosω1cos ω2).
B. Adaptation of the Passband Widths
Once ωi,i=1,...,nare identified in Stage I, the
next step is to adaptively tune the passband width para-
meter vector θW=[αl
1,...,α
l
n]Tthat minimizes the PES.
Recalling (18), (21), and the fact that J(q1)depends5on
N(q1), one can obtain
y(k)=(1qmQ(q1))dF(k)
=
n
i=1
Nli
l
i,q1)JW,q1)dF(k). (35)
4The selection of the compensator can be done manually or by adopting an
initial adaptation process [18].
5When m=1, J(q1)=1. However, for more gen-
eral case, J(q1), as a solution of the Diophantine equation
in (13), depends on the parameter
vector θW.
Similarly, if we replace αl
iand θWwith their adaptive versions
ˆαl
i(k1)and ˆ
θW(k1)in (35), then y(k)will also serve as
an aprioriestimation error (defined as 0(k)). Therefore, the
adaptive system is given by
0(k)=
n
i=1
Nlωi,ˆαl
i(k1), q1Jˆ
θW(k1), q1dF(k)
(36)
where ˆαl
i(k1), i=1,...,nare the aprioriestimates of
the unknown passband width parameters and ˆ
θW(k1)=
[αl
1(k1),...,α
l
n(k1)]T.
In regard of the nonlinear relationships with respect to
the unknown parameter vector θW, recursive prediction-error
method (RPEM; [30, Ch. 11]) approach is adopted for unbi-
ased local convergence. The recursive PAA is given as
ˆ
θW(k)=ˆ
θW(k1)+F(k1)φ(k1)e0(k)(37)
F(k)=1
λ(k)
×F(k1)F(k1)φ(k1)φT(k1)F(k1)
λ(k)+φT(k1)F(k1)φ(k1)
(38)
where φ(k1)is the negative gradient of 0(k)with
respect to the latest parameter estimate ˆ
θW(k1),
i.e., φ(k1)=[φ1(k1),...,φ
n(k1)]T
−[((∂0(k))/(∂ ˆαl
1(k1))), . . . , ((∂0(k))/(∂ ˆαl
n(k1)))]T.
To get φi(k1), i=1,...,n, notice that
φi(k1)
=− ∂0(k)
ˆαl
i(k1)
=−
n
i=1Nlωi,ˆαl
i(k1)J(ˆ
θW(k1))
ˆαl
i(k1)dF(k)
=−
n
i=1Nlωi,ˆαl
i(k1)
ˆαl
i(k1)J(ˆ
θW(k1))dF(k)
n
i=1
Nlωi,ˆαl
i(k1)J(ˆ
θW(k1))
ˆαl
i(k1)dF(k)
φi,1(k1)+φi,2(k1). (39)
Note that rewriting (36) yields
dF(k)=n
i=1
Nlωi,ˆαl
i(k1), q1J(ˆ
θW(k1), q1)
1
×0(k). (40)
Substituting (40) into (39), one can obtain
φi,1(k1)
=−
n
i=1Nlωi,ˆαl
i(k1)
ˆαl
i(k1)J(ˆ
θW(k1))dF(k)
=−
Nlωi,ˆαl
i(k1)
ˆαl
i(k1)
n
j=i
Nlωj,ˆαl
j(k1)J(ˆ
θW(k1))
3301813 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 5, MAY 2017
×n
i=1
Nlωi,ˆαl
i(k1)J(ˆ
θW(k1))1
0(k)
=−
Nlωi,ˆαl
i(k1)
ˆαl
i(k1)Nl1ωi,ˆαl
i(k1)0(k)
=q1(q1cos ωi)
11αl
i(k1)cos ωiq1αl
i(k1)q20(k).
(41)
Proof of the last equality in (41) is provided in
Appendix A.
For the second term φi,2(k1)in (39), note that with
given nand m, the analytical expression of J(ˆ
θW(k1), q1)
can be found by solving the Diophantine equation in (13). It
yields a polynomial of order m1 whose coefficients are func-
tions of ˆ
θW(k1). Hence, J(ˆ
θW(k1), q1)/∂ ˆαl
i(k1),
i=1,2,...,ncan be explicitly calculated, all of which are
polynomials of order m1.
To further improve the estimation performance and increase
the convergence rate, the a posteriori estimation error
(k)n
i=1Nli,ˆαl
i(k), q1)J(ˆ
θW(k), q1)dF(k)is used
to update φi(k1)in (39) and the PAA can be summarized
as follows.
1) Initialization:
ˆαi(0)=0.9,i=1,...,n
F(0)=30In/2(0)
φ(0)=φ(1)=[0,...,0]TRn
(1)=(2)=0.
2) Main Loop: For k=1,2,...
0(k)=
n
i=1
Nlωi,ˆαl
i(k1), q1
×J(ˆ
θW(k1), q1)dF(k)(42)
ˆ
θW(k)=ˆ
θW(k1)+F(k1)φ(k1)0(k)(43)
F(k)=1
λ(k)F(k1)
F(k1)φ(k1)φT(k1)F(k1)
λ(k)+φT(k1)F(k1)φ(k1)
(44)
φ(k1)=[φ1(k1),...,φ
n(k1)]Tgiven by (39)(45)
(k)=
n
i=1
Nlωi,ˆαl
i(k), q1J(ˆ
θW(k), q1)dF(k)(46)
Update φ(k1)by ˆαl(k)and (k). (47)
In the above 2-DOF adaptation algorithm, suppose that
there are nunknown frequency peaks in the disturbances,
then by explicitly utilizing the affine relationships between
aiand bivalues in (22) and (23), one can see that only
nparameters (a1,a2,...) need to be identified in the center
frequency adaptation stage. Meanwhile, in the passband width
adaptation process, the number of adaptive parameters is also
n,i.e.,αl
1
l
2,.... Moreover, the analytical expression of φi(k)
can be found and calculated via (39)–(41) in a recursive way,
where in each time step, a system with a maximum order of
2n+m1 is handled.
Remark:
1) For implementation, both terms in (39) can be trans-
formed into state-space forms such that in (47), all the
states are updated with the a posterior error and the
a posterior parameter estimate.
2) RPEM algorithm is advantageous in finding the local
minima under noisy environments. Initial values of αl
i
will thus influence the adaptation results. It is common
practice to set reasonable lower and upper bounds of
the parameters, based on the engineering problem in
practice.
V. STABILITY OF THE CONTROLLER
This section discusses the stability of the proposed con-
trol scheme. Based on the previous discussion, we have the
following.
1) The baseline controller C(z1)stabilizes the closed-loop
system, i.e., all baseline closed loop poles are within the
unit circle.
2) At frequencies where disturbance attenuation is desired,
P(ejω)ejmωPn(ejω)holds, i.e., the mismatch
between the plant and its nominal model is negligible in
the interested spectral regions.
Based on these, when P(z1)zmPn(z1)holds, the
structure in Fig. 2 is essentially a pseudo Youla parame-
terization [31] and the closed-loop stability depends on the
stability of the Qfilters. Recall the fact that Q(z1)=
BQ(z1)/AQ(z1)where AQ(z1)=AN(z1)=n
i=1(1
(1+αl
i)cos ωiz1+αl
iz2). Hence, the poles of the Qfilter
are located at
zi
1,2=1+αl
icos ωi
2±j1+αl
i2cos2ωi4αl
i
2i=1,...,n
with |zi
1,2|<1 if (proof provided in Appendix B)
1+αl
i2cos2ωi<4αl
i,i=1,...,n.(48)
Therefore, (48) provides a sufficient condition for the sta-
bility of the Qfilters as well as the closed-loop stability. For
implementation, it is easy to satisfy (48) in practice. In Stage I,
for the adaptation of ωivalues, αl
i1 is set, which makes
the inequality automatically satisfied, since we focus on the
disturbance whose spectral peaks are well below the Nyquist
frequency, i.e., cos2ωi<1. In Stage II, with the converged
ωivalues, (48) is utilized to calculate the lower and upper
bounds on αl
iduring adaptation.
When P(z1)=zmPn(z1), we apply robust stability
analysis to the problem. Let the plant model be subject
to a bounded uncertainty (z1), such that P(z1)=
zmPn(z1)(1+(z1)). Substituting the actual dynamics
into (7) yields the closed-loop characteristic equation
1+P(z1)C(z1)+Q(z1)zm(z1)=0.(49)
SUN et al.: ADAPTIVE LOOP SHAPING FOR WIDEBAND DISTURBANCES ATTENUATION 3301813
Fig. 8. Experimental setup of the VCFP system.
TAB L E I
RESONANCE PARAMETERS OF THE VCFP SYSTEM
Therefore, a sufficient condition for the robust stability of
the closed-loop system is
|Q(ejω)|<
1+P(ejω)C(ejω)
(ejω)
=
1
S0(ejω)(ejω)
ω. (50)
VI. EXPERIMENTS
A. Experimental Setup and Baseline Performance
In this section, the proposed adaptive loop shaping approach
for wideband disturbance attenuation is conducted on a VCFP
system for verification. As shown in Fig. 8, the experimental
setup includes a VCFP simulator board, a dSPACE real-
time DS1104 controller, and a user interface developed via
ControlDesk.6
The main component of the experimental setup is the VCFP
simulator board, which contains a complete analog circuit that
is designed to mimic the mechanical dynamics of HDDs based
on a lumped parameter model with three masses of inertia and
two springs with dampings [32]. The model and parameters
of the VCFP simulator are given as follows (see Appendix C
for detailed parameters):
P(s)
=2.03×1016 (s+1.25×105)2
s(s+620)(s+9407)(s2+1478s+9.33×107)(s2+1873s+1.70×108)
and the resonance parameters are listed in Table I.
With a sampling frequency of Fs=8000 Hz, Fig. 9 presents
the frequency responses of the plant and the identified discrete-
time nominal model z3Pn(z1). It shows that via the lumped
6ControlDesk is a software package developed by dSPACE that allows the
users to develop controllers in MATLAB Simulink and deploys them to the
dSPACE real-time controller.
Fig. 9. Frequency responses of the plant and its nominal model.
Fig. 10. Magnitude response of the sensitivity function S0(z1).
parameter model, the input–output frequency response of the
VCFP system shares similar characteristics with HDDs, with
a damped inertia type of main dynamics and a collection of
high-frequency resonance modes.
Fig. 10 shows the magnitude response of the baseline
sensitivity function S0(z1)achieved with a PID controller.
One can see that the bandwidth of the closed-loop system is
about 1/20 of the Nyquist frequency, which is typical in digital
control. Above that, disturbance attenuation is quite limited.
As a matter of fact, most high-frequency disturbances will be
amplified with the baseline controller.
Fig. 11 gives an example baseline closed-loop performance
in the presence of wideband disturbances.7Clearly, strong
vibrations remain in the PES. Moreover, such vibrations gen-
erate two wide spectral peaks at frequencies around 1000 and
1500 Hz, which are located above the closed-loop bandwidth
and largely amplified.
Hence, we focus on the attenuation of the wideband distur-
bances that generate the two spectral peaks (i.e., n=2) above
the bandwidth. From Fig. 9, it can be seen that, within our
interested frequency range, the nominal model approximately
equals to the actual system, i.e., P(z1)z3Pn(z1)holds.
7The wideband disturbance data used in experiments is a scaled and resam-
pled version from actual test data of audio vibrations in HDDs. Typically,
the frequency peaks of such audio vibrations are within 1000–4000 Hz.
Considering the relatively low sampling rate, a two-peak audio vibration
profile is chosen.
3301813 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 5, MAY 2017
Fig. 11. Baseline control performance with wideband disturbances.
Fig. 12. Adaptation to the center frequencies of the spectral peaks.
B. Performance With Adaptive Compensation
1) Adaptation of the Center Frequencies: To attenuate the
remaining vibrations in Fig. 11, as discussed in Section IV, the
first step is to correctly identify the center frequency parame-
ters f1and f2. With a fixed notch width θW=[0.92,0.92]T,
the estimates of f1,2are shown in Fig. 12, where we can see
that both center frequencies have been correctly identified.
Fig. 13 shows the frequency response of the converged Q
filter. As discussed in Section III, it is of bandpass prop-
erty with two passbands, the center frequencies of which
are located at the identified frequencies. The corresponding
compensation performance is shown in Fig. 14. We can see
that, compared to the baseline performance, the large spectral
peaks in the PES have been largely attenuated, although the
cancelation is not strong enough due to the narrow notch
width.
2) Adaptation of the Notch/Passband Widths: To s h ow
the influence of notch/passband widths to the compensation
performance, a series of experiments with different notch
widths is conducted with the identified frequencies f1and f2.
As shown in Fig. 15, even if we set an identical width
parameter for both peaks, there is an “optimal” width (αl),
such that the steady-state PES can be minimized. If αlgrows
too large, the generated notch will be too narrow to be
Fig. 13. Frequency response of the Qfilter from center frequency adaptation.
Fig. 14. Performance comparison with and without the add-on adaptive
compensator with fixed θW.
Fig. 15. Relationship between the compensation performance and the notch
width parameters.
effective. On the other hand, if αldecreases too much, the
notch becomes too wide; due to the “waterbed” effect, the
influence at other frequencies can be significant and eventually
degrade the overall servo performance. In fact, if the notch
width becomes too wide, the robust stability condition in (50)
can be violated, leading to system instability.
To adaptively find such “optimal” notch widths, the PAA
discussed in Stage II (Section IV) is implemented and the
SUN et al.: ADAPTIVE LOOP SHAPING FOR WIDEBAND DISTURBANCES ATTENUATION 3301813
Fig. 16. Evolution of the width parameters.
Fig. 17. Frequency response of the converged Qfilter with widths adaptation.
Fig. 18. Baseline error and compensated error with adaptively tuning widths.
results are shown in Figs. 16–18. The width parameters
for different spectral peaks converge to optimal values in
Fig. 16, such that the resulted Qfilter in Fig. 17 generates
two passbands that are wide enough to cancel the major
vibration components, but still not too aggressive to amplify
the disturbances at other frequencies, i.e., the gains of the Q
filter at other frequencies are reasonably small.
With the converged width parameters, the compensation
performance of the proposed structure is given in Fig. 18.
It shows that, compared to the baseline error and the one with
TAB L E I I
MODEL PARAMETERS OF THE VCFP SYSTEM
preselected width parameters (Fig. 14), the proposed algorithm
can automatically tune for better passband width parameters to
minimize the error. The large spectral peaks around 1000 and
1500 Hz have been effectively attenuated in the compensated
error spectrum, as shown in Fig. 18.
Remark:
1) Note that although the lumped-parameter-model-based
VCFP system has difference with the structured dynam-
ics of HDDs, yet it captures the key input–output
frequency features of that of HDDs (a damped iner-
tia type of main dynamics and a collection of high-
frequency resonances). Moreover, from an algorithmic
viewpoint, the adaptation algorithm is greatly decoupled
from the model information, due to the nearly “model-
free” formulation in (7). As a result, the key results
on Qfilter design and the 2-DOF adaptation algorithm
addressed above will remain representative in real HDD
applications.
2) In determining the number of frequency bands, i.e.,
hyperparameter n, two ways can be adopted: 1) empir-
ical estimate based on system model and typical fre-
quency peaks of the audio vibrations (our experience
from a large amount of actual test data is that two
bands are commonly sufficient) and 2) simultaneously
run the proposed algorithm with different n(the number
of center frequencies) and choose the minimum one that
can achieve acceptable PES. More detailed discussion
about this point can be found in [18].
VII. CONCLUSION
In this paper, the problem of wideband disturbance atten-
uation at high frequencies in precision information storage
systems is addressed. An adaptive loop shaping approach is
proposed. Control parameters with respect to both the center
frequencies and the widths of the spectral peaks are devised in
an “optimal” way, such that the PES can be minimized. The
PAAs for both are discussed. Experiments are conducted on
3301813 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 5, MAY 2017
d¯x(t)
dt =
d1d12
J1
c12
J1
d12
J1
c12
J100
kM
J1
10 0 0 000
d12
J2
c12
J2
d12 d23 d2
J2
c12 c23
J2
d23
J2
c23
J20
00 1 0 000
00 d23
J3
c23
J3
d23 d3
J3
c23
J30
00 0 0 100
kE
L00 000
R
L
¯x(t)+
0
0
0
0
0
0
1/L
u(t)
y(t)=[
00000krkx0x(t)(54)
a VCFP system. The results show that the proposed adaptive
approach can effectively attenuate wideband disturbances with
correctly identified center frequencies and automatically tuned
optimal notch widths.
APPENDIX A
PROOF OF THE LAST EQUALITY IN (41)
Given
Nlωi
l
i,q1=12cosωiq1+q2
11+αl
icos ωiq1+αl
iq2
Bi,q1)
Aωi
l
i,q1
we have
Nlωi,ˆαl
i(k1)
ˆαl
i(k1)=
Bi,q1)
ˆαl
i(k1)Aωi,ˆαl
i(k1), q1
A2ωi,ˆαl
i(k1), q1
Bi,q1)Aωi,ˆαl
i(k1),q1
ˆαl
i(k1)
A2ωi,ˆαl
i(k1), q1(51)
where Bi,q1)/∂ ˆαl
i(k1)=0and
Ai,ˆαl
i(k1), q1)/∂ ˆαl
i(k1)=−cosωiq1+q2.
Substituting them into (51), one can obtain
Nlωi,ˆαl
i(k1)
ˆαl
i(k1)=Bi,q1)(cos ωiq1q2)
A2ωi,ˆαl
i(k1), q1.(52)
Note that Nl1i,ˆαl
i(k1), q1)=
((Ai,ˆαl
i(k1), q1))/(Bi,q1))), hence, we have
Nli,ˆαl
i(k1))
ˆαl
i(k1)Nl1i,ˆαl
i(k1), q1)
=(q2cos ωiq1)
Ai,ˆαl
i(k1), q1)(53)
which yields the final result in (41).
APPENDIX B
PROOF OF CONDITION IN (48)
Given AQ(z1)=n
i=1(1(1+αl
i)cos ωiz1+αl
iz2),
if (1+αl
i)2cos2ωi<4αl
i,i=1,...,n[i.e., (48) holds], then
the poles of the Qfilter will be located at
zi
1,2=1+αl
icos ωi
2±j1+αl
i2cos2ωi4αl
i
2i=1,...,n
whose magnitudes are given by
1+αl
i2cos2ωi
4+1+αl
i2cos2ωi4αl
i
4,i=1,...,n
=21+αl
i2cos2ωi4αl
i
4=1+αl
i2cos2ωi2αl
i
2.
Therefore, |zi
1,2|<1,i=1,...,nis equivalent to
1+αl
i2cos2ωi<2+2αl
i,i=1,...,n.
1+αl
icos2ωi<2,i=1,...,n
which obviously holds with αl
i(0,1)and cos2ωi<1for
all i=1,...,n.
APPENDIX C
PLANT PARAMETERS OF THE VCFP SYSTEM
The VCFP system is built based on a lumped parameter
model with two masses of inertia connected by three springs
with dampings. Its state-space dynamics representation is
given in (54) as shown at the top of this page, with parameters
listed in Table II.
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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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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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