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DECOUPLED DISTURBANCE OBSERVERS FOR DUALINPUTSINGLEOUTPUT
SYSTEMS WITH APPLICATION TO VIBRATION REJECTION IN DUALSTAGE HARD
DISK DRIVES
Xu Chen
Department of Mechanical Engineering
University of California, Berkeley
Berkeley, California 94720
Email: maxchen@me.berkeley.edu
Masayoshi Tomizuka
Department of Mechanical Engineering
University of California, Berkeley
Berkeley, California 94720
Email: tomizuka@me.berkeley.edu
ABSTRACT
The disturbance observer (DOB) has been one eﬀective
robust control approach for servo enhancement in single
inputsingleoutput systems. This paper presents a new
extension of the DOB idea to dualinputsingleoutput sys
tems, and discusses an optimal Qﬁlter design. The pro
posed decoupled disturbance observer (DDOB) provides
a ﬂexible approach to use the most suitable actuators for
compensating disturbances at diﬀerent frequencies. Such
a scheme is helpful, e.g., for modern dualstage hard disk
drives, where enhanced servo design is becoming more and
more essential in the presence of audio vibrations.
1 INTRODUCTION
With the ever increasing demand of larger capacity in
hard disk drive (HDD) systems, dualstage actuation has
become an essential technique to break the bottleneck of the
servo performance in singleactuator HDDs [1, 2]. In this
structural conﬁguration, the microactuator has enhanced
mechanical performance in the highfrequency region, pro
viding the capacity to greatly increase the servo bandwidth.
Among diﬀerent choices of the secondary actuators, piezo
electric (PZT) microactuators have various creditable prop
erties and have been the research focus since their appear
ance (see, e.g., [3, 4] and the references therein). In this
dualinputsingleoutput (DISO) system, the two actuators
receive respectively current and voltage inputs, while only
the position error of the read/write head is measurable for
servo control.
Despite the mechanical advantages, compared to
singlestage HDDs, much less research has been conducted
to the servo control of a dualstage HDD. One algorithm
that is useful for singlestage HDDs [5, 6] but not well de
veloped for dualstage HDDs, is the disturbance observer
(DOB) [7]. The DOB counteracts the disturbance by its es
timate, which is generated by utilizing an inverse model of
the plant and a socalled Q ﬁlter. As a ﬂexible and power
ful addon element for servo enhancement, DOB has had
broad applications in ﬁelds other than HDDs, including
but not limited to: optical disk drives [8], linear motors [9],
positioning tables [10], robot arms [11], and automotive
engines [12]. These results are also restricted to single
inputsingleoutput (SISO) systems, while the generaliza
tion to DISO plants has not been fully addressed. Among
the related literature, [13] applied one conventional DOB
to each actuator in a dualstage HDD. The ﬁnal position of
the DISO system here is the summation of the outputs of
the two actuators: VoiceCoilMotor (VCM) actuator and
Microactuator. The conventional DOB for the microactua
tor treated the ﬁrststage output and the actual disturbance
as an eﬀective total disturbance, and tried to cancel it. In
the mean time, for the VCM actuator, the actual distur
bance was also regarded (together with the microactuator
output) as part of the eﬀective disturbance in the VCM
DOB. What is unclear here is that what portion of the ac
tual disturbance is canceled by each DOB. If the cancel
lation of the lowfrequency components relied too heavily
on the microactuator, a complicated consequence may arise
since the moving range of the microactuator is very limited.
Safety concerns will also arise as DOB for the microactua
tor treated the ﬁrststage output as an internal disturbance,
and tended to undo the achieved (longrange) movement
of the VCM actuator. References [14] and [15] discussed
statespace designs to implement the idea of disturbance
observers in special classes of MIMO systems. Within this
framework, the transferfunction approach of model inver
sion and Qﬁlter design was replaced with an observertype
statespace construction. A customized treatment of DISO
systems was not given. Independent application of each
actuator for disturbance rejection (decoupled disturbance
compensation) was not achieved.
Two main limitations in the generalization of DOB to
DISO system have been the nontrivial model inversion and
the distribution of the compensation eﬀorts. This paper
proposes a new decoupled disturbance observer (DDOB)
to simplify the above obstacles. Diﬀerent from previous
literature, the coupling of individual channels is directly
considered in the structural DDOB design, resulting in sev
eral advantageous properties. First, a full separation of the
actual external disturbances can be achieved. No cross
channel coupling eﬀects enter as internal disturbances. Sec
ond, the proposed scheme has clear time and frequency
domain design intuitions, which were lacking in [13–15].
Finally, we have the design ﬂexibility to distribute the com
pensation eﬀort according to the mechanical properties of
each actuator and the frequency range of the disturbances.
These features make DDOB beneﬁcial for the compensation
of audio vibrations in HDDs. This type of problem is faced
more and more in modern HDDs, as highpower speakers
in multimedia applications (such as allinone computers
and digital TVs) generate large amounts of external dis
turbances. Such vibrations are extremely diﬃcult to han
dle in a costeﬀective way for HDDs, due to their intrinsic
properties of (a) environmental dependence; (b) appear
ing in a wide frequency range (from 300 Hz to as high as 4
kHz); and (c) having multiple resonances and wide spectral
peaks [16].
In the remainder of the paper, we will discuss the de
sign of DDOB for general DISO systems. The practical
implementation on HDDs will be incorporated as a design
example throughout the discussion. Sections 2 and 3 pro
vide respectively the controller structure and the stability
conditions. The central design of the Q ﬁlter is shown in
Section 4 (the proposed optimal Qﬁlter construction is also
suitable for SISO DOBs). Section 5 provides a detailed case
study. Section 6 concludes the paper.
2 DDOB STRUCTURE
We start by considering a general discretetime linear
timeinvariant DISO system P(z−1)=[P1(z−1),P2(z−1)], with
the inputoutput relation:
y(k)=P1(z−1)u1(k)+P2(z−1)u2(k)+d(k).(1)
Here y(k),ui(k)(i=1,2) and d(k)represent respectively
the plant output, the plant inputs, and the lumped external
disturbance. We will slightly abuse the notation Pi(z−1)
to denote a transfer function and pulse transfer function,
so that Pi(z−1)ui(k)represents the timedomain output of
Pi(z−1). Fig. 1 shows the structure of the proposed DDOB
for the second channel P2(z−1). The idea is to apply the
compensation signal c2(k)to one actuator (in this case the
second actuator), such that the overall lumped disturbance
d(k)is compensated. Throughout the paper, we denote
ˆ
Pi(z−1) as the nominal model of Pi(z−1) (i=1,2), and mias
the relative degree of ˆ
Pi(z−1). In this way, although ˆ
P−1
2(z−1)
may not be realizable/causal, z−m2ˆ
P−1
2(z−1) in Fig. 1 is proper
in its minimal realization.
c
2
(k
)
Q
2
(z
¡
1
)
P
1
(z
¡
1
)
P
2
(z
¡
1
)
d
(k
)
y(k
)
u
1
(k
)
u
2
(k
)
z
¡
m
2
^
P
1
(z
¡
1
)
z
¡
m2^
P¡1
2
(z
¡
1
)
u
¤
2
(k
)
y
1
(k
)
Figure 1: Block diagram of DDOB for P2(z−1).
2.1 Timedomain Disturbancerejection Criteria
From Fig. 1, the output of Q2(z−1) is given by1c2=
Q2[z−m2ˆ
P−1
2(y−ˆ
P1u1)−z−m2u2]. Substituting (1) into this
result, we have
c2=Q2hz−m2ˆ
P−1
2(P1−ˆ
P1)u1+z−m2(ˆ
P−1
2P2−1)u2
+z−m2ˆ
P−1
2di.(2)
If Pi(z−1)=ˆ
Pi(z−1), (2) indicates that
c2(k)=Q2(z−1)z−m2ˆ
P−1
2(z−1)d(k).(3)
Notice that ˆ
P−1
2(z−1)d(k)can be regarded as an equiv
alent input disturbance for P2(z−1), and that c2(k)in (3) is
a delayed and ﬁltered version of ˆ
P−1
2(z−1)d(k). This ”ob
served” disturbance, after being multiplied by −1 and then
added into u2(k)in Fig. 1, gets ﬁltered through P2(z−1) and
1For simplicity, the indexes kand z−1are omitted here.
cancels d(k) (−P2c2(k)≈ −z−m2Q2d(k)≈ −d(k) if z−m2Q2≈1),
without inﬂuencing the output of P1(z−1) (the position out
put of the ﬁrst actuator).
2.2 Modelfollowing Property
One can remark that when Pi(z−1) diﬀers from ˆ
Pi(z−1),
the model mismatch is absorbed as an internal disturbance
in (2) (see the ﬁrst two terms in the square brackets). In this
subsection we explore this observation in greater details.
Notice that c2(k)=u2(k)−u∗
2(k) in Fig. 1. Combining this
information with (2), we can solve for u2(k)and substitute
the result to (1), to get: y(k)=Gyd(z−1)d(k)+Gyu1(z−1)u1(k)+
Gyu∗
2(z−1)u∗
2(k),where the three transfer functions are Gyd =
1−ˆ
P−1
2P2z−m2Q2
1+(ˆ
P−1
2P2−1)z−m2Q2,Gyu1=P1−ˆ
P−1
2P2(P1−ˆ
P1)z−m2Q2
1+(ˆ
P−1
2P2−1)z−m2Q2, and
Gyu∗
2=P2
1+(ˆ
P−1
2P2−1)z−m2Q2.
If z−m2Q(z−1)=1, we have
Gyd(z−1)=0,Gyu1(z−1)=ˆ
P1(z−1),Gyu∗
2(z−1)=ˆ
P2(z−1).(4)
Here Gyd(z−1)=0 explains the disturbancerejection
result in Section 2.1. Additionally, we observe that the
dynamics between the nominal inputs (u1and u∗
2) and
the output now is forced to follow the nominal model
ˆ
Pi(z−1) (i=1,2)–thus the rejection of modeling mismatch
within the DDOB loop. DDOB hence has the nominal
modelfollowing property. Notice that (4) equally holds
if one replaces z−1with e−jω, in which case the nomi
nal model following is enforced at the frequencies where
e−m2jωQ(e−jω)=1.
2.3 Operation of Two DDOBs
Swapping every applicable subindex between 1 and 2
in the preceding discussions, we get the DDOB for P1(z−1).
By linearity and (3), if two DDOBs operate simultaneously,
the disturbance compensation is achieved by
d(k)−P1(z−1)c1(k)−P2(z−1)c2(k)=
1−z−m1Q1(z−1)−z−m2Q2(z−1)d(k).(5)
One can remark that if a single DDOB already achieves
canceling the disturbance, say, d(k)−P1(z−1)c1(k) already
approximates 0, then the second DDOB is not necessary
and we should set Q2(z−1)=0. This is the ideal situation
when one actuator alone can eﬀectively handle all the dis
turbances. In practice, this may not always be feasible due
to the mechanical limitation of the actuators. In addition,
notice that d(k)−P1(z−1)c1(k)=1−z−m1Q1(z−1). It is the
oretically not possible for z−m2Q(z−1)=1 (using a causal
Q(z−1)) to hold over the entire frequency region.2The sec
ond DDOB can then be used to reduce the residual errors
of the ﬁrst DDOB.
We propose to apply frequencydependent DDOBs
based on the actuator dynamics and disturbance proper
ties. For example, in HDD applications, the VCM actu
ator (P1(z−1) in Fig. 1) has a large actuation range and
the microactuator (P2(z−1) in Fig. 1) suits only for small
range positioning. Additionally, ˆ
P−1
1(z−1) has properties
similar to a double diﬀerentiator in the highfrequency
region [5, 6], yielding large highfrequency noises in the
output of ˆ
P−1
1(z−1). Such actuator dynamics renders VCM
DDOB to have increased diﬃculties as the disturbance fre
quency gets higher and higher. The microactuator on the
other hand has a model of a DC gain plus resonances above
4 kHz, and a better signaltonoise ratio during implemen
tation of ˆ
P−1
2(z−1). From the above considerations, in the
lowfrequency region, we can apply DDOB to the large
stroke VCM actuator, by assigning Q1(z−1) to be a low
pass/bandpass ﬁlter and Q2(e−jω)≈0. At middle and high
frequencies, the precise and fasterresponse microactuator
can be more eﬀectively used. This is achieved by assign
ing Q1(e−jω)≈0 and Q2(z−1) to have a bandpass structure.
Throughout this paper, unless otherwise stated, we assume
the above decoupled disturbancerejection scheme.
3 STABILITY AND LOOPSHAPING CRITERIA
This section discusses the design criteria and the
closedloop stability when DDOB is applied to a closed
loop consisting of the DISO plant and a baseline feedback
controller C(z−1)=[C1(z−1),C2(z−1)]T. Fig. 2 shows the pro
posed controller implementation. We will present analysis
of DDOB for the secondary actuator. The result for the ﬁrst
actuator is obtained by interchanging the subindexes in
the transfer functions.

c
2
(k
)
Q
2
(z
¡
1
)
P
1
(z
¡
1
)
P
2
(z
¡
1
)
d
(k
)
y(k
)
+
u
1
(k
)
u
2
(k
)
e
(k
)
+


r
=
0
z
¡
m
2
C
2
(z
¡
1
)
C
1
(z
¡
1
)
+

^
P
1
(z
¡
1
)
z
¡
m
^
P
¡1
2
(z
¡
1
)
+
+++
+
Figure 2: Closedloop block diagram with DDOB for
P2(z−1).
2There are also stability constrains when the plant model ˆ
Pi(z−1) does
not fully capture Pi(z−1).
It can be shown (see Appendix A) that the block dia
gram in Fig. 2 is equivalent to that in Fig. 3, where DDOB
aﬀects the secondary actuator via the following series and
parallel addon components:
C2,s(z−1)=1
1−z−m2Q2(z−1)(6)
C2,p(z−1)=h1+ˆ
P1(z−1)C1(z−1)iz−m2ˆ
P−1
2(z−1)Q2(z−1).(7)
P
1
(z
¡
1
)
P
2
(z
¡
1
)
d
(k
)
y(k
)
C
1
(z
¡
1
)
C
2
(z
¡
1
)
C
2
;
s
(z¡1
)
C
2
;
p
(z¡1
)
Figure 3: An equivalent block diagram of the system in Fig.
2: DDOB is decomposed to series and parallel modules.
3.1 Nominal Stability and Loopshaping Criteria
From Fig. 3, the loop transfer function (obtained by
cutting oﬀthe feedback line of y(k)) is
L=P1C1+P2C2,s(C2+C2,p) (8)
=P1C1+P2
C2+1+ˆ
P1C1z−m2ˆ
P−1
2Q2
1−z−m2Q2
.(9)
If Pi=ˆ
Pi, (9) simpliﬁes to L=P1C1+P2C2+z−m2Q2
1−z−m2Q2, and the
sensitivity function of the closedloop system is
S=1
1+L=1−z−m2Q2
1+P1C1+P2C2
.(10)
Notice that 1/(1+P1C1+P2C2)is the baseline closed
loop sensitivity function. Stherefore is stable as long as
Q2is stable. In addition, 1 −z−m2Q2can be applied as a
frequencydomain design criteria for the desired loop shap
ing. Speciﬁcally, from (10), the complementary sensitivity
function is
T=1−S=P1C1+P2C2+z−m2Q2
1+P1C1+P2C2
.(11)
In the frequency regions where z−m2Q2is approximately 1,
S≈0 in (10) and T≈1 in (11), i.e., the closedloop system
has enhanced performance of disturbance rejection and ref
erence following. When z−m2Q2is approximately 0, Sand
Tare close to their baseline versions (without DDOB) and
the original system response is preserved. One can remark
that the proposed algorithm inherits the beneﬁt of the SISO
DOB [7] in that it uses a single ﬁlter Q2to ﬂexibly enhance
the system performance at the desired frequencies.
With slightly more algebra, we can obtain the closed
loop internal stability condition:
Theorem 1. (nominal stability) Given an internally stable
baseline feedback system, if the exact model of the plant is avail
able; and ˆ
P−1
2(z−1)is stable, then the closedloop system in Fig.
2 is internally stable as long as Q2(z−1)is stable.
Proof. From (9), under the stated conditions, the closed
loop characteristic polynomial is given by
DQ2Nˆ
P2×
DP1DP2DC1DC2+NP1NC1DP2DC2+NP2NC2DP1DC1
where N(·)and D(·)denote respectively the numerator
and denominator of a transfer function. Notice that
DP1DP2DC1DC2+NP1NC1DP2DC2+NP2NC2DP1DC1is the
characteristic polynomial for the baseline system. The in
ternal stability follows readily from the assumptions.
3.2 Robust Stability
Since plant uncertainty always exist in reality, actual
implementation of the Q ﬁlter is constrained by the robust
stability condition. Consider the plant being perturbed
to ˜
Pi(e−jω)=Pi(e−jω)1+Wi(e−jω)∆i(e−jω)(i=1,2), where
Wi(e−jω)’s are weighting functions, and the multiplicative
disk uncertainties satisfy
∆i(e−jω)
∞≤1. Since the DISO
system is a special multipleinputmultipleoutput (MIMO)
system, the µanalysis (see, e.g., [17]) tool can be applied to
derive the robust stability condition.
Theorem 2. The closedloop system in Fig. 2 is stable w.r.t. the
perturbed plant if and only if the following structured singular
value µis strictly less than 1.
µ=1−z−m2Q2P1C1 W1
1+P1C1+P2C2
+P2C2+(1+P1C1)z−m2Q2W2
1+P1C1+P2C2.(12)
Proof. See Appendix B.
Remark: Overall (12) infers that in the regions where a
good model is available for the plant (i.e., Wiis small),
the structured singular value is small and we have ﬂexible
design freedom in Q2(z−1). If e−m2jωQ2(e−jω)=0, DDOB is
turned oﬀat this frequency and (12) is simply the structured
singular value of the baseline feedback system. This infers
that the baseline system needs to be robustly stable. In
the frequency region where e−m2jωQ2(e−jω) is close to unity,
µ≈ W2and the robust stability depends on the model un
certainty of the secondary actuator. For dualstage HDDs,
accuracy of the model is usually preserved up to at least
5kHz, providing a large range for safe Qﬁlter design.
4 DESIGN OF Q FILTERS
From (10), forming Q(z−1) as a lowpass ﬁlter yields
the enhanced lowfrequency servo performance similar to
conventional SISO DOBs [7]. Various researches have been
conducted w.r.t. designing such Q ﬁlters [18–20]. For vi
bration rejection, the disturbance is not restricted to occur
at low frequencies. In this case, it is more beneﬁcial to
assign to Q(z−1) a bandpass property. This section pro
vides an optimal design of Q(z−1) to achieve loop shaping
at selective frequency locations. By using convex optimiza
tion techniques, we are able to design Q(z−1) with arbitrary
magnitude (upper) bounds and at the same time minimize
the disturbance ampliﬁcation in the closedloop system.
Consider the following construction in (10):
1−z−mQ(z−1)=Fn f (z−1)K(z−1),(13)
K(z−1)=k1+k2z−1+...knk+1z−nk.(14)
Here z−mQ(z−1) can be either z−m1Q1(z−1) or
z−m2Q2(z−1); Fn f (z−1) is a notch ﬁlter that provides the de
sired low gains (in a range of frequencies) to (10) (see Fig.
6); K(z−1) is essential for realizability of Q(z−1) and provides
additional optimal properties to Q(z−1).
Consider the general notchﬁlter structure Fn f (z−1)=
Bn f (z−1)/An f (z−1) with Bn f (z−1)=b1+b2z−1+···+bnb+1z−nb
and An f (z−1)=a1+a2z−1+···+ana+1z−na. Solving (13) gives
Q(z−1)=zmAn f (z−1)−Bn f (z−1)K(z−1)
An f (z−1)=:zmX(z−1)
An f (z−1).
Since zmis not causal, to have a realizable Q(z−1), the co
eﬃcients of z−i(i=0,1,...,m−1) need to be zero in X(z−1).3
Expanding the convolution Bn f (z−1)K(z−1) and grouping
the coeﬃcients in An f (z−1)−Bn f (z−1)K(z−1), we obtain the
3If m=0, causality is automatically satisﬁed.
causality condition in the following matrix form:
a1
a2
.
.
.
am
−
b10 0 0 01,nk+1−m
b2b10 0 01,nk+1−m
.
.
.......0 01,nk+1−m
bm... b2b101,nk+1−m
k1
k2
.
.
.
.
.
.
knk+1
=0.(15)
If nk+1=m, the mequations in (15) deﬁne a unique
solution for K(z−1). Additionally, nkcan be set to be larger
than m−1 so as to allow more design freedom in Q(z−1).
First, we can minimize the inﬁnity norm of 1 −z−mQ(z−1)
(maximum magnitude in frequency response), which will
in turn minimize the disturbance ampliﬁcation in the sen
sitivity function (10). This can be achieved by minimizing
K(z−1)∞in (13). Second, as discussed in Section 3.2, to
keep the system robustly stable, the magnitude of Q(z−1)
should be small at frequencies outside its passband, es
pecially at the frequencies where large model uncertainty
exists (normally in the highfrequency region). This corre
sponds to conﬁning Q(e−jωi) ≤ i, where iis some user
deﬁned bound and ωiis the frequency at which the mag
nitude constraint is required (there can be multiple of such
constraints).
By applying the boundedreal lemma, the H∞
performance objective (i.e., min K(z−1)∞) can be trans
lated to a linear matrix inequality (LMI) [21, 22]. The
causality constraint (15) is a set of linear equations.4In
addition, due to the FIR construction of K(z−1) in (13),
the gain constraint Q(e−jωi) ≤ ican be transformed to
a convex quadratic constraint. To see this point, no
tice ﬁrst that Q(e−jωi) ≤ iis equivalent to Q(e−jωi)2≤
2
i. Denoting θ=[k1,k2, . .. knk+1]Tas the coeﬃcient vec
tor of K(z−1) in (1314), we can express the inequality
Q(e−jωi)2=zm(1 −Fn f (z−1)K(z−1))z=ejωi
2≤2
iin the follow
ing quadratic form of θ:
θThψr(ωi)ψT
r(ωi)+ψm(ωi)ψT
m(ωi)iθ
−2ψT
r(ωi)θ+1≤2
i(16)
4Explicitly nk>m−1 is assumed in this case, since if nk=m−1 then
the solution of K(z−1) is unique from (15).
with
ψT
r(ωi)=Fr(ωi)φT
r(ωi)−Fm(ωi)φT
m(ωi)
ψT
m(ωi)=Fr(ωi)φT
m(ωi)+Fm(ωi)φT
r(ωi)
Fn f (e−jωi)=Fr(ωi)−jFm(ωi)
φT
r(ωi)=h1,cos(ωi), .. . cos (nkωi)i
φT
m(ωi)=h0,sin(ωi), .. . sin (nkωi)i.
Here φr(ωi) and φm(ωi) come from coeﬃcients of
the real and the imaginary parts of K(e−jωi),φT
r(ωi)θ−
jφT
m(ωi)θ;ψr(ωi) and ψm(ωi) are from Fn f (e−jωi)K(e−jωi),
ψT
r(ωi)θ−jψT
m(ωi)θ; and (16) is obtained by substituting
Fn f (e−jωi)K(e−jωi) into Q(e−jωi)2=1−Fn f (e−jωi)K(e−jωi)2.
Notice that (16) is a convex constraint in θ, since
ψr(ωi)ψT
r(ωi)+ψm(ωi)ψT
m(ωi) is positive semideﬁnite.
Summarizing the above discussions, we obtain the fol
lowing constrained optimization problem
min
θ,P,γ :γ
s.t.: causality constraint (17)
ATPA −P ATPB CT
BTPA BTPB −γI DT
C D −γI
0,P0 (18)
θThψr(ωi)ψT
r(ωi)+ψm(ωi)ψT
m(ωi)iθ
−2ψT
r(ωi)θ+1≤2
i,i=1,2,... (19)
where ’s.t.’ denotes ’subject to’ and we have translated the
objective of ’min : K(z−1)∞’ to ’min : γs.t. (18)’ (using
the boundedreal lemma). Here A,B,Cand Dare the
statespace matrices of K(z−1). We choose the controllable
canonical form
A="0nk−1,1Ink−1
0 01,nk−1#,B="0nk−1,1
1#
C=hk2,...,knk+1i,D=k1
such that (18) is linear in the decision variables θand P.
The entire optimization problem is now convex, and can
be eﬃciently solved using the interiorpoint method (see,
e.g., [23]) in modern optimization.
5 CASE STUDY
The proposed DDOBs are applied in this section to a
simulated example that uses the system conﬁguration on
page 195 of the book [24]. The plant model comes from
identiﬁcation of an actual experimental setup. A set of
disturbance data is obtained from audiovibration tests on
an actual HDD. The top plot of Fig. 7 shows the resulting
spectrum of the position error signal, where it is observed
that large spectral peaks appear at 1000 Hz and 3000 Hz. We
will design DDOBs for both the VoiceCoilMotor (VCM)
and microactuator (MA) actuators. The former is denoted
as VCM DDOB, and the latter as MA DDOB.
Two major resonances exist in the VCM plant, and
are compensated via two notch ﬁlters at 3.0 kHz and 6.5
kHz. The 11order resonancecompensated VCM model is
treated as a generalized plant P1(z−1). Such a design enables
a reducedorder and minimumphase ˆ
P1(z−1). As the notch
ﬁlters introduced some additional phase loss to P1(z−1),
overall ˆ
P1(z−1) contains a 2step delay, i.e., m1=2. After
the above construction, one nonminimumphase zero ap
pears in ˆ
P1(z−1) near the Nyquist frequency. This zero is
replaced with a stable one that lies strictly inside the unit
circle, yielding the ﬁnal nominal model shown in Fig. 4.
Through the above design, both the magnitude and the
phase of P1(z−1) is well captured by ˆ
P1(z−1) up to around 6
kHz.
The microactuator also contains two resonances that
are compensated by notch ﬁlters (at 6.5 kHz and 9.6 kHz).
This actuator is a minimumphase system (DC gain plus
resonances) by nature, which simpliﬁes the ˆ
P2(z−1) design.
We directly model ˆ
P2(z−1) to include the resonances and
have m2=1 in Fig. 1.
The decoupled sensitivity design (see, e.g., [4]) is
used to form the baseline feedback loop. The baseline
closedloop sensitivity function has the magnitude re
sponse shown by the solid line in Fig. 5. We can recognize
that such a loop shape is quite standard in feedback control.
10
2
10
3
10
4
100
50
0
50
100
Gain (dB)
10
2
10
3
10
4
180
90
0
90
180
Phase (degree)
Frequency (Hz)
plant
plant w/ notch filters
nominal model
Figure 4: ˆ
P1(z−1) design for DDOB in VCM actuation stage.
101102103104
−160
−140
−120
−100
−80
−60
−40
−20
0
20
Frequency (Hz)
Magnitude (dB)
w/o compensator
w/ compensator
Figure 5: Magnitude responses of the sensitivity functions
with and without DDOBs.
VCM DDOB is conﬁgured to work at the lowfrequency
region, with a bandpass Q ﬁlter that is centered at 1000 Hz
with a passband of 400 Hz. MA DDOB aims at rejecting
the highfrequency disturbances, with a 600 Hzpassband
bandpass Q ﬁlter centering at 3000 Hz. From the resulting
magnitude response of the sensitivity function (the dashed
line in Fig. 5), it is observed that two deep notches are cre
ated at the corresponding Qﬁlter center frequencies, indi
cating the strong disturbance rejections there. In the Qﬁlter
design, we have constructed Fn f (z−1)=1−2 cos ω0z−1+z−2
1−2αcosω0z−1+α2z−2in
(13), and constrained the optimal Q(z−1) to have its gains
at DC and Nyquist frequency to be lower than 50dB. This
contributes to the strongly retained baseline loop shape in
Fig. 5. Due to Bode’s Integral Theorem, some magnitude
increase occurs around 5 kHz. Such ampliﬁcation can be
reduced by designing the notch ﬁlter Fn f (z−1) to be not as
sharp as presented.
The ﬂexibility of DDOB is explained in Fig. 6, where
we ﬁx z−miand ˆ
Pi(z−1) in DDOBs and vary the center fre
quency of the Q ﬁlter. Six Q ﬁlters are evaluated, with
the resulting six sensitivity functions plotted in an overlaid
fashion. The ﬁrst three gain reductions come from VCM
DDOB with Q1(z−1) centered at 500 Hz, 900 Hz, and 1500
Hz respectively. The remaining three are generated by MA
DDOB (Q2(z−1) centered at 2300 Hz, 3100 Hz, and 3900
Hz). It is observed that by simple alternation of Q ﬁlters,
the servo loop can be customized to a great extension.
Figs. 7 and 8 present respectively the frequency and
timedomain servo performances using a modiﬁed version
of the disturbance data from actual vibration tests. It can
be observed that the spectral peaks at the corresponding
frequencies are signiﬁcantly reduced by DDOBs, and that
the magnitudes of position errors are decreased to be less
than one half of the original values.
101102103104
−180
−160
−140
−120
−100
−80
−60
−40
−20
0
20
Frequency (Hz)
Magnitude (dB)
w/o DDOB
w/ MA DDOB3
w/ VCM DDOB1
w/ VCM DDOB3
w/ VCM DDOB2
w/ MA DDOB1
w/ MA DDOB2
Figure 6: Magnitude responses of the sensitivity functions
with diﬀerent Qﬁlter conﬁgurations.
0 5000 10000 15000
0
1
2
3
Amplitude (%Track)
0 5000 10000 15000
0
0.5
1
1.5
2
2.5
Frequency (Hz)
Amplitude (%Track)
DDOB(s) off
DDOB(s) on
Figure 7: Spectra of the position error signals using a pro
jected disturbance proﬁle from actual experiments.
6 CONCLUSIONS
This paper has presented a ﬂexible feedback control ap
proach for servo enhancement in dualinputsingleoutput
control systems. Speciﬁcally we have discussed the ap
plication of the algorithm to compensate disturbances in
dualstage hard disk drives. The advantages of decou
pled disturbance rejection, extended highfrequency dis
turbance rejection in HDD systems, and optimal Qﬁlter
design, have been discussed in details.
ACKNOWLEDGMENT
This work was supported in part by the Computer Me
chanics Laboratory (CML) in the Department of Mechani
cal Engineering, University of California, Berkeley and by
0 20 40 60 80 100
−100
−50
0
50
100
PES (%Track)
0 20 40 60 80 100
−100
−50
0
50
100
Revolution
PES (%Track)
DDOB(s) off
DDOB(s) on
Figure 8: Time traces of the position error signals in Fig. 7.
a research grant from Western Digital Corporation.
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Appendix A: Equivalence between Figs. 2 and 3:
In Fig. 2, splitting the output of Q2(z−1) into two parts,
and relocating the summing junction after ˆ
P1(z−1), we get
Fig. 9a. Since the reference is zero, Fig. 9a is equivalent to
Fig. 9b. Finally noting that the ˆ
P1(z−1)C1(z−1) block in Fig.
9b does not inﬂuence the path from C1(z−1) to P1(z−1), we
obtain Fig. 3, the equivalent block diagram for analysis,
with the addon serial and parallel terms given by (6) and
(7).
Q
2
(z
¡
1
)
P
1
(z
¡
1
)
P
2
(z
¡
1
)
d
(k
)
y(k
)
z
¡
m
2
^
P
1
(z
¡
1
)
z
¡
m
2^
P
¡
1
2
(z
¡
1
)Q
2
(z
¡
1
)
C
2
(z
¡
1
)
C
1
(z
¡
1
)
(a) An equivalent block diagram of the system in Fig. 2: the input
to z−m2ˆ
P2(z−1) is relocated; the summing junction before Q2(z−1) is
separated.
z
¡
m2^
P
¡
1
2
(z
¡
1)Q
2
(z
¡
1
)
Q
2
(z
¡
1
)
P
1
(z
¡
1
)
P
2
(z
¡
1
)
d
(k
)
y(k
)
C
1
(z
¡
1
)
z
¡
m
2
^
P
1
(z
¡
1
)C
1
(z
¡
1
)
C
2
(z
¡
1
)
(b) An equivalent block diagram of the system in Fig. 3: the signs of
the signals are changed, after another relocation of block diagrams.
Figure 9: Block diagram transformation for Fig. 2.
Appendix B: Proof of Theorem 2
Consider ﬁrst the general closedloop system for DISO
plants under perturbation, as shown in Fig. 10, wherein
˜
Ci’s are the equivalent feedback controller. To obtain the
robust stability condition, we ﬁrst transform Fig. 10 to the
generalized representation in Fig. 11.
~
C
2
~
C
1
P
1
P
2
W
1
¢
1
W
2
¢
2
Figure 10: The general closedloop system for DISO plants
under perturbations.
¢
1
0
0 ¢
2
G
Figure 11: The generalized block diagram of Fig. 10
From µanalysis, the closedloop system is stable w.r.t.
the plant perturbations if and only if Gis stable and the
structured singular value of Gsatisﬁes: ∀ω,µ∆G(e−jω)<1.
In Fig. 11, consider the smallest (in the sense of H∞norm)
perturbation ∆such that the following stability boundary
is attained:
detI+ ∆(e−jω)G(e−jω)=0.(20)
After standard blockdiagram analysis, the generalized
plant Gcan be shown to be
G=W˜
CP
1+P˜
C=1
1+P1˜
C1+P2˜
C2"W1˜
C1
W2˜
C2#hP1P2i.(21)
Substituting ∆ = diag{∆1,∆2}and (21) to (20) yields (for
simpliﬁed notation, the frequency index e−jωis omitted)
det(I+ ∆G)
=det
I+"∆10
0∆2#
1+P1˜
C1+P2˜
C2"W1˜
C1
W2˜
C2#hP1P2i
=1+P1∆1W1˜
C1+P2∆2W2˜
C2
1+P1˜
C1+P2˜
C2
,(22)
where the last equality used the determinant identity
det(I+AB)=det (I+BA).
Combining (22) and (20), the minimumH∞norm per
turbation is obtained if ∆1=∆2=:∆0and the following
equality holds
1−
P1W1˜
C1
1+P1˜
C1+P2˜
C2
∆0−
P2W2˜
C2
1+P1˜
C1+P2˜
C2
∆0=0.
By deﬁnition (see, e.g., [17]), the structured singular
value is
µ=1
∆0=P1˜
C1W1+P2˜
C2W2
1+P1˜
C1+P2˜
C2
.(23)
When DDOB is in the feedback loop as shown in Fig.
3, ˜
C1=C1and ˜
C2=C2,sC2+C2,p, where C2,sand C2,pare
given by (6) and (7). Therefore, assuming ˆ
Pi=Pi, after
simpliﬁcation, one can get the explicit form of (12).