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DECOUPLED DISTURBANCE OBSERVERS FOR DUAL-INPUT-SINGLE-OUTPUT
SYSTEMS WITH APPLICATION TO VIBRATION REJECTION IN DUAL-STAGE 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-
input-single-output systems. This paper presents a new
extension of the DOB idea to dual-input-single-output sys-
tems, and discusses an optimal Q-filter 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 dual-stage 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, dual-stage actuation has
become an essential technique to break the bottleneck of the
servo performance in single-actuator HDDs [1, 2]. In this
structural configuration, the microactuator has enhanced
mechanical performance in the high-frequency 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
dual-input-single-output (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
single-stage HDDs, much less research has been conducted
to the servo control of a dual-stage HDD. One algorithm
that is useful for single-stage HDDs [5, 6] but not well de-
veloped for dual-stage 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 so-called Q filter. As a flexible and power-
ful add-on 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-
input-single-output (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 dual-stage HDD. The final position of
the DISO system here is the summation of the outputs of
the two actuators: Voice-Coil-Motor (VCM) actuator and
Microactuator. The conventional DOB for the microactua-
tor treated the first-stage 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 low-frequency 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 first-stage output as an internal disturbance,
and tended to undo the achieved (long-range) movement
of the VCM actuator. References [14] and [15] discussed
state-space designs to implement the idea of disturbance
observers in special classes of MIMO systems. Within this
framework, the transfer-function approach of model inver-
sion and Q-filter design was replaced with an observer-type
state-space 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 high-power speakers
in multimedia applications (such as all-in-one computers
and digital TVs) generate large amounts of external dis-
turbances. Such vibrations are extremely difficult to han-
dle in a cost-effective 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 Q-filter 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 discrete-time linear
time-invariant DISO system P(z−1)=[P1(z−1),P2(z−1)], with
the input-output 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 time-domain 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 Time-domain Disturbance-rejection 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 Model-following 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 disturbance-rejection
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-
model-following 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 sub-index 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 frequency-dependent 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 high-frequency
region [5, 6], yielding large high-frequency 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 signal-to-noise ratio during implemen-
tation of ˆ
P−1
2(z−1). From the above considerations, in the
low-frequency region, we can apply DDOB to the large-
stroke VCM actuator, by assigning Q1(z−1) to be a low-
pass/band-pass filter and Q2(e−jω)≈0. At middle and high
frequencies, the precise and faster-response microactuator
can be more effectively used. This is achieved by assign-
ing Q1(e−jω)≈0 and Q2(z−1) to have a band-pass structure.
Throughout this paper, unless otherwise stated, we assume
the above decoupled disturbance-rejection scheme.
3 STABILITY AND LOOP-SHAPING CRITERIA
This section discusses the design criteria and the
closed-loop 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 inter-changing the sub-indexes 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: Closed-loop 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 add-on 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 Loop-shaping 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 closed-loop 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
frequency-domain 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 closed-loop 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 closed-loop 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 multiple-input-multiple-output (MIMO)
system, the µ-analysis (see, e.g., [17]) tool can be applied to
derive the robust stability condition.
Theorem 2. The closed-loop 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−m2Q2||P1C1| |W1|
|1+P1C1+P2C2|
+|P2C2+(1+P1C1)z−m2Q2||W2|
|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., |Wi|is 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≈ |W2|and the robust stability depends on the model un-
certainty of the secondary actuator. For dual-stage HDDs,
accuracy of the model is usually preserved up to at least
5kHz, providing a large range for safe Q-filter design.
4 DESIGN OF Q FILTERS
From (10), forming Q(z−1) as a low-pass filter yields
the enhanced low-frequency 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 band-pass 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 closed-loop 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 notch-filter 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 high-frequency 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 bounded-real 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 (13-14), 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 semi-definite.
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 bounded-real lemma). Here A,B,Cand Dare the
state-space 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 interior-point 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 audio-vibration 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 Voice-Coil-Motor (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 11-order resonance-compensated VCM model is
treated as a generalized plant P1(z−1). Such a design enables
a reduced-order and minimum-phase ˆ
P1(z−1). As the notch
filters introduced some additional phase loss to P1(z−1),
overall ˆ
P1(z−1) contains a 2-step delay, i.e., m1=2. After
the above construction, one non-minimum-phase 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 minimum-phase 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
closed-loop 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 low-frequency
region, with a band-pass Q filter that is centered at 1000 Hz
with a passband of 400 Hz. MA DDOB aims at rejecting
the high-frequency disturbances, with a 600 Hz-passband
band-pass 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 Q-filter center frequencies, indi-
cating the strong disturbance rejections there. In the Q-filter
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
time-domain 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 Q-filter 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 dual-input-single-output
control systems. Specifically we have discussed the ap-
plication of the algorithm to compensate disturbances in
dual-stage hard disk drives. The advantages of decou-
pled disturbance rejection, extended high-frequency dis-
turbance rejection in HDD systems, and optimal Q-filter
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 add-on 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 closed-loop 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 closed-loop system for DISO plants
under perturbations.
¢
1
0
0 ¢
2
G
Figure 11: The generalized block diagram of Fig. 10
From µ-analysis, the closed-loop 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 block-diagram 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 minimum-H∞-norm per-
turbation is obtained if |∆1|=|∆2|=:|∆0|and 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˜
C1|W1|+P2˜
C2|W2|
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).
