Conference PaperPDF Available

Decoupled Disturbance Observers for Dual-Input-Single-Output Systems With Application to Vibration Rejection in Dual-Stage Hard Disk Drives

Authors:

Abstract and Figures

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 systems, and discusses an optimal Q-filter design. The proposed 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.
Content may be subject to copyright.
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 eective
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 dierent 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 dierent 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 eective 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 eective 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 eorts. This paper
proposes a new decoupled disturbance observer (DDOB)
to simplify the above obstacles. Dierent 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 eects 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 eort 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 dicult to han-
dle in a cost-eective 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(z1)=[P1(z1),P2(z1)], with
the input-output relation:
y(k)=P1(z1)u1(k)+P2(z1)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(z1)
to denote a transfer function and pulse transfer function,
so that Pi(z1)ui(k)represents the time-domain output of
Pi(z1). Fig. 1 shows the structure of the proposed DDOB
for the second channel P2(z1). 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(z1) as the nominal model of Pi(z1) (i=1,2), and mias
the relative degree of ˆ
Pi(z1). In this way, although ˆ
P1
2(z1)
may not be realizable/causal, zm2ˆ
P1
2(z1) in Fig. 1 is proper
in its minimal realization.
c
2
(k
)
Q
2
(z
¡
1
)
1
(z
¡
1
)
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(z1).
2.1 Time-domain Disturbance-rejection Criteria
From Fig. 1, the output of Q2(z1) is given by1c2=
Q2[zm2ˆ
P1
2(yˆ
P1u1)zm2u2]. Substituting (1) into this
result, we have
c2=Q2hzm2ˆ
P1
2(P1ˆ
P1)u1+zm2(ˆ
P1
2P21)u2
+zm2ˆ
P1
2di.(2)
If Pi(z1)=ˆ
Pi(z1), (2) indicates that
c2(k)=Q2(z1)zm2ˆ
P1
2(z1)d(k).(3)
Notice that ˆ
P1
2(z1)d(k)can be regarded as an equiv-
alent input disturbance for P2(z1), and that c2(k)in (3) is
a delayed and filtered version of ˆ
P1
2(z1)d(k). This ”ob-
served” disturbance, after being multiplied by 1 and then
added into u2(k)in Fig. 1, gets filtered through P2(z1) and
1For simplicity, the indexes kand z1are omitted here.
cancels d(k) (P2c2(k)≈ −zm2Q2d(k)≈ −d(k) if zm2Q21),
without influencing the output of P1(z1) (the position out-
put of the first actuator).
2.2 Model-following Property
One can remark that when Pi(z1) diers from ˆ
Pi(z1),
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(z1)d(k)+Gyu1(z1)u1(k)+
Gyu
2(z1)u
2(k),where the three transfer functions are Gyd =
1ˆ
P1
2P2zm2Q2
1+(ˆ
P1
2P21)zm2Q2,Gyu1=P1ˆ
P1
2P2(P1ˆ
P1)zm2Q2
1+(ˆ
P1
2P21)zm2Q2, and
Gyu
2=P2
1+(ˆ
P1
2P21)zm2Q2.
If zm2Q(z1)=1, we have
Gyd(z1)=0,Gyu1(z1)=ˆ
P1(z1),Gyu
2(z1)=ˆ
P2(z1).(4)
Here Gyd(z1)=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(z1) (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 z1with ejω, in which case the nomi-
nal model following is enforced at the frequencies where
em2jωQ(ejω)=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(z1).
By linearity and (3), if two DDOBs operate simultaneously,
the disturbance compensation is achieved by
d(k)P1(z1)c1(k)P2(z1)c2(k)=
1zm1Q1(z1)zm2Q2(z1)d(k).(5)
One can remark that if a single DDOB already achieves
canceling the disturbance, say, d(k)P1(z1)c1(k) already
approximates 0, then the second DDOB is not necessary
and we should set Q2(z1)=0. This is the ideal situation
when one actuator alone can eectively 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(z1)c1(k)=1zm1Q1(z1). It is the-
oretically not possible for zm2Q(z1)=1 (using a causal
Q(z1)) 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(z1) in Fig. 1) has a large actuation range and
the microactuator (P2(z1) in Fig. 1) suits only for small-
range positioning. Additionally, ˆ
P1
1(z1) has properties
similar to a double dierentiator in the high-frequency
region [5, 6], yielding large high-frequency noises in the
output of ˆ
P1
1(z1). Such actuator dynamics renders VCM
DDOB to have increased diculties 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
P1
2(z1). From the above considerations, in the
low-frequency region, we can apply DDOB to the large-
stroke VCM actuator, by assigning Q1(z1) to be a low-
pass/band-pass filter and Q2(ejω)0. At middle and high
frequencies, the precise and faster-response microactuator
can be more eectively used. This is achieved by assign-
ing Q1(ejω)0 and Q2(z1) 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(z1)=[C1(z1),C2(z1)]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(z1).
2There are also stability constrains when the plant model ˆ
Pi(z1) does
not fully capture Pi(z1).
It can be shown (see Appendix A) that the block dia-
gram in Fig. 2 is equivalent to that in Fig. 3, where DDOB
aects the secondary actuator via the following series and
parallel add-on components:
C2,s(z1)=1
1zm2Q2(z1)(6)
C2,p(z1)=h1+ˆ
P1(z1)C1(z1)izm2ˆ
P1
2(z1)Q2(z1).(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 othe feedback line of y(k)) is
L=P1C1+P2C2,s(C2+C2,p) (8)
=P1C1+P2
C2+1+ˆ
P1C1zm2ˆ
P1
2Q2
1zm2Q2
.(9)
If Pi=ˆ
Pi, (9) simplifies to L=P1C1+P2C2+zm2Q2
1zm2Q2, and the
sensitivity function of the closed-loop system is
S=1
1+L=1zm2Q2
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 zm2Q2can be applied as a
frequency-domain design criteria for the desired loop shap-
ing. Specifically, from (10), the complementary sensitivity
function is
T=1S=P1C1+P2C2+zm2Q2
1+P1C1+P2C2
.(11)
In the frequency regions where zm2Q2is approximately 1,
S0 in (10) and T1 in (11), i.e., the closed-loop system
has enhanced performance of disturbance rejection and ref-
erence following. When zm2Q2is 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
P1
2(z1)is stable, then the closed-loop system in Fig.
2 is internally stable as long as Q2(z1)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(ejω)=Pi(ejω)1+Wi(ejω)i(ejω)(i=1,2), where
Wi(ejω)’s are weighting functions, and the multiplicative
disk uncertainties satisfy
i(ejω)
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=|1zm2Q2||P1C1| |W1|
|1+P1C1+P2C2|
+|P2C2+(1+P1C1)zm2Q2||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(z1). If em2jωQ2(ejω)=0, DDOB is
turned oat 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 em2jωQ2(ejω) 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(z1) 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(z1) a band-pass property. This section pro-
vides an optimal design of Q(z1) to achieve loop shaping
at selective frequency locations. By using convex optimiza-
tion techniques, we are able to design Q(z1) 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):
1zmQ(z1)=Fn f (z1)K(z1),(13)
K(z1)=k1+k2z1+...knk+1znk.(14)
Here zmQ(z1) can be either zm1Q1(z1) or
zm2Q2(z1); Fn f (z1) is a notch filter that provides the de-
sired low gains (in a range of frequencies) to (10) (see Fig.
6); K(z1) is essential for realizability of Q(z1) and provides
additional optimal properties to Q(z1).
Consider the general notch-filter structure Fn f (z1)=
Bn f (z1)/An f (z1) with Bn f (z1)=b1+b2z1+···+bnb+1znb
and An f (z1)=a1+a2z1+···+ana+1zna. Solving (13) gives
Q(z1)=zmAn f (z1)Bn f (z1)K(z1)
An f (z1)=:zmX(z1)
An f (z1).
Since zmis not causal, to have a realizable Q(z1), the co-
ecients of zi(i=0,1,...,m1) need to be zero in X(z1).3
Expanding the convolution Bn f (z1)K(z1) and grouping
the coecients in An f (z1)Bn f (z1)K(z1), 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+1m
b2b10 0 01,nk+1m
.
.
.......0 01,nk+1m
bm... b2b101,nk+1m
k1
k2
.
.
.
.
.
.
knk+1
=0.(15)
If nk+1=m, the mequations in (15) define a unique
solution for K(z1). Additionally, nkcan be set to be larger
than m1 so as to allow more design freedom in Q(z1).
First, we can minimize the infinity norm of 1 zmQ(z1)
(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(z1)||in (13). Second, as discussed in Section 3.2, to
keep the system robustly stable, the magnitude of Q(z1)
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(ejωi)| ≤ 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(z1)||) 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(z1) in (13),
the gain constraint |Q(ejωi)| ≤ ican be transformed to
a convex quadratic constraint. To see this point, no-
tice first that |Q(ejωi)| ≤ iis equivalent to |Q(ejωi)|2
2
i. Denoting θ=[k1,k2, . .. knk+1]Tas the coecient vec-
tor of K(z1) in (13-14), we can express the inequality
|Q(ejωi)|2=zm(1 Fn f (z1)K(z1))z=ejωi
22
iin the follow-
ing quadratic form of θ:
θThψr(ωi)ψT
r(ωi)+ψm(ωi)ψT
m(ωi)iθ
2ψT
r(ωi)θ+12
i(16)
4Explicitly nk>m1 is assumed in this case, since if nk=m1 then
the solution of K(z1) 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 (ejωi)=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 coecients of
the real and the imaginary parts of K(ejωi),φT
r(ωi)θ
jφT
m(ωi)θ;ψr(ωi) and ψm(ωi) are from Fn f (ejωi)K(ejωi),
ψT
r(ωi)θjψT
m(ωi)θ; and (16) is obtained by substituting
Fn f (ejωi)K(ejωi) into |Q(ejωi)|2=|1Fn f (ejωi)K(ejωi)|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
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+12
i,i=1,2,... (19)
where ’s.t.’ denotes ’subject to’ and we have translated the
objective of ’min : ||K(z1)||’ to ’min : γs.t. (18)’ (using
the bounded-real lemma). Here A,B,Cand Dare the
state-space matrices of K(z1). We choose the controllable
canonical form
A="0nk1,1Ink1
0 01,nk1#,B="0nk1,1
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 eciently 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(z1). Such a design enables
a reduced-order and minimum-phase ˆ
P1(z1). As the notch
filters introduced some additional phase loss to P1(z1),
overall ˆ
P1(z1) contains a 2-step delay, i.e., m1=2. After
the above construction, one non-minimum-phase zero ap-
pears in ˆ
P1(z1) 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(z1) is well captured by ˆ
P1(z1) 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(z1) design.
We directly model ˆ
P2(z1) 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(z1) 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 (z1)=12 cos ω0z1+z2
12αcosω0z1+α2z2in
(13), and constrained the optimal Q(z1) 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 (z1) to be not as
sharp as presented.
The flexibility of DDOB is explained in Fig. 6, where
we fix zmiand ˆ
Pi(z1) 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(z1) centered at 500 Hz, 900 Hz, and 1500
Hz respectively. The remaining three are generated by MA
DDOB (Q2(z1) 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 dierent 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.
References
[1] Abramovitch, D. Y., and Franklin, G. F., 2002. “A brief
history of disk drive control”. IEEE Control Syst. Mag.,
22(3), pp. 28–42.
[2] Al Mamun, A., and Ge, S. S., 2005. “Precision control
of hard disk drives”. IEEE Control Syst. Mag., 25(4),
pp. 14–19.
[3] Suthasun, T., Mareels, I., and Al Mamun, A., 2004.
“System identification and controller design for dual
actuated hard disk drive”. Control Engineering Practice,
12(6), pp. 665–676.
[4] Horowitz, R., Li, Y., Oldham, K., Kon, S., and Huang,
X., 2007. “Dual-stage servo systems and vibration
compensation in computer hard disk drives”. Control
Engineering Practice, 15(3), pp. 291–305.
[5] White, M., Tomizuka, M., and Smith, C., 2000. “Im-
proved track following in magnetic disk drives using a
disturbance observer”. IEEE/ASME Trans. Mechatron-
ics, 5(1), Mar., pp. 3–11.
[6] Jia, Q.-W., 2009. “Disturbance rejection through distur-
bance observer with adaptive frequency estimation”.
IEEE Trans. Magn., 45(6), June, pp. 2675–2678.
[7] Ohnishi, K., 1993. “Robust motion control by distur-
bance observer”. Journal of the Robotics Society of Japan,
11(4), pp. 486–493.
[8] Yang, K., Choi, Y., and Chung, W. K., 2005. “On
the tracking performance improvement of optical disk
drive servo systems using error-based disturbance ob-
server”. IEEE Trans. Ind. Electron., 52(1), Feb., pp. 270–
279.
[9] Tan, K. K., Lee, T. H., Dou, H. F., Chin, S. J., and
Zhao, S., 2003. “Precision motion control with dis-
turbance observer for pulsewidth-modulated-driven
permanent-magnet linear motors”. IEEE Trans. Magn.,
39(3), pp. 1813–1818.
[10] Kempf, C. J., and Kobayashi, S., 1999. “Disturbance
observer and feedforward design for a high-speed
direct-drive positioning table”. IEEE Trans. Control
Syst. Technol., 7(5), pp. 513–526.
[11] Eom, K. S., Suh, I. H., and Chung, W. K., 2001. “Dis-
turbance observer based path tracking control of robot
manipulator considering torque saturation”. Mecha-
tronics, 11(3), pp. 325 – 343.
[12] Bohn, C., Cortabarria, A andHärtel, V., and Kowal-
czyk, K., 2004. “Active control of engine-induced vi-
brations in automotive vehicles using disturbance ob-
server gain scheduling”. Control Engineering Practice,
12(8), pp. 1029 – 1039.
[13] Nie, J., and Horowitz, R., 2009. “Design and imple-
mentation of dual-stage track-following control for
hard disk drives”. In Proc. 2nd Dynamic Systems and
Control Conf., Vol. 2, pp. 565–572.
[14] Guo, L., and Chen, W.-H., 2005. “Disturbance atten-
uation and rejection for systems with nonlinearity via
dobc approach”. International Journal of Robust and
Nonlinear Control, 15(3), pp. 109–125.
[15] Zheng, Q., Chen, Z., and Gao, Z., 2009. “A practical
approach to disturbance decoupling control”. Control
Engineering Practice, 17(9), pp. 1016–1025.
[16] Deller, J. R., Hansen, J. H. L., and Proakis, J. G., 1999.
Discrete-Time Processing of Speech Signals. Wiley-IEEE
Press, Sept.
[17] Zhou, K., and Doyle, J. C., 1998. Essentials of robust
control. Prentice Hall New Jersey, Oct.
[18] Wang, C.-C., and Tomizuka, M., 2004. “Design of
robustly stable disturbance observers based on closed
loop consideration using h-infinity optimization and
its applications to motion control systems”. In Proc.
2004 American Control Conf., Vol. 4, pp. 3764 –3769.
[19] Kemp, C. C., and Kobayashi, S., 1996. “Discrete-time
disturbance observer design for systems with time de-
lay”. In Proc. 1996 4th International Workshop on Ad-
vanced Motion Control, Vol. 1, pp. 332–337.
[20] Choi, Y., Yang, K., Chung, W. K., Kim, H. R., and
Suh, I. H., 2003. “On the robustness and performance
of disturbance observers for second-order systems”.
IEEE Trans. Autom. Control, 48(2), Feb., pp. 315–320.
[21] Scherer, C., Gahinet, P., and Chilali, M., 1997. “Mul-
tiobjective output-feedback control via LMI optimiza-
tion”. IEEE Trans. Autom. Control, 42(7), July, pp. 896–
911.
[22] Boyd, S. P., El Ghaoui, L., Feron, E., and Balakrishnan,
V., 1994. Linear matrix inequalities in system and control
theory. Society for Industrial Mathematics.
[23] Grant, M., and Boyd, S., 2011. CVX: Matlab soft-
ware for disciplined convex programming, version
1.21. http://cvxr.com/cvx, Feb.
[24] Al Mamun, A., Guo, G., and Bi, C., 2007. Hard disk
drive: mechatronics and control. CRC Press.
Appendix A: Equivalence between Figs. 2 and 3:
In Fig. 2, splitting the output of Q2(z1) into two parts,
and relocating the summing junction after ˆ
P1(z1), we get
Fig. 9a. Since the reference is zero, Fig. 9a is equivalent to
Fig. 9b. Finally noting that the ˆ
P1(z1)C1(z1) block in Fig.
9b does not influence the path from C1(z1) to P1(z1), 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 zm2ˆ
P2(z1) is relocated; the summing junction before Q2(z1) 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,µG(ejω)<1.
In Fig. 11, consider the smallest (in the sense of Hnorm)
perturbation such that the following stability boundary
is attained:
detI+ ∆(ejω)G(ejω)=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 ejωis omitted)
det(I+ ∆G)
=det
I+"10
02#
1+P1˜
C1+P2˜
C2"W1˜
C1
W2˜
C2#hP1P2i
=1+P11W1˜
C1+P22W2˜
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).
... Finally, Section 9 concludes the paper. A short version of the paper was presented in [15]. The present paper is a substantially extended version that provides detailed proofs and derivations (in Sections 2, 3, and 4), newly developed extensions (in Sections 1, 3.4, 6, and 7), and implementation details (Section 8). ...
... However, for the special case of DISO plants, we have a closed-form solution for µ, as described in the following theorem. Theorem 2. The closed-loop system in Fig. 5 is stable w.r.t. the perturbed plant in Eq. (15) and Eq. (16), if and only if nominal stability holds and the following structured singular value satisfies µ < 1: ...
... Fig. 15 shows the magnitude of the perturbations [W i ∆ i in Eqs. (15) and (16)] at different frequencies. This is a very typical uncertainty set in precision control, where the plant dynamics has large variations only at high frequencies. ...
Article
Full-text available
The disturbance observer (DOB) has been a popular robust control approach for servo enhancement in single-input single-output systems. This paper presents a new extension of the DOB idea to dual-and multi-input single-output systems, and discusses an optimal filter design technique for the related loop-shaping. The proposed decoupled disturbance observer (DDOB) provides the flexibility to use the most suitable actuators for compensating disturbances with different spectral characteristics. Such a generalization 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 vibration disturbances.
... Such a delay greatly affects the performance of disturbance rejection. This paper applies a new Q-filter design method [5] to improve the disturbance rejection. ...
... The DOB design provides the disturbance cancellation property by incorporating the term (1 − ( )) (z ) in Eq. (5). Consider the following construction: Solving (6) gives ...
... The μ-analysis tool can be applied to derive the robust stability condition [5]. The closed-loop system in Fig. 25 is stable if and only if the structured singular value of M(s) is strictly less than 1. ...
Conference Paper
Full-text available
Variable-gear-ratio steering is an advanced feature in automotive vehicles. As the name suggest, it changes the steering gear ratio depending on the speed of the vehicle. This feature can simplify steering for the driver, which leads to various advantages, such as improved vehicle comfort, stability, and safety. One serious problem, however, is that the variable gear-ratio system generates unnatural torque to the driver whenever the variable-gear-ratio control is activated. Such unnatural torque includes both low-frequency and steering speed-dependent components. This paper proposes a control method to cancel this unnatural torque. We address the problem by using a tire sensor and a set of feedback and feedforward algorithms. Effectiveness of the proposed method is experimentally verified using a hardware-in-the-loop experimental setup. Stability and robustness under model uncertainties are evaluated.
... No large sensitivity amplifications are observed due to: (i), a gain scheduling in the Q filter with Q implement = k Q Q(z −1 ), k Q ≤ 1 and (ii), the use of a damped notch filter F n f (z −1 ) = A(βz −1 )/A(αz −1 ) (α < β < 1) in the design of Q(z −1 ). Auxiliary zeros and convex optimization can be applied for specific magnitude constraints on Q(z −1 ) [8]. ...
Conference Paper
Full-text available
We have many servo systems that require nano/micro level positioning accuracy. This requirement sets a number of interesting challenges from the viewpoint of sensing, actuation, and control algorithms. This paper considers the control aspect for precision positioning. In motion control systems at nano/micro levels, it has been noted that the control algorithm should be customized as much as possible to the spectrum of disturbances. We will examine how prior knowledge about the disturbance spectrum should be utilized in the design of control algorithms and what are advantages of such prior knowledge. The algorithms will be evaluated on a simulated hard disk drive (HDD) benchmark problem and a laboratory setup for a wafer scanner that is equipped with a laser interferometer for position measurement.
Article
Full-text available
This paper presents an overview of a linear matrix inequality (LMI) approach to the multiobjective synthesis of linear output-feedback controllers. The design objectives can be a mix of H<sub>∞</sub> performance, H<sub>2</sub> performance, passivity, asymptotic disturbance rejection, time-domain constraints, and constraints on the closed-loop pole location. In addition, these objectives can be specified on different channels of the closed-loop system. When all objectives are formulated in terms of a common Lyapunov function, controller design amounts to solving a system of linear matrix inequalities. The validity of this approach is illustrated by a realistic design example
Article
Full-text available
This paper discusses the design and implementation of two track-following controllers for dual-stage hard disk drive servo systems. The first controller is designed by combining an outer loop sensitivity-decoupling (SD) controller with an inner loop disturbance observer (DOB). The second is designed by combining mixed H2/H∞ synthesis techniques with an add-on integral action. The designed controllers were implemented and evaluated on a disk drive with a PZT-actuated suspension-based dual-stage servo system. Position error signal (PES) for the servo system was obtained by measuring the slider displacement with an LDV and injecting a simulated track runout.
Article
Full-text available
In this paper, a new adaptive disturbance rejection scheme is introduced. In the proposed scheme, adaptive frequency estimation technique is incorporated into the traditional disturbance observer method. The frequency of the dominant disturbance is estimated online by a fast stable adaptive algorithm, and the estimated frequency is used to adaptively tune the bandwidth of the disturbance observer. Application to a HDD system shows that the proposed method exhibits good capability of disturbance rejection. Comparing with the traditional disturbance observer method, the new method incurs much less drop in phase margin.
Book
The hard disk drive is one of the finest examples of the precision control of mechatronics, with tolerances less than one micrometer achieved while operating at high speed. Increasing demand for higher data density as well as disturbance-prone operating environments continue to test designers’ mettle. Explore the challenges presented by modern hard disk drives and learn how to overcome them with Hard Disk Drive: Mechatronics and Control. Beginning with an overview of hard disk drive history, components, operating principles, and industry trends, the authors thoroughly examine the design and manufacturing challenges. They start with the head positioning servomechanism followed by the design of the actuator servo controller, the critical aspects of spindle motor control, and finally, the servo track writer, a critical technology in hard disk drive manufacturing. By comparing various design approaches for both single- and dual-stage servomechanisms, the book shows the relative pros and cons of each approach. Numerous examples and figures clarify and illustrate the discussion. Exploring practical issues such as models for plants, noise reduction, disturbances, and common problems with spindle motors, Hard Disk Drive: Mechatronics and Control avoids heavy theory in favor of providing hands-on insight into real issues facing designers every day.
Article
This paper proposes a path-tracking algorithm to compensate for path deviation due to torque limits. The algorithm uses a disturbance observer with an additional saturation element at each joint of n degrees of freedom (DOF) manipulator to obtain a simple equivalent robot dynamic (SERD) model. This model is represented as an n independent double integrator system and is designed to ensure stability under input saturation. For an arbitrary trajectory generated for a given path in Cartesian space whenever any of the actuators is saturated, the desired acceleration of the nominal trajectory in Cartesian space is modified on-line by using SERD. An integral action with respect to the difference between the nominal and modified trajectories is utilized in the nonsaturated region of actuators to reduce the path error. To verify the effectiveness of the proposed algorithms, real experiments were performed for a two DOF SCARA-type direct-drive arm.
Article
In this paper the disturbance attenuation and rejection problem is investigated for a class of MIMO nonlinear systems in the disturbance-observer-based control (DOBC) framework. The unknown external disturbances are supposed to be generated by an exogenous system, where some classic assumptions on disturbances can be removed. Two kinds of nonlinear dynamics in the plants are considered, respectively, which correspond to the known and unknown functions. Design schemes are presented for both the full-order and reduced-order disturbance observers via LMI-based algorithms. For the plants with known nonlinearity, it is shown that the full-order observer can be constructed by augmenting the estimation of disturbances into the full-state estimation, and the reduced-order ones can be designed by using of the separation principle. For the uncertain nonlinearity, the problem can be reduced to a robust observer design problem. By integrating the disturbance observers with conventional control laws, the disturbances can be rejected and the desired dynamic performances can be guaranteed. If the disturbance also has perturbations, it is shown that the proposed approaches are infeasible and further research is required in the future. Finally, simulations for a flight control system is given to demonstrate the effectiveness of the results. Copyright © 2005 John Wiley & Sons, Ltd.