Robust disturbance observer for the track-following

ARTICLE IN PRESS
Control Engineering Practice 12 (2004) 577–585
Robust disturbance observer for the track-following control system
of an optical disk drive
Jung Rae Ryooa, Tae-Yong Dohb, Myung Jin Chunga,*
a
Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong,
Yuseong-gu, Daejeon 305-701, South Korea
b
Department of Control and Instrumentation Engineering, Hanbat National University, San 16-1 Duckmyong-dong, Yuseong-gu,
Daejeon 305-719, South Korea
Received 25 January 2002; received in revised form 28 February 2003; accepted 27 June 2003
Abstract
For track-following control in an optical disk drive, a compensator should be designed to satisfy conﬂicting speciﬁcations. The
compensator design procedure usually requires numerous trial and errors. Utilization of a disturbance observer (DOB) for
enhancing track-following precision simpliﬁes the compensator design procedure by removing the requirement for disturbance
attenuation from the compensator design speciﬁcations. However, the absence of DOB design guideline considering plant modelling
uncertainty has hindered practical application of the DOB. In this paper, a robust stability condition for the DOB-based trackfollowing control system is presented. A graphical design guideline based on frequency response analysis is proposed, and a DOB is
designed to preserve the overall stability. To show the validity, simulation and experimental results are presented.
r 2003 Elsevier Ltd. All rights reserved.
Keywords: Disturbance observer; Track-following control; Optical disk drive; Multiplicative uncertainty; Robust stability; Nyquist criterion
1. Introduction
For read-out/recording of data from/to an optical
disk, a laser spot should be positioned precisely on a
data track by the track-following control system of an
optical disk drive. However, track-following precision is
severely affected by various external disturbances, the
dominant component of which is radial directional
oscillation of tracks due to inevitable disk eccentricity.
Moreover, plant uncertainty threatens overall system
stability, and measurement noise in the high-frequency
region causes deterioration of tracking performance.
Therefore, disturbance attenuation, robust stability, and
noise rejection are the major design speciﬁcations of the
track-following controller. Note that the three requirements conﬂict with each other. For example, a high loop
gain for disturbance attenuation with a restriction on
*Corresponding author. Tel.: +82-42-869-3429; fax: +82-42869-3410.
E-mail addresses: [email protected] (J.R. Ryoo),
[email protected] (T.-Y. Doh), [email protected]
(M.J. Chung).
0967-0661/$ - see front matter r 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/S0967-0661(03)00140-0
control bandwidth usually results in a reduced stability
margin, which produces an unstable response. For this
reason, the controller design procedure requires numerous trial and errors and the order of the designed
controller becomes too high to be applied to a practical
system.
A disturbance observer (DOB)-based control structure can be utilized to resolve the mutually conﬂicting
speciﬁcations by virtue of the ﬂexibility of two degreeof-freedom structure (Umeno & Hori, 1991). If a DOB
is added to the conventional track-following control
system, the disturbance attenuation performance is
greatly enhanced. In addition, the disturbance attenuation requirement can be removed from the design
speciﬁcations of the feedback compensator, which
results in simpliﬁed design speciﬁcations for robust
stability and noise rejection. Resultant decrease of the
compensator order as well as relative ease of design
procedure is achieved.
The DOB has been widely utilized in various motion
control ﬁelds (White, Tomizuka, & Smith, 2000;
Fujiyama, Katayama, Hamaguchi, & Kawakami,
2000; Iwasaki, Shibata, & Matsui, 1999). Most previous
ARTICLE IN PRESS
578
J.R. Ryoo et al. / Control Engineering Practice 12 (2004) 577–585
DOB preserving robust stability is obtained by analyzing the frequency response based on Nyquist plots.
Simulations and experiments are conducted to validate
the proposed design guideline and results are presented
with experimental environment.
This paper is organized as follows. Section 2 presents
a general description of a track-following control system
with a DOB and controller design speciﬁcations. In
Section 3, a DOB design procedure is explained based
on a robust stability condition for the DOB-based trackfollowing control system. The proposed DOB design
procedure is applied to simulations and experiments,
and results are presented in Section 4. Finally, some
concluding remarks are given in Section 5.
works, however, have focused on performance enhancement with qualitative analysis of stability. Because
performance enhancement by DOB may reduce stability
margin (Komada, Machii, & Hori, 2000), consideration
of robust stability is essential for design of DOB. To
take some previous works related to the robust stability
issue, Kempf and Kobayashi (1999) extended DOB
design to account for time delay in plants. Meantime,
Eom, Suh, and Chung (2000) presented a robust
stability condition for a DOB-based motion control
system and applied a DOB to a manipulator control to
simplify complicated nonlinear dynamics to a constant
inertia system. Guven@
.
and Guven@
.
(2001) extended
robustness issue to real parametric uncertainty by
structured singular value method. Although there have
been various trials and great contributions regarding the
robustness issue in DOB-based control system, the
absence of systematic design guideline based on
quantitative analysis of unknown plant dynamics has
hindered practical application of DOB. Because a DOB
is usually used with an outer loop compensator, a
proper guideline for DOB design considering both plant
uncertainty and the feedback compensator is required
for appropriate application of a DOB.
In this paper, a graphical design method of DOB is
presented for the track-following control system with
plant modeling perturbation. The real plant is represented as a multiplicative uncertainty model, and a
feedback compensator guaranteeing robust stability of
the conventional control loop is assumed to be given.
Then a DOB design guideline that preserves the robust
stability is proposed. A robust stability condition for the
track-following control system with a DOB is presented,
and the maximum bound on the cutoff frequency of the
Rotation Axis
2. System description and DOB properties
2.1. System description
As depicted in Fig. 1, the most common trackfollowing mechanism for positioning the laser spot
along the radial direction of the disk is a dual-stage
actuator with both ﬁne and coarse actuator components. The ﬁne actuator, which is mounted on the
optical pickup, is a linear voice-coil motor (VCM) and is
characterized by fast response. On the other hand, the
coarse actuator is usually composed of a DC or stepping
motor with much slower response, the use of which is
restricted to DC offset rejection from the ﬁne actuator
displacement. Thus, track-following performance is
almost completely dependent on the ﬁne actuator. For
this reason, only the ﬁne actuator is herein considered as
the plant of track-following control system and, without
xs
xc
xf
Vertical
Direction
Recorded
Layer
Radial
Direction
Disk
Spindle Motor
Objective
Lens
Laser Spot
kf
f
mf
df
dc
PD
Optical
Element
Fine
Actuator
LD
Coarse
Actuator
Fig. 1. Track-following control mechanism in optical disk drive.
ARTICLE IN PRESS
J.R. Ryoo et al. / Control Engineering Practice 12 (2004) 577–585
e
C (s)
u
P(s)
xf
+
d
ub
−
u
+
ud
K PD
579
P( s)
+
xf
+
−
Q(s)
+
+
Fig. 2. Block diagram of track-following control system.
d
K PD
Q( s )
K PD Pn ( s)
e
Fig. 3. General structure of DOB.
loss of generality, the laser spot position xs is assumed to
be identical with the ﬁne actuator displacement xf :
A typical track-following control system is shown in
Fig. 2, where PðsÞ is the ﬁne actuator and CðsÞ is the
feedback compensator. The target track position on
which the laser spot should be located during trackfollowing control is represented as an external input d:
Due to disk eccentricity and irregular shape of data
tracks, d is a signal that oscillates radially with the
frequency of disk rotation, but the phase and amplitude
of the oscillation are not known. Hence, d is regarded as
an unknown external disturbance. In addition, due to
inherent limitations of opto-mechanical structure,
neither d nor the spot position xf is measurable. The
only available value is the position error between the
target track and the laser spot, and the error is ampliﬁed
by the photo detector gain KPD : Therefore, a trackfollowing control system is a representative error feedback control system, and controller design is formulated
as a disturbance attenuation problem to ﬁnd a controller
reducing the magnitude of the tracking error below an
acceptable level.
The ﬁne actuator determines radial position of the
laser spot by moving the objective lens of mass mf ; to
which a spring with spring constant kf and a damper
with damping ratio df is attached. Thus, a second-order
system of mass–spring–damper is adequate to model the
plant dynamics. The relationship between the ﬁne
actuator force f and displacement xf is described as
F ðsÞ ¼ ðmf s2 þ df s þ kf ÞXf ðsÞ;
ð1Þ
where F ðsÞ and Xf ðsÞ are the Laplace transformations of
f and xf ; respectively. f is dependent on the control
input u with the relationship shown as
Kt
F ðsÞ ¼
ðUðsÞ sKemf Xf ðsÞÞ;
ð2Þ
Ra þ sLa
where Ra and La are the resistance and inductance of the
VCM armature, respectively, Kt is the force constant,
Kemf is the back EMF constant, and UðsÞ is the Laplace
transformation of u: As La is sufﬁciently small compared
with Ra ; (2) is usually simpliﬁed by ignoring La : Finally,
a transfer function from u to xf is obtained as below
Kt =Ra
Pn ðsÞ ¼
;
ð3Þ
2
mf s þ ðdf þ ðKt Kemf =Ra ÞÞs þ kf
where Pn ðsÞ is the nominal plant model of the trackfollowing control system. For simplicity, (3) is represented as
Pn ðsÞ ¼
K
:
s2 þ as þ b
ð4Þ
Although (4) is nearly exact in the low-frequency region,
Pn ðsÞ differs from the actual plant PðsÞ in the highfrequency region where the effect of La is not negligible.
In addition to the unmodeled inductance component,
resonances at high frequencies are barely considered in
the plant model. To take into account the effect of the
unmodeled dynamics, PðsÞ should be modeled with
consideration of the uncertainty. For the purpose, a
multiplicative uncertainty model is adequate (Zhou &
Doyle, 1998) and PðsÞ is given as
PðsÞ ¼ ð1 þ DðsÞW ðsÞÞPn ðsÞ;
ð5Þ
where W ðsÞ is a ﬁxed stable transfer function for weight
and DðsÞ is a variable stable transfer function satisfying
jjDjjN p1:
2.2. Properties of DOB
The structure of a typical DOB is presented in Fig. 3.
The general form of QðsÞ is given by
QðsÞ ¼
qn
sn
qm s m þ ? þ q0
;
þ qn1 sn1 þ ? þ q0
ð6Þ
where n > m is the order of denominator, m is the order
of numerator, and qi ; i ¼ 0; y; n are ﬁlter coefﬁcients.
Usually, qi ’s are determined with the binomial coefﬁcients described as qi ¼ n!=ððn iÞ!i!Þoni
(Kwon, 2002)
c
and QðsÞ is modiﬁed as below
Pm
qi s i
QðsÞ ¼ i¼0 n ;
ð7Þ
ðs þ oc Þ
where oc is the cutoff frequency of QðsÞ:
As well-investigated in the previous works (Komada
et al., 2000; Kempf & Kobayashi, 1999; Eom et al.,
2000), the advantages of DOB are effective only within
the bandwidth of QðsÞ and it is thus desirable to increase
oc for disturbance attenuation performance. The
property is clearly explained by the sensitivity function
ARTICLE IN PRESS
J.R. Ryoo et al. / Control Engineering Practice 12 (2004) 577–585
580
e
C ( s)
ub
u
+
P(s)
+
xf
Q( s )
+
ud
+
DOB
+
d
e
−
C ( s)
ub
+
+
1
1 − Q( s )
u
P(s)
Q( s )
K PD Pn ( s )
K PD
xf
+
d
−
K PD
Ceq (s)
Q( s )
K PD Pn ( s )
Fig. 5. Equivalent block diagram of DOB-based track-following
control system.
Fig. 4. DOB-based track-following control system.
SDOB ðsÞ which is given by
SDOB ðsÞ ¼ 1 QðsÞ ¼
s
Pn
mþ1
i¼mþ1 qi s
ðs þ oc Þn
From (9), the overall nominal loop transfer function
Leq;n ðsÞ is expressed as
im1
;
ð8Þ
Leq;n ðsÞ ¼ KPD Ceq ðsÞPn ðsÞ ¼
where PðsÞ ¼ Pn ðsÞ is assumed for simplicity. Note that
ðm þ 1Þ zeros at the origin enable tracking error to be
reduced in the low-frequency region and result in zero
steady-state error for some types of disturbances like
step, ramp, etc.
To take the advantage of the disturbance attenuation,
a DOB is added to a conventional track-following
control system as shown in Fig. 4. Since various design
methods for CðsÞ have been proposed in previous works
(Lee, Moon, Jin, & Chung, 1998; Moon, Lee, & Chung,
1998), CðsÞ is assumed to be given without loss of
generality. Then, the DOB is used to improve the trackfollowing precision based on the stability provided by
CðsÞ: The tracking performance is enhanced by increasing the order and the bandwidth of QðsÞ: However,
excessive order and bandwidth of QðsÞ can lead to
resultant destruction of the stability condition by plant
uncertainty in the high-frequency region. Therefore,
QðsÞ should be designed carefully with a compromise
between performance and stability. The design of a
DOB-based track-following control system results in
proper selection of QðsÞ to guarantee better tracking
performance without system instability for all plants
represented as (5).
3. Design of a DOB with robust stability
3.1. Robust stability conditions
Because the system is an error feedback control
system, Fig. 4 is modiﬁed to an equivalent block
diagram as shown in Fig. 5 (Ueda, Imagi, & Tamayama,
1999) where the feedback compensator and DOB are
merged into an equivalent feedback compensator Ceq ðsÞ
described as
Ceq ðsÞ ¼
CðsÞ þ QðsÞðKPD Pn ðsÞÞ1
:
1 QðsÞ
ð9Þ
Ln ðsÞ þ QðsÞ
;
1 QðsÞ
ð10Þ
where Ln ðsÞ is the nominal loop transfer function
without DOB and is described as
Ln ðsÞ ¼ KPD CðsÞPn ðsÞ:
The following proposition is helpful to DOB design
under plant uncertainty.
Proposition 1. The DOB-based track-following control
system depicted in Fig. 5 is internally stable for a
given PðsÞAP ¼ fð1 þ DðsÞW ðsÞÞPn ðsÞ : jjDjjN p1g if and
only if,
jW ðjoÞLeq;n ðjoÞjoj1 þ Leq;n ð joÞj
8oX0:
Remark 1. Note that robust stability of the system is not
ensured by separate robust stability conditions of inner
and outer loop, i.e.,
jW ðjoÞLn ðjoÞjoj1 þ Ln ðjoÞj 8oX0
and
jW ðjoÞQð joÞjo1 8oX0:
Proposition 1 has a simple graphical interpretation
that is similar to the representation in Doyle, Francis,
and Tannenbaum (1992). Because CðsÞ; QðsÞ; and Pn ðsÞ
are stable and no unstable pole-zero cancellation exists
in Ceq ðsÞ; there is no unstable pole in Leq;n ðsÞ: Therefore,
the Nyquist plot of Leq;n ðsÞ does not encircle the critical
point, ð1; j0Þ; in the clockwise direction. And the
DOB-based track-following control system is robustly
stable for all plants of P if and only if the critical point
lies outside the disk centered at Leq;n ðjoÞ with a radius of
jW ðjoÞLeq;n ðjoÞj at every frequency oX0: The condition
is graphically represented in Fig. 6. In addition to robust
stability, the ﬁgure illustrates the minimum phase
margin fmin that the system guarantees, even for the
worst plant.
ARTICLE IN PRESS
J.R. Ryoo et al. / Control Engineering Practice 12 (2004) 577–585
Im
Uncertainty
Disk
-1
Re
min
WLeq, n
L eq,n( j)
Fig. 6. Graphical representation of the robust stability condition.
Proposition 1 is easily converted to a more general
stability condition for a DOB-based control system by
substituting Leq;n ðsÞ of (10) into the condition of the
proposition, which is rewritten as
jW ð joÞðLn ðjoÞ þ QðjoÞÞjoj1 þ Ln ð joÞj
8oX0:
ð11Þ
Thus, the following proposition can be obtained.
Proposition 2. In the DOB-based track-following control
system shown in Fig. 4, if the nominal closed-loop system
without the DOB, i.e., Ln ðsÞð1 þ Ln ðsÞÞ1 is stable, then
the system is internally stable for a given PðsÞAP ¼
fð1 þ DðsÞW ðsÞÞPn ðsÞ : jjDjjN p1g if and only if,
jW ð joÞðLn ðjoÞ þ QðjoÞÞjoj1 þ Ln ð joÞj
8oX0:
Remark 2. The nominal complementary sensitivity
function Tn ðsÞ of the overall system shown in Fig. 4 is
represented as
Ln ðsÞ þ QðsÞ
:
Tn ðsÞ ¼
1 þ Ln ðsÞ
Thus, the robust stability condition of Proposition 2 is
equivalent to the condition from Doyle et al. (1992)
shown as
jjWTn jjN o1:
Remark 3. For both proofs of Propositions 1 and 2,
stability of the nominal feedback system Tn ðsÞ must
be satisﬁed. Though it is not explicitly mentioned in the
propositions, as both Ln ðsÞð1 þ Ln ðsÞÞ1 and QðsÞ are
stable rational transfer functions with no unstable polezero cancellation, Tn ðsÞ is also stable.
581
to choose the order and cutoff frequency of QðsÞ: A
transfer function is said to be proper if the degree of
denominator is greater or equal to that of numerator.
The order of QðsÞ should be determined to ensure that
the block containing P1
n ðsÞ in Fig. 4 is proper. If the
block is not proper, pure differentiators are resultantly
included in the block causing ampliﬁcation of highfrequency noise. In addition to overall properness,
disturbance attenuation performance is also considered
while determining the order of QðsÞ: As mentioned in
(8), the numerator order of QðsÞ determines both the
number of integrator in the loop and the roll-off of loop
gain in control bandwidth. In track-following control
systems, the frequency spectrum of the external disturbance is known to be a maximum at the disk
rotational frequency and decreases with roll-off value
of 40 db=dec at frequencies higher than the disk
rotation (Akkermans & Stan, 2001). Therefore, the
numerator order of QðsÞ should be selected to provide
sufﬁcient loop gain at the frequency of disk rotation. In
addition to the ﬁlter order, the cutoff frequency of QðsÞ
should be as large as possible without violating the
robust stability condition. Therefore, a DOB design
procedure guaranteeing robust stability under plant
modeling perturbation is obtained as follows:
Step 1: Set the relative degree of QðsÞ to the degree of
Pn ðsÞ:
Step 2: Select the numerator order of QðsÞ with
consideration of performance.
Step 3: Construct the Nyquist plot of Leq;n ðjoÞ with
uncertainty disks having radii of jW ð joÞLeq;n ðjoÞj:
Step 4: Find the cutoff frequency oc so that the
critical point ð1; j0Þ lies outside the disk centered at
Leq;n ðjoÞ at every frequency oX0 subject to jLeq;n ðjoÞj
> 1: The minimum phase margin can be assured in the
similar manner.
Step 5: Increase the relative degree of QðsÞ to satisfy
the condition of Proposition 1 inside the unit circle
centered at the origin, if needed.
Remark 4. Increasing the numerator order of QðsÞ is an
effective method of improving the disturbance attenuation performance. However, increased loop gain can
lead to saturation of control input during a transient
response. Although the stability problem of saturation is
not considered in detail herein, excessive loop gain due
to increased numerator order of QðsÞ is not desirable.
4. Simulations and experiments
3.2. Design of DOB
4.1. Plant model
Based on Proposition 1, a graphical design method
for the DOB is presented, which is actually a procedure
A DVD-ROM drive with a disk rotating at 2400 rpm
was used for all simulations and experiments. The
ARTICLE IN PRESS
J.R. Ryoo et al. / Control Engineering Practice 12 (2004) 577–585
582
−40
between the track center and the laser spot to the
electrical error signal. The track pitch of the disk is
0:74 mm and the measurable range is 70:37 mm; which
corresponds to the tracking error signal amplitude of
72:0 V: Therefore, KPD is 5:4 106 V=m:
−60
−80
Mag (dB)
−100
−120
4.2. Design example
−140
−160
−180
−200
1
10
2
10
3
10
Freq (Hz)
4
10
5
10
Fig. 7. Bode magnitude plots of the nominal plant (– – –), a real plant
with a resonance and phase lag at high frequency (—), and the upper
and lower uncertainty bounds (y).
Information about the extraneous disturbance, such
as maximally injectable magnitude and bandwidth, is
speciﬁed for the controller design (Akkermans & Stan,
2001). Based on the data, required loop gain and
bandwidth for sufﬁcient disturbance rejection and the
minimum stability margin for stable track-following
pull-in are calculated, and the resultant feedback
compensator is described as
CðsÞ ¼
nominal model of the ﬁne actuator is represented by a
second-order transfer function with a resonance frequency of 62 Hz and a damping ratio of 0.08 as
Pn ðsÞ ¼
s2
76:35
þ 61:98s þ 153675:98
ðm=VÞ:
Due to various unmodeled dynamics, such as phase lag
and resonances at high frequencies, the real plant model
PðsÞ must be represented as a multiplicative uncertainty
model as
PðsÞ ¼ ð1 þ DðsÞW ðsÞÞPn ðsÞ;
where jjDjjN p1: The weighting function W ðsÞ; which
represents the amount of uncertainty with respect to
frequency, is selected to cover every unmodeled
dynamics. Modeling perturbation is usually negligible
in the low-frequency region and increases as frequency
does, which leads to a high-pass ﬁlter type W ðsÞ:
According to the optical pickup actuator speciﬁcation,
there exist subresonances at approximately 20 kHz with
peak magnitudes less than 10 dB: In addition, some
phase lag due to the ignored inductance is also present in
the high frequency region. Finally, W ðsÞ is determined
as
W ðsÞ ¼
2s2
:
ðs þ 40;000pÞ2
ð12Þ
The magnitude bode plots of the nominal plant model
and a perturbed model are shown in Fig. 7. Note that
the uncertainty boundary determined by (12) covers the
perturbed model.
The loop gain is also dependent on KPD ; which is the
conversion ratio of the position sensor from the distance
26:25ðs þ 1257Þðs þ 1885Þðs þ 12; 566Þðs þ 25; 133Þ
:
ðs þ 314Þðs þ 377Þðs þ 75; 398Þðs þ 157; 080Þ
ð13Þ
QðsÞ is designed to preserve robust stability and
enhance the tracking precision. First, the relative degree
of QðsÞ is set at the value of Pn ðsÞ: Then, the order of
numerator is selected based on the analysis of the
external disturbance. Because the external disturbance
due to eccentric rotation of the disk is characterized with
40 dB=dec of roll-off at frequencies higher than the
rotational frequency, a ﬁrst-order polynomial is sufﬁcient for the numerator of QðsÞ; from which a double
integrator is generated in the control loop. In the next
step, the Nyquist plot of Leq;n ðsÞ is constructed to ﬁnd an
appropriate value for oc : For a track-following control
system, minimum phase margin should be assured to
suppress overshoot in the transient response. Finally, a
low-pass ﬁlter is obtained as
QðsÞ ¼
3o2c s þ o3c
;
ðs þ oc Þ3
ð14Þ
where oc ¼ 2000p rad=s: In Fig. 8(a), the graphical
representation of the robust stability condition of
Proposition 1 is shown to be satisﬁed. Because the
robust stability condition is satisﬁed inside the unit
circle centered at the origin, it is not necessary to
increase the relative degree of QðsÞ: Note that the system
is not only robustly stable but also guarantees 30 of
minimum phase margin, which is the recommended
amount for a stable track-following pull-in operation
(Suzuki, Tanaka, & Miura, 1990). Compared with
the case without DOB shown in Fig. 8(b), the
stability margin is not changed after the designed
DOB is added.
Use of a DOB removes one requirement of disturbance rejection from the three speciﬁcations for the
ARTICLE IN PRESS
J.R. Ryoo et al. / Control Engineering Practice 12 (2004) 577–585
The compensator of reduced order is represented as
1
CðsÞ ¼
0.5
Im
φ
min
= 30°
−1
−1.5
−2
−2
−1.5
−1
−0.5
Re
(a)
0
0.5
Fig. 10(a) illustrates the external disturbance d
composed of three harmonics of the disk spindle
frequency with different phases. The tracking error,
the output of compensator ub ; and the output of DOB
ud are shown in Fig. 10(b)–(d), respectively, where the
feedback compensator of (13) and the DOB with (14)
are used. To show the effect of DOB, DOB is turned on
at 100 ms; and the tracking error is abruptly reduced. In
addition, due to the DOB, the output from CðsÞ is
diminished.
Use of the DOB is effective to reduce the order of the
feedback compensator, as illustrated in (15). Simulation
results for the same disturbance of Fig. 10(a) are shown
in Fig. 11. Even though the tracking error of Fig. 11(b)
is slightly larger than the error of Fig. 10(b), the
compensator design procedure is much easier than that
of Fig. 10.
0.5
Im
0
φ
min
= 30°
−1
−1.5
−2
−2
−1.5
−1
−0.5
(b)
0
0.5
1
Re
Fig. 8. Graphical representation of robust stability condition: (a) with
DOB; (b) without DOB.
50
0
Mag (dB)
ð15Þ
4.3. Computer simulations
1
1
−0.5
9:1ðs þ 12; 566Þðs þ 25; 133Þ
;
ðs þ 62; 832Þðs þ 75; 398Þ
which is designed to satisfy the two remaining speciﬁcation of control bandwidth and robust stability. The
same DOB with QðsÞ of (14) satisﬁes the robust stability
condition. At the same time, the frequency responses of
sensitivity functions with the DOB are low enough to
ensure sufﬁcient disturbance attenuation as depicted in
Fig. 9. Because the plant uncertainty is negligible in the
low-frequency region, four sensitivity functions with
perturbed plant models demonstrate that the trackfollowing performance is not affected by the plant
uncertainty.
0
−0.5
583
−50
−100
−150
−200
0
10
1
10
2
10
3
10
4
10
5
10
Freq (Hz)
Fig. 9. Sensitivity functions with the nominal plant model and four
perturbed models (—: with DOB, – – –: without DOB).
feedback compensator design. Therefore, the compensator design procedure becomes simple and the order of
the resultant feedback compensator can be decreased.
4.4. Experiments
The designed controller with the DOB was applied to
the experimental apparatus with DSP shown in Fig. 12.
The TMS320C6701, which is a 32 bit ﬂoating point DSP
with the core architecture of Very Long Instruction
Word (VLIW), performs the key role in the experiments.
Due to the excellent computing power of one Giga
ﬂoating-point operations per second (GFLOPS), it is
useful for implementation of a digital control system
requiring a very high sampling frequency. The DOBbased track-following control system ran at a sampling
frequency of 200 kHz; which is the usual sampling
frequency for track-following control in commercial
optical disk drives. The controller designed in continuous-time domain was transformed to a discrete-time
controller by the pole-zero mapping method, which
generates a reasonable result with a sufﬁciently high
sampling frequency compared with control bandwidth.
ARTICLE IN PRESS
J.R. Ryoo et al. / Control Engineering Practice 12 (2004) 577–585
200
0.1
100
0.05
0
−100
−200
(a)
0
20
40
60
80
100
120
140
160
180
200
time (msec)
0
−0.05
−0.1
(b)
0.4
0.2
0.2
0
−0.2
−0.4
0
20
40
60
80
(c)
100
120
140
160
180
20
40
60
80
120
140
160
180
200
120
140
160
180
200
−0.2
0
20
40
60
80
(d)
time (msec)
100
0
−0.4
200
0
time (msec)
0.4
DOB output (V)
compensator output (V)
tracking error (µm)
disturbance (µm)
584
100
time (msec)
Fig. 10. Simulation results with high-gain feedback compensator: (a) external disturbance; (b) tracking error; (c) feedback compensator output;
(d) DOB output.
0.1
tracking error (µm)
disturbance (µm)
200
100
0
−100
−200
0
20
40
60
80
100
120
140
160
180
0
−0.05
−0.1
200
0.4
0.4
0.2
0.2
0
−0.2
−0.4
0
20
40
60
(c)
80
100
0
20
40
60
80
(b)
time (msec)
DOB output (V)
compensator output (V)
(a)
0.05
120
140
160
180
200
120
140
160
180
200
120
140
160
180
200
0
−0.2
−0.4
(d)
time (msec)
100
time (msec)
0
20
40
60
80
100
time (msec)
Fig. 11. Simulation results with low-gain feedback compensator: (a) external disturbance; (b) tracking error; (c) feedback compensator output;
(d) DOB output.
Actuator driver
PD signal
Focus
Fine
LPF
Coarse
DVD-ROM Drive
Amp
DAC
TMS320C
6701
RF
Amp
ADC
Focus error
Tracking error
DSP based digital control system
Fig. 12. Experimental environment.
Fig. 13 illustrates an experimental result. The DOB
was turned on at 100 ms; as in the simulations.
Although the results are slightly affected by measurement noise, the effect of DOB is evident in the results.
After the DOB is turned on, the feedback compensator
output is reduced while the DOB output in Fig. 13(c) is
increased. Therefore, the external disturbance of the
disk rotational frequency of 40 Hz is almost completely
attenuated by DOB. The tracking error illustrated in
Fig. 13(b), which is calculated from the tracking error
signal with consideration of KPD ; reveals that the
tracking precision is enhanced after the DOB is turned
on. For reliable transfer of data from/to an optical disk,
70:037 mm of tracking error is known to be tolerable in
industrial ﬁeld. The DOB enabled track-following
control system to reduce the tracking error to a value
below the boundary, resulting in more reliable
reading/writing of data from/to the optical disk. The
improved performance is clearly illustrated by the
Fast Fourier Transform (FFT) results shown in
Fig. 14. The peaks between 0 and 100 Hz are the
component of the disk rotation, and the magnitude in
Fig. 14(a) is less than the magnitude in Fig. 14(b) more
than 10 dB:
ARTICLE IN PRESS
J.R. Ryoo et al. / Control Engineering Practice 12 (2004) 577–585
design for the feedback compensator by removing the
requirement for disturbance attenuation. The designed
DOB was applied to a track-following control system.
Simulation and experimental results were presented to
validate the effectiveness.
0.1
tracking error (µm)
0.05
0
−0.05
−0.1
(a)
585
0
20
40
60
80
100
time (msec)
120
140
160
180
200
Acknowledgements
compensator output (V)
0.4
The authors would like to thank the anonymous
reviewers for useful comments and suggestions.
0.2
0
References
−0.2
−0.4
0
20
40
60
80
100
time (msec)
120
140
160
180
200
0
20
40
60
80
100
time (msec)
120
140
160
180
200
(b)
DOB output (V)
0.4
0.2
0
−0.2
−0.4
(c)
Fig. 13. Experimental results: (a) tracking error; (b) feedback
compensator output; (c) DOB output.
−10
mag (dBV)
−20
−30
−40
−50
−60
0
100
200
300
400
(a)
500
600
700
800
900
1000
600
700
800
900
1000
freq. (Hz)
−10
mag (dBV)
−20
−30
−40
−50
−60
0
100
(b)
200
300
400
500
freq. (Hz)
Fig. 14. FFT results of the tracking error signal: (a) with DOB;
(b) without DOB.
5. Conclusion
In this paper, a design procedure for a DOB used in a
track-following control system was presented. A robust
stability condition under uncertain plant dynamics was
introduced. A graphical design method based on
analysis of the frequency response was proposed to
preserve the robust stability. The procedure is simple
and intuitive because it uses a Nyquist plot and
uncertainty disks. Use of DOB results in ease of
Akkermans, T. H., & Stan, S. G. (2001). Digital servo IC for optical
disc drives. Control Engineering Practice, 9(11), 1245–1253.
Doyle, J. C., Francis, B. A., & Tannenbaum, A. R. (1992). Feedback
control theory. New York, NY: Macmillan.
Eom, K. S., Suh, I. H., & Chung, W. K. (2000). Disturbance observer
based path tracking control of robot manipulator considering
torque saturation. Mechatronics, 11(3), 325–343.
Fujiyama, K., Katayama, R., Hamaguchi, T., & Kawakami, K. (2000).
Digital controller design for recordable optical disk player using
disturbance observer. Proceedings of the International Workshop on
Advanced Motion Control, Nagoya, Japan (pp. 141–146).
.
.
Guven@,
B. A., & Guven@,
L. (2001). Robustness of disturbance
observers in the presence of structured real parametric uncertainty.
Proceedings of the American Control Conference, Arlington,
Virginia, USA (pp. 4222–4227).
Iwasaki, M., Shibata, T., & Matsui, N. (1999). Disturbance-observerbased nonlinear friction compensation in table drive system. IEEE
Transactions on Mechatronics, 4(1), 3–8.
Kempf, C. J., & Kobayashi, S. (1999). Disturbance observer and
feedforward design for a high-speed direct-drive positioning table.
IEEE Transactions on Control Systems Technology, 7(5), 513–526.
Komada, S., Machii, N., & Hori, T. (2000). Control of redundant
manipulators considering order of disturbance observer. IEEE
Transactions on Industrial Electronics, 47(2), 413–420.
Kwon, S. J. (2002). Robust tracking control and state estimation of
mechanical systems: A perturbation compensator based approach.
Ph.D dissertation, POSTECH.
Lee, M.-N., Moon, J.-H., Jin, K. B., & Chung, M. J. (1998). Robust
HN control with multiple constraints for the track-following
systems of an optical disk drive. IEEE Transactions on Industrial
Electronics, 45(4), 638–645.
Moon, J.-H., Lee, M.-N., & Chung, M. J. (1998). Repetitive control
for the track-following servo system of an optical disk drive. IEEE
Transactions on Control Systems Technology, 6(5), 663–670.
Suzuki, M., Tanaka, H., & Miura, Y. (1990). Pull-in condition and
method of tracking servo in optical disk drive. Journal of the
Institute of Television Engineers of Japan, 44(10), 1391–1397.
Ueda, J., Imagi, A., & Tamayama, H. (1999). Track following control
of large capacity ﬂexible disk drive with disturbance observer using
two position sensors. Proceedings of the International Conference
on Advanced Intelligent Mechatronics, Atlanta, Georgia, USA
(pp. 144–149).
Umeno, T., & Hori, Y. (1991). Robust speed control of DC
servomotors using modern two degrees-of-freedom controller
design. IEEE Transactions on Industrial Electronics, 38(5), 363–368.
White, M. T., Tomizuka, M., & Smith, C. (2000). Improved track
following in magnetic disk drives using a disturbance observer.
IEEE Transactions on Mechatronics, 5(1), 3–11.
Zhou, K., & Doyle, J. C. (1998). Essentials of robust control. Upper
Saddle River, NJ: Prentice-Hall.

Download Report