Conference PaperPDF Available

Selective Iterative Learning Control to Deal with Iteration-dependent Disturbances

Authors:
Proceedings of ISFA 2014
2014 International Symposium on Flexible Automation
Awaji-Island, Hyogo, Japan, 14-16 July, 2014
SELECTIVE ITERATIVE LEARNING CONTROL WITH NON-REPETITIVE
DISTURBANCE REJECTION
Liting Sun
Department of Mechanical Engineering
University of Science
and Technology of China
Hefei, Anhui, 230027
Email: litingsun@me.berkeley.edu
Xu Chen
Department of Mechanical Engineering
University of California at Berkeley
Berkeley, California, 94720
Email: maxchen@me.berkeley.edu
Masayoshi Tomizuka
Department of Mechanical Engineering
University of California at Berkeley
Berkeley, California, 94720
Email: tomizuka@me.berkeley.edu
ABSTRACT
In precision systems that repeatedly execute the same task,
iterative learning control (ILC) may be adopted for rejection of
iteration-independent repetitive disturbances and improvement
of trajectory tracking. However, in practice, non-repetitive dis-
turbances also exist and may overlap with the repetitive ones in
the frequency domain. Such non-repetitive disturbances greatly
limit or even degrade the achievable performance of the stan-
dard ILC algorithm. In this paper, we discuss a new ILC strat-
egy with a disturbance observer (DOB) and a time-varying Q
filter for improved learning. In the proposed learning scheme,
repetitive disturbances are selectively learned and attenuated, and
non-repetitive disturbances are either largely rejected or retained
without undesired amplification. The new strategy also provides
us more flexibility in designing the Q filters in ILC and DOB.
Algorithm verification is provided by simulation for precision
motion control of a wafer scanner system.
1INTRODUCTION
Iterative learning control (ILC) is a well-known and effective
control technique for servo improvement in systems that repeat-
edly perform the same task. In the learning process, tracking er-
rors from past iterations are incorporated to generate a new feed-
forward control signal to improve the performance in the next
iteration [1, 2]. However, in practice, two kinds of disturbances
enter the system: repetitive ones which are independent of the
iteration number and non-repetitive ones that vary from iteration
to iteration. Standard ILC only learns and attenuates the repeti-
tive errors, while non-repetitive errors entering into the learning
loop will greatly limit or even degrade the achievable ILC per-
formance [3, 4].
Agreatdealofeffortshavebeenmadetoattainmorerobust
ILC performance in the presence of non-repetitive disturbances.
One approach is to prefilter the input error signal and decompose
it into a repetitive part and a non-repetitive part. Only the repeti-
tive component is allowed to enter the learning loop of ILC. For
example, Lee et. al [5] and Phan and Longman [6] used Kalman
filters to remove the iteration-independent components for more
effective learning, while Merry et. al [7] employed a wavelet fil-
ter to filter out the non-repetitive disturbances. Other approaches
focus on adjusting the ILC scheme itself. Examples of this class
include high-order ILC, segmented ILC, and ILC with a time-
varying robustness filter (Q filter). More specifically, assuming
that the patterns of the non-repetitiveness are known, Chen and
Moore [8] constructed an iteration-domain disturbance observer
(a special high-order ILC) for non-repetitive disturbance rejec-
tion. Sandipan et. al [4] used a time-domain segmented ILC strat-
egy for the precision motion control of a wafer scanner, where the
learning process is turned on and off based on the magnitudes
of the repetitive and non-repetitive disturbances in each itera-
tion. A further generalization is to equip ILC with a time-varying
Qfilterwhosecut-offfrequencyisiterativelytunedonlineac-
cording to the time-frequency analysis of the positioning errors.
When repetitive disturbances are located mainly at low frequen-
cies, the Q-filter bandwidth is decreased to reduce the influence
of the non-repetitive disturbances. Otherwise the bandwidth is
increased for maximum learning ability and better performance.
1Copyright c
2014 by ASME
ISFA2014-11L
Zhang et. al [9] and Rotariu et. al [10] both constructed such ILC
schemes; [9] used a wavelet transform for time-frequency anal-
ysis and [10] adopted the Wigner distribution algorithm. The
aforementioned ILC algorithms with enhanced robustness can
effectively avoid undesired error amplifications caused by the
non-repetitive disturbances, yet little attention has beenpaidto
the non-repetitive disturbance rejection performance.
In this paper, a new ILC scheme combining a disturbance
observer (DOB) and a time-varying Q filter is proposed for per-
formance improvement in the presence of non-repetitive distur-
bances, especially when their frequencies overlap with the repet-
itive disturbances. Compared to [5]- [10], one advantage of the
proposed algorithm is the enhanced non-repetitive disturbance
rejection ability, particularly for discrete-time systemswithtime
delays. This is achieved by a new method to design the Q filter
in DOB. Another benefit (compared with previous ILC schemes
using DOB [11]) is that by introducing a time-varying Q filter,
non-repetitive disturbances at frequencies above the bandwidth
of DOB can also be filtered out from the learning loop. This
maximally increases the repetitive-disturbance-attenuation band-
width in different phases of the trajectory without causing unde-
sired amplification of non-repetitive disturbances. Additionally,
the new algorithm allows us more flexibility in the design of the
QfiltersinbothILCandDOB.
The remainder of the paper is organized as follows. Section
2describesawaferscannersystemandformulatestheproblem
of using standard ILC when both repetitive and non-repetitive
disturbances exist. Section 3 presents the proposed ILC struc-
ture. Simulation results are provided in Section 4. Section 5
concludes the paper.
2EXPERIMENTALHARDWAREANDPROBLEMFOR-
MULATION
2.1 Description of the Wafer Scanner System
Awaferscannerisamachinethatperformstheessential
photolithography steps in the manufacture of integrated circuits.
It consists of a light source, a reticle stage, several projection
lenses and a wafer stage, as shown in Fig. 1. The wafer stage
and the reticle stage are both high precision motion systems that
carry a silicon wafer and a mask with designed circuits patterns.
The tolerable positioning errors of the two stages are in the order
of nanometers so that the patterns can be accurately printed on
the wafer. A laboratory testbed wafer scanner is shown in Fig.
2. The stages here are both driven by three-phase linear motors
and positions of the stages are measured by a laser interferometry
system at a sampling frequency of 2.5kHz.
In this paper, we consider the reference tracking problem
for the wafer stage. An example of the scanning trajectory is
shown in Fig. 3. The trajectory consists of two distinct phases.
The first phase is the acceleration phase, where the speed of the
stage is increased as quickly as possible to a specific value so
Light source
Reticle stage
Wafer stage
Projection lenses
Mask
Figure 1:Schematic of the photolithography process
Figure 2:The experimental hardware of the wafer scanner
0 0.5 1 1.5 2 2.5
0
0.05
0.1
Reference trajectory
Time(s)
Position(m)
Acceleration Phase
Constantspeed Phase
Figure 3:Reference trajectory of the wafer stage in one scan
that the scanning can be performed. In this phase, repetitive
errors caused by trajectory tracking dominate and contain rich
spectral components. The second phase is the constant-speed
phase, where the force ripple of the linear motor and environ-
mental vibrations become the major sources of position errors.
Although their frequency characteristics are fixed, their ampli-
tudes and initial phases are normally iteration-dependent.Thus,
using standard ILC in this phase cannot effectively improve the
positioning accuracy.
Figure 4 shows the measured and the identified closed-loop
frequency response from the reference input yd(k)to the out-
put y(k)of the wafer scanner (plant P(z1))withabaselinePID
feedback controller C(z1)(See Fig. 5). The identified nominal
model P
n(z1)of the plant P(z1)is:
P
n(z1)=3.4766 ×107z21+0.8z1
12z1+z2(1)
where z1denotes the one-step delay operator.
2Copyright
c
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−80
−60
−40
−20
0
20
Magnitude (dB)
100101102103
−1440
−1080
−720
−360
0
360
Phase (deg)
Frequency (Hz)
Experiment data
Identified model
Figure 4:Measured and identified closed-loop frequency re-
sponse from yd(k)to y(k)of the system
2.2 Problem Formulation with Standard ILC
In this section, we review the standard ILC algorithm and re-
veal the influence caused by non-repetitive disturbances. Figure
5showsaserialILCstructureaddedtothefeedbackcontrolloop.
yd(k),rj(k),yj(k),ej(k),uj(k)and dj(k)represent, respectively,
the reference signal, the feedforward signal from ILC, the output
position signal, the position error signal, the input control signal
and the disturbance signal in the j-th iteration.
e
j
(k)
L(z
¡1
)
memory
Q(z
¡1
)
rj(k
C(z
¡1
)
u
j
(k)
y
j
(k
)
P(z
¡1
)
y
d
(k
)
dj(k
)
memory
Figure 5:AserialILCaddedtothefeedbackloop
Decompose the disturbance dj(k)in Fig. 5 to a repetitive
disturbance dr(k)and a non-repetitive one dnj(k).Thesystem
output can then be written as:
Yj(z1)= P(z1)C(z1)
1+P(z1)C(z1)[Yd(z1)+Rj(z1)]
+P(z1)
1+P(z1)C(z1)[Dr(z1)+Dnj(z1)] (2)
where the sensitivity function and complementary sen-
sitivity function of the feedback loop are S0(z1)=
1/[1+P(z1)C(z1)] and T0(z1)=P(z1)C(z1)/[1+
P(z1)C(z1)];capitalizedsymbolsYj,Yd,Rj,Drand Dnj
are used for expressing the signals in the zdomain. Then the
feedforward command rj(k)generated in a standard first-order
ILC is:
rj(k)=Q(q)[rj1(k)+L(q)ej1(k+m)] (3)
where kis the time index within each iteration, mis the relative
degree of T0(z1),andqrepresents the forward time-shift opera-
tor, i.e., qx(k)=x(k+1).Q(q)and L(q)are the robustness filter
and the learning filter in ILC, respectively. Substituting Eq. (3)
into Eq. (2) yields the closed-loop iteration-domain dynamics:
Rj+1=Q(1zmLT0)Rj+zmQL[(1T0)YdPS0(Dr+Dncj)]
Ej+1=Q(1zmLT0)Ej+(1Q)[(1T0)Yd+PS0Dr]
+PS0(Dnj+1QDnj)(4)
Thus, a sufficient condition for stability of the iteration process
in Eq. (4) is that Q(1zmLT0)satisfies:
||Q(1zmLT0)||<1(5)
where || • ||=max
0
ω
π
|•|
z=ej
ω
.WithEq.(5)satised,perfor-
mance of the standard ILC is evaluated by the asymptotic errors
which can be expressed as:
E=1Q
1Q(1zmLT0)[(1T0)Yd+PS0Dr]
+1
1Q(1zmLT0)PS0(Dnj+1QDnj)(6)
Equation (5) reveals that an optimal choice for the learning fil-
ter Lis the inverse of the closed-loop complementary sensitiv-
ity function, namely, L=zmT1
0.Herezmguarantees that L
is realizable. More details about this will be given in Section
2.3. Equation (6) also indicates that for perfect error rejection,
namely, for eliminating the repetitive errors in one iteration, Q
should be equal to one at all frequencies. However, due to the
model mismatches at high frequencies, the Q filter is normally
set as a lowpass filter whose bandwidth is determined by uncer-
tainties of the system model. For example, if the actual system
is:
P(z1)=P
n(z1)(1+(z1)) (7)
where P
n(z1)is the identified nominal model in Eq. (1) and
(z1)is the multiplicative uncertainty term, then T0can be ex-
pressed as:
T0(z1)=T0n(z1)(1+T(z1)) (8)
where T0n(z1)=P
n(z1)C(z1)/(1+P
n(z1)C(z1)) and
T(z1)is the equivalent uncertainty in the complementary sen-
sitivity function. The learning filter then becomes L=zmT1
0n.
Recall Eq. (5) and we will get the following condition for stabil-
ity robustness:
|Q(ej
ω
)|<1
|
T(ej
ω
))|,
ω
(9)
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Suppose the cut-off frequency of the Q filter is
ω
c(Q),then
only repetitive errors at frequencies lower than
ω
c(Q)will be
learned and attenuated by ILC, while for those at higher fre-
quencies, learning is essentially cut off. However, in practice,
repetitive errors may be distributed over a wide frequency range
(for example, the case for the position errors of the wafer scan-
ner in the acceleration phase). Also, non-repetitive errorscontain
components at frequencies lower than
ω
c(Q)(such as position er-
rors caused by the force ripple in the wafer scanner). Therefore,
standard ILC may fail to learn some repetitive errors and in the
meantime mistakenly learn some non-repetitive ones, resulting
in error amplifications. Consider, for instance, a pure sinusoidal
non-repetitive periodic disturbance with fixed frequency
ω
0that
is smaller than
ω
c(Q)but with a random initial phase in each
iteration, i.e., dnj(k)=sin(
ω
0k+
φ
j).ThenQ(ej
ω
0)is approxi-
mately 1 and the error dynamics are:
ej+1(k)=[1qmL(q)T0(q)]ej(k)+P(q)S0(q)[dnj+1(k)dnj(k)]
where
dnj+1(k)dnj(k)=2cos(
ω
0k+
φ
j+
φ
j+1
2)sin(
φ
j+1
φ
j
2)
Thus, in the worst case, amplification of disturbance by a factor
of two may be caused in the learning process.
2.3 Experimental Results Using Standard ILC in
Wafer Scanner System
Figure 6 and Fig. 7 show the experiment results of perform-
ing a standard ILC on the wafer scanner. The system has a non-
repetitive disturbance at about 18.32Hz caused by the force rip-
ple of the linear motor. The L and Q filters are designed accord-
ing to Eq. (5) and Eq. (9), L(z1)=zmT1
0n(z1)=zm[1+
P
n(z1)C(z1)]/[P
n(z1)C(z1)].Inthissystem,T0n(z1)is
minimum phase, so L(z1)can be directly derived, otherwise
L(z1)should be designed using stable inversion method such as
the ZPET algorithm [12]. The Q filter is a lowpass filter with cut-
off frequency
ω
c=300
π
rad/s.ItcanbeseeninFig.6thatfrom
the first to the second iteration, the positioning performance is
significantly improved. However, no improvement is apparentin
the following iterations. Figure 7 shows the frequency spectrum
of the position errors in iterations 2 to 5. The figure shows that
the non-repetitive disturbance at about 18.32Hz is greatly ampli-
fied, which has limited the performance improvement of ILC, as
confirmed in Fig. 8.
3PROPOSED SELECTIVE ILC WITH NON-
REPETITIVE DISTURBANCE REJECTION
3.1 Proposed ILC Structure
To e n hance the perfo r m a n c e robu s t n e ss of I L C i n t h e p r e s-
ence of non-repetitive disturbances, a new ILC scheme is pro-
0 0.5 1 1.5 2 2.5
2
0
2x 104
Errors(m)
0 0.5 1 1.5 2 2.5
5
0
5
x 106
Errors(m)
0.5 1 1.5 2 2.5
5
0
5
x 106
Time(s)
Errors(m)
iteration 2
iteration 3
iteration 4
iteration 5
iteration 1
Figure 6:Positioning error signals in different iterations
100102
0
2
4
6
8x 107
Frequency(Hz)
Amplitude
100102
0
2
4
6
8x 107
Frequency(Hz)
Amplitude
100102
0
2
4
6
8x 107
Frequency(Hz)
Amplitude
100102
0
2
4
6
8x 107
Frequency(Hz)
Amplitude
Iteration 5
Iteration 3
Iteration 2
Iteration 4
18.32Hz
18.32Hz
18.32Hz
18.32Hz
Figure 7:Frequency-domain tracking errors for iteration 2 to 4
2 4 6 8 10 12 14
0
5
10
x 107
Iterations
2norm of the position errors
Figure 8:2-norm of the tracking errors from iteration 2 to 16
posed in this section, as shown in Fig. 9. It contains two fea-
tures: the selective iterative learning process (the dashed box)
and a DOB for non-repetitive disturbance rejection (the dotted
box). QIand QDrepresent the Q filters in ILC and in DOB, re-
spectively. P1
mis a stable and realizable inverse of the nominal
4Copyright
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model P
n(z1)such that P1
m(z1)=zmP1
n(z1).
The first benefit of this ILC scheme is enhanced attenua-
tion of both repetitive disturbance and non-repetitivedisturbance,
even if their frequencies overlap with each other. On one hand,
non-repetitive disturbances at frequencies within the bandwidth
of the DOB is greatly rejected, generating a more ”clear sig-
nal” in time domain. On the other hand, the time-varying Q
filter further controls the remaining non-repetitive errorsfrom
entering the learning loop of ILC in iteration domain and maxi-
mally increases the repetitive-disturbance-attenuation bandwidth
in different phases of the trajectory. Moreover, the proposed ILC
sheme provides more design flexibility to the design of QIand
QDas discussed in the following section.
ej(k
)
L(z
¡1
)
memory
Q
I
(z
¡1
)
rj(k
)
C(z¡1
)
u(k
)
y
j
(k
)
P(z
¡1
)
z
¡m
Q
D
(z
¡1
)
P
¡1
m
(z
¡1
)
d
r
(k
)
yd(k
)
dn
j
(k
)
memory
Figure 9:The proposed selective ILC with DOB
3.2 Closed-loop Dynamics and Design of QIand QD
As shown in Fig. 9, the closed-loop positioning error in the
first iteration is:
e1(k)=yd(k)y1(k)
=1QD(qmPP1
m)
1+PC QD(qmPP1
m)yd(k)
(1qmQD)P
1+PC QD(qmPP1
m)[dr(k)+dn1(k)] (10)
where we have the new sensitivity function S(z1)and comple-
mentary sensitivity function T(z1)as:
S=1qmQD
1+PC QD(qmPP1
m)(11)
T=PC +QDPP1
m
1+PC QD(qmPP1
m)(12)
Based on Eq. (10) and recalling Eq. (2) - Eq. (6), we can then
derive the new closed-loop iteration-domain ILC dynamics:
Rj+1=QI(1zmLT )Rj+QIzmL[(1T)YdPS(Dr+Dnj)]
Ej+1=QI(1zmLT )Ej+(1QI)[(1T)Yd+PSDr]
+PS(Dnj+1QIDnj)(13)
The goal of ILC is to cancel all repetitive disturbances by
learning. Recalling Eq. (5), by choosing L=zmT1
n,wecan
design a lowpass filter QIwith cut-off frequency
ω
cthat sat-
isfies |QI(ej
ω
)|<1
|
T(ej
ω
)|,
ω
.Noting1QI(ej
ω
)0at
low frequencies (
ω
<
ω
c)inEq. (13),thelow-frequencyrepeti-
tive disturbances can be almost canceled in one iteration. Inthe
meantime, the non-repetitive component of the error in Eq. (13)
caused by the non-repetitive disturbances is
enj+1(k)=P1qmQD
1+PC QD(qmPP1
m)nj(k)(14)
where nj=dnj+1(k)QIdnj(k).
As discussed in Section 2, for the case where the non-
repetitive disturbances dnj(k)are concentrated at particular fre-
quencies with iteration-dependent amplitudes and phases, unde-
sired amplification will occur with QI(ej
ω
)1atthosefrequen-
cies. Thus, more consideration is required for the design of QI
and will be given in Section 3.2.2.
3.2.1 Non-repetitive Disturbances at Frequencies
Within the Bandwidth of QDIn this case, the non-repetitive
disturbance nj(k)at frequencies within the bandwidth of QDwill
be rejected by DOB if QDis designed to satisfy the following
constraints:
|QD(ej
ω
)|<1
|
T(ej
ω
)|,
ω
|1zmQD|z=ej
ω
nj1
ω
nj
ω
c(QD)
(15)
where T(ej
ω
)is the multiplicative uncertainty term of the com-
plementary sensitivity function in Eq. (8). The first constraint
guarantees the closed-loop stability with DOB [13] and makes
QDalow-passfilterbecauseT(ej
ω
)is normally large at high
frequencies. The second constraint comes from Eq. (14) and
gives us an additional guideline for designing QDproperly. Here,
to be able to compensate disturbances in discrete-time systems
with time delays (e.g., the wafer scanner system in this paper),
QDshould be designed carefully to estimate the amplitude of the
disturbance as well as to compensate for the phase delay. For
example, if njis concentrated at low frequencies, then a proper
choice for Q(q1)is set 1 qmQD(q)=HD(q)to be of high-
pass characteristics. Assume that
1qmQD(q)=HD(q)= B(q)
A(q)J(q)(16)
which is equivalent to the Diophantine equation, A(q)=
B(q)J(q)+qmBQD(q),ifweletQD(q)share the same denom-
inator with HD(q),namely,QD(q)=BQD(q)/A(q).Solvingthis
Diophantine Equation gives us a minimum-order QD(q)satisfy-
ing Eq. (16).
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As an example, let
HD(q)= 0.9481 1.896q1+0.9481q2
11.894q1+0.899q2(17)
This filter has a cut-off frequency at 30Hz and the resultant Dio-
phantine Equation yields
QD(q)= 0.1118 0.1064q1
11.894q1+0.899q2
with J(q)=1.055 +0.1122q1.Thefrequencyresponseof
QD(z1)and HD(z1)are shown in Fig. 10.
101100101102103
80
60
40
20
0
Magnitude(dB)
101100101102103
90
0
90
180
Frequency(Hz)
Phase
QD(z1)
HD(z1)
Figure 10:The designed QD(z1)and the resultant HD(z1)
3.2.2 Non-repetitive Disturbances at Frequencies
Above the Bandwidth of QDNon-repetitive disturbances
at frequencies above the bandwidth of QDcannot be rejected by
DOB, and a proper QIis required which can selectively prevent
the non-repetitive disturbances from entering the ILC learning
scheme and maximally preserve its ability to reject repetitive dis-
turbances.
Suppose that there is a sinusoidal non- repetitive periodic dis-
turbance component dn(k)at frequency
ω
0which is much higher
than the bandwidth of QD,i.e.,QD(
ω
0)1inEq. (14). Then
the tracking error in the ( j+1)-th iteration will become
enj+1(k)=PS[dnj+1(k)QIdnj(k)] (18)
Therefore, to prevent the disturbance in the j-th iteration from
entering the ( j+1)-th iteration, QI(ej
ω
0)0shouldbesatised.
Namely, QIshould contain a notch filter as follows:
QI(q)=QI0(q)12cos(2
πω
0)q1+q2
12
α
cos(2
πω
0)q1+
α
2q2(19)
where QI0(q)is a baseline Q filter in the standard ILC.
As we discussed in Section 2.1, the position error of a pre-
cision system has different characteristics in different phases of
the trajectory. In the acceleration phase, the error mainly comes
from tracking of the trajectory and contain rich frequency com-
ponents. Thus, the higher the bandwidth of QI(
ω
c), the better the
learning performance. However, in the constant-speed phase, the
error components caused by the non-repetitive disturbances(for
example, force ripple) dominate and the ILC should selectively
filter them out to avoid undesired amplification; namely, QIin
this phase should contain notches at those frequencies wherethe
non-repetitive disturbances appear. Consequently, QIbecomes a
time-varying filter which has a wider bandwidth in the accelera-
tion phase (for better trajectory following) and multiple notches
in the constant-speed phase. To guarantee the performance dur-
ing the switching process, a smooth switching algorithm may be
used; for example,
QI=
QI0,ttae
(1
α
(t))QI0+
α
(t)QI1,tae <t<tae +t
QI1,ttae +t
(20)
where QI0is the QIfilter in the acceleration phase and QI1in the
constant-speed phase designed based on Eq. (19). tae is the time
instant when the acceleration phase ends and tis the switching
period.
α
(t)gradually varies from 0 to 1 as
α
(t)=(ttae)/t.
4SIMULATIONRESULTS
Simulations have been performed to verify the proposed al-
gorithm. The reference trajectory is as shown in Fig. 3. Both
repetitive and non-repetitive disturbances are introduced. Repet-
itive disturbances are distributed over a wide frequency range
[0, 100Hz] and non-repetitive periodic disturbances appearonly
at specific frequencies (18.32Hz and 50Hz) with initial phases
varying from iteration to iteration.
Figure 11 shows the error signal and its frequency spectrum
with the baseline PID controller.
0 0.5 1 1.5 2 2.5
2
0
2x 104
Time(s)
Errors(m)
100101102
0
1
2x 105
Frequency(Hz)
Amplitude
18.32Hz
50Hz
Figure 11:The position error with the baseline PID controller
4.1 Using Standard ILC Algorithm
The simulation result using a standard ILC algorithm is
shown in Fig. 12. The Q filter and learning filter are configured
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as described in Section 2.3. Figure 12(a) shows that ILC can
effectively reduce the position error. After the second iteration,
however, no consistent improvement can be seen. For example,
from the second iteration to the third iteration, the position er-
ror norm is significantly amplified by ILC. Figure 12(b) shows
the frequency spectrum of the error signals in the first three it-
erations of ILC. It can be seen that the repetitive disturbances
are effectively eliminated, but the error becomes significant at
the non-repetitive disturbance frequencies (at about 18.32Hz and
50Hz).
2 4 6 8 10 12 14 16
0
0.5
1
1.5
2
2.5
3x 10
-5
Iterations
2 norm of the position errors
10 20 30 40 50 60 70 80
0
0.2
0.4
0.6
0.8
1
1.2
1.4x 1 0
-5
Frequency(Hz)
Amplitude
iteration 1
iteration 2
iteration 3
Without ILC
WIth ILC
(a)
(b)
49 49.5 50 50.5
0
1
2
3
4
5
6
x 10
-6
Figure 12:The position error using standard ILC without DOB
The same phenomena are observed in experiments. As
shown in Fig. 6 to Fig. 8 in Section 2.3, the non-repetitive distur-
bance at 18.32Hz is also amplified and the ILC performance is
degraded. Additionally, in experiments, we are subjected tomore
constraints in designing QIwithout DOB. The robust stability
condition in Eq. (5) requires QI0athighfrequencies,which
will further degrade the ILC performance. As shown in Fig. 7,
disturbances with frequencies higher than 150Hz (
ω
c(QI))can-
not be attenuated by learning.
4.2 Using only DOB without ILC
Figure 13 shows the simulation results with and without
DOB in one iteration; no ILC is used. We observe that in the
constant-speed phase, DOB works well and significantly reduces
the position errors by suppressing the disturbances at frequencies
lower than the cut-off frequency of QD(
ω
c(QD)=60
π
rad/s).
The transient performance in the acceleration and deceleration
phases, however, is minimally improved. Note that in these
phases, the position error caused by the tracking of the reference
trajectory dominates and DOB has little effect to reduce the er-
ror. Additionally, the non-repetitive disturbance at 50Hz (larger
than
ω
c(QD))remainsunchanged.
4.3 Using the proposed ILC structure
Figure 14 and Fig. 15 show a comparison of the simulation
results using different algorithms. The QDand QIin the pro-
100101102103
0
0.5
1x 105
Frequency(Hz)
Amplitude
Error spectrum in the constantspeed phase
0 0.1 0.2 0.3 0.4
1
0
1
2
x 104
Time(s)
Position errors (m)
w/ DOB
w/o DOB
1 1.2 1.4 1.6
2
0
2
4x 105
Time(s)
Position errors (m)
In the accelaration phase In the constantspeed phase
Figure 13:Position errors with and without DOB (simulation)
posed ILC scheme are designed, respectively, as a lowpass fil-
ter with cut-off frequency
ω
c=60
π
rad/sand as a time-varying
filter in Eq. (20), where t=100Ts=0.04s,andQD(z1),
QI0(z1)and QI1(z1)are given by
QD(z1)= 0.1118 0.1064z1
11.894z1+0.899z2
QI0(z1)= 0.02792 +0.05583z1+0.02792z2
11.475z1+0.5866z2
QI1(z1)=QI0(z1)1.0372(12cos(2
π
Ts18.32)z1+z2)
11.98cos(2
π
Ts18.32)z1+0.9801 z2
0.9963(12cos(2
π
Ts50)z1+z2)
11.98cos(2
π
Ts50)z1+0.9801z2
Also, in the simulation, QI(z1)in the standard ILC is set to be
QI0(z1)in each iteration.
0 2 4 6 8 10 12 14 16
0
0.5
1
1.5
2
2.5
3x 105
Iterations
2norm of the position errors
Standard ILC
Standard ILC with DOB ([11])
ILC with timevarying QL
ILC with timevarying QL and DOB (proposed scheme)
Figure 14:The position errors using different ILC algorithms
It can be seen that the proposed ILC attains the best perfor-
mance with effective disturbance rejection and enhanced robust-
ness to non-repetitive periodic disturbances. Consistent with the
experimental result shown in Fig. 8, the position error obtained
by standard ILC varies greatly in different iterations due tounex-
pected amplifications of the non-repetitive errors. The algorithm
7Copyright
c
2014 by ASME
10 20 30 40 50 60 70
108
106
104
Frequency(Hz)
Amplitude
15 20 25
0.5
1
1.5
2x 105
Frequency(Hz)
Amplitude
48 49 50 51 52
0
0.5
1
1.5 x 105
Frequency(Hz)
Amplitude
Standard ILC
Standard ILC with DOB ([11])
ILC with timevarying QL and DOB (proposed scheme)
Figure 15:Spectrum of the position error in the 5-th iteration
using different ILC algorithms
in [11] can significantly reduce the position error. The perfor-
mance robustness to non-repetitive periodic disturbances,how-
ever, is still not good. This is because the DOB can only reduce
the non-repetitive error components with frequencies lowerthan
ω
c(QD)(e.g.,the error component at 18.32Hz in the simulation),
but has little effect on those higher than
ω
c(QD)(the one at 50Hz
in the simulation) and undesired amplifications of those signals
cannot be avoided. By introducing a time-varying QL,thepro-
posed ILC scheme not only effectively suppresses disturbances
with frequencies lower than
ω
c(QD),butalsoflexiblycontrols
the remaining non-repetitive errors from entering the learning
loop of ILC. Thus, undesired amplifications of the non-repetitive
disturbances have been effectively avoided. Compared to stan-
dard ILC and the algorithm in [11], it can be seen clearly in Fig.
15 that no amplification of the non-repetitivedisturbance at50Hz
occurs by using the proposed ILC algorithm.
5CONCLUSION
In this paper, a new ILC scheme with robust performance
in the presence of non-repetitive disturbances is proposed.By
integrating a DOB and a time-varying Q filter in ILC, the pro-
posed algorithm not only maximally preserves its repetitive-
error-rejection ability, but also provides enhanced attenuation to
non-repetitive disturbances at low frequencies. Moreover,these-
lective learning characteristic of the time-varying Q filteravoids
undesired amplification of the error caused by non-repetitive pe-
riodic disturbances.
ACKNOWLEDGMENT
The authors gratefully acknowledge the kind support pro-
vided by Nikon Research Corporation of America, Agilent Tech-
nologies and National Instruments.
REFERENCES
[1] Arimoto, S., Kawamura, S., and Miyazaki, F., 1984, Better-
ing operation of Robots by learning, J. Robotic Syst, 1(2),
pp. 123-140.
[2] Bristow, D. A., Tharayil, M., and Alleyne, A. G., 2006,
Asurveyofiterativelearningcontrol,IEEEControlSyst.
Mag., 26(3), pp. 96-115.
[3] Merry, R., van de Molengraft, R., and Steinbuch, M.,
2005, The influence of disturbances in iterative learning con-
trol, Proc. 2005 IEEE Conference on Control Applications,
Toronto, pp. 974-979.
[4] Sandipan, M., Joshua, C., and Tomizuka, M., 2007, Pre-
cision Positioning of Wafer Scanners Segmented Iterative
Learning Control for Nonrepetitive Disturbances, IEEE Con-
trol Syst. Mag., 27(4), pp. 20-25.
[5] Lee, J. H., Lee, K. S., and Kim, W. C., 2000, Model-based
iterative learning control witha quadratic criterion for time-
varying linear systems, Automatica, 36, pp. 641657.
[6] Phan, M. Q., and Longman, R. W., 2002, Higher-order iter-
ative learning control by pole placement and noise Filtering,
Proc. 15th IFAC world congress, Barcelona, Spain, 15(1),
pp. 988-995.
[7] Merry, R., van de Molengraft, R., and Steinbuch, M., 2008,
Iterative learning control with wavelet filtering, Int. J. Robust
Nonlin. Control, 18, pp. 1052-1071.
[8] Chen, Y. and Moore, K. L., 2002, Harnessing the nonrepet-
itiveness in iterative learning control, Proc. 41st IEEE Con-
ference on Decision and Control, 3, pp. 3350-3355.
[9] Zhang, B., Wang, D., and Ye, Y., 2005, Wavelet transform-
based frequency tuning ILC, IEEE Trans. Syst. Man Cyb. -
Part B, 35(1), pp. 107-114.
[10] Rotariu, I., Steinbuch, M., and Ellenbroek, R., 2008, Adap-
tive Iterative Learning Control for High Precision Motion
Systems, IEEE Trans. Control Syst. Techn, 16(5), 1075-
1082.
[11] Yu, S., and Tomizuka, M., 2009, Performance Enhance-
ment of Iterative Learning Control System Using Distur-
bance Observer, Proc. IEEE/ASME International Confer-
ence on Advanced Intelligent Mechatronics, Singapore, July
14-17.
[12] Tomizuka, M., 1987, Zero Phase Error Tracking Algorithm
for Digital Control, ASME J. Dyn. Syst. Meas. Control.,
109(1), pp. 65-68.
[13] Kempf, C. and Kobayashi, S., 1996, Discrete-Time Distur-
bance Observer Design for Systems With Time Delay, Proc.
International Workshop on Advanced Motion Control, 1, pp.
332-337.
8Copyright
c
2014 by ASME
... Iterative learning control (ILC) is an effective technique to improve the tracking performance of the systems which operate in a repetitive manner. It has been applied to a variety of industrial problems such as robot manipulators [1,2], micro positioning stages [3], hard disk drives [4], wafer scanning systems [5][6][7], etc. One main challenge in ILC is to design learning filters with guaranteed convergence and robustness. ...
... The effectiveness of the proposed ILCs is demonstrated based on the simulations of a wafer scanning system. More detailed descriptions of this system can be found in [6,7]. The sampling frequency of the system is 2500 Hz. ...
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