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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

Constant−speed 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(z−1))withabaselinePID

feedback controller C(z−1)(See Fig. 5). The identiﬁed nominal

model P

n(z−1)of the plant P(z−1)is:

P

n(z−1)=3.4766 ×10−7z−21+0.8z−1

1−2z−1+z−2(1)

where z−1denotes the one-step delay operator.

2Copyright

c

⃝2014 by ASME

−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(z−1)= P(z−1)C(z−1)

1+P(z−1)C(z−1)[Yd(z−1)+Rj(z−1)]

+P(z−1)

1+P(z−1)C(z−1)[Dr(z−1)+Dnj(z−1)] (2)

where the sensitivity function and complementary sen-

sitivity function of the feedback loop are S0(z−1)=

1/[1+P(z−1)C(z−1)] and T0(z−1)=P(z−1)C(z−1)/[1+

P(z−1)C(z−1)];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)[rj−1(k)+L(q)ej−1(k+m)] (3)

where kis the time index within each iteration, mis the relative

degree of T0(z−1),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(1−zmLT0)Rj+zmQL[(1−T0)Yd−PS0(Dr+Dncj)]

Ej+1=Q(1−zmLT0)Ej+(1−Q)[(1−T0)Yd+PS0Dr]

+PS0(Dnj+1−QDnj)(4)

Thus, a sufﬁcient condition for stability of the iteration process

in Eq. (4) is that Q(1−zmLT0)satisﬁes:

||Q(1−zmLT0)||∞<1(5)

where || • ||∞=max

0≤

ω

≤

π

|•|

z=ej

ω

.WithEq.(5)satisﬁed,perfor-

mance of the standard ILC is evaluated by the asymptotic errors

which can be expressed as:

E∞=1−Q

1−Q(1−zmLT0)[(1−T0)Yd+PS0Dr]

+1

1−Q(1−zmLT0)PS0(Dnj+1−QDnj)(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=z−mT−1

0.Herez−mguarantees 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(z−1)=P

n(z−1)(1+△(z−1)) (7)

where P

n(z−1)is the identiﬁed nominal model in Eq. (1) and

△(z−1)is the multiplicative uncertainty term, then T0can be ex-

pressed as:

T0(z−1)=T0n(z−1)(1+△T(z−1)) (8)

where T0n(z−1)=P

n(z−1)C(z−1)/(1+P

n(z−1)C(z−1)) and

△T(z−1)is the equivalent uncertainty in the complementary sen-

sitivity function. The learning ﬁlter then becomes L=z−mT−1

0n.

Recall Eq. (5) and we will get the following condition for stabil-

ity robustness:

|Q(ej

ω

)|<1

|△

T(ej

ω

))|,∀

ω

(9)

3Copyright

c

⃝2014 by ASME

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)=[1−qmL(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(z−1)=z−mT−1

0n(z−1)=z−m[1+

P

n(z−1)C(z−1)]/[P

n(z−1)C(z−1)].Inthissystem,T0n(z−1)is

minimum phase, so L(z−1)can be directly derived, otherwise

L(z−1)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 10−4

Errors(m)

0 0.5 1 1.5 2 2.5

−5

0

5

x 10−6

Errors(m)

0.5 1 1.5 2 2.5

−5

0

5

x 10−6

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 10−7

Frequency(Hz)

Amplitude

100102

0

2

4

6

8x 10−7

Frequency(Hz)

Amplitude

100102

0

2

4

6

8x 10−7

Frequency(Hz)

Amplitude

100102

0

2

4

6

8x 10−7

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 10−7

Iterations

2−norm 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. P−1

mis a stable and realizable inverse of the nominal

4Copyright

c

⃝2014 by ASME

model P

n(z−1)such that P−1

m(z−1)=z−mP−1

n(z−1).

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)

=1−QD(q−m−PP−1

m)

1+PC −QD(q−m−PP−1

m)yd(k)

−(1−q−mQD)P

1+PC −QD(q−m−PP−1

m)[dr(k)+dn1(k)] (10)

where we have the new sensitivity function S(z−1)and comple-

mentary sensitivity function T(z−1)as:

S=1−q−mQD

1+PC −QD(q−m−PP−1

m)(11)

T=PC +QDPP−1

m

1+PC −QD(q−m−PP−1

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(1−zmLT )Rj+QIzmL[(1−T)Yd−PS(Dr+Dnj)]

Ej+1=QI(1−zmLT )Ej+(1−QI)[(1−T)Yd+PSDr]

+PS(Dnj+1−QIDnj)(13)

The goal of ILC is to cancel all repetitive disturbances by

learning. Recalling Eq. (5), by choosing L=z−mT−1

n,wecan

design a lowpass ﬁlter QIwith cut-off frequency

ω

cthat sat-

isﬁes |QI(ej

ω

)|<1

|△

T(ej

ω

)|,∀

ω

.Noting1−QI(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)=P1−q−mQD

1+PC −QD(q−m−PP−1

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

|1−z−mQD|z=ej

ω

nj≪1

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-passﬁlterbecause△T(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(q−1)is set 1 −q−mQD(q)=HD(q)to be of high-

pass characteristics. Assume that

1−q−mQD(q)=HD(q)= B(q)

A(q)J(q)(16)

which is equivalent to the Diophantine equation, A(q)=

B(q)J(q)+q−mBQD(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).

5Copyright

c

⃝2014 by ASME

As an example, let

HD(q)= 0.9481 −1.896q−1+0.9481q−2

1−1.894q−1+0.899q−2(17)

This ﬁlter has a cut-off frequency at 30Hz and the resultant Dio-

phantine Equation yields

QD(q)= 0.1118 −0.1064q−1

1−1.894q−1+0.899q−2

with J(q)=1.055 +0.1122q−1.Thefrequencyresponseof

QD(z−1)and HD(z−1)are shown in Fig. 10.

10−1100101102103

−80

−60

−40

−20

0

Magnitude(dB)

10−1100101102103

−90

0

90

180

Frequency(Hz)

Phase

QD(z−1)

HD(z−1)

Figure 10:The designed QD(z−1)and the resultant HD(z−1)

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)≈0shouldbesatisﬁed.

Namely, QIshould contain a notch ﬁlter as follows:

QI(q)=QI0(q)1−2cos(2

πω

0)q−1+q−2

1−2

α

cos(2

πω

0)q−1+

α

2q−2(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,t≤tae

(1−

α

(t))QI0+

α

(t)QI1,tae <t<tae +△t

QI1,t≥tae +△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)=(t−tae)/△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 10−4

Time(s)

Errors(m)

100101102

0

1

2x 10−5

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

6Copyright

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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 QI≈0athighfrequencies,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 10−5

Frequency(Hz)

Amplitude

Error spectrum in the constant−speed phase

0 0.1 0.2 0.3 0.4

−1

0

1

2

x 10−4

Time(s)

Position errors (m)

w/ DOB

w/o DOB

1 1.2 1.4 1.6

−2

0

2

4x 10−5

Time(s)

Position errors (m)

In the accelaration phase In the constant−speed 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(z−1),

QI0(z−1)and QI1(z−1)are given by

QD(z−1)= 0.1118 −0.1064z−1

1−1.894z−1+0.899z−2

QI0(z−1)= 0.02792 +0.05583z−1+0.02792z−2

1−1.475z−1+0.5866z−2

QI1(z−1)=QI0(z−1)1.0372(1−2cos(2

π

Ts∗18.32)z−1+z−2)

1−1.98cos(2

π

Ts∗18.32)z−1+0.9801 ∗z−2

∗0.9963(1−2cos(2

π

Ts∗50)z−1+z−2)

1−1.98cos(2

π

Ts∗50)z−1+0.9801z−2

Also, in the simulation, QI(z−1)in the standard ILC is set to be

QI0(z−1)in each iteration.

0 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

Standard ILC

Standard ILC with DOB ([11])

ILC with time−varying QL

ILC with time−varying 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

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10 20 30 40 50 60 70

10−8

10−6

10−4

Frequency(Hz)

Amplitude

15 20 25

0.5

1

1.5

2x 10−5

Frequency(Hz)

Amplitude

48 49 50 51 52

0

0.5

1

1.5 x 10−5

Frequency(Hz)

Amplitude

Standard ILC

Standard ILC with DOB ([11])

ILC with time−varying 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.

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