M. van Haren et al, "A Frequency-Domain Approach for Enhanced Performance and Task Flexibility in Finite-Time ILC,"
published in European Journal of Control, 2024, uploaded June 21, 2024

A Frequency-Domain Approach for Enhanced
Performance and Task Flexibility in Finite-Time ILC
Max van Haren1 , Kentaro Tsurumoto2 , Masahiro Mae2 ,Lennart Blanken1,3 , Wataru Ohnishi2 and Tom Oomen1,4

Abstract— Iterative learning control (ILC) techniques arecapable of improving the tracking performance of control
systems that repeatedly perform similar tasks by utilizing datafrom past iterations. The aim of this paper is to achieve boththe task flexibility enabled by ILC with basis functions and theperformance of frequency-domain ILC, with an intuitive design
procedure. The cost function of norm-optimal ILC is determined that recovers frequency-domain ILC, and consequently,
the feedforward signal is parameterized in terms of basis
functions and frequency-domain ILC. The resulting method has
the performance and design procedure of frequency-domain
ILC and the task flexibility of basis functions ILC, and arecomplimentary to each other. Validation on a benchmark
example confirms the capabilities of the framework.

I. I NTRODUCTION
The increasing requirements for precision mechatronics
result in a situation where both tracking performance andtask flexibility, which is the ability to have high performancefor different references, are important. Feedforward control
is effective in compensating known disturbances for systems,
leading to improved performance. Feedforward control isoften based on models [1], which is generally achieved inindustrial applications by means of basis functions feedforward control. In basis functions feedforward control, thefeedforward signal is a linear combination of basis functions that relate to physical quantities, such as acceleration
feedforward for the inertia [2–4]. Due to modeling and tuning
inaccuracies, the increasing requirements for performance are
generally not achieved.
This work is part of the research programme VIDI with project number
15698, which is (partly) financed by the Netherlands Organisation for
Scientific Research (NWO). In addition, this research has received funding
from the ECSEL Joint Undertaking under grant agreement 101007311(IMOCO4.E). The Joint Undertaking receives support from the European
Union Horizon 2020 research and innovation programme. Furthermore, it
is partially funded by the joint JSPS-NWO funding programme Research
Network on Learning in Machines, with grant numbers JPJSBP220234403
and 040.040.060. It is also partly supported by JSPS KAKENHI grant
numbers 23H01431, and 24KJ0959.
1 Max van Haren, Lennart Blanken and Tom Oomen are with the Control Systems Technology Section, Department of Mechanical Engineering,
Eindhoven University of Technology, Eindhoven, The Netherlands, e-mail:m.j.v.haren@tue.nl.
2 Kentaro Tsurumoto, Masahiro Mae and Wataru Ohnishi are withthe department of Electrical Engineering and Information Systems, theUniversity of Tokyo, Tokyo, Japan.
3 Lennart Blanken is with Sioux Technologies, Eindhoven, The Netherlands.
4 Tom Oomen is with the Delft Center for Systems and Control, Delft
University of Technology, Delft, The Netherlands.

Iterative Learning Control (ILC) can improve tracking
performance with respect to model based feedforward control[5], and hence, can fulfill the increasing requirements for
performance. ILC utilizes information from past iterations
to improve the tracking performance in the current iteration.
For ILC to be industrially applicable, it is required that ILC(R1) is task flexible;(R2) has high tracking performance; and(R3) has an intuitive design procedure.
In this paper, two types of ILC are considered, and arereferred to as frequency-domain and norm-optimal ILC.
First, frequency-domain ILC uses infinite-time frequencydomain system representations to iteratively update the feedforward signal [6]. Frequency-domain ILC has the advantagethat convergence can be verified and tuned using frequency
response functions (FRFs), that are accurate and inexpensive
to obtain [7]. As a result, frequency-domain ILC leads toan intuitive frequency-domain design procedure consistingof manual loop shaping and has high tracking performance.
Frequency-domain ILC is typically implemented in finitetime, where convergence can still be analyzed [8]. However,conventional frequency-domain ILC is not directly capableof task flexibility, and hence, does not satisfy requirement
R1.
Second, norm-optimal ILC utilizes a finite-time cost function to iteratively optimize the feedforward signal [9]. The
main advantage of norm-optimal ILC is that the feedforward
signal can be parameterized into basis functions, that enables
task flexibility [10, 11]. However, if the basis functions
are not sufficiently rich to describe the inverse system, theperformance is significantly worse compared to frequencydomain ILC, and therefore does not achieve requirement R2.
Important developments have been made to combine thetask flexibility of ILC with basis functions and the performance of frequency-domain ILC. In [12, 13], frequencydomain ILC is projected on basis functions, resulting in task
flexibility, but reducing tracking performance. Furthermore,in [14], frequency-domain ILC is combined with ILC withbasis functions using a sequential optimization problem, that
results in the performance of frequency-domain ILC andthe task flexibility of ILC with basis functions. However,
the approach results in an unintuitive design procedure, notsatisfying requirement R3.
Although ILC methods with high performance and task
flexibility are investigated, a method that achieves both high

fj
rj

ej

uj
K

goal is to reduce the reference induced error signal ej ∈RN over multiple trials j with the trial-varying feedforward
signal fj ∈ RN .

yj
P

B. Classes of ILC

Fig. 1: Control structure considered.

performance and task flexibility, with an intuitive design
procedure, is currently lacking. In this paper, high performance (R2) is achieved by deliberately overparameterizing
the feedforward signal in a low number of basis functions toaccomplish task flexibility (R1). Additionally, a complementary signal is optimized via an intuitive frequency-domain
ILC design (R3). The key contributions in this paper include
the following.
(C1) Determining the equivalent norm-optimal finite-time
representation of frequency-domain ILC by specificchoice of weighting matrices, that enables intuitive
tuning in the frequency-domain (R3) (Section III-A).
(C2) Achieving both task flexibility (R1) and high performance (R2) by exploiting an overparameterized feedforward signal, building upon C1 (Section III-B).
(C3) Validation of the framework on an example (Section IV).
Notation: Let H(z) denote a discrete-time, Linear
Time-Invariant (LTI), single-input, single-output system. The
frequency response function of H(z) is obtained by substituting z = ejω ∀ω ∈ [0, 2π), and is denoted by H(ejω ).
Signals are of length N . Vectors are denoted as lowercase
letters and matrices as uppercase letters,Pe.g., x and X. The∞
z-transform of signal x(k) is X (z) = k=0 x(k)z −1 . Leth(k) ∀k ∈ Z be the impulse responsePcoefficients of H(z),∞with infinite impulse response y(k) = τ =−∞ h(τ )u(k−τ ).
Let u(k) = 0 for k < 0 and k ≥ N to obtain the finite-time
convolution






y(0)
h(0)
h(−1)
···
 y(1)   h(1)
h(0)
···

 

=
..
..
..
..



.
.
.
.
y(N − 1)h(N − 1) h(N − 2) · · ·
|
{z
} |
{z
y

H

N





h(−N + 1)u(0)h(−N + 2)
  u(1) 
,

..



h(−1)
.
h(0)
u(N − 1)
}|
{z
}
u

N ×N

with y, u ∈ R and H ∈ R
is the finite-time convolution matrix corresponding to H(z).
II. P ROBLEM F ORMULATIONIn this section, the problem that is dealt with in this paper
is formulated. First, the problem setup is presented. Second,the different classes of ILC considered in this paper aredescribed. Finally, the problem that is addressed in this paper
is defined.
A. Problem Setup
The control structure is seen in Fig. 1. The LTI system
P is stabilized by LTI feedback controller K. The finitetime reference signal rj ∈ RN can be trial varying. The

In this section, the three considered classes of ILC, that isnorm-optimal ILC, ILC with basis functions and frequencydomain ILC, are presented.
1) Norm-Optimal ILC: Norm-optimal ILC is a type of
ILC that minimizes a finite-time cost function, typically
2

2

2

NO
NO
− fjN O W
+ fj+1min ∥êj+1 ∥We + fj+1
W
NO
fj+1

f

∆f

,

(1)

where ∥x∥2W = x⊤ W x, We , Wf and W∆f are symmetric positive (semi)definite
weighting matrices [9], êj+1 =NO
ej − Jˆ fj+1
− fjN O , and finite-time convolution matrix−1Jˆ = P̂ I + K P̂, with convolution matrix Pb ∈ RN ×N
b of system P, and identity matrix
representing model P
I ∈ RN ×N . The cost function in (1) is quadratic in theNO
optimization variables fj+1
, and hence, has a minimizer thatis analytically computed asNO
fj+1= QN O fj + LN O ej ,(2)with norm-optimal ILC robustness and learning matrices
QN O and LN O .
2) ILC with Basis Functions: ILC with basis functions
achieves reference flexibility by minimizing a cost function
and parameterizing the feedforward signal in basis functionsasfj = ψθj ,(3)with feedforward parameters θj ∈ Rnθ ×1 and basis functionsψ ∈ RN ×nθ . For ILC with basis functions, the cost function222

min ∥êj+1 ∥We + ∥fj+1 ∥Wf + ∥fj+1 − fj ∥W∆f ,
θj+1

(4)

is minimized. Similarly to norm-optimal ILC in (2), theoptimal solution to (4) is of the form
θj+1 = QBF θj + LBF ej ,(5)
BFwith basis functions robustness and learning matrices Q
and LBF .
3) Frequency-Domain ILC: Frequency-domain ILC iteratively improves the tracking performance by utilizing
infinite-time frequency-domain representations. Frequencydomain ILC is designed by the infinite-time update lawfFj+1(z) = Qf (z)(Fjf (z) + αLf (z)Ej (z)),(6)

with Qf the robustness filter, that is used to enforce convergence and filter out unwanted effects, Lf the learning
filter and α the learning gain. Frequency-domain ILC is
implemented in finite-time as [8]ffj+1= Qf (fjf + αLf ej ),fwith finite-time convolution matrices Q
sponding to Qf (z) and Lf (z).

(7)
fand L , corre-

y

k
u

106
2

m1

m2

103
1.5

100

d12
d2
Fig. 2: System that is used for validation. The system is
discretized with zero-order hold, and has one sample delay.

10-5

10-3
10-6
0

50

100

150

200

250

300

(a) Error 2-norm for frequency-domain ILC ( ) and norm-optimal
ILC with Wf = 1.39·10−8 I ( ) and Wf = 5.2·10−9 I ( ).

C. Problem Definition
The aforementioned classes of ILC have several design
problems and limitations, and are as follows.
• For norm-optimal ILC, it is highly complex to robustly
choose the weighting matrices, see for example [15, 16],and W = wI severely limits performance. Furthermore,
norm-optimal ILC does not have task flexibility.
• ILC with basis functions reduces performance if ψ does
not accurately describe the inverse system P −1 [13, 17].
• Frequency-domain ILC does have an intuitive design
procedure consisting of manual loop-shaping [5, 13] andresults in high performance, but conventionally does nothave task flexibility.
In Example 1, the limited performance of norm-optimal ILC
with identity weighting matrices and model uncertainty isillustrated, since robust monotonic convergence is difficult
to achieve.
Example 1: A simulation study illustrates that frequencydomain ILC performs better than norm-optimal ILC for
an inaccurate model and identity weighting matrices. The
system is a mass-spring-damper as seen in Fig. 2,where
further elaboration is provided in Section IV. The error
2-norm during 300 trials of norm-optimal and frequencydomain ILC is shown in Fig. 3a, and the maximum error
2-norm during these trials and the steady state error 2-norm∥e∞ ∥2 are shown in Fig. 3b. The results in Fig. 3 illustratethat for norm-optimal ILC to reduce the steady-state error∥e∞ ∥2 beyond frequency-domain ILC,
 it first increases the
maximum error 2-norm maxj ∥ej ∥2 at least a factor 3·106
due to non-monotonic convergence, which is unacceptable inindustrial applications.
Hence, the problem addressed in this paper is to develop
an ILC algorithm that simultaneously satisfies all three requirements for industrial applicability of ILC, that combines
the advantages of current ILC techniques.
III. M ETHOD
In this section, the developed method is presented. The
overparameterized feedforward signal
 

 θj
fj = ΨΘj = ψ IN
= ψθj + fjf ,(8)
fjf
that consists of basis functions and frequency-domain ILC,
is exploited to achieve both task flexibility and high performance, leading to (R1) and (R2). Frequency-domain ILC is

1010

100
10-6

10-5

5·10

-5

(b) Steady-state and maximum error 2-norm for norm-optimal ILC
with Wf = wf I ∀ wf ∈ [10−14 , 10−7 ] after 75 ( ), 150 ( ) and300 ( ) trials, Wf = 5.2 · 10−9 I ( ) and Wf = 1.9 · 10−8 I ( )correspond to Fig. 3a, frequency-domain ILC ( ), and maximum
error 2-norm for fj = 0 ( ).

Fig. 3: Illustration that norm-optimal ILC with We = I
and W∆f = 0 converges slowly and non-monotonically forinaccurate models.

used since it has better performance than norm-optimal ILC
for inaccurate models, as shown in Section II, and due to its
intuitive design procedure (R3).
First, the norm-optimal description of frequency-domain
ILC is determined, such that second, the overparameterized
feedforward signal in (8) consisting of frequency-domain and
ILC with basis functions can be jointly optimized. Finally, a
procedure summarizes the developed method.
A. Norm-Optimal Representation of Frequency-Domain ILC
In this section, it is shown that the finite-time implementation of frequency-domain ILC in (7) is equivalent to (1) fora very specific choice of weighting matrices We , Wf andW∆f . First, the finite-time frequency-domain ILC update is
written using the cost function
2fmin ∥êj+1 ∥Wef + fj+1
f
fj+1

2
Wff

f
+ fj+1
− fjf

2
f
W∆f

,

(9)

that is minimized by


f
f −1
f
fj+1
=(Jˆ⊤ Wef Jˆ + Wff + W∆f)
Jˆ⊤ Wef Jˆ + W∆ffjf
f −1 ˆ⊤+(Jˆ⊤ Wef Jˆ + Wff + W∆f) J Wef ej=Qf (fjf + αLf ej ),(10)fwhere the last identity is found by substituting fj+1
from thefinite-time update law of frequency-domain ILC in (7). The

contributions of fjf and ej in (10) are separated to achieve


f −1
f
Qf = (Jˆ⊤ Wef Jˆ + Wff + W∆f
Jˆ⊤ Wef Jˆ + W∆f),(11)
αQf Lf = (Jˆ⊤ Wef Jˆ + W f + W f )−1 Jˆ⊤ W f .
(12)
f

∆f

must be symmetric and positive semidefinite,
but the product of two symmetric matrices is not necessarilyfsymmetric, Jˆ⊤ Wef Jˆ + W∆fin (13) is chosenfJˆ⊤ Wef Jˆ + W∆f
= I.
(14)

By substituting (14) and Wff from (13) into (12), thefollowing identity is found(15)

leading to the main result in this section in Theorem 1.
Theorem 1: Let Jˆ be invertible and Lf = Jˆ−1 , then theminimizer of the cost function in (9) with
Wef = αJˆ−⊤ Lf ,−1
Wff = Qf
− I,

(16b)

f
W∆f
= (1 − α)I,

(16c)

(16a)

is equal to finite-time frequency-domain ILC in (7).
Proof: Straightforward manipulation of (15) lead to Wein (16a). Substituting (14) into (13) leads to Wf in (16b).
Finally, the resulting We in (16a) is substituted into (14) toresult in W∆f in (16c).
Remark 1: The conditions that Jˆ is invertible and Lf =−1ˆJ
in Theorem 1 are not restrictive, since the weighting
matrix Wef can be approximated as the symmetric matrix
⊤

2

Θj+1

2

(18)

+ ∥Θj+1 − Θj ∥W∆
with

(13)

Because Wff

Wef = αLf Lf ,2

min V (Θj+1 ) = min ∥êj+1 ∥Wef + ∥Θj+1 ∥Wθ,f

Θj+1

e

From (11), Wff is derived as



−1
f
Wff = Jˆ⊤ Wef Jˆ + W∆fQf−I .

αQf Lf = Qf Jˆ⊤ Wef ,by utilizing the overparameterized feedforward signal in (8)fand preserving the weighting Wff and W∆fexclusively on
the frequency-domain component as(17)

Wθ,f =


Wθ
0


0
f ,
Wf

W∆ =


W∆θ
0


0
,
f
W∆f

(19)

where Wθ , W∆θ ∈ Rnθ ×nθ are the weighting matrices on
the feedforward parameters θ, that are typically chosen as
Wθ = wθ ψ ⊤ ψ and W∆θ = w∆θ ψ ⊤ ψ, resulting in equivalentweighting as conventional ILC with basis functions. The
minimizer of (18) is given by−1ˆ + Wθ,f + W∆
Θj+1 = Ψ⊤ Jˆ⊤ Wef JΨˆ + W∆ Θj + Ψ⊤ Jˆ⊤ Wef ej .
· Ψ⊤ Jˆ⊤ Wef JΨ(20)
Remark 2: If the weighting on θ is chosen sufficiently
small Wθ << Wff , the parameterization (8) consisting of
frequency-domain and ILC with basis functions is naturally complimentary and the parameterization will assign asmuch information in the basis functions feedforward signal
as possible. Additionally, a targeted regularization on thefrequency-domain component can be done by using theimage of ψ, similarly to [18],2
min V (Θj+1 ) + λ 0 U1 Θj+1 2 ,Θj+1

with singular value decomposition  Σ 0 V1⊤
ψ = U1 U2.
0 0 V2⊤
C. Procedure
In this section, the developed method for achieving task
flexibility, high performance and an intuitive design procedure by combining frequency-domain ILC with basis function ILC is summarized in Procedure 1.

since Lf is similar to J −1 , and is shown in Section IV-A.
Assumption 1: To ensure that Wff in (16b) is symmetric,
there is assumed that the robustness filter is designed withzero-phase, i.e., Qf (z) = Qf1 ( z1 )Qf1 (z) [9].
To summarize, finite-time frequency-domain ILC in (7) isrecovered by specifically choosing the weighting matrices(16a), (16b) and (16c) and optimizing (9), resulting in
an intuitive frequency-domain design procedure (R3) for
norm-optimal ILC. In the next section, the norm-optimal
description of frequency-domain ILC is used when overparameterizing the feedforward signal.

In this section, the developed method is validated and compared with frequency-domain and ILC with basis functions.
First, the validation setup is shown, including the system
and model, the reference signals and the ILC designs. Second, the norm-optimal equivalent description of frequencydomain ILC is validated. Finally, the developed method withbasis functions is validated and compared for a trial-varying
reference signal.

B. Inclusion of Basis-Function in Frequency-Domain ILC

A. Simulation Setup and Approach

In this section, both task flexibility and high performance
are achieved by deliberately overparameterizing the feedforward in terms of basis functions and frequency-domain ILC
by utilizing the norm-optimal representation of finite-timefrequency-domain ILC. The cost function in (9) is adjusted

A two-mass-spring-damper system with one sample delay
and a sampling time of 1 ms, with inaccurate model, is simulated to validate the developed ILC technique. The massspring-damper system represents the dominant dynamics ofmechatronic systems [3, 4], and is for example an actuator

IV. S IMULATION E XAMPLE

10 -3

Procedure 1: (Norm-optimal frequency-domain ILC withbasis functions)
1) Design learning filter Lf (z), α and zero-phase robustness filter Qf (z) as in frequency-domain ILC.
2) Derive Lf and Qf , that are the finite-time convolution
matrices of Lf (z) and Qf (z).
3) Choose the basis functions ψ in (8).
4) Compute equivalent norm-optimal weighting matricesf
Wef , Wff and W∆fusing (16a), (16b) and (16c).
a) If We is non-symmetrical, follow Remark 1.
5) Initialize Θ1 , e.g., as Θ1 = 0.
6) For j ∈ {1, 2, 3, . . . , Ntrials }.
a) Calculate fj = ΨΘj in (8).
b) Apply fj to closed-loop system and record ej .
c) Calculate Θj+1 using (20).

TABLE I: Parameters used for the system seen in Fig. 2 forthe true system and model.
Parameter

True

Model

Unit

m1
m2
k
d2
d12

0.072
0.01
1000
0.031
1

0.09
0.006
1800
0
0.915

[kg]
[kg]
[N/m]
[Ns/m][Ns/m]−3−4−5

2.80z + 12.4z − 0.65z − 1.58z,
1 − 3.78z −1 + 5.46z −2 − 3.56z −3 + 0.89z −4
−2
−3
−4
−5
4.00z + 21.4z + 5.85z − 1.25zbP(z)= 10−7 ·.
1 − 3.56z −1 + 4.98z −2 − 3.26z −3 + 0.85z −4P(z) = 10−7 ·(21)b jω ) are seen in Fig. 4.
The FRFs of P(ejω ) and P(eAdditionally, the FRF of the true system P(ejω ) is available
for stability analysis, but not for the design of learning filters.
The feedback controller is a lead filter and a first-order lowpass filter, that achieves a closed-loop bandwidth of 10 Hz

100
50
0
-80

-100 -100
-150
10-2

2
0
0

0.05

0.1

0.15

0.2

Fig. 5: The first ( ) and second ( ) reference signals thatare used during validation.

with sufficient robustness margins, and is given by
K(z) =

108.6 + 112.9z −1 − 100z −2 − 104.3z −3.
1 − 0.65z −1 − 0.95z −2 + 0.70z −3(22)

The learning filter Lf (z) is designed by approximating theinverse process sensitivity using ZPETC [19]. The robustness filter Qf (z) is a zero-phase second order Butterworth
lowpass filter with a cutoff frequency of 40 Hz, that was
manually tuned to achieve convergence according to∀ω ∈ [0, 2π].
(23)
Two fourth-order polynomial reference signals are designed
using the approach in [3] with N = 229 samples and areseen in Fig. 5. Following 10 consecutive trials with the firstreference signal, it is then switched to the second reference
signal for another 10 consecutive trials.
a) Basis Function Design: The basis functions ψ arebbased on the inverse model of P(z)and are chosen as....ψ = r̈ r ,(24)
where the derivatives of the reference signal are readily
available by design of the reference signal. Wθ and W∆θin (19) are chosen as 0, since no additional robustness isnecessary.
b) Recovering
Norm-Optimal
FormulationoffFrequency-Domain ILC: Wff and W∆f
are calculatedfusing respectively (16b) and (16c), resulting in W∆f= 0ff
since α = 1. We is computed using (17), since L ̸= Jˆ−1
due to the use of ZPETC, as indicated in Remark 1. A
surface plot of the weighting matrix Wff is seen in Fig. 6.
Frequency-domain ILC and its norm-optimal equivalentachieve the same tracking error for every trial, where theerror signal during trial 5 and 10 are shown in Fig. 7,
validating their equivalence.
B. Validation Results

-60
-50

4

Qf (ejω )(1 − αJ (ejω )Lf (ejω )) < 1,with a flexible coupling and mass attached. The system isseen in Fig. 2, and the parameters of the true system P(z)band the model P(z)are seen in Table I, and are given by
−2

6

10-1

100

101

102

Fig. 4: FRF of the system P(ejω ) ( ) and of the modelb jω ) ( ).
available for ILC P(eThe error 2-norm and signal for 20 trials of ILC are seenin Fig. 8 and Fig. 9. The plant estimate using the basis
function feedforward filter is seen in Fig. 10. The following
observations are made.
•Fig. 8 shows frequency-domain ILC is converged attrials 10 and 20. However, its error 2-norm significantly
increases at trial 11, showing that it lacks task flexibility.

10 -2

10 -3

10 -4

10 -5
2

Fig. 6: Surface plot of the weighting matrix Wff ( ), withcross section of the central values ( ), that if used in the
norm-optimal cost function (1) results in the frequencydomain ILC update.

1

4

6

8

10

12

14

16

18

20

Fig. 8: Error 2-norm for 20 trials of frequency-domain
ILC ( ), basis function ILC ( ) and developed combined
frequency-domain and basis function ILC ( ). At trial 11,
the reference signal is changed ( ).
5

10 -5

10 -5

0.5

0
0
-0.5
-1
0

-5
0
0.05

0.1

0.15

Fig. 7: Tracking performance after 5 trials of ILC for
frequency-domain ILC ( ) and norm-optimal equivalent( ) and after 10 trials for frequency-domain ILC ( ) andnorm-optimal equivalent ( ) using the first reference signal,
showing the same error signal.

Though ILC with basis functions enables reference
flexibility as seen in Fig. 8, its higher error 2-norm
compared to the other methods stems from lacking ṙ inits basis function ψ to compensate the viscous friction
d2 .
• The error 2-norm in Fig. 8 and the time-domain error
signal for trial 20 in Fig. 9 demonstrate the developed
approach’s superior performance against both basis
function and frequency-domain ILC. It surpasses basis
functions ILC by compensating damping d2 with thefrequency-domain component, which is lacking in thebasis function ψ. The developed approach outperforms
frequency-domain ILC by capturing high-frequency effects with ψ, that for frequency-domain ILC is filteredout by robustness filter Qf (z).
• Similar error 2-norm to ILC with basis functions underreference change illustrates the method’s task flexibility.
• From Fig. 10 it becomes clear that the basis functions
feedforward parameters θj are estimated consistently
with the inverse model, which is enabled since Wθ =0 << Wff as described in Remark 2.
•

0.05

0.1

0.15

0.2

0.2

Fig. 9: Tracking error after 20 trials e20 of frequency-domain
ILC ( ), basis function ILC ( ) and developed combined
frequency-domain and basis function ILC ( ).

V. C ONCLUSIONSIn this paper, both task flexibility and performance areachieved through the use of an overparameterized feedforward signal consisting of frequency-domain and basis
functions ILC. The finite-time norm-optimal representationof frequency-domain ILC is derived, that is consequently
used in overparameterizing the feedforward signal. The basis functions and frequency-domain ILC components arecomplimentary by appropriately regularizing the frequencydomain component. An example validates the equivalentnorm-optimal representation, and by exploiting the overparameterized feedforward signal, the performance is significantly increased. Hence, the developed method is a key
enabler for improving performance and task flexibility incontrol.
Ongoing research focuses on rigorously validating themethod through experimental testing on a real-world setup,
in addition to computationally efficient implementations andvalidating robustness of the developed approach.
R EFERENCES[1]

J. Butterworth, L. Pao, and D. Abramovitch, “Analysis and comparison of three discrete-time feedforward model-inverse control
techniques for nonminimum-phase systems,” Mechatronics, vol. 22(5), 2012.

100
50
0
-60
-50

-80

-100 -100
-150
10-2

10-1

100

101

102

Fig. 10: FRFs of system P(ejω ) ( ) and of estimate using−1 jω
the inverse basis function feedforward filter FBF(e , θj )( ), with finite-time representation FBF (θj )r = ψθj .

[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]

[14]

[15]
[16]
[17]
[18]

[19]

M. Boerlage, R. Tousain, and M. Steinbuch, “Jerk derivative feedforward control for motion systems,” in Am. Control Conf., vol. 5(1), 2004.
P. Lambrechts, M. Boerlage, and M. Steinbuch, “Trajectory planningand feedforward design for electromechanical motion systems,”
Control Eng. Pract., vol. 13 (2), 2005.
T. Oomen, “Control for Precision Mechatronics,” in Encycl. Syst.
Control, London: Springer London, 2020, pp. 1–10.
D.A. Bristow ; M. Tharayil ; A.G Alleyne., D. A. Bristow, and
M. Tharayil, “A survey of iterative learning control,” IEEE Control
Syst., vol. 26 (3), 2006.
S. Arimoto, S. Kawamura, and F. Miyazaki, “Bettering operation of
Robots by learning,” J. Robot. Syst., vol. 1 (2), 1984.
R. Pintelon and J. Schoukens, System Identification: A Frequency
Domain Approach. John Wiley & Sons, 2012.
M. Norrlöf and S. Gunnarsson, “Time and frequency domain convergence properties in iterative learning control,” Int. J. Control, vol. 75(14), 2002.
S. Gunnarsson and M. Norrlöf, “On the design of ILC algorithms
using optimization,” Automatica, vol. 37 (12), 2001.
M. Phan and J. Frueh, “Learning control for trajectory tracking usingbasis functions,” in Proc. 35th IEEE Conf. Decis. Control, vol. 3,
1996.
J. van de Wijdeven and O. Bosgra, “Using basis functions in iterative
learning control: analysis and design theory,” Int. J. Control, vol. 83(4), 2010.
S. Mishra and M. Tomizuka, “Projection-Based Iterative Learning
Control for Wafer Scanner Systems,” IEEE/ASME Trans. Mechatronics, vol. 14 (3), 2009.
F. Boeren, A. Bareja, T. Kok, and T. Oomen, “Frequency-Domain
ILC Approach for Repeating and Varying Tasks: With Application
to Semiconductor Bonding Equipment,” IEEE/ASME Trans. Mechatronics, vol. 21 (6), 2016.
K. Tsurumoto, W. Ohnishi, and T. Koseki, “Task Flexible andHigh Performance ILC: Preliminary Analysis of Combining a Basis
Function and Frequency Domain Design Approach,” in IFAC World
Congr., 2023.
J. X. Xu and Y. Tan, “On the robust optimal design and convergence
speed analysis of iterative learning control approaches,” Automatica,
vol. 15 (1), 2002.
B. D. Gorinevsky, “Loop shaping for iterative control of batchprocesses,” IEEE Control Syst., vol. 22 (6), 2002.
J. van Zundert, J. Bolder, and T. Oomen, “Optimality and flexibility
in Iterative Learning Control for varying tasks,” Automatica, vol. 67,
2016.
J. Kon, N. de Vos, D. Bruijnen, J. van de Wijdeven, M. Heertjes,
and T. Oomen, “Learning for Precision Motion of an Interventional
X-ray System: Add-on Physics-Guided Neural Network Feedforward
Control,” in 22nd IFAC World Congr., 2023.
M. Tomizuka, “Zero Phase Error Tracking Algorithm for Digital
Control,” ASME. J. Dyn. Sys., Meas., Control, vol. 109 (1), 1987.