2024 European Control Conference (ECC)
June 25-28, 2024. Stockholm, Sweden

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) is capable of improving the tracking performance of repetitive control systems
by utilizing data from past iterations. The aim of this paper is toachieve both task flexibility, which is often achieved by ILC withbasis functions, and the performance of frequency-domain ILC,
with an intuitive design procedure. The cost function of normoptimal 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 are complimentary to each other. Validation on abenchmark 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 be flexible for different
references, are important. Feedforward control is effective
in compensating known disturbances for systems, leading to
improved performance. Feedforward control is often basedon models [1], which is generally achieved in industrial
applications by means of basis functions feedforward, where
the feedforward signal is a linear combination of basis
functions that relate to physical quantities, such as acceleration feedforward for the inertia [2]–[4]. Due to modelingand tuning inaccuracies, the increasing requirements for
performance are generally not achieved.
Iterative Learning Control (ILC) can improve tracking
performance with respect to model based feedforward control[5], and hence, can fulfill the increasing requirements forThis 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.
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.

Copyright ©2024 EUCA

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 [6]. Frequency-domain ILC has the advantage thatconvergence can be verified and tuned using frequency
response function (FRF), that are accurate and inexpensive toobtain [7], leading to an intuitive design procedure and hightracking performance. Frequency-domain ILC is typically
implemented in finite-time, where convergence can still beanalyzed [8]. However, conventional frequency-domain ILC
is not directly capable of task flexibility, and hence, does notsatisfy 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 basis functions ILC with the performanceof frequency-domain ILC. In [12], [13], frequency-domain
ILC is projected on basis functions, resulting in task flexibility, but reducing tracking performance. Third, in [14],frequency-domain ILC is combined with basis functions ILC
using a sequential optimization problem, that results in theperformance of frequency-domain ILC and the task flexibility of basis functions ILC, but resulting in an unintuitive
design procedure, not satisfying requirement R3.
Although ILC methods with high performance and task
flexibility are investigated, a method that achieves both highperformance and task flexibility, with an intuitive design
procedure, is currently lacking. In this paper, high performance (R2) is achieved by deliberately overparameterizing
the feedforward in a low number of basis functions toaccomplish task flexibility (R1) and a complementary signalthat is optimized via an intuitive frequency-domain ILC

1189

fj
rj

ej

B. Classes of ILC

ujK

In this section, the three considered classes of ILC, thatis norm-optimal ILC, basis functions ILC and frequencydomain ILC, are presented.
1) Norm-Optimal ILC: Norm-optimal ILC is a type of ILC
that minimizes a finite-time cost function, typicallyyj
PFig. 1: Control structure considered.

2

2

NO
fj+1

design (R3). The key contributions in this paper include thefollowing.
(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 for norm-optimal ILC(R3) (Section III-A).
(C2) Achieving both task flexibility (R1) and high performance (R2) by exploiting an overparameterized feedforward signal, building upon the norm-optimal descriptionof frequency-domain ILC from 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
x and matrices asPuppercase X. The z-transform of signal∞−1x(k) is X (z) =. Let h(k) ∀k ∈ Z be the
k=0 x(k)z
impulse response P
coefficients 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 ) h(N − 1) · · ·
|
{z
} |
{z
y

H

N

2

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





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

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.

∆f

,

(1)

where is used that ∥x∥2W = x⊤ W x, We , Wf and W∆f
are symmetric positive (semi)definite weighting matrices [9],êj+1 = ej − Jˆ (fj+1 − fj ), and finite-time convolution−1matrix Jˆ = P̂ I + K P̂, with model P̂ ∈ RN ×N and
identity matrix I ∈ RN ×N . The optimal solution to (1) is of
the form(2)
fj+1 = QN O fj + LN O ej ,with norm-optimal ILC robustness and learning matrices
QN O and LN O .
2) Basis Functions ILC: Basis functions ILC achieves reference flexibility by minimizing a cost function and parameterizing the feedforward signal in basis functions as
fj = ψθj ,(3)with feedforward parameters θj ∈ Rnθ ×1 and basis functionsψ ∈ RN ×nθ . For basis functions ILC, the cost function222

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

(4)

is minimized. The optimal solution to (4) is of the form
θj+1 = QBF θj + LBF ej ,(5)with basis functions ILC robustness and learning matrices
QBF 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)),

N ×Nwith y, u ∈ R and H ∈ R
is the finite-time convolution matrix corresponding to H(z)f(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 ),(7)with finite-time convolution matrices Qf and Lf , corresponding to Qf (z) and Lf (z).
C. Problem Definition

A. Problem Setup
The control structure is seen in Fig. 1. The LTI system P
is stabilized by LTI feedback controller K. The finite-timereference rj ∈ RN can be trial varying. The goal is to reduce
the reference induced error ej ∈ RN over multiple trials jwith the trial-varying feedforward signal fj ∈ RN .

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.

1190

y

k
u

106
2

m1

10-5

103

m2

1.5

100

d12

10-3

d2

10-6
0

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

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 ( ).

Basis functions ILC reduces performance if ψ does notaccurately describe the inverse system P −1 [13], [17].
• Frequency-domain ILC does have an intuitive design
procedure consisting of loop-shaping [5], [13] and results 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
details are given in Section IV. The error 2-norm during 300
trials of norm-optimal and frequency-domain ILC is shownin Fig. 3a, and the maximum error 2-norm during thesetrials and the steady state error 2-norm ∥e∞ ∥2 are shownin Fig. 3b. The results in Fig. 3 illustrate that for normoptimal ILC to reduce the steady-state error ∥e∞ ∥2 beyond
frequency-domain ILC, it first increases the error 2-norm atleast a factor 3 · 106 , which is unacceptable in industrial
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.

1010

100
10-6

10-5

Fig. 3: Illustration that norm-optimal ILC with We = I
and W∆f = 0 converges slowly and non-monotonically forinaccurate models, leading to significantly higher maximum
errors maxj (∥ej ∥2 ).

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)for a very specific choice of weighting matrices We , Wf
and W∆f . First, the frequency-domain ILC update is written
using the cost function
2

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
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 basis functions ILC can be jointly optimized. Finally,a procedure summarizes the developed method.

-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 fj = 0 ( ).

III. M ETHOD
In this section, the developed method is presented. The
overparameterized feedforward signal
 

 θj
fj = ΨΘj = ψ IN
= ψθj + fjf ,
(8)
fjf

5·10

f
min ∥êj+1 ∥Wef + fj+1
f
fj+1

2

f
+ fj+1
− fjf
f

Wf

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
f −1
,
Jˆ⊤ Wef Jˆ + W∆f)
Qf = (Jˆ⊤ Wef Jˆ + Wff + W∆f(11)
αQf Lf = (Jˆ⊤ W f Jˆ + W f + W f )−1 Jˆ⊤ W f .
(12)

1191

e

f

∆f

e

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

with
(13)

Because Wff must be symmetric positive semidefinite, but
the product of two symmetric matrices is not necessarilyfin (13) is chosen
symmetric, Jˆ⊤ Wef Jˆ + W∆f


f
= I.
(14)
Jˆ⊤ Wef Jˆ + W∆f
By substituting (14) and Wff from (13) into (12), thefollowing identity is foundαQf Lf = Qf Jˆ⊤ Wef ,(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 =−1
Jˆ in Theorem 1 are not restrictive, since the weighting
matrix Wef can be approximated as the symmetric matrix⊤
Wef = αLf Lf ,WθWθ,f =
0(17)

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.

2

2

Θj+1

2

+ ∥Θj+1 − Θj ∥W∆

(18)


0
,
f
W∆f

(19)

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.
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).

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
by utilizing the overparameterized feedforward signal in (8)fand preserving the weighting Wff and W∆fexclusively on
the frequency-domain component as
min V (Θj+1 ) = min ∥êj+1 ∥Wef + ∥Θj+1 ∥Wθ,f


W∆θ
W∆ =
0

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 inequivalent weighting as conventional basis functions ILC.
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) consistingof frequency-domain and basis functions ILC is naturally
complimentary and the parameterization will assign as muchinformation in the basis functions feedforward as possible.
Alternatively, a targeted regularization on the frequencydomain component can be done by using the image of ψ,
similarly to [18],2
min V (Θj+1 ) + λ 0 U1 Θj+1 2 ,B. Inclusion of Basis-Function in Frequency-Domain ILC

Θj+1


0
,
Wff

IV. S IMULATION E XAMPLE
In this section, the developed method is validated andcompared with frequency-domain and basis functions ILC.
First, the validation setup is shown, including the system and

1192

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

10 -3
6
4

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]

2
0
0

0.05

0.1

0.15

0.2

Fig. 5: The first ( ) and second ( ) references that are usedduring validation.

100
50

5

10 -6

0
-60

-50

0

-80

-100

-100

-150
10-2

10-1

100

101

102

-5
0

Fig. 4: FRF of the system P(ejω ) ( ) and of the modelb jω ) ( ).
available for ILC P(emodel, the reference and the ILC designs. Second, the normoptimal equivalent description of frequency-domain ILC isvalidated. Finally, the developed method with basis functions
is validated and compared for a trial-varying reference.
A. Simulation Setup and Approach
A two-mass-spring-damper system with one sample delay
and a sampling time of 1 ms, with inaccurate model, issimulated to validate the developed ILC technique. The
system is seen in Fig. 2, and the parameters of the true systemb
P(z) and the model P(z)are seen in Table I, and are givenby
2.80z −2 + 12.4z −3 − 0.65z −4 − 1.58z −5,
1 − 3.78z −1 + 5.46z −2 − 3.56z −3 + 0.89z −44.00z −2 + 21.4z −3 + 5.85z −4 − 1.25z −5bP(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
with sufficient robustness margins, and is given by
108.6 + 112.9z −1 − 100z −2 − 104.3z −3.
K(z) =
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
Qf (ejω )(1 − αJ (ejω )Lf (ejω )) < 1,∀ω ∈ [0, 2π].
(23)
Two different references are used to show task flexibility,
where both references are performed for 10 consecutive

0.05

0.1

0.15

0.2

Fig. 6: Tracking performance after 10 trials of ILC for
frequency-domain ILC ( ) and norm-optimal equivalent ( )using the first reference, showing the same error profile.

trials. The references are fourth-order polynomial references
designed using the approach in [3] with N = 229 samples
and are seen in Fig. 5.
a) Basis Function Design: The basis functions ψ arebbased on the inverse model of P(z)and are chosen as... ....r ,ψ = r̈ r(24)
where the derivatives of the reference are readily available
by design of the reference trajectory. Wθ and W∆θ in (19)are chosen as 0, since no additional robustness is necessary.
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.
Frequency-domain ILC and its norm-optimal equivalentachieve the same tracking error after 10 trials as seen in
Fig. 6, validating their equivalence.
B. Validation Results
The error for 20 trials of ILC is seen in Fig. 7 and Fig. 8.
The plant estimate using the basis function feedforward isseen in Fig. 9. The following observations are made.
• Fig. 7 shows frequency-domain ILC converges at trials
10 and 20, but its error increases at trial 11, showingthat it lacks task flexibility.
• Though basis functions ILC enables reference flexibility
as seen in Fig. 7, its higher error compared to the othermethods stems from lacking ṙ in its basis function ψ tocompensate the viscous friction d2 .
• The error at trial 20 in Fig. 7 and Fig. 8 demonstrate thedeveloped approach’s superior performance against bothbasis function and frequency-domain ILC. It surpasses1193

10 -2

10 -3
10 -4
10 -5
2

4

6

8

10

12

14

16

18

20

Fig. 7: 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 is changed ( ).
10 -5
5
0
-5
-10
0

0.05

0.1

0.15

0.2

Fig. 8: Tracking error after 20 trials e20 of frequency-domain
ILC ( ), basis function ILC ( ) and developed combined
frequency-domain and basis function ILC ( ).

basis functions by handling friction with the frequencydomain component, and outperforms frequency-domain
ILC by capturing high-frequency effects with ψ, thatfor frequency-domain ILC is filtered out by Qf (z).
• Similar error 2-norm to basis functions ILC underreference change illustrates the method’s task flexibility.
• From Fig. 9 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.
V. C ONCLUSIONSIn this paper, both task flexibility and performance isachieved through the use of an overparameterized feedforward signal consisting of frequency-domain and basis
functions ILC. The finite-time norm-optimal representation
100
50
0

-60

-50

-80

-100

-100

-150
10-2

40

80

10-1

120

100

101

102

of frequency-domain ILC is derived, that is consequently
used in overparameterizing the feedforward signal. The basis
functions and freqency-domain ILC components are complimentary by appropriately regularizing the frequency-domaincomponent. An example validates the equivalent normoptimal 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 in control.
Ongoing research is aimed at verifying the developed
method on an experimental setup and computationally efficient implementations.
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Fig. 9: FRFs of system P(ejω ) ( ) and of estimate using−1 jω
the inverse basis function feedforward model FBF(e , θj )( ), with finite-time representation FBF (θj )r = ψθj .
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