1

Rational Basis Functions in Iterative Learning Control- With Experimental Verification on a Motion System

Joost Bolder and Tom Oomen

Abstract—Iterative Learning Control (ILC) approaches oftenexhibit poor extrapolation properties with respect to exogenoussignals, such as setpoint variations. This paper introduces rationalbasis functions in ILC. Such rational basis functions havethe potential to both increase performance and enhance theextrapolation properties. The key difficulty that is associatedwith these rational basis functions lies in a significantly morecomplex optimization problem when compared with using pre-existing polynomial basis functions. In this paper, a new iterativeoptimization algorithm is proposed that enables the use ofrational basis functions in ILC for single-input single-outputsystems. An experimental case study confirms the advantagesof rational basis functions compared to pre-existing results, aswell as the effectiveness of the proposed iterative algorithm.

Index Terms—ILC, basis functions, optimal control.

I. INTRODUCTION

LEARNING control is used in many motion systems.Examples include: additive manufacturing machines [1],

[2], robotic arms [3], printing systems [4], pick and placemachines, electron microscopes, and wafer stages [5]–[7]. Theoften repetitive tasks for these systems typically vary to somedegree to address tolerances in the products being processed.

Iterative Learning Control (ILC) [8] can significantly en-hance the performance of systems that perform repeatedtasks. After each repetition the command signal for the nextrepetition is updated by learning from past executions. Akey assumption in ILC is that the task of the system isinvariant under the repetitions. As a consequence, the learnedcommand signal is optimal for the specific task only. Ingeneral, extrapolation of the learned command signal to othertasks leads to a significant performance deterioration [6].

Several approaches have been proposed to enhance theextrapolation properties of ILC to a class of reference signals.In [5] a segmented approach to ILC is presented and applied toa wafer stage. This approach is further extended in [2], wherethe complete task is divided into subtasks that are learnedindividually. The use of such a signal library is restricting inthe sense that tasks are required to consist of standardizedbuilding blocks. Instead of using a signal library, in [9]–[11],the extrapolation properties of ILC are enhanced through theuse of basis functions. These basis functions can be used toparameterize the ILC command signal in terms of the task.

The pre-existing results [4], [6] employ so-called polyno-mial basis functions. These polynomial basis functions can beinterpreted as parameterizing the command signal in terms of

The authors are with the Eindhoven University of Technology, Dept. ofMechanical Engineering, Control Systems Technology group, P.O. Box 513,5600 MB Eindhoven, The Netherlands, j.j.bolder@tue.nl

the reference using a Finite Impulse Response (FIR) filter. Im-portantly, such polynomial basis functions retain the analyticsolution of the ILC as obtained in [8]. In [6], the polynomialbasis functions in [12] are implemented in the ILC frameworkof [9] and successfully applied to an industrial wafer stagesystem, whereas in [4] an application to a wide-format printerapplication is reported. Finally, extensions of the approachtowards input shaping are presented in [13].

Although the use of polynomial basis functions enhancesthe extrapolation properties of ILC algorithms, the polynomialnature of the basis functions severely limits the achievableperformance and extrapolation properties. The basis functionstypically constitute an approximate model inverse of the truesystem [14], [15]. The use of polynomial basis functionsimplies that a perfect inverse can be obtained only if thesystem has a unit numerator. Since many physical systemsare modeled using rational models, containing both polesand zeros, this implies that existing results necessarily leadto undermodeling in the model inverses. Consequently, boththe achievable performance and extrapolation properties arelimited.

This paper aims to introduce a new ILC framework that canachieve improved performance and extrapolation properties forthe class of single-input single-output rational systems. To thisend, this paper introduces rational basis functions in ILC.The key feature is that both the numerator and denominatorof the rational structure are parameterized, hence allowingboth zeros and poles in obtaining the model inverse. Thetechnical difficulty associated with these basis functions isthat the analytic solution of standard optimal ILC [8] and thebasis function approach in [9] is lost. In fact, the resultingoptimization problem is non-convex, in general.

The main and novel contribution of this paper is theintroduction of rational basis functions in iterative learningcontrol, and a new parameter update solution that resortsto a sequence of optimization problems with an analyticsolution. Interestingly, the results in [9], [6], [13] are directlyrecovered as a special case of the novel results. The proposedsolution has strong connections to common algorithms in bothtime domain system identification [16] and frequency domainsystem identification [17], [18].

The notation that is used in this paper is introduced in thenext section. In Section III, the problem formulation is for-mally stated. Then, in Section IV, the new parameterization isproposed, followed by an analysis of its consequences for theoptimization problem (Section IV-A). Section IV-B contains anovel iterative solution to the optimization problem, whichconstitutes the main contribution of this paper. Section Vestablishes connections to pre-existing results that employ

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basis functions in ILC. In Section VI, an experimental casestudy is presented that reveals the advantages of employingrational basis functions and efficacy of the proposed iterativesolution.

II. PRELIMINARIES

A discrete-time transfer function is denoted as H(z), withz a complex indeterminate. The ith element of a vector θ isexpressed as θ[i]. A matrix B ∈ Rn×n is defined positive(semi-)definite iff xTBx ≥ 0,∀x 6= 0 ∈ Rn and is denotedas B � 0. For a vector x, the weighted 2-norm is ||x||W =xTWx.

All signals and systems are discrete time and often implic-itly assumed of length n. Given a system H(z), and input andoutput vectors u, y ∈ Rn×1. Let h(t), t ∈ Z be the infinite-time impulse response vector of H(z). Then, the finite-timeresponse of H to u is given by the truncated convolution

y[t] =

t∑l=1−n

h(l)u[t− l],

with 0 ≤ t < n, and zero initial conditions. This finite-timeconvolution is written as:

y[0]y[1]

...y[n− 1]

︸ ︷︷ ︸

y

=

h(0) h(−1) ... h(1−n)

h(1) h(0) ... h(2−n)

.... . .

...h(n−1) h(n−2) ... h(0)

︸ ︷︷ ︸

H

u[0]u[1]

...u[n− 1]

︸ ︷︷ ︸

u

,

with H the convolution matrix corresponding to H(z), andu, y the input and output vectors. Note that H(z) is notrestricted to be a causal system.

Given a transfer function with parametric coefficients:

H(θ, z) =

m∑i=1

ξi(z)θ[i],

with parameters θ ∈ Rm×1, and basis functions ξi(z), herei = 1, 2, ... ,m. The finite-time response of H to input u isgiven by

y = ΨHuθ (1)

with ΨHu = [ξ1u, ξ2u, ... , ξmu] ∈ Rn×m, here ξi ∈ Rn×n arethe convolution matrices corresponding to ξi(z). Note that (1)is equivalent to y = H(θ)u, with H(θ) the convolution matrixof H(θ, z).

As an example, let n = 4, ξ1(z) = 1 and ξ2(z) = z−1,then H(θ, z) = θ[1] + z−1θ[2], and accordingly:

H(θ) =

θ[1] 0 0 0θ[2] θ[1] 0 00 θ[2] θ[1] 00 0 θ[2] θ[1]

,ΨHu =

u(1) 0u(2) u(1)u(3) u(2)u(4) u(3)

III. PROBLEM FORMULATION

In this section, the problem addressed in this paper isdefined in detail. First, in Section III-A, the general ILC setupis introduced. Then, in Section III-B, optimization-based ILCis introduced. This is further tailored towards polynomial basisfunctions in Section III-C, followed by a definition of theproblem that is addressed in the present paper.

C P0r ej

uj

yj

−

fj

Fig. 1. ILC setup.

A. Problem setup

The considered ILC setup is shown in Fig. 1. The setupconsists of a feedback controller C, and system P0. Both areassumed linear time invariant (LTI), causal, and single-inputsingle-output. During an experiment with index j and lengthn, the reference r and system output yj are measured. Thefeedforward signal is denoted fj . Note from Fig. 1 that

ej = S0r − P0S0fj , (2)

with S0 := (I + CP0)−1 the sensitivity. In ILC, the feedfor-ward is generated by learning from measured data of previousexperiments, also called trials. The objective is to minimizeej+1, i.e., the predicted tracking error for the next experiment.From (2), it follows that

ej+1 = S0r − P0S0fj+1. (3)

Since r is constant, S0r is eliminated from (2) and (3), yieldingthe error propagation from trial j to trial j + 1:

ej+1 = ej − P0S0(fj+1 − fj). (4)

B. Norm-optimal ILC

Norm-optimal ILC is an important class of ILC algorithms,where fj+1 is determined from the solution of an optimizationproblem, see [19], [20], and [8]. Further extension, namelyconstrained optimization is considered in [21], [22], whichcan for instance be used to prevent actuator saturation.

The optimization criterion in norm-optimal ILC is typicallydefined as follows.

Definition 1 (Norm-optimal ILC). The optimization criterionfor norm-optimal ILC algorithms is given by

J (fj+1) := ||ej+1||We+ ||fj+1||Wf

+ ||fj+1 − fj ||W∆f, (5)

with We � 0, and Wf ,W∆f � 0.

In (5), We � 0, and Wf ,W∆f � 0 are user-defined weight-ing matrices to specify performance and robustness objectives,including: i) robustness with respect to model uncertainty(Wf ), and ii) convergence speed and sensitivity to trial varyingdisturbances (W∆f ). The corresponding feedforward update isgiven by

fj+1 = arg minfj+1

J (fj+1). (6)

The solution to (6) can be computed analytically from mea-surements ej and fj , given a model PS, since (5) is a quadraticfunction in fj+1. The advantage of using a model in ILCin comparison to model-based feedback approaches is the

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significant performance improvements enabled by non-causalfiltering operations in the time-domain as explained in [23].

In view of (3), the norm-optimal ILC computes a commandsignal fj that is optimal in (5) for a specific referencetrajectory r. As a result, changing r implies that the commandsignal fj is not optimal in general. To introduce extrapolationcapabilities in ILC, basis functions are introduced in the nextsection.

C. Norm-optimal ILC with polynomial basis functions

In [9], basis functions have been introduced in ILC that areof the form

fj = Ψθj , (7)

where fj is a linear combination of user-selected vectors Ψ =[ψ1, ψ2, ... , ψm]. Notice that the basis functions Ψ in (7) are inso-called lifted notation. The basis functions in (7) encompassstandard norm-optimal ILC with Ψ = I . Only specific choicesenhance the extrapolation properties. The essence of enhancingextrapolation of the ILC command signal to different taskslies in choosing fj to be a function of r. Therefore, let fj =F (θj)r. Subsequent substitution into (2) yields

ej = S0r − P0S0F (θj)r (8)= (I − P0F (θj))S0r.

Equation (8) reveals that if the feedforward is parameterizedin terms of the reference r, then the error in (8) can be madeinvariant under the choice of r, given that F (θj) is selectedas F (θj) = P−1

0 .In order to retain the analytic solution to (6), the filter F (θj)

is typically chosen as a polynomial function that is linear inθj , e.g., a FIR filter, see [6], [7]. Hence, for this particularchoice, the basis functions are referred to as polynomial basisfunctions.

Consequently, F (θj) = P−10 , can only be achieved if P0

is restricted to be a rational function with a unit numerator,i.e., no zeros. In case this condition is violated, the achievableperformance and extrapolation properties of ILC are severelydeteriorated. Since typical physical systems are modeled usingrational models that contain both poles and zeros, a unitnumerator imposes a significant restriction.

D. Paper contribution: ILC with rational basis functions forenhancing performance and extrapolation properties

In view of the limitations imposed by the polynomial basisfunctions in Section III-C, this paper aims to investigate moregeneral parameterizations that enhance: i) tracking perfor-mance, and ii) extrapolation properties of the learned feed-forward command signal. This paper contains the followingcontributions:

I. general rational basis functions are proposed,II. the consequences of a more general rational param-

eterization on the resulting ILC optimization problemare investigated, revealing a significantly more complexoptimization problem,

C P0

rej uj

yj

−

Learning Update θjfj

F (θj)

ucj

Fig. 2. Controller structure with rational basis

III. a solution strategy is proposed to deal with the moredifficult optimization problem that is introduced by thegeneral rational parameterization,

IV. the results are experimentally validated on a benchmarkmotion system and compared with pre-existing results.

Preliminary research related to I and III appeared in [24]. Thepresent paper extends these initial findings with more theoryand explanations, and includes an experimental validation.

IV. A NEW FRAMEWORK FOR ITERATIVE LEARNINGCONTROL WITH RATIONAL BASIS FUNCTIONS

In this section, the main contribution of this paper is pre-sented: the formulation, analysis, and synthesis of an optimalILC with rational basis functions, i.e., aspects I, II, and III inSection III-D. As is argued in Section III-C, the motivationfor using such basis functions stems from (8), which revealsthat parameterizing the feedforward command signal in termsof the reference signal enables extrapolation of the learnedfeedforward command signal to other reference trajectories.

Definition 2 (Rational basis for optimal ILC). The rationalbasis functions are defined as

fj = F (θj)r, (9)

where F ∈ F ,

F ={B(θj)

−1A(θj)∣∣θj ∈ Rma+mb

}, (10)

and

A(θj) =

ma∑i=1

ξAi θj [i],

B(θj) = I +

mb∑i=1

ξBi θj [i+ma].

Here, ξAi and ξBi are the convolution matrices correspond-ing to user-chosen polynomial transfer functions ξAi (z) andξBi (z), respectively. The matrix B(θj)

−1 is the convolutionmatrix corresponding to B(θj , z)

−1 with B(θj , z) = 1 +∑mb

i=1 ξBi (z)θj [i+ma]. The parameters θj = [θAj , θ

Bj ]T .

The ILC command fj for ILC with rational basis functionsin Definition 2 is implemented in the ILC setup of Fig. 1, seeFig. 2 for the resulting block diagram.

Remark 1. The underlying transfer function F (θj , z) ofF (θj) can be computed and analyzed in the frequency do-main, e.g., by its frequency response function F (θj , e

iω),

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using: A(θj , z) =∑ma

i=1 ξAi (z)θj [i], B(θj , z) = 1 +∑mb

i=1 ξBi (z)θj [i+ma], and F (θj , z) = B(θj , z)

−1A(θj , z).

Remark 2. The classical ILC with polynomial basis functionsapproach, see Section III-C and [6], [7], is recovered bysetting mb = 0. Indeed, this leads to B(θj) = I , and henceF (θj) = A(θj) with A linear in θj .

A. Analysis of the resulting optimization problem

Aspect II in Section III-D is elaborated on in this section.The difficulty associated with the rational basis functionparameterization (10) involves the complexity of solving thecorresponding optimization problem. In fact, the rational basis(10) in general prevents an analytic solution to (6). This isrevealed by the following theorem.

Theorem 3. Let Wf = W∆f = 0 and consider the parame-terization (10) and (9). Then, J (fj+1), see (5), is nonlinearin θj+1.

Proof: Substitution of (10) and (9) into (4) yields

ej+1 =ej + P0S0fj −B(θj+1)−1A(θj+1)P0S0r,

substituting ej+1 in (5)

J (θj+1) =

eTj Weej + fTj ST0 P

T0 WeP0S0fj+

rTST0 PT0 B(θj+1)−TA(θj+1)WeB(θj+1)−1A(θj+1)P0S0r+

2eTj WeP0S0fj − 2eTj WeB(θj+1)−1A(θj+1)P0S0r−2fTj S

T0 P

T0 WeB(θj+1)−1A(θj+1)P0S0r.

(11)

Theorem 3 reveals that B(θj+1)−1 in (11) leads to a perfor-mance criterion (5) that is nonlinear in the parameters θj+1.As a result, no analytic solution is available in general andthe performance criterion is typically non-convex in θj+1. Inthe next section, an iterative solution is proposed to calculateθj+1, constituting contribution III, see Section III-D.

B. Synthesis of optimal iterative learning controllers withrational basis functions

In this section, an ILC algorithm is developed that enablesoptimal controller synthesis using rational basis functions. Themain idea is to solve a sequence of least-squares problems andto consider the nonlinear terms as a priori unknown weightingfunctions. The basic concept is to recast (5) to:

J (θj+1) =∥∥B(θj+1)−1[B(θj+1)ej+1]

∥∥We

+ (12)∥∥B(θj+1)−1[B(θj+1)fj+1]∥∥Wf

+∥∥B(θj+1)−1[B(θj+1)fj+1]− fj∥∥W∆f

.

In (12), J (θj+1) is nonlinear in θj+1 due to the termB(θj+1)−1. However, J (θj+1) is linear in θj+1 in the termsB(θj+1)ej+1 and B(θj+1)fj+1. In view of this distinction,an auxiliary index k is introduced, i.e., θ<k>j+1 and θ<k−1>

j+1 .

These are substituted into (12), yielding

Jk(θ<k>j+1 ) =∥∥B(θ<k−1>

j+1 )−1[B(θ<k>j+1 )e<k>j+1

]∥∥We

+∥∥B(θ<k−1>j+1 )−1

[B(θ<k>j+1 )f<k>j+1

]∥∥Wf

+∥∥B(θ<k−1>j+1 )−1

[B(θ<k>j+1 )f<k>j+1

]− fj

∥∥W∆f

,

(13)

where

e<k>j+1 =ej + P0S0fj −B(θ<k>j+1 )−1A(θ<k>j+1 )P0S0r.

Notice that (12) is recovered by setting θj+1 = θ<k>j+1 =

θ<k−1>j+1 . In addition, notice that if θ<k−1>

j+1 is known, thenJk(θ<k>j+1 ) is a quadratic function of θ<k>j+1 . Consequently,θ<k>j+1 can be calculated analytically. The basic principle isto fix the nonlinear B(θ<k−1>

j+1 )−1 at iteration k and interpretit as an iteratively adjusted weighting function. By iteratingover θ<k>j+1 , it is aimed that the a priori unknown weighting byB(θ<k−1>

j+1 )−1 is effectively compensated after convergence ofthe iterative procedure. Clearly, this necessitates a solution to(13) for θ<k>j+1 , given θ<k−1>

j+1 . The following theorem providesthe analytic solution to θ<k>j+1 that minimizes Jk(θ<k>j+1 ).

Theorem 4. Given θ<k−1>j+1 , fj and ej . Then Jk(θ<k>j+1 ), see

(13), is minimized by

θ<k>j+1 = L<k>ej +Q<k>fj , (14)

with

L<k> =[ΨT

1 WeΨ1 + ΨT2 (Wf+

W∆f )Ψ2]−1

ΨT1 WeB(θ<k−1>

j+1 )−1

Q<k> =[ΨT

1 WeΨ1 + ΨT2 (Wf +W∆f )Ψ2

]−1 (ΨT

2 W∆f+

ΨT1 WeB(θ<k−1>

j+1 )−1PS)

where

Ψ1 = B(θ<k−1>j+1 )−1

[ΨAPSr, −ΨB

ej −ΨBPSfj

],

Ψ2 = B(θ<k−1>j+1 )−1

[ΨAr , 0

].

Proof: Note that (13) is quadratic in θ<k>j+1 . A necessarycondition for optimality is ∂Jk

∂θ<k>j+1

= 0. Solving this linear

equation for θ<k>j+1 yields the parameter update in (14).Note that Theorem 4 in itself does not lead to the optimal

solution of (6) in general, since B(θ<k−1>j+1 )−1 is unknown.

The proposed solution is to iteratively solve for θ<k>j+1 in(13), given an θ<k=0>

j+1 , for increasing k. In this approach,B(θ<k−1>

j+1 )−1 can be interpreted as an a priori unknownweighting in the cost function. This weighting is compensatedfor during each iteration over k by updating L<k> and Q<k>.These steps are formulated in the following parameter updatealgorithm that addresses aspect III, see Section III-D.

Algorithm 5. Given fj and ej , set k = 0 and initializeθ<k−1>j+1 = θj . Then, perform the following sequence of steps.1) Determine L<k>, Q<k>.2) Determine θ<k+1>

j+1 = Q<k>fj + L<k>ej .3) Set k → k + 1 and go back to (1) until an appropriate

convergence condition is met: θ<k→∞>j+1 = θj+1.

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Theorem 4 and Algorithm 5 provide a new solution andalgorithm to minimize J (θj+1) in (12), constituting the mainresult of this paper. The key novelty of these results lies intheir use in optimal ILC algorithms. Indeed, related algorithms,see [16], [17], and [18] are successfully used in system iden-tification. Despite the fact that the objective function is non-convex in general, practical use has revealed good convergenceproperties and in fact global convergence has been establishedfor specific cases, e.g., see [25] and [26].

Remark 3. In case Theorem 4 involves a non-minimumphase B(θ<k−1>

j+1 , z) and hence unstable B(θ<k−1>j+1 , z)−1,

the filtering operations cannot be performed in the usualmanner, since time domain computation leads to unboundedresults. Several approaches to calculate the filtered signals canbe pursued, including: i) approximations, e.g., see [15], [27],[28], and ii) exact methods, for instance, the stable inversionapproach in [14]. In the latter, the filter is seen as a non-causaloperator instead of an unstable one, see also [29, Section 1.5].

V. CONNECTIONS TO PRE-EXISTING APPROACHES

As mentioned throughout the preceding sections, specificbasis choices can be recovered as special cases, including theresults in [6], [7], and [13]. In this section, these specificparameterizations are compared with respect to the unifiedframework and solution as proposed in the present paper.

A. FIR structure

In [6], a polynomial basis F (θj) is used that is linear inthe parameters θj . Given its close resemblance to the well-known FIR basis, the parameterization is referred to as FIRparameterization. This FIR parameterization connects to therational basis functions in Definition 2 by setting mb = 0 andma = m. As a result, B(θj) = I . Furthermore, in [6], We = Iand Wf = W∆f = 0. Finally, let ξ(z) = 1 − z−1, then thebasis functions used are ξA1 = ξ, ξA2 = ξ2, ..., ξAm = ξm.

B. Extended FIR

In [13], a more general polynomial basis is presented thatextends the FIR parameterization in [6]. By exploiting thecommutativity property for SISO LTI systems, the frameworkin [13] can be recast in the form of Fig. 2 by selecting:We = I,Wf = W∆f = 0; let ξ(z) = 1 − z−1, then thebasis functions used are: ξA1 = ξ, ξA2 = ξ2, ..., ξAma

= ξma ,ξB1 = ξ, ξB2 = ξ2, ..., ξBmb

= ξmb , and B(θj+1)−1 := I .This approach coincides with Algorithm 5, but without step3. Hence, there is no compensation of the a priori un-known weighting function. As a result, an a priori unknownweighting B(θj+1) is introduced in the performance criterionJERR(θj+1) = ‖[B(θj+1)ej+1]‖We

.

VI. EXPERIMENTAL RESULTS

In this section, the proposed algorithm is experimentallydemonstrated, revealing the increased performance and extrap-olation properties in comparison with pre-existing results. Inparticular, the following approaches are compared:

Fig. 3. Two-mass spring motion system: optical encoder (a), motor (b),mass-spring-mass (c), optical encoder (d).

1) the proposed norm-optimal ILC with rational basis func-tions,

2) the pre-existing norm-optimal ILC with polynomial basisfunctions (FIR), see, e.g., [6], and

3) standard norm-optimal ILC.

A. Experimental Setup

The experimental two-mass spring motion system is pre-sented in Fig. 3. The system consists of a current-controlledDC-motor (Fig. 3b) driving a mass (inertia) m1 that isconnected to mass (inertia) m2 (Fig. 3) via a flexible shaft(Fig. 3c). The positions of m1 and m2 are measured by opticalencoders (Fig. 3a and d). This system is well-suited to use asa benchmark system in order to examine prototype controlalgorithms [30]. In the results presented in this paper onlythe position measurement of m1 is used for control, and theposition measurement of m2 is ignored.

B. Iterative learning controller design

Step 1, system identification: open-loop system identifi-cation is performed to identify the system P0. The systemis excited with random phased multisines at a samplingfrequency fs of 1 kHz. Both a parametric (P (z)) and a non-parametric model (Pfrf) is estimated using the measurementdata. The corresponding Bode diagrams are shown in Fig. 4,accompanied with the 3σ (99.7%) confidence interval of themeasured frequency response function Pfrf.Step 2, basis functions and weighting matrices: The se-lection of basis functions ξAi , ξ

BI determines the model set

F (θ), see Definition 2. Ideally, the structure of A(θj) andB(θj) should include the structure of the true inverse-systemP−1

0 = B0(θ)−1A0(θ). Here, the structure of A(θj) andB(θj) is selected as follows: let ξ(z) = 1

Ts(1 − z−1) be

a differentiator, then ξA1 = ξ, ξA2 = ξ2, ..., ξAma= ξma and

ξB1 = ξ, ξB2 = ξ2, ..., ξBmb= ξmb . This structure ensures the

DC-gain A(θj , z = 1) = 0, and B(θj , z = 1) = 1, thelatter guarantees that the rational structure F is well definedfor all θ. This basis is expected to work well for all systemswith infinite gain at zero frequency. If desired, it can easily

6

Magnitude

[dB

]

−100

−50

0

50

Frequency [Hz]

Phase

[◦ ]

100 101 102

−180

−90

0

90

180

Fig. 4. Frequency response measurement Pfrf (solid black), 3σ confidenceinterval of Pfrf (shaded gray), model P (eiω) (dashed red).

be changed for systems with a finite DC-gain by adding aparameter such that A(θj , z = 1) = θj [1]. In the FIR case,this basis selection allows θAj to be directly interpreted as thefeedforward parameters compensating for effects related to:velocity, acceleration, jerk and snap, see [12]. The scaling inξ(z) with the sampling time Ts is to improve the numericalconditioning, and is related to the δ-operator approach in [31,Section 12.9].

Clearly, the optimal solution for the feedforward filter isgiven by F (θ) = P−1

0 , since this choice leads to minimalJ(θ). Using the identified model as a guideline, this choicecorresponds to ma = 6 and mb = 3. However, to enablea fair comparison, also for the proposed rational basis, arestricted complexity parameterization is pursued such thatunder-modeling is present. In particular:

• proposed ma = 4,mb = 2,• FIR ma = 4,mb = 0.

The rational filter is an extension of the FIR filter, wheretwo zeros are added by setting mb = 2. The selection of thenumber of parameters is similar to the problem of model orderselection in system identification, see [32]. On the one hand,additional parameters increase the size of the model-set andtherefore reduce bias, on the other hand, the variance on theparameters typically increases. Both aspects are a source oferror between F (θ) and P−1

0 , and as such, manipulating thistrade-off is part of the controller design. Notice that standardnorm-optimal ILC can be viewed as using a FIR structurewith ma = n and mb = 0 parameters in the ILC with basisfunctions framework. Hence, the basis functions affect theperformance only and do not affect the convergence speed.

The weighting matrices in Definition 1 specify the perfor-mance and robustness objectives. In the results for both theFIR and Rational structure presented in this paper We =I · 103,Wf = W∆f = 0. This leads to an inverse model ILC,where the inverses in Theorem 4 are well-defined for the par-ticular basis functions. For standard norm-optimal ILC, a smallweighting on the learning speed is introduced: W∆f = I ·0.05.

Magnitude

[dB

]

−100

−50

0

50

Frequency [Hz]

Phase

[◦ ]

100 101 102−180

−90

0

90

180

Fig. 5. Simulation results: P0(eiω) and F (θ∞, eiω)−1, P0(eiω) (solidgrey), proposed (solid black), FIR (dashed-dotted blue).

C. Preliminary simulation with a fixed referenceThe proposed rational structure is compared with the FIR

structure. The reference used is r1, depicted in Fig. 7. Thecorresponding ILC algorithms are invoked. To interpret theconverged feedforward, the parameterized feedforward filtersare visualized using a Bode diagram. The results are depictedin Fig. 5, where F (θ∞, e

iω)−1 for the FIR and rationalstructures, and P0(eiω) are compared. Fig. 5 reveals that forfrequencies up to 5 Hz the dynamics of P0 are captured well byboth approaches. The anti-resonance (i.e., complex conjugateszeros) around 16.3 Hz is only captured by the proposedapproach. The FIR structure does not have poles, hence itsinverse does not have zeros and cannot accurately representthe anti-resonance. Summarizing, from visual inspection itis concluded that for the proposed approach in this paperF (θ∞, e

iω)−1 has the closest resemblance with the systemP0(eiω). It is therefore expected that the proposed approachhas the best extrapolation capabilities if r changes, and thiswill be validated in the next section in an experimental testcase on the benchmark motion setup in Fig. 3.

D. Experimental resultsIn this section, an experimental case study is presented

where the extrapolation capabilities of the proposed ratio-nal and FIR feedforward parameterizations are verified andcompared with standard norm-optimal ILC. The model, basisfunctions, and weighting matrices obtained in the previoussection are used in the experiments. First, a preliminaryexperiment with reference r1 establishes correspondence withthe simulations followed by a case study on the extrapolationcapabilities of the different approaches.

1) Experiment with a fixed reference: 15 trials are per-formed. The Bode diagram of the converged parameterizedfeedforward filters is shown in Fig. 6. Here, F (θ14, e

iω)−1

for the FIR and rational structures and Pfrf are compared.A visual comparison with the simulation, see Fig. 5, revealssimilar results, except for the proposed approach, that showsa slightly better correspondence with Pfrf, in particular at theanti-resonance at 16.3 Hz.

7

Magnitude

[dB

]

−100

−50

0

50

Frequency [Hz]

Phase

[]

100 101 102−180

−90

0

90

180

◦

Fig. 6. Experimental results: Pfrf and F (θ14, eiω)−1, Pfrf (solid grey),proposed (solid black), FIR (dashed-dotted blue).

1.7 1.8 1.9 2 2.1 2.20

5

10

15

Time [s]

r[rad]

Fig. 7. The different references: r1 (solid black), r2 (dashed red), r3 (dashed-dotted blue).

2) Extrapolation of references: three 4th order polynomialreferences are defined, see Fig. 7, where:• r2 is equal to r1 with 0.01s delay• r3 has 5% extra distance with identical maximal velocity

as r1 and r2.In total, 15 trials are performed, where the reference is changedfrom r1 to r2 to r3 at trials 4 and 9, respectively. Thefeedforward signal, or parameter vector θj , is not re-initializedwhen changing the reference in order to demonstrate theextrapolation capabilities.

The results are presented in Fig. 8 and Fig. 9. The costfunction values, see Fig. 8, show that all approaches improveperformance compared to feedback only (f0 = 0). At j = 4and j = 9 the reference is changed without re-initializingthe feedforward signals. The key result is that both the FIRand proposed parameterization are insensitive to the change inthe task, in contrast to standard norm-optimal ILC, where thereference changes results in a large increase in cost functionvalue. The results in Fig. 8 also confirm that the proposed ra-tional feedforward parameterization leads to improved trackingperformance in comparison with the FIR structure. In addition,identical extrapolation capabilities are achieved.

The time domain tracking errors for trials 8 and 9 are shownin Fig. 9. These correspond with the time domain signals priorto the reference change r2 → r3 (Fig. 9, left column) andafter the change (Fig. 9, right column), respectively. Theseresults confirm earlier conclusions, since: i) the sensitivity of

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14101

102

103

104

105

106

Trial

J(θ

j)

Fig. 8. Cost function values J(θj): where the proposed approach (×) andthe pre-existing FIR approach (©) are insensitive to the reference changesr1 → r2 (black dashed) at j = 4 and from r2 → r3 at j = 9 (black dotted),in contrast to standard norm-optimal (4).

−0.3−0.2−0.1

00.10.20.3

Pro

pose

d

e8, [rad] e9, [rad]

−0.3−0.2−0.1

00.10.20.3

FIR

1.5 2 2.5−0.3−0.2−0.1

00.10.20.3

Norm

-optim

al

Time [s]1.5 2 2.5

Time [s]

Fig. 9. Comparison of time domain tracking errors: prior to reference change(left column, j = 8), and after the reference change (right column, j = 9).Proposed approach (top row, black), FIR (middle row, blue), norm-optimal(bottom row, red).

the tracking error when using standard norm-optimal ILC withrespect to a small, i.e., 5% reference change is demonstrated,ii) the tracking performance of the FIR and proposed param-eterizations are insensitive to the change in reference, and iii)the proposed approach outperforms the approach with the FIRstructure.

VII. CONCLUSION

In this paper, a novel framework for ILC with rational basisfunctions is presented. Herein, basis functions are adoptedto enhance the extrapolation properties of learned command

8

signals to other tasks. Indeed, in pre-existing approaches,polynomial basis functions are employed that are only optimalfor systems that have a unit numerator, i.e., no zeros.

The difficulty associated with rational basis functions liesin the synthesis of optimal iterative learning controllers, sincethe analytic solution to optimal ILC algorithms is lost. Inthis paper, an iterative algorithm is proposed that effectivelysolves the optimization problem. It has close connections towell-known and powerful iterative solution methods in systemidentification.

The advantages of using rational basis functions in ILCare confirmed in a relevant experimental study: i) improvedperformance with respect to pre-existing methods that addressextrapolation in ILC is demonstrated, and ii) the performanceis insensitive for the presented changes in the reference. Theconvergence aspects of the proposed method are experiencedto be good, as is also confirmed in studies on related algo-rithms in system identification [18].

Ongoing research is towards: extending the approach tomultiple-input multiple-output systems, investigating other so-lutions to the optimization problem, and addressing non-measurable performance variables, see [33] and [3].

ACKNOWLEDGMENT

The authors would like to acknowledge the fruitful discus-sions, guidance, and the contributions of the late professorOkko Bosgra and professor Maarten Steinbuch, both with theEindhoven University of Technology. In addition, Sjirk Koeke-bakker, Oce Technologies, is also gratefully acknowledged.

This work is supported by Oce Technologies, and by theInnovational Research Incentives Scheme under the VENIgrant “Precision Motion: Beyond the Nanometer” (no. 13073)awarded by NWO (The Netherlands Organisation for ScientificResearch) and STW (Dutch Science Foundation).

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