Robust stability of iterative learning control schemes

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust Nonlinear Control 2008; 18:1018–1033Published online 8 October 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/rnc.1236

Robust stability of iterative learning control schemes

Mark French*,y

School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, U.K.

SUMMARY

A notion of robust stability is developed for iterative learning control in the context of disturbanceattenuation. The size of the unmodelled dynamics is captured via a gap distance, which in turn is related tothe standard H2 gap metric, and the resulting robustness certificate is qualitatively equivalent to thatobtained in classical robustH1 theory. A bound on the robust stability margin for a specific adaptive ILCdesign is established. Copyright # 2007 John Wiley & Sons, Ltd.

Received 3 May 2007; Accepted 4 May 2007

KEY WORDS: iterative learning control; adaptive control; robust stability; gap metric

1. INTRODUCTION

The field of iterative learning control (known as ILC) is concerned with procedures for learningcontrol tasks (typically in the context of high precision tracking) over a number of ‘learningtrials’ of finite time duration, whereby the process is reset between trials, and the controller isaltered from trial to trial to improve performance. This area was motivated originally in thecontext of robotics [1], and has subsequently developed a more abstract footing in generalcontrol theory, and applications in a variety of domains have been considered: see for examplethe surveys [2, 3], and see [4, 5] for representative recent application examples. It is pertinent toobserve that both ILC and the related area of repetitive control have achieved considerablesuccess in applications, perhaps in contrast to the other major theory of learning in control,namely adaptive control.

Underpinning the practical success of any control design is the requirement for robustness. Itis surprising therefore that there is little theoretical understanding of robustness of iterativelearning schemes, particularly considering their successful application strongly suggests inherentrobustness properties. This should be tempered by the fact that in many ILC schemes ‘long-term

*Correspondence to: Mark French, School of Electronics and Computer Science, University of Southampton,Southampton SO17 1BJ, U.K.yE-mail: [email protected]

Copyright # 2007 John Wiley & Sons, Ltd.

stability’ problems have been observed in practical implementations and simulations, see e.g. [6].This phenomena result in divergence of the error profile after a large number of trials includinga period of apparent convergence, and has been attributed variously to the effects ofdisturbances, numerical errors, and/or un-modelled dynamics. Indeed, to the best knowledge ofthe author, studies of robustness of ILC are limited to plants with multiplicative perturbations.It is well known that in general multiplicative perturbations alone do not capture all the featuresof closed-loop uncertainties, for example, such perturbations can never describe the differencebetween an open-loop stable and an open-loop unstable plant, and such differences are notnecessarily significant for control (see e.g. [7]). In particular, [8], asserts robustness to positivereal multiplicative uncertainties (limited by �908 phase variations over all frequencies) and[9, 10] also consider multiplicative perturbations, and relate robust stability of an underlyingfeedback controller to the related ILC algorithm.

These earlier results are of great interest, but the conditions are too strong for practical use.The aim of this paper is to demonstrate that for a class of ILC designs, provable robustnessproperties hold. In particular, the class of uncertainties permitted are the classical co-primefactor uncertainty structures studied in robust linear control, and the size of the robustnessmargin is computed.

In the context of linear systems, the classical robust stability margin bP;C is defined to be themaximum radius of a ball centred on P in the H2 gap metric for which the controller C isguaranteed to stabilize all plants P1 within [7, 11]. The standard H2 gap dðP;P1Þ measures thesize of the smallest stable co-prime factor perturbation between normalized co-prime factorrepresentations of a pair of plants P and P1: This paper develops a similar framework for theanalysis of iterative learning schemes, and constructs a robustness margin for a particular ILCdesign.

In particular, we consider the classical class of relative degree one linear systems with knownhigh frequency gain, coupled with an adaptive ILC scheme (see e.g. [12, 13]). The control task isstabilization, i.e. tracking of the zero output trajectory, and is to be achieved in the face of Lp

input and output disturbances acting over all the trials. The resulting robust stability marginBP;CðrÞ is shown to be strictly positive, and dependent on the size of the disturbances r > 0: Thatis, the objective of tracking zero can be realized asymptotically for all plants within the robuststability margin. The resulting margin can be interpreted relative to the standard H2 gap [7, 11].

At present, the analysis is restricted to stabilization problems, namely the tracking of the zerotrajectory. Conventional ILC designs aim to achieve high precision tracking of more generaltrajectories.z The extension of the current robustness results to tracking is the topic of currentresearch; however, we observe that we note three reasons for the interest in the stabilizationspecifications considered:

1. Tracking of the zero trajectory is a particular case of the general tracking problem,therefore it is necessary that good robustness results can be achieved in this special case asa precursor to more general tracking results.

2. The problem studied is actually that of disturbance attenuation: stabilization has to beachieved in the presence of input and output disturbances and gain function bounds aregiven; this a legitimate ILC problem in its own right.

zThe ILC designs considered also have tracking properties (when the output disturbance is taken to be the reference),but we do not consider this point further in this paper.

ROBUST STABILITY OF ITERATIVE LEARNING CONTROL SCHEMES 1019

Copyright # 2007 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2008; 18:1018–1033

DOI: 10.1002/rnc

3. The analogous adaptive control problem has been extensively studied, and gap robustnessresults are available [14, 15]. A comparison of the nature of robust stability margin betweenthe adaptive and the ILC settings shows the importance of the stabilizing effect of re-initializing the process at the start of every ILC trial. In the adaptive problem, the statemaintains a memory of the past which builds up and makes a large contribution toreducing the size of the robustness margin.

Thus, we will establish provable conditions under which an iterative learning scheme works withno long-term stability problems in the presence of a realistic class of un-modelled dynamics anddisturbances.

2. STATEMENT OF THE ROBUST ITERATIVE LEARNING PROBLEMAND MAIN RESULT

We let U; Y denote normed signal spaces corresponding, respectively, to the input and outputsignals. Specifically, all signal spaces considered are taken to be R valued Lp½T� spaces definedover one of the following three domains: T ¼ ½0;T �;Rþ;N� ½0;T �: The first two domains havea natural total ordering, whilst the 2D domain is totally ordered as follows: ðk1; t1Þ ¼ t15t2 ¼ðk2; t2Þ if and only if k15k2 or if k1 ¼ k2 and t15t2: A truncation operator is defined as follows:

TtðvÞ ¼vðtÞ; t5t

0; t5t

(

The plant and controller operators S; X will be assumed to be causal, in the sense that ifTtu ¼ Ttv then TtSu ¼ TtSv and TtXu ¼ TtXv: Given a signal space U (resp. Y), the extendedspace Ue (resp. Ye), is defined

Ue ¼ fu 2 mapðT;RÞjjjTtujjU51 8t > 0g ð1Þ

where mapðT;RÞ denotes the set of all functions T! R:Throughout this paper, we consider feedback loop ½S;X� given by the equations:

½S;X�:y1 ¼ Su1; y0 ¼ y1 þ y2

u2 ¼ Xy2; u0 ¼ u1 þ u2

where u0; u1; u2 2 Ue; y0; y1; y2 2 Ye and S : Ue ! Ye; X : Ye ! Ue for appropriate signalspaces U;Y; see Figure 1. Here ðu0; y0Þ 2We; (where W ¼ U�Y), are the external signals (ordisturbances), and u1; u2; y1; y2 are the internal signals.

It is implicit in this definition that the feedback loop is globally well posed, namely thatif the external signals ðu0; y0Þ lie in the extended space We; then the internal signals arealso locally bounded, that is u1; u2 2 Ue; y1; y2 2 Ye: Equivalently, the following operator isdefined:

PS;X :We !We :u0

y0

!/

u1

y1

!ð2Þ

since it is straightforward to observe that if u0; u1 2 Ue; y0; y1 2 Ye then u2 2 Ue; y2 2 Ye: Notethat this global well-posedness property (which is equivalent to excluding finite escape times),will be verified for all the concrete cases considered later in the paper.

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The closed loop ½S;X� is said to be stable if PS;XðWÞ �W; that is the bounded externalsignals imply bounded internal plant signals u1 2 U; y1 2 Y (and consequently bounded internalcontroller signals u2 2 U; y2 2 Y; similar to the previous observation).

The graph of an plant operator S : Ue! Ye is the collection of all bounded input, outputpairs compatible with the plant equations:

GWS ¼ fðu1; y1Þ

T2 Ue �Ye : y1 ¼ Su1; u1 2 U; y1 2 Yg ð3Þ

Similarly, the controller operator graph is defined as follows:

GWX ¼ fðu2; y2Þ

T2 Ue �Ye : u2 ¼ Xy2; u2 2 U; y2 2 Yg ð4Þ

Stability of ½S;X� is then equivalent to the relation W ¼ GWS � GW

X : Finally, we observe that if Sis stabilizable in the sense that for all u 2 Lp½0;T � there exists v 2 Lp½Rþ� such that vj½0;T � ¼ u andðv;SvÞT 2 G

Lp½Rþ�S ; then

GLp½0;T �S ¼ G

Lp½Rþ�S j½0;T � ð5Þ

2.1. Plant dynamics along the trial

We first consider the case where U ¼ Lp½0;T �; Y ¼ Lp½0;T �: The nominal plant

S : Lpe ½0;T � ! Lp

e ½0;T � ð6Þ

is assumed to be linear, time invariant, single input, single output, relative degree one and wherethe sign of the high frequency gain b is known. Such a plant can be expressed in the followingform:

d

dt

y1

z

!¼�a11 a12

a21 A22

" #y1

z

!þ

b

0

!u1; xð0Þ ¼

y1ð0Þ

zð0Þ

" #¼ 0

Sðu1Þ ¼ y1

where a11 2 R; a12; a21 2 Rn�1; A22 2 Rðn�1Þ�ðn�1Þ; and without loss of generality b ¼ 1 2 R: Thefollowing property for S is critical to the analysis that follows, namely that there exists aconstant MT > 0 such that

jja12TTzjj4MT jjTTy1jj ð7Þ

The set of all such plants is denoted by M1MT: Note that if S is minimum phase then M can be

chosen to be independent of T > 0:A second plant operator S1 : Lp

e ½0;T � ! Lpe ½0;T � will also be considered. This represents the

‘actual’ plant (thus containing the unmodelled dynamics), and is assumed to be linear, timeinvariant and single input, single output.

Figure 1. The closed-loop system ½S;X�:



DOI: 10.1002/rnc

2.2. Dynamics over the trials

The iterative learning problem consists of applying the control repeatedly over a number oftrials, each of length T ; with the aim of improving performance as the trial progress. The signalspaces are taken to be U ¼ Y ¼ LpðN� ½0;T �Þ; with norm:

jjxð� ; �Þjj ¼X1i¼1

jjxði; �ÞjjpLp½0;T �

!1=p

ð8Þ

The first argument is interpreted as the trial index, the second the time elapsed along the trial.The nominal and actual plants are unchanged between the trials:

PS : LpeðN� ½0;T �Þ ! Lp

eðN� ½0;T �Þ : u1/y1; y1ðk; �Þ ¼ Su1ðk; �Þ ð9Þ

PS1: Lp

eðN� ½0;T �Þ ! LpeðN� ½0;T �Þ : u1/y1; y1ðk; �Þ ¼ S1u1ðk; �Þ ð10Þ

where S and S1 are defined in Section 2.1.The controller Cðk; tÞ : Lp

eðN� ½0;T �Þ ! LpeðN� ½0;T �Þ will represent the map y2/u2 and is

required to be causal.The specific iterative learning controller considered in this paper is defined as follows:

Cy0 : LpeðN� ½0;T �Þ ! Lp

eðN� ½0;T �Þ : y2/u2

u2ðk; �Þ ¼ �yky2ðk; �Þ

yk ¼ yk�1 þ ajjy2ðk� 1; �ÞjjpLp½0;T �

y0 2 ð0;1Þ; y2ð0; �Þ ¼ 0 ð11Þ

The learning gain a > 0 is taken to be fixed and positive throughout the paper. This controllerforms a simple variant on the controllers previously in ILC, for example, this is a special case ofthe ILC design considered in [13] (the feedforward terms have been suppressed) and see also [12]for related designs.

It is simple to observe that the closed loop ½PS1;Cy0 � is globally well posed for any LTI plant

S1; since it forms a linear system along each trial.

2.3. Gap metrics and the robust stability margin

We give two definitions of variants of the gap metric from [16]. Let

OWS1;S2¼ fF : GS1

! GS2jF is causal; surjective; and Fð0Þ ¼ 0g ð12Þ

where

dWðS1;S2Þ ¼ infF2OS1 ;S2

supx2GS1 \f0g;t2T\0

jjTtðF� IÞjGS1xjjW

jjTtxjjW

� �ð13Þ

This defines a directed gap appropriate for global applications.

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Let Br �W denote the ball centred at 0 and of radius r > 0 in W: Then let

dWðrÞðS1;S2Þ :¼ infF2OS1 ;S2

supx2GS1\Br\f0g;

t2T\0

jjTtðF� IÞjGS1xjjW

jjTtxjjW

� �ð14Þ

where

OWðrÞS1;S2

:¼ F : GS1\ Br! GS2

F is causal; Fð0Þ ¼ 0 and

TtðF� IÞTt is compact for all t > 0

��( )

ð15Þ

This defines a directed gap appropriate for regional applications.In the case of the signal space L2½Rþ�; the gap can be directly related to the classical notion of

uncertainty considered in H1 control [7, 17] as follows. Suppose S1;S2 2 R; where Rrepresents the set of all proper transfer functions. We let N1;D1;N2;D2 2 RH1 formnormalized coprime factorizations of S1; S2:

Si ¼ NiD�1i ; Nn

i Ni þDn

i Di ¼ 1; i ¼ 1; 2} ð16Þ

Then the directed gap between S1 and S2 is given by

d0ðS1;S2Þ :¼ inffjjðDN ;DDÞjjH1jDN ;DD 2 RH1;S2 ¼ ðN1 þ DNÞðD1 þ DDÞ�1g

The H2 gap between S1 and S2 is then given by

d0ðS1;S2Þ � d0ðS2;S1Þ ¼ maxfd0ðS1;S2Þ; d0ðS2;S1Þg ð17Þ

and measures the size of the smallest stable co-prime factor difference between normalized co-prime factorizations of the two plants. If d0ðS1;S2Þ51 then it is shown in [16] that

dL2½Rþ�ðrÞðS1;S2Þ ¼ dL2½Rþ�ðS1;S2Þ ¼ d0ðS1;S2Þ ð18Þ

Finally, we define the robustness margin at disturbance level d > 0 as follows:

BPS ;CðdÞ ¼ supfr50jdðS;S1Þ4r ) jjPPS1 ;Cðu0; y0ÞjjLpðN�½0;T �Þ51

for all jjðu0; y0ÞjjLpðN�½0;T �Þ4dg ð19Þ

In the H2; LTI setting, it is known that BP;CðdÞ ¼ bP;C ¼ jjPP;Cjj�1 for all d > 0: Our aim is to

show that the controllers considered in the following section have the property that BP;CðdÞ > 0for all d > 0; and further to provide explicit bounds for the robustness margins.

3. ROBUST STABILITY THEORY FOR ILC

3.1. Gap metrics for ILC

We now establish relations between the LpðN� ½0;T �Þ gap between two plants PS1; PS2

and theLp½Rþ�; Lp½0;T � gap distances between S1 and S2:

}Here NnðsÞ :¼ Nð�%sÞT:



DOI: 10.1002/rnc

Theorem 3.1Let 14p41 and suppose S1;S2 2 R: Then

dLpðN�½0;T �ÞðPS1;PS2Þ4dLp½0;T �ðS1;S2Þ4dLp½Rþ�ðS1;S2Þ ð20Þ

and for r > 0;

dLpðN�½0;T �ÞðrÞðPS1;PS2Þ4dLp½0;T �ðrÞðS1;S2Þ4dLp½Rþ�ðS1;S2Þ ð21Þ

ProofLet F 2 O

Lp½0;T �S1;S2

and define

C : LpðN� ½0;T �Þ � LpðN� ½0;T �Þ ! LpðN� ½0;T �Þ � LpðN� ½0;T �Þ

ðCwÞðk; tÞ ¼ ðFwðk; �ÞÞðtÞ ð22Þ

It is straightforward to observe that C 2 OLpðN�½0;T �ÞPS1 ;PS2

: Then,

dLpðN�½0;T �ÞðPS1;PS2Þ4 sup

w2GPS1\f0g

t2N�½0;T �\f0g

jjTtðI �CÞwjjLpðN�½0;T �Þ

jjTtwjjLpðN�½0;T �Þ

4 supw2GPS1

\f0g

k51;T5t>0

Pk�1i¼1 jjðI � FÞwði; �ÞjjLp½0;T � þ jjTtðI � FÞwðk; �ÞjjLp½0;T �Pk�1

i¼1 jjwði; �ÞjjLp½0;T � þ jjTtwðk; �ÞjjLp½0;T �

4 supw2GPS1

\f0g

T5t>0

maxfjjðI � FÞjG

Lp ½0;T �S1

jjLp½0;T �; jjTtðI � FÞjG

Lp ½0;T �S1

jjLp½0;T �g

4 supT5t>0

jjTtðI � FÞjG

Lp ½0;T �S1

jjLp½0;T �

¼ dLp½0;T �ðS1;S2Þ ð23Þ

Since F 2 OLp½0;T �S1;S2

was arbitrary, it follows that

dLpðN�½0;T �ÞðPS1;PS2Þ4dLp½0;T �ðS1;S2Þ ð24Þ

as required. The inequality dLpðN�½0;T �ÞðrÞðPS1;PS2Þ4dLp½0;T �ðrÞðS1;S2Þ follows by a similar bound

once we have shown that F 2 OLp½0;T �ðrÞS1;S2

implies C 2 OLpðN�½0;T �ÞðrÞPS1 ;PS2

: Note that C is causal and

Cð0Þ ¼ 0 as previously, thus it remains to show that if TtðF� IÞTt is compact for all t > 0; thenTðk;tÞðC� IÞTðk;tÞ is also compact for all ðk; tÞ 2 N� ½0;T �\f0g:

Let ðk; tÞ 2 N� ½0;T �\f0g: Tðk;tÞðC� IÞTðk;tÞ is continuous since Tðk;tÞðC� IÞTðk;tÞ is linear andbounded. Let O � LpðN� ½0;T �Þ � LpðN� ½0;T �Þ be a bounded set and let fwngn51 be asequence in O: Since Qt ¼ TtðF� IÞTt is compact, hence maps bounded sets to relativelycompact sets for all t > 0; it follows that there exists a bounded subsequence fwni ðk; �Þgi51 � Osuch that QTwni ð1; �Þ; . . . ;QTwni ðk� 1; �Þ and Qtwni ðk; �Þ all converge in Lp½0;T � as i!1:

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Hence since

Tðk;tÞðC� IÞTðk;tÞwð� ; �Þ ¼ ðQTwð1; �Þ; . . . ;QTwðk� 1; �Þ;Qtwðk; �ÞÞ ð25Þ

it follows that Tðk;tÞðC� IÞTðk;tÞwni converges in Tðk;tÞðC� IÞTðk;tÞO as i!1: Hence

Tðk;tÞðC� IÞTðk;tÞ is compact for all ðk; tÞ 2 N� ½0;T �\f0g; and C 2 OLpðN�½0;T �ÞðrÞS1;S2

as required.

We now show that dLp½0;T �ðS1;S2Þ4dLp½Rþ�ðS1;S2Þ: Since TTFTT 2 OLp½0;T �S1;S2

for all F 2 OLp½Rþ�S1;S2

;and since S 2 R is stabilizable,

GLp½Rþ�S1j½0;T � ¼ G

Lp½0;T �S1

ð26Þ

it follows that

dLp½Rþ�ðS1;S2Þ ¼ infF2OLp ½Rþ�

S1 ;S2

supx2G

Lp ½Rþ�

S1\f0g;

t>0

jjTtðF� IÞjLp½Rþ�GS1

xjj

jjTtxjj

0@

1A ð27Þ

5 infF2OLp ½Rþ�

S1 ;S2

supx2G

Lp ½Rþ�

S1j½0;T �\f0g;

T5t>0

jjTtðTTFTT � IÞjG

Lp ½Rþ�

S1j½0;T �

xjj

jjTtxjj

0@

1A ð28Þ

5 infU2OLp ½0;T �

S1 ;S2

supx2G

Lp ½0;T �S1

\f0g;

T5t>0

jjTtðU� IÞjG

Lp ½0;T �S1

xjj

jjTtxjj

!ð29Þ

¼ dLp½0;T �ðS1;S2Þ ð30Þ

as required. The inequality dLp½0;T �ðrÞðS1;S2Þ4dLp½Rþ�ðrÞðS1;S2Þ follows in a similar manner, since

TTFTT 2 OLp½0;T �ðrÞS1;S2

for all F 2 OLp½Rþ�ðrÞS1;S2

(since TtðF� IÞTt ¼ TtðTTFTT � IÞTt for all T5t > 0;

the compactness of TtðTTFTT � IÞTt follows from the compactness of TtðF� IÞTt). &

3.2. Robust stability theory for ILC

We now establish the following global robust stability result. It should be noted that the stabilitymargin is determined by the gain of the 2D closed-loop operator PPS ;C; whilst the gap ismeasured by the classical 1D measure of uncertainty dLp½Rþ�ðS;S1Þ:

Theorem 3.2Let 14p41 and suppose S;S1 2 R; C is causal, Cð0Þ ¼ 0; ½PS;C� is stable and ½PS1

;C� isglobally well posed. Then

BPS ;CðdÞ51

jjPPS;CjjLpðN�½0;T �Þfor all d50 ð31Þ

If dLp½Rþ�ðS;S1ÞjjPPS;CjjLpðN�½0;T �Þ ¼ e51; then:

jjPPS1 ;CjjLpðN�½0;T �Þ4jjPPS;CjjLpðN�½0;T �Þ

1þ dLp½Rþ�ðS;S1Þ

1� e

� �ð32Þ



DOI: 10.1002/rnc

ProofLet U : LpðN� ½0;T �Þ ! Lp½Rþ� denote the isomorphism ðUuÞðk; sÞ ¼ uððk� 1ÞT þ sÞ: Define

P : Lpe ½Rþ� ! Lp

e ½Rþ�; Pu ¼ ðU 8 PS 8 U�1Þu

P1 : Lpe ½Rþ� ! Lp

e ½Rþ�; P1u ¼ ðU 8 PS1 8 U�1Þu

K : Lpe ½Rþ� ! Lp

e ½Rþ�; Ku ¼ ðU 8C 8 U�1Þu ð33Þ

Since U is isometric, and by Theorem 3.1, we have

dLp½Rþ�ðP;P1Þ ¼ dLpðN�½0;T �ÞðPS;PS1Þ4dLp½Rþ�ðS;S1Þ ð34Þ

Noting that P;P1;K are causal and satisfy Pð0Þ ¼ P1ð0Þ ¼ Kð0Þ ¼ 0; [16, Theorem 1] thenestablishes:

jjPP1;K jjLp½Rþ�4jjPP;K jjLp½Rþ�

1þ dLp½Rþ�ðP;P1Þ

1� dLp½Rþ�ðP;P1ÞjjPP;K jj½Rþ�

� �ð35Þ

Since U is isometric, we also have:

jjPP;K jjLp½Rþ� ¼ jjPPS;CjjLpðN�½0;T �Þ

jjPP1;K jjLp½Rþ� ¼ jjPPS1 ;CjjLpðN�½0;T �Þ ð36Þ

Inequality (32) now follows from inequality (35) by (34) and (36) as required. &

The specific application to the adaptive ILC algorithm considered in this paper requires afurther robust stability result which is regional in nature.

Theorem 3.3Let 14p41 and suppose S;S1 2 R; C is causal, Cð0Þ ¼ 0; ½PS;C� is Br-stable and ½PS1

;C� isglobally well posed. Let 04e51: Then

BPS ;Cðrð1� eÞÞ5e

jjPPS ;CjBrjjLpðN�½0;T �Þ

ð37Þ

If dLp½Rþ�ðS;S1ÞjjPPS;CjBrjjLpðN�½0;T �Þ4e; then

jjPPS1 ;CjBrð1�eÞjjLpðN�½0;T �Þ4jjPPS;CjBr

jjLpðN�½0;T �Þ1þ dLp½Rþ�ðS;S1Þ

1� e

� �ð38Þ

ProofThe proof establishes the result from Theorem 3.1 and [16, Theorem 4] analogously to thededuction of Theorem 3.2 from Theorem 3.1 and [16, Theorem 1]. &

4. PROOF OF THE L2 RESULT

Let p ¼ 2; so U ¼ Y ¼ L2ðN� ½0;T �Þ: We first establish an intermediate result for a first-ordersystem. In the final analysis, this result will play the role of providing bounds on the internal

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signals from the trial index at which the closed loop has achieved a certain 1D stable polelocation in the dynamics along the trial.

Proposition 4.1Let M > 0; � > 0; b > 0 and suppose a11 þ b >M: Consider the system:

@

@ty1ðk; tÞ ¼ � a11y1ðk; tÞ þ u1ðk; tÞ þ vðk; tÞ

u2ðk; tÞ ¼ � yky2ðk; tÞ

yk ¼ bþ aXk�1i¼0

jjy2ði; �Þjj2L2½0;T �

y1ðk; 0Þ ¼ 0 for all k 2 N

y2ð0; tÞ ¼ 0 for all t 2 ½0;T � ð39Þ

where jjvði; �ÞjjL2½0;T �4Mjjy1ði; �ÞjjL2½0;T �; i51:Then

u1

y1

!��

��U�Y

4n �; �;u0

y0

!��

��U�Y

!ð40Þ

where � ¼ a11 þ ��M and

nð�; �; rÞ ¼ g1r2 þ 3 1þ

g2r2

�

� �2

r2 þ 3 2þb�

� �2

g22r6

!1=2

ð41Þ

and

g1 ¼3

�2þ 3 1þ

b�

� �2

ð42Þ

g2 ¼ bþ 3a 1þ 3 1þb�

� �2 !

þ9a�2

ð43Þ

ProofFirstly, we note that

@

@ty1ðk; tÞ ¼ � a11yðk; tÞ þ u0ðk; tÞ � u2ðk; tÞ þ vðk; tÞ

¼ � a11y1ðk; tÞ þ u0ðk; tÞ þ ykðy0ðk; tÞ � y1ðk; tÞÞ þ vðk; tÞ

¼ � ða11 þ ykÞy1ðk; tÞ þ u0ðk; tÞ þ yky0ðk; tÞ þ vðk; tÞ ð44Þ



DOI: 10.1002/rnc

Let V : R! Rþ be defined as Vðy1Þ ¼12y21: Then

@

@tV ¼ y1ðk; tÞ

@

@ty1ðk; tÞ ¼ �ða11 þ ykÞy21ðk; tÞ þ y1ðk; tÞðu0ðk; tÞ þ yky0ðk; tÞ þ vðk; tÞÞ ð45Þ

Integrating on ½0;T �; and using the inequality jjvðk; �Þjj4Mjjy1ðk; �Þjj we obtain

12 y

21ðk;TÞ4 � ða11 þ ykÞjjy1ðk; �Þjj2L2½0;T �

þ jjy1ðk; �ÞjjL2½0;T �ðjju0ðk; �ÞjjL2½0;T � þ ykjjy0ðk; �ÞjjL2½0;T � þ jjvðk; �ÞjjL2½0;T �Þ

4 � ða11 þ yk �MÞjjy1ðk; �Þjj2L2½0;T �

þ jjy1ðk; �ÞjjL2½0;T �ðjju0ðk; �ÞjjL2½0;T � þ ykjjy0ðk; �ÞjjL2½0;T �Þ ð46Þ

Hence,

jjy1ðk; �Þjj2L2½0;T �4

1

a11 þ yk �Mðjju0ðk; �ÞjjL2½0;T � þ ykjjy0ðk; �ÞjjL2½0;T �Þ

� �2

4 31

ða11 þ b�MÞ2jju0ðk; �Þjj

2L2½0;T � þ 3 1þ

ba11 þ b�M

� �2

jjy0ðk; �Þjj2L2½0;T � ð47Þ

By inequality (47), we obtain

jjy1jj2Y4

3

ða11 þ b�MÞ2jju0jj

2U þ 3 1þ

ba11 þ b�M

� �2

jjy0jj2Y ð48Þ

so

jjy1jj2Y4g1jjðu0; y0Þ

Tjj2U�Y ð49Þ

To bound jju1jjU; we first observe that

yk ¼ bþ aXk�1i¼1

jjy2ði; �Þjj2L2½0;T �4bþ 3a

Xk�1i¼1

1þ 3 1þb

a11 þ b�M

� �2 !

jjy0ði; �Þjj2L2½0;T �

þ3

ða11 þ b�MÞ2jju0ði; �Þjj

2L2½0;T �

!

4bþ 3a 1þ 3 1þb

a11 þ b�M

� �2 !

jjy0jj2Y

þ9a

ða11 þ b�MÞ2jju0jj

2U

¼ g2jjðu0; y0ÞTjj2U�Y ð50Þ

M. FRENCH1028


DOI: 10.1002/rnc

Then,

jju1ðk; �ÞjjL2½0;T �4 jju0ðk; �ÞjjL2½0;T � þ ykjjy2ðk; �ÞjjL2½0;T �

4 jju0ðk; �ÞjjL2½0;T � þ ykjjy0ðk; �ÞjjL2½0;T � þ ykjjy1ðk; �ÞjjL2½0;T �

4 1þyk

a11 þ b�M

� �jju0ðk; �ÞjjL2½0;T �

þ yk 2þb

a11 þ b�M

� �jjy0ðk; �ÞjjL2½0;T � ð51Þ

By inequality (51), we see that

jju1jj2U ¼

X1k¼1

jju1ðk; �Þjj2L2½0;T �

4 3X1k¼1

1þyk

a11 þ b�M

� �2

jju0ðk; �Þjj2L2½0;T �

þ y2k 2þb

a11 þ b�M

� �2

jjy0ðk; �Þjj2L2½0;T �

!

4 3 1þsupk51 yk

a11 þ b�M

� �2

jju0jj2U þ 3 2þ

ba11 þ b�M

� �2

supk51

yk

� �2

jjy0jj2Y ð52Þ

so if jjðu0; y0ÞTjjU�Y4r then

jju1jj2U43 1þ

g2r2

a11 þ b�M

� �2

r2 þ 3 2þb

a11 þ b�M

� �2

g22r6 ð53Þ

and the result follows. &

We now give the main result, which establishes a quadratic lower bound on the robuststability margin.

Theorem 4.2Let the plant PS and the controller C0 be defined by Equations (9), (11) where S 2M1

M : Thenfor all � > 0

BPS;CðrÞ5min

�r

nð�þM � a11 þ 3�r2ðð2�þ 1Þ2 þ 1Þ; �; 2rÞ;1

2�

�ð54Þ

where n is defined as in Proposition 4.1 and � ¼ sup04�4�þM�a11

k��;�kL2½0;T �

ProofLet r > 0; and suppose w0 ¼ ðu0; y0Þ

T; jjw0jj4r: Suppose S1 is such that

dL2½Rþ�ðS;S1Þ4min

�r

nð�þM � a11 þ 3�r2ðð2�þ 1Þ2 þ 1Þ; �; 2rÞ;1

2�

�



DOI: 10.1002/rnc

Consider ½PS1;C0�: Let K 2 ½1;1� be the smallest integer such that

yK5M � a11 þ � ð55Þ

if the above inequality is satisfied and let K ¼ 1 if not. Then by the well posedness of ½PS1;C0�;

we have yK51 and

jjTK ;0y2jj2 ¼

XK�1k¼1

jjy2ðk; �Þjj2L2½0;T � ¼

1

asup

14k4K

yk ¼yKa

jjTK ;0u2jj2 ¼

XK�1k¼1

jjyky2ðk; �Þjj2L2½0;T �4XK�1k¼1

jjykjj2L1½0;T �jjy2ðk; �Þjj2L2½0;T �4

y3Ka

ð56Þ

and so

jjTK ;0PPS1 ;C0w0jj

243

�1

aðy3K þ yK Þ þ r2

�51 ð57Þ

Thus if K ¼ 1 we are done.On the other hand, suppose K51: By shifting the trial index, we obtain

jjPPS1 ;C0w0jj ¼ jjTK ;0PPS1 ;C0

w0jj þ jjðI � TK ;0ÞPPS1 ;C0w0jj

¼ jjTK ;0PPS1 ;C0w0jj þ jjPPS1 ;CyK

v0jj ð58Þ

where v0ðk; tÞ ¼ w0ðk� K ; tÞ; ðk; tÞ 2 N� ½0;T �; and thus it suffices to bound jjPPS1 ;CyKv0jj:

Hence consider ½PS;CyK �:Note that ½PS1;CyK � is globally well posed. We first bound �K . Since by

Theorem 3.1, dL2½0;T �ð�;�1Þ4dL2½Rþ�ð�;�1Þ4 12�, it follows from [16, Theorem 1] that

k��1;�K�1kL2½0;T �4k��;�K�1kL2½0;T �

1þ dL2½0;T �ð�;�1Þ

1� dL2½0;T �ð�;�1Þk��;�K�1kL2½0;T �

� �42�þ 1

Hence

�K ¼ �K�1 þ �ky2ðk� 1; �Þk2L2½0;T �

4�þM � a11 þ 3�r2ðkð0 1Þ��1;�K�1k2L2½0;T � þ 1Þ

4�þM � a11 þ 3�r2ðð2�þ 1Þ2 þ 1Þ

By Proposition 4.1, we see that

jjPPS;CyKjB2rjj4

nð�; �; 2rÞ2r

ð59Þ

where n is defined by Equation (41) with b ¼ �þM � a11 þ 3�r2ðð2�þ 1Þ2 þ 1Þ: By inequality(59), it follows that

dL2½Rþ�ðS;S1Þ51

2jjPPS ;CyKjB2rjj

ð60Þ

Hence by Theorem 3.3, with e ¼ 12; if follows that jjPPS1 ;CyK

jBrjj51 and in particular

jjPPS1 ;CyKv0jj51 since jjv0jj4jjw0jj4r: This completes the proof. &

M. FRENCH1030


DOI: 10.1002/rnc

5. SIMULATION EXAMPLE

As an example, we consider the nominal plant to be an integrator:

S ¼1

sð61Þ

and the true plant to be the integrator with a multiplicative all pass factor perturbation:

SN1 ¼

N � s

N þ s�1

sð62Þ

Straightforward estimations, see [14, 15], show

d0ðS;SN1 Þ ! 0 as N !1 ð63Þ

Furthermore, note that there is an 1808 phase difference between the two plants at highfrequencies, thus taking this example beyond the previous ILC robustness theory [8] as discussedin the Introduction.

Simulations of the closed-loop system with the IlC controller with a learning gain given bya ¼ 0:1; a trial length of T ¼ 2 and input and output disturbances given by

u0ðk; tÞ ¼ 0; y0ðk; tÞ ¼20

k2expð�20ðt� 1Þ2Þ; ðk; tÞ 2 N� ½0; 2� ð64Þ

are shown in Figure 2, where the logarithm of the square of the L2½0; 2� output error isplotted against the trial number (for the first 25 trials) for integer values of N; 54N430: Thisclearly illustrates instability for small values of N (54N411; whereby the unmodelled dynamics

0 5 10 15 20 2510-6

10-5

10-4

10-3

10-2

10-1

100

101

102

Trial number

Squ

are

outp

ut e

rror

siz

e

Figure 2. L2½0;T � output error evolution against ILC trial index for 54N430:



DOI: 10.1002/rnc

is significant), and long-term stability for larger values of N; as predicted qualitatively bythe theory.

6. DISCUSSION AND CONCLUSION

The results in this paper demonstrate that explicit robustness guarantees for a class of ILCdesigns can be given. The uncertainty classes considered are of a classical type, thus the designshave the same (qualitative) robustness certificates as are obtained by H1 controllers. Weobserve that such classical robustness guarantees have, to date remained elusive in theliterature}for example, the previous result [8], required the (multiplicative) unmodelleddynamics to have a maximum of �908 phase deviations from the nominal model over the entirefrequency spectrum, and this is too restrictive to capture many seemingly mild forms ofunmodelled dynamics encountered in any application. The results presented are alsointerpretable ‘in the frequency domain’, and the example considered in detail is representativeof the class of system which lies outside the scope of [8]; note however, that in contrast theresults presented here, [8] establishes monotone convergence. More generally the gap or co-prime factor uncertainty model adopted is more general than the other previously consideredresults in ILC which are restricted to uncertainty models of a multiplicative type, e.g. [9, 10].

In contrast to the situation for LTI plants and controllers, the robustness margins aredependent on the disturbance level: this appears to be a feature of ‘learning’ systems, seefor example the recent related results in adaptive control [14, 15] where the margin also is ofthis form.

The results are limited to a specialized class of ILC design, namely those of an adaptivenature, and are restricted to tracking of the zero trajectory in the presence of disturbances. It isof interest to determine whether the approach can be extended to encompass other approachesto ILC design, to tracking problems and in alternative signal space settings. We have notattempted to consider the optimization of the resulting margins; the problem of optimizingrobustness margins and/or performance in ILC is clearly an important area for futureconsideration, this will involve obtaining tight upper and lower bounds on the margins. We havedefined the margin in terms of the 1D gap measure of plant uncertainty. Further work on theinterpretation of the 2D gap is clearly important, especially in the cases of short trial lengthswhen the 1D gap may be unduly conservative.

We consider the results in this paper to be a significant step towards realizing a usefultreatment of robustness in the iterative learning context.

ACKNOWLEDGEMENTS

The author would like to thank R. Bradley and the reviewers for a number of useful comments on the draftmanuscript.

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DOI: 10.1002/rnc

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Robust stability of iterative learning control schemes