Internal-model-based design of repetitive and iterative learning controllers for linear multivariable systems

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Internal-model-based design of repetitiveand iterative learning controllers for linearmultivariable systemsDick De Roover a , Okko H. Bosgra b & Maarten Steinbuch ca SC Solutions Inc., 3211 Scot Boulevard, Santa Clara, CA, 95054, USAb Mechanical Engineering Systems and Control Group, Delft University ofTechnology, Mekelweg 2, Delft, 2628, CD, The Netherlandsc Faculty of Mechanical Engineering, Systems and Control Group, EindhovenUniversity of Technology, P.O. Box 513, Eindhoven, 5600, MB, The Netherlands

Version of record first published: 08 Nov 2010.

To cite this article: Dick De Roover, Okko H. Bosgra & Maarten Steinbuch (2000): Internal-model-based design ofrepetitive and iterative learning controllers for linear multivariable systems, International Journal of Control,73:10, 914-929

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Internal-model-based design of repetitive and iterative learning controllers for linear multivariablesystems

DICK DE ROOVER{*, OKKO H. BOSGRA{ and MAARTEN STEINBUCH}

Repetitive and iterative learning control are two modern control strategies for tracking systems in which the signals areperiodic in nature. This paper discusses repetitive and iterative learning control from an internal model principle point ofview. This allows the formulation of existence conditions for multivariable implementations of repetitive and learningcontrol. It is shown that repetitive control can be realized by an implementation of a robust servomechanism controllerthat uses the appropriate internal model for periodic distrubances. The design of such controllers is discussed. Next it isshown that iterative learning control can be implemented in the format of a disturbance observer/compensator. It isshown that the resulting control structure is dual to the repetitive controller, and that both constitute an implementationof the internal model principle. Consequently, the analysis and design of repetitive and iterative learning control can begeneralized to the powerful analysis and design procedure of the internal model framework, allowing to trade-o� theconvergence speed for periodic-disturbance cancellation versus other control objectives, such as stochastic disturbancesuppression.

1. Introduction

In practice, many tracking systems have to deal withperiodic reference and/or disturbance signals, for ex-ample computer disk drives, rotating machine tools, orrobots that have to perform their tasks repeatedly. It iswell known that any periodic signal can be generated byan autonomous system consisting of a time-delay ele-ment inside a positive feedback loop. Therefore, inview of the internal model principle (Francis andWonham 1975) it might be expected that accommoda-tion of these periodic signals can be achieved by dupli-cating this model inside a feedback loop. In theliterature, two types of compensators can be foundwhich accomplish this: the repetitive controller (see,e.g. Inoue et al. 1981, Hara et al. 1988, Tomizuka et al.1989, Sadegh 1991) , and the iterative learning controller(see, e.g. Arimoto et al. 1984, Bondi et al. 1988, Mooreet al. 1992).

Although it has been recognized that both schemesdi� er in the way periodic compensation is performed(Hara et al. 1988, Horowitz 1993) , still the impressionexists that both schemes are equivalent. However, in arecent paper it was shown that the schemes are notequivalent but are related by duality, which is a conse-quence of the di� erence in location of the internal model

inside the compensator (de Roover and Bosgra 1997) . Itwas shown that a repetitive controller has the structureof a servo compensator Ð with the internal modellocated at the system outputÐ while a learning control-ler has the structure of a disturbance observer, with theinternal model located at the system input.

In this paper we use the general framework given inde Roover and Bosgra (1997) to set up a general frame-work for the synthesis of (MIMO) repetitive and learn-ing controllers. It is shown that a number of existingrepetitive and learning control schemes can be put intothis framework according to speci® c modi® cations in theinternal model.

The remainder of this paper is organized as follows.In the next section we describe existing repetitive andlearning control approaches. In } 3 we give a discussionof the properties of existing approaches and we de® nethe robust periodic control problem. In } 4 we show howthe repetitive and learning control problem can be for-mulated and solved in an internal-model-based frame-work, which allows the joint formulation of periodic-disturbance rejection and otherÐ equally important Ðcontrol objectives. This is illustrated with an exampleof a MIMO aircraft model in } 5.

Throughout this paper, R denotes the ® eld of realnumbers. Let nu denote the dimension of the vector u,then R u denotes the set of all nu-vectors with elements inR . Likewise, R u£y denotes the set of all nu £ ny matriceswith elements in R , and Iu denotes the nu £ nu identitymatrix. Furthermore, z denotes the discrete-time delayoperator, and R …z† denotes the set of all rational func-tions with real coe� cients in z. Let M 2 R n£m then»…M† denotes the rank of M , and »…M† µ min fn ;mg.Discrete time is indicated by tk ; k ˆ . . . ;¡1 ;0 ;1 ; . . ..Signals as a function of discrete time are indicated eitheras x…tk† or as xk for shorthand notation (xr…tk† and xr;k

International Journal of Control ISSN 0020± 7179 print/ISSN 1366 ± 5820 online # 2000 Taylor & Francis Ltdhttp ://www.tandf.co.uk/journals

INT. J. CONTROL, 2000 , VOL. 73, NO. 10, 914 ± 929

Received 1 February 1999. Revised 1 December 1999.* Author for correspondence : e-mail : roover@scsolutions.

com{ SC Solutions Inc., 3211 Scott Boulevard, Santa Clara,

CA 95054, USA.{ Mechanical Engineering Systems and Control Group,

Delft University of Technology, Mekelweg 2, 2628 CD Delft,The Netherlands.

} Faculty of Mechanical Engineering, Systems and ControlGroup, Eindhoven University of Technology, P.O. Box 513,5600 MB Eindhoven, The Netherlands.

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for signals with subscripts, respectively) . A recursion oriteration is indicated as xi , i ˆ 1 ;2 ; . . ..

2. Existing repetitive and learning control algorithms

Let the system to be controlled be given by adiscrete-time time-invariant state-space realizationfA ;B ; C ;Dg having transfer function matrix

P…z† ˆ C…zI ¡ A†¡1B ‡ D

In iterative learning control, the response of the systemover a ® nite horizon of N samples is considered for a® xed initial state:

x…tk‡1† ˆ Ax…tk†‡ Bu…tk†; x…0† ˆ x0

y…tk† ˆ Cx…tk†‡ Du…tk†

)

…1†

Without loss of generality, x0 is taken as zero. The ® nite-interval response of (1) is

y…0†

y…1†

..

.

y…N ¡ 1†

266666664

377777775

ˆ

D 0 0 ¢ ¢ ¢ 0

CB D 0 ¢ ¢ ¢ 0

CAB CB D ¢ ¢ ¢ 0

..

. ... ..

. . .. ..

.

CAN¡2B CAN¡3B CAN¡4B ¢ ¢ ¢ D

266666666664

377777777775

u…0†

u…1†

..

.

u…N ¡ 1†

266666664

377777775

…2†or

y ˆ Gu …3†As repeated trials are considered in repetitive control,we write yi ˆ Gui for the ith trial (iteration),i ˆ 0 ;1 ;2 ; . . .. Note that each trial assumes the sameinitial condition for the system to be controlled. Let

r ˆ

r…0†r…1†

..

.

r…N ¡ 1†

266666664

377777775

be the reference vector over the time horizon of one trail,and let similarly ei ˆ r ¡ yi be the error vector. Then aprototype update law that implements iterative learningcontrol by updating the past inputs on the basis of thepast error is

ui‡1 ˆ ui ‡ L ei …4†where L is a matrix of appropriate dimensions. Using (3)this can be rewritten to

ui‡1 ˆ ui ‡ L …r ¡ Gui†

ˆ …IN ¡ L G†ui ‡ L r

and the recursion for the error is

ei‡1 ˆ …IN ¡ GL †ei

If the recursion algorithm converges for i ! 1, then

L Gu1 ˆ L r

which implies L e1 ˆ 0. Several approaches exist in theliterature to design L such that guaranteed and con-trolled convergence conditions exist (see, e.g. Amannet al. 1996).

In repetitive control one assumes that the referencesignal r…tk† is periodic with period N, i.e. r…tk‡N† ˆ r…tk†,k ˆ 0 ;1 ; . . .. A prototype algorithm for repetitive con-trol is to select the input u…tk‡N† as

u…tk‡N† ˆ u…tk†‡ R…z†e…tk†; k ˆ . . . ;0 ;1 ;2 ; . . . …5†where R…z† is a rational matrix in the shift operator z.This implies

y…tk‡N† ˆ y…tk†‡ P…z†R…z†e…tk†or

y…tk† ˆ P…z†R…z†F…z†e…tk†where F…z† :ˆ …zNIy ¡ Iy†¡1. This relation is representedin ® gure 1. Considering r…tk† as external input

y…tk‡N† ˆ ‰Iy ¡ P…z†R…z†Šy…tk†‡ P…z†R…z†r…tk†

Repetitive and iterative learning controllers 915

Figure 1. Repetitive control as feedback system.

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One di� erence between iterative learning control andrepetitive control is the fact that iterative learning con-trol assumes a ® xed initial condition for the system atthe beginning of each period, whereas repetitive controlassumes the system to have an initial condition at thebeginning of a period that is the result of the previousperiods. Thus iterative learning control considersmotions such as a repeated pick and place operationof a robot, whereas repetitive control considers periodi-cities such as those occurring in rotating equipment hav-ing constant speed of rotation.

In many applications one neglects the e� ects of the® xed initial condition in iterative learning. Then theupdate law (4) assumes the form

u…tk‡1† ˆ u…tk†‡ L …z†e…tk† …6†

where L …z† is a rational matrix in the shift operator z.Then

u…tk† ˆ F…z†L …z†e…tk†

and this relation is shown in ® gure 2. Note that in® gure 1 and ® gure 2 the block F…z† has the internalrepresentation as given in ® gure 3, where z¡NI repre-sents a delay of one period of N samples. If F…z† isregarded as a causal dynamic system, it will have Nstate variables in each channel. Note that in repetitivecontrol ( ® gure 1) there are ny channels as F…z† operatesin the output space, whereas in learning control (® gure2) F…z† has nu channels as it operates in the input space.The characteristic polynomial of F…z† is zN ¡ 1, ny or nutimes repeated. This characteristic polynomial has all itsroots evenly distributed on the unit circle. Asymptoticstability of the loop in ® gure 1 or ® gure 2 implies that allthese roots have to be moved inside the unit circle by theaction of the feedback loop.

The stability analysis suggested in the literature pro-ceeds with isolating the delay chain of the internal model( ® gure 3) in an equivalent system representation (see, e.g.Hara et al. 1988) . For the repetitive controller of ® gure 1this is shown in ® gure 4. Since the delay chain has mag-nitude equal to one the small gain theorem can be used,which states that the following condition is su� cient forthe equivalent system to be stable

kIy ¡ P…z†R…z†ki < 1 …7†

for some induced i-norm. Equation (7) motivates tochoose the repetitive control feedback as R…z† ˆP¡1…z†, i.e. equal to the (right) inverse of the systemP…z†. Consequently, the dimension of R is now deter-mined by the system P, and not by the number N of theperiod length anymore.

In the literature on iterative learning control,schemes like (6) are called past error feedforward (see,e.g. Padieu and Su 1990, Moore et al. 1992, Ammanet al. 1996) . An alternative is to use current-error feed-back (see, e.g. Owens 1993, Goh 1994, Goh and Yan1996) where

u…tk‡N† ˆ u…tk†‡ L …z†e…tk‡N†

Now

u…tk† ˆ …Iu ¡ z¡NIu†¡1L …z†e…tk† …8†

where the transfer function F 0…z† ˆ …Iu ¡ z¡NIu†¡1 hasinternal representation as shown in ® gure 5.

916 D. de Roover et al.

Figure 2. Iterative learning control as feedback systems.

Figure 3. Internal representation of F…z†.

Figure 4. Equivalent system representation of repetitivecontrol system.

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In this case a su� cient condition for stability of thesystem (8) is

k…Iu ¡ L …z†P…z††¡1ki < 1 …9†which leads to high-gain solutions for L …z†, whereas asu� cient condition for stability of the system (6) is

kjIu ¡ L …z†P…z†ki < 1 …10†This shows the advantage of past-error feedforwardschemes over current-error feedback schemes if one isonly interested in stabilization of the loop, i.e. in con-vergence of the recursion. In practical situations neitherequation (9) nor equation (10) can be realized, and fre-quency weighted norms have to be introduced. The fre-quency up to which (10) holds in practical situations, isin general two to three times larger than the frequencyup to which (9) is valid (see, e.g. de Roover 1996, deRoover et al. 1996).

3. Discussion and problem de® nition

3.1. Discussion of properties of repetitive and learningcontrollers

The existing repetitive and learning control schemeshave several limitations that will be discussed.

. Repetitive and learning control are assumed tocontribute to the performance of a control systemby identifying the most useful feedforward inputsignal that suppresses the periodic disturbances. Inthis paper the recursion of the repetitive or learn-ing part of the controller will be represented asfeedback. Convergence of this recursion thentranslates into stability of the complete feedbacksystem. In this case the feedback system can bethought of to contain a compensator C…z† usedfor disturbance suppression, and in addition arepetitive or learning part used for suppressingthe periodic disturbances. In general, the repetitiveor learning part will in¯ uence the sensitivity func-tion of the complete loop and thus will modify thedisturbance suppressing properties of the control-ler C…z†. In this respect, fast convergence of the

recursion is not necessarily the best solution. Fastconvergence of a repetitive controller might implythat stochastic disturbances occurring in the pastare e� ectively contributing to the formation of theperiod disturbance waveform. In that case thefeedforward signal is corrupted by stochastic dis-turbances and a� ects the sensitivity function of theloop negatively. Rather, the speed of convergenceshould be part of an overall feedback controldesign where a single goal of disturbance suppres-sion is pursued. The internal model approach con-sidered in the sequel provides a suitable jointdesign of controller C…z† and the repetitive orlearning controller part.

. In many cases a controller is already available forthe plant P…z†, and a learning controller is addedto the existing scheme, see ® gure 6. Then a su� -cient small-gain-based condition for stability is

kIu ¡ L …z†…Iy ‡ P…z†C…z††¡1P…z†ki < 1

i.e. P…z† is left multiplied by the output sensitivityfunction, which itself is stable if C…z† stabilizesP…z†. This shows that in principle it is still possibleto design L …z† given C…z†. However, L …z† has to bechosen such that it operates within the stabilitymargin of the loop closed by C…z†. It must beconcluded that better results could be expected ifL …z† and C…z† are designed simultaneously on thebasis of a common design goal (i.e. common con-trol performance).

. The discussion of using past-error feedforward orcurrent-error feedback should be discussed alsofrom the point of view that one should utilizethe form of the error feedback that eliminatesthe periodic disturbances and simultaneously hasthe best performance in terms of an overall controlperformance criterion.

In the following section it will be shown that aninternal-model-based framework can be formulatedthat allows the joint design of C…z† and a learning orrepetitive controller having favourable properties with


Figure 5. Internal representation of F 0…z†.Figure 6. Learning controller added to existing feedback

controller C…z†.

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respect to a single performance criterion. First, therobust periodic control problem will be formulated.

3.2. The robust periodic control problem

Any periodic signal can be generated by an autono-mous system consisting of a time-delay inside a positivefeedback loop with appropriate initial conditions, see® gure 5. For example, a discrete time periodic signalof length N can be generated by

xw…tk‡1† ˆ Awxw…tk†; xw…t0† ˆ xw0

w…tk† ˆ Cwxw…tk†

9=

; …11†

with

Aw ˆ

0 1 0 ¢ ¢ ¢ 0

..

. ... ..

. . .. ..

.

0 0 0 ¢ ¢ ¢ 1

1 0 0 ¢ ¢ ¢ 0

266666664

377777775

2 R N£N

Cw ˆ ‰1 0 0 ¢ ¢ ¢ 0Š

Note that the spectrum of Aw consists of N roots equallyspaced on the unit disk, i.e. these are the roots ofdet …zI ¡ Aw† ˆ zN ¡ 1. Next, consider a discrete timelinear time-invariant (LTI) plant P…z† 2 R …z†y£u withinput signal u ˆ up ‡ wu 2 R u, and output signaly ˆ Pu 2 R y, according to ® gure 7. Given a desiredperiodic output signal r…tk† ˆ r…tk‡N†;tk ˆ 0, D T ,2 D T ; . . . ; with D T denoting the sampling time. Lete ˆ r ¡ y be the tracking error. Then we de® ne therobust periodic control problem as:

De® nition 1: The robust periodic control problem isto ® nd a feedback compensator C…z† for the systemP…z† such that :

1. The resulting compensated system is exponen-tially stable.

2. The tracking error e tends to zero asymptotically,for all periodic references r and periodic disturb-ances wu satisfying (11). (Note that there is nofundamental di� erence between an error resultingfrom a disturbance at the output or from a refer-ence input r. We will only consider r. )

3. Properties 1 and 2 are robust, i.e. they also hold incase the dynamics of P are perturbed

The solution to the robust periodic control problemis provided by the internal model principle (Francis andWorham 1975) which states:

Internal model principle: Suppose that the controllerC…z† in ® gure 7 contains in each channel a realization

of the disturbance generating system, driven by the errore…z†. Further, let the controller C…z† be such that thefeedback connection of C…z† and P…z† is internallystable. Then C…z† solves the robust periodic controlproblem.

As repetitive and learning control attempt to solvethe (robust) periodic control problem, it follows that theinternal model principle provides a solution for repeti-tive and learning control. Moreover, the internal modelprinciple can be formulated in the format of a servocom-pensator where the disturbance model is realized in eachchannel of the output space, or in a dual format wherethe disturbance model is realized in each channel of theinput space. The ® rst format corresponds to the mannerin which repetitive control is implemented. The secondformat utilizes the structure of a disturbance observerand corresponds with iterative learning control. Bothstructures will be discussed in the next section.

4. Synthesis in an internal-model-base d framework

This section extends the approach given in deRoover and Bosgra (1997). Introducing disturbancedynamics in the controller and assuming the feedbacksystem to behave asymptotically allows for state feed-back of the disturbance dynamics as those states areavailable for feedback. This would lead to current-error feedback instead of past-error feedforward. Weact according to the following points of view:

. We assume that the feedback controller must havea loop feedback control performance in additionto the requirements of asymptotic rejection of per-iodic distrubances. In this case there is an advan-tage in applying current-error feedback instead ofpast-error feedforward as the latter introduces adelay of one period in each channel of C…z†. Thisdelay only obstructs the stabilization of the dis-turbance dynamics at the cost of control perform-ance.

. As the state of the disturbance dynamics in C…z† isavailable for feedback, it is worthwhile to applystate feedback of this state for the purpose ofstabilization and control performance enhance-ment.


Figure 7. General periodic control problem; up and wudenote a periodic control signal and periodic inputdisturbance, respectively.

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. In addition, control performance will be realizedby using an estimated-state- feedback compensatoradded to the servo part containing the disturbancemodel. The state variables of the disturbancemodel act as memory variables for periodic errors.

. Instead of concentrating on the stabilization of thememory variables, as is automatically done in clas-sical approaches to repetitive and learning control,we here apply feedback to the memory variableswith feedback loop control performance as under-lying goal.

4.1. Repetitive control

De® ne

Aw :ˆ

0 1 0 ¢ ¢ ¢ 0

..

. ... ..

. . .. ..

.

0 0 0 ¢ ¢ ¢ 1

1 0 0 ¢ ¢ ¢ 0

26666664

37777775

2 R N£N

Bw :ˆ

0

..

.

0

1

26666664

37777775

2 R N£1

Cw :ˆ ‰1 0 0 ¢ ¢ ¢ 0Š 2 R 1£N

and

Ar :ˆ

Aw 0

Aw

. ..

0 Aw

26666664

37777775

Br :ˆ

Bw 0

Bw

. ..

0 Bw

26666664

37777775

Cr :ˆ

Cw 0

Cw

. ..

0 Cw

26666664

37777775

where each diagonal block is repeated ny times. In asimilar way, de® ne fAl ;Bl ;Clg where each diagonalblock is repeated nu times. Then

Cr…zINy ¡ Ar†¡1Br ˆ z¡NIy…Iy ¡ z¡NIy†¡1

Cl…zIu ¡ Al†¡1Bl ˆ z¡NIu…Iu ¡ z¡NIu†¡1

Also, de® ne Kr 2 R u£Ny , L l 2 R Nu£y. An implemen-tation for repetitive control with current-error feedback,based upon state feedback of memory variablesand estimated-state feedback of the plant P…z† ˆC…zI ¡ A†¡1B ‡ D, is

System: xk‡1 ˆ Axk ‡ Buk

yk ˆ Cxk ‡ Duk

Observer : x̂k‡1 ˆ Ax̂k ‡ Buk ‡ L °k

Feedback error: ek ˆ rk ¡ yk

Observer error: °k ˆ ek ‡ Cx̂k ‡ Duk

Disturbance memory: xr ;k‡1 ˆ Arxr;k ‡ Brek

Control input: uk ˆ Krxr;k ‡ Kx̂k

9>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>;

…12†Here L is the plant state observer gain, and Kr and K

are the state feedback laws for disturbance memory stateand plant state, respectively. The repetitive controllercan be represented in the block diagram of ® gure 8where

-A :ˆ A ‡ BK‡ L …C ‡ DK†. The structure con-

tains the estimated-state feedback controller C…z†x̂k‡1 ˆ -Ax̂k ‡ L ek

uk ˆ Kx̂k

9=

;ˆ K…zI ¡ -A†¡1L ˆ: C…z† …13†

acting between ek and uk, plus in addition the disturb-ance accommodating part having transfer functionKr…zINy ¡ Ar†¡1Br, operating on ek and feeding an addi-tional input in the controller C…z†. We observe that thememory variables are linked to the output space ek . Bybringing the memory variables in a structure displayingthe delay N, and feeding into the uk signal, we can com-pletely separate the memory variables from the feedbackcontroller C…z†

Kr…zINy ¡ Ar†¡1Br ˆ: R 0…z†…Iy ¡ z¡NIy†¡1

where R 0…z† ˆ R 01z

¡1 ‡ R 02z¡2 ‡ ¢ ¢ ¢‡ R 0

Nz¡N , is one-to-one related to Kr in the sense that the column…Ni ‡ j ‡ 1† of Kr equals column …jny ‡ i ‡ 1† of

‰R 01jR 0

2j ¢ ¢ ¢ jR 0NŠ; i ˆ 0;1;. . . ;ny ¡1; j ˆ 0;1;. . . ;N ¡1

which directly follows from R 0j‡1 ˆ KrA

jrBr, j ˆ 0;1; . . . ;

N ¡ 1. The structure of the repetitive controller, writtenwith separate memory block, and completely equivalentto ® gure 8, is shown in ® gure 9. The block

-GR…z† follows

from the path between the output of the blockKr…zINy ¡ Ar†¡1Br and signal uk in ® gure 8, and equals

-GR…z† ˆ Iu ‡ K…zI ¡ -A†¡1…B ‡ L D†


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Figure 9 shows that internal-model-based repetitivecontrol can be implemented in a form resembling theaddition of a memory-variable block to a feedback con-troller C…z†, where C…z† is the estimated-state- feedbackcontroller de® ned by (13).

The classical stability analysis of repetitive controlconsiders the feedback path around z¡NINy . This pathis de® ned between the signals in and out in ® gure 10.From this ® gure it follows that

out…z† ˆ ‰I ‡ …I ‡ P…z†C…z††¡1P…z† -GR…z†R 0…z†Š¡1in…z†

ˆ ‰I ‡ -GK…z†R 0…z†Š¡1in…z† …14†

where -GK follows from substituting the expressions andsimplifying

-GK…z† ˆ ‰D ‡ …C ‡ DK†…zI ¡ A ¡ BK†¡1BŠ …15†

Thus the memory variables experience a feedback struc-ture as shown in ® gure 11. The feedback condition of® gure 11 can be described by the equations

xr;k‡1 ˆ Arxr;k ‡ Brek

yr;k ˆ Krxr;k

zk‡1 ˆ …A ‡ BK†zk ‡ Byr;k

ek ˆ ¡…C ‡ DK†zk ¡ Dyr;k

or equivalently

xr;k‡1

zk‡1

" #ˆ

Ar ¡BrC

0 A

" #

|‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚}system matrix

xr;k

zk

" #

‡¡BrD

B

" #

|‚‚‚‚‚‚{z‚‚‚‚‚‚}input matrix

‰Kr KŠ|‚‚‚‚{z‚‚‚‚}state feedback

xr ;k

zk

" # …16†

This shows the underlying state feedback mechanism.The feedback laws Kr and K are to be designed for acommon control goal that assures the control perform-ance of the feedback loop shown in ® gure 11.


Figure 8. Repetitive controller based on internal model principle.

Figure 9. Internal-model-based repetitive control as add-on to C…z†.

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The implementation of the repetitive controllerneeds the design of an observer gain L in addition tothe feedback gains Kr and K. If L , Kr and K have beendesigned, the implementation of the control structure in® gure 8 or ® gure 9 is uniquely determined.

4.2. L earning control

In learning control the memory variables are linkedto the input space. In de Roover and Bosgra (1997) it hasbeen shown that learning controllers can be formulatedin an internal-model framework by dualizing the resultsof the repetitive controller. The underlying structurethen is generated by a disturbance observer, which hasbeen shown in de Roover and Bosgra (1997) to be theexact dual of a servo compensator. Now we assume forthe time being a periodic disturbance at the plant input,generated by a system fAl ;Clg with non-zero initial con-dition, as shown in ® gure 12. An observer for this systemnow results by duplicating the system and applying feed-back L r, L to the estimated states from an observererror. The estimate of the disturbance, d̂k , is used tocompensate the disturbance dk. In addition, estimated

plant state feedback is applied through the feedbackgain K. Finally, the assumed error d at the plant inputis replaced by an actual error resulting from the refer-ence input r. Thus the disturbance estimator compen-sates for an input disturbance equivalent to the controlerror in output space. The structure is shown in ® gure 13.Observe that the memory variables in the observerdirectly are linked to the input space. We now have

System: xk‡1 ˆ Axk ‡ Buk

yk ˆ Cxk ‡ Duk

Observer: x̂k‡1 ˆ Ax̂k ‡ B…uk ‡ d̂k†‡ L °k

Observer error: °k ˆ ek ‡ Cx̂k ‡ D…uk ‡ d̂k†Feedback error: ek ˆ rk ¡ yk

Disturbance memory: xl;k‡1 ˆ Alxl;k ‡ L lek

d̂k ˆ Clxl ;k

Control input: uk ˆ Kx̂k ‡ Clxl;k

9>>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>>;

…17†By combining several of these equations, the feed-

back structure underlying the scheme in ® gure 13 can be


Figure 10. Feedback path in ! out around memory delay z¡N in IM-based repetitive control.

Figure 11. Equivalent feedback structure around memory variables in repetitive control.

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made transparent. The equivalent structure is shown in® gure 14. Inspection of ® gure 14 shows that the struc-ture is dual to the repetitive structure of ® gure 8. Thedelay structure becomes apparent by de® ning

Cl…zINu ¡ Al†¡1L l ˆ: …Iu ¡ z¡NIu†¡1L 0…z†where L 0…z† ˆ L 0

1z¡1 ‡ L 02z¡2 ‡ ¢ ¢ ¢‡ L 0

Nz¡N , de® ned byL 0

j‡1 :ˆ ClAjlL l , j ˆ 0 ;1 ; . . . ;N ¡ 1. Alternatively, L l

and L 0…z† are related one-to-one by the fact that eachrow …Ni ‡ j ‡ 1† of L l equals the row …jnu ‡ i ‡ 1† of

‰L 01

TjL 02

Tj ¢ ¢ ¢ L 0N

TŠT ; i ˆ 0 ;1 ; . . . ;nu ¡ 1;

j ˆ 0 ;1 ; . . . ;N ¡ 1

Then the structure of ® gure 14 is equivalent to the struc-ture of ® gure 15. This ® gure directly shows duality withthe repetitive controller of ® gure 9. Note that the con-troller C…z† equals C…z† ˆ K…zI ¡ -A†¡1L , i.e. C…z† is theresult of the design of the state feedback law K and the

state observer gain L , and equals the expression asshown previously in the repetitive controller structureof ® gure 9.

An analysis of the feedback around the memoryvariables dual to the results in ® gure 11, is shown in® gure 16, where, after some algebraic manipulation,-GI…z† can be computed as

-GI…z† ˆ ‰D ‡ C…zI ¡ A ¡ L C†¡1…B ‡ L D†ŠNote the duality with the expression (15). The feedbackstructure in ® gure 16 can be realized by the system

xl ;k‡1 ˆ Alxl ;k ‡ L l°k

d̂k ˆ Clxl ;k

zk‡1 ˆ …A ‡ L C†zk ‡ …B‡ L D†d̂k

°k ˆ Czk ‡ Dd̂k


Figure 12. State representation of plant …A;B ;C ;D† with periodic input disturbance.

Figure 13. Distrubance observer-based learning controller with estimated-state feedback K.

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Figure 14. Disturbance-observer-based learning control showing (dual) internal model principle.

Figure 15. Internal-model-based learning control as add-on to C…z†.

Figure 16. Equivalent feedback structure around memory variables in learning control.

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or equivalently

xl ;k‡1

zk‡1

" #ˆ

Al 0

BCl A

" #

|‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚}system matrix

xl ;k

zk

" #

‡L l

L

" #

|‚‚{z‚‚}observer gains

‰DCl CŠ|‚‚‚‚‚‚{z‚‚‚‚‚‚}output matrix

xl ;k

zk

" # …18†

Thus L l and L must be designed such that the closedloop system (18) has favourable properties (damping,loop gain, sensitivity) . Note that (18) is dual to (16).

4.3. Design of internal-model-based repetitive andlearning control

The design of the repetitive controller (12) and of thelearning controller (17) involves the speci® cation of thegain matrices fL ;Kr ;Kg and fK;L l ;L g, respectively.These controllers are required to solve the robustperiodic control problem. The following theorem de® nesnecessary and su� cient conditions for the internal-model-based repetitive and learning controllers tosolve the robust periodic control problem.

Theorem 1: Consider the repetitive controller (12) andassume that L is chosen such that A ‡ L C is asymptoti-cally stable, and assume that fKr ;Kg is chosen such thatthe system (16) is asymptotically stable. Then the con-troller (12) solves the robust periodic control problem asde® ned in De® nition 1, if and only if

»¶I ¡ A B

¡C D

" #

… †ˆ nx ‡ ny 8¶ 2 ¼…Aw† …19†

Proof : This result can be directly derived from the re-sults available for the general servo compensator bynoting that a repetitive controller is a servo compensa-tor for periodic signals, which is a special class of per-sistent signals. A full proof for the general case can befound in de Roover (1997) (see also de Roover andBosgra 1997) &

Due to the established duality, a dual version holdsfor the learning controller.

Theorem 2: Consider the learning controller (17) andassume that K is chosen such that A ‡ BK is asymptoti-cally stable, and assume that fL l ;L g is chosen such thatthe system (18) is asymptotically stable. Then the con-troller (17) solves the robust periodic control problem asde® ned in De® nition 1, if and only if

»¶I ¡ A B

¡C D

" #

… †ˆ nx ‡ nu ; 8¶ 2 ¼…Aw† …20†

and

»…B†‡ ny ˆ »BD… † …21†

Proof : Follows by similarÐ yet dualÐ reasoning fromthe proof of Theorem 1, noting that a learning con-troller is a special class of disturbance observers forpersistent periodic signals. &

Theorem 1 requires that P…z† does not have trans-mission zeros located at the spectrum of Aw and thatP…z† has at least as many inputs as outputs. Theorem 2requires that P…z† does not have transmission zeroslocated at the spectrum of Aw and that P…z† has atleast as many outputs as inputs. These conditions followby requiring the controllability of the series connectionof plant P…z† and memory dynamics, and by requiringthe observability of the series connection of memorydynamics and plant P…z†, respectively. However, thesecond condition in Theorem 2Ð which results fromthe requirement of asymptotic tracking of r…z† in theoutput space, as opposed to cancelling an estimated dis-turbance in the input spaceÐ is true only if P…z† has atleast as many inputs as outputs. Thus the learning con-troller (17) will show asymptotic tracking of r…z† only ifP…z† is square and invertible.

A further question is whether it is feasible to designthe repetitive controller as shown in ® gure 9 in twoseparate steps: First, determine C…z† as an estimated-state feedback compensator by choosing K and L ,given fA ;B ;C ;Dg ; second, determine Kr which speci® esthe key properties of the `add-on’ periodic disturbanceaccommodating part, which is the series connection of-GR…z† and Kr…zINy ¡ Ar†¡1Br. Stability of the composite

system resulting from the interconnection of both partswith the system can be investigated by a su� cient con-dition provided by the small-gain theorem. This requiresthat the transfer function between in and out in ® gure 10is smaller than unity in terms of an induced norm (seerelation (14))

k…I ‡ …I ‡ P…z†C…z††¡1P…z† -GR…z†R 0…z††¡1ki < ˆ 1

which is equivalent to

k…I ‡ -GK…z†R 0…z††¡1ki < ˆ 1

where -GK…z† is given by (15). This condition implies thatthe sensitivity function of the loop in ® gure 17Ð thetransfer function between in and out Ð has inducednorm smaller than 1. Here, -GK…z†, which is ® xed by K,L and R 0…z†, is to be designed and will determine Kr. Asit is impossible for linear systems to have a sensitivityfunction which has induced norm smaller than oneÐwhereas the strict inequality is required as z¡N attainsthe absolute value 1 for all harmonics of the basic fre-quencyÐ a design of Kr must be precisely tuned in con-


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junction with the properties of K and L . This requires ajoint, simultaneous, design of K and Kr. The separationprinciple allows fKr ;Kg and L to be designed sepa-rately.

One possibility is to use an LQG approach. Considerthe system (16), written as

²k‡1 ˆAr ¡BrC

0 A

" #²k ‡

¡BrD

B

" #·k ²0 given

·k ˆ ‰Kr KŠ²k

Determine ‰Kr KŠ such that

X1

kˆ0

²Tk

Q1 0

0 Q2

" #²k ‡ ·T

k R·k

is minimal. Then ‰Kr KŠ follows from the appropriatediscrete Riccati equation. By selecting Q1 relativelysmall, we can express the desire that the memory vari-

ables should contribute to the control performance asspeci® ed for fA ;B;C ;Dg in terms of a choice of Q2 andR. In addition, L can be determined on the basis of poleassignment or as a Kalman gain, based on assumednoise statistics for system and measurement noise forfA ;B ;C ;Dg.

The approach for learning control can follow a com-pletely similarÐ yet dualÐ approach.

5. Example: MIMO aircraft model

Consider the aircraft model described in theAppendix of Maciejowski (1989). This model has threeinputs, three outputs and ® ve states. The speci® cation isto achieve a high bandwidth of approximately 10 rad/sfor each loop. In Maciejowski (1989) this example isused to compare di� erent MIMO feedback design tech-niques. For this example we use LQG with integral con-trol as the MIMO feedback design method. By selectingthe appropriate weighting functions, we can duplicatethe high-bandwidth feedback design. Figure 18 showsthe maximum and minimum singular values of the sen-sitivity and complementary sensitivity function. Figure19 shows the corresponding step responses to unit stepdemands for each loop. The responses are fast (settlingwithin 2 s), well damped (overshoot < 25%), and inter-action is greatly reduced, compared to the open-loopplant.


Figure 17. Sensitivity function of separate design.

Frequency [Hz]

1 100.1 100

Gai

n [d

B]

-40

-20

0

-60

20Singular Values Closed Loop

Figure 18. Maximum (thick curves) and minimum (thin curves) singular values of nominal sensitivity function (dark grey) andcomplementary sensitivity function (black).

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Next, consider the following disturbance acting oninput 1

w…tk† ˆ 10 sin …5tk†‡ 5 sin …10tk ‡ ¿1†

‡ 2 sin …20tk ‡ ¿2†‡ 2n…tk†;

tk ˆ 0 ; D T ;2 D T ; . . .

with D T ˆ 5e ¡ 3s (0.005s) sampling time, ¿1, ¿2 phaseshifts of 45 and 30 degrees, respectively, and n…tk† aGaussian distributed random noise with mean value 0and 3¼ value 1. Basically, this disturbance constitutes a5 Hz periodic signal with two harmonics. Figure 20shows the closed-loop step responses to step demandson all three outputs with the disturbance acting on input1. Clearly, because of closed-loop interaction at 5, 10and 20 Hz, signi® cant oscillations occurs in all threechannels.

To accommodate this disturbance, either a repetitiveor learning controller can be designed. As the plant issquare (3 inputs, 3 outputs) and does not have transmis-sion zeros at 5, 10 or 20 Hz, both conditions inTheorem 1 and Theorem 2 are satis® ed, i.e. either arepetitive or learning controller exists that can accom-modate this disturbance. As the disturbance is a contin-uous periodicity, as opposed to a repeated operation, wewill design a repetitive controller.

For this, the plant with integral states is augmentedwith an internal model for 5 Hz periodic signals accord-ing to (11). With D T ˆ 5e-3 sec, this results in a 40 stateinternal model in each loop. Note that this internalmodel is capable to compensate for all 20 harmonicsof the 5 Hz base frequency; in case there are only afew harmonics to be suppressed a reduced internalmodel can be used consisting of a few oscillators. Bysolving the LQG problem for the augmented system, a3-input, 3-output, 120-state repetitive controller isdesigned in conjunction with the original observer-state feedback controller. Figure 21 shows the maximumand minimum singular values of the sensitivity functionand complementary sensitivity function of the closed-loop system with repetitive controller. This ® gureshows how the repetitive controller introduces highgain at frequency 5 Hz, and all its higher harmonics.Figure 22 shows the closed-loop step responses to stepdemands on all outputs. Note that the oscillation isgreatly reduced, especially in the interaction. It takesthe repetitive controller a few seconds to accommodatethis disturbance (or to learn it), but the disturbance willeventually vanish completely with time. The speed ofaccommodation (convergence) is directly related to themagnitude of the weighting function in the LQG controlproblem. The trade-o� is the increase in overshoot (30to 35%), which can be explained from comparison of


Time [sec]

0.5 1 1.5 2 2.50 3

0

0.2

0.4

0.6

0.8

1

1.2

-0.2

1.4Output step response

Figure 19. Closed-loop step responses to step demand on output 1 (black curves), output 2 (dark grey curves) and output 3 (lightgrey curves).

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Time [sec]

0.5 1 1.5 2 2.50 3

0

0.2

0.4

0.6

0.8

1

1.2

-0.2


Figure 20. Closed-loop step responses to step demand on output 1 (black curves), output 2 (dark grey curves) and output 3 (lightgrey curves) with periodic disturbance on input 1.

Frequency [Hz]

1 100.1 100

Gai

n [d

B]

-40

-20

0

-60

20Singular Values Closed Loop

Figure 21. Maximum (thick curves) and minimum (thin curves) singular values of the sensitivity function (dark grey) and com-plementary sensitivity function (black) of the closed loop with repetitive controller.

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® gure 18 with ® gure 21: the gain reduction at 5 Hz andits higher harmonics causes a gain increase at other,surrounding, frequencies. The peak in the sensitivityfunction in ® gure 21 is larger than in ® gure 18, whichcauses a less damped step response.

This trade-o� shows exactly the reason why it is soimportant to design the repetitive controller in con-junction with the observer-state feedback controller:damping, periodic disturbance rejection, speed ofconvergence, and other control objectives can betraded-o� in one general powerful framework. Forexample, if the overshoot is not acceptable, the conver-gence speed of the repetitive controller can be decreased,or the integral gain at low frequencies can be reduced.Note that the resulting MIMO 120-state repetitivecontroller can still be implemented as an add-on deviceaccording to ® gure 9.

6. Conclusions and preview on further research

This paper gives a general framework for the analy-sis and design of repetitive and learning controllersexplicitly derived from results available for the internalmodel principle. The internal model framework givesnecessary and su� cient conditions for existence of asolution to the problem of robust asymptotic trackingand rejection of periodic signals. The existence con-ditions allow for a proper choice between a repetitive

or learning controller, dependent on location of zerosand number of inputs and outputs of the plant. Onceexistence of a repetitive or learning controller has beenveri® ed, the design of such a controller boils down to thedesign of a stabilizing compensator for the seriesconnection of the plant and an internal model of theperiodic signal, using any model-based control designtechnique, such as LQG, H1 and/or ·-synthesis. Also,model-based predictive control (MBPC) ILC schemes,as used and exploited by, e.g. Amman et al. (1998), canbe straightforwardly brought into this frameworkbecause of its model-based nature.

Consequently, the analysis and design of thesemodel-based repetitive and ILC approaches can be gen-eralized to the powerful analysis and design procedureof the internal model framework, allowing to trade-o�the convergence speed for periodic-disturbance cancella-tion versus other control objectives, such as stochasticdisturbance suppression by using appropriate weightingfunctions and design parameters. An example for aMIMO aircraft model showed the importance of thesetrade-o� s.

Further research will focus on the actual use in con-trol design, for which it will be necessary to address thecomputational complexity. This is because the internalmodel of the periodic signal can have large dimensions:the state dimension equals the number of samples N of


Time [sec]

0.5 1 1.5 2 2.50 3

0

0.2

0.4

0.6

0.8

1

1.2

-0.2


Figure 22. Closed-loop step responses with repetitive controller to step demand on output 1 (black curves), output 2 (dark greycurves) and output 3 (light grey curves) with periodic disturbance on input 1.

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one period. In real life applications this number caneasily exceed 1000. For this reason it would be interest-ing to investigate the use of basis functions to arrive atreduced order internal models. Note that for the aircraftexample, the order could have been reduced by onlyincluding an internal model for the oscillators at 5, 10and 20 Hz, which boils down to including an 18-statecompensator. If only one or two harmonics are contri-buting to the output response, there is an advantage innot using the full repetitive controller: the gain increasein the unimportant harmonics can be used to balancethe trade-o� in favor of other performance require-ments. From implementation point of view, the repeti-tive controller might still be favourable: its internalmodel can be implemented as a First-In, First-Out(FIFO) bu� er or memory loop, as opposed to an inter-nal model for oscillators. Further investigation in theseareas is necessary to balance all aspects in a systematicway.

References

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ROOVER, D. DE, BOSGRA, O. H., SPERLING, F . B., andSTEINBUCH, M ., 1996, High-performance motion controlof a ¯ exible mechanical servomechanism. Selected Topicsin Identi® cation Modelling and Control, 9, 69± 78.

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Documents

Internal-model-based design of repetitive and iterative learning controllers for linear multivariable systems