a39_Automatica

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Automatica 35 (1999) 1185}1199

Repetitive control of MIMO systems using H design

George Weiss*, Martin HaK fele

Department of Electrical and Electronic Engineering, Imperial College of Science, Technology and Medicine, Exhibition Road,

London SW7 2BT, UK

Institut fuKr Systemdynamik und Regelungstechnik, UniversitaKt Stuttgart, 70550 Stuttgart Vaihingen, Germany

Received 10 September 1997; received in "nal form 3 November 1998

Abstract

In many engineering problems, periodic signals which should be either tracked (reference signals) or rejected (disturbances) occur.

A successful way of solving such problems is repetitive control, a well-known technique based on the internal model principle. Here,

we extend repetitive control theory to plants with several outputs, of which only some have to track reference signals. The other

outputs are used to supply additional information to the controller. We analyze stability, robustness and give estimates of the size ofthe error in such a feedback system. Our approach to this analysis is new, based on the recent theory of regular linear systems. We

introduce a correction to the amount of delay used in the internal model and this leads to a signi"cant improvement in performance

(i.e., to a smaller error). 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Repetitive control; Internal model; Delay line; Regular linear system; Exponential stability; w-Stability; Hcontrol theory

1. Introduction

Many signals in engineering are periodic, or at least

they can be well approximated by a periodic signal over

a large time interval. This is true, for example, for mostsignals associated with engines, electrical motors and

generators, power converters. Thus, it is a natural control

problem to try to track a periodic signal with the output

of a plant, and/or (what is almost the same), to try to

reject a periodic disturbance acting on the same plant.

Repetitive control achieves this goal by the internal

model principle introduced by Francis and Wonham

(1975), with an in"nite-dimensional internal model. Such

a model is obtained by connecting one or more delay

lines into a feedback loop. This approach was "rst pro-

posed 18 years ago, see Inoue et al. (1981a, b). The subject

has reached maturity with the papers by Hara et al.

(1988) and Yamamoto (1993). We refer to HillerstroKm

(1996) and to HillerstroKm and Sternby (1996) for the

discrete-time version of repetitive control and to Hiller-

stroKm and Walgama (1996) for a survey and further

references. Robustness of sampled-data repetitive control

systems (with respect to structured time-varying per-

*****

* Corresponding author. E-mail: [email protected].

This paper was not presented in any IFAC meeting. This paper

was recommended for publication in revised form by Associate Editor

A. L. Tits under the direction of Editor T. Bas'ar.

turbations) was considered in Langari and Francis

(1995, 1996). An important recent reference on robust-

ness issues in repetitive control is Lee and Smith (1998),

which is a part of the thesis of R.C.H. Lee. A closely

related area of control theory is iterative learning control,see for example the books of Moore (1993), Rogers and

Owens (1992) and the thesis of Amann (1996).

In this paper, we consider the plant to be a "nite-

dimensional, linear time-invariant (LTI) system, work-

ing in continuous time. This plant is MIMO (multiple-

input multiple-output). We assume that the period

of the external signals (references and disturbances) is

known.

We assume that one part of the plant output has to

track a reference (possibly zero, if we only consider the

disturbance rejection problem), while another part of the

output is available to supply additional information to

the controller, and thus (hopefully) improve the perfor-

mance of the feedback system. If the controller contains

an internal model based on delay lines and if it uses the

second part of the output as described above, then we call

the closed-loop system a repetitive control system with

additional measurement information(see Fig. 1). This gen-

eralization of repetitive control, and the numerical design

of the corresponding controllers, have been made pos-

sible by the emergence ofHcontrol theory and by the

availability of reliable software tools which can solve the

standardH problem.

0005-1098/99/$- see front matter 1999 Elsevier Science Ltd. All rights reservedPII: S 0 0 05 - 1 0 9 8 ( 9 9 ) 0 0 0 3 6 - 9


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Fig. 1. A repetitive control system with additional measurement in-

formation y. P is the transfer function of the plant to be controlled,

M denotes the transfer function of the internal model and C is the

transfer function of the compensator that stabilizes the feedback sys-

tem. The external signal w contains both the disturbances and the

references, and e is the error which should be kept small.

In the literature on repetitive control that we are aware

of, the amount of delay used in the internal model is

equal to the period of the external signals. We found that

actually, a slight adjustment of this delay leads to better

results. This correction is explained in Section 2, after the

general structure of our repetitive control systems has

been presented.

One important issue which we must address is the

stability and robustness of the repetitive control system.

We show in Section 4 that under certain realistic assump-

tions, repetitive control systems designed as proposed

here are exponentially stable and also w-stable in the

sense of Georgiou and Smith (1993). Our analysis uses

the recently developed theory of regular linear systems

and their dynamic stabilization: see Weiss (1994a) and

Weiss and Curtain (1997). This approach to repetitivecontrol originates in the short paper of Weiss (1997)

which deals with the repetitive control of SISO plants,

but with several distinct periods (multi-periodic repeti-

tive control).

Another important issue is to estimate the size of the

error signal in a repetitive control system, when periodic

external signals are present. Using techniques from regu-

lar linear systems, we show in Section 5 that the error can

be decomposed into a periodic steady-state part and

a transient part, and the transient component converges

to zero in a suitable sense (the norm over one period

decays exponentially). If the external signals have deriva-

tives in

(as we would expect in engineering applica-

tions), then we get exponential convergence to zero of

the transient part. We estimate the norm over one

period of the steady-state error and draw conclusions

about how to design the controller to make this norm

small.

In Section 6 we give a design procedure for the control-

ler and in Section 7 we present a simple design example

(the control of a DC motor) with simulation results. In

this example, we demonstrate the bene"ts of using an

additional measurement signal from the plant and also

the bene"ts of the correction to the amount of delay in

the internal model.

2. The structure of a repetitive control system

The general structure of the control systems which we

study is shown in Fig. 1. In this "gure P denotes thetransfer function of the plant, which is a "nite-dimen-

sional LTI systemP, C is the transfer function of thestabilizing compensatorC, which is also a "nite-dimen-sional LTI system, andMdenotes the transfer function of

the internal model M, which is an in"nite-dimensionalregular linear system. &&Stabilizing'' for C means that it

satis"es certain conditions listed in Theorem 5. These

conditions imply that if we would choose M"I, then the

feedback system in Fig. 1 (which would become "nite-

dimensional with this M) would be stable.

The plant has two inputs, w (t)3 and u (t)3, and

two outputs,e

(t)3

and y

(t)3

. The external inputw contains disturbances, which have to be rejected,

and/or reference signals, which should be tracked. The

control input ofPis u. The"rst outpute is the trackingerror, andy is the additional measurement information.

MandCare the two parts of the controller. Neither ofthe three subsystems in Fig. 1 is stable in general, but the

controller must be designed such that the whole repeti-

tive control system is exponentially stable.

It is not di$cult to compute that the transfer function

fromw to e is

G"SMG, (1)

where, after partitioning

P"P

P

P

P and C"[C C],

SM"(I!P(I!C

P

)C

M), (2)

G"(P

#P

(I!C

P

)C

P

).

Using the formula (I!)"I#(!),SMcan be rewritten as

SM"I#P(I!C

P!C

MP

)C

M.

Note thatGwould be the transfer function fromwtoeifM were zero.

The remainder of this section is devoted to the design

ofM. The design ofCneeds a lot more preparation and it

will be discussed in Section 6. The internal model should

be capable of generating signals very similar to w(precise

equality is not necessary, and in fact it is not possible in

our case). We assume thatw is periodic with period, sothat the Fourier expansion

w(t)"

we (3)

1186 G. Weiss, M. HaKfele/Automatica 35 (1999) 1185}1199


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Fig. 2. (a) The structure of the internal model M, which consists of

qcopies of a feedback connection of a delay line and a "lter W . Here

(s)"exp(!s)I. (b) The location of the poles of the internal model

M. A cross denotes the actual location of a pole and a dot denotes an

ideal pole location, corresponding to" and W"1. For low fre-

quencies the two almost overlap.

holds, where"2/is called the fundamental frequencyof w. We choose M to be the following qq diagonalmatrix:

M(s)"I/(1!eW(s)), (4)

whereW is a real-rational, stable SISO transfer function

with

W

41. (5)

In Eq. (5), W

denotes the H norm ofW . The time

delay'0 can be chosen equal to the periodofwbut,

as we shall see, it is actually better to choose it slightly

smaller. A transfer function M as in Eq. (4) can be

obtained by connecting q delay lines (s)"eI (hereI is the qq identity matrix) with q copies of the "lterW into a feedback loop, as shown in Fig. 2a.

We would like to choose

and the"lterW such as to

keep the error e as small as possible. Ideally, we would

like to take " and W"1 for the following reason:with this choice, the internal model M would be

M(s)"I

1!e.

and this function has in"nitely many poles on the imagi-

nary axis, namely ik, wherek3. Thus, assuming rankP

(ik)"qand a suitableC, we getG(ik)"0, as can beseen from Eqs. (1) and (2). Ifw is as in Eq. (3), then its

Laplace transformwL has simple poles at a subset of the

points ik. However, the zeros ofG cause that eLwill nothave poles at the points ik, so that there will be no

steady-state error. In practice we cannot choose W"1,for three reasons:

(i) One reason is technological: We cannot realize a de-

lay line with in"nite bandwidth, and thus any delay

line that we can build can be modelled as an ideal

delay line in series with a low-pass or a band-pass

"lter.

(ii) Our su$cient conditions on C which imply that

C stabilizes the whole system (see Theorem 5) are

such that for most plants, they would become im-

possible to satisfy if we had W"1.

(iii) IfW"1, then the resulting feedback system, even if

it is stable, is not robustly stable with respect to

delays. This means that arbitrarily small delays con-

nected in cascade withM will destabilize the system.

We shall say more about the di$culties (ii) and (iii) in

Remark 6. We can overcome the three problems above

by imposingW(R)"0. To avoid the robustness prob-

lem described in (iii), an additional condition on P andC is needed, see the second part of Theorem 5.

The above considerations show that Wmust be a non-

constant"lter. However, we try to choose it as close to

1 as possible in the frequency band of interest. Ifwis as in

Eq. (3), then we say that it is con,ned (or limited) to

a frequency band [,

]L(0,R) if w

"0 for

k[,

]. Often the lower limit is

"0, but the

upper limit must satisfy(R. All signals appearing in

engineering are con"ned to some frequency band, or we

may assume that they are con"ned to one ifw is very

small for allk outside this band.

We assume that is much smaller than !, sothat there are many relevant frequencies in [

,

].

Indeed, otherwise it is not worth using repetitive control,

since a "nite-dimensional internal model might be more

appropriate. Ifw is con"ned to a certain frequency band

[,

], then we chooseW such thatW(i) is very close

to 1 for3[,

]. Then, with an appropriate choice of

, M will have poles very close to ikfor thosek3 for

whichk3[,

]. Thus, G will have zeros very close

to these points ik, causing the steady-state part ofeto besmall (as will be shown in Section 5).

Often"0, and then the simplest choice for W is

W(s)"1/(1#s). (6)

Denoting"1/, we choose such that

, (7)

where means&&much smaller''. But we must be carefulnot to choose

too large, because this might lead to an

unsolvable H problem when we try to design C, or it

might lead to a very large gain ofC. Some trial and error

may be necessary for the right choice.

Now we get to the problem of choosing the delay

. If

Wis a band-pass"lter, then the best choice we can think

of is". However, ifWis a low-pass "lter as in Eq. (6),

then we write eW(s)"eW(s), where W(i)should be as close as possible to 1 for4

. This is in

order to get the poles ofM as close as possible to ik, asexplained earlier in this section. Since the Taylor expan-

sions of e and of W(s) are well known, we cancompute the"rst terms of the Taylor expansion ofW

:

W

(s)"1#(!!)s#O(s).

Thus, a good way of gettingW

(i) close to 1 for lowis to choose

"!. (8)

G. Weiss, M. HaKfele/Automatica 35 (1999) 1185}1199 1187


4/15

The same argument remains valid for any "lterW whose

Taylor expansion is W(s)"1!s#O(s). Since byEq. (7) , formula (8) can be understood as a smallcorrection subtracted from the valueused in previousreferences. We shall see in the design example in Sec-

tion 7 that this correction actually helps to reduce the

steady-state error signi"cantly. The poles of M will be

located close to the imaginary axis, slightly to the left.For(

, they will be very close to ik. The location

of the poles can be imagined as in Fig. 2b, where a cross

denotes the actual placement of the poles ofM and a dot

indicates the ideal placement corresponding to "

andW"1.

3. Some background

In this section we recall some facts about regular linear

systems and their dynamic stabilization. For a more

detailed discussion on well-posedness and regularity we

refer to Weiss (1994a, b) and to Weiss and Curtain (1997).

We make the following convention: If a meromorphic

function is de"ned on some right half-plane and can be

extended meromorphically to a greater right half-plane,

we will not make any distinction between the initial

function and its extension. This will not lead to con-

fusions, since the extension, if it exists, is unique. Indeed,

an analytic function on a connected domain is uniquely

de"ned by its series expansion at one point.

For each 3, H

denotes the space of bounded

and analytic functions on the right half-plane "

s3Re(s)'. With the norm

G"sup

G(s), (9)

H

is a Banach space. With our earlier convention, we

have that

HLH

if4.

For"0 we use the notationHinstead ofH

. By the

maximum modulus principle, the Hnorm of a transfer

function G3H, denotedG

, can be interpreted as

the peak value in the Bode amplitude plot of G. The

space H

has natural generalizations for vector-valued

and for matrix-valued functions. We shall use the nota-

tion H also for spaces of vector- and matrix-valued

functions, and we shall often indicate the range space in

parentheses. In formula (9), absolute values should now

be replaced by the natural vector and matrix norms

(Euclidean norm and greatest singular value).

De5nition 1. A -valuedwell-posed transfer function

is an element of one of the spaces H

(), for some

3. Such a function G is called regular if the limit

D" lim

G() (3)

exists. In this case, D3 is called the feedthrough

matrixofG.

For example, any well-posed transfer function obtain-

able from rational functions and delays by "nitely many

algebraic operations is regular (this includes all the trans-

fer functions which arise in this work).

Let be an LTI system with input space

, statespace X and output space . We assume that X is

a Hilbert space. For a vector-valued function u de"ned

on [0,R), we denote by Puits restriction to [0,]. The

systemis calledwell-posed, if on any"nite time interval[0,], the operator from the initial state x(0) and theinput function P

u to the "nal state x () and the output

functionPy is bounded. Thus, we can write

x ()

Py"

)

x(0)

Pu ,

where x(0)3X, P

u3([0,], ), x()3X,

Py3([0,],) and the 22 matrix consists ofbounded operators between the appropriate spaces. The

precise de"nition of a well-posed linear system is more

complicated, since the families of operators ,,, must satisfy functional equations expressing causality

and time-invariance (one of these is Eq. (10) below). For

the details we refer to Salamon (1989) or Weiss (1994a).

We write"(,,, ).We recall some facts about well-posed linear systems

which will be used later. First of all, is a strongly

continuous semigroup of operators onX. The system iscalled exponentially stable if the growth bound of is

negative. For any50, we denote by S the right-shiftoperator by , acting on

([0,R), ). Similarly,S*

will denote the left-shift operator on the same space. If

u,v3

([0,R), ), their-concatenationis de"ned by

u

v"Pu#S

v.

The following functional equation holds for all50:

(u

v)"u#

v. (10)

There exist two linear operators

: XP

([0,R), ) and

:

([0,R), )P

([0,R), )

such that for every50,

"P

, "P

.

For every 3, we denote by

([0,R), ) the

space of all functions of the form v (t)"eu(t), whereu3([0,R), ). By de"nition, v

"u

. Let us

denote bythe growth bound of. Then for every',we have that

3L(X,

([0,R), )) and

3L(

([0,R), ),

([0,R), )). The operator

is shift-invariant, i.e.,

S"S

for all 50.This implies that

can be represented by a well-

posed transfer function G in the following sense: If



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u3

([0,R), ) andy"

u, then the Laplace trans-

forms ofu andyare related by

yL(s)"G(s)uL(s) for Re (s)'max,.

G is called the transfer function of . We haveG3H

() for all'.

The system is called regular if G

is regular. Suchsystems have a simple description via their generating

operators A, B, C, D, which are the analogues of the

matrices appearing in the usual representation of"nite-

dimensional linear systems. The operators A,BandCare

unbounded in general. A is the generator of the semi-

group . The following result has been proved by

Rebarber (1993): If a regular linear system is stabilizable,

detectable and its transfer function is in H(), then

the system is exponentially stable. For a precise de"ni-

tion of stabilizability and detectability in this framework

we refer to Rebarber (1993) and to Weiss and Curtain

(1997).

The simplest example of a regular linear system withunboundedA, Band C is a delay line. Its transfer func-

tion is e, with '0. Its natural state-space is

[0,

], and for x3D(A),

(Ax)()"dx()

d , B"

, Cx"x(0) and D"0.

The domain of the generator is

D(A)"x3H[0,] x(

)"0,

where H[0,

] denotes the subspace of [0,

]

consisting of absolutely continuous functions whosederivative is in [0,

]. The growth bound of is!R.

C is bounded from D(A) (with the graph norm) to

and B is bounded from to D(A*). (These A,B,C,Dwill not play any role in this paper.) Basically, this

delay line is the only regular linear system needed in this

paper. More precisely, all the regular systems which we

shall meet, are obtained by connecting delay lines and

"nite-dimensional LTI systems in feedback loops.

Such feedback systems, if at all well-posed, are always

regular.

In the sequel we turn our attention to the feedback

connection. LetP and C be the transfer functions of twowell-posed linear systems P andC. We connect thesesystems like in Fig. 3 (for this, we assume that the dimen-

sions of P and C are such that PC and CP exist). We

denote the closed-loop transfer function from [r d] to

[e u] by L. This transfer function is de"ned where

I#PC is invertible, or equivalently, where I#CP is

invertible, and

L" I

!C

P

I"

(I#PC) !P(I#CP)

C(I#PC) (I#CP) .

Fig. 3. The standard feedback connection of two well-posed linearsystems. We may think ofPas the plant,Cas the compensator, ras the

reference signal,das the disturbance signal ande as the tracking error.

De5nition 2. With P, C and L as above,

(i) C is an admissible feedback transfer function for P if

L is well-posed,

(ii) C stabilizes P ifL3H,

(iii) Cexponentially stabilizesPifL3H

for some(0.

We now recall a result from"nite-dimensional systems

theory. For matrix-valued rational functions, &&well-

posed'' is equivalent to &&proper''. Proper rational func-

tions are always regular. For the concept of pole-zero

cancellation and other details we refer to the survey

paper of Logemann (1993).

Proposition 3. Suppose that P and C are proper rational

matrix-valued transfer functions. hen C is an admissible

feedback transfer function for P if and only if

det(I#P(R)C(R))O0. C stabilizes P (equivalently,

C exponentially stabilizes P) if and only if

(I#PC)3Hand there are no unstable pole-zero can-

cellations in PC.

Now letP andC be well-posed linear systems withtransfer functions P and C. We assume that C is an

admissible feedback transfer function for P. Then the

feedback connection in Fig. 3 de"nes a new well-posed

linear system called theclosed-loopsystem corresponding

toP andC.The results of Rebarber mentioned earlier imply the

following theorem. For the proof, the reader may consult

Weiss and Curtain (1997, Section 4).

Theorem 4. etP andC be stabilizable and detectableregular linear systems, with transfer functions P and C,

respectively. =e assume that C is an admissible feedback

transfer function forP.hen the corresponding closed-loop

system is exponentially stable if and only ifC stabilizesP.

IfP andC are exponentially stable, then the state-ment of the above theorem becomes very simple. Indeed,

there is no need to mention stabilizability and detectabil-

ity, since they follow from stability. Moreover, C stabil-

izes P if and only if (I#PC)3H, or equivalently

(I#CP)3H.



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4. Stability and robustness

We consider two "nite-dimensional LTI systems

PandCwith transfer functionsPandC, partitioned asin Section 2:

P"

P

P

P P

and C"[C

C

].

The dimensions of P and C are (q#p)(r#m) andm(q#p). The regular linear systemM is obtained byconnecting q delay lines (realized as explained in Sec-

tion 3) into a feedback loop, as shown in Fig. 2a, whereW is the transfer function of a stable "nite-dimensional

LTI system W and W41. The transfer function

MofMis given in Eq. (4). We interconnect the systemsP,Cand Mas shown in Fig. 1, obtaining a repetitivecontrol system with the general structure studied here.

We now state our main result about its stability and

w-stability.

Theorem 5. =ith P, C and M as described above,assume that the following conditions are satis,ed:

(i) he interconnection ofP andC shown in Fig. 4a isstable.

(ii) =e haveS(1, where

S"W(I!P

(I!C

P

)C

). (11)

hen the repetitive control system shown in Fig. 1 is

exponentially stable.

If moreover the conditions

W(R)"0 and P

(R)

P

(R) ) C(R)(1 (12)hold, then the feedback system in Fig. 1 is w-stable, in

particular,it is robustly stable with respect to multi-delays.

For the de"nition ofw-stability we refer to Georgiou

and Smith (1989, 1993). Roughly, it means that the feed-

back system remains stable if the high frequency behav-

ior of P, M and C is changed. For the de"nition to be

applicable, the blocks P and M have to be combined

into a single block. Alternatively, we could combineC and M into a single block, and this view would lead

to the same result, as stated in Theorem 5. The concept

of w-stability is stronger than robustness with respect

to multi-delays, as de"ned in Logemann et al. (1996,

Section 6).

Note that the system in Fig. 4a is"nite dimensional. Its

stability is of course independent of the "lter W. Note

also thatSfrom Eq. (11) is the transfer function from ato

bin Fig. 4a. Like S

after Eq. (2), S can be rewritten as

S"W(I#P

(I!C

P!C

P

)C

).

Fig. 4. (a) The interconnection of the plant

(with transfer function

P) and the controller

(with transfer function C) as in Fig. 1, but

without the internal model (M"1) and with a "lter W. The transfer

function froma to b is denoted by S. (b) A feedback system equivalent

to the one in Fig. 1 when no inputs are present, i.e., w"0. The

subsystem with transfer functionScontains everything except the delay

lines, and it is shown in detail in Fig. 4a. Recall that the diagonal matrix

(s)"eI represents the q delay lines.

The above theorem does not make any mention of the

signal win Fig. 1 (its possible periodicity), nor of the error

e. It is concerned only with stability and w-stability. The

w-stability conditions (12) can sometimes be relaxed, see

Example 7. Note that all the conditions are independent

of the delay

.

Proof. First we prove the exponential stability of the

feedback system. For this, we transform the control sys-

tem of Fig. 1 into the one shown in Fig. 4b, with no input

and no output. Here, the systemSwith transfer functionScontains the plant, the compensator and the "lter, but

not the delay lines. As in Section 2,(s)"eI. It is notdi$cult to see that S is exactly the system shown inFig. 4a. By assumption (i) of Theorem 5 we have that

S is stable. Its transfer function from a to b is S .According to Theorem 4 and the comments after it, the

system in Fig. 4b is exponentially stable if and only if

(I!S)3H. Since "1, it follows from as-

sumption (ii) of Theorem 5 that this is indeed the case.

Now we assume that Eq. (12) holds as well. Then,

according to Georgiou and Smith (1993, Section 8), the

feedback system in Fig. 1 is w-stable if

M(R) 0

0 IP

(R)

P

(R) ) C(R)(1.Indeed, since W(R)"0, P , C and M all have limits assPR in

, uniformly with respect to the argument of

s. Since M(R)"I, the above condition for w-stability

reduces to the second part of the conditions (12).

Remark 6. As promised in Section 2, we now comment

on the di$culties which would result from choosing the



7/15

"lter W"1. If W were allowed to be 1, then for most

plants the conditionS(1 would become impossible.

Indeed, ifW"1 and rankP

(s)(qfor somes3

, or

for s"R, then we see from Eq. (11) that S51.

Moreover, with W"1 the repetitive control system

would not be robustly stable with respect to delays, in the

sense of Logemann et al. (1996). This means that arbitrar-

ily small delays in the feedback loop (e.g., on the signal u)would destabilize the system. This follows from the main

result of Logemann et al. (1996). In particular, the feed-

back system would not be w-stable. The lack of robust-

ness for in"nitely many poles ofM on the imaginary axis

can be understood also in the context of the analysis of

Lee and Smith (1996, 1998), who use the gap metric

robustness margin to evaluate the system.

The problem now is to "nd a compensator C such

that the conditions (i) and (ii) of Theorem 5 are satis"ed.

We shall formulate this as a standardHproblem. This

class of problems has been extensively studied in therecent robust control literature and good algorithms (and

programs) for their solution are available, see for

example Green and Limebeer (1995) or Zhou et al. (1996).

It may happen that the problem has no solution. In our

speci"c case, we may overcome this by choosing a less

restrictive weight function W, which means that the

frequency band [,

] in which W(i) is very close

to 1 gets narrower, and thus the steady-state error

may grow.

Example 7 (SISO control system). Now we restrict our-

selves to a plant with transfer functionP

which is SISO,

and hence to a compensator with transfer functionC which is SISO as well. Thus, there is no additional

measurement informationy, so that the closed-loop sys-

tem of Fig. 1 reduces to the one in Fig. 5. Apart from this,

our analysis is quite general, so that this discussion is not

really an example, it concerns a class of repetitive control

systems. Repetitive control systems as in Fig. 5 have

been analyzed in several earlier papers, and our notation

follows Weiss (1997).

To simplify the exposition, we assume that the plant

and the compensator are minimal (and hence we may

represent them by their transfer functions). We have

w"

[r d]

, where r is the reference and d is the dis-turbance, so that here P"[I !P!P

]. Thus,

P"[I !P

], P

"!P

and P

and P

do not

exist. We haveC"Cand C

does not exist. The condi-

tions (i) and (ii) of Theorem 5 become simpler. Indeed, the

stability of the feedback system from Fig. 4a required in

(i) is equivalent to the fact that C stabilizes P

. The

transfer function S from (11) is the weighted sensitivity:S"W(1#P

C), and condition (ii) has now the

simpler form

W(1#P

C)(1. (13)

Fig. 5. SISO repetitive control system, the classical structure.M is the

internalmodel, C is the stabilizing compensator andP

is the plant. The

reference r and the disturbance d are periodic with period , and theerror e should be kept small.

The problem of"nding a stabilizing compensatorC that

satis"es the condition (13) is called the weighted sensitiv-

ity H problem.

The su$cient conditions onw-stability in Eq. (12) may

be relaxed to

W(R)(1 and P

(R)C(R)(1!W(R),

and they appear in this form in Weiss (1997). Theseresults have a partial MIMO extension in which P

and

Cbecome MIMO systems, but we still use the (somewhat

restrictive) block diagram of Fig. 5. The stability part of

Theorem 5 in this context (withoutw-stability) is due to

Hara et al. (1988) (see their Corollary 1).

5. Estimates for the errore

In this section we give some estimates of the size of the

error e in the repetitive control system of Fig. 1. We

decompose the errore into two parts, a steady-state and

a transient part, e"e#e, where e is periodic ande3

([0,R), ) for some (0. The reader should

recall the notation and the material on well-posed linear

systems from Section 3.

Lemma 8. If"(,,, ) is an exponentially stablewell-posed linear system and u3 ([0,R), ), then

lim

u"0.

Proof. For each t50, we split u into two parts,

u"u

v,

wherev"S*

u. Then by the functional equation (10),

u"

(u

v)"

u#

v.

Using the fact that if is exponentially stable, then

4Mfor allt50, as proved, e.g., in Weiss (1989), we

get

u4Mu#Mv

.

SinceP0 andv

P0, we obtain the claim to be

proved.



8/15

Lemma 9. et G3H

(pm) for some (0. Assumethat the input function u3

([0,R), ) has the

decomposition

u"u#u

, (14)

where u

is periodic with periodand u3

([0,R), )

with440. If y is the corresponding output function,yL"GuL,then y3

([0,R), )has a similar decomposi-

tion:

y"y#y

, (15)

where y

is periodic with period and y3

([0,R), ).

Moreover,if the vector-valued sequences(u)and(y

)are

the Fourier coe.cients of u

and y

, i .e.,

u"

ue and y

"

ye

with"2/ ( is the fundamental frequency), then

y"G(ik)u

. (16)

If is an exponentially stable well-posed linear systemwith transfer functionG, and if u from Eq. (14) is its input

function, then for any initial state, the state trajectory of approaches a limit cycle.

We remark that (u) and (y

) are vector-valued l se-

quences and the two Fourier series appearing in the

lemma converge in the -sense on any"nite time inter-

val. Note that ifu"0, we can take ", so that the

transient output signal y

is in

([0,R), ). This case

will be used later.

Proof. We know from Salamon (1989) (see also Sta!ans,

1999) that G has a realization"(,,, ) such thatthe growth bound of the semigroup is and

3L(X,

([0,R), )), (17)

where X is the state space of. We show that the statetrajectoryx corresponding to any initial state x(0) and

the input u from Eq. (14),

x(t)"x(0)#

u (18)

approaches a limit cycle x

. More precisely, there exists

a continuous X-valued periodic function with period ,denoted by x

, such that

lim

x(t)!x

(t)"0. (19)

To prove Eq. (19) (i.e., the last part of the lemma), we do

not need Eq. (17), only the exponential stability of. The

functionx

must satisfy

x

(t)"x

(0)#u

. (20)

Because of the periodicity (x

()"x

(0)), we must have

(I!) x

(0)"

u

. (21)

Since is exponentially stable, the spectral radius ofis

less than one. Thus, I!

is invertible, and we can

de"ne the initial state ofx

by

x(0)"

(I!

)

u .For all other t, x

is now de"ned by Eq. (20), and it is

easy to see that this function is indeed periodic. Using

Eqs. (18) and (20), we get

x(t)!x

(t)"x(0)#

u!

x

(0)!u

.

Since is exponentially stable, Eq. (19) holds if

lim

u!

u

"lim

u

(t)"0

and according to Lemma 8 this is the case.

Now we have to prove that also y can be decomposed

into a periodic part y and a transient part y. Theoutput function y is given by

y"

x(0)#

u.

We de"ne

y"

x

(0)#

u

,

y"

[x (0)!x

(0)]#

u

, (22)

so that clearly Eq. (15) holds. First we show that y

is

indeed periodic with period . Recall that by S*

we

denote the left-shift operator by . Then the functionalequations for and (see Weiss, 1994a) can be com-bined into

S*y"

x()#

S*u.

This equation applied to x

and u

yields

S*y

"

x

()#

S*u

"

x

(0)#

u"y

.

The fact that S*y

"y

means that y

is periodic with

period.Now we turn toy

from Eq. (22). The"rst summand of

y

is in

([0,R), ), because of Eq. (17). Since 4,

this implies that it is also in ([0,R),

). SinceG3H

(), we know that for every 5,

is

a bounded operator from

([0,R), ) to

([0,R), ). Since u3

([0,R), ), the second

summand ofy

in Eq. (22) also in

([0,R), ). Thus

y3

([0,R), ) and we have proved the "rst part of

the lemma.

According to the Fourier series expansion of u

, its

Laplace transform is

uL

(s)"

u

1

s!ik.



9/15

Since (u)3l, this series converges absolutely for all

s3 which are not of the form ik. Thus,uL

has poles at

ik for every k3, and u

is the corresponding residue,

i.e., u"Res(uL

, ik). Since uL

is analytic in

(because

u3

([0,R), )), the residues ofuL"uL

#uL

can be

computed: u"Res(uL, ik). Similarly, using Eq. (15),

y"Res(yL, ik). Since yL"GuL,

y"Res(GuL, ik)

"G(ik) Res(uL, ik)"G(ik)u.

We remark that the Laplace transforms of y

and

y

can be computed using the last lemma:

yL

(s)"

G(ik)s!ik

u,

yL

(s)"

G(s)!G(ik)s!ik

u#G(s)uL

(s).

A decomposition similar to Eq. (15), although in adi!erent context, has appeared in Langari and Francis

(1995, 1996).

We apply this lemma to the repetitive control

system in Fig. 1. We assume that the external signal

w3

([0,R), ) is periodic with period , as in (3).We show that the error signale can be decomposed into

e

ande

and we give the formula for the computation of

the Fourier coe$cients ofe

.

Proposition 10. =ith the notation ofheorem 5, assume

that the conditions(i), (ii)and (12)are satis,ed. hen there

exists an(0 such that the following properties hold:(I) he transfer functionGfrom wtoe in the system from

Fig. 1, which was given in Eq. (1), satis,es

G3H

().

(II) Suppose that w3

([0,R), ) is periodic with

period , as in Eq. (3). hen the error e in the re-petitive control system from Fig. 1, for any initial

state, can be decomposed as e"e#e

, where

e3

([0,R), ) and e

is periodic with period.

Moreover,let (e)be the sequence of Fourier coe.cients of

e

, similarly as in Eq. (3), and let "2/. hen for all

k3,

e"G(ik) w

. (23)

Proof. According to Theorem 5, the feedback system in

Fig. 1 is exponentially stable. Thus, the growth bound of

its semigroup is negative:

()(0. The statements in

the above proposition hold for any '

(). Indeed,

property (I) holds since the growth bound of G is less

than the growth bound of.

Now we prove property (II). Sincew3

([0,R), )

is periodic with period, it is clear that the transient part

of this input signal is zero. If the initial state of the system

is zero, theneL"GwL . According to Lemma 9 with",ecan be decomposed as claimed in the proposition. If the

initial state is not zero, then it generates a signal which

belongs to

([0,R), ), so that it can be absorbed into

e

. Applying formula (16) to this particular situation, we

obtain Eq. (23).

We want to emphasize the strength of the conclusion

e3

([0,R), ) in the last proposition. This implies

that there is an M'0 such that

e

()d4Me for allt50.

Ifwis su$ciently smooth, then the above conclusion can

be strengthened:

Proposition 11. =ith the assumptions and the notation

of Proposition 10, if the derivative wR belongs to

([0,R), ), then

lim

ee

(t)"0.

Proof. We know from Proposition 10 that e"e#e

,

with e

periodic and e3

([0,R), ). If we

apply Proposition 10 to the derivatives, it follows that

also eR3

([0,R), ). De"ne v (t)"ee

(t), so that

v(t)3([0,R), ). Then from

vR(t)"ee

(t)#eeR

(t)

we see that vR3

([0,R

),

). By Barbalat's Lemma,v3 and vR3 imply that lim

v(t)"0, which is

what we had to prove.

Lemma 12. he transfer function SMfrom Eq. (2) can be

written as

SM"(I!W)(I!SW)S

, (24)

where

(s)"eI and S"(I!P

(I!C

P

)C

)

(this is like the transfer function SM but with the internal

model M"I).

Proof. Using that M"I!W, we have the follow-ing chain of equalities:

SM"M[M!P(I!C

P

)C

]

"M[S !W]

"M[S

(I!SW)]

"M(I!SW)S

.



10/15


11/15

wherex consists of the states ofP andWand

AI BI

BI

CI

DI

DI

CI

DI

DI

"

A 0 0 0 B

B

B

C

A

B

0 B

D

B

D

D

C

C

D

0 D

D

D

D

0 0 0 0 0

C

0 0 D

D

C

0 0

D

D

. (26)

The algorithm hinfopt of MATLAB ( The Math-

Works) which solves the standard H problem needs

DI

to have full column rank, respectively DI

to have

full row rank. To ensure this, the additional blocks

and

have been added. If the rank conditions for

DI

and DI

are already satis"ed without those blocks,

then of course we can omit them.

Minimizing the transfer function from wJ to zJ is not

the problem we really want to have solved. We want

a stabilizing compensatorC for whichS(1 (theseare conditions (i) and (ii) of Theorem 5) and we want the

ratio

/(1!) to be as small as possible (because of theestimate (25)). Note thatS"WS

is the transfer function

fromato bandS

G

is the transfer function from wto e,

all in Fig. 6. The arti"cial parameterhas been introduc-ed in order to search for the best ratio

/(1!) by

varying. By solving the Hproblem for various values

of,and, we are looking for the solution for which

/(1!) is positive and as small as possible.In our design examples (one of which is presented

in the next section) we have usually considered

and

temporarily "xed and made iterations over . Occa-sionally, we have changed the parameters

and

by

a factor of 10 or 0.1. We had to watch not only the ratio

/(1!) but also the Bode plots of the compensator C,to avoid compensators with too high gains at certain

frequencies, since these would be di$cult to implement.

7. A design example

In this section we apply the results of the previous

sections to the control of a DC-motor with "xed stator

"eld (e.g., a permanent magnet), whose movements (its

rotation angle) should track a periodic signal, without

being signi"cantly in#uenced by the periodic load torque.

For example, this motor might be driving (via some

gears) a joint in a robot arm, which has to perform

a periodic task. The equations which model the motor

have been taken from Ogata (1992, Chapter 5). By wedenote the angle of the motor shaft, by"Rthe angular

velocity, byMthe load torque, byuthe input voltage and

by i the armature current. Then

d

dt

i"0 1 0

0 0

0 ! !

i

#

0

!

0

M#

0

0

u.

We denote by r the reference angle and by e"r!the tracking error. For the sake of comparison, "rst we

consider that there is no additional measurement avail-

able. Then the plant P from Fig. 1 (with the signal y not

present) has the realization

xR"

0 1 0

0 0

0 ! !

x#

0 0

0 !

0 0

w#

0

0

u,

e"[!1 0 0]x#[1 0]w, (27)

wherex"[ i]

and w"[r M]

.Assuming that we have the additional measurement

information, the set of ordinary di!erential equationsremains the same, but we have a second output y of the

plant. Now the output ofP is given by

e

y"!1 0 0

0 1 0x#1 0

0 0w#0

0u. (28)We consider the following numerical values: the resist-

anceR"0.2, the inductance "20 mH, the momentof inertia of the rotor, with gears and joint

J"1.510kg m and the motor torque constant

(same as the back-electromagnetic force constant)k"910Vs/rad. For computing the compensator,we choose the following low-pass "lter:

W(s)"200

s#200. (29)

First we have considered the plant with the output as

in Eq. (27) (no additional measurement information).

After some trial and error we have chosen "10,

while

does not exist. As a result of some iterations, the

value of has been chosen to be 30, which has led to"0.69 and

/(1!)"75.59. The transfer function of

the corresponding compensator isC(s)"3.11710

(s#10.94)(s!241s#1.8510)

(s#4.3110) (s#200)(s!1.2210s#7.1810).

The factor 3.11710 might look very large, but if wedraw the Bode plot of this compensator, we see that its

gain is never larger than 58 dB. In fact, the factor

s#4.3110 in the denominator can be approximatedby the constant 4.3110 without causing any notice-able change in the frequency range of interest, and it is



12/15

Fig. 7. The referencer (left) and the disturbance load torque M (right). The horizontal axis shows the time in seconds.

this reduced controller which would be used in a practi-

cal implementation. The simulation results (see Fig. 8) are

not a!ected by this crude model reduction.

When we have used the additional measurement asin Eq. (28), then the search over the parameters has led us

to"10,

"10and "40. Now"0.48 and

/(1!)"3.16. The transfer function C"[C

C

] of

the corresponding compensator is given by

C

(s)"1.2110 (s#10)(s#7.2710)

(s#3.8710)(s#200)(s#5.0110),

C

(s)"!1.8510

(s#990)(s#10)

(s#6.6710)(s#3.8710)(s#5.0110).

Again we have very large factors in the expressions ofC

and C

. The Bode plot of C

shows that its gain is

always less than 70 dB and we consider this to be feasible.

We see that one pole and one zero ofC

are practically

equal to zero and so we eliminate the corresponding

factors from the expression ofC

. Additionally, we divide

the large pole ofC

by 500 and divide alsoC

by 500, so

that for relatively low frequencies, it remains practically

unchanged. Thus we obtain the approximation C

for the

component C

. The Bode plot ofC

is more problematic:

its gain gets close to 89 dB for frequencies between 10

and 10rad/s. However, this frequency range is far from

the frequency band of interest and therefore we can apply

some rudimentary approximation techniques, leading to

the approximation denoted by C

: We divide the two

large poles ofC

by 500 and at the same time, we divide

the whole transfer function by 500, so that the values of

C for low frequencies remain practically unchanged.Additionally, we approximate the pole which is very

close to zero by zero. We obtain the reduced compen-

sator C"[C

C

] de"ned by

C

(s)"2.4210 (s#10)

(s#7.7410)(s#200),

C

(s)"!7.410 (s#10)(s#990)

s (s#1.3310)(s#7.7410).

The gain of C

is obviously unbounded for very low

frequencies, because of the pole at zero, but for frequen-

cies above 2 rad/s it remains less than 35 dB. Thus the

reduced compensator is realistic for implementation. The

simulations have been carried out both with and without

compensator approximation, and the results have been

indistinguishable. Thus, using more sophisticated model-

reduction techniques for C would not make any notice-

able di!erence.

For the simulation, we assume that the reference signalris obtained by passing a rectangular signal of frequency

0.25 Hz and amplitude 20 rad through the low-pass"lter

F(s)"(1#0.25s)(1#0.04s). The disturbance signal

M is the superposition of three sinusoids:

M(t)"1.5sin t#1sin5t#0.7sin 6.5t.

Both r and M are plotted in Fig. 7. The fundamental

frequency of these signals is 0.25 Hz, and so "4 s. Tochoose

, we make the correction described in Section 2.

From Eq. (8) we obtain "3.995 s. The controller

consists of the internal model (4) with the "lter (29) and

one of the stabilizing compensators C that were com-

puted before.

If we simulate the repetitive control system without

additional measurement information, we get the results

shown in Fig. 8. On the left-hand side we see the graph of

the erroreover the "rst 20 s of the simulation, and on the

right-hand side we have zoomed into the time interval

from 55 to 59 seconds. Note that the scales of the plots

are di!erent. During the "rst 3.995 s, the internal model

behaves like the identity. After that, the error decreases

sharply as the signal from the delay line becomes active.

After approximately 3

the error has practically reached

its steady-state shape, and this is what we see in detail in

the right picture. The peaks of the steady-state error areapproximately 0.17 rad.

If we do the simulation with additional measurement

information with the compensator from Eq. (30), we

obtain the results shown Fig. 9, with the same time scales

as in Fig. 8. We can see that the results are much better.

Like before, during the"rst 3.995 s the error is relatively

large, but compared to the previous simulation, it is

already much smaller. The error decreases again sharply

after each period. When we look at the steady-state error

on the right, we see that practically only some peaks

remain. Compared to the case before, the peaks of the



13/15

Fig. 8. The errorebetweenrand for the controlled DC-motor, when the compensator hasnoadditional measurement information. The right pictureshows a portion of the curve where practically only the steady-state error is left. The horizontal axis shows the time in seconds. The period of the

external signals is 4 s.

Fig. 9. The error e between r and for the controlled DC-motor, when the compensator has additional measurement information. As inFig. 8, the right picture corresponds to the steady-state error and we see that this is much smaller than in Fig. 8, especially if we think in terms of

norms.

Fig. 10. The errore betweenr and for the controlled DC-motor, when the compensator has additional measurement information (as in Fig. 9) butwithout the correction of the time-delay. As in Figs. 8 and 9, the right picture corresponds to the steady-state error and we see that this is much larger

than in Fig. 9. The transient error is practically not a!ected by the correction.

steady-state error are approximately 6 times smaller, and

this signal is almost zero between the peaks (which occur

where the reference signal switches). Thus the improve-

ment is by a much higher factor if measured in terms of

the norm over one period.

Finally, we verify that the correction in Eq. (8) leads to

a smaller steady-state error. If we redo the simulation

with the compensator from Eq. (30) but with""4 s,

we get the graphs from Fig. 10. We can see that at the

beginning, the result is similar to the simulation result

with the same compensator and with the correction in

the time-delay. But when we look at the peaks of the

steady-state error, we see that they are approximately 12

times higher than previously. In the norm their ratio

would be even larger.

8. Conclusions and directions for future research

We have presented a fairly general design procedure of

repetitive controllers for MIMO linear plants in continu-

ous time. Our framework allows the use of additional

measurement information from the plant to improve

performance. Another improvement is achieved by a



14/15

correction of the delay time, making it slightly less than

the period of the external signals. The closed-loop system

is in"nite-dimensional, and we employ the theory of

regular linear systems. We can guarantee exponential

stability and w-stability of the feedback system under

fairly simple, checkable conditions, which do not depend

on the amount of delay. We have introduced the de-

composition of the error signal into a steady-state anda transient part, and have derived estimates for the for-

mer. These estimates have been used to build the design

procedure.

An important issue for further research is to allow

external signals of slowly varying shape, including vari-

ations of the period. Variations of the shape with un-

changed period are automatically taken care of by the

repetitive control system, and the transient response dies

down at an exponential rate. On the other hand, a signal

of varying period (strictly speaking, this is a contradic-

tion in terms) necessitates a mechanism for tracking the

period, and various ideas for this are available. The most

widespread approach is to use a phase-locked-loop, and

for some other approaches we refer to HillerstroKm (1996),

Reinke (1994), Russell (1986) and Tsao and Qian (1993).

An entirely di!erent direction for further research is to

allow external signals which are superpositions of peri-

odic signals of various (arbitrary) periods. This area,

called multi-periodic repetitive control, has been explored

in Garimella and Srinivasan (1994) and in Weiss (1997)

and the authors will present a more detailed investigation

of this in the near future (see also the preliminary report

Weiss and HaKfele (1998)).

Another technical question which is worth thinking

about is how to minimize the expression /(1!)appearing in our estimate (25) of the steady-state error,

over all controllers which satisfy the conditions in

Theorem 5.

Note. This work was carried out at the University of

Exeter, UK, Whilst Martin HaKfele was there as an Eras-

mus eschange student studying towards the MEng (Eur)

degree under the supervision of George Weiss.

References

Amann, N. (1996). Optimal algorithms for iterative learning control.Ph.D. thesis. School of Engineering, University of Exeter, UK.

Francis, B. A., & Wonham, W. M. (1975). The internal model principle

for linear multivariable regulators.Applied Mathematics and Optim-

ization, 2, 170}194.

Garimella, S. S., & Srinivasan, K. (1994). Application of repetitive

control to eccentricity compensation in rolling. Proceedings

of the American Control Conference, Baltimore, Maryland

(pp. 2904}2908).

Georgiou, T., & Smith, M. C. (1989). w-stability of feedback systems.

Systems and Control etters, 2 , 271}277.

Georgiou, T., & Smith, M. C. (1993). Graphs, causality and stabili-

zability: Linear, shift-invariant systems on [0,R). Mathematics

of Control, Signals, and Systems, 6, 195}223.

Green, M., & Limebeer, D. J. N. (1995). inear robust control. Informa-

tion and System Sciences Series. Englewood Cli!s, NJ: Prentice-

Hall.

Hara, S., Yamamoto, Y., Omata, T., & Nakano, M. (1988). Repeti-

tive control system: A new type servo system for periodic exo-

genous signals. IEEE ransactions on Automatic Control, 33,

659}668.

HillerstroKm, G. (1996). Adaptive suppression of vibrations *a repeti-

tive control approach. IEEE ransactions on Control Systems

echnology, 4, 72}78.

HillerstroKm, G., & Sternby, J. (1996). Robustness properties of repetitive

controllers. International Journal of Control, 65, 939}961.

HillerstroKm, G., & Walgama, K. (1996). Repetitive control theory and

applications *A survey. Proceedings of the IFAC 13th Triennial

World Congress, San Francisco, California.

Inoue, T., Nakano, M., & Iwai, S. (1981a). High accuracy control of

servomechanism for repeated contouring. Proceedings of the 10th

Annual Symposium on Incremental MotionControl,Systems and Devi-

ces(pp. 258}292).

Inoue, T., Nakano, M., Kubo, T., Matsumoto, S., & Baba, H. (1981b).

High accuracy control of a proton synchrotron magnet

power supply. Proceedings of the IFAC 8th World Congress

(pp. 216}221).

Langari, A., & Francis, B. (1995). Design of optimal sampled-data

repetitive control systems. Proceedings of the 33rd Allerton

Conference on Communication, Control and Computing, Monticello,

Illinois.

Langari, A., & Francis, B. (1996). Robustness analysis of sampled-data

repetitive control systems. Proceedings of the IFAC 13th Triennial

World Congress, San Francisco, California (pp. 19}24).

Lee, R. C. H., & Smith, M. C. (1996). Some remarks on repetitive

control systems. Proceedings of the Conference on Decision and

Control, Kobe, Japan.

Lee, R. C. H., & Smith, M. C. (1998). Robustness and trade-o!s in

repetitive control. Automatica, 34, 889}896.

Logemann, H. (1993). Stabilization and regulation of in"nite-dimen-

sional systems using coprime factorizations. In A. Bensoussan, R. F.

Curtain, & J. L. Lions (Eds.), Analysis and optimization of systems:

State and frequency domain approaches for in,nite-dimensional sys-tems, Lecture Notes in Computer Science (Vol. 185, pp. 102}139).

Berlin: Springer.

Logemann, H., Rebarber, R., & Weiss, G. (1996). Conditions for robust-

ness and nonrobustness of the stability of feedback systems with

respect to small delays in the feedback loop. SIAM Journal of

Control and Optimizations, 34, 572}600.

Moore, K. L. (1993).Iterative learning control for deterministic systems.

Advances in Industrial Control. London: Springer.

Ogata, K. (1992). System dynamics (2nd ed.). Englewood Cli!s, NJ:

Prentice-Hall.

Rebarber, R. (1993). Conditions for the equivalence of internal and

external stability for distributed parameter systems.IEEEransac-

tions on Automatic Control, 34, 994}998.

Reinke, R. D. (1994). Adaptive Regeln zum ernen und Repro-

duzieren periodischer Signale mit dynamischen Netzwerken. Ph.D.thesis. Fachbereich Mathematik, UniversitaKt Kaiserslautern,

Germany.

Rogers, E., & Owens, D. H. (1992).Stability analysis for linear repetitive

processes. Lecture Notes in Computer Science (Vol. 175). Berlin:

Springer.

Russell, D. L. (1986). Frequency/period estimation and adaptive

rejection of periodic disturbances. SIAM Journal of Control and

Optimization, 24, 1276}1308.

Salamon, D. (1989). Realization theory in Hilbert space. Mathematical

Systems heory, 21, 147}164.

Sta!ans, O. J. (1999). Admissible factorizations of Hankel operators

induce well-posed linear systems. Systems and Control etters (to

appear).



15/15

Tsao, T.-C., & Qian, Y.-X. (1993). An adaptive repetitive control scheme

for tracking periodic signals with unknown period. Proceedings

of the American Control Conference, San Francisco, California

(pp. 1736}1740).

Weiss, G. (1989). Admissibility of unbounded control operators.SIAM

Journal of Control and Optimization, 27, 527}545.

Weiss, G. (1994a). Transfer functions of regular linear systems, part I:

Characterizations of regularity. ransactions of the American

Mathematical Society, 342, 827}854.

Weiss, G. (1994b). Regular linear systems with feedback.Mathematics of

Control, Signals, and Systems, 7, 23}57.

Weiss, G. (1997). Repetitive control systems: Old and new ideas. In C.

Byrnes, B. Datta, D. Gilliam, & C. Martin (Eds.), Systems and

control in the twenty-,rst century (Vol. 22, pp. 389}404). PSCT.

Boston: BirkhaKuser.

Weiss, G., & Curtain, R. F. (1997). Dynamic stabilization of regu-

lar linear systems. IEEE ransactions on Automatic Control, 42,

4}21.

Weiss, G., & HaK fele, M. (1998). The exponential stability of multi-

periodic repetitive control systems, Proceedings of the MNS

Symposium, Padova, Italy.

Yamamoto, Y. (1993). Learning control and related problems in in"-

nite-dimensional systems. In H. Trentelman, & J. Willems (Eds.),

Essays on control: Perspectives in the theory and its applications.

(pp. 191}222) Boston: BirkhaKuser.

Zhou, K., Doyle, J. C., & Glover, K. (1996).Robust and optimal control.

Englewoods Cli!s, NJ: Prentice-Hall.

George Weissreceived the Control Engin-eering degree from the Polytechnic Insti-tute of Bucharest, Romania, in 1981 andthe Ph.D. degree in Applied Mathematicsfrom the Weizmann Institute, Rehovot,Israel, in 1989. He has been working atthe Weizmann Institute, at Ben-GurionUniversity in Beer Sheva, Israel, at theUniversity of Exeter, UK, and currently heis with Imperial College in London, UK.

His research interests are distributedparameter systems, operator semigroups,

power electronics, repetitive control and sampled-data systems.

Martin HaKfele was born in Tettnang,Germany in 1971. He received his M.Eng.(Eur) degree in 1997 from the Universityof Exeter. In the same year he receivedhis Diploma Degree in Mechanical Engin-eering from the University of Stuttgart.Since 1998 he is Research Assistant atthe Max Planck Institute for Dynamics ofComplex Mechanical Systems in Mag-

deburg and studying towards the Ph.D.His research area there is modeling, con-trol and optimization of polymerization

processes. He is also interested in H control.


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