An op imal an i-windup s a egy o epe i i e con ol sys ems
Ge m´an A. Ramos, Ramon Cos a-Cas ell´o
Abs ac — Repe i i e Con ol includes an In e nal Model
wi h high gain and slow ime esponse cha ac e is ics which
make i p one o he windup e ec . A solu ion o his p oblem
is he inclusion o an An i-Windup compensa o . Al hough
he e exis many gene al An i-Windup syn hesis me hods in
li e a u e, some p oblems can a ise as a esul o a s aigh -
o wa d applica ion o hem in Repe i i e Con ol. This pape
p esen s he analysis and adap a ion o he Model Reco e y
An i-Windup s a egy in he Repe i i e Con ol ame. Thus, an
op imal LQ design is p oposed ha looks o a deadbea eco e
beha iou a e sa u a ion and a global asymp o ic s abili y o
he closed loop sys em. Th ough simula ion i is shown ha he
p opounded scheme achie es be e acking pe o mance han
o he simila LQ designs.
I. INTRODUCTION
As an In e nal Model P inciple (IMP) [1] based con ol
s a egy, Repe i i e Con ol (RC) [2], [3] uses an In e nal
Model (IM) ha cha ac e izes he signal o be acked o
ejec ed. In his way, he IM o he RC p o ides in ini e
o e y high gain a a gi en equency an i s ha monics. I
is well known ha , in sys ems wi h ac ua o sa u a ion, a
con olle wi h hese cha ac e is ics may p oduce a wind-
up e ec in which he s a es o he con olle can g ow
unbounded. E en i he gain is no in ini e bu high, he
s a es can o e g ow signi ican ly making ha de o eco e
he sys em o he linea ideal one. Some condi ions ela ed
o he boundedness o he s a e o he RC wi h ac ua o
sa u a ion a e s a ed in [4].
As i is known, ma ginally s able o uns able con olle s
a e p one o o igina e he unbounded g owing o he con-
olle s a e. Thus, using he pole analysis, i can be no ed ha
he IM used in s anda d RC is ma ginally s able and hose
used in High O de Repe i i e Con ol (HORC) [5], [6] ha e
poles o e he uni ci cle wi h mul iplici y equal o g ea e
han wo which can yield Bounded Inpu Bounded Ou pu
(BIBO) uns able IM [7]. Addi ionally, he IM gene ally
imposes a slow ansien esponse o he closed loop which
wo sens he ac ua o sa u a ion e ec . The e o e, since he
linea design o he epe i i e con ol does no include he
sa u a ion in he ac ua o , i is con enien o include an An i-
Windup (AW) compensa o . A ecen e iew o s anda d AW
echniques can be ound in [8] and [9].
This wo k is pa ially suppo ed by he spanish Minis e io de Educaci´on
y Ciencia (MEC) unde p ojec DPI2010-15110
G. A. Ramos is wi h he Depa men o Elec ical and Elec onic
Enginee ing, Uni e sidad Nacional de Colombia, Bogo ´a DC, Colombia
[email p o ec ed]
R. Cos a-Cas ell´o is wi h he Ins i u e o Indus ial and Con ol Engi-
nee ing, Uni e si a Poli `ecnica de Ca alunya, A . Diagonal, 647, 08028
Ba celona, Spain [email p o ec ed]
In [9], mode n AW p oposals ha e been classi ied in wo
g oups: Di ec Linea An i-Windup (DLAW) and Model Re-
co e y An i-Windup (MRAW). The DLAW app oach seeks
o ind an AW compensa o ha assu es speci ic pe o mance
and s abili y p ope ies o he closed loop sys em. The
MRAW app oach selec s he AW il e in such a way ha i
makes in a ian he compensa o -plan sys em. Howe e , due
o he cha ac e is ics o he RC, mos o he s anda d AW
designs should no be applied s aigh o wa d since some
di icul ies migh appea du ing design o implemen a ion.
As a esul , i would be necessa y o adap he gene ic AW
s a egies in o de o be applied in RC.
In he DLAW scheme, some s a egies a e based on
sol ing a Linea Ma ix Inequali y (LMI) p oblem [10];
howe e , complica ions usually a ise since he size o his
LMI depends mos ly on he IM o de which is usually la ge.
Thus, o he RC case, he implemen a ion o his scheme
depends on whe he he LMI is compu a ionally sol able o
no . Al hough he DLAW scheme allows us o ob ain an AW
compensa o o o de 0, he solu ion includes elemen s ha
yield a la ge numbe o on-line calcula ions, hus inc easing
he compu a ional bu den.
The MRAW scheme uses he model o he plan in i s
s uc u e. Al hough he o de o he plan could be la ge, i is
usually signi ican ly smalle han he IM o de . Fu he mo e,
he p ocedu e o ind he eedback gains does no depend on
he con olle dynamics, he e o e he ela ed LMI is always
sol able. The compu a ional load o he MRAW scheme
implemen a ion is he lowes one in compa ison wi h he
o he s a egies.
The p oposals in [11], [4] and [12], a e h ee examples
o AW design o epe i i e con ol. In [11], an AW law
is de i ed o I e a i e Lea ning Con ol (ILC) and also
a ex ension o RC is b ie ly desc ibed. Howe e , he AW
s a egy is de i ed o a speci ic plan and he desc ibed
epe i i e con olle does no co espond o he s anda d
a chi ec u e since he il e s o s abili y and obus ness
a e no included. In [4], he AW scheme cancels ou he
dynamics o he IM du ing sa u a ion and adds a s uc u e o
shape he ansien s when he sys em sa u a es and ge s back
om sa u a ion. Howe e , he IM cancella ion implies ha ,
in addi ion o he epe i i e con olle o de , i is necessa y
o implemen an AW il e which has a leas he o de o he
IM. The e o e, his scheme will be cos es ic i e since i
depends on a sui able implemen a ion pla o m. The wo k in
[12], can be ca ego ized as a DLAW design. The s a egy is
de i ed in con inuous ime domain. I is an ex ension o RC
o he gene al AW design in [10], whe e he case o delayed
sys ems is desc ibed. Also in his app oach, unlike he RC
R(z)E(z)U(z)Y(z)
+
++
+−
−
+
+
W(z)Gx(z)
Gp(z)
H(z)
G (z)
Gc(z)
D(z)
Repe i i e con olle
Fig. 1. Block-diag am o he epe i i e con olle plug-in app oach.
design ha will be desc ibed he e, he il e s o obus ness
and s abili y a e designed oge he wi h he DLAW syn hesis.
In iew o he analysis abo e, he MRAW appea s as a
good s a egy when aken in o accoun he compu a ional
sol abili y and load in he design and implemen a ion o RC.
Thus, in his wo k a MRAW scheme is p esen ed in which he
eco e y o he sys em is achie ed using he app oxima ion
o a deadbea ansi ion (see [13] o he deadbea concep ).
The ad an age o selec ing a deadbea o e o he designs is
shown as well as he design me hod.
The pape is o ganized as ollows, Sec ion II p esen s he
basics o he epe i i e con ol including some design issues
and he s abili y condi ions. Sec ion III desc ibes he gene al
MRAW scheme. Sec ion IV analyses he s abili y o he
sys em and p opounds an op imal design. In Sec ion V he
expe imen al esul s a e shown and inally he conclusions
a e p esen ed in Sec ion VI.
II. DIGITAL REPETITIVE CONTROL
Digi al epe i i e con ol uses an IM which in oduces
in ini e/high gain a a selec ed undamen al equency and i s
ha monics [5]. This IM has he ollowing ans e unc ion:
G (z) = W(z)H(z)
1−W(z)H(z),(1)
whe e W(z) = z−Nand H(z)is a null-phase FIR low-pass
il e in cha ge o p o ide obus ness a high equencies.
Wi h H(z) = 1, IM (1) p o ides in ini e gain a equencies
ω= (2k−1)2π/N, wi h k= 1,2,...,(N/2) + 1, whe e
N=Tp
Tsis he disc e e pe iod o he signal, Tpbeing he
pe iod o he signal o be acked/ ejec ed and Tsbeing he
sampling pe iod.
Also, in o de o p o ide obus ness agains equency
unce ain y/ a ia ion a HORC echnique has been de el-
oped. This e sion o RC uses he IM (1) wi h W(z) =
PM
k=1 wkz−kN and PM
k=1 wk= 1 [6].
Besides he IM, which assu es s eady s a e pe o mance,
epe i i e con olle s include a s abilizing il e , Gx(z),
which assu es closed-loop s abili y. T adi ionally, epe i i e
con olle s a e implemen ed in a “plug-in” ashion, i.e. he
epe i i e compensa o is used o augmen an exis ing nomi-
nal con olle , Gc(z)(Figu e 1). This nominal compensa o
is designed o s abilize he plan , Gp(z), and p o ides
dis u bance a enua ion ac oss a b oad equency spec um.
The closed-loop sys em o Figu e 1, using (1) as he IM,
is s able i he ollowing condi ions a e ul illed ([14]):
1) The closed loop sys em wi hou he epe i i e con-
olle is s able, i.e. Go(z) = Gc(z)Gp(z)
1+Gc(z)Gp(z)is s able.
I is ad isable o design he con olle Gc(z)wi h a
high enough obus ness ma gin.
2) kW(z)H(z) (1 −To(z)Gx(z)) k∞<1, whe e
H(z)and Gx(z)mus be selec ed o mee his
condi ion. A i ial s uc u e1which is o en used
o minimum phase sys ems is ([15]): Gx(z) =
k .(Go(z))−1. As a gued in [16], k mus be designed
looking o a ade-o be ween obus ness and an-
sien esponse.
The ans e unc ion o he comple e con olle (see
Figu e 1) esul s:
G c(z) = U(z)
E(z)= (1 + G (z)Gx(z))Gc(z)(2)
Addi ionally, he ansien o con e gence ime is domi-
na ed by he IM dynamics, which is in gene al much slowe
han he closed loop only wi h he con olle Gc(z)[17],
[18].
III. THE GENERAL MRAW SCHEME
Figu e 2 shows he MRAW s uc u e, whe e Gp(z)is he
plan , G c(z)is he con olle (2), sa (·)is he sa u a ion
unc ion and Caw(z)is he AW compensa o . In he MRAW
s a egy, he misma ch be ween he sa u a ed con ol ac ion
and he non-sa u a ed one is ed back o he con olle by
means o he AW compensa o , which is designed o be he
model o he plan , σ1,k being he ou pu ha is used wi h
his pu pose. Addi ionally, ano he eedback signal, σ2,k,
is added wi h he aim o imp o ing he beha iou o he
sys em when i ge s ou om sa u a ion. Thus, he design
o his eedback in ol es di e en app oaches. The In e nal
Model Con ol (IMC) AW s a egy [19], u ns ou o be he
pa icula case whe e σ2,k = 0. This causes ha when ge ing
ou o sa u a ion he sys em eco e y elies on he plan poles,
which can yield a non app op ia ed pe o mance. A s a egy
based on P edic i e Con ol which seeks an l2pe o mance
c i e ion can be ound in [20], an op imiza ion p ocedu e
using he Linea Quad a ic (LQ) app oach is p oposed in
[21] and a ully nonlinea s a egy is desc ibed in [22].
In his wo k, he signal σ2,k is designed o be a linea
eedback o he AW compensa o s a e. This is aimed a
inding a simple linea solu ion o he AW p oblem in case
o RC, also a oiding he algeb aic loop ha can be c ea ed
using he eedback o he con ol ac ion misma ch, as in [21].
Fu he mo e, we analyse he bene i s o designing a deadbea
beha iou in he AW il e in case o epe i i e con ol. Also,
as p e iously men ioned, he ad an age o using he MRAW
scheme o he epe i i e con ol case is ha he design
does no depend on he IM o de . Addi ionally, as will be
desc ibed la e on, he e o and con ol signals a e he ideal
ones (as i he sys em had no sa u a ion in he ac ua o ),
which isola es he con olle om he sa u a ion e ec s.
1The e is no p oblem wi h he imp ope ness o Gx(z)because he IM
p o ides he epe i i e con olle wi h a high posi i e ela i e deg ee.
+_
+
+
++
+_
Caw (z)
G c (z)sa () Gp(z)
kekuk¯ukˆukyk
σ2,k σ1,k
ηk
Fig. 2. The MRAW scheme in RC.
A. Selec ed MRAW scheme
Conside he MRAW scheme depic ed in Figu e 2. Le he
disc e e- ime and asymp o ically s able linea plan Gp(z)be
xk+1 =Axk+Bsa (¯uk)
yk=Cxk
(3)
whe e
sa (¯uk) = 


umin ¯uk< umin
¯ukumin ≤¯uk≤umax
umax ¯uk> umax
(4)
wi h umin <0and umax >0.
The s a e-space ep esen a ion o he epe i i e con olle
G c(z)is:
¯xk+1 =A c ¯xk+B cek
uk=C c ¯xk+D cek
(5)
The AW il e Caw(z)is de ined om he plan model (3)
as:
χk+1 =Aχk+B(uk−sa (uk+σ2,k))
σ1,k =Cχk
(6)
and
σ2,k =Kχk(7)
whe e Kis he design pa ame e o he AW il e .
I can be no iced ha while he inpu in sys em (3) is
he sa u a ed con ol ac ion, he inpu in sys em (6) is he
di e ence be ween he sa u a ed and non-sa u a ed con ol
ac ion. This ac , oge he wi h
ηk=yk+σ1,k,(8)
helps o de e mine he sys em in a iance. Thus, de ining
ξk=xk+χk, no icing ha ¯uk=uk+σ2,k and adding
equa ions (3) wi h (6) we ha e:
ξk+1 =Aξk+Buk
ηk=Cξk
(9)
In his way, om he inpu uk o he ou pu ηk, he sys em
in Figu e 3 can be seen as a Linea Time In a ian (LTI) one
wi h he dynamics o he plan .
This means ha ηkis he ideal plan ou pu in he sense
ha i would be he plan ou pu in a sys em wi hou ac ua o
sa u a ion. Fu he mo e, in he closed loop o Figu e 2, he
con ol ac ion ac ion ukis he ideal con ol ac ion, i.e. uk
is he same con ol signal as he one in a sys em wi hou
ac ua o sa u a ion. This ac isola es he con olle om he
sa u a ion e ec s, allowing us o educe he analysis o he
+_
+
+
++
+_
Caw(z)
G c(z)sa () Gp(z)
kekuk¯ukˆukyk
σ2,k σ1,k
ηk
Fig. 3. The in a ian pa o he MRAW scheme.
beha iou o he in a ian pa shown in Figu e 3, including
i s in e nal s abili y.
Rema k 1: In his scheme he de ia ion om he ideal
pe o mance can be measu ed ough σ1,k, since σ1,k is he
di e ence be ween he ideal beha iou and he plan ou pu
σ1,k =ηk−yk.
P oposal 1: Gi en a RC design, he smalles possible σ1,k
co esponds o he bes possible pe o mance in case o
sa u a ion ( he smalles de ia ion om he ideal beha iou ).
The e o e, he p oblem o mula ion is o ind he design
pa ame e Ksuch ha σ1,k is small enough o ob ain a good
acking pe o mance.
I is impo an ha he AW design aims a achie ing
good acking pe o mance since RC is a echnique which
is in ended o ob ain null s eady-s a e acking e o . Also
due o his RC ea u e, we a e in e es ed in he sa u a ion
e ec p oduced in s eady s a e e en hough i also can occu
in ansien s a e.
IV. MRAW PROPOSAL FOR RC: DESIGN AND STABILITY
The p oposal is based on he idea o ha ing a deadbea
eco e once he sys em ge s back om sa u a ion. The goal
is o ob ain a Caw(z)AW il e such ha du ing sa u a ion
akes he o m o he plan model, and addi ionally, when
he con ol ac ion ge s back om sa u a ion, he ou pu s o
Caw(z) anish in a ini e numbe o samples.
To ob ain a deadbea beha iou du ing eco e y i is
needed ha he eedback loop c ea ed by σk,2 eloca es all
he poles o Caw(z) o z= 0, which can be done using he
pole placemen p ocedu e, hus ob aining he gains ec o K.
Howe e , he in e nal s abili y o he sys em mus be e i ied.
A. S abili y
Rema k 2: The closed loop s abili y o he sys em in Fig-
u e 2 is es ablished by he design o he RC and addi ionally
by he in e nal s abili y o he sys em in Figu e 3.
Mo eo e , om he ac s ha : 1) we a e assuming an
asymp o ically s able plan and 2) om inpu uk o ou pu
ηk he sys em in Figu e 3 can be seen as a LTI one wi h he
dynamics o he plan , we ha e ha he in e nal s abili y o
his sys em can be es ablished analysing only he s abili y o
he in e connec ion be ween he sa u a ion block and Caw(z).
As a esul , in o de o check he in e nal s abili y o he
sys em in Figu e 3 he ollowing condi ion should be e i ied
o he sys em in equa ions (6) and (7):
V(χk+1)−V(χk) + Ψ <0,(10)
V(χk) = χT
kPχkbeing he candida e Lyapuno unc ion
and Ψ = 2(uk−sa (uk))W(sa (uk)) he sec o condi ion2
(see [23]), P > 0and W > 0being symme ic ma ices o be
ound. Since Ψ≥0, ollowing he S-p ocedu e we can ind
ha e i ying condi ion (10) implies V(χk+1)−V(χk)<0
and as a consequence he in e nal s abili y is es ablished.
B. Op imal design
In o de o ob ain a design as close as possible o he
deadbea beha iou explained p e iously and also assu ing
global asymp o ic s abili y, condi ion (10) can be pu oge he
wi h an op imal LQ design in a LMI o m.
Rema k 3: In his case he LQ design is used o ind he
deadbea gain Kwhich is shown o be an op imal when he
weigh ma ix Qp=TTT,Tbeing he linea ans o ma ion
o he sys em (6) in o he con ollable canonical o m (see
[24]).
O he esul s ela ed wi h he deadbea design as an
op imal LQ solu ion can be ound in [25]. Thus, he p oblem
o mula ion is o ind Ksuch ha he s abili y o he
in e connec ion be ween Caw, equa ions (6) and (7), and
he sa u a ion block is p ese ed and addi ionally, sol e he
cons ained LQ p oblem:
minKP∞
k=0 χT
kQpχk
subjec o
χk+1 =Aχk+Bσ1,k
σ1,k =Kχk.
The comple e p oblem can be o mula ed as an LMI
minimiza ion p oblem:
min γ
s. .


−Q AQ BU
⋆−Q XT
1
⋆ ⋆ −2U

<0
γI I
I−Q>0


−Q ⋆ ⋆
AQ +BX10⋆
QpQ0−Qp

<0
whe e Q=QT>0,U=UT>0,γ > 0and X1=KQ.
Is i wo h o say ha he e exis s some conse a i eness
in he sec o condi ion Ψwhich is applied o non-linea i ies
belonging o he sec o [0,1]. In gene al, his ac yields a
gain K ha is an app oxima ion o he deadbea solu ion.
V. SIMULATIONS RESULTS
This sec ion shows he esul s ound by simula ion using
he AW design p esen ed p e iously and a compa ison wi h
o he op imal LQ design oge he wi h he IMC AW s a egy.
2In his case he memo yless unc ion sa () is said o belong o he
sec o [0,1] since sa ( , u) [u−sa ( , u)] ≥0, which is called he sec o
condi ion.
A. Simula ion se up
Wi h he pu pose o compa ing he AW s a egies de-
sc ibed in his wo k, a linea epe i i e con olle design will
be gi en. Thus, conside he ollowing disc e e- ime linea
s able plan :
Gp(z) = 2.146z+ 0.7585
z2−0.9945z+ 0.03498 (11)
The con olle is cons uc ed om model (11), o N=
100 and sampling pe iod o Ts= 5 ms. Acco ding o Sec ion
II, he ollowing design issues ha e been aken in o accoun :
•Gc(z) = 0.5p o ides a e y obus inne loop.
•The i s o de linea -phase FIR il e
H(z) = 0.02z+ 0.96 + 0.02z−1
p o ides su icien obus ness in he p esen case.
•The ac ha Gp(z)is minimum-phase allows Gx(z) =
k G−1
0(z), wi h k = 0.75.
Also, a second o de HORC has been designed o com-
pa ison pu poses. Thus, M= 2,w1= 2 and w2=−1ha e
been selec ed.
Gi en he s a e space disc e e- ime sys em: (A, B, C, D),
and i s eachabili y ma ix WA=B AB ··· An−1B,
hen he ma ix T= T
n T
nA··· T
nAn−1wi h T
n
he las ow o W−1
A. Thus, o his example:
A=0.0244 −0.1251
0.0903 0.9701 , B =0.0903
0.0216,
C=0 99.5450, D = 0,
T=−8.1858 34.2736
2.8936 34.2736.
Using he op imal MRAW app oach desc ibed in Sec ion
IV-B, he pa ame e s ha ha e been ound o be a easible
solu ion a e: Kdb =3.1823 24.8267, using Qp=TTT
o a deadbea app oxima ion and Ks =1.0747 8.1270,
using Qp= 105Ias he LQ design used o compa ison
pu poses. The idea behind he las LQ design is o ind a
solu ion ha keeps small he s a e o he AW compensa o .
B. Simula ion esul s
This sec ion analyses he sa u a ion in s eady s a e. The
sa u a ion limi s ha e been chosen o be umin =−5and
umax = 2. The e e ence signal kis depic ed in Figu e 4
oge he wi h he sys em plan ou pu ykand con ol signal
ukwhen he se ings o HORC ha e been applied wi hou
ac ua o sa u a ion. As can be seen, in his case, he epe i i e
con olle success ully acks he e e ence signal.
Figu e 5 shows he con ol ac ion uk, i.e. he con ol
ac ion p o ided by he epe i i e con olle . The sys em wi h
ac ua o sa u a ion bu wi hou AW mechanism is called SAT
and he sys em wi hou sa u a ion is called Ideal. Fo he
SAT scheme wo op ions a e depic ed, SAT RC and SAT
HORC, o s anda d and HORC espec i ely. As can be
seen, bo h SAT RC and SAT HORC con ol signals p esen
an undesi able wind-up e ec , he SAT HORC being he
wo se case. This phenomena is due o he pole mul iplici y
in he IM o HORC which also makes i slowe han he
RC one. As has been poin ed ou , when he MRAW scheme
is included, he con ol signal ukco esponds wi h he ideal
one. The es o he examples will be ca ied ou using only
he second o de HORC.
9 9.2 9.4 9.6 9.8 10
−100
−50
0
50
100
Ou pu and e e ence
Time (s)
9 9.2 9.4 9.6 9.8 10
−5
0
5
Con ol signal
Time (s)
k
yk
Fig. 4. S eady s a e e e ence k, ou pu ykand con ol signal ukwi hou
sa u a ion.
012345678910
0
500
1000
1500
2000
2500
3000
3500
Con ol ac ion
Con ol signal uk
Time (s)
1 1.2 1.4 1.6 1.8
0
50
100
150
200 SAT RC
SAT HORC
Ideal
Fig. 5. Con ol ac ion uk o RC and HORC wi h ac ua o sa u a ion and
wi hou AW il e .
Figu e 6 depic s he plan ou pu ykand sa u a ed con ol
signal ˆukusing he p oposed AW design. Thus, he op imal
design p oposed he e deno ed by Kdb a e compa ed wi h
Ks and K= 0 which co esponds o he esponse using a
s anda d LQ design and K= 0 (IMC AW) espec i ely, as
desc ibed in he p e ious sec ion. I is shown ha o K= 0
he sys em eco e is oo much slow; in ac , in his example,
he ideal ou pu is ha dly eached again be o e ge ing in o
sa u a ion again. On he o he hand, he sa u a ed con ol
ac ion app oxima es e y well he ideal one, excep when
uk> umax, bu as can be seen, his is no he desi able
beha iou . Fo Ks , he plan ou pu is close o he ideal
one and also i is seen ha i s co esponding con ol ac ion
emains sa u a ed longe han in he p e ious case. Finally,
using Kdb o app oxima e a deadbea beha iou , i can be
no iced ha he plan ou pu ge s close o he ideal ou pu
and i s con ol ac ion emains sa u a ed longe .
Figu e 7, shows he ou pu o he AW compensa o σ1,k,
which, as men ioned be o e, can be seen as he de ia ion
om he ideal esponse. I is shown ha du ing he ime
he h ee sys ems a e in sa u a ion he de ia ion is simila ;
howe e he esponse is qui e di e en once he sys ems ge
ou o sa u a ion, he esponse o Kdb being he smalles
one. I is wo h o no ice ha , since o his example he RC
and he HORC design ha e he same acking pe o mance,
σ1,k in Figu e 7 is he same in bo h cases.
9 9.2 9.4 9.6 9.8 10
−100
−50
0
50
100
Plan Ou pu yk
Time (s)
9 9.2 9.4 9.6 9.8 10
−4
−2
0
2
4
Time (s)
Ideal
Kdb
Ks
K=0
Con ol signal ˆuk
Fig. 6. S eady s a e sa u a ion beha iou .
9 9.2 9.4 9.6 9.8 10
0
10
20
30
40
50
60
Time (s)
Kdb
Ks
K=0
σ1,k
Fig. 7. E olu ion o σ1,k ( he de ia ion om he ideal beha iou ).
VI. CONCLUSIONS
In his pape , he Model Reco e y An i-Windup scheme
is s udied and adap ed o he Repe i i e Con ol case. An
op imal LQ design has been p oposed aimed a inding a
deadbea eco e beha iou and assu ing he global asymp-
o ic s abili y o he closed loop sys em. Th ough simula ion
esul s i is shown ha he p oposed AW scheme ge s be e

pe o mance in he de ia ion om he ideal plan ou pu
compa ed wi h o he simila LQ designs. The u u e esea ch
includes he inclusion o less- es ic i e sec o condi ions o
he nonlinea sa u a ion unc ion in o de o be e app oxi-
ma e he deadbea design.
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