TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.6, Jun
e
201
4, pp. 4222 ~ 4
2
2
9
DOI: 10.115
9
1
/telkomni
ka.
v
12i6.553
5
4222
Re
cei
v
ed
No
vem
ber 1
0
, 2013; Re
vi
sed
De
cem
ber 1
8
,
2013; Accep
t
ed Jan
uary 2
2
, 2014
Output Regulation for Saturated Systems with
Nonlinear Exosystem
Huan
g w
e
i
Schoo
l of Auto
mation, Ha
ngz
hou D
i
anz
i Uni
v
ersit
y
,
Han
g
zho
u
, 310
012, Ch
in
a.
email: h
w
@
h
d
u
.edu.cn
A
b
st
r
a
ct
In this brief, we investi
gate the outp
u
t regu
lati
o
n
prob
le
m of saturated li
near syste
m
u
nder th
e
action of nonlinear ex
osystem
. P
a
rticularly, for saturat
ed system
s
wi
th periodic
a
lly tim
e
-dependent
exosyte
m
, the
K-step asy
m
ptotically
re
gul
atabl
e
re
gio
n
is give
n,
w
h
ich is
a s
e
t of a
ll
ini
t
ial states
of th
e
pla
n
t and th
e e
c
osystem. I
m
pr
oved i
n
tern
al
mo
de
l pri
n
cip
l
e
s
are construct
ed a
nd the fe
e
dback c
ontrol
l
e
r
is
desi
gne
d to
e
n
sure
exp
o
n
e
n
tial
outp
u
t re
gul
ation
i
n
the
reg
u
lata
bl
e r
egi
on w
i
th
dist
urba
nce r
e
j
e
cti
on.
Simulati
on
exa
m
p
l
es
are
giv
en to
il
lustrate
the
e
ffectiven
ess of
prop
os
ed
method, t
h
e syste
m
s g
o
int
o
stable ra
pid
l
y a
nd per
iod
i
ca
lly.
Ke
y
w
ords
: saturatio
n
constra
i
nt, output reg
u
l
atio
n,
intern
al
mo
de
l princ
i
pl
e
s
, feedback co
ntroll
er
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
Saturation co
nstrai
nt is a kind of nonli
near
co
n
s
trai
nt in many practi
cal condi
tions. In
this p
ape
r
we con
s
ide
r
th
e regulatio
n
probl
em
of
li
near
system
subj
ect to
a
c
tuator
satu
rat
i
on
unde
r the
act
i
on of n
online
a
r exo
s
ystem
.
This a
d
d
r
e
s
se
s the
pro
b
l
e
m of de
sig
n
i
ng a fe
edb
ack
controlle
r for
an un
ce
rtain
plant so that t
he cl
os
ed-l
o
o
p
syste
m
is i
n
ternally st
abl
e and th
e out
put
of
the clo
s
ed
-loop syste
m
can
a
s
ympt
oti
c
ally
track a class
of refe
re
nce
inp
u
ts i
n
the p
r
e
s
en
ce
o
f
a cla
s
s of disturban
ce
s. Franci
s
a
nd Wo
nham [1
-3
] propo
sed th
e in
ternal m
ode
prin
ciple,
whi
c
h
aims to co
nvert the output
regulatio
n problem of
a given plant into a stabilizatio
n probl
em of an
augme
n
ted system com
p
o
s
ed of the giv
en plant an
d a well defin
ed
dynamic co
mpen
sato
r.
For th
e
case
s
whe
r
e th
e
exogen
ou
s si
gnal
s a
r
e
co
n
s
tant, Fran
cis [3] de
signe
d
a line
a
r
robu
st re
gula
t
or ba
sed o
n
the linea
r ap
proximatio
n o
f
the plant ca
n solve
t
he l
o
cal
st
ru
ct
ur
a
lly
stable o
u
tput
regul
ation p
r
oblem for th
e
nonline
a
r pl
ant. Hua
ng a
nd Ru
gh [4]
made a fu
rth
e
r
work a
nd put
the solution
to nonline
a
r
plant und
er n
o
rmal di
stu
r
b
ance. Self-Adaptive meth
od
and optimal f
eedb
ack co
ntrol [5-7] we
re use
d
in
so
lving the pro
b
lem of glob
e robu
st out
put
regul
ation fo
r n
online
a
r
system
di
stu
r
bed
by
un
certai
n exog
enou
s sig
nal
s.
Di
stu
r
ba
n
c
e
sup
p
re
ssion
of a class of nonlin
ear
systems was
stu
d
ied in [8-1
0].
Ho
wever, it
sho
u
ld b
e
po
inted that m
o
st
of the
studie
s
are ca
rrie
d
with
se
mi-sta
ble
exosystem, t
he p
r
oble
m
o
f
output re
gul
ation for
satu
rated
system
s un
de
r the a
c
tion of
nonli
nea
r
exosystem
h
a
s received relatively less
attention.
The few works
motivate our
c
u
rrent res
e
arc
h
are [11, 12]. In [11], robust adaptive
con
s
tr
ain
e
d
motion tracking control methodol
ogy
was
derived fo
r b
ound
ed n
onli
near
effect
s
and external
disturban
ce within
the clo
s
ed
-loo
p
sy
stem.
Output regula
t
ion for pe
rio
d
ic
sign
al of
con
s
tr
ai
ned
MIMO syste
m
subj
ect to
actuato
r
s
sat
u
rate
d
is
studie
d
in
[12-1
4
] p
r
op
o
s
ed
ad
aptive
fuzzy
s
lidi
ng mode
co
ntrol
approa
ch
to
solve
the co
ntrol
probl
em of
X-Z inve
rted
pend
ulum i
n
the p
r
e
s
e
n
ts
of syste
m
un
cert
ainties and externa
l
disturban
ce
s.
The
obje
c
tive of this pa
pe
r i
s
to
study
t
he p
r
oble
m
of output
re
g
u
lation fo
r
sa
turated
system
s un
d
e
r the a
c
tion
of nonlin
ear
exosystem.
B
a
se
d on
our
earlie
r results in [15], a si
mple
feedba
ck co
ntrolle
r base
d
on a stabil
i
zing la
w
wa
s achi
eved for output re
g
u
lation of linear
system
with input co
nst
r
ai
ns. Un
der th
e
action of
a n
online
a
r exo
system actio
n
, the probl
em to
be ad
dresse
d
in this
pap
er is the foll
owi
ng: (1
)
Cha
r
a
c
teri
ze
of the
reg
u
latabl
e region. T
he first
task of thi
s
p
aper i
s
to
ch
ara
c
teri
ze
the
set
of
initial
con
d
ition
s
fro
whi
c
h
the
r
e
exist ad
missi
b
le
controls to
ke
ep the state
b
ound
ed an
d to drive t
he tracking e
r
ror t
o
0 asym
ptotically. (2
) De
si
gn
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Output Re
gul
ation for Satu
rated System
s with Nonlin
ear Exosyste
m
(Huang
we
i)
4223
of con
s
traine
d state fee
d
back
cont
roll
er. Find
a st
ate feedb
ack law a
nd
co
nstru
c
t the
state
controlle
r.
2. Problem Statement a
n
d Preliminaries
Con
s
id
er the
system:
)
(
)
1
(
)
(
)
(
)
(
)
(
)
(
)
(
)
1
(
k
S
k
k
Q
k
Cx
k
e
k
P
k
Bu
k
Ax
k
x
ω
ω
ω
ω
(1)
Whe
r
e
A
R
n
×
n
,
B
R
n
×
m
,
P
R
n
×
r
,
C
R
p
×
n
,
Q
R
p
×
r
. Th
e first pl
ant d
e
scrib
e
s a pl
ant, with
state
n
R
x
, input
m
R
u
and
u
1, subje
c
t to the effect of distu
r
ban
ce
rep
r
e
s
ente
d
by
k
P
ω
.
The e
r
ror b
e
twee
n the a
c
t
ual outp
u
t
k
Cx
a
nd a
referen
c
e sig
nal
k
Q
ω
is de
fined a
s
k
e
by
the se
con
d
e
quation. The t
h
ird eq
uation
descri
b
e
s
the
exosystem
with state
r
R
ω
and
S
R
r
×
r
.
Due to the con
s
trai
nt input, it’s well know
n that the initial state of the
plant and
exosystem
ca
n not be in the whol
e spa
c
e. We sh
oul
d
characte
ri
ze
the set of all initial states (
x
0
,
ω
0
)
R
n+r
,
o
n
which th
e p
r
oble
m
of
co
nstrai
ned
o
u
tput regul
ation
is
solvable.
This
set
is
ca
lled
regul
atable re
gion. If we can con
s
tru
c
t a state feedba
ck law,
u
=
(
x
,
ω
),
(
x
,
ω
)
1 and
(0,
0) =0, by which follo
wing
condition
s are satisfie
d:
A. Plant
x
(
k
+1)
=
Ax
(
k
)+B
(
x
,0) is a
s
ymptotically stabl
e
on the equili
brium p
o
int
x
=0.
B. For all i
n
itial state
s
(
x
0
,
ω
0
)
R
n
+
r
in
regul
atable
region, the
cl
ose
-
loo
p
syst
em ha
s
lim
k
→∞
e
(
k
)=0.
To begin
with
, some ne
ce
ssary a
s
sumpt
i
ons a
r
e ma
d
e
:
A1. The pair (
A
,
B
) is stabili
zabl
e.
A2. S has all its eigenval
ue
s on the unit
circle an
d dia
gonali
z
abl
e.
A3.
S
P
A
Q
C
0
,
is measu
r
able.
A4. There exi
s
t matrices
Π
and
Γ
solve the linea
r matrix equation
Q
C
P
B
A
S
0
(2)
In this paper,
we con
s
id
er
two kind
s of nonlin
ear ext
e
rnal di
sturba
nce: the sq
ua
re wave
and trian
g
le
wave. The square wave i
s
disco
n
tinuo
us
an
d unde
rivable, can b
e
descri
bed
as
ω
(
k
+1
)
=
S
ω
(
k
), S is an unit matrix. Let
ω
(0)
=
[
m
m
]’, when
k
=
nT
/2 (
n
=0, 1, 2 …),
ω
(
k
)=
(-
1)
n
ω
(0
).
There’re t
w
o
step
sign
als
o
f
different am
plitude in
one
cycle, a
nd th
e step
sig
nal
is line
a
r. As t
he
perio
d
T
i
s
lo
ng en
oug
h, the a
c
tion
of exosystem
can be
viewe
d
as to
w
con
s
tant distu
r
ba
nce
that works al
ternatively. Review ou
r ea
rler
work
s in
[15], it is p
o
ssible to de
sign a
n
ea
sil
y
impleme
n
tabl
e state
co
ntroller to
ma
ke
the cl
ose
loo
p
syste
m
sta
b
le a
s
ymptoti
c
ally, sim
u
lati
on
results a
r
e shown in se
ction 5. Detaile
d study
on o
u
tput regul
ation pro
b
lem i
s
fouces on t
he
influen
ce of p
e
riodi
c
tri
angl
e wave.
3. The Regul
atable
Ragio
n
The tri
angl
e
wave i
s
conti
nuou
s
but
un
derivabl
e. T
r
i
angle
with
pe
riod
T
and
a
m
plitude
m is de
scribe
d as follo
ws,
whe
r
e
0
)
0
(
ω
:
3
,
2
,
1
,
0
1
2
2
1
n
T
n
k
T
nT
a
k
T
nT
k
nT
a
k
k
ω
ω
ω
(
3
)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4222 – 4
229
4224
At the equilibrium point, let
Ga
k
k
u
,
k
k
x
, by (1) then:
0
k
Q
k
C
k
Q
k
Cx
k
e
3
,
2
,
1
,
0
1
2
2
n
T
n
k
T
nT
a
BGa
k
P
k
B
k
A
k
T
nT
k
nT
a
BGa
k
P
k
B
k
A
k
(4)
If
B
has full row ran
k
, then
G exists ma
d
e
:
3
,
2
,
1
,
0
1
2
2
n
T
n
k
T
nT
BG
T
nT
k
nT
BG
3
,
2
,
1
,
0
1
2
2
n
T
n
k
T
nT
k
P
k
B
k
A
k
T
nT
k
nT
k
P
k
B
k
A
k
(5)
Due to
0
k
, by (4), (5), the internal mode of tr
iangle
wave a
c
tion is
rep
r
e
s
ent
s by (6):
0
Q
C
P
B
A
(
6)
Con
s
id
er sy
stem (1), a
con
t
rol sign
al
u
is said to be ad
missi
ble if
u
(
k
)
1.
Defini
tion 3.
1
: For
some
K
>0,(
x
0
,
ω
0
)
R
n
×R
r
i
s
s
a
i
d
t
o
b
e
K
-ste
p reg
u
latabl
e
if there
exists a
n
ad
missi
ble u
m
a
ke
s (1)
sati
sfy
e
(
K
)=0. Th
e set of all
re
gulatabl
e pai
r (
x
0
,
ω
0
) i
s
K
-s
tep
regul
atable
re
gion, den
oted
by
R
g
(
K
)
.
Acco
rdi
ng to
cla
ssi
cal
re
g
u
lation th
eory
,
there
exist
s
matrix
Π
R
n
×
r
and matrix
Γ
R
m
×
r
make
s the
equatio
n (6
) solvable, a
nd (6
) is a
zero state e
quation
whi
c
h describe
s
the
equilib
rium p
o
int
a
s
Ga
k
k
u
,
k
k
x
an
d
whe
r
e
e
=0.
Due to the restriction tha
t
u
(
k
)
1,
e
(
k
)
will go to zero a
s
ymptoti
c
ally at the equilibri
um poi
nt only if:
0
sup
k
Γ
ω
(
k
)+
Ga
1
(
7
)
So, the exosystem initial con
d
ition
s
co
rre
sp
ondi
ng to this eq
uilibrium point a
r
e
rest
ricted i
n
the
comp
act set
W
0
={
ω
0
R
r
:
Γ
aT
/2+
Ga
1,
k
0}.
Defini
tion 3.
2
: For some
K
>0, a state
0
x
is sai
d
to be
null co
ntroll
a
b
le if there ex
ists a
n
admissibl
e
u
make
s the
sy
stem
state
transfo
rm
s from
0
0
x
x
and sat
i
sfies
0
lim
k
x
k
.
The
set of a
ll the null
co
ntrollabl
e reg
i
on
x
0
i
s
n
u
ll
cont
rolla
ble
regi
on, de
n
o
ted by
B
A
C
,
.
Specially, th
e set of
null
cont
rollabl
e
regio
n
i
s
called
K
-step
null controll
a
b
le re
gion
when
0
K
x
, denoted by
B
A
C
K
,
.
By s
i
milarity transformation, we may assume:
m
n
n
n
n
n
n
R
B
B
B
R
A
A
A
)
(
2
1
)
(
)
(
2
1
2
1
2
1
2
1
,
0
0
(
8
)
Whe
r
e
A
1
h
a
s
all
eigenval
ues i
n
si
de o
r
on the
unit ci
rcle a
nd
A
2
h
a
s
all
eigenval
ues
outsi
de t
he
unit ci
rcl
e
. So, the null
co
ntrollabl
e regi
on
C
(
A
,
B
)=
1
n
R
C
(
A
2
,
B
2
). We co
nsi
d
e
r
th
e condition
about all
the
eigenvalu
e
s of
A
are o
u
tside the
unit
circle.
Gen
e
ral
l
y, if
K
is la
rg
e
eno
ugh
(i.e.
K
=10
~
30
),
B
A
C
K
,
is fairly approxi
m
ate to
B
A
C
,
.
Corre
s
p
ondin
g
ly, let:
2
1
2
1
2
1
,
,
Q
Q
Q
P
P
P
x
x
x
(9)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Output Re
gul
ation for Satu
rated System
s with Nonlin
ear Exosyste
m
(Huang
we
i)
4225
No
w, we will
descri
be the regulata
b
le re
gion
R
g
in terms of
B
A
C
K
,
and
W
0
.
Lemma 1 [
1
6]
.
Let
r
n
R
V
2
0
be the
uniqu
e soluti
on to the linear matrix equ
ation
V
0
S
-
A
2
V
0
=
P
2.
Then the
K
-step regulata
b
le
re
gion
R
g
(
K
) is
given by:
R
g
(
K
)
2
2
0
20
0
0
20
10
,
:
,
,
2
1
B
A
C
V
x
W
R
R
x
x
K
n
n
(
1
0
)
For th
e first
se
mi-cycle
of trian
g
le
wave, let
2
1
T
T
, by
carrying
out
a
simila
rity
transfo
rmatio
n.
i
P
A
i
Bu
A
x
A
T
x
T
i
i
T
T
i
i
T
T
1
0
1
1
0
1
0
1
1
1
1
1
1
We get:
1
2
1
2
1
1
1
1
1
2
1
1
T
Q
T
Cx
T
Q
T
Cx
T
e
T
e
Since
1
2
T
Q
is bou
nded for all
k
and
K
A
2
→∞
whe
n
k
→
1
T
,
0
lim
1
k
e
T
k
stand
s on:
0
1
1
0
2
1
2
0
2
1
2
20
T
i
i
T
i
i
ia
P
A
i
u
B
A
x
Den
o
te
1
0
2
1
2
0
T
i
i
i
P
A
V
,
V
0
satisfies
V
0
-
A
2
V
0
= (
A
-
I
)
-1
P
2 .
Let (
A
-
I
) =
D
, then
D
(
V
0
-
A
2
V
0
)=
P
2.
For the
se
co
nd semi-cycl
e
of triangl
e wave, whi
c
h
can
be viewe
d
as the
re
su
lt of half a
cycle p
a
rall
el transl
a
tion to
wards the
rig
h
t directio
n o
n
the time axis.
1
0
,
1
aT
a
k
k
The re
gulato
r
equation:
BG
Q
C
P
B
A
0
Similarly, let
1
0
1
2
1
2
0
T
i
i
i
T
P
A
V
,
-(A
-I)
=
D
，
we
get
D(V
0
-A
2
V
0
)=P
2
.
4. State Fe
e
dback
Con
t
r
o
ller Design
In this se
ction
,
we will co
nstruct a st
ate f
eedb
ack cont
rolle
r for abov
e system.
Lemma 2
[17]. Let
λ
(0, 1
)
, for any initial con
d
ition
0
~
x
C
λ
=
C
(
λ
-1
A
2
,
λ
-1
B
2
), there
exists
a state fee
dba
ck l
a
w
u
(
k
)=
h
[
x
(
k
)] su
ch
the solution
of
x
(
k
+1
)=
A
2
x
(
k
)+
B
2
u
(
k
) sati
sfies
x
(
k
)
λ
k
ρ
c
λ
(
x
0
)
C
λ
and the
co
ntrol sig
nal
u
(
k
)
λ
k
ρ
c
λ
(
x
0
)
λ
k
Lemma
2 giv
e
s
a bal
an
ce
betwe
en the
state
conv
e
r
g
ence rate a
n
d
the
cont
rol
of all the
initial state in
C
, denoted by
λ
k
. The const
r
uction of this
state feedba
ck co
ntrolle
r can be fund in
[14], base
d
o
n
whi
c
h,
we
will con
s
tru
c
t
a revi
sed
co
ntrolle
r la
w f
o
r regul
ation
probl
em in
th
is
pape
r.
Theorem
2.
Assu
me there exi
s
ts a matrix
V
0
which
sati
sfies
D
(
V
0
-
AV
0
)=
P
2,
for every
initial pair (
x
0
,
ω
0
) in the re
gulatabl
e regi
on, unde
r the
following
con
t
roller:
u
(
k
)=
h
[
x
(
k
)-
λ
k
V
2
ω
(
k
)
-(1
-
λ
k
)
Π
2
ω
(
k
)]
+ (1
-
λ
k
)(
Γ
ω
(
k
)+
Ga
)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4222 – 4
229
4226
The clo
s
e
d
-l
o
op sy
ste
m
sa
tisfies lim
k
→∞
e
(
k
)=
0.
Proof.
Co
rre
s
po
ndin
g
to (8), we
can di
vide system (1) in to two subsy
s
tems.
k
P
k
u
B
k
x
A
k
x
k
P
k
u
B
k
x
A
k
x
2
2
2
2
1
1
1
1
1
1
Den
o
te
2
,
1
,
1
~
i
k
k
V
k
x
k
x
i
k
i
k
i
i
By Lemma 1, for i =1, 2, we get:
a
a
V
k
P
D
I
k
B
k
u
B
k
x
A
k
x
i
k
i
k
i
k
i
k
i
i
i
i
1
)
1
(
)
(
~
1
~
Base on the
controlle
r defin
ed in Lemm
a
2, we co
nst
r
u
c
t a cont
rolle
r:
Ga
k
k
x
h
k
u
k
1
~
2
Apply it to the
two subsystems:
a
V
k
P
D
I
k
x
h
B
k
x
A
k
x
a
V
k
P
D
I
k
x
h
B
k
x
A
k
x
k
k
k
k
2
2
2
2
2
2
2
1
1
2
1
1
1
1
~
~
1
~
~
~
1
~
We can get
k
T
T
k
k
x
h
k
x
1
1
2
2
~
[
,
0
~
lim
by Lemma 2. Since
A
1
is semi-stable a
n
d
k
T
k
x
h
1
2
~
[,
k
x
1
~
also
conve
r
gen
ce
s to the origin.
1
1
~
1
1
2
T
k
T
Ga
k
k
x
h
k
u
The clo
s
e
d
-lo
op sy
stem
sa
tisfies
0
lim
1
k
e
T
k
. Simila
r cont
rolle
r can be co
nstructed
for the se
con
d
semi
-cy
cle
of a triangle cycle.
5. Numerical
Examples
Example 1
. A semi-sta
ble system
as follo
ws u
nder the a
c
t
i
on of squ
a
re sign
al
(T/2=100
0)
k
k
x
k
e
k
k
k
k
u
k
x
k
x
1
0
0
1
1
0
0
1
)
(
1
0
0
1
1
)
(
1
.
0
0
0
1
.
0
)
(
1
0
0
1
2
.
1
2
.
0
0
4
.
1
1
With
x
0
=[
-1.5 -0.8]
T
,
ω
(0)=
[1.5 1.5]
T
, the regul
ation eq
uation ha
s so
lutions.
1
0
0
1
,
3
.
0
2
.
0
0
5
.
0
P
A
S
,
5
.
0
25
.
0
0
25
.
0
V
Applying the controlle
r pro
v
ided in [13], we get:
u
(
k
)=
h
[
x
(
k
)-0.
97
k
V
ω
(
k
)
-(1
-
0
.97
k
)
Π
ω
(
k
)]
+
(1-0.9
7
k
)
Γ
ω
(
k
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Output Re
gul
ation for Satu
rated System
s with Nonlin
ear Exosyste
m
(Huang
we
i)
4227
The clo
s
e
d
-l
o
op state tra
cking sh
own in Figure 1.
Figure 1. Clo
s
ed
-loo
p State Tra
cki
ng un
der the Squ
a
re Signal Di
sturba
nce in Example 1
Example 2.
The followi
ng system unde
r the actio
n
of triangle
signal
(T=100
0)
k
k
x
k
e
k
k
u
k
x
k
x
1
0
0
1
1
0
0
1
)
(
1
.
0
0
0
1
.
0
)
(
1
0
0
1
2
.
1
2
.
0
0
4
.
1
1
In the fist
semi-cycle,
x
0
=[-0.1 -0.0
1]
T
,
ω
0
=[0 0]
T
, a=[0.00
3
0.0
04]
T
. T
he
re
gulation
equatio
n ha
s solutio
n
s
1
0
0
1
,
3
.
0
2
.
0
0
5
.
0
P
A
S
,
1
0
0
1
G
D
(
V
-
AV
)=
P
h
a
s a uni
que
solution:
5
.
2
875
.
1
0
625
.
0
V
We get
t
he state
f
eedb
ack co
ntrolle
r
u
(
k
)=
h
[
x
(
k
)-0.95
k
V
ω
(
k
)-
(1
-0.95
k
)
Π
ω
(
k
)]+
(
1
-
0.95
k
)(
Γ
ω
(
k
)+
Ga
)
The clo
s
e
d
-l
o
op state tra
cking are pl
otted in Figure 2.
Figure 2. Clo
s
ed
-loo
p State Tra
cki
ng in
First Semi
-cy
cle in Exampl
e
2
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4222 – 4
229
4228
In the last se
mi-cy
cle,
x
0
=[
1.5 2.0]
T
，
ω
0
=[1.5 2.0]
T
，
a
=
[0.003 0.0
0
4
]
T
1
0
0
1
,
3
.
0
2
.
0
0
5
.
0
P
A
S
,
1
0
0
1
G
There exists t
he uniq
ue sol
u
tion to
D
(
V
-
AV
)=
P
5
.
2
875
.
1
0
625
.
0
V
The state fee
dba
ck
cont
rol
l
er
u
(
k
)=
h
[
x
(
k
)-0.9
5
k
V
ω
(
k
)-(1-0.95
k
)
Π
ω
(
k
)]
+ (1
-0.95
k
)(
Γ
ω
(
k
)+
Ga
). The
clo
s
ed
-loo
p state tracking
s are plotted in
Figure 3.
Figure 3. Clo
s
ed
-loo
p State Tra
cki
ng in
Last Semi-cy
c
le in Exampl
e
2
In each
cycl
e perio
d, two different in
ternal mo
de
prin
ciple
s
a
r
e applie
d for a semi
-
cycle respectively,
thus
G
and
V
are go
t and the stat
e-feed
ba
ck
controlle
r
u(
k)
are con
s
tru
c
t
ed.
State trackin
g
in t
w
o
cy
cl
es
are
sho
w
n in
Figu
re
4
,
with
x
0
=
[-0.1 -0.01]
T
，
ω
0
=
[0 0]
T
，
a=
[0.
003
0.004]
T
.
Figure 4. State Tra
cki
ng in
Two Cy
cle
s
in Example 2
6. Conclusio
n
In this p
ape
r,
we
studie
d
th
e output
re
gu
lation p
r
obl
e
m
of suture
d l
i
ne
system
u
nder the
action of no
nlinea
r exco
system. At the equilib
ri
u
m
point, initial state of the plant an
d
the
exosystem
are re
stri
cted i
n
a
comp
act
set
W
0
. The
K-
Step
a
s
ymptotically reg
ualatabl
e re
gi
on
R
g
(
K
) i
s
de
scrib
ed by
W
0
and
K-
Ste
p
null contro
llable regio
n
B
A
C
K
,
. Segmente
d
co
ntrol
strategi
es a
r
e applied to
external di
stu
r
ban
ce
s like the squ
a
re
si
gnal and tri
a
ngle sig
nal. The
internal
pri
n
ci
ples fo
r ea
ch
semi
-cy
cle of
the
exosy
s
te
m are
given.
Based
on the
state feed
ba
ck
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Output Re
gul
ation for Satu
rated System
s with Nonlin
ear Exosyste
m
(Huang
we
i)
4229
laws propo
sed, the cont
rolle
r is
co
n
s
tru
c
ted
re
sp
ectively and
example
s
show th
at it can
sup
p
re
s
s
so
me ex
ternal n
online
a
r di
stu
r
ban
ce effe
ctiv
ely
.
Ackn
o
w
l
e
dg
ements
This wo
rk
is sup
porte
d
by Scien
c
e and
Tech
nolo
g
y Proje
c
t
of
Zh
ejiang
P
r
ovin
ce (G
rant
No. 201
2C21
095), China.
Referen
ces
[1
]
BA
Fra
n
c
is.
T
he l
i
ne
ar
mu
lti
v
aria
ble r
e
g
u
la
tor prob
le
m
. I
EEE Conf
erence on
Dec
i
sion and Contro
l
inclu
d
i
ngs t
he
15t
h S
y
mpos
iu
m on Adapt
iv
e
Processes.
NJ.
1976;
1
5
:
873-
878.
[
2
]
BA Francis,
WM Wonham.
I
n
t
e
rnal mod
e
l pr
inc
i
p
l
e f
o
r linear mu
lt
iv
aria
ble re
gul
at
ors.
Appli
e
d
Mathem
atics and Optim
i
z
a
tion
.
1975;
2(
2):
170-1
94.
[
3
]
BA Francis,
WM Wonham.
T
he i
n
t
e
rn
al m
o
del
princ
i
p
l
e
of
cont
rol
t
h
e
o
r
y
.
Automatica
.
197
6;
12(
5):
457-
465.
[
4
]
J Hua
ng,
Z C
h
en.
A g
e
n
e
ral
framew
ork for
output r
egu
lati
on pr
ob
le
m
.
Pr
ocee
din
g
s of
t
he Amer
ica
n
Cont
ro
l Conf
er
ence.
Anch
ora
ge.
200
2;
1:
10
2-10
9.
[
5
]
Q Gong,
W Lin
.
A not
e on glo
bal o
u
t
put
regu
lat
i
o
n
of
nonl
in
ear s
y
st
ems i
n
t
he out
put
f
e
e
dback f
o
rm.
IEEE Transactions on Aut
o
m
a
tic Control
.
20
0
3
;
48(6):
10
49-
105
4.
[
6
]
Z Ding.
Glo
b
a
l
out
put
r
egu
lat
i
on of
uncert
a
in
non
lin
ear s
y
st
ems
w
i
t
h
e
x
o
g
eno
us si
gna
ls.
Autom
a
tica
.
200
1;
37(1):
11
3-11
9.
[
7
]
Z Ding.
Glo
b
a
l st
ab
iliz
at
io
n
and
dist
ur
ba
nce su
ppr
essi
on of
a class
of
no
nli
near
s
y
st
ems
w
i
t
h
uncert
a
in int
e
rnal mo
del.
Aut
o
matica
.
20
03;
39(3):
471-
47
9.
[
8
]
VO Nikif
o
rov.
Adapt
iv
e n
o
n
l
i
near t
r
ackin
g
w
i
t
h
com
p
l
e
t
e
compe
n
sat
i
on
of
unk
no
w
n
dist
urb
ances
.
Europ
e
a
n
Jour
nal of Co
ntrol.
199
8;
4:
132-1
39.
[
9
]
Z Ding.
Adapt
i
v
e st
abil
i
zat
i
o
n
of
a class of
non
lin
ear s
y
st
ems
w
i
t
h
unst
abl
e int
e
rn
al d
y
n
a
mics.
IE
EE
T
r
ansactio
n
s o
n
Auto
matic C
ontrol.
20
03;
4
8
(10):
17
88-
17
92.
[
10]
Z Ding.
Adapt
i
v
e dist
urb
anc
e reject
i
on of
no
nlin
ear s
y
st
em
s in an e
x
t
e
nd
ed out
p
u
t
f
eed
back f
o
rm.
IET
Contro
l T
heory
and App
lic
atio
n
.
2007;
1(
1):
298-3
03.
[
11]
HC L
i
a
w
,
B S
h
irinz
a
d
eh.
Ro
bust
a
d
a
p
t
i
ve
const
r
ai
ned
m
o
t
i
on
t
r
ackin
g
cont
rol
of
Pi
e
z
o-act
uat
e
d
f
l
exure-
base
d
mecha
n
isms f
o
r micro/
nan
o m
ani
pul
at
io
n.
IEEE Transactions on Industrial Electronics
.
201
1;
58(4):
14
06-1
415.
[
12]
JV Flores,
JM
Gomes Da Silva,
LFA Pere
ira,
DG
Sbar
b
a
ro.
R
e
p
e
t
i
t
i
ve
cont
ro
l desi
g
n
f
o
r
m
i
mo
sy
ste
m
s
w
i
th
sa
tu
ra
ting
a
c
tu
ato
r
s.
IEEE
Transactions on A
u
tomatic Control
.
201
2;
57(1):
192-1
98.
[
13]
WWang Qi,
Gao T
i
an,
He
H
e
.
An Ad
apt
iv
e Fuzz
y C
ont
r
o
l Met
h
od f
o
r
Spacecr
a
f
t
s
Based
on
T
-
S
Mode
l.
Te
l
k
omn
i
ka
In
do
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Jo
u
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HHen
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Cont
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r a c
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vert
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l.
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lk
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m
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an J
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ont
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t
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o
f
t
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t
h
I
EEE
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an
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