I
n
d
on
e
s
ian
Jou
r
n
al
of
E
lec
t
r
ical
E
n
gin
e
e
r
in
g
a
n
d
Com
p
u
t
e
r
S
c
ience
Vol.
25
,
No.
3
,
M
a
r
c
h
2022
,
pp.
1328
~
1343
I
S
S
N:
2502
-
4752,
DO
I
:
10
.
11591/i
jee
c
s
.
v
25
.i
3
.
pp
1328
-
1343
1328
Jou
r
n
al
h
omepage
:
ht
tp:
//
ij
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c
s
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D
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omput
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in
g, C
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pe
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it
y of
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c
hnol
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a
pe
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out
h A
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ic
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Ar
t
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I
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o
AB
S
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RA
CT
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ti
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le
h
is
tor
y
:
R
e
c
e
ived
M
a
r
8
,
2021
R
e
vis
e
d
De
c
23
,
2021
Ac
c
e
pted
J
a
n
7
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202
2
O
n
e
o
f
t
h
e
mo
s
t
ch
al
l
en
g
i
n
g
as
p
ec
t
s
i
n
t
h
e
n
o
n
l
i
n
ear
co
n
t
ro
l
o
f
a
mag
n
et
i
c
l
ev
i
t
a
t
i
o
n
(Ma
g
l
e
v
)
s
y
s
t
em
i
s
t
o
fi
n
d
a
n
effi
c
i
en
t
c
o
n
t
ro
l
al
g
o
r
i
t
h
m
t
o
ach
i
e
v
e
t
h
e
s
t
a
b
i
l
i
t
y
an
d
accu
racy
o
f
t
h
e
cl
o
s
e
d
-
l
o
o
p
s
y
s
t
em.
T
h
e
ch
a
l
l
e
n
g
e
i
s
t
h
e
n
t
o
d
e
v
e
l
o
p
a
l
i
n
ear
i
zi
n
g
co
n
t
r
o
l
al
g
o
r
i
t
h
m
t
o
mai
n
t
a
i
n
a
s
t
ee
l
b
al
l
at
a
d
es
i
red
p
o
s
i
t
i
o
n
.
In
t
h
i
s
p
ap
er,
a
n
o
v
el
l
i
n
ear
i
zi
n
g
co
n
t
ro
l
al
g
o
ri
t
h
m
i
s
p
ro
p
o
s
ed
,
w
h
i
c
h
c
o
n
s
i
s
t
s
o
f
t
h
e
L
y
a
p
u
n
o
v
d
i
rec
t
met
h
o
d
(L
D
M)
an
d
t
h
e
mo
d
e
l
referen
ce
c
o
n
t
ro
l
(MRC).
T
h
e
L
y
a
p
u
n
o
v
fu
n
ct
i
o
n
i
s
d
ev
e
l
o
p
ed
u
s
i
n
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t
h
e
n
o
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i
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ear
eq
u
at
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o
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h
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mag
n
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n
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l
i
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ear
s
ec
o
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d
er
s
y
s
t
em.
T
w
o
co
n
t
ro
l
me
t
h
o
d
s
are
d
ev
e
l
o
p
ed
t
o
g
u
ara
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t
ee
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t
em
ro
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s
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n
es
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o
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t
p
u
t
s
t
ab
i
l
i
t
y
.
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rs
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l
y
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a
n
ew
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n
t
e
g
ral
l
i
n
ear
q
u
ad
ra
t
i
c
re
g
u
l
at
o
r
(IL
Q
R)
i
s
d
e
s
i
g
n
e
d
fo
r
t
h
e
ref
eren
ce
mo
d
el
.
T
h
en
,
an
ad
d
i
t
i
o
n
a
l
i
n
n
o
v
a
t
i
v
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p
ro
p
o
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t
i
o
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a
l
g
a
i
n
i
s
co
m
b
i
n
ed
w
i
t
h
t
h
e
l
i
n
ear
i
zi
n
g
co
n
t
ro
l
l
er
t
o
mak
e
t
h
e
n
o
n
l
i
n
ear
co
n
t
r
o
l
s
i
g
n
a
l
s
t
r
o
n
g
er.
T
h
e
s
i
m
u
l
a
t
i
o
n
res
u
l
t
s
i
n
d
i
cat
e
t
h
a
t
t
h
e
p
ro
p
o
s
ed
l
i
n
eari
z
i
n
g
c
o
n
t
r
o
l
l
er
h
as
e
x
cel
l
en
t
s
et
-
p
o
i
n
t
t
rack
i
n
g
,
n
o
t
i
me
d
el
a
y
,
fas
t
ri
s
i
n
g
a
n
d
s
et
t
l
i
n
g
t
i
me
s
,
an
d
ach
i
ev
e
s
s
t
a
t
es
s
t
ab
i
l
i
t
y
.
K
e
y
w
o
r
d
s
:
I
ntegr
a
l
li
ne
a
r
qua
dr
a
ti
c
r
e
gulator
L
inea
r
iza
ti
on
L
ya
punov
M
a
gne
ti
c
levitation
M
ode
l
r
e
f
e
r
e
nc
e
c
ontr
ol
Nonlinea
r
S
tabili
ty
Th
i
s
i
s
a
n
o
p
en
a
c
ces
s
a
r
t
i
c
l
e
u
n
d
e
r
t
h
e
CC
B
Y
-
SA
l
i
ce
n
s
e.
C
or
r
e
s
pon
din
g
A
u
th
or
:
Yoha
n
Da
r
c
y
M
f
oumboul
ou
De
pa
r
tm
e
nt
of
E
lec
tr
ica
l,
E
lec
tr
on
ic,
a
nd
C
omput
e
r
E
nginee
r
ing
,
C
a
pe
P
e
nins
ula
Unive
r
s
it
y
of
T
e
c
hn
ology
B
e
ll
vil
le
C
a
mpus
,
P
.
O
B
ox
7530,
C
a
pe
T
own,
S
ou
th
Af
r
ica
E
mail:
f
a
bolous
86yo@ya
hoo.
f
r
1.
I
NT
RODU
C
T
I
ON
R
a
pid
ur
ba
niza
ti
on
due
to
r
ur
a
l
e
xodus
ha
s
br
ough
t
a
c
r
is
is
in
the
tr
a
ns
por
tation
s
e
c
tor
be
c
a
us
e
a
lot
mor
e
pe
ople
now
make
us
e
of
tr
a
ns
por
t
in
the
ur
ba
n
e
nvir
onment.
Additi
ona
ll
y,
the
e
nvir
onment
ha
s
be
e
n
im
pa
c
ted
ne
ga
ti
ve
ly
due
to
the
e
xc
e
s
s
ive
us
e
of
f
o
s
s
il
f
ue
ls
in
the
tr
a
ns
por
tation
s
e
c
tor
to
mee
t
the
de
mands
[
1]
–
[
4]
.
T
he
f
a
s
t
de
pletion
of
non
-
r
e
ne
wa
ble
r
e
s
our
c
e
s
ha
s
highl
ight
e
d
the
ne
e
d
f
or
c
lea
n,
e
f
f
ici
e
nt,
a
nd
s
us
taina
ble
mea
n
s
of
tr
a
ns
por
t
[
4]
.
T
he
M
a
glev
s
ys
tem
ha
s
be
e
n
identif
ied
a
s
a
s
olut
ion
to
the
c
r
is
is
in
the
tr
a
ns
por
tation
s
e
c
tor
be
c
a
us
e
it
is
a
n
e
lec
tr
omec
h
a
nica
l
s
ys
tem,
a
nd
ther
e
f
o
r
e
it
doe
s
not
make
us
e
of
f
os
s
il
f
ue
ls
[
5]
.
T
he
m
a
gne
ti
c
levitation
(
M
a
glev)
s
ys
tem
is
ope
n
loop
ins
table
a
nd
highl
y
non
li
ne
a
r
,
whic
h
make
s
it
a
ve
r
y
c
ha
ll
e
nging
c
ontr
ol
p
r
oblem
[
6
]
–
[
8]
.
T
h
e
e
f
f
icie
nt
c
ontr
ol
of
a
M
a
glev
s
ys
tem
c
a
n
r
e
duc
e
the
ope
r
a
ti
ng
c
os
t,
f
ue
l
e
c
onomy,
dr
ivi
ng
r
a
nge
a
nd
p
e
r
f
or
manc
e
in
va
r
ious
indus
tr
ies
[
9
]
,
[
10
]
.
One
of
the
mos
t
e
f
f
icie
nt
methods
to
s
tabili
z
e
a
nd
e
ns
ur
e
r
obus
tnes
s
of
the
M
a
glev
s
ys
tem
is
t
he
li
ne
a
r
iza
ti
on
tec
hniq
ue
[
11]
–
[
13]
.
T
he
li
ne
a
r
iza
ti
on
method
dr
a
ws
de
duc
ti
ons
a
bou
t
the
loca
l
s
tabili
ty
of
a
nonli
ne
a
r
s
ys
tem
a
r
ound
a
n
ope
r
a
ti
ng
point
f
r
om
the
s
tabili
ty
c
ha
r
a
c
ter
is
ti
c
s
of
the
s
ys
tem’
s
li
ne
a
r
e
s
ti
mation.
T
he
s
tabili
ty
o
f
dyna
mi
c
s
ys
tems
c
a
n
be
a
na
lyze
d
in
a
ve
r
y
pr
e
c
is
e
wa
y
with
L
ya
punov
methods
if
the
e
quivale
nt
math
e
matica
l
models
a
r
e
e
xpr
e
s
s
e
d
a
s
s
ys
tems
of
nor
mal
dif
f
e
r
e
nti
a
l
e
qua
ti
ons
[
14]
,
[
15]
.
I
n
the
pa
pe
r
,
W
ong
[
16]
,
the
de
s
ign
of
a
pha
s
e
lea
d
c
ompens
a
tor
to
s
tabili
z
e
nonli
ne
a
r
a
nd
li
ne
a
r
ize
d
models
of
a
magne
ti
c
l
e
vit
a
ti
on
s
ys
tem
wa
s
pr
opos
e
d.
T
he
c
ontr
ol
ler
s
howe
d
s
a
ti
s
f
a
c
tor
y
r
e
s
ult
s
on
both
models
o
f
the
magne
ti
c
l
e
vit
a
ti
on
Evaluation Warning : The document was created with Spire.PDF for Python.
I
ndone
s
ian
J
E
lec
E
ng
&
C
omp
S
c
i
I
S
S
N:
2502
-
4752
De
v
e
lopme
nt
of
a
ne
w
l
inear
iz
ing
c
ontr
oll
e
r
us
ing
L
y
apunov
s
tabi
li
ty
…
(
Y
ohan
Dar
c
y
M
foumboulo
u)
1329
s
ys
tem.
B
ut
the
c
ontr
oll
e
r
wa
s
not
r
obu
s
t,
whic
h
wa
s
a
major
dr
a
wba
c
k.
T
he
li
ne
a
r
a
nd
nonli
ne
a
r
s
t
a
te
-
s
p
a
c
e
r
e
gulator
s
to
c
ontr
ol
a
nonli
ne
a
r
dyna
mi
c
model
of
a
magne
ti
c
levitation
s
ys
tem
wa
s
pr
opos
e
d
[
17]
.
T
he
c
ontr
oll
e
r
s
c
ould
gua
r
a
ntee
the
s
tabili
ty
of
the
c
lo
s
e
d
-
loop
s
y
s
tem
only
in
s
mall
int
e
r
va
ls
.
I
n
lar
ge
i
nter
va
ls
,
the
c
ontr
oll
e
r
s
c
ould
not
br
ing
the
s
ys
tem
to
e
quil
i
br
ium
.
Z
ha
ng
e
t
al.
[
18]
,
the
f
e
e
dba
c
k
li
ne
a
r
iza
ti
on
tec
hnique
wa
s
a
ppli
e
d
to
im
pr
ove
the
pe
r
f
or
manc
e
of
a
magne
ti
c
levitation
s
ys
tem.
F
e
e
dba
c
k
li
ne
a
r
iza
ti
o
n
s
howe
d
be
tt
e
r
r
e
s
ult
s
c
ompar
e
d
to
T
a
ylor
l
inea
r
iza
ti
on
tec
hnique.
A
hybr
id
e
xc
it
a
ti
on
c
on
t
r
ol
a
lgor
it
hm
b
a
s
e
d
on
T
a
ylor
s
e
r
ies
e
xpa
ns
ion
a
r
ound
a
n
ope
r
a
ti
ng
point
to
br
ing
a
magne
ti
c
levitation
s
ys
tem
to
s
tabili
ty
wa
s
s
ugge
s
ted
[
19]
.
T
his
method
s
howe
d
that
whe
n
the
pos
it
ion
of
the
levitation
s
ys
tem
is
f
a
r
a
wa
y
f
r
om
it
s
e
quil
ibr
ium
point
,
the
c
ont
r
oll
e
r
c
a
nnot
gua
r
a
ntee
the
s
tabili
ty
of
the
c
los
e
d
-
loop
s
ys
tem.
J
inquan
e
t
al.
[
20]
,
a
n
a
da
pti
ve
r
obus
t
r
e
gulato
r
to
c
ontr
ol
a
n
onli
ne
a
r
magne
ti
c
levitation
tr
a
in
s
us
pe
ns
ion
s
ys
tem
wa
s
de
s
igned.
T
he
c
ont
r
oll
e
r
s
howe
d
s
a
ti
s
f
a
c
tor
y
r
e
s
ult
s
to
va
r
iation
of
pa
r
a
mete
r
s
,
but
only
if
the
s
tate
s
of
th
e
magne
ti
c
levitation
s
ys
tem
we
r
e
s
ubje
c
ted
to
c
o
ns
tr
a
int
s
.
T
o
im
pr
ove
the
pe
r
f
o
r
manc
e
of
a
hybr
id
e
xc
it
a
ti
on
magne
ti
c
levitation
s
ys
tem,
[
21]
de
s
igned
a
li
ne
a
r
c
ontr
oll
e
r
ba
s
e
d
on
r
obus
t
f
e
e
dba
c
k
li
ne
a
r
iza
ti
on
method.
T
his
method
pr
ov
ided
a
li
mi
ted
d
e
gr
e
e
of
r
obus
tnes
s
be
c
a
us
e
the
tot
a
l
mas
s
of
the
s
us
pe
ns
io
n
c
ould
not
be
a
c
c
ur
a
tely
mea
s
ur
e
d.
C
laudio
e
t
al.
[
22]
,
the
s
tudy
a
nd
de
s
ign
of
a
magne
ti
c
levitator
s
ys
tem
ba
s
e
d
on
e
lec
tr
onic
c
omponents
we
r
e
pr
opos
e
d.
T
he
e
lec
tr
onic
c
ir
c
uit
de
ve
loped
pr
ovided
a
c
e
r
tain
de
gr
e
e
of
s
tabili
ty
whe
n
a
tr
iode
o
r
t
r
iode
a
lt
e
r
na
ti
n
g
c
ur
r
e
nt
(
T
R
I
AC
)
wa
s
us
e
d
to
r
e
gulate
the
f
low
of
c
ur
r
e
nt
in
the
c
ir
c
uit
.
A
s
igni
f
ica
nt
dr
a
wba
c
k
of
thi
s
tec
h
nique
is
that
it
only
wo
r
ks
f
o
r
the
li
ne
a
r
ize
d
model
of
th
e
magne
ti
c
levitation
s
ys
tem
a
nd
c
a
nnot
be
a
ppli
e
d
to
it
s
nonli
ne
a
r
model
whic
h
is
the
na
tur
e
of
the
s
ys
tem.
T
he
de
ve
lopm
e
nt
of
a
de
c
oupli
ng
c
ontr
ol
s
olut
ion
to
s
olve
the
ins
tabili
ty
o
f
a
modul
e
s
us
pe
ns
ion
s
ys
tem
wa
s
done
[
23]
.
T
he
s
olut
ion
ha
d
pos
it
ive
r
e
s
ult
s
,
but
the
a
uthor
s
did
no
t
c
ons
ider
the
a
c
tual
e
f
f
e
c
ts
o
f
t
he
tr
a
c
k
ir
r
e
gular
i
ti
e
s
dur
ing
r
e
a
l
-
ti
me
ope
r
a
ti
o
n
of
the
s
us
pe
n
s
ion
modul
e
,
whic
h
ha
s
a
s
igni
f
ica
nt
im
pa
c
t
on
the
nonli
ne
a
r
be
ha
vior
of
the
s
ys
tem.
A
s
olut
ion
to
s
olve
the
f
luctua
ti
ons
of
the
s
us
pe
ns
ion
s
ys
tem
wh
e
n
a
magne
ti
c
levitation
s
ys
tem
pa
s
s
e
s
a
t
low
s
pe
e
d
ove
r
a
tr
a
c
k
s
tep
wa
s
pr
opos
e
d
[
24]
.
T
he
a
uthor
s
de
ve
loped
a
f
e
e
dba
c
k
li
ne
a
r
iza
ti
on
c
ontr
oll
e
r
ba
s
e
d
on
a
de
c
oupli
ng
tec
hnique.
I
n
c
ompar
is
on
to
a
tr
a
dit
i
ona
l
de
r
ivative
a
nd
int
e
gr
a
l
(
P
I
D
)
c
ont
r
ol
a
lgor
i
t
hm,
the
de
c
oupli
ng
c
ontr
oll
e
r
r
e
duc
e
d
the
f
luctua
ti
ons
by
a
maximum
of
49.
6
%
.
Unf
o
r
tunate
ly,
the
c
ontr
ol
s
olut
ion
s
howe
d
li
mi
tations
be
c
a
us
e
it
c
a
nnot
be
a
ppli
e
d
t
o
wide
r
ope
r
a
ti
ng
r
a
nge
s
s
uc
h
a
s
medium
a
nd
hi
gh
-
s
pe
e
d
s
it
ua
ti
ons
.
T
his
pa
pe
r
p
r
opos
e
s
a
li
n
e
a
r
izing
c
ontr
ol
a
lgor
i
th
m
a
s
pos
s
ibl
e
s
olut
ion
to
the
indus
tr
ial
c
ha
ll
e
nge
s
of
a
c
hieving
a
c
c
ur
a
te
c
ontr
ol
of
a
nonli
ne
a
r
magne
ti
c
levitation
s
ys
tem.
L
ya
punov
s
tabili
ty
theor
y
ba
s
e
d
on
the
model
r
e
f
e
r
e
nc
e
c
ontr
ol
tec
hnique
is
a
ppli
e
d
to
the
nonli
ne
a
r
magne
ti
c
levitation
s
ys
tem.
T
he
r
e
s
e
a
r
c
h
ga
p
a
nd
mer
it
o
f
the
li
ne
a
r
izing
c
ontr
o
ll
e
r
de
ve
loped
in
thi
s
pa
pe
r
c
ompar
e
d
to
the
other
c
ontr
oll
e
r
s
r
e
view
e
d
in
the
li
ter
a
tur
e
is
that
the
p
r
opos
e
d
c
ontr
oll
e
r
c
a
n
s
tabili
z
e
a
ll
the
s
tate
s
of
c
los
e
d
-
loop
s
ys
tem
a
t
qui
c
k
e
r
r
a
te
a
nd
s
igni
f
ica
ntl
y
im
pr
ove
their
pe
r
f
or
manc
e
s
.
T
o
gua
r
a
ntee
a
n
ove
r
s
hoot
be
low
2%
,
f
a
s
t
r
is
e
ti
me,
pe
r
f
e
c
t
s
e
t
-
point
tr
a
c
king
a
nd
r
obus
tnes
s
of
the
c
los
e
d
-
lo
op
to
pa
r
a
mete
r
s
unc
e
r
tainti
e
s
,
two
innovative
a
p
pr
oa
c
he
s
a
r
e
pr
opos
e
d:
(
a
)
An
a
ddit
ional
p
r
opor
ti
ona
l
c
on
tr
oll
e
r
ga
in
is
mu
lt
ipl
ied
to
the
nonli
ne
a
r
c
ont
r
ol
ler
.
T
his
c
ombi
na
ti
on
of
c
ontr
oll
e
r
s
gua
r
a
ntee
s
the
r
obus
tnes
s
of
the
r
e
s
ult
ing
nonli
ne
a
r
c
ontr
ol
s
ignal
;
a
nd
(
b)
A
r
e
f
e
r
e
nc
e
model
made
of
the
c
ombi
na
ti
on
of
a
li
ne
a
r
model
c
ontr
oll
e
d
by
a
n
it
e
r
a
ti
ve
li
ne
a
r
qua
dr
a
ti
c
r
e
gulator
(
I
L
QR
)
c
ont
r
oll
e
r
is
de
s
igned.
T
his
c
om
bination
make
s
the
s
tate
s
of
the
r
e
f
e
r
e
nc
e
model
s
table
to
c
ha
nge
s
of
the
be
ha
vior
o
f
the
nonli
ne
a
r
model
in
r
e
a
l
-
ti
me.
2.
RE
S
E
AR
CH
M
E
T
HO
D
T
he
r
e
s
e
a
r
c
h
de
s
ign
will
be
e
xplaine
d
in
two
s
e
c
t
ions
,
na
mely,
f
ir
s
t
the
theor
y
be
hind
de
s
igni
ng
a
li
ne
a
r
izing
c
ontr
ol
a
lgo
r
it
hm
ba
s
e
d
on
the
L
ya
p
unov
s
tabili
ty
theor
y
(
s
ubs
e
c
ti
on
2.
1
)
.
T
he
n
,
the
s
teps
to
de
s
ign
the
li
ne
a
r
izing
c
ontr
ol
ler
to
s
tabili
z
e
the
M
a
glev
s
ys
tem
will
be
pr
ov
ided
(
s
ubs
e
c
ti
on
2.
2)
.
2.
1.
T
h
e
or
y
t
o
d
e
s
ign
a
li
n
e
ar
izin
g
c
on
t
r
ol
algo
r
it
h
m
b
as
e
d
on
t
h
e
L
yap
u
n
ov
s
t
ab
il
i
t
y
T
he
nonli
ne
a
r
magne
ti
c
levitation
s
ys
tem
is
c
ha
r
a
c
ter
ize
d
by
the
non
li
ne
a
r
s
tate
a
s
(
1
)
a
nd
(
2)
[
25]
:
̇
=
(
)
+
(
)
(
1)
=
(
2)
w
he
r
e
∈
ℜ
is
the
s
tate
ve
c
tor
(
n
-
ve
c
tor
)
;
∈
ℜ
is
the
c
o
ntr
ol
ve
c
tor
;
∈
ℜ
is
the
ve
c
tor
va
lued
f
unc
ti
on;
∈
ℜ
1
is
the
plant
output
;
∈
ℜ
1
×
is
the
ou
tput
mat
r
ix.
T
he
nonli
ne
a
r
model
of
the
magne
ti
c
levitation
s
ys
tem
is
us
e
d
to
de
ve
lop
the
e
qua
ti
on
of
the
li
ne
a
r
izing
c
ontr
oll
e
r
.
M
ode
l
r
e
f
e
r
e
nc
e
c
ontr
ol
(
M
R
C
)
a
nd
L
ya
punov
s
e
c
ond
method
f
o
r
s
tabili
ty
a
r
e
us
e
d
to
de
s
ign
a
li
ne
a
r
izing
c
ontr
oll
e
r
.
I
n
the
ne
xt
s
e
c
ti
ons
,
the
s
teps
to
de
ve
lop
the
c
ontr
oll
e
r
a
r
e
de
s
c
r
ibed.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2502
-
4752
I
ndone
s
ian
J
E
lec
E
ng
&
C
omp
S
c
i
,
Vol.
25
,
No.
3
,
M
a
r
c
h
20
22
:
1328
-
1343
1330
2.
1.
1.
Re
f
e
r
e
n
c
e
m
od
e
l
T
o
p
r
ope
r
ly
make
the
output
of
the
M
a
glev
s
ys
tem
a
c
c
ur
a
te,
it
is
ne
c
e
s
s
a
r
y
to
de
s
ign
a
n
idea
l
r
e
f
e
r
e
nc
e
model
s
ys
tem.
T
he
idea
is
to
make
the
e
r
r
or
ve
c
tor
be
twe
e
n
s
tate
ve
c
tor
of
the
r
e
f
e
r
e
nc
e
m
ode
l
a
nd
the
s
tate
ve
c
tor
o
f
the
M
a
glev
s
ys
tem
go
to
z
e
r
o
a
s
t
he
ti
me
tends
to
inf
ini
ty.
T
he
de
s
ign
pr
obl
e
m
is
to
de
ve
lop
a
c
ontr
o
ll
e
r
that
a
lwa
ys
pr
oduc
e
s
a
s
ign
a
l
that
f
or
c
e
s
the
s
tate
o
f
the
magne
ti
c
levitation
s
ys
tem
towa
r
d
the
r
e
f
e
r
e
nc
e
model
s
tate
[
26]
–
[
28]
.
F
ig
ur
e
1
s
hows
the
block
diagr
a
m
of
the
c
los
e
d
-
loop
M
R
C
s
ys
tem
c
onf
igur
a
ti
on,
whe
r
e
v
is
the
c
ontr
ol
input
o
f
the
r
e
f
e
r
e
nc
e
model.
F
igur
e
1.
M
ode
l
-
r
e
f
e
r
e
nc
e
c
ontr
ol
s
ys
tem
a
lgor
it
h
m
T
he
r
e
f
e
r
e
nc
e
model
c
a
n
be
di
f
f
e
r
e
nt
,
li
ne
a
r
,
o
r
no
nli
ne
a
r
,
ti
me
invar
iant
or
ti
me
va
r
iant
,
a
nd
s
o
on.
I
n
thi
s
pa
pe
r
,
it
is
a
s
s
umed
that
the
r
e
f
e
r
e
nc
e
mode
l
is
li
ne
a
r
a
nd
de
s
c
r
ibed
by
(
3)
:
̇
=
+
=
(
3)
w
he
r
e
∈
ℜ
is
the
s
tate
ve
c
tor
o
f
the
model;
∈
ℜ
is
the
c
ontr
ol
ve
c
tor
f
or
the
r
e
f
e
r
e
nc
e
model;
∈
ℜ
×
is
the
c
ons
tant
s
tate
matr
ix;
∈
ℜ
×
is
the
c
ons
tant
c
on
tr
ol
matr
ix
a
nd
∈
ℜ
1
×
is
the
c
ons
tan
t
output
matr
ix
.
I
t
is
a
s
s
umed
that
the
e
igenva
lues
of
ha
ve
ne
ga
ti
ve
r
e
a
l
pa
r
ts
s
o
that
the
model
-
r
e
f
e
r
e
nc
e
s
ys
tem
ha
s
a
n
a
s
ympt
oti
c
a
ll
y
s
table
s
tate
of
e
quil
i
br
ium
.
T
he
c
ontr
ol
input
c
a
n
be
s
e
lec
ted
in
s
uc
h
a
wa
y
that
f
oll
ows
s
ome
de
s
ir
e
d
tr
a
jec
tor
y,
whic
h
then
will
be
f
oll
owe
d
by
the
nonli
ne
a
r
mag
ne
ti
c
levi
tation
s
ys
tem.
T
he
e
r
r
o
r
ve
c
tor
is
de
f
ined
by
(
4)
:
=
−
(
4)
w
he
r
e
∈
ℜ
,
is
the
a
c
tual
s
tate
of
the
plant
.
T
he
r
e
qui
r
e
ments
towa
r
ds
the
c
los
e
d
-
loop
s
ys
tems
a
r
e
that
the
e
r
r
or
mus
t
be
r
e
duc
e
d
to
z
e
r
o
by
a
s
uit
a
ble
c
ontr
ol
ve
c
tor
.
T
o
include
the
model
e
qua
ti
on
a
nd
the
plant
e
qua
ti
on
in
the
e
r
r
o
r
(
4)
it
is
ne
c
e
s
s
a
r
y
to
di
f
f
e
r
e
nti
a
te
the
e
r
r
o
r
(
4)
a
c
c
or
ding
t
o
the
ti
me:
̇
=
̇
−
̇
=
+
−
(
)
−
(
)
=
−
+
+
−
(
)
−
(
)
=
(
−
)
+
+
−
(
)
−
(
)
∴
=
−
,
then
the
a
bove
e
qua
ti
on
c
a
n
be
s
im
pli
f
ied
a
s
(
5)
.
̇
=
+
+
−
(
)
−
(
)
(
5)
T
he
(
5)
is
a
dif
f
e
r
e
nti
a
l
e
qua
ti
on
f
or
the
e
r
r
o
r
ve
c
t
or
.
T
he
n
a
li
ne
a
r
izing
c
ontr
oll
e
r
c
a
n
be
de
s
igned
s
uc
h
that
a
t
s
tea
dy
s
tate
=
a
nd
̇
=
̇
,
or
=
̇
=
0
.
T
hus
,
the
e
quil
ibr
ium
=
0
will
be
the
or
igi
n
of
the
c
oor
dinate
s
ys
tem.
2.
1
.
2.
L
in
e
ar
izin
g
c
on
t
r
oll
e
r
B
a
s
e
d
on
the
unde
r
s
tanding
of
the
L
ya
punov
dir
e
c
t
method,
the
pos
it
ive
de
f
ini
te
L
ya
punov
f
unc
ti
on
f
or
the
s
ys
tem
is
c
ons
tr
uc
ted
a
nd
i
ts
ti
me
de
r
ivat
ive
̇
is
e
xa
mi
ne
d.
I
f
̇
is
ne
ga
ti
ve
de
f
ini
te
,
that
me
a
ns
that
the
e
ne
r
gy
c
ontaine
d
in
the
s
ys
tem
is
c
onti
nuous
ly
dis
s
ipating.
T
he
s
ys
tem
is
movi
ng
towa
r
ds
t
he
s
table
e
quil
ibr
ium
.
T
he
f
oll
owing
s
ub
-
s
tep
s
pr
e
s
e
nt
the
pr
oc
e
dur
e
s
of
how
the
li
ne
a
r
izing
c
ontr
oll
e
r
de
s
ign
is
ba
s
e
d
on
the
L
ya
punov
dir
e
c
t
method
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
ndone
s
ian
J
E
lec
E
ng
&
C
omp
S
c
i
I
S
S
N:
2502
-
4752
De
v
e
lopme
nt
of
a
ne
w
l
inear
iz
ing
c
ontr
oll
e
r
us
ing
L
y
apunov
s
tabi
li
ty
…
(
Y
ohan
Dar
c
y
M
foumboulo
u)
1331
S
tep
1:
C
ons
tr
uc
ti
on
of
a
L
ya
punov
f
unc
ti
on
f
or
t
he
s
ys
tem
a
nd
de
ter
mi
na
ti
on
of
it
s
f
ir
s
t
de
r
ivative
.
An
idea
l
point
to
s
tar
t
the
de
s
ign
of
the
c
ontr
ol
ve
c
tor
is
to
c
ons
tr
uc
t
a
L
ya
punov
f
unc
ti
on
s
ys
tem.
I
n
thi
s
pa
pe
r
,
the
L
ya
punov
f
unc
t
ion
is
a
s
s
umed
to
be
in
q
ua
dr
a
ti
c
f
or
m
(
6)
.
(
)
=
(
6)
w
he
r
e
∈
ℜ
×
is
a
pos
it
ive
-
de
f
ini
te
He
r
mi
ti
a
n
o
r
r
e
a
l
s
ymm
e
tr
ic
matr
ix
.
B
e
c
a
us
e
the
f
unc
ti
on
(
)
is
in
qua
dr
a
ti
c
f
or
m
a
nd
the
matr
ix
is
pos
it
ive
de
f
ini
te
,
it
is
t
r
ue
that
(
)
is
pos
it
ive
de
f
ini
te.
Dif
f
e
r
e
nti
a
ti
ng
the
pos
it
ive
de
f
ini
te
f
unc
ti
on
(
)
a
lon
g
the
s
ys
tem
tr
a
jec
tor
y
,
it
s
ti
me
de
r
ivative
is
obtaine
d
a
s
:
̇
(
)
=
̇
+
̇
=
[
+
+
−
(
)
−
(
)
]
+
[
+
+
−
(
)
−
(
)
]
=
[
+
+
−
(
)
−
(
)
]
+
[
+
+
−
(
)
−
(
)
]
=
+
+
−
(
)
−
(
)
+
+
+
−
(
)
−
(
)
̇
(
)
=
[
+
]
+
2
(
7)
w
he
r
e
:
2
=
+
−
(
)
−
(
)
−
(
)
−
(
)
+
+
=
+
−
(
)
−
(
)
−
(
)
−
(
)
+
+
2
=
2
[
−
(
)
−
(
)
+
]
(
8)
s
ince
is
a
s
ymm
e
tr
ica
l
matr
ix
a
nd
=
:
=
[
−
(
)
−
(
)
+
]
(
9)
is
a
s
c
a
lar
qua
nti
ty
.
S
tep
2:
C
a
lcula
ti
on
of
the
li
ne
a
r
izing
c
ontr
ol
a
lgor
it
hm
.
(
)
is
a
s
s
umed
to
be
a
L
ya
punov
f
unc
ti
on,
if
it
s
f
ir
s
t
de
r
ivative
is
ne
ga
ti
ve
de
f
ini
te
then
the
s
ys
tem
e
xpr
e
s
s
e
d
in
(
7)
is
s
table
.
T
he
f
ir
s
t
de
r
i
va
ti
ve
of
(
)
is
the
s
um
o
f
two
e
xpr
e
s
s
ions
:
̇
(
)
=
[
+
]
+
2
(
10)
f
or
̇
(
)
to
be
ne
ga
ti
ve
de
f
ini
te
,
the
two
ter
ms
o
f
(
10)
mus
t
be
ne
ga
ti
ve
de
f
ini
te
:
[
+
]
<
0
+
=
−
w
he
r
e
is
a
pos
it
ive
de
f
ini
te
matr
ix
a
nd
̇
(
)
=
[
+
]
+
2
(
f
ir
s
t
c
on
dit
ion)
.
≤
0
(
S
e
c
ond
c
ondit
ion
)
.
B
a
s
e
d
on
(
9)
a
nd
(
10
)
,
it
c
a
n
be
c
onc
luded
that
c
a
n
be
made
ne
ga
ti
ve
or
e
qua
l
to
z
e
r
o
thr
ough
s
uit
a
ble
s
e
lec
ti
on
of
the
plant
c
ont
r
ol
ve
c
to
r
whic
h
is
pa
r
t
of
the
f
ir
s
t
de
r
ivative
o
f
the
L
ya
punov
f
u
nc
ti
on
̇
(
)
.
T
he
n
f
r
om
not
ing
that
(
)
→
∞
a
s
‖
‖
→
∞
,
the
e
quil
ib
r
ium
s
tate
=
0
is
a
s
ympt
oti
c
a
ll
y
s
table
in
the
lar
ge
r
r
a
nge
.
T
he
f
ul
f
il
ment
of
c
ondit
ion
(
1)
c
a
n
be
a
c
hieve
d
by
a
n
idea
l
c
hoice
of
the
matr
ix
s
i
nc
e
the
e
igenva
lues
of
the
s
tate
matr
ix
a
r
e
s
e
lec
ted
to
be
with
ne
ga
ti
ve
r
e
a
l
pa
r
ts
.
T
he
pr
oblem
to
s
olve
now
is
to
s
e
lec
t
a
n
a
ppr
opr
iate
ve
c
tor
s
o
that
is
e
it
he
r
z
e
r
o,
or
ne
ga
ti
ve
s
c
a
lar
qua
nti
ty.
T
he
de
ter
mi
na
ti
on
of
the
li
ne
a
r
izing
c
ontr
oll
e
r
c
a
n
be
done
wi
th
p
r
ope
r
s
e
lec
ted
va
lues
o
f
the
matr
ix
o
r
the
mat
r
ix
.
T
he
obtaine
d
li
ne
a
r
izing
c
ontr
oll
e
r
make
s
the
s
ys
tem
s
table
a
nd
f
oll
ows
the
de
s
ir
e
d
tr
a
jec
tor
y
de
ter
m
ined
by
the
r
e
f
e
r
e
nc
e
model
.
2
.
2
.
D
e
s
i
g
n
o
f
a
l
ya
p
u
n
ov
-
b
as
e
d
an
d
M
RC
-
b
as
e
d
l
i
n
e
a
r
i
z
i
n
g
c
o
n
t
r
ol
ler
f
or
t
h
e
m
a
gn
e
t
i
c
le
v
it
a
t
i
on
s
ys
t
e
m
B
a
s
e
d
on
the
s
tudy
a
nd
unde
r
s
tanding
o
f
the
M
R
C
theor
y,
the
L
ya
punov
s
tabili
ty
theor
y,
the
L
ya
punov
dir
e
c
t
method
a
nd
the
L
QR
c
ontr
ol
met
hod,
the
f
ol
lowing
s
ub
-
s
e
c
ti
ons
c
ove
r
e
xpli
c
it
ly
th
e
de
s
ign
of
the
li
ne
a
r
izing
c
ontr
oll
e
r
to
s
tabili
z
e
the
nonli
ne
a
r
magne
ti
c
levitation
s
ys
tem.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2502
-
4752
I
ndone
s
ian
J
E
lec
E
ng
&
C
omp
S
c
i
,
Vol.
25
,
No.
3
,
M
a
r
c
h
20
22
:
1328
-
1343
1332
2.
2.
1.
T
h
e
n
o
n
li
n
e
ar
m
o
d
e
l
of
t
h
e
m
agn
e
t
ic
levit
at
ion
s
ys
t
e
m
F
igur
e
2
s
hows
the
s
c
he
matic
diagr
a
m
o
f
the
ma
gne
ti
c
levitation
s
ys
tem
de
ve
loped
by
[
16]
.
T
he
magne
ti
c
levitation
s
ys
tem
is
a
n
e
lec
tr
o
-
mec
ha
nica
l
s
ys
tem
made
of
the
f
oll
owing
c
ompon
e
nts
:
a
n
e
lec
tr
omagne
t
,
a
c
ur
r
e
nt
c
ontr
oll
e
r
,
a
s
e
ns
or
,
phot
o
-
e
mi
tt
e
r
s
,
a
photo
-
r
e
c
e
iver
,
a
nd
a
s
tee
l
ba
ll
.
T
he
goa
l
o
f
the
s
ys
tem
is
to
c
ontr
ol
the
pos
it
ion
o
f
the
s
tee
l
b
a
ll
by
r
e
gulating
the
c
ur
r
e
nt
in
the
e
lec
tr
omagne
t
thr
ough
the
input
volt
a
ge
.
T
he
dyna
mi
c
o
f
the
s
ys
tem
is
de
r
ived
ba
s
e
d
on
the
f
ir
s
t
p
r
inciples
of
ba
s
ic
e
lec
tr
ica
l
a
nd
mec
ha
nica
l
law
s
.
F
igur
e
2.
M
a
gne
ti
c
levitation
s
ys
tem
[
16
]
T
he
nonli
ne
a
r
r
e
duc
e
d
or
de
r
o
f
the
magne
ti
c
levi
t
a
ti
on
s
ys
tem
de
s
c
r
ibed
in
F
igur
e
2
is
s
e
lec
ted
f
or
the
inves
ti
ga
ti
on
[
29]
.
T
his
r
e
duc
e
d
o
r
de
r
model
is
de
f
ined
a
s
:
[
̇
1
̇
2
]
=
[
2
−
−
2
]
+
[
0
1
(
1
+
)
4
]
,
(
0
)
=
0
(
11
)
=
(
12)
t
he
s
tate
s
of
the
nonli
ne
a
r
magne
ti
c
levitation
s
ys
tem
a
r
e
de
f
ined
a
s
:
1
=
2
=
ℎ
w
he
r
e
:
=
[
1
0
]
;
,
a
nd
a
r
e
c
ons
tants
r
e
late
d
with
the
magne
ti
c
c
oil
pr
ope
r
ti
e
s
.
T
he
va
lues
of
th
e
pa
r
a
mete
r
s
of
the
p
r
oc
e
s
s
a
r
e
:
=
9
.
81
/
=
0
.
12
(
13)
=
0
.
95
=
6
.
28
=
0
.
15
/
(
14)
t
he
nonli
ne
a
r
model
r
e
pr
e
s
e
nted
in
(
11)
a
nd
(
12
)
c
a
n
be
r
e
wr
it
ten
in
the
c
omm
on
f
or
m
a
s
(
15)
.
̇
=
(
)
+
(
)
=
(
15)
F
igur
e
3
s
hows
the
be
ha
viour
of
the
nonli
ne
a
r
r
e
duc
e
d
or
de
r
model
of
the
magne
ti
c
levitation
s
ys
tem.
T
he
s
im
ulation
is
done
with
the
f
oll
owing
pa
r
a
mete
r
s
:
−
I
nit
ial
c
ondit
ions
:
[
0.
05m
0]
’
−
S
tep
input
:
0
.
3[
volt
s
]
T
he
pos
it
ion
s
tate
r
e
s
pons
e
of
the
nonli
ne
a
r
mo
de
l
of
the
magne
ti
c
levitation
s
ys
tem
s
hows
that
unde
r
s
tep
c
onti
nuous
f
o
r
c
e
,
the
ba
ll
pos
it
ion
mov
e
s
towa
r
d
inf
ini
ty
a
s
the
ti
me
goe
s
.
T
his
a
na
lys
is
c
onf
ir
ms
tha
t
the
magne
ti
c
ba
ll
levitation
is
a
nonli
ne
a
r
ope
n
loop
uns
table
s
ys
tem
that
ne
e
ds
to
be
c
ontr
oll
e
d
e
f
f
icie
ntl
y.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
ndone
s
ian
J
E
lec
E
ng
&
C
omp
S
c
i
I
S
S
N:
2502
-
4752
De
v
e
lopme
nt
of
a
ne
w
l
inear
iz
ing
c
ontr
oll
e
r
us
ing
L
y
apunov
s
tabi
li
ty
…
(
Y
ohan
Dar
c
y
M
foumboulo
u)
1333
F
igur
e
3.
Ope
n
loop
r
e
s
pons
e
of
the
nonli
ne
a
r
mod
e
l
of
magne
ti
c
levitation
s
ys
tem
whe
n
the
s
tep
inp
ut
is
a
t
0.
3
volt
s
2.
2.
2.
M
od
e
l
o
f
t
h
e
d
e
s
ire
d
li
n
e
ar
s
ys
t
e
m
(
r
e
f
e
r
e
n
c
e
m
od
e
l)
T
he
li
ne
a
r
r
e
f
e
r
e
nc
e
model
c
a
n
be
wr
it
ten
in
the
f
o
ll
owing
f
or
m:
̇
=
+
,
(
0
)
=
0
(
16)
=
(
17)
w
he
r
e
∈
ℜ
2
is
the
de
s
ir
e
d
s
tate
s
pa
c
e
ve
c
tor
,
∈
ℜ
1
is
the
c
ontr
ol
ve
c
tor
f
o
r
the
r
e
f
e
r
e
nc
e
model
,
∈
ℜ
2
×
2
a
nd
∈
ℜ
2
×
1
a
r
e
the
s
tate
a
nd
c
ontr
ol
matr
ice
s
of
the
r
e
f
e
r
e
nc
e
model
in
the
s
tate
-
s
pa
c
e
f
or
m,
0
is
the
ini
ti
a
l
s
tate
.
T
he
model
of
the
magne
ti
c
levi
t
a
ti
on
is
of
s
e
c
ond
or
de
r
.
T
he
r
e
f
o
r
e
,
the
de
s
ir
e
d
model
is
s
e
lec
ted
to
be
of
s
e
c
ond
or
de
r
too
.
T
he
e
igenva
lues
of
the
s
tate
matr
ix
a
r
e
s
e
lec
ted
to
be
wi
th
ne
ga
ti
ve
r
e
a
l
pa
r
ts
to
e
ns
ur
e
s
tabili
ty
of
the
r
e
f
e
r
e
nc
e
model
.
2.
2.
3.
De
t
e
r
m
in
a
t
ion
o
f
t
h
e
e
r
r
or
b
e
t
we
e
n
t
h
e
r
e
f
e
r
e
n
c
e
m
od
e
l
an
d
t
h
e
m
aglev
s
t
at
e
s
T
h
e
e
r
r
o
r
b
e
t
w
e
e
n
t
h
e
r
e
f
e
r
e
n
c
e
m
o
d
e
l
a
n
d
t
h
e
n
o
n
l
i
n
e
a
r
m
o
d
e
l
o
f
t
h
e
m
a
g
n
e
t
i
c
l
e
v
i
t
a
t
i
o
n
s
y
s
t
e
m
i
s
(
1
8
)
.
=
−
,
∈
ℜ
2
×
2
(
18)
T
he
e
r
r
or
s
ignal
mus
t
be
r
e
duc
e
d
to
z
e
r
o
by
a
s
uit
a
ble
c
ontr
ol
ve
c
tor
.
T
he
dif
f
e
r
e
nti
a
l
e
qua
ti
on
of
the
e
r
r
or
is
(
9)
.
̇
=
̇
−
̇
=
+
−
(
)
−
(
)
=
+
−
+
−
(
)
−
(
)
=
[
−
]
+
+
−
(
)
−
(
)
=
+
+
−
(
)
−
(
)
(
19)
T
he
pr
oblem
is
to
de
s
ign
a
c
ontr
o
l
ve
c
tor
,
s
uc
h
t
ha
t
a
t
the
e
quil
ibr
ium
s
tate
=
,
̇
=
̇
,
=
̇
=
0
is
a
c
hieve
d.
−
De
s
ign
of
of
the
li
ne
a
r
izing
c
ontr
oll
e
r
S
tep
1:
C
ons
tr
uc
ti
on
of
L
ya
punov
f
unc
ti
on
T
he
c
ons
tr
uc
ti
on
of
the
L
ya
punov
f
unc
t
ion
f
o
r
the
e
r
r
or
di
f
f
e
r
e
nti
a
l
s
hown
in
(
18)
is
:
(
)
=
(
20)
W
he
r
e
is
a
s
ymm
e
tr
ica
l
pos
it
ive
de
f
ini
te
mat
r
ix,
∈
ℜ
2
×
2
.
S
tep
2:
C
a
lcula
ti
on
o
f
the
f
i
r
s
t
de
r
ivative
o
f
the
L
ya
punov
f
unc
ti
on
T
he
c
a
lcula
ti
on
of
the
f
ir
s
t
de
r
ivative
of
the
L
ya
punov
fu
nc
ti
on
is
the
f
ol
lowing:
̇
(
)
=
̇
+
̇
=
[
+
+
−
(
)
−
(
)
]
+
[
+
+
−
(
)
−
(
)
]
0
1
2
3
4
5
6
7
8
9
10
-
1
0
0
10
20
30
40
50
60
70
80
T
i
m
e
P
o
s
i
t
i
o
n
[
m
]
b
a
l
l
p
o
s
i
t
i
o
n
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2502
-
4752
I
ndone
s
ian
J
E
lec
E
ng
&
C
omp
S
c
i
,
Vol.
25
,
No.
3
,
M
a
r
c
h
20
22
:
1328
-
1343
1334
=
[
+
+
−
(
)
−
(
)
]
+
[
+
+
−
(
)
−
(
)
]
=
+
+
−
(
)
−
(
)
+
+
+
−
(
)
−
(
)
=
[
+
]
+
2
[
+
−
(
)
−
(
)
]
=
−
+
2
=
+
w
he
r
e
the
matr
ix
i
s
s
ymm
e
tr
ica
l
a
nd
pos
it
ive
de
f
i
nit
e
.
=
[
+
−
(
)
−
(
)
]
(
21
)
T
he
de
r
ived
(
21)
is
the
e
xpr
e
s
s
ion
o
f
the
f
ir
s
t
de
r
ivative
o
f
the
L
ya
punov
f
unc
ti
on.
T
o
make
the
e
r
r
or
in
the
c
los
e
d
loop
s
ys
tem
to
go
t
o
z
e
r
o
a
s
ti
me
goe
s
to
inf
in
it
y
(
→
∞
)
,
it
is
f
unda
menta
l
f
or
thi
s
e
qua
ti
on
to
be
ne
ga
ti
ve
de
f
ini
te.
T
he
f
ir
s
t
e
xpr
e
s
s
ion
of
th
is
e
qua
ti
on
is
ne
ga
ti
ve
de
f
ini
te
a
s
is
s
e
le
c
ted
to
be
pos
it
ive
de
f
ini
te
.
T
he
n
the
s
e
c
ond
e
xpr
e
s
s
ion
c
a
n
be
made
z
e
r
o
or
ne
ga
ti
ve
≤
0
by
a
c
onve
nient
s
e
lec
ti
on
of
the
c
ontr
ol
.
S
tep
3:
C
a
lcula
ti
on
o
f
the
li
ne
a
r
izing
c
ont
r
oll
e
r
T
he
c
a
lcula
ti
on
o
f
the
li
ne
a
r
izing
c
ontr
ol
ler
is
done
by
s
ome
tr
a
ns
f
or
mations
o
f
the
e
xpr
e
s
s
ion
f
or
:
=
[
+
−
(
)
−
(
)
]
≤
0
=
[
+
−
(
)
]
−
[
(
)
]
≤
0
[
+
−
(
)
]
≤
[
(
)
]
(
22)
T
he
e
xpr
e
s
s
ions
f
r
om
both
s
ides
of
the
e
qua
ti
on
a
r
e
s
c
a
lar
s
,
whic
h
de
pe
nd
on
ti
me.
T
ha
t
is
the
r
e
a
s
on
why
it
is
pos
s
ibl
e
to
d
ivi
de
both
s
ides
by
[
(
)
]
a
nd
obtain:
≥
[
+
−
(
)
]
[
(
)
]
(
23)
S
tep
4:
R
e
pr
e
s
e
ntation
of
the
diag
r
a
m
o
f
the
c
los
e
d
-
loop
s
ys
tem
B
a
s
e
d
on
(
23)
,
a
diagr
a
m
of
the
c
los
e
d
loop
s
ys
tem
c
a
n
be
dr
a
wn.
T
he
e
xpr
e
s
s
ion
of
the
li
ne
a
r
izing
c
ontr
oll
e
r
is
mul
ti
pli
e
d
by
a
ne
w
pr
opo
r
ti
ona
l
ga
i
n
>
to
make
the
r
e
a
li
z
a
ti
on
in
the
(
23
)
s
tr
onge
r
.
T
he
nonli
ne
a
r
c
on
tr
oll
e
r
de
ve
loped
make
s
the
f
i
r
s
t
de
r
ivative
of
the
L
ya
punov
f
unc
ti
on
ne
ga
ti
ve
.
T
he
n,
it
li
ne
a
r
ize
s
the
c
los
e
d
loop
s
ys
tem
c
ons
is
ti
ng
of
the
li
ne
a
r
izing
c
ontr
oll
e
r
a
nd
the
magne
ti
c
levitation
s
ys
tem.
T
his
c
ombi
na
ti
on
make
s
the
be
ha
viour
o
f
the
c
los
e
d
-
loop
s
ys
t
e
m
f
oll
ow
the
be
ha
viour
o
f
the
r
e
f
e
r
e
nc
e
model.
T
he
block
diagr
a
m
of
the
c
los
e
d
loop
s
ys
tem
is
s
hown
in
F
igur
e
4
.
F
igur
e
4.
B
lock
diagr
a
m
of
the
c
los
e
d
loop
s
ys
tem
Evaluation Warning : The document was created with Spire.PDF for Python.
I
ndone
s
ian
J
E
lec
E
ng
&
C
omp
S
c
i
I
S
S
N:
2502
-
4752
De
v
e
lopme
nt
of
a
ne
w
l
inear
iz
ing
c
ontr
oll
e
r
us
ing
L
y
apunov
s
tabi
li
ty
…
(
Y
ohan
Dar
c
y
M
foumboulo
u)
1335
2.
2.
4.
De
s
ign
of
a
l
in
e
ar
c
on
t
r
ol
f
or
t
h
e
li
n
e
ar
ize
d
c
los
e
d
-
loop
s
ys
t
e
m
F
igur
e
4
s
hows
that
the
de
s
ir
e
d
ve
c
tor
de
pe
nds
on
the
input
c
ontr
ol
ve
c
tor
f
o
r
the
r
e
f
e
r
e
nc
e
model.
Dif
f
e
r
e
nt
va
lues
of
will
give
di
f
f
e
r
e
nt
v
a
lues
of
.
F
r
om
the
e
xpr
e
s
s
ion
of
the
nonli
ne
a
r
c
ontr
ol
given
by
(
23
)
,
the
va
lues
of
de
pe
nd
on
the
pa
r
a
mete
r
s
of
the
nonli
ne
a
r
magne
ti
c
levitation
mod
e
l.
T
he
im
pleme
ntation
of
the
li
ne
a
r
izing
c
ontr
oll
e
r
then
c
a
nnot
be
ve
r
y
s
uc
c
e
s
s
f
ul
be
c
a
us
e
of
the
in
f
luenc
e
o
f
the
dis
tur
ba
nc
e
s
,
a
nd
the
c
ha
nge
s
of
the
plant
pa
r
a
mete
r
s
.
T
he
li
ne
a
r
izing
a
nd
s
tabili
z
ing
e
f
f
e
c
ts
c
ould
be
los
t
a
nd
c
ould
make
the
s
ys
tem
un
s
table
.
T
his
mea
n
s
that
a
n
a
ddit
ional
innovative
I
ntegr
a
l
L
inea
r
Qua
dr
a
ti
c
R
e
gulator
(
I
L
QR
)
c
ontr
ol
ler
mus
t
be
de
s
igned
to
make
the
c
los
e
d
loop
s
ys
tem
mor
e
r
obus
t
a
nd
it
s
output
e
xa
c
tl
y
to
f
oll
ow
the
de
s
ir
e
d
be
ha
viour
of
the
r
e
f
e
r
e
nc
e
model.
S
tep
1:
S
pe
c
if
ica
ti
on
of
the
c
los
e
d
-
loop
s
ys
tem
with
the
r
e
f
e
r
e
nc
e
model
a
nd
f
or
the
l
inea
r
ize
d
c
lo
s
e
d
-
loop
s
ys
tem
L
e
t
a
s
s
ume
that
the
de
s
ir
e
d
output
of
the
e
nti
r
e
c
lo
s
e
d
loop
s
ys
tem
is
a
s
e
t
po
int
va
lue
.
B
e
c
a
us
e
of
thi
s
a
s
s
umpt
ion,
it
is
c
r
uc
ial
to
de
ter
mi
ne
the
opti
mal
c
ontr
ol
law
s
uc
h
that:
=
or
=
−
whe
n
→
∞
.
S
tep
2:
De
s
ign
o
f
the
li
ne
a
r
qua
dr
a
ti
c
c
ontr
oll
e
r
T
he
e
xpr
e
s
s
ion
of
the
e
r
r
o
r
s
ignal
be
twe
e
n
the
s
e
t
-
point
a
nd
the
ou
tput
o
f
the
p
lant
is
de
f
ined
a
s
:
̇
=
+
,
0
=
−
0
t
he
a
im
is
to
de
s
ign
a
li
ne
a
r
int
e
gr
a
l
qua
dr
a
ti
c
c
ontr
oll
e
r
to
make
the
e
r
r
or
be
twe
e
n
the
s
e
t
-
point
a
nd
the
c
ur
r
e
nt
va
lue
of
the
s
ys
tem
output
to
go
to
z
e
r
o
.
T
he
n
the
e
xtende
d
ve
r
s
ion
of
the
model
is
buil
t
a
s
(
2
4)
.
̇
=
+
̇
+
1
=
−
,
(
0
)
=
0
(
24)
=
I
t
is
ne
c
e
s
s
a
r
y
to
de
s
ign
the
li
ne
a
r
int
e
gr
a
l
qua
dr
a
ti
c
c
ontr
oll
e
r
in
the
s
tate
-
s
pa
c
e
f
or
m:
=
̅
∅
̅
=
+
+
1
+
1
,
̅
∅
∈
ℜ
×
(
+
1
)
(
25)
w
he
r
e
:
̇
̅
=
̅
̅
+
̅
+
[
0
2
×
1
1
]
̅
=
[
0
−
0
]
,
̅
=
[
0
]
,
̅
=
[
1
]
(
26)
T
he
f
unda
menta
l
idea
of
(
24)
a
nd
(
25
)
is
that
the
s
e
r
vo
pr
ob
lem
is
c
onve
r
ted
to
a
pr
oblem
f
or
de
s
ign
of
a
li
ne
a
r
qua
dr
a
ti
c
r
e
gulator
in
whic
h
the
s
e
t
-
poi
nt
is
z
e
r
o
.
T
he
p
r
oblem
to
f
ind
the
mat
r
ix
c
ont
r
oll
e
r
̅
c
a
n
be
(
27)
:
=
∫
[
‖
̅
‖
2
̅
+
‖
‖
2
̅
]
∞
0
,
̅
∈
ℜ
(
+
1
)
×
(
+
1
)
,
̅
∈
ℜ
×
(
27)
w
he
r
e
:
̅
>
a
nd
̅
≥
a
r
e
we
ight
ing
matr
ice
s
[
30
]
,
[
31]
.
E
qu
a
ti
on
(
2
7)
is
mi
nim
ize
d
unde
r
the
model
(
24
)
S
tep
3:
S
olut
ion
o
f
the
li
ne
a
r
qua
d
r
a
ti
c
r
e
gulator
p
r
oblem
T
he
r
e
s
olut
ion
o
f
li
ne
a
r
qua
dr
a
ti
c
r
e
gulator
p
r
oble
m
is
given
by
(
28)
.
=
−
̅
∅
̅
=
−
̅
+
1
=
−
̅
−
1
̅
̅
̅
(
28)
I
n
(
28
)
,
̅
is
the
s
olut
ion
of
the
R
icc
a
ti
as
:
̅
∅
=
[
1
]
∈
ℜ
1
×
2
t
he
s
olut
ion
of
the
pr
ob
lem
c
a
n
be
f
ound
in
M
AT
L
AB
us
ing
the
‘
lqr
’
f
unc
ti
on,
it
s
s
tr
uc
tur
e
is
a
s
f
oll
o
w:
[
̅
∅
,
̅
,
]
=
(
̅
,
̅
,
̅
,
̅
)
̅
∅
is
the
mat
r
ix
o
f
the
r
e
gulator
;
̅
is
the
matr
ix
of
th
e
R
icc
a
ti
e
qua
ti
on
a
nd
is
the
ve
c
tor
o
f
the
poles
of
the
c
los
e
d
-
loop
matr
ix
[
̅
−
̅
̅
∅
]
.
T
o
make
the
s
ys
tem
s
table
,
a
ll
the
poles
mus
t
be
with
r
e
a
l
ne
ga
ti
ve
pa
r
ts
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2502
-
4752
I
ndone
s
ian
J
E
lec
E
ng
&
C
omp
S
c
i
,
Vol.
25
,
No.
3
,
M
a
r
c
h
20
22
:
1328
-
1343
1336
T
he
c
ontr
ol
is
obtaine
d
a
s
(
29
)
.
=
−
+
1
(
29)
T
he
a
ugmente
d
matr
ice
s
with
the
a
ddit
ional
int
e
gr
a
tor
s
tate
s
c
a
n
be
e
xpr
e
s
s
e
d
a
s
:
̅
=
[
0
−
0
]
=
[
0
1
0
−
2
−
3
0
−
1
0
0
]
;
̅
=
[
0
]
=
[
0
1
0
]
w
he
r
e
:
̅
=
[
∈
ℜ
2
×
2
0
∈
ℜ
2
×
1
∈
ℜ
1
×
2
0
∈
ℜ
1
×
1
]
;
̅
=
[
∈
ℜ
2
×
1
0
∉
ℜ
1
×
1
]
T
he
va
lues
of
the
we
ight
ing
mat
r
ice
s
̅
a
nd
̅
a
r
e
s
umm
a
r
ize
d
in
T
a
ble
1.
T
a
ble
1
a
ls
o
s
hows
the
dif
f
e
r
e
nt
va
lues
o
f
̅
a
t
d
if
f
e
r
e
nt
s
e
t
point
s
.
S
tep
4:
Applica
ti
on
of
the
li
ne
a
r
int
e
g
r
a
l
c
ontr
oll
e
r
to
the
c
los
e
d
-
loop
s
ys
tem
with
the
li
ne
a
r
izing
c
ontr
oll
e
r
a
nd
the
r
e
f
e
r
e
nc
e
model
T
he
s
tr
uc
tur
e
of
the
block
diagr
a
m
wi
th
the
li
ne
a
r
izing
M
R
C
ba
s
e
d
on
L
ya
punov
s
e
c
ond
method
is
s
hown
in
F
igur
e
5
.
F
or
the
im
pleme
ntation
o
f
the
li
ne
a
r
c
ontr
ol
ler
in
the
c
los
e
d
-
loop
s
ys
tem,
it
is
i
mpor
tant
that
the
f
e
e
dba
c
k
is
not
take
n
f
r
om
the
output
of
t
he
r
e
f
e
r
e
nc
e
model
but
f
r
om
the
output
of
the
nonli
ne
a
r
model
of
the
magne
ti
c
levitation
pr
oc
e
s
s
.
Us
ing
the
pr
oc
e
s
s
r
e
a
l
output
will
lea
d
to
be
tt
e
r
r
e
s
ul
ts
a
s
the
int
e
gr
a
l
L
QR
c
ontr
oll
e
r
c
ompens
a
tes
f
or
dis
tur
ba
n
c
e
s
ove
r
the
r
e
a
l
p
r
oc
e
s
s
.
T
a
ble
1.
P
a
r
a
me
ter
s
obtaine
d
f
or
the
I
L
QR
S
e
t
poi
nt
s
I
ni
ti
a
l
c
ondi
ti
ons
M
a
tr
ix
̅
M
a
tr
ix
̅
F
e
e
dba
c
k c
ont
r
ol
le
r
ga
in
̅
0.01m
[
0.05 0 0]
’
[
91000
0
0
0
300
0
0
0
10
]
0.1
[
952
.
67
67
.
1
−
10
]
0.09m
[
0.05 0 0]
’
[
69000
0
0
0
750
0
0
0
25
]
1
[
261
.
37
31
.
8
−
5
]
F
igur
e
5.
B
lock
diagr
a
m
of
the
L
ya
punov
s
tabili
ty
ba
s
e
d
on
model
r
e
f
e
r
e
nc
e
c
ont
r
ol
s
ys
tem
3.
S
I
M
UL
AT
I
ON
RE
S
UL
T
S
AN
D
AN
AL
YS
I
S
T
he
s
im
ulation
is
done
in
M
AT
L
AB
/S
im
uli
nk
e
nvir
onment.
T
he
c
los
e
d
-
loop
diagr
a
m
ba
s
e
d
on
L
ya
punov
dir
e
c
t
method
is
s
hown
in
F
igu
r
e
6
.
T
his
c
los
e
d
loop
diagr
a
m
c
omp
r
is
e
s
f
our
im
po
r
t
a
nt
s
ub
-
s
ys
tems
:
−
R
e
f
e
r
e
nc
e
model
(
F
igur
e
7)
−
I
ntegr
a
l
L
QR
c
ontr
oll
e
r
(
F
igu
r
e
8)
−
M
a
gne
ti
c
levitation
nonli
ne
a
r
model
(
F
igu
r
e
9)
−
L
inea
r
izing
c
ontr
oll
e
r
(
F
igur
e
10)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
ndone
s
ian
J
E
lec
E
ng
&
C
omp
S
c
i
I
S
S
N:
2502
-
4752
De
v
e
lopme
nt
of
a
ne
w
l
inear
iz
ing
c
ontr
oll
e
r
us
ing
L
y
apunov
s
tabi
li
ty
…
(
Y
ohan
Dar
c
y
M
foumboulo
u)
1337
F
igur
e
6.
S
im
ul
ink
diagr
a
m
of
the
L
ya
punov
dir
e
c
t
method
ba
s
e
d
on
M
R
C
F
igur
e
7.
S
im
ul
ink
block
d
iagr
a
m
o
f
the
li
ne
a
r
r
e
f
e
r
e
nc
e
model
a
nd
it
s
c
ont
r
oll
e
r
F
igur
e
8.
S
im
ul
ink
block
d
iagr
a
m
o
f
the
I
L
QR
c
ontr
oll
e
r
Evaluation Warning : The document was created with Spire.PDF for Python.