TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.5, May 2014, pp
. 3690 ~ 36
9
6
DOI: http://dx.doi.org/10.11591/telkomni
ka.v12i5.4255
3690
Re
cei
v
ed Au
gust 28, 20
13
; Revi
sed
De
cem
ber 2
4
, 2013; Accepte
d
Jan
uary 6, 2014
Partial-state Finite-time Stabilization Control of Chaos
in PMSM with Uncertain Parameters
Chua
nshe
ng
Tang
1
*, Yuehong Dai
2
, Hongbing Yan
g
3
Schoo
l of Mechatron
i
cs Engi
neer
ing, Un
iver
sit
y
of
Electro
n
i
c
Science a
nd
T
e
chnolog
y
of Chin
a,
Che
ngd
u 61
17
31, Chi
n
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: tcs111@1
63.
com
A
b
st
r
a
ct
In this p
aper,
a nov
el
partia
l
-
s
tate finite-ti
m
e c
ontro
l sch
e
m
e
is pr
ese
n
te
d for stabi
li
z
i
n
g
chaos
in
per
ma
nent
ma
gnet sync
h
ro
n
ous
motor w
i
t
h
unc
ertai
n
p
a
r
ameters. F
i
rst, a new
conc
e
p
t of parti
al-st
a
te
finite-ti
m
e
stab
ility is
i
n
troduc
ed. Bas
e
d
on
the casc
ad
e-c
onn
ected
system the
o
ry, a
control
l
er
is th
en
desi
gne
d
in
d
e
tail
to
improv
e the
p
e
rfor
mance
of
th
e sy
stem. Th
e sta
b
ility
of th
e
propos
ed
sche
m
e
i
s
verified acc
o
rding to Ly
apunov stab
ility.
This m
e
thod is demonstrated
to be hi
ghly robust again syst
em
para
m
etric vari
ations. Fina
lly,
numeric
al si
mulati
on resu
lts are pres
ente
d
to illustrate the
effectiveness
of
the prop
ose
d
meth
od.
Ke
y
w
ords
:
per
ma
nent
ma
gnet
sy
nchr
on
ous motor,
fin
i
te-tim
e sta
b
il
ity, cha
o
s c
ontrol,
casca
de-c
onn
ecte
d
system
theory
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
Since the lat
e
1980
s, re
search h
a
s
co
nfirm
ed that
cha
o
s i
s
a re
al phen
omen
on in all
motor d
r
ive
systems,
su
ch
as in
du
ction
motors
,
DC motors,
and switch
ed
relu
ctance motors [1].
Cha
o
tic be
ha
vior in perm
a
nent magn
et DC moto
r op
en drive sy
st
ems
was first
addre
s
sed b
y
Hemati [2]. Li
has foun
d th
at cha
o
s was also
existe
d
in pe
rman
ent
magn
et syn
c
hron
ou
s mot
o
r
(PMSM) [3]. With its ch
an
ging ope
ratin
g
para
m
eters, PMSM exhibits com
p
lex
behavior th
at
redu
ce
s
sy
st
em p
e
rf
o
r
ma
nce,
i
n
cl
udin
g
limit
cy
cl
e
s
and
chao
s
o
scill
ation. Th
us, thi
s
b
eha
vior
sho
u
ld
be
suppresse
d o
r
eve
n
elimi
nated. T
o
a
ddre
s
s thi
s
probl
em, Li
prop
osed
ge
neri
c
mathemati
c
al
model
s and
con
d
u
c
ted in
-depth theo
re
ti
cal an
alysi
s
, whi
c
h a
r
e the
foundation
s
f
o
r
controlling ch
aos.
De
spite th
e
nume
r
ou
s
m
e
thod
s u
s
e
d
to co
ntrol
ch
aos, fe
w can
be
directly a
pplied
to
PMSM. The main method
s inclu
de de
couplin
g cont
rol [4],
feedba
ck
control [5,
6], back-ste
p
p
ing
control [7], passivity control [8], sliding mode
co
ntrol [9], adaptive control [1
0, 11], and fuzzy
control [12, 13]. Howeve
r, deco
upling
control,
feed
back, ba
ck-st
eppin
g
cont
rol, and passi
vity
control, all o
f
which
dep
end on th
e mathemati
c
al
model of th
e system,
cannot gu
ara
n
tee
dynamic pe
rforma
nce be
cause of
u
n
ce
rtain
system
para
m
eters.
PMSM req
u
ires a
pa
ram
e
ter
adaptive me
chani
sm for th
e adaptive control te
chni
que. Thi
s
re
quire
ment in
cre
a
ses the
co
st
and
com
p
lexity of the syst
em an
d
re
du
ce
s its
re
spo
n
se
ca
pa
city. Sliding mo
d
e
co
ntrol
req
u
ire
s
uncertain
terms to
me
et
spe
c
ific mat
c
h conditi
o
n
s and
exhibit
s
in
he
rent
chattering.
Fu
zzy
control is
usu
a
lly base
d
on
Taka
gi–Su
g
eno fu
zzy mo
dels
of the sy
stem. Li
et al.
[13] prop
ose
d
the fuzzy fee
dba
ck contro
l schem
e, which
exhibi
t
s
slo
w
re
spo
n
s
ivene
ss. Li
et al. [12] th
en
prop
osed opti
m
al fuzzy gu
arante
ed cost
control,
which exhibits bet
ter re
spon
siv
ene
ss but ha
s a
stru
cture that is too
co
mple
x for applicati
on.
The above
m
entione
d co
n
t
rol method
s can
only e
n
su
re the a
symptotic exponential
stability of the system and not its time
optimality
(i.e., the shortest adjustm
ent time). Finite-time
control a
c
hie
v
es sy
stem
stability in finite time an
d contain
s
fra
c
ti
onal p
o
wer, g
i
ving this m
e
thod
highe
r
rob
u
st
ness th
an th
e ab
ove met
hod
s [14].
With the a
d
van
t
ages of fa
st re
spo
n
se, hi
gh
tracking p
r
e
c
ision a
nd strong ro
bu
stne
ss fo
r
uncert
a
in paramete
r
s, this meth
od ha
s bee
n
su
ccessfully
applie
d to control
all kin
d
s
of syste
m
s,
su
ch
as roboti
c
ma
nipulato
r
s [1
5],
spa
c
e
c
r
a
f
t
sy
st
em
s [
16]
,
A
C
s
e
rv
o
sy
st
ems [
17]
and
cha
o
tic
syst
ems [18,
19]. Wei et al. [2
0]
applie
s this m
e
thod to control a cha
o
tic p
e
rma
nent ma
gnet syn
c
h
r
o
nou
s motor
(PMSM) syste
m
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Partial-state Finite-tim
e Stab
ilization Co
ntrol of Cha
o
s
in PMSM wi
th… (Chuan
sheng Ta
ng
)
3691
Ho
wever,
the
sp
eed
state
equatio
n h
a
s
only on
e exte
rnal
controll
a
b
le va
riable
(i
.e., load to
rqu
e
),
whi
c
h i
s
gen
erally n
o
t a
r
b
i
trarily
cont
rol
l
able. T
h
u
s
, it is difficult to
impleme
n
t in
pra
c
tice
a
nd i
s
not con
s
ide
r
e
d
an
un
ce
rtai
n pa
ram
e
ter for th
e meth
o
d
s i
n
[8,
20].
To
solve th
ese p
r
oble
m
s,
we
prop
ose a
no
vel partial
-sta
te finite-time
cha
o
tic
co
ntroller (PSFT
C
C)
th
at
ac
counts
for parameter
uncertainties
in PMSMs. With this
controller,
the
system exhibit
s
not only rap
i
d re
spo
n
se but
also robu
stne
ss u
nde
r un
certain pa
ram
e
ters.
This pa
per i
s
o
r
g
ani
zed
as foll
ows.
Se
ction
2 i
n
trodu
ce
s the
basi
c
co
nce
p
ts
and
lemma
s of the ca
scade
-conne
cted
system theory
(CCST
)
and t
he finite-time
stability theory
(FTST). Se
ction 3 prese
n
t
s the cha
o
s model of
PMSM. Section 4 describe
s
in detail th
e
PSFTCC d
e
s
ign an
d verifies the sta
b
ility of
t
he controlle
r according to L
y
apunov sta
b
ility.
Section
5
pre
s
ent
s the
sim
u
lation
re
sult
s to
illust
rate
the effective
ness of
the
method.
Fi
na
lly,
Section 6 con
c
lud
e
s.
2. Basic Con
cepts and Le
mmas
Important con
c
ept
s and le
mmas n
e
cessary for
cont
roller de
sig
n
a
r
e given bel
o
w
.
Con
s
id
er a cascad
e-con
n
e
cted
system
describ
ed by
:
(,
)
()
x
fx
z
zg
z
,
(1)
Whe
r
e
n
x
R
and
m
zR
are
system st
ates,
(,
)
f
xz
and
()
g
z
are
1
C
vectors, an
d
(0
,
0
)
0
f
and
(0)
0
g
.
Defini
tion 1
[21]
.
Con
s
ide
r
the dyn
a
mi
c sy
stem
()
x
fx
, where
n
x
R
is th
e
system
state.
This sy
stem i
s
finite
-time
stable if it
ha
s
a con
s
tant
0
T
(whi
ch
may d
e
pend
on
the
i
n
itial state
)
that meets th
e followin
g
co
ndition
s:
①
li
m
(
)
0
tT
xt
and
②
()
0
xt
, if
tT
.
Defini
tion 2.
Con
s
id
er
syst
em (1), if th
ere exist
s
cont
rol inp
u
t
12
(
,
,
...
,
)
m
uu
u
u
,
mn
, s
u
ch that
partial
state
s
of the sy
stem
(1) are finite-time st
abl
e a
nd othe
r
states a
r
e
glob
ally asymptotically
stable, then
system (1
) is called pa
rtial-state finite-time stable.
Lemma 1
[22]
.
If system
(1) meet
s th
e followi
ng
condition
s,
①
(,
0
)
x
fx
and
()
zg
z
are
globally a
s
ymptotically st
able at
0
x
and
0
z
, respe
c
tively, and
②
all st
ates of sy
ste
m
(1)
are bo
und
ed,
then system
(1) i
s
glob
ally asymptot
icall
y
stable at eq
uilibriu
m
(x, z) = (0, 0
)
.
Lemma 2
[21]
.
If a continuou
s po
sitive definite funct
i
on
()
Vt
s
a
tis
f
ies
the following differential
inequ
ality,
()
()
Vt
m
V
t
for
0
tt
, and
0
()
0
Vt
, where
0
m
and
01
are
consta
nts,
then
()
Vt
sati
sfie
s the ine
qual
ity
①
11
00
()
(
)
(
1
)
(
)
Vt
Vt
m
t
t
for
0
[,
]
m
tt
t
and
②
()
0
Vt
for
1
tt
at any initial time
0
t
.
1
t
is given a
s
1
10
(1
)
V
tt
m
,
whi
c
h is th
e resp
on
se
time of the system.
Lemma 3.
For positive real
numbe
rs a, b
,
and c, if
(0
,1
)
c
, then
()
cc
c
ab
a
b
.
3. Contr
o
ller Design fo
r PMSM Chao
tic Sy
stem
In this sectio
n, it is given the chao
s mo
del
of the PM
SM drive sy
stem and
at the sa
m
e
time the cont
rolle
r is de
sig
ned in detail.
Then,
The
stability of the
prop
ose
d
co
ntrol sch
e
me
is
verified via Lyapun
ov stabl
e theory.
3.1. D
y
manic Model and Chao
tic Ch
a
r
act
eristic
s
of PMSM Ch
aotic Sy
stem
The tran
sformed mod
e
l of PMSM with
the
smooth ai
r gap can be
expre
s
sed a
s
follows
[3]:
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 5, May 2014: 3690 – 36
96
3692
()
dd
q
d
qq
d
q
qL
ii
w
i
v
ii
w
i
w
v
wi
w
T
,
(2)
Whe
r
e
d
v
,
q
v
,
d
i
, and
q
i
a
r
e t
he tra
n
sfo
r
m
ed
stator vo
ltage
comp
o
nents an
d curren
t
comp
one
nts i
n
the
d-q
fra
m
e,
w
an
d
L
T
are the tran
sfo
r
med
an
gle
speed
and
external
loa
d
torque re
spe
c
tively,
and
γ
and
σ
are the motor pa
ram
e
ters.
Con
s
id
erin
g the ca
se that,
after an ope
ration
of the
system, the e
x
ternal input
s are set
to z
e
ro, namely,
0
dq
L
vv
T
, system (2) be
com
e
s a
n
autonom
ou
s syste
m
:
()
dd
q
qq
d
q
ii
w
i
ii
w
i
w
wi
w
,
(3)
The m
ode
rn
nonlin
ear the
o
ry such a
s
b
i
furcatio
n a
n
d
ch
ao
s h
a
s b
een
used to
study the
nonlin
ear
ch
ara
c
teri
stics
of PMSM drive system
in
[3].
It has found that, wi
th the operat
ing
para
m
eters
γ
and
σ
falli
ng
into
a
cert
ain a
r
ea, PMS
M
will
exhibit
co
mplex
dynamic b
ehavi
o
r,
su
ch a
s
p
e
rio
d
ic, qu
asi
periodic
and
cha
o
tic be
haviors. In ord
e
r to
make
an ove
r
all inspe
c
tion
of
dynamic b
e
h
a
vior of the PMSM, the bi
furcation dia
g
ram of the angle spee
d
w
with incre
a
si
ng of
the pa
ramete
r
is illustrate
d in Fig
u
re
1(a). We can
see that the
system sho
w
s abun
dant a
n
d
compl
e
x dyn
a
mical
be
haviors with i
n
creasi
ng p
a
ra
meter
. The
typical chaoti
c
attra
c
to
r i
s
s
h
ow
n
in
F
i
gu
r
e
1(
b
)
w
i
th
0
dq
L
vv
T
,
25
, and
5.
4
6
.
(a)
(b)
Figure 1. Bifurcatio
n Dia
g
ram and the Chara
c
te
ri
zatio
n
s of Ch
ao
s in PMSM (a) Bifurcatio
n
diagram of st
ate variable
w
with the para
m
eter
(b) typical chaoti
c
attracto
r.
With un
certai
n param
eters, the dynamic
model of the system
can be de
scri
bed as
follows
:
()
()
(
)
dd
q
qq
d
q
ii
w
i
ii
w
i
w
wi
w
,
(4)
Whe
r
e
∆
γ
an
d
∆
σ
represe
n
t
the unce
r
tainty of
γ
and
σ
resp
ectively and are both
boun
ded.
0
50
10
0
15
0
200
-10
0
10
20
30
40
γ
w
0
20
40
60
-5
0
0
50
-20
0
20
i
d
i
q
w
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Partial-state Finite-tim
e Stab
ilization Co
ntrol of Cha
o
s
in PMSM wi
th… (Chuan
sheng Ta
ng
)
3693
Followi
ng
a
n
actual ope
rati
on,
this articl
e
as
su
me
s th
at the fluctu
ation rang
e of
system
para
m
eters is 30%, that is,
1
0.3
,
2
0.3
.
3.2. Contr
o
ller Desig
n
System (3) indi
cat
e
s three equilibrium points:
0
(0
,
0
,
0
)
S
and
2,
3
(
1
,1
,1
)
S
. Given that
25
,
0
S
is lo
cally st
able, a
nd
1
S
a
nd
2
S
are
both
locally un
stab
le [3] . Assuming that one
e
quilibri
um poi
nt of system (3) is
(,
,
)
dd
q
d
d
Si
i
w
, then:
0
0
()
0
dd
d
d
d
q
d
qd
q
d
d
d
d
d
dq
d
d
ii
w
i
ii
w
i
w
wi
w
.
(5)
To qui
ckly
stabilize to equilibrium point
(,
,
)
dd
q
d
d
Si
i
w
,
1
u
and
2
u
are u
s
ed to control t
h
e
system (4). Under the
cont
rol efforts
1
u
and
2
u
, the control
l
ed system
ca
n be rep
r
e
s
e
n
ted as:
1
2
()
()
(
)
dd
q
qq
d
q
ii
w
i
u
ii
w
i
w
u
wi
w
.
(6)
Let
1
dd
d
ei
i
,
2
qq
d
ei
i
and
3
d
ew
w
, we can
obtain the
d
y
namic e
r
ror
equatio
ns of the syste
m
:
11
2
3
2
3
1
22
1
3
1
3
3
3
2
32
3
()
()
(
)
dq
d
dd
d
d
ee
e
e
e
w
e
i
u
ee
e
e
e
w
e
i
e
e
w
u
ee
e
.
(7)
The control objective
i
s
to stabili
ze
the
system (6) at t
he equilibrium point
(,
,
)
dd
q
d
d
Si
i
w
,
that is,
we
de
sign
the
co
ntroller to
stabili
ze th
e e
r
ror system (7) at
e
=0. S
o
we
will focus on
the
controlle
r de
signing for
system (7).
System analy
s
is i
s
req
u
ire
d
prio
r to con
t
roller d
e
si
gn.
If
the states
1
e
and
2
e
are clo
s
e to
z
e
ro after time
t
m
, namely,
1
0
e
and
2
0
e
for
t
>
t
m
, then system
(6)
can b
e
given as:
11
2
3
2
3
1
22
1
3
1
3
3
2
33
dq
d
dd
d
eb
e
e
e
e
w
e
i
u
ee
e
e
e
w
e
i
c
e
u
ea
e
(8)
System (8
) i
s
a typical
ca
sc
ade
d-con
n
e
cted
sy
stem
. The th
i
r
d e
quation
of system (8)
indicates that
error
e
3
i
s
gl
obally asym
p
t
otically stabl
e at
3
0
e
. Thus, i
f
the erro
rs
1
e
and
2
e
are
stabili
ze
d
at zero, th
e
system
be
comes glo
bal
l
y
stable.
Nex
t, we d
e
sig
n
the controll
er to
st
abili
ze
1
e
and
2
e
at (0, 0).
Based o
n
CCST and FTST
,
the controlle
r is de
sign
ed
as follo
ws:
Theorem 1.
Con
s
id
er dyn
a
mic e
rro
r sy
stem (8
). If th
e controlle
r is desig
ned a
s
:
13
1
1
23
3
2
2
2
()
qd
dd
ue
i
k
e
ue
i
e
L
w
s
i
g
n
e
k
e
(9)
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TELKOM
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Vol. 12, No. 5, May 2014: 3690 – 36
96
3694
Then, syste
m
(8) exhibits p
a
rtia
l-state fin
i
te-time stabili
ty at
equilibrium point O (0
, 0, 0),
whe
r
e k
1
and
k
2
are th
e coefficient
s of
the termin
al attracto
rs th
at are p
o
sitiv
e
real
numb
e
r
s,
q
p
(whe
re
pq
and both term
s are positive od
d integers), and
1
0.3
L
.
Proof.
Co
ntrolling
1
u
and
2
u
transfo
rm
s the first and
se
co
nd equ
ation
s
of Equation (8) into:
11
2
3
2
1
1
22
1
3
1
2
2
2
()
d
d
ee
e
e
e
w
k
e
ee
e
e
e
w
w
L
w
s
i
g
n
e
k
e
(9)
If the candida
te Lyapunov functio
n
is defi
ned a
s
22
1
11
2
2
()
Ve
e
, then the time deriv
ative
of
1
V
along the traje
c
tory of (9
) is:
11
1
2
2
Ve
s
e
e
s
e
11
2
3
2
1
1
2
2
1
3
1
2
2
2
()
(
(
)
)
dd
ee
e
e
e
w
k
e
e
e
e
e
e
w
w
L
w
s
i
g
n
e
k
e
22
1
1
12
1
1
2
2
2
2
()
ee
k
e
k
e
L
w
e
w
e
11
11
2
2
ke
k
e
0.5
(
1
)
2
0
.5
(
1
)
0
.5
(
1
)
2
0.5
(
1
)
11
11
11
2
2
22
22
()
(
)
()
(
)
ke
k
e
20
.
5
(
1
)
2
0
.
5
(
1
)
11
12
1
22
((
)
(
)
)
me
e
m
V
,
Whe
r
e
0
.
5(
1
)
0
.
5(
1
)
11
12
22
mi
n
(
(
)
,
(
)
)
0
mk
k
an
d
1
2
(1
)
. If
01
, then
01
. Lemma
2 in
dicate
s that t
he state
s
1
e
an
d
2
e
come
c
l
ose to zero wit
h
in finite time
t
m
,
that is
, s
u
bs
ys
tem (9) is
finite-time s
t
able.
After
t
m
,
1
0
e
an
d
2
0
e
. Subs
tituting
1
0
e
and
2
0
e
into the third
e
quation o
f
system (9) yi
elds:
33
()
ee
(10)
Thus,
sub
s
ystem (10
)
is gl
obally asympt
otically stabl
e
.
Based
on the
above e
quati
ons,
con
d
itio
n
①
i
s
met f
o
r Le
mma
1. More
over, th
e PMSM
cha
o
tic
syste
m
is bo
und
e
d
; that i
s
,
co
ndition
②
is
also
met fo
r
Lemma
1. T
h
us,
system
(8
) i
s
globally asym
ptotically stab
le.
1
e
a
nd
2
e
a
r
e f
i
nite-time
sta
b
le, an
d
3
e
is gl
obally asymptotically
stable. Definition 2 indi
cate
s that system
(8) is
p
a
rtial
-
state finite-ti
m
e stable at
O (0, 0, 0).
4. Simulation results
We
use SIM
U
LINK
of MA
TLAB to verif
y
t
he fea
s
ibili
ty of the p
r
o
posed PSFT
C
C for
a
PMSM chaoti
c
syste
m
. In the simul
a
tion
, the f
ourth-o
rder Rung
e–K
utta method is used to sol
v
e
the system
s
with time ste
p
size of 0.0
01. T
he pa
ra
metric valu
es of PMSM are the sam
e
as
those
in Se
ct
ion 3.
Witho
u
t
loss of
gen
erality, we
se
lect
0
(
0
,0
,0
)
S
as the
desi
r
ed
eq
uili
brium
point. The c
o
ntrol method tak
e
s
effec
t
af
ter t=
15
s
.
The sim
u
latio
n
sho
w
s the result of the p
r
opo
se
d met
hod with the
uncertain p
a
rameters
disrega
rd
ed
(Figure 2
)
. T
he
sele
ct
ive
para
m
eters i
n
theo
ry a
r
e
12
50
kk
k
,
7
9
, and
10
L
(
0.3
6
L
). With
parameter un
ce
rtainties a
s
su
med to
be
the same
as those i
n
Section 3, th
e re
sults of t
he propo
se
d
control law
are give
n in Figure 3 (
12
1
kk
k
) a
nd
Figure 4 (
12
10
kk
k
).
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TELKOM
NIKA
ISSN:
2302-4
046
Partial-state Finite-tim
e Stab
ilization Co
ntrol of Cha
o
s
in PMSM wi
th… (Chuan
sheng Ta
ng
)
3695
Figure 2. The
States Re
sp
onse of the
Propo
se
d Scheme with
out
Uncertain
Parameters
Figure 3. The
States Re
sp
onse of the
Propo
se
d Scheme with
Co
ntrol Para
met
e
rs
k=1 Co
nsi
dering Un
ce
rtain
Paramete
rs
Figure 4. The
States Re
sp
onse of the Propo
s
e
d Sc
heme with Control Parameters
k
=
10
Con
s
id
erin
g Un
certai
n Parameters
Figure 4
shows that sy
stem states
can
qui
ckly
stabili
ze to t
he equilibri
um point
0
(
0
,0
,0
)
S
. When u
n
ce
rtain pa
ramet
e
rs
are
con
s
i
dere
d
, increa
sing the
cont
rol pa
ramete
rs ca
n
enha
nce syst
em re
spo
n
se.
5. Conclusio
n
In this paper, a novel partial-state finite-time co
ntro
l sche
m
e is
prop
osed for PMSM
cha
o
tic
syste
m
in the presence of
pa
ra
meter u
n
cert
ainties. Thi
s
method a
ppli
e
s the finite
-time
stability theory to casca
d
e
-
con
n
e
c
ted
system
s to
imp
r
ove their
perf
o
rma
n
ce. Simulation
re
su
lts
verify that the
pro
p
o
s
ed
co
ntrolle
r exhibi
ts qui
ck resp
onsive
n
e
s
s a
nd
stron
g
rob
u
stne
ss. Addi
ng
the control voltage to the state equati
on of t
he system maintai
n
s sta
b
ility.
More
over, the
stru
cture of this controll
er
is
ea
sy to design a
nd impl
ement. Future re
sea
r
ch should inve
sti
gate
the imple
m
en
tation of the
prop
osed
co
n
t
rol sch
e
me
usin
g an
exp
e
rime
ntal
set
up. The
sch
e
m
e
can al
so b
e
e
x
tended to sy
nch
r
oni
ze PM
SM c
haoti
c
systems
with u
n
ce
rtain pa
ra
meters.
Ackn
o
w
l
e
dg
ements
This wo
rk wa
s
supp
orted by
the
Nation
al
Scien
c
e
an
d Technol
ogy
Major P
r
oje
c
t of the
Ministry of Science and Te
chn
o
logy
of China (Proje
ct No. 200
9ZX0
4001
).
0
5
10
15
20
25
30
-5
0
0
50
t
id
0
5
10
15
20
25
30
-2
0
0
20
t
iq
0
5
10
15
20
25
30
-2
0
0
20
t
w
0
5
10
15
20
25
30
-5
0
0
50
t
i
d
0
5
10
15
20
25
30
-5
0
0
50
t
iq
0
5
10
15
20
25
30
-2
0
0
20
t
w
0
5
10
15
20
25
30
-50
0
50
t
id
0
5
10
15
20
25
30
-50
0
50
t
iq
0
5
10
15
20
25
30
-20
0
20
t
w
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02-4
046
TELKOM
NI
KA
Vol. 12, No. 5, May 2014: 3690 – 36
96
3696
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