TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.4, April 201
4, pp. 2667 ~ 2
6
7
6
DOI: http://dx.doi.org/10.11591/telkomni
ka.v12i4.4096
2667
Re
cei
v
ed Au
gust 7, 201
3; Re
vised O
c
to
ber 17, 20
13;
Accept
ed No
vem
ber 5, 20
13
CLF Based Stabilization of Chaos in PMSM with
Uncertain Parameters
Chua
nshe
ng
Tang*, Yueh
ong Dai, Hua
Sun
Schoo
l of Mechatron
i
cs Engi
neer
ing, Un
iver
sit
y
of El
ectro
n
i
c
Science a
nd
T
e
chnolog
y
of Chin
a, Che
n
g
d
u
611
73
1, Chin
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: tcs111@1
63.
com
A
b
st
r
a
ct
T
he rob
u
st stabil
i
z
a
ti
on pr
o
b
le
m of ch
ao
s suppr
essio
n
for per
man
e
n
t ma
gn
et synchro
nou
s
m
o
tors (PMS
M) drive system
is investigated in
pres
enc
e
of parametric uncertainties
. Based on c
o
ntrol
Lyap
un
ov func
tion (C
LF
) app
roach, a
new
state feed
back
control
l
er is
de
sign
ed to re
al
i
z
e
t
he state
of th
e
system
gl
oba
ll
y asy
m
ptotic
all
y
stabl
e.
The st
abil
i
ty of th
e pr
opos
ed c
ontro
l
sche
m
e is v
e
ri
fied vi
a
Lyap
un
o
v
stable the
o
ry. F
i
nally, si
mu
lat
i
on res
u
lts ill
us
tr
ate the effectiveness of
the p
r
esente
d
meth
od.
Ke
y
w
ords
: p
e
rm
an
en
t ma
gne
t syn
ch
ron
ous m
o
to
r, ch
a
o
s
control, co
n
t
rol Lya
p
u
nov
function, stat
e
feedb
ack contr
o
l
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
With the adv
antage
s of hi
gh torq
ue/ine
r
tia ratio,
high torque/weight ratio, c
o
mpac
t
s
i
z
e
and no
roto
r l
o
ss, permane
nt magnet
synch
r
on
ou
s
m
o
tor
(PMSM
)
drive
sy
stem has been wid
e
ly
use
d
in indu
strial appli
c
atio
n, such as in
robot
i
c
syste
m
, CNC
syst
em, disk
drive
systems, an
d so
on [1-3]. Duri
ng the past
years,
the
st
ability of the motor drive system, whi
c
h is an essent
ial
requi
rem
ent f
o
r i
ndu
strial
a
u
tomation
ma
nufactu
ring,
h
a
s re
ceive
d
consi
derable
a
ttenuation. Up
to now, it ha
s bee
n found
that chao
s
wa
s wid
e
ly
existed in all kinds of moto
r drive syste
m
s,
su
ch a
s
indu
ction moto
rs,
DC moto
rs, and switched
relu
ctan
ce
motors [4]. Chaotic b
ehavi
o
r in
perm
ane
nt magnet
DC m
o
tor ope
n d
r
ive syste
m
wa
s
first add
re
ssed by Hemati
[5]. Li in [6] has
found that
ch
aos
wa
s
also
existed i
n
p
e
rma
nent m
a
gnet syn
c
h
r
o
nou
s moto
r (PMSM). With
out
con
s
id
erin
g power ele
c
tri
c
swit
chin
g, PMSM
dr
ive
system
can
be tran
sform
ed into
a typ
i
cal
Lore
n
z
syst
em, whi
c
h i
s
well
kno
w
n exhibiting
chaoti
c
be
havior. In most en
gine
ering
appli
c
ation
s
, this unde
si
ra
ble cha
o
tic o
scill
ation,
whi
c
h will extrem
ely destroy th
e stabilization
of
the system o
r
even indu
ce
system
collap
s
e,
sh
ould b
e
supp
re
ssed
or even elimi
nated.
Re
cently, nu
mero
us meth
ods h
a
ve be
en su
cc
e
ssfu
lly used to control PMSM
chaoti
c
system,
su
ch
as feedb
ack co
ntrol [7,
8], passivi
ty control [9], d
y
namic
su
rfa
c
e
co
ntrol [1
0],
adaptive cont
rol [11, 12], sliding
mo
de control [13], finite time cont
rol [14], Lyap
unov expon
e
n
ts
approa
ch [1
5
,
16] an
d fu
zzy control [1
7, 18]. Ho
we
ver, mo
st of
those
metho
d
s
either do
not
con
s
id
er un
certain (such
as
fe
edb
ack control
an
d pa
ssivity
control)
pa
ra
meters o
r
h
a
ve
compli
cate
d control
st
ru
cture (such
as Lyapun
ov expone
nts ap
proach and fu
zzy co
ntrol
)
, which
have preve
n
ted the appli
c
ation of
those
methods in p
r
acti
ce.
Control Lyap
unov fun
c
tio
n
is o
ne of t
he mo
st po
werful tool to
desi
gn
controller fo
r
nonlin
ear
systems, which
wa
s first intro
duced by
Art
s
tein [19]
an
d Sontag [2
0
]. This meth
o
d
,
converting st
ability descri
p
tions into t
ools fo
r solv
ing
stabili
zati
ons
, has made tremendous
impact o
n
sta
b
ilizatio
n the
o
ry. Up to
no
w, CL
F
meth
od ha
s b
een
su
ccessfully
applie
d to co
ntrol
manipul
ators [21], converter
syst
em
s [
22], high
-volt
age
dire
ct
current
(HV
D
C) sy
stems [
23],
swit
che
d
sy
st
ems [
2
4]
and
som
e
no
nlin
ear
sy
st
em
s
with time d
e
l
a
y [25]. Wan
g
et al. [26,
27]
first u
s
ed thi
s
method to
a
c
hieve
ch
aoti
c
syn
c
h
r
o
n
ization of two
chaotic
sy
ste
m
s. But it ha
s on
e
disa
dvantag
e
that the
re
sp
onse
spe
ed
o
f
the p
r
op
os
e
d
control
syst
em i
s
n
o
t tun
able, b
e
cau
s
e it
has n
o
adju
s
table
contro
l param
eters in the cont
roller. In fac
t, it is
nec
es
sary to provide
adju
s
table
co
ntrol pa
ram
e
ters fo
r u
s
ers.
To so
lve thi
s
problem, Y
ang et al. [28] prop
osed
an
improve
d
co
ntrolle
r with
time-vary p
a
ram
e
ters fo
r nonli
nea
r
system. Th
e
introdu
ction
of
adju
s
table ti
me-vary pa
rameter h
a
s
not onl
y lessened the
co
ntrolle
r’s d
e
p
ende
nce on
th
e
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046
TELKOM
NI
KA
Vol. 12, No. 4, April 2014: 2667 – 2
676
2668
sele
ction
of
CLF,
but al
so
made
the
co
ntrolle
r o
p
timal for some
a
d
justa
b
le
co
n
t
rol pe
rfo
r
ma
nce
indexe
s
. To t
he be
st of ou
r kno
w
ledg
e, there
ha
s
n
o
re
sult repo
rted in the lite
r
ature so far
to
apply this me
thod to stabili
ze chaoti
c
system
s. In
this paper, we attempt to
use this ap
pro
a
ch to
control PMSM chaoti
c
sy
stem with uncertain pa
ram
e
ters.
This p
ape
r is organi
zed
as follo
ws.
Section 2 int
r
odu
ce
s the
basi
c
con
c
e
p
ts and
lemma
s of CLF. In Section 3, we pre
s
e
n
t the c
hao
s model of PMSM driv
e syst
em first. And then
the cont
rolle
r is desi
gne
d based on
CL
F theory and
the the stab
ility of the controlled
clo
s
e
d
system
s i
s
verified a
c
co
rding to Lya
p
unov sta
b
ility theory. Sect
ion 4 p
r
e
s
ent
s the
simul
a
tion
results to illustrate the effectiveness of
the method. Fin
a
lly, Section 5 con
c
lu
de
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
.
Defini
tion1:
[
29]
Con
s
id
er
the followin
g
affine nonlin
e
a
r sy
stem.
()
(
)
x
fx
g
x
u
,
(1)
Whe
r
e
n
x
R
denotes the state vector of the system,
m
uR
denotes the control
input vector,
()
:
nn
f
xR
R
and
()
:
nm
g
xR
R
are sm
ooth vector fi
elds
with nonl
inear
(0
)
0
f
. A positive
definite fun
c
tion
V
(x) i
s
a
CLF of the
syst
em (1
) if it is
smooth,
pro
p
er, an
d satisfi
e
s the foll
owi
ng
condition:
()
0
,
0
(
)
0
gf
LV
x
x
L
V
x
,
(2)
Whe
r
e
()
g
LV
x
and
()
f
LV
x
denotes the Lie de
rivative of
()
Vx
along
()
g
x
and
()
f
x
,
r
e
spec
tively.
Rem
a
r
k
1
：
()
Vx
p
o
sitive me
an
s that
(0
)
0
V
an
d
()
0
Vx
for
0
x
, and
()
Vx
prope
r mean
s
that
(0
)
V
as
x
.
Defini
tion 2
:
[29] Assume
that
()
kx
:
n
RR
is a fu
nction
with
(0
)
0
k
.
()
uk
x
is alm
o
st
smooth o
n
n
R
if it is smooth a
w
ay from the
orig
in a
nd co
ntinuou
s at the origin of
n
R
.
Lemma 1:
[29]
Let
(,
)
Vx
is a CLF of system (1). The
n
there exist
s
an almo
st smooth
feedba
ck con
t
rol
la
w
()
uk
x
such that
syste
m
(1
) i
s
glob
a
lly
asym
ptotically stable. And
t
h
e
control law
is
:
()
()
()
T
uk
x
p
x
x
,
(3)
22
4
[(
)
(
)
(
(
(
)
)
]
(
)
,
(
)
0
()
0,
(
)
0
xx
x
x
x
px
x
,
(4)
Whe
r
e
()
()
f
x
LV
x
and
()
()
g
x
LV
x
.
Proof:
If
()
0
x
, note that
(,
)
Vx
is a CLF of system
(1), then
(,
)
(
)
(
)
0
f
Vx
x
L
Vx
;
If
()
0
x
,
2
4
(
,
)
(
)
(
)
(
,
)
()
(
(
()
)
(
)
0
Vx
x
x
k
x
x
x
x
.
Thus,
(,
)
0
Vx
for all
states
x
.
More
over,
()
()
f
x
LV
x
is continu
ous a
nd
2
4
((
)
)
((
)
)
xo
x
, s
o
()
uk
x
is co
ntinuou
s at
the origin. Th
us
()
uk
x
an almost
smooth
cont
rol and it is gl
obally asymp
t
otically stabl
e at the
equilibrium
x
=0.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
CLF Based S
t
abilization of Chaos in PM
SM wi
th Uncertain Param
e
ters
(Chuansheng Tang)
2669
Rem
a
r
k
2
：
I
n
co
ntrol the
o
ry, a co
ntro
l Lyapunov f
unctio
n
(,
)
Vx
is a gene
rali
zati
on of the
notion of Lya
punov fun
c
tio
n
()
Vx
used in the
stability anal
ysis. The o
r
di
nary Lyapu
no
v function
is u
s
e
d
to
test wheth
e
r a
d
y
namical
syst
em i
s
stable
(more re
stri
ctively,
asympt
otically
stable
)
.
That is,
wh
ether th
e
syste
m
sta
r
ting in
a state
0
x
in some domain
will remain in
, or for
asymptotic
st
ability will eventually return to
0
x
. The control-Lyapu
no
v function is
use
d
to tes
t
wheth
e
r a sy
stem
i
s
fe
ed
back stabl
e,
that
is, whet
her
for any state
x
th
ere exists a cont
rol
()
uk
x
su
ch that the system
can b
e
brou
ght
to the ze
ro state
with the co
ntrol
.
3. Contr
o
ller Design fo
r PMSM Chao
tic Sy
stem
In this
se
ctio
n, it is given
the chao
s m
o
del
of the PM
SM drive
syst
em an
d at th
e sa
me
time the co
ntrolle
r is
desi
g
ned in
detail. Then,
Th
e stability of the prop
osed
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 tra
n
sfo
r
med mod
e
l o
f
PMSM with the smo
o
th ai
r gap
can
be
expre
s
sed a
s
follows
[3]:
()
dd
q
d
qq
d
q
qL
ii
w
i
v
ii
w
i
w
v
wi
w
T
,
(5)
Whe
r
e
d
v
,
q
v
,
d
i
, and
q
i
are the tran
sform
ed stator
vo
ltage com
p
o
nents an
d current
comp
one
nts i
n
the d-q fra
m
e,
w
and
L
T
are the transfo
rmed an
gle speed an
d external lo
ad
torque re
spe
c
tively,
and
γ
and
σ
are the motor pa
ram
e
ters.
Con
s
id
erin
g the case that, after an o
p
e
r
ation
of the
system, the e
x
ternal inp
u
ts are
set
to z
e
ro, namely,
0
dq
L
vv
T
, system (5) 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
,
(6)
The mode
rn
nonlin
ear the
o
ry su
ch a
s
bifurcatio
n and
chao
s ha
s b
een used to study the
nonlin
ear ch
ara
c
teri
stics
of PMSM dri
v
e system
in
[6]. It has f
ound th
at ,
with the
ope
rating
para
m
eters
γ
and
σ
falling
into a certai
n area, PMS
M
will exhibit complex dy
namic b
ehavi
o
r,
su
ch a
s
peri
o
dic, qu
asi
pe
riodic
and
cha
o
tic be
havio
rs. In o
r
de
r to
make
an
overall inspe
c
tion
of
dynamic b
e
h
a
vior of the PMSM, the bi
furcation dia
g
ram of the angle spe
ed
w
wit
h
incre
a
si
ng
of the p
a
ra
m
e
ter
is illust
rated in Figure 1(a).
W
e
ca
n see that the sy
st
em shows
abundant
and complex
dynamical be
haviors with i
n
crea
sing p
a
rameter
. The typical ch
aoti
c
attra
c
tor 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.
46
.
Acco
rdi
ng to
cha
o
s th
eory, the Lyapu
nov
expon
en
ts and
po
we
r spect
r
um a
r
e two
effective met
hod
s to dete
r
mine
whethe
r a continuo
us dynami
c
sy
stem is
cha
o
tic. In ge
neral, a
three
-
dime
nsi
onal no
nlinea
r system h
a
s one po
sitive Lyapunov e
x
ponent
s, implying that it
i
s
cha
o
tic.
Fig
u
r
e
1
(
c)
a
nd (d) sho
w
the
Lyapun
ov expone
nts a
n
d
power
sp
ect
r
um of PMS
M
cha
o
t
i
c sy
st
e
m
(6
) wit
h
25
, and
5.
46
. Wh
e
n
the
param
eters a
r
e
set
as ob
ove,
cal
c
ulate
d
Lyapun
ov exponents a
r
e:
1
0.
47
9453
E
L
,
2
0
.
0
249
05
E
L
,
3
7.91
454
8
E
L
,and
the Lyapun
ov dimensi
on is
2.0574
32
L
D
, which me
an
s the syste
m
is ch
aotic.
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TELKOM
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Vol. 12, No. 4, April 2014: 2667 – 2
676
2670
(a)
(b)
(c
)
(d)
Figure 1. Bifurcatio
n De
ag
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
cha
o
tic
attracto
r, (c)
Lyapun
ov
expone
nts, (d
) power
spe
c
t
r
um of state variabl
e
w
With un
ce
rtai
n paramete
r
s, the dynamic
mod
e
l of the syste
m
can be d
e
scri
bed a
s
follows
:
()
()
(
)
dd
q
qq
d
q
ii
w
i
ii
w
i
w
wi
w
,
(7)
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.
Be
c
a
us
e th
e
p
a
r
ame
t
e
r
s
and
are
related to the parameters
of PMS
M
drive
sy
ste
m
,
su
ch a
s
re
sistors, in
duct
o
rs, mag
netic,
whi
c
h
w
ill ch
ange i
n
a
certain the rang
e of tempe
r
at
ure.
Therefore, thi
s
a
r
ticle
assu
mes th
at the
fluct
uation ra
nge
of syste
m
param
ete
r
s is
30%, tha
t
is,
1
0.3
,
2
0.3
.
3.2. Contr
o
ller Desig
n
System (6) i
ndicates th
ree equilibrium
points:
0
(0
,
0
,
0
)
S
,
1
(1
,
1
,
1
)
S
, and
2
(
1
,1
,1
)
S
. Given that
25
,
0
S
is l
o
cally
stable,
and
1
S
and
2
S
are b
o
th
locally un
stab
le [6] . Assuming that one
e
quilibri
um poi
nt of system (5) is
(,
,
)
dd
q
d
d
Si
i
w
, then:
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
0
20
40
60
80
10
0
120
-2
0
-1
5
-1
0
-5
0
5
10
Ti
m
e
Ly
apunov
ex
ponen
ts
0
200
0
400
0
60
00
800
0
10000
0
0.
2
0.
4
0.
6
0.
8
1
f
P
o
w
e
r
S
pec
tr
um
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
CLF Based S
t
abilization of Chaos in PM
SM wi
th Uncertain Param
e
ters
(Chuansheng Tang)
2671
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
.
(8)
To qui
ckly st
abilize to equilibrium
point
(,
,
)
dd
q
d
d
Si
i
w
,
1
u
and
2
u
are u
s
ed to
cont
rol t
he
system (7). 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
.
(9)
Let
1
dd
d
ei
i
,
2
qq
d
ei
i
and
3
d
ew
w
, we can
obtain the d
y
namic e
rro
r
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
.
(10)
In orde
r to de
sign the
controller
，
system
(10
)
ca
n be rewritten in a
comp
act form
as
：
()
(
)
ef
e
g
e
B
u
(11)
Whe
r
e
12
3
(,
,
)
T
ee
e
e
，
(,
)
T
and
,
12
3
2
3
21
3
1
3
3
23
()
()
dq
d
dd
d
ee
e
e
w
e
i
f
ee
e
e
e
w
e
i
e
ee
,
3
23
00
()
0
0
d
ge
e
w
ee
,
12
(,
)
uu
u
,
10
01
00
B
is cont
rol i
nput matrix.
The
cont
rol
obj
ective
is
to
stabil
i
ze
the
syst
em (9) at t
he equilibrium poi
nt
(,
,
)
dd
q
d
d
Si
i
w
, that is,
we
desi
gn the
controller to stabilize the
error system
(10) at
e
=0.
S
o
we
will focus on t
he controller
des
i
gning for system (11).
Theorem 1.
Con
s
id
er e
rro
r dynami
c
system (11
)
. If th
e positive fun
c
tion
V
(e
) is d
e
fined by:
22
2
1
12
3
2
()
(
)
V
e
eee
,
(12)
Then
there e
x
ists
a
n
almo
st smooth
fe
edba
ck cont
rol
la
w
(,
)
uk
e
such
that sy
stem
(11) is
globally a
s
ymptotically st
ab
le. And the co
ntrol law
u
is
:
(,
)
(
,
)
(
)
T
uk
e
p
e
e
,
(13)
2
24
[
(
)
(
)
(
()
()
)
(
(
(
)
)
]
(
)
,
(
)
0
(,
)
0,
(
)
0
ee
e
e
e
e
e
pe
e
,
(14)
Whe
r
e
()
()
f
eL
V
e
,
()
()
B
eL
V
e
,
()
()
g
eL
V
e
and
，
μ
is an adju
s
table p
a
ra
m
e
ter.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 4, April 2014: 2667 – 2
676
2672
Proof.
12
3
(
)
()
[
]
()
f
eL
V
e
e
e
e
f
e
,
12
3
3
23
0
0
()
(
)
[
]
0
0
gd
eL
V
e
e
e
e
e
w
ee
,
12
3
1
2
()
()
[
]
[
0
]
B
eL
V
e
e
e
e
B
e
e
.
Cas
e
1) If
()
()
0
B
eL
V
e
and
0
e
, we c
a
n obtain that
12
0
ee
and
3
0
e
.
Then,
2
3
(
(
)
(
)
)
()
()
(
)
0
f
V
fe
g
e
L
V
e
e
e
e
, s
o
()
Ve
is one
CLF
of
system
(11
)
.
2
3
()
(
(
)
(
)
)
()
()
(
)
0
V
Ve
f
e
g
e
B
u
e
e
e
e
.
Cas
e
2) If
()
0
e
,
()
(
(
)
(
)
)
()
()
()
(
,
)
V
Ve
f
e
g
e
B
u
e
e
e
k
e
e
24
()
()
(
(
)
(
)
)
(
(
()
)
ee
e
e
e
24
((
)
(
)
)
(
(
(
)
)
0
ee
e
Thus
, for all (
x
,
μ
), the posi
t
ive and prop
er functio
n
V
(e) de
cre
a
se a
l
ong the traje
c
tory of
the error
syst
em (11
)
.
Moreove
r
,
()
()
f
eL
V
e
and
()
()
g
eL
V
e
are bot
h continu
o
us, and
2
4
((
)
)
(
(
)
)
eo
e
, s
o
(,
)
uk
e
is
co
ntinuou
s at th
e
origi
n
. Thu
s
(,
)
uk
e
an al
mo
st
smooth
cont
rol and it is glo
bally asympto
t
ically
stable f
o
r sy
stem (1
1
)
unde
r control effect
u
.
4. Simulation results
We u
s
e SIM
U
LINK of MA
TLAB to verify the f
easibili
ty of the prop
ose
d
controll
er for a
PMSM cha
o
tic sy
stem. In the sim
u
lation
, the fourth-order
Run
ge–K
utta method i
s
u
s
ed to
sol
v
e
the sy
stem
s
with time
ste
p
si
ze
0.00
1. The
pa
ram
e
tric val
ues of
PMSM a
r
e t
he
same
a
s
in
Section
3.
Without l
o
ss of g
ene
ralit
y, we
sele
ct
S
1
(
1
,
1
,
1
) a
s
the
desi
r
e
d
equilib
rium p
o
int. When
γ
=25, the desi
red equilibri
um point is S
1
(24,
23
,
23
). The control
method takes effect after
t
=20
s and the
adjusta
ble p
a
ram
e
ter
μ
=5.
Figure 2. State Traje
c
tori
es (
d
i
,
q
i
and
w
) and
Control Input
u
of System (9) without
Con
s
id
erin
g Un
certai
n Parameters.
0
10
20
30
40
0
20
40
60
t
i
d
0
10
20
30
40
-20
0
20
40
t
i
q
0
10
20
30
40
-2
0
-1
0
0
10
20
t
w
0
10
20
30
40
-15
-10
-5
0
5
t
u
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
CLF Based S
t
abilization of Chaos in PM
SM wi
th Uncertain Param
e
ters
(Chuansheng Tang)
2673
Figure 3. State Traje
c
tori
es (
d
i
,
q
i
and
w
) and
Control Input
u
of System (9) Con
s
ide
r
ing
Un
certai
n Parameters
The
simul
a
tio
n
results
with
only
one
co
ntrol i
nput, th
at is,
[0
1
0
]
T
B
, is shown in
Figure 2
(with
out co
nsi
d
e
r
ing un
ce
rtain
para
m
eter
s)
and Fi
gure 3
(witho
ut con
s
i
derin
g un
ce
rt
ain
para
m
eters). It can be se
e
n
from Figu
re 2 and Fig
u
r
e 3 that the state traje
c
to
ries a
nd cont
rol
input of the cl
ose
d
loop PM
SM chaoti
c
system with
on
ly one control
input can q
u
ickly sta
b
ilize to
its unstabl
e equilibri
um poi
n
t S
1
. We
ca
n seen f
r
om
Figure 2
and
Figure 3 th
at
it take
s ab
out
2s
to stabli
z
e
t
hem to
S
1
,
but the
r
e
are st
ron
g
ch
attering
ph
e
nomen
on
wh
en the
u
n
ce
rtain
para
m
eters is con
s
ide
r
ed.
Figure 4
and
Figu
re 5
sh
ow th
e the
st
ate traje
c
to
ries
and
co
ntrol input
s of t
he PMSM
cha
o
tic
syste
m
with
two
control i
nput
s.
We
ca
n
se
e
from Figu
re 4 that
t
he
state
s
of th
e
syste
m
stabili
ze to S1 within 1
s
, and there a
r
e
no over
sho
o
t in the contro
l inputs. Moveover, wh
en th
e
uncertain
parameters a
r
e take
n into con
s
ide
r
ation,
th
e state
s
of the syst
em
s ca
n
also st
abili
ze
to their equil
i
brium
with n
o
chatte
ring.
So, t
he perfo
rman
ce of th
e PMSM cha
o
tic syste
m
with
two control i
n
puts i
s
si
gnifi
cantly better than that
with
just on
e control input. It ha
s fast, a
c
cura
te
and ro
bu
st pe
rforma
nce of the se
con
d
method (with two control in
puts).
Figure 4. State Traje
c
tori
es (
d
i
,
q
i
and
w
) and
Control Input (
1
u
and
2
u
)
of Syst
em (9)
withou
t
Con
s
id
erin
g Un
certai
n Parameters
0
5
10
15
20
0
20
40
60
t
i
d
0
5
10
15
20
-2
0
0
20
40
t
i
q
0
5
10
15
20
-2
0
-1
0
0
10
20
t
w
0
5
10
15
20
-6
0
-4
0
-2
0
0
t
u
0
10
20
30
40
0
20
40
60
t
i
d
0
10
20
30
40
-2
0
0
20
40
t
i
q
0
10
20
30
40
-2
0
-1
0
0
10
20
t
w
0
10
20
30
40
0
5
10
15
t
u
u
1
u
2
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 4, April 2014: 2667 – 2
676
2674
Figure 5. State Traje
c
tori
es (
d
i
,
q
i
and
w
) and
Control Input (
1
u
and
2
u
)
of Syst
em (9)
Con
s
id
erin
g Un
certai
n Parameters
Figure 6
sh
ows the
stat
e traje
c
tori
es
i
d
of the p
r
opo
se
d
con
t
roller
with d
i
fferent
adju
s
table pa
ramete
r
μ
. It
sho
w
s that with the incre
a
s
ing of pa
ra
meter
μ
, the transitio
n time is
redu
ce
d acco
rdingly. So we can
choo
se
the paramet
er
μ
accordin
g to the desig
n requi
rem
e
n
t
of
the system p
e
rform
a
n
c
e.
(a)
(b)
(c
)
(d)
Figure 6. State Traje
c
tori
es of PMSM C
haotic System
with Differe
nt Paramete
r
μ
: a)
μ
=
0
.5, b)
μ
=2,
c)
μ
=4 d)
μ
=10
0
5
10
15
20
0
20
40
60
t
i
d
0
5
10
15
20
-20
0
20
40
t
i
q
0
5
10
15
20
-20
-10
0
10
20
t
w
0
5
10
15
20
-
200
-
100
0
100
t
u
u
1
u
2
0
2
4
6
8
10
12
14
16
18
20
0
10
20
30
40
50
60
t
i
d
0
2
4
6
8
10
12
14
16
18
20
0
10
20
30
40
50
60
t
i
d
0
2
4
6
8
10
12
14
16
18
20
0
10
20
30
40
50
60
t
i
d
0
2
4
6
8
10
12
14
16
18
20
0
10
20
30
40
50
60
t
i
d
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
CLF Based S
t
abilization of Chaos in PM
SM wi
th Uncertain Param
e
ters
(Chuansheng Tang)
2675
5. Conclusio
n
We devel
op
a novel nonli
near fee
dba
ck co
nt
rol
sch
eme that acc
ounts fo
r parameter
uncertaintie
s
in a PMSM chaotic
syste
m
. This
cont
roller i
s
de
sig
ned b
a
sed o
n
CLF
theo
ry. The
advantag
es o
f
the propo
se
d controlle
r are as follo
ws:
1) It has a
unified form for PMSM cha
o
ti
c
sys
tem with
or without parametric
uncertaintie
s
.
So the exact mathematic model of
th
e syste
m
is
not req
u
ired.
If there are
no
uncertain p
a
rameters, the para
m
eter
δ
meets
δ
=0 in
the cont
rol la
w (14
)
;
2) The
re
spo
n
se
spe
ed of
the clo
s
ed
-l
oop a
r
e tuna
ble. It has b
een verifie
d
that an
approp
riate in
cre
a
se of the gain
μ
can eff
e
ctively impro
v
e the re
sp
on
sivene
ss of the system;
3) Its
stru
ctu
r
e is
ea
sy to d
e
sig
n
an
d im
plement. Th
e
pre
s
e
n
ted m
e
thod i
s
e
qua
l to only
add the
cont
rol voltages to
the state eq
uation of PSM
S. So it can solve the p
r
oblem exi
s
tin
g
in
[8] and [11] that there is no
controllabl
e variabl
e to
con
t
rol in the spe
ed equ
ation o
f
the system.
Future
re
se
arch
sho
u
ld i
n
vestigate th
e i
m
pleme
n
tatio
n
of the
pro
p
o
se
d control
scheme
by usi
ng
an
experim
ental
setu
p. The
scheme
c
an
also
be
exte
nded
to
synchroni
ze
PMS
M
cha
o
t
i
c sy
st
e
m
s wit
h
un
ce
rt
ain pa
ramet
e
rs.
Ackn
o
w
l
e
dg
ements
This
wo
rk wa
s
sup
porte
d
by the Nation
al Scie
nce an
d Te
chn
o
logy
Major Proj
ect of the
Ministry of Science and Te
chn
o
logy
of China (Proje
ct No. 200
9ZX0
4001
).
Referen
ces
[1]
F
a
rzad T
,
Hamed N. Ma
xim
u
m torqu
e
per
amper
e
contro
l
of perman
ent
magn
et s
y
nchr
ono
us motor
usin
g g
e
n
e
tic
a
l
gorit
hm.
T
E
LK
OMNIKA T
e
lec
o
mmunic
a
tio
n
Co
mp
uting
El
e
c
tronics
an
d C
ontrol.
20
11;
9(2): 237-
44.
[2]
Hua
ng
X,
Li
n
R. Nov
e
l
desi
g
n for
direct
tor
que
co
ntrol s
ystem of PMSM
.
T
E
LKOMNIKA Indo
nes
ia
n
Journ
a
l of Elec
trical Eng
i
ne
eri
ng.
201
3; 11(4)
: 2102-2
1
0
9
.
[3]
Song
X. Des
i
g
n
and sim
u
l
a
ti
on of PMSM
feed
back li
ne
ali
near
izatio
n co
ntrol s
y
st
em.
TELKOMNIKA
Indon
esi
an Jou
r
nal of Electric
al Eng
i
ne
eri
ng.
2013; 1
1
(3): 1
245-
125
0.
[4]
Cha
u
KT
, W
a
n
g
Z
.
Chaos
in
eletric
drive s
ystem
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