Internati
o
nal
Journal of P
o
wer Elect
roni
cs an
d
Drive
S
y
ste
m
(I
JPE
D
S)
V
o
l.
5, N
o
. 4
,
A
p
r
il
201
5, p
p
.
51
2
~
51
9
I
S
SN
: 208
8-8
6
9
4
5
12
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJPEDS
Adaptive Fuzzy Integral Sliding-
Mode Regulator for Induction
Motor Using Nonlinear Sliding Surface
Yo
ng
-Ku
n
Lu
School of
Electr
onic Information
and Au
tomation
,
Tianjin Univ
ersity
of
Scien
c
e an
d Technolog
y
,
Tianjin
, Ch
ina
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Sep 21, 2014
Rev
i
sed
Jan 26, 201
5
Accepted
Feb 10, 2015
An adaptive fu
z
z
y
in
tegr
al slidi
ng-m
ode controller using nonlin
ear sliding
s
u
rface is
des
i
gned for the s
p
eed regula
t
or of a field-orien
t
ed indu
ction m
o
tor
drive in this pap
e
r. Com
b
ining t
h
e convent
iona
l integr
al sliding s
u
rface wi
th
fract
ional-ord
e
r
integr
al
, a
no
nline
a
r s
lid
ing
s
u
rface
is
prop
os
ed for th
e
integr
al sliding-
mode speed cont
rol, which can overcome the windup
problem and the
convergen
ce speed problem
. An adaptive fuzzy
control term
is util
iz
ed to
ap
proxim
a
te
the
u
n
cert
a
in
t
y
.
The
stabili
t
y
of
the
controll
er is
analy
z
ed b
y
Lyapunov stability theor
y
.
The
eff
ectiveness of th
e proposed
speed regul
ator i
s
dem
onstrated b
y
th
e sim
u
latio
n results in com
p
arison wit
h
the conventional
integral
slid
ing-
mode controller
based on bo
undar
y
lay
e
r.
Keyword:
Ada
p
tive
fuzzy control term
I
ndu
ctio
n m
o
to
r
In
teg
r
al slid
i
n
g-m
o
d
e
con
t
ro
ller
Non
lin
ear slid
i
n
g surface
Sp
eed
regu
lator
Copyright ©
201
5 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Y
ong
-K
un
Lu,
Sch
ool
o
f
El
ec
t
r
o
n
i
c
I
n
f
o
rm
at
i
on a
n
d
Aut
o
m
a
t
i
on,
Tian
j
i
n
U
n
i
v
ersity o
f
Scien
ce
an
d Tech
no
logy,
1
038
Dagun
an Ro
ad, Hex
i
Di
strict,
Tianj
i
n
Mu
n
i
cip
a
lity, PR
Ch
in
a.
Em
a
il: au
to
m
a
t
i
o
n
c
n
@
12
6.com
1.
INTRODUCTION
I
ndu
ctio
n
m
o
to
r
(
I
M
)
h
a
s
b
e
en
w
i
d
e
ly ap
p
l
ied
in
th
e industr
ial f
i
eld
o
w
i
n
g
t
o
its less-
main
ten
a
n
ce,
lo
wer-co
s
t and ex
cellen
t
-reliab
ility. Hi
g
h
v
a
riab
le sp
eed
p
e
rfo
r
m
a
n
ce o
f
i
n
du
ction
m
o
to
r is ach
iev
e
d
t
h
rou
g
h
fi
el
d-
ori
e
nt
ed
cont
rol
(F
OC
)
.
In fi
el
d
-
o
r
i
e
nt
ed co
nt
r
o
l
(
o
r vect
or c
ont
rol
)
, t
h
e i
n
d
u
c
t
i
on m
o
t
o
r can be
co
n
t
ro
lled
i
n
a
m
a
n
n
e
r sim
ila
r to
th
e con
t
ro
l
o
f
sep
a
rately ex
cited
DC m
o
tor. Th
e m
a
j
o
r
p
r
ob
lem
o
f
FOC is
th
e sen
s
itiv
ity to
larg
e
un
cert
a
in
ties wh
ich
are du
e to
m
a
g
n
e
tization
satu
ration
,
tem
p
eratu
r
e
v
a
riatio
n
,
load
di
st
ur
ba
nces,
et
c [1]
.
I
n
o
r
der t
o
i
m
prov
e t
h
e pe
rf
orm
a
nce of s
p
eed regulato
r und
er u
n
certain
ti
es
in
mechanical pa
ra
m
e
ters and load torq
ue
, m
a
ny
im
prov
ed
sp
eed re
g
u
l
a
t
o
r
o
f
F
O
C
we
re
pr
op
ose
d
fo
r i
n
d
u
ct
i
o
n
m
o
to
r d
r
iv
es
[1
]-[4
]. Du
e to th
e g
ood
robu
stn
e
ss, fast
dyn
amics resp
on
se and
easy
i
m
p
l
e
m
en
tatio
n
,
th
e
sl
i
d
i
n
g
-
m
ode c
ont
rol
has
bee
n
use
d
i
n
t
h
e
cont
rol
o
f
indu
ctio
n m
o
to
r
[5
]-[7
]. B
u
t slid
ing
-
m
o
d
e
con
t
ro
l is
suf
f
eri
ng f
r
o
m
t
h
e chat
t
e
ri
ng p
h
e
nom
enon
. O
n
e effec
t
i
v
e sol
u
t
i
o
n i
s
repl
aci
ng t
h
e si
gn f
u
nct
i
on
by
co
n
tinuo
us satu
ration
fun
c
tio
n
s
[8
]. Bou
n
d
a
ry layer is a p
opu
lar saturatio
n
fun
c
tion
at th
e co
st
o
f
t
h
e
increase
d
steady-state tracki
ng e
r
r
o
r
.
The
i
nvest
i
g
at
i
o
ns
on i
n
t
e
gral
sl
i
d
i
n
g-m
ode co
nt
r
o
l
l
e
r fo
r i
n
duct
i
o
n
m
o
tor can be fou
n
d
in [9]
-
[
1
1
]
. In [9]
,
an int
e
gral s
l
i
d
i
n
g-m
ode c
ont
r
o
l
st
r
a
t
e
gy
usi
ng sat
u
rat
i
o
n f
unct
i
o
n was
used to stabilize speed t
r
acki
n
g
of each
induction m
o
tor while synchronizing
its speed wi
th the speed of
other
m
o
to
rs. A slid
ing
-
m
o
d
e
con
t
ro
ller was presen
ted
fo
r sen
s
o
r
less FOC o
f
ind
u
c
tion
m
o
to
r with
m
o
d
e
l
refe
rence
ada
p
t
i
ve sy
st
em
i
n
[10]
,
w
h
ere
an i
n
t
e
g
r
al
sl
i
d
i
n
g-
m
ode cont
rol
u
s
i
ng
b
o
u
n
d
ary
l
a
y
e
r was
desi
gne
d.
An
in
teg
r
al slid
ing
-
m
o
d
e
contro
l u
s
ing
boun
d
a
ry laye
r wa
s ado
p
t
e
d f
o
r speed c
ont
rol
l
e
r
of i
n
duct
i
o
n
m
o
t
o
r
dri
v
es
wi
t
h
ref
e
rence
m
odel
and
a L
u
e
nbe
rg
er
obse
r
ver i
n
[
11]
.
Ho
we
ver
,
t
h
e wi
n
d
u
p
p
r
o
b
l
e
m
and t
h
e con
v
er
ge
nce spee
d pr
o
b
l
e
m
are not
di
sc
usse
d i
n
t
h
e ab
ove
in
teg
r
al slid
ing-m
o
d
e
co
n
t
ro
l strateg
i
es for sp
eed
regu
la
to
r
o
f
FOC. As men
tio
n
e
d
in
[12], th
e in
teg
r
al actio
n
may lead
to
win
d
u
p
pro
b
l
em, and
sign
if
ican
t ov
er
shoo
t
may o
ccu
r th
at
r
e
q
u
i
r
e
s long
ti
m
e
f
o
r
r
ecover
y
. To
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Ad
apt
i
ve F
u
zzy
I
n
t
e
gr
al
Sl
i
d
i
n
g-M
o
de Re
g
u
l
a
t
o
r
f
o
r
I
n
d
u
ct
i
on
Mot
o
r
Usi
n
g
N
o
nl
i
n
e
a
r
…
(
Y
on
g-K
u
n
Lu)
51
3
el
im
i
n
at
e t
h
e wi
n
d
u
p
p
h
e
n
o
m
enon
fo
r an
i
n
t
e
gral
sl
i
d
i
n
g-m
ode co
nt
r
o
l
,
t
h
e i
n
t
e
gral
act
i
on was t
u
r
n
ed
on
onl
y
whe
n
t
h
e
n
o
rm
of t
r
ack
i
ng e
r
r
o
rs
was
l
o
we
r t
h
an
a
pre
d
et
erm
i
ned
val
u
e i
n
[1
2]
.
M
o
re
ove
r,
i
t
i
s
wel
l
-
kn
o
w
n t
h
at
t
h
e
i
n
t
e
gral
act
i
o
n
m
a
y
sl
ow d
o
w
n t
h
e co
n
v
er
gence
spee
d
of
t
r
acki
n
g e
r
r
o
r
.
The
deri
vat
i
v
e
act
i
on
may speed
up
the convergence spee
d
of
track
ing
erro
r, bu
t as we kno
w th
e tim
e
deri
vative
of m
echanica
l
spee
d (accelerated
m
echanic
al speed)
is se
nsitive to the noise a
nd
diffi
cult to obtain
at present e
v
e
n
usi
n
g
im
pro
v
ed
di
f
f
e
r
ent
i
a
t
o
r
s
su
ch
as n
onl
i
n
ea
r
di
ffe
re
nt
i
a
t
o
r [
13]
a
nd sl
i
d
i
n
g m
ode di
f
f
er
ent
i
a
t
o
r
[1
4]
.
Hence
,
accelerated m
e
chanical spee
d
is seldom
e
m
p
l
oyed in practical speed re
gula
t
or of FOC.
On the other
hand, the
in
v
e
stig
ation
of
fraction
a
l-o
r
der
co
n
t
ro
l h
a
s attracted
m
o
re
and
m
o
re inter
e
sts. T
h
e
fracti
onal
-
o
r
de
r c
o
n
t
roller
is th
e ex
ten
s
ion
o
f
in
teg
e
r-o
rd
er co
n
t
ro
ller
[15
]
, wh
ich int
r
oduces extra
degrees of free
dom
. The fract
ional-
or
der
sl
i
d
i
n
g-
m
ode cont
rol
were
di
sc
usse
d i
n
[
16]
a
n
d
[1
7]
.
A f
r
act
i
o
nal
-
or
der
i
n
t
e
g
r
al
sl
i
d
i
n
g-m
ode
fl
u
x
obs
er
ver
was
pr
o
v
i
d
e
d
t
o
es
t
i
m
a
t
e
t
h
e d- a
nd
q-a
x
i
s
fl
ux
es i
n
t
h
e st
at
i
onary
re
fere
nce
fram
e
for se
n
s
orl
e
ss
vect
o
r
co
nt
r
o
l
l
ed i
n
duct
i
o
n
m
o
t
o
rs i
n
[
1
6
]
. A f
r
act
i
o
na
l
-
or
de
r sl
i
d
i
n
g
-
m
ode co
nt
r
o
l
schem
e
based o
n
p
a
ram
e
ters au
to
-t
u
n
i
n
g
fo
r the v
e
lo
city contro
l of
p
e
rm
an
en
t m
a
g
n
e
t syn
c
hro
nou
s m
o
to
r
was
p
r
op
osed
in
[1
7]
.
Th
e m
a
in
co
n
t
rib
u
tion
of this p
a
p
e
r lies in
th
e fo
llo
wi
ng
th
ree asp
ects: (1
) An
ad
ap
tiv
e fu
zzy
sl
i
d
i
n
g
-
m
ode c
ont
rol
l
e
r i
s
pr
o
pos
ed a
n
d su
c
cessf
ul
l
y
a
p
p
lied
to
t
h
e sp
eed
regu
lato
r
o
f
in
du
ctio
n m
o
t
o
r. (2
)
Com
b
ining the
conventional in
tegral sliding
surface with fractional-or
der
integral, a nonl
inear sliding surface
i
s
pro
p
o
sed
fo
r
t
h
e i
n
t
e
gral
sl
i
d
i
n
g-m
ode spe
e
d re
gul
at
o
r
, w
h
i
c
h can
ove
rc
om
e
t
h
e wi
nd
u
p
p
h
en
om
enon
an
d
spee
d up c
o
nverge
nce. (3)
T
h
e a
d
aptiv
e
fuzzy cont
rol term
based on the
nonlinea
r sli
d
ing s
u
rface is
applied
to
approx
im
ate
th
e
u
n
c
ertain
t
y
.
2.
DY
N
A
MI
C M
O
DEL
O
F
I
N
DU
CTIO
N M
O
TOR
Th
e m
a
th
e
m
ati
c
s m
o
d
e
l o
f
an in
du
ction
m
o
to
r can
b
e
written
in
t
h
e
ro
t
o
r ro
tating
referen
ce fram
e
(d
-q
) [1
0]
as f
o
llows:
r
qL
r
md
q
qr
d
q
d
dq
d
d
=-
-
d
d
=-
+
d
d
=-
-
-
+
d
d
=-
+
+
+
d
ω
ρψ
i
β
T
αω
t
ψ
a
ψ
aL
i
t
i
δ
i
υω
ψ
ω
ib
u
t
i
δ
i
υ
a
ψω
ib
u
t
(
1
)
Whe
r
e
p
mm
r
=/
(
)
ρ
nL
J
L
,
rr
=/
aR
L
,
σ
S
=1
/
(
)
bL
L
,
mm
=/
α
BJ
,
m
=1
/
β
J
,
m
σ
Sr
=/
(
)
υ
LL
L
L
,
22
2
Sr
r
m
σ
Sr
=(
+
)
/
(
)
δ
R
LR
L
L
L
L
;
r
ω
and
ω
are t
h
e
rot
o
r m
echanical spee
d a
n
d th
e sy
nc
hr
on
o
u
s
spee
d;
ψ
is
the rot
o
r flu
x
;
m
L
is the m
u
tual
inductance;
2
σ
mr
S
=1
-
/
(
)
L
LL
L
is the m
o
tor leakage i
n
ductance
;
d
i
and
q
i
are the
d,
q-a
x
is stator c
u
rrents;
r
R
and
r
L
are the rotor
re
sistance and inducta
nce;
S
R
and
S
L
are the stator res
i
stance and inductance;
p
n
is the num
ber
of p
o
le
pair
s;
d
u
and
q
u
are the
d, q-a
x
is
stato
r
volta
ges;
L
T
is the external load torque;
m
J
and
m
B
are the
m
echanical inertia of m
o
m
e
nt a
n
d
the
da
m
p
ing tor
q
ue
coef
ficient; electrom
a
gnetic tor
q
ue
of
an
in
duct
i
o
n
m
o
tor is de
fine
d as:
ep
m
q
r
=/
Tn
L
ψ
iL
(
2
)
From
(1
) a
n
d (
2
)
,
one
has:
r
re
d
=-
-
+
d
ω
αω
h
β
T
t
(
3
)
Whe
r
e
Lm
=/
hT
J
. Consider the uncertai
n
ties in
(3), one gets:
r
re
d
=-
(
+
Δ
)-
(
+
Δ
)+
(
+
Δ
)
d
ω
αα
ω
hh
ββ
T
t
(
4
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-86
94
I
J
PEDS
Vo
l. 5
,
No
. 4
,
Ap
r
il 2
015
:
51
2
–
51
9
51
4
Whe
r
e
Δ
α
,
Δ
h
and
Δ
β
are the tim
e-varying
value
of
α
,
h
and
β
,respectively.
The s
p
ee
d trac
kin
g
e
r
r
o
r
is de
fine
d as:
*
rr
()
=
(
)
-
()
et
ω
t
ω
t
(
5
)
Whe
r
e
*
r
ω
is the s
p
eed
refe
rence
.
The tim
e derivative of
Equation (5) is:
d(
)
=
-
()
-
(
)
+
()
+
(
)
d
et
α
et
β
ut
ξ
t
η
t
t
(
6
)
Whe
r
e
*
*
r
r
d(
)
()
=
(
)
+
+
(
)
d
ω
t
ξ
t
αω
th
t
t
, uncertainty term
re
()
=
Δ
()
-
Δ
()
+
Δ
()
η
t
αω
t
β
Tt
h
t
,
e
=
uT
.
The c
o
ntrol
objective is t
o
fi
nd a s
p
eed re
gulato
r u
s
in
g ad
a
p
tiv
e fu
zzy
sliding-m
ode cont
roller i
n
rot
o
r flu
x
orie
n
t
ed
re
fere
nce fr
am
e
for
the
t
r
a
c
kin
g
of
spee
d in
p
r
ese
n
ce of
m
odel
unce
r
tainty
.
Th
e
ov
er
all
b
l
o
c
k d
i
ag
r
a
m
fo
r a
d
i
r
ect f
i
eld
-
o
r
ien
t
ed indu
ctio
n m
o
to
r
dr
iv
e is shown
in
Figur
e
1
,
whic
h co
nsists
of a
n
in
ducti
on m
o
tor
(IM
)
,
a SP
WM
v
o
ltage so
urce i
n
verter
, tw
o current controllers, two
coo
r
dinate tra
n
slators,
a c
u
r
r
e
n
t m
odel, a
n
d
a spee
d
re
gula
t
or
usi
n
g
ada
p
tive f
u
zzy
slid
ing
-
m
ode co
nt
roller
base
d
on a
novel nonlinea
r sli
d
ing s
u
rface.
*
r
ω
e
*
sq
i
*
e
T
r
p
m
L
nL
*
sd
i
*
sq
u
*
sd
u
*
sa
u
*
sc
u
*
sb
u
sq
i
sd
i
r
ω
e
θ
_
_
_
÷
ψ
Figu
re
1.
O
v
er
all block
dia
g
r
a
m
for
a
dir
ect
field-oriented i
n
duction m
o
tor drive
3.
DESIGN OF ADAPTIVE
FUZZ
Y
SLIDING-MODE CONTROLLER
The nonlinear sliding
surf
ace
can be defi
ned as:
11
1
22
1
2
2
+|
|
=+
<
|
|
||
>
ea
e
e
γ
Se
a
e
γ
e
γ
ee
γ
(
7
)
Whe
r
e
1
a
,
2
a
,
1
γ
and
2
γ
are positive desi
gn
param
e
ters,
1
0
=(
)
d
t
ee
ττ
,
20
=D
(
)
ν
t
ee
,
0
D(
)
ν
t
e
is the fractional-
order integral
operat
or,
-1
<
<
0
ν
, the
conve
ntional
integral slidi
n
g
surface is
use
d
in the s
m
all speed
tr
ack
ing
er
ro
r in
terv
al to
elim
i
n
ate th
e
windup
p
h
e
no
m
e
n
o
n
.
According to the expon
ential reachi
ng
law:
d
=-
-
s
a
t
(
/
)
d
S
fS
K
S
φ
t
(
8
)
Whe
r
e
f
and
K
are positive design param
e
ters,
|(
)
|
K
η
t
. T
h
e c
ont
rol la
w
can
be
desig
n
e
d
as:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PEDS
I
S
SN:
208
8-8
6
9
4
Ad
aptive F
u
zzy
I
n
tegr
al Slidi
n
g-M
o
de Re
g
u
lator
for
I
n
d
u
ct
ion
Mot
o
r
Usi
n
g
N
o
nline
a
r
…
(
Y
on
g-K
u
n
Lu)
51
5
mf
m
m
1
+1
mf
m
0
m
1
2
mm
m
f
m
m
2
[+
+
+
s
a
t
(
/
)
]
+
(
-
)
|
|
=[
+
+
+
s
a
t
(
/
)
]
+
D
-
<
|
|
-+
ξ
++
+
s
a
t
(
/
)
|
|
>
ν
t
J
ξ
uf
S
K
S
φ
JB
e
e
γ
uJ
ξ
uf
S
K
S
φ
Je
B
e
γ
e
γ
Be
J
J
u
f
J
S
K
J
S
φ
e
γ
(
9
)
Whe
r
e
f
u
is the adaptive
fuzzy cont
rol term to
approxim
a
te the uncertainty ter
m
,
sa
t(
)
is the saturation
fu
nctio
n defi
ne
d
as:
/|
|
sa
t(
/
)
=
sgn(
)
|
|>
S
φ
S
φ
S
φ
SS
φ
(
1
0
)
Whe
r
e
sgn(
)
is the sign
function,
φ
is the width
of
bounda
ry
layer whic
h can re
duce the chattering
phe
n
o
m
e
non.
The f
r
actional
-
o
r
de
r de
rivati
ve co
ntr
o
l term
+1
m0
D
ν
t
J
e
in (
9
) is use
d
to spee
d
up
con
v
er
ge
nce
o
f
spee
d tracki
n
g
err
o
r
.
The a
p
pr
o
x
im
ation of
fractio
nal-
or
d
e
r de
rivate an
d
integral
play
s an im
portant
role i
n
the fracti
onal-
or
der c
o
ntr
o
l.
We ad
o
p
t the
integer-order
m
odel to approxi
m
a
te the fr
actional-
or
der
deri
vate
and i
n
tegr
al in a suitable f
r
eq
ue
ncy
inter
v
al [1
8]
.
T
h
e
fractio
nal-
or
d
e
r de
rivative
use
d
in the p
r
op
ose
d
cont
roller is not sensitive t
o
the noise
beyond the selected frequency interval.
The fuzzy input va
riables
of the ada
p
tive fuzzy cont
r
o
l term
[19]
, [
20]
are
S
and
e
. B
y
using the
singleto
n
fuzz
ification, pr
o
d
u
ct
in
fere
nce engi
ne
a
n
d ce
nter a
v
era
g
e
defuzzifi
cation, the ad
ap
tiv
e f
u
z
z
y
cont
rol term
is give
n as:
F
=1
T
f
=1
F
=1
=1
ˆ
==
ij
ij
n
j
m
i
n
m
j
j
i
μ
u
u
μ
bw
(
1
1
)
Whe
r
e
T
12
ˆˆ
ˆ
=,
,
,
m
uu
u
b
is the c
o
nse
que
nt
para
m
e
ter vecto
r
,
w
is the
vecto
r
of
f
u
zzy
ba
sis f
u
n
c
tions,
=2
n
,
F
ij
μ
are the m
e
m
b
ershi
p
f
u
nctio
ns
of
in
put
varia
b
les,
ˆ
j
u
is the
point in
out
put space
of t
h
e
fuzzy syste
m
at
whic
h t
h
e m
e
m
b
ership
fu
nct
i
on
o
f
out
put
varia
b
le ac
hieves its m
a
xim
u
m
value,
m
i
s
the num
b
er of
fuzzy
rules.
The
param
e
ter vector is a
d
apt
e
d acc
o
rding t
o
the following updating law:
T
11
(
(
||
||<
)
o
r
(
|
|
||=
a
n
d
0
)
)
d
=
d
0o
t
h
e
r
s
rS
M
M
S
t
wb
b
b
w
b
(
1
2
)
Whe
r
e
r
and
1
M
are
t
h
e positive
design param
e
ters
.
4.
STABILITYANALYSIS
The
o
p
tim
a
l param
e
ter vector
is defi
ned
as:
1
0f
||
|
|
=a
r
g
m
i
n
[
s
u
p
|
(
)
-
|
]
Ω
N
η
u
b
x
bx
(
1
3
)
And
λ
is
de
fine
d as
the m
i
nim
a
l app
r
oxim
a
tion
er
ro
r.
Ch
oo
se th
e Ly
ap
uno
v fun
c
tion
s
as:
2
1
1
=(
)
2
VS
t
(
1
4
)
2T
2
11
=(
)
+
(
)
(
)
22
VS
t
t
t
r
qq
(
1
5
)
Whe
r
e
0
=-
qb
b
.
The deri
vative of
Eq
uatio
n (1
4)
with respect
to tim
e is:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-86
94
I
J
PEDS
Vo
l. 5
,
No
. 4
,
Ap
r
il 2
015
:
51
2
–
51
9
51
6
1
d
d
==
(
-
-
s
a
t
(
/
)
+
(
)
)
dd
V
S
SS
f
S
K
S
φη
t
tt
(
1
6
)
If
||
>
S
φ
, then:
2
1
d
=(
-
-
s
a
t
(
/
)
+
(
)
)
-
-
|
|
+
|
(
)
|
|
|
d
V
Sf
S
K
S
φη
tf
S
K
S
η
tS
t
(
1
7
)
Thus, i
f
the condition
of
|(
)
|
K
η
t
is satisfied,
1
d
0
d
V
t
hol
ds,
an
d
1
d
0
d
V
t
only
w
h
e
n
=0
S
.
On
the othe
r h
a
nd
, If
||
S
φ
, co
nsi
d
erin
g (
1
3)
, the
deri
vative
of
E
quatio
n
(
1
5
)
w
ith res
p
ect to
tim
e
is:
T
2
d
d1
d
=+
dd
d
V
S
S
tt
r
t
b
q
(
1
8
)
Fr
o
m
(
10)
, (11
)
,
(
12)
an
d (18
)
, th
en
:
T
2
d
d1
d
=
+
=
[
--
s
a
t
(
/
)
+
]
=
[
--
/
+
]
dd
d
V
S
SS
f
S
K
S
φλ
Sf
S
K
S
φ
λ
tt
r
t
b
q
(
1
9
)
If the ada
p
tive
fuzzy
co
ntrol
term
is prope
rly
designe
d,
λ
is sufficiently small,
then
2
d
0
d
V
t
hol
d
s
,
and
2
d
0
d
V
t
on
ly wh
en
=0
S
. That m
eans Lyapunov function
2
V
will decrease gradually and the sliding
surface will conve
rge to zero. If the syst
em
of sliding s
u
rfa
ce is stable, th
e speed trac
king error will conve
rge
to zero.
5.
SIMULATION RESULTS
Sim
u
lations ar
e carried
o
u
t u
s
ing the
Sim
u
link
p
acka
g
e
of MATLAB
.
T
h
e ov
erall cont
rol structure
for the sim
u
lat
i
on is shown in Fi
gure
1.T
h
e
specificatio
ns
an
d
nom
inal param
e
ters of
m
o
tor o
p
erate
d
usin
g
direct
r
o
to
r field orie
ntation
a
r
e give
n i
n
Ta
bl
e 1
[
1
]
.
Tab
l
e
1
.
Sp
ecificatio
n
s
an
d No
m
i
n
a
l Par
a
m
e
ter
s
of
an
I
nductio
n
M
o
to
r
Motor para
m
e
te
r
Value
Output power
(
H
P)
50
Rated voltage (
V
)
460
Nu
m
b
er
of pole pair
s (
P
)
2
Rated fr
equency
(Hz)
60
Stator resistance
(
Ω
)
0.
087
Rotor resistance (
Ω
)
0.
228
Stator
inductance (m
H)
35.
5
Rotor
inductance (
m
H)
35.
5
M
u
tual inductance (m
H)
34.
7
Mechanical iner
tia
of
m
o
m
e
nt(kg•
m
2
) 2
Dam
p
ing tor
que coefficient(
N•m
•
s
)
0.
2
The operating sequences
are descri
be
d as follows. The initia
l load torque is co
nstant (0N•m
)
.Aft
er
the initial constant speed refe
rence
of 90rad/s from
ti
m
e
t
=0 to 0.
1s. Fr
om
tim
e
t
=0.1 to 0.25
s, the
spe
e
d
refe
rence is i
n
creased linearl
y
from
90 t
o
120ra
d
/s, a
nd t
h
en from
t
=0.6 to
0.9s spee
d refere
nce
is dec
r
eased
from
120 to
90rad/s. At tim
e
t
=1.
1
s c
o
nstant
load t
o
r
q
ue
(190N•m
)
is applied.
The val
u
es of
m
e
chanical inertia of m
o
m
e
nt
m
J
and d
a
m
p
ing tor
que c
o
ef
ficient
m
B
ar
e 0.831
kg•
m
2
an
d 0.5N•m
•s
d
u
r
i
ng
th
e simu
latio
n
,
i.e., there are
uncert
ai
nties in the
mechanical
param
e
ters. Si
mulation
tests have
bee
n
per
f
o
rm
ed in
or
der
to c
o
m
p
are the
dy
nam
i
c per
f
o
rm
ance o
f
the
pr
o
p
o
s
ed
spee
d re
g
u
lator
with
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PEDS
I
S
SN:
208
8-8
6
9
4
Ad
aptive F
u
zzy
I
n
tegr
al Slidi
n
g-M
o
de Re
g
u
lator
for
I
n
d
u
ct
ion
Mot
o
r
Usi
n
g
N
o
nline
a
r
…
(
Y
on
g-K
u
n
Lu)
51
7
the conventional integral
sl
iding
-
m
ode co
ntr
o
ller base
d
on b
o
u
n
d
a
r
y
lay
e
r, i.e., the pr
op
ose
d
co
ntr
o
ller
without the
nonlinea
r sliding
surface
of E
q
uation
(7) a
n
d the ada
p
tive
fuzz
y cont
rol term
of Equation
(11).
In t
h
e
fre
que
nc
y
dom
ain, the
f
r
actional
-
o
r
de
r
deri
vative
of
+1
0
D
ν
t
e
can
be e
x
presse
d as
+1
ν
s
, w
h
ere
s
is
the Laplace va
riable. Figure
2 and Figure
3 show th
e bode diagram
of the fractional-orde
r de
rivative
s
0.2
in
the sim
u
lation and
the
b
ode
di
agram
of t
h
e i
n
teger-
o
r
de
r
der
i
vative
s
.
Figu
re
2.
B
o
de
diag
ram
of
s
0.2
in the
sim
u
lation
Figu
re
3.
B
o
de
diag
ram
of
s
The de
sig
n
pa
ram
e
ters of th
e pr
op
ose
d
s
p
eed re
gulat
or
are
=-
0
.
8
ν
,
=1
.
5
φ
,
1
=1
γ
,
2
=5
γ
,
=1
0
0
r
,
1
=2
0
M
,
1
=1
a
,
2
=1
a
,
=1
f
,
=
100
K
an
d the
f
u
zz
y
m
e
m
b
ership
fu
nctio
ns
of
e
are
desi
gne
d as:
11
F
=
m
i
n
(1
,
m
a
x
(0
,
1
-
(
4
+
6)/
3
)
)
μ
e
;
12
F
=
m
a
x
(
0
,
m
in
(
1
+
(
4
+
3)
/3,
1
-
(
8
+
6)
/3
)
)
μ
ee
;
13
F
=
m
ax(0,
m
i
n
(1
+
(
8
+
3)/
3
,
1
-
(
8
+
3)/3
)
)
μ
ee
;
14
F
=
m
a
x
(
0
,
m
in
(
1
+
8
/3
,
1
-
8
/3
)
)
μ
ee
;
15
F
=
m
a
x
(
0
,
m
in
(
1
+
(
8
-
3)
/3
,
1
-
(
8
-
3
)
/3
)
)
μ
ee
;
16
F
=
m
a
x
(
0
,
m
in
(
1
+
(
8
-
6
)
/3
,
1
-
(
4
-
3
)
/3
)
)
μ
ee
;
17
F
=
m
i
n
(1,
m
ax(0,
1
+
(
4
-
6)/
3
))
μ
e
.
The
fuzzy
m
e
m
b
ership f
u
nc
tions
of
S
are the sam
e
as those
of
e
. The sliding surface of the
com
p
ared c
o
ntroller is
selected as
1
=+
Se
e
.
Figu
re 4 sh
o
w
s
the desir
e
d
m
o
tor
spee
d (
D
ash
-
dot
li
ne)
,
the
r
o
to
r
s
p
e
e
d base
d o
n
t
h
e
c
o
m
p
ared
cont
roller
(
D
as
hed
line) a
n
d t
h
e r
o
to
r s
p
ee
d
base
d o
n
t
h
e
proposed controller (Solid lin
e
)
.
It is clear t
h
at the
r
o
to
r
sp
eed
p
e
rf
or
m
a
n
ce o
f
the pr
opo
sed co
ntr
o
ller
is
b
e
tte
r than that
of the com
p
ared
controller after the step
change
of e
x
ternal l
o
ad torque.
Figure
4. Reference s
p
ee
d a
n
d rotor s
p
eed res
p
onse
The
pe
rf
orm
a
nces o
f
t
h
e m
o
tor
tor
q
ue a
r
e i
llustrated
by
F
i
gu
re
5 a
n
d
Fi
gu
re
6.
It is
se
en th
at the
m
o
tor
tor
que
s are within reas
ona
ble ran
g
es
i
n
Fi
gu
re 5-
6.
(
r
a
d
/s
e
c
)
(r
a
d
/
se
c)
AF
I
S
M
Re
f
IS
M
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-86
94
I
J
PEDS
Vo
l. 5
,
No
. 4
,
Ap
r
il 2
015
:
51
2
–
51
9
51
8
Figu
re 5.
To
r
q
ue resp
o
n
se of
the
p
r
op
ose
d
c
ont
roller
Figu
re 6.
To
r
q
ue resp
o
n
se of
the
com
p
are
d
c
ont
roller
The sim
u
latio
n res
u
lts re
vea
l
that the pres
ented
m
e
thod
has better t
r
ac
kin
g
pe
rf
orm
a
nce tha
n
the
con
v
e
n
tional
integ
r
al slidin
g
-
m
ode co
ntrolle
r
base
d
on
bounda
ry layer
under unce
rtainties in t
h
e m
echanical
param
e
ters and load torque.
6.
CO
NCL
USI
O
N
I
n
th
is
p
a
p
e
r,
an
ad
ap
tiv
e
f
u
zz
y
sliding
-
m
o
d
e
vect
or
co
ntr
o
l has
be
e
n
pres
ented for s
p
ee
d
regulator
of in
d
u
ctio
n m
o
tor. It is pr
op
ose
d
as a sl
iding-m
ode controller whic
h has a nonlinear sliding surface to
ove
rc
om
e the win
d
u
p
ph
en
o
m
enon
o
f
c
o
n
v
e
ntional i
n
teg
r
al
sliding
-
m
ode spee
d c
ontr
o
l
l
er strategy
a
n
d s
p
eed
up convergence by fractional-
order derivati
ve control
term which is not sensitive to the noise beyond the
selected f
r
eq
ue
ncy
inter
v
al.
M
o
re
ove
r,
the
pr
o
pose
d
sliding
-
m
ode c
ont
r
o
ller inc
o
rp
ora
t
es a f
r
actio
nal-o
r
de
r
adaptive
fuzzy cont
rol term
based
on
t
h
e nonlinear sliding surface
to appr
oxim
a
te the uncertainty. The
n
the
closed loop stability of the presented
design has been proved
by Lyapunov stability th
eory. Finally, by
means
of sim
u
lation e
x
am
ples, it has been
sh
ow
n t
h
at the p
r
op
os
ed co
ntr
o
l m
e
tho
d
im
pro
v
es tracki
ng
per
f
o
r
m
ance
of speed in com
p
arison
with the co
nve
ntio
nal inte
gral sli
d
in
g-m
ode c
o
ntr
o
ller
based
on
b
o
u
n
d
a
r
y
lay
e
r in
prese
n
ce
o
f
e
x
ternal l
o
ad
dist
ur
ba
n
ce a
n
d m
echanical
para
m
e
ter variations.
ACKNOWLE
DGE
M
ENTS
This w
o
r
k
is sup
p
o
rte
d
by
the Science an
d Tech
n
o
lo
gy
Devel
opm
ent Fo
un
datio
n o
f
the Highe
r
Education Institutions of Tianj
i
n M
u
nicipality of Chin
a (No. 20130722).
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BIOGR
AP
H
Y
O
F
AUTH
O
R
Yong-Kun Lu was born in Yanbian, Chin
a,
in 1976.
He received the B.S.
degr
ee in Industrial
Autom
a
tion from
Dalian Institu
te of Ligh
t Industr
y,
China
,
in 1
999, the M
.
S. d
e
gree
in Power
Ele
c
tronics and
Power Drives from
Dalian M
a
r
itim
e Univ
ersity, Chin
a,
in 2
002, and Ph.D.
degree
in Contro
l Scien
c
e and
En
gineer
ing from
Ti
anjin University
, Chin
a, in
201
0. In 2002, h
e
joined th
e S
c
ho
ol of El
ectron
i
c
Inform
ation and
A
u
tom
a
tion, T
i
anjin U
n
iv
ers
i
t
y
of S
c
ienc
e and
Techno
log
y
,
Ch
ina.
H
i
s
t
each
i
n
g
is
e
l
e
c
tri
c
al
and electron
ic technique
.
Hi
s
c
u
rrent
rese
arc
h
inter
e
sts ar
e fo
cused on adv
a
nced contro
ller
for
el
ec
tric
al
drives
a
nd el
ectron
i
c
ins
t
rum
e
ntat
ion.
Evaluation Warning : The document was created with Spire.PDF for Python.