TELKOM
NIKA
, Vol. 13, No. 4, Dece
mb
er 201
5, pp. 1263
~1
269
ISSN: 1693-6
930,
accredited
A
by DIKTI, De
cree No: 58/DIK
T
I/Kep/2013
DOI
:
10.12928/TELKOMNIKA.v13i4.1898
1263
Re
cei
v
ed Au
gust 30, 20
14
; Revi
sed O
c
t
ober 2
6
, 201
5; Acce
pted
No
vem
ber 1
0
,
2015
Adaptive Fuzzy Sliding Mode Control for a Class of
Nonlinear System
Xue Xiao
1, 2
, Zheng
3
, Don
g
Haobin
4,
*
1
Institute of Ge
oph
ys
ics an
d Geomat
ics, Ch
ina U
n
ivers
i
t
y
of Geoscienc
e
s
, W
uhan 43
00
74, Hub
e
i, Chi
n
a
2
Electrical & El
ectronic En
gin
eeri
ng Institue,
Nan
y
a
ng Instit
ute of T
e
chnol
og
y,
Nan
y
a
n
g
473
00
4, Hena
n,
Chin
a
3
Ph
y
s
ics & El
e
c
tronic Eng
i
ne
erin
g Col
l
eg
e, Nan
y
a
ng N
o
rm
al Univ
ersit
y
, N
a
n
y
a
ng 4
730
6
1
, Hena
n, Chi
n
a
4
School of Aut
o
matio
n
, Chin
a
Universit
y
of
Geoscie
n
ces, W
uhan 4
3
0
074
, Hubei, Ch
ina
e-mail: d
ong
hb
@cug.e
du.cn
A
b
st
r
a
ct
For a class of non
lin
ear system w
i
th par
a
m
eter
perturb
ati
on an
d extern
al distur
banc
e,
adaptiv
e
fu
z
z
y
co
ntrol c
an be us
ed to
appr
oach th
e system u
n
know
n functions to r
educ
e t
he cont
rol in
put an
d the
steady-state e
rror. And an
a
daptiv
e
sw
itch control g
a
i
n
w
hose ad
aptiv
e law
is decr
e
asin
g functio
n
is
desi
gned to w
eaken the syst
em
c
hatteri
ng,
the sw
itch ga
i
n
of esti
m
a
te
w
ill increase
on the basis
of the
original without decreas
ing
with t
he elim
ination
of interf
erenc
e.If system
is
interfer
enced
m
a
ny times.
Agai
nst the shortcomin
gs, this paper pro
p
o
s
es an
i
m
prov
ed ad
aptive l
a
w
that
can
w
eaken the syste
m
chatterin
g
effe
ctively w
h
ile
mainta
ini
ng th
e strong ro
bustn
ess. T
he si
mul
a
tion res
u
lts b
y
tests show
tha
t
this met
hod is
correct and eff
e
ctive.
Ke
y
w
ords
: Ad
aptive F
u
zz
y
C
ontrol, Integr
al
Slid
i
ng Mo
de, Nonl
in
ear System, Ro
bustn
es
s
Copy
right
©
2015 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
As a co
ntrol
method for
system
s whi
c
h
retain
stron
g
robu
stne
ss with the cha
r
a
c
teristics
of parametri
c uncertai
n
ties and
di
stu
r
ba
nce, slidin
g mode co
ntrol
techn
o
logy i
s
hig
h
lighte
d
by
control re
se
a
r
ch. In orde
r to achieve e
x
cellent
control effect, many research
e
r
s a
pply vari
ous
control theo
ri
es to
sliding
mode
control, su
ch
a
s
adaptive sli
d
i
ng mod
e
co
ntrol [1-3], fuzzy
slidin
g mod
e
control [4] a
nd neu
ral n
e
t
work sli
d
ing
mode
control [5]. Tracto
r stee
rin
g
an
gle
controlle
r is desig
ned t
h
rou
gh a
dap
tive sliding
mode
contro
l method in
quotation [
6
].
Outstan
d
ing
traje
c
tory tracking
an
d h
a
n
d
ling
st
ability
is
obtain
ed
and th
e effe
cts of p
a
ramet
e
r
pertu
rbatio
ns and exte
rnal
distu
r
ban
ce
s on the
sy
ste
m
ope
rability
can
be ove
r
come effe
ctively
[7]. Adaptive
integral sli
d
i
ng mode i
s
applie
d to
system
s with u
n
ce
rtain pa
ra
meters and t
he
effects of co
n
t
rol are a
c
hi
e
v
ed su
ccessf
ully
in quotation [8]. Feedb
ack linea
rization method a
nd
postu
re
co
ntrol hold
e
r co
mbining
with
variable
sli
d
i
ng mo
de
stru
cture
a
r
e d
e
signed
ba
sed
on
differential g
e
o
metry for
he
avy equipme
n
t aird
rop
pro
c
e
ss
motion
model
with th
e ch
aracte
rist
ics
of stron
g
cou
p
ling, stro
ng
nonlin
earity a
nd larg
e distu
r
ban
ce [9
-12]
.
Ho
wever, the
r
e i
s
a p
r
obl
e
m
in the a
b
o
v
e re
sea
r
ch
es. Th
e tradit
i
onal a
daptiv
e law i
s
non de
crea
si
ng function, the uncertai
n
ty is estimate
d to incre
a
se
in the origin
al basi
s
, not with
the estimate
d value of disturb
a
n
c
e an
d the di
sa
pp
eara
n
ce of gradu
ally decreasi
ng, buffe
ting
increa
sed
wit
h
the time
p
r
olongi
ng. Th
e metho
d
of
dead
time
ch
ara
c
teri
stic a
nd a
daptive l
a
w
nonlin
ear
ch
ara
c
teri
stics i
n
com
b
inatio
n, can be
tte
r solve the problem. The
basi
c
ide
a
of the
method i
s
, in
rea
c
hi
ng
m
ode
of the t
r
aditional
slidi
ng mo
de
co
ntrol, ad
aptive la
w g
uaran
tee
system
ca
n q
u
ickly re
ach the sli
d
ing
su
rface; when
th
e switchi
ng fu
nction val
ue
reache
s the
set
value, the ad
aptive law to
cha
nge, the
switchi
ng
fun
c
t
i
on a
s
the val
ue of the
switchin
g gai
n, the
slidin
g mod
e
cont
rol of
switchi
ng g
a
in
decrea
s
e
d
with the d
e
crease of switching fun
c
tion,
the
final driven b
y
the steady system e
n
ters the slidi
ng
mode.
This p
ape
r use
d
fuzzy slidin
g mode
cont
rol to solve a cla
s
s of strongly
cou
p
le
d
nonlin
ear sy
stem pa
ramet
e
r
pertu
rbatio
n an
d exte
rn
a
l
distu
r
ba
nce
by usi
ng th
e
adaptive i
n
te
gral
type. Throu
g
h
the de
sig
n
of integral ty
pe sli
d
i
ng
su
rface b
a
sed o
n
the e
rro
r, u
s
ing
ada
ptive to
the un
kno
w
n
function
s in
the sy
stem
of fuzzy
app
rox
i
mation of th
e switchi
ng
g
a
in ad
aptive l
a
w
desi
gn of
swi
t
ching
fun
c
tio
n
imp
r
oved f
u
zzy infe
ren
c
e sy
stem, we
ake
n
the
ch
a
ttering, an
d L
e
e
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No
. 4, Decem
b
e
r
2015 : 126
3 – 1269
1264
Jaa
p
Andria
n
o
f direct met
hod to prove
the stab
ility
of the system, the simulation re
sults
sho
w
that, the improved metho
d
effectively
wea
k
e
n
ing th
e adaptive la
w is non de
crea
sing
control
input ch
atteri
ng, so that the system
h
a
s good dynami
c
perfo
rma
n
ce and ro
bu
stn
e
ss.
2. Sy
stem Descriptio
n
a
nd Integral ty
pe
Sliding
Mode Surfac
e Design
Con
s
id
erin
g the SISO nonl
inear
system
(,
)
(
,
)
(
)
x
fx
t
g
x
t
u
t
d
t
(1)
In the a
bove
formula,
(,
)
f
xt
，
(,
)
g
xt
are
un
kno
w
n
no
nlinea
r fun
c
ti
ons,
and
0
(,
)
gx
t
,
()
dt
external inte
rfere
n
ce.Defi
n
ing the
system tra
c
kin
g
error i
s
et
x
t
r
t
, usin
g
the
system tra
c
ki
ng error fee
d
back
buil
d
sli
d
ing mod
e
su
rface
12
s
tK
E
t
e
t
k
e
t
k
e
t
So:
12
1
2
00
tt
s
te
k
e
k
e
d
x
t
r
t
k
e
k
e
d
(2)
Can
be
see
n
from the typ
e
(2
), the
system
h
a
s
co
nstru
c
ted
the
integral sli
d
ing mo
de
surf
a
c
e,
t
h
e
sy
st
em t
r
a
cki
ng e
rro
r d
e
p
end
s on
the
state fee
d
b
a
ck matrix
12
1,
,
Kk
k
, by
determi
ning suitable
1
k
and
2
k
, the tra
cki
ng e
r
ror
et
will be
cl
ose to
ze
ro,
and the
syste
m
will
have a goo
d dynamic p
e
rf
orma
nce.
3. Design of
an Ada
p
tiv
e
Fuzz
y
Slidin
g Mode Co
ntroller
3.1. Algorith
m
Design
For type (1
) nonlin
ear
system, said if
(,
)
f
xt
,
(,
)
g
xt
and
()
dt
as is
known,
0
st
st
can
be ba
sed o
n
the slidi
ng mo
de in the idea
l state of
the control law fo
r the cal
c
ulati
on of su
rface:
*1
12
(,
)
,
ut
g
x
t
f
x
t
d
t
r
t
k
e
t
k
e
t
(
3
)
If
g
(
x,
t
)an
d
d(t)
is un
kn
own
,
*
ut
is difficult to achi
eve, an
d a fuzzy system approa
ch
*
ut
is
use
d
.
The switchin
g
functio
n
s
t
is as the i
nput
of fuzzy
cont
rolle
r, which form
s a
singl
e
input fuzzy
approximatio
n system, fuzzy rule
s,
and
the fuzzy con
t
roller for:
Rule
i
：
IF
s
is
i
s
F
, THEN
u
is
i
(
4
)
And
i
=1
，，
，
，
，
23
…m
i
and
i
s
F
are
fuzzy
set
s
; by
centroid
met
hod to defuz
z
i
fic
a
tion the fuz
z
y
controller out
put will be:
TT
f
11
,/
mm
zi
i
i
ii
us
(
5
)
And
i
is the
i
-th rule’s
weight,
12
3
,
,
,
...,
m
,
12
3
,
,
,
...
,
m
,
i
is defined
to be:
1
/
m
ii
i
i
(6)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Adaptive F
u
zzy Slidin
g Mo
de Co
ntrol
for a Class of Nonline
a
r Syst
em
(Don
g Ha
obin)
1265
On the basi
s
of fuzzy app
roximation the
o
ry, there is
an optimal fu
zzy sy
stem
*
fz
,
us
that can
approa
ch
*
ut
.
**
*
T
fz
,(
)
ut
u
s
(
7
)
And the
is ap
proximate e
r
ror that meets
||
D
.If fuz
z
y
s
y
s
t
em
f
z
u
a
p
p
r
o
ac
hes
*
ut
, s
o
T
fz
ˆ
ˆ
,
us
(
8
)
And
ˆ
is the e
s
timated value
of
*
.The swit
ching control l
a
w to compe
n
sate the
error bet
ween
swit
chin
g con
t
rol law
ro
bu
stne
ss,
stro
n
g
on t
he app
roximation error,
d
e
fined as
the switch
ing
control law:
vs
sgn
ut
s
t
(
9
)
So the total c
ontrol rule of s
y
s
t
em (1) is
:
fv
s
z
ut
u
u
(10
)
In the swit
chi
ng co
ntrolle
r, becau
se of the uncertain
p
a
ram
e
ters
of the system a
nd the existe
nce
of interferen
ce, re
sulting in
the switchi
n
g gain
t
is diffi
cult to dete
r
mine, the a
c
t
ual control to
determi
ne, if
t
the value selected is too large, will
produce buff
e
ting larger,
if too small,
robu
st
syste
m
de
cline
a
n
d
tend
to
be
unsta
ble. In
o
r
de
r to
re
du
ce the
amo
unt
of
cal
c
ulatio
n can
be used to co
ntrol sy
stem from the la
w o
f
use to desi
g
n
t
, definition:
vs
ˆ
sgn
ut
s
t
(11
)
2
ˆ
ts
t
(
1
2
)
ˆ
t
is the swit
ch gain of estimate,
2
is the adaptive factor of the real nu
mber to
cha
r
a
c
teri
zet
he sp
eed of the adaptive
law with
spe
ed. The ada
ptive estimation error
can
be
defined a
s
:
ˆ
tt
D
(
1
3
)
Define
*
ˆ
, so formula (7)
can
be tran
sform
ed as:
**
T
ff
f
f
ˆˆ
zz
z
z
u
uuuu
(14
)
Put formula (2) into formul
a (3), an
d ca
n get:
1
*
1
,,
,,
ut
g
x
t
f
x
t
d
t
r
t
e
t
s
t
gx
t
g
x
t
u
t
s
t
(
1
5
)
So:
**
fv
s
,,
z
s
tg
x
t
u
t
u
t
g
x
t
u
u
u
t
(
1
6
)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No
. 4, Decem
b
e
r
2015 : 126
3 – 1269
1266
3.2. The Stabilit
y
Proof
Define Lya
p
u
nov function
as:
22
12
,,
0.5
22
T
gx
t
g
x
t
Vt
s
t
t
So:
12
*
f
12
T
12
T
vs
12
T
12
,,
,,
,
,,
,
,
ˆˆ
,,
,
ˆ
,
T
T
z
T
gx
t
g
x
t
V
t
st
st
t
t
gx
t
g
x
t
st
g
x
t
u
u
t
t
t
gx
t
g
x
t
st
g
x
t
t
t
gx
t
gx
t
s
t
s
t
g
x
t
u
t
D
t
gx
t
gx
t
s
t
t
ˆˆ
,,
Dt
t
s
t
g
x
t
s
t
g
x
t
(17
)
For sy
stem st
ability, fuzzy approximatio
n coeffi
ci
ent estimation u
s
i
ng the followi
ng algo
rithm
1
ˆ
s
t
(
1
8
)
Put the type (12) an
d (1
8) i
n
to type (17),
available:
ˆ
ˆ
,,
,
,,
,,
,0
V
t
t
s
tg
x
t
s
t
g
x
t
t
D
s
tg
x
t
s
t
g
xt
Ds
t
g
xt
s
t
g
x
t
D
s
t
g
x
t
Ds
t
g
x
t
(19
)
Becau
s
e
the type
(12
)
cha
r
acte
ri
zation of
t
he ad
aptive estim
a
tion l
a
w i
s
n
on d
e
c
re
asi
n
g
function, nam
ely the adapti
v
e law
ˆ
t
doe
s
not cha
nge
with the wea
k
e
n
ing or di
stu
r
ban
ce, ca
n
only in
cre
a
se
in the
o
r
igin
a
l
ba
sis,
and
t
he a
c
tual
sy
stem
subj
ect to
distu
r
b
a
n
c
es or pa
ram
e
ter is
variable,
with
the exten
s
io
n of time, the
switch
in
g a
d
aptive law is
deci
ded
by th
e co
ntrol
will
be
more a
nd mo
re large, chat
tering
will strengthe
n.
In orde
r to re
duce buffeting st
rength
syste
m
,
the method of
adaptive law
is used.
3.3. Impro
v
e
d
Adaptiv
e
C
ontrol La
w
Here d
r
a
w
on
the exp
e
rie
n
c
e
of metho
d
of
the
n
onlin
e
a
r cha
r
a
c
teri
stics of
de
ad zone
to
desi
gn ad
apti
v
e law to improve the form
ula (12
)
such as follo
ws:
2
ˆ
,
ˆ
,
ts
t
s
ts
n
s
(
2
0
)
Her
e
0
,
0
n
. The theoreti
c
al an
al
ysis is a
s
foll
ows:
(a)
Whe
n
the sy
stem state, fa
r distan
ce
sli
d
ing
surfa
c
e, namely the a
ppro
a
ch se
cti
on of sliding
mode
control, need to use
the cont
rol la
w ca
n
gua
ran
t
ee the syste
m
stability an
d the larg
er
th
e
s
y
s
t
e
m
to
th
e s
lid
in
g s
u
r
f
ac
e
,
th
en
th
e
ada
ptive law (12),
whi
c
h
ca
n g
uara
n
tee
th
e
system'
s
con
v
ergen
ce i
n
a larg
e ra
nge
, and the
con
t
rol input
stro
ng, ca
n qui
ckly force th
e
system
state into the slid
ing surface
set.
The festiv
al has proved the stability.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Adaptive F
u
zzy Slidin
g Mo
de Co
ntrol
for a Class of Nonline
a
r Syst
em
(Don
g Ha
obin)
1267
(b)
Whe
n
the
sta
t
e of the syst
em, has
retu
rned to
the sli
d
ing surfa
c
e, a
modified ad
aptive
law
to
control, can
make the
swi
t
ch gain decreased wi
th t
he decrease of t
he error. The stability
proof:
The Lyapu
no
v function
2
1
,
0.5
2
T
gx
t
Vt
s
t
(
2
1
)
By formula (1
7) and
(18
)
a
v
ailable:
T
1v
s
2
,/
,
ˆ
,s
g
n
,s
g
n
s
g
n
,
Vt
gx
t
s
t
s
t
g
x
t
u
st
g
x
t
t
s
t
st
g
x
t
s
t
s
t
n
s
t
g
x
t
s
t
n
st
st
(
2
2
)
The app
roxim
a
tion error g
o
e
s to ze
ro, so
t
he type (22) with the following
cha
nge
s:
22
,,
0
Vt
gx
t
s
t
n
s
t
s
t
gx
t
s
t
n
s
t
(23
)
(c)
Multiple di
stu
r
ban
ce
by t
he system,
ch
ange
s in the
system e
r
ror perfo
rman
ce
in S. Erro
r
increa
se o
r
decrea
s
e S wi
ll be in (a) a
nd (b
)
the bo
unda
ry line a
nd switchi
ng of two state
rep
r
e
s
entatio
n, eventually drove
conve
r
ge to zero, switchi
ng esti
mation value
of control g
a
i
n
also d
e
crea
sed.
(d)
In the ada
ptive law exp
r
e
ssi
on
(20
)
, d
e
term
ini
ng th
e gain
of swi
t
ch control th
e sp
eed
of
cha
nge, is b
enefici
a
l for the syste
m
to unce
r
tain in
terfere
n
ce su
ppre
s
sion, o
n
the oth
e
r
hand, takes
a long time to sup
p
re
ss the dist
u
r
ba
n
c
e. For
a small positive
numbe
r, ca
n
conve
r
ge into
the sliding m
ode to en
sure
system. As a
bound
ary value.
4. Verificatio
n and Simulation An
aly
s
is
Based
o
n
th
e tra
c
king
error state
fee
dba
ck
integ
r
al sli
d
ing
mo
de
su
rface,
use
the
followin
g
5 ki
nds of mem
b
er
ship func
tion of fuz
z
y
:
2
exp
/
6
/
/
2
4
NM
ss
(24
)
2
exp
/
12
/
/
24
NS
ss
(25
)
2
exp
/
/
2
4
ZO
ss
(26
)
2
ex
p
/
12
/
/
24
PS
ss
(27
)
2
ex
p
/
6
/
/
2
4
PM
ss
(28
)
Method to ve
rify this, con
s
ider the i
n
vert
ed pen
dulum
system, the f
o
llowin
g
eq
u
a
tion of
state are:
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No
. 4, Decem
b
e
r
2015 : 126
3 – 1269
1268
12
2
12
1
1
1
2
22
11
sin
c
os
sin
/
c
o
s
/
4/3
c
o
s
/
4
/3
c
o
s
/
cc
cc
xx
gx
m
l
x
x
x
m
m
x
m
m
x
ut
d
t
lm
x
m
m
l
m
x
m
m
(29
)
And in
(29
)
1
x
is
swi
ng a
n
g
le an
d
2
x
is swi
ng spe
ed,
2
9.8m
/
g
s
,
1k
c
mg
vehicle
quality
0.1
k
mg
as the
pend
ulum ro
d quality
0.5m
l
half the length
of pend
uluma
s
t
he control
input. Follow
the sinu
soid
a
l
signal, the positio
n co
m
m
and, the initial state
of the system, when
the system is: interferen
ce
; follo
w the step sign
al, the initial stat
e is applie
d for 0 se
cond
s, the
interferen
ce o
f
0.2 seco
nd
s.
The co
ntrol l
a
w by type (10), (1
8) an
d
(20), the con
t
roller p
a
ra
m
e
ters
1
1000
,
2
10
,
0.01
,
1
n
the simulation cu
rve as
shown fromFig
u
re 1 to 3.
(a) Po
sition tracking
sine
si
gnal waveform
(b)
cont
rol inp
u
t wav
e
form
Figure 1. Sinusoi
dal si
gnal
system to
follow an
d co
ntrol the input waveform
(a) T
he conventional meth
od pro
c
e
s
sin
g
step
waveform signal tracking
(b) T
he impro
v
ed method p
r
ocessin
g
ste
p
waveform s
i
gnal track
i
ng
Figure 2. The
system ste
p
with sig
nal waveform
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Adaptive F
u
zzy Slidin
g Mo
de Co
ntrol
for a Class of Nonline
a
r Syst
em
(Don
g Ha
obin)
1269
(a) Ada
p
tive para
m
eter
co
nventional
wav
e
for
m
(b) Ada
p
tive para
m
eters i
m
prove
d
wav
e
form
Figure 3. Adaptive trajecto
ry
5. Conclusio
n
Based
on th
e analysi
s
of
the adaptive
law of
traditi
onal ad
aptive
fuzzy slidi
n
g
mode
control m
e
th
od i
s
ove
r
co
me, imp
r
ovin
g meth
od i
s
put forwa
r
d.
The
nonlin
ea
r
cha
r
a
c
teri
sti
c
s of
applie
d to th
e ad
aptive l
a
w, effe
ctively solv
e th
e
adaptive l
a
w is
non
de
creasi
ng
defe
c
ts.
Simulation
re
sults sho
w
th
at the im
prov
ed meth
od
is
corre
c
t a
nd
effective. Whil
e
maintaini
ng t
he
origin
al ad
ap
tive fuzzy sli
d
ing mo
de
control
b
a
sed
on adva
n
tag
e
of su
ppressing distu
r
ba
nce,
new meth
od
obviously weaken the chattering,
re
duces the control inp
u
t, and achieve
the
adaptive la
w to estimate
the parame
t
ers
cha
nge
with the di
sturba
nce cha
nge
s, effecti
v
ely
solve
s
the problem
s of the previou
s
met
hod
s.
Referen
ces
[1]
Shui-C
hu
n Li, Cing-
Chic
h
T
s
ai,
Hs
u-Ch
ih H
uan
g.
No
nl
ine
a
r
Ada
p
tive S
l
i
d
i
ng-Mo
de
Co
ntrol D
e
si
gn for
T
w
o-W
heeled
Hu
ma
n T
r
ans
portatio
n
Ve
hi
cle
. Proceedings of IEEE Inte
rnational Co
nferenc
e o
n
S
y
stems, Man
and C
y
b
e
rn
etic
s (SMC 2009).
200
9, 196
5-19
70.
[2]
F
a
rzin Piltan.
Desig
n
Gradi
e
n
t Descent Op
timal Slid
in
g Mode C
ontro
l of
Contin
uum R
obots.
IAES
Internatio
na
l Journ
a
l of Ro
bo
tics and Auto
mation.
20
12, 1(
4): 175-1
89.
[3]
Gouich
iche A
b
delm
adj
id, Bo
ucher
it Moha
med Se
ghir, S
a
fa Ahme
d, Messlem Y
oucef
. Sensorl
e
s
s
Slidi
ng M
ode V
e
ctor Co
ntrol o
f
Induction M
o
tor Drives.
Inter
natio
nal J
our
n
a
l of Pow
e
r Ele
c
tronics a
n
d
Drive Syste
m
s
. 2012, 2(3): 2
7
7
-28
4
.
[4]
Haitao Liu, T
i
e Zhang. Fuzz
y
Sliding Mode Control of Robotic
Manipulators
w
i
th Kinematic and
D
y
namic
U
n
ce
rtainties.
J
our
nal
of
Dyna
mi
c Syste
m
s, M
easur
e
m
ent
a
nd
Contr
o
l
. 2
012,
13
4(6):
061
00
7 (Article
number).
[5]
Hua
ng
SJ, Hu
a
ng
KS, Chi
o
u
KC.
Dev
e
lo
pm
ent
a
nd ap
plic
a
t
ion of
a nov
el radi
al basis
fun
c
tion
s
lid
ing
mode co
ntroll
e
r
.
Mechatronic
s
. 2003, 13(
4): 313-
329.
[6]
Z
hou L
iji
e, W
ang Ne
ng
jia
n, Z
hang
Defu. A
dapt
iv
e sli
d
in
g
mode of tract
o
r aircraft s
y
stem.
Contro
l
theory an
d ap
p
licatio
n
. 20
12, 29 (4): 529-
53
4.
[7]
W
enlo
ng S
o
n
g
,
Yaqi
u
Liu,
Li
p
iing
Su
n.
Mo
de
l Ref
e
renc
e A
d
aptive
Integr
al-
t
ype Sl
idi
n
g
M
ode
Co
ntrol
Design for a class of Uncertain System
s
. Pro
c
eed
ing
of the
6th W
o
rld C
o
n
g
ress on Inte
lli
gent C
ontro
l
and Autom
a
tio
n
. 2006: 2
056-
206
0.
[8]
Li Dad
o
n
g
, Sun Xiu
x
i
a
, Do
ng W
enh
an, Xu Gu
angz
hi.
A sliding mo
de varia
b
l
e
structure he
av
y
equ
ipme
nt aird
rop lo
ng
itudi
na
l
control l
a
w
d
e
s
ign b
a
se
d o
n
feed
back l
i
ne
ari
z
ation.
C
ontrol theory
a
n
d
app
licati
ons
. 2
013, 30 (1): 5
4
-
60.
[9]
Liu W
enj
ia
ng, Sui Qingm
ei, Z
hou F
e
n
g
y
u, Xi
ao Hair
on
g. Adaptive fuzz
y
s
l
i
d
in
g mode co
n
t
rol for ship
course co
ntroll
er desi
gn b
a
se
d on.
Informati
on an
d contro
l
. 2012, 4
1
(2): 1
36-1
41.
[10]
Huo lo
ng,
L
e
Guiga
o
,
Ma D
a
w
e
i, Hu
J
i
a
n
.
T
he app
licati
on
of ad
aptiv
e
fuzz
y sli
d
in
g
mode
in t
h
e
positi
on servo
s
y
stem of rock
et launc
her.
F
i
re and co
mman
d
control
. 20
12
. 37(9): 137-1
4
0.
[11]
Liu Yu
nfen
g, Peng Yu
nh
ui
, Yang
Xi
aog
an
g, Miao Don
g
.
F
u
zz
y
slidi
n
g
mode control
of adaptiv
e
non
lin
ear
s
y
ste
m
s bas
ed
on
hig
h
g
a
i
n
o
b
se
rver.
Syste
m
s eng
ine
e
ri
ng an
d
e
l
ectron
ics
. 200
9,
3
1
(7):
172
3-17
27.
[12]
Liu Ji
nkun. Mat
l
ab sim
u
lati
on for slidi
ng mo
de
control. Beij
in
g:
T
s
inghu
a
Un
iversit
y
press,
200
5.
Evaluation Warning : The document was created with Spire.PDF for Python.