TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 12, Decembe
r
2014, pp. 81
9
3
~ 819
9
DOI: 10.115
9
1
/telkomni
ka.
v
12i12.63
32
8193
Re
cei
v
ed Ma
y 14, 201
4; Revi
sed O
c
tob
e
r 23, 201
4; Acce
pted No
vem
ber 1
2
, 2014
Improved Reaching Law Sliding Mode Control
Algorithm Design for DC Motor Based on Kalman Filter
Li Liu
Schoo
l of mechan
ical En
gi
ne
erin
g and A
u
to
mation,
Xih
ua
Univers
i
t
y
, C
h
i
n
a
E-mail: rub
y
l
l
@
163.com
A
b
st
r
a
ct
Aiming
at the inaccur
a
tely
m
o
deling and some
unc
ertain ex
isting
in servo system serious
ly
affected the c
ontrol q
u
a
lity and th
e
insta
b
ility
pro
b
le
m, slidi
ng mo
de
control a
l
g
o
rith
m w
i
th i
m
prov
e
d
reach
i
ng
l
a
w
i
s
pro
pose
d
in
this
pa
per. T
he
i
m
prov
ed
r
each
i
ng
l
a
w
is
use
d
to
w
eak
en th
e c
hatteri
ng
prob
le
m existi
n
g
in the sli
d
in
g mo
de co
ntrol. Also the kal
m
a
n
filter is us
ed
to inhi
bit the int
e
rferenc
e, w
h
ich
mak
e
the s
e
rv
o system
hav
e
strong a
n
ti-int
erferenc
e
ab
ilit
y and th
e ab
ili
ty of w
eakenin
g
the ch
atterin
g
prob
le
m
existin
g
in
the
sli
d
in
g
mo
de c
ontro
l.
T
he si
mu
la
ti
on
results s
how
that the
al
gorit
h
m
c
an
effective
l
y
inhi
bit th
e exte
rnal
distur
banc
e a
nd
nois
e
ex
isting
in
th
e sy
stem, a
nd
mak
e
the syste
m
h
a
ve stro
ng
anti
-
interfere
n
ce
ab
ility. At the
sa
me
ti
me, th
e c
hatterin
g
also
i
s
obv
ious
ly i
n
h
i
bite
d, an
d th
e
meth
od
makes
t
h
e
system stab
ility
and contro
l qu
a
lity be
en furth
e
r improv
ed.
Ke
y
w
ords
: DC m
o
tor, SMC, Kalm
an filter, im
prov
e
d
reach
i
ng l
a
w
,
chattering, si
mu
latio
n
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
Train control system, abb
reviated as T
C
S and it
s task i
s
to control the velocity [1] and
the distan
ce [2] betwee
n
trains in the rai
l
way and
to p
r
otect the safe and high ef
ficien
cy of train
runni
ng [3,
4]. The train
po
sition i
s
ve
ry
importa
nt in t
he T
C
S
simu
lation envi
r
on
ment. An id
e
a
l
TCS testing
a
nd sim
u
lation
platform sho
u
ld sim
u
late the re
al rail
wa
y operatin
g e
n
vironm
ent a
n
d
study the
TCS theory. In
testing
and
simulati
on
plat
form, differen
t
peopl
e h
a
ve the
differe
nt
con
c
e
r
n a
b
o
u
t the train,
su
ch a
s
the
policym
a
k
e
r
and researcher, etc. T
h
erefo
r
e, the
3D
visuali
z
ation
of train can b
e
sep
a
rate
d two a
s
pe
cts.
Sliding mod
e
variable st
ru
cture
cont
rol
is a sp
eci
a
l n
online
a
r cont
rol metho
d
, whi
c
h is
actually the d
i
scontinuity of cont
rol. The
control strate
gy is diffe
rent
from other
control meth
o
d
,
becau
se the
stru
cture
of system
is
mainly unfixed, but the system ca
n b
e
cha
nge
d in the
dynamic p
r
o
c
e
s
s a
c
cordi
ng to th
e
cu
rre
nt stat
e
of
the
system
whi
c
h
is
ch
ange
d by
so
me
destin
a
tion.
And the m
e
thod m
a
kes
system move
ac
cording
to
the state t
r
aj
ectory
of slid
ing
mode. Be
ca
use th
e
slidi
ng mo
de
ca
n be d
e
si
gn
ed an
d ha
s
nothing to
d
o
with th
e o
b
ject
para
m
eter an
d dist
urb
a
n
c
e, whi
c
h
ma
kes th
e va
ri
ab
le structu
r
e
contro
l
have q
u
ick respon
se,
and be n
o
t sensitive to the distu
r
ban
ce
and pa
ra
met
e
r chang
e [1]. But due to the influen
ce
of
factors
su
ch
as tim
e
d
e
lay
switch,
spati
a
l lag
swit
ch
and th
e
syste
m
ine
r
tia [2],
it is
difficult f
o
r
the sy
stem to
stri
ctly slidi
n
g alo
ng the
sl
iding
m
ode
fo
r the
bala
n
ce, but
cro
s
sing
back
and
fort
h
on b
o
th
side
s of the
sli
d
in
g mo
de
su
rfa
c
e to
p
r
od
uc
e chatteri
ng.
Due
to vib
r
ati
on, it is e
a
sy
to
motivate the
high frequ
e
n
cy mo
delin
g dynami
c
i
n
the
syste
m
, and d
a
m
age the
sy
ste
m
perfo
rman
ce.
In o
r
de
r to
wea
k
e
n
the
chatteri
ng
of
the
system,
Gao
Weibi
n
g p
r
op
osed t
h
e
rea
c
hin
g
law
to wea
k
en
ch
attering p
r
obl
em pro
d
u
c
ed
by variable st
ructu
r
e
control. The pape
r [3]
desi
gne
d a
variable
st
ru
cture
controll
er
with
a
filter, whi
c
h
can effe
ctively eliminate t
h
e
chatteri
ng of
control
sign
al. With the developm
ent
of artificial intelligen
ce, schol
ars have
pu
t
forwa
r
d the
method b
a
se
d on artifici
al
intelligen
ce
t
o
solve
the chattering pro
b
lems existin
g
in
the slidin
g
mode
contro
l, Such a
s
the fu
zzy method, neu
ra
l netwo
rk m
e
thod; Ge
ne
tic
optimizatio
n algorith
m
, etc. [1].
This p
ape
r h
a
s p
r
op
osed
a slidi
ng mo
d
e
co
ntrol al
g
o
rithm
with i
m
prove
d
re
a
c
hin
g
la
w
and kalman f
ilter techn
o
lo
gy for the DC se
rvo
syst
em, which ca
n make the
system have fast
tracking performance
and i
m
prove the
ability of weakening t
he chattering.
The si
mulation
results
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 12, Decem
ber 20
14 : 8193 – 81
99
8194
sho
w
that the
propo
se
d co
ntrol sche
me
can effe
ct
ively improve the
dynamic cha
r
acte
ri
stic of the
system, an
d has
stron
g
an
ti-interferen
c
e
ability and weaken the ch
attering abilit
y.
2. Sy
stem Descriptio
n
2.1. Mathem
atic Model
With the developme
n
t of the modern indu
st
ry, DC
motor is u
s
e
d
as the Executive
Termin
ation for the se
rvo system widely.
The ma
them
atical mod
e
l of the magnet
ic bru
s
hl
ess
DC
motor i
s
e
s
ta
blish
ed
ba
se
d on
the
working
p
r
inci
ple
of ma
gneti
c
bru
s
hle
s
s
DC motor [4]. T
he
workin
g pri
n
ciple and the
equivalent
circuit diag
ram
of DC moto
r
wa
s depi
cted
in the followi
ng
figure (entitle
d Figure 1).
Figure 1. Wo
rking p
r
in
ciple
and the eq
uivalent circuit di
agra
m
of DC
motor
From th
e
working
pri
n
ci
ple
of mag
netic
bru
s
hle
s
s
DC motor,
we
can obtai
n the
voltage
balan
ce e
qua
tion of DC mo
tor arm
a
ture
circuit:
aa
a
a
E
n
a
a
E
a
a
UE
R
I
C
I
R
K
n
R
I
(1)
The DC moto
r’s dyna
mic e
quation of is
given as follo
ws:
a
aE
a
a
a
di
uK
n
R
i
L
dt
(2)
In the Equation (2), th
e paramete
r
s
a
R
is the lo
op resi
stan
ce
，
a
I
is
the loop
cu
rrent
，
a
E
is the ind
u
cti
on ele
c
tro
m
ot
ive force
，
a
U
is t
he voltage of
the circuit
，
E
C
is the ele
c
tro
m
otive
forc
e c
o
ns
tant
，
n
is the motor’s spee
d
，
E
K
is the electromo
t
ive force whi
c
h is p
r
od
uce
d
by unit
spe
ed.
Balanced e
quation of ele
c
trodyn
a
mi
cs is as follo
ws:
eL
E
a
T
a
dn
TB
n
T
J
C
i
K
i
dt
(3)
Whe
r
e
e
T
is the instanta
neo
us electro
m
ag
n
e
tic torqu
e
，
L
T
is the load torque
，
B
is
dampin
g
co
efficient
，
J
is the moment of in
ertia
，
T
K
is
torque c
o
ns
tant.
Assu
ming the
initial conditi
ons i
s
ze
ro, the motor lo
ad
is con
s
tant, the equ
ation
s
are
transfo
rme
d
by the Lapla
c
e. The tran
sfer func
tio
n
of DC moto
r ca
n be obtain
e
d
as follows:
2
()
()
()
T
aa
a
a
T
E
K
ns
Gs
Us
L
J
s
R
J
L
B
s
R
B
K
K
(
4
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Im
proved
Re
achi
ng La
w Sliding Mod
e
Control Algo
rithm
Design F
o
r DC Motor… (Li Liu
)
8195
2. Design
for Discrete Sliding Mode Co
ntroller
2.1. Reachin
g La
w
Domestic exp
e
rts propo
sed
the reaching
law appr
oa
ch
to reduce or
inhibit the chatterin
g
of SMC in the
premise of e
n
suring
the
condition of sli
d
ing existence
0
S
S
has been
meet.
An
d
given four different meth
ods of reach
i
ng law,
such as con
s
tant reaching law, exponential
approa
ch law, power reach
i
ng law and general reac
hing law, where the expone
ntial approach is
applied widel
y [2], but the discrete
form
of exponent
ial reaching la
w’s switching
zone is zo
nal. So
which
can not
be clo
s
e to the origin ultimately but
a
chattering ne
ar the origin d
u
ring the pro
c
ess
moving [2
].
In order to solve the
chatteri
ng pheno
menon of exponential reaching law ne
ar the
origin [5, 6] propose
d
a va
riable rate re
aching la
w fo
r continuou
s
system, it’s discrete form
as
follows
:
))
(
sgn(
)
(
)
(
k
S
X
T
k
S
k
S
1
1
(
5
)
Where:
1
X
-- 1 norm of x.
Reachin
g
spe
eds of variable rate reachin
g
law is
1
X
, and is proportion
al to
1
X
.
T
he
switching
zo
ne pass thro
u
gh the origin
with two
rays, which
can
m
a
ke
S =
0 in t
he middle of t
h
e
two rays, can
be stabilized
at the origin, Howeve
r, when the syst
em just
entered to switchi
ng
zone
1
X
will ge
t a large val
ue, and have a big chattering in SMC. In
order overcome the
problem of th
e variable
rate rea
c
hing la
w and
expon
ential approa
ch law,[7]has propo
sed
a
new
reaching law
as follows:
))
(
sgn(
(
tan
)
(
)
(
)
(
)
k
S
X
sig
T
k
S
Tq
k
S
1
1
1
(
6
)
Whe
r
e:
1
1
1
1
1
1
2
1
X
X
e
e
X
sig
X
sig
))
(
)
(
(
tan
2.2. Design
of Sliding Model Con
t
rol
l
er
In this paper, in order to enhance th
e abilit
y of
anti-interference, and red
u
ce the
chattering of
VSC, we mad
e
use
improv
ed rea
c
hing
law an
d
kalma
n
filter to desi
gn the
control
l
er,
the structure of the controller was sho
w
n
in Figure 4
[8].
Figure 4. The
diagra
m
of controlle
r
Assu
ming G
(
s) i
s
the tra
n
sfer functio
n
of
servo
syste
m
, and its sta
t
e spa
c
e e
q
u
a
tion as
follow:
)
(
)
(
)
(
k
Bu
k
Ax
k
x
1
(
7
)
Whe
r
e:
T
k
x
k
x
k
x
(
)
(
)
(
2
1
)
(
k
x
1
-- The a
c
tual velocity.
)
(
k
x
2
-- The
ch
angin
g
rate of
velocity.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 12, Decem
ber 20
14 : 8193 – 81
99
8196
Suppo
sed
)
(
k
r
,
)
(
k
dr
a
s
the
velo
city orde
r
and
it’s chan
ge
rate.
)]
(
),
(
[
k
dr
k
r
k
R
,
predi
cted by l
i
near extrapol
ation
1
k
R
is:
)
(
)
(
)
(
1
2
1
k
r
k
r
k
r
(
8
)
)
(
)
(
)
(
1
2
1
k
dr
k
dr
k
dr
(
9
)
Define
swit
ch
ing functio
n
is:
))
(
)
(
(
k
x
R
C
E
C
k
S
k
e
e
(
1
0
)
)]
(
)
(
[
))
(
(
)
(
k
Bu
k
Ax
R
C
k
x
R
C
E
C
k
S
k
e
k
e
e
1
1
1
1
(
1
1
)
The novel rea
c
hin
g
law’
s di
screte form is:
))
(
sgn(
(
tan
)
(
)
(
)
(
)
k
S
X
sig
T
k
S
Tq
k
S
1
1
1
(
1
2
)
From the a
b
o
v
e equation.
The slidi
ng m
ode control la
w of se
rvo sy
stem is:
))]
(
sgn(
(
tan
)
(
)
(
)
(
[
)
(
)
(
)
k
S
X
sig
T
k
S
Tq
k
Ax
C
R
C
B
C
k
u
e
k
e
e
1
1
1
1
(13)
Where:
]
,
[
1
c
C
e
3. Kalman Filtering Oper
ator
The co
ntinuo
us mod
e
l of control sy
stem
is c
onve
r
ted
into a discret
e
model, the discrete
state equ
atio
n and mea
s
u
r
ement equ
ation are give
n as:
)
(
)
(
)
(
))
(
)
(
(
)
(
)
(
k
v
k
Cx
k
y
k
w
k
u
B
k
Ax
k
x
v
1
(14)
Whe
r
e
)
(
k
x
and
)
(
k
y
v
re
spectively are
the state ve
ctor an
d ob
se
rvation vecto
r
.
A
is the
state mat
r
ix,
B
is the
co
ntrol
matrix,
C
is th
e output
ob
servation m
a
trix,
)
(
k
w
is the
process
noise sig
nal,
)
(
k
v
i
s
the ob
serva
t
ion noise sig
nal.
The flow
cha
r
t of kalman filtering alg
o
rith
m is sh
own in
Figure 5 [9]:
|1
ˆ
kk
x
ˆ
k
x
k
y
Figure 5. The
flow cha
r
t of kalma
n
filteri
n
g
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Im
proved
Re
achi
ng La
w Sliding Mod
e
Control Algo
rithm
Design F
o
r DC Motor… (Li Liu
)
8197
The re
cu
rsiv
e algorithm o
f
the corresp
ondin
g
ka
lm
a
n
filter of flo
w
ch
art is giv
en as [9,
10]:
The be
st esti
mation is:
1
1
1
k
k
k
k
x
A
x
ˆ
ˆ
|
；
(
1
5
)
The erro
r vari
ance of expe
ct estimate
s i
s
:
T
k
k
T
k
k
k
k
k
QB
B
A
P
A
P
1
1
1
1
1
1
|
；
(
1
6
)
The kal
m
an g
a
in is:
1
]
)
(
[
)
(
R
C
k
Cp
C
k
p
K
T
T
k
；
(
1
7
)
The upd
ate e
s
timation is
：
)
ˆ
(
ˆ
ˆ
1
1
1
1
k
k
v
k
k
k
k
x
CA
y
K
x
A
x
；
(
1
8
)
The
upd
ated estimate covarian
ce
i
s
：
1
k
k
k
k
P
C
K
I
P
|
)
(
;
(
1
9
)
The output of
filter is
：
k
e
x
C
y
ˆ
；
(
2
0
)
Whe
r
e Q, R a
r
e the covaria
n
ce mat
r
ix of
rand
om noi
se
w(k),v(k) respectively.
4. Numerical
Simulation
In orde
r to verify the effectiveness of the
controller, u
s
e MATLAB to make
simul
a
tion for
the DC. the servo sy
stem’s status eq
uati
on [8]:
)
(
)
(
)
(
))
(
)
(
(
)
(
)
(
k
v
k
Cx
k
y
k
w
k
u
B
k
Ax
k
x
1
(21)
Whe
r
e:
9753
0
0
0010
0
0
1
.
.
.
A
；
1314
0
000
0
.
.
B
；
0
1
C
；
0
D
；
)
(
k
w
is the
pro
c
e
s
s white noi
se
sign
al;
)
(
k
v
is the
observatio
n
white n
o
ise signal. In
orde
r to
prove the
effect of
sy
stem,
the
system
with
kalman filte
r
a
nd the
sy
ste
m
witho
u
t kal
m
an
filter are
re
spe
c
tive
ly simulate
d. The
simulation p
a
ram
e
ters
are
given
as:
40
,
130
,
280
cq
,
[
0
.5
,
0
.5
]
x
,
05
0
.
.
The sim
u
latio
n
results a
r
e
given in the Figure 6
-
Fig
u
re 9
：
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 12, Decem
ber 20
14 : 8193 – 81
99
8198
Figure 6. Position trajecto
ry without filter
Figure 7. Position trajecto
ry with filter
Figure 8. Phase traje
c
to
ry without filter
Figure 9. Phase traje
c
to
ry with filter
The Figu
re 6
-
Figu
re 7 is t
he po
sition trajec
to
ry of the Figure 8-Fi
gure
9
is the
pha
se
trajecto
ry. From the simul
a
tion re
sults (Figure 6-Fi
gu
re 9)
we ca
n obtain the co
nclu
sio
n
s that
the
external
distu
r
ban
ce
an
d n
o
ise
of sy
ste
m
after a
ddin
g
kalman
filter can
be effe
ctively restrain
ed,
the ch
atterin
g
of the slidi
ng mod
e
vari
able st
ru
cture co
ntrol i
s
also in
hibited
,
and ca
n be
tter
reali
z
e the fu
nction of the
controlle
r.
5. Conclusio
n
Aim at the in
accurately modelin
g and some un
certai
n existing in servo syste
m
seri
ou
sly
affected the
control qu
ality and the in
stability pro
b
l
e
m, the slidi
ng mod
e
vari
able st
ru
cture is
applie
d to th
e
se
rvo
syste
m
. Co
nsid
eri
ng the
cha
tte
ring problem
of
se
rvo slidi
ng
m
ode
vari
able
stru
cture co
n
t
rol and the existing interf
eren
ce,
which affect the quality and stability of system
control, a rea
c
hin
g
la
w ap
p
r
oa
ch i
s
u
s
e
d
to wea
k
e
n
th
e ch
attering
p
r
oble
m
existi
ng in the
slidi
n
g
mode control. At the same time, kalman
filter is us
e
d
to inhibit the interferen
ce, thus to imp
r
ov
e
the quality and stability of servo sy
stem. Syst
em simulation
s sho
w
that the schem
e can
effectively su
ppre
s
sed ext
e
rnal di
stu
r
b
ance and
n
o
i
s
e, whi
c
h m
a
ke
s the sy
stem have strong
anti-inte
rfere
n
ce
ability. And the
chatte
ring of
the
sli
d
ing mo
de v
a
riabl
e st
ru
cture
cont
rol al
so
had
bee
n o
b
v
iously in
hibi
ted, the
syst
em
stab
ility
and co
ntrol quality
ha
s a
l
so bee
n
furt
her
improve
d
.sta
bility and cont
rol quality be
en furthe
r improved.
Referen
ces
[1]
LIU Jin-k
un, S
UN F
u
-chu. R
e
searc
h
an
d d
e
ve
l
opme
n
t on
theor
y a
nd
al
gorithms
of sli
d
in
g mod
e
control.
Co
ntrol
T
heory and A
pplic
atio
ns.
20
07,24(
3):40
7
[2]
W
B
Gao.
T
heor
y
fou
ndati
on o
f
Variabl
e Structur
e Contro
l, Beiji
ng: Scie
nc
e and T
e
chno
l
o
g
y
Pr
ess of
Chin
a,19
90.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Im
proved
Re
achi
ng La
w Sliding Mod
e
Control Algo
rithm
Design F
o
r DC Motor… (Li Liu
)
8199
[3]
YANADA H, OHNISHI H. F
r
eque
nc
y-sha
p
e
d
slid
in
g
mod
e
control of a
n
electro
h
y
dra
u
li
c servo-motor
.
J of Systems a
nd Co
ntrol a
n
d
Dyna
mics.
19
99; 213(
1): 441
– 448.
[4]
HL, FQW, HCX
. Design
for
Mecha
n
trical In
tegratio
n S
y
ste
m
. Beiji
ng: Hi
g
her e
ducati
on
press. Bei
jin
g.
201
1.
[5]
MSS Ahme
d, Z
Ping, W
Y
J
i
e, Mod
i
fie
d
sl
idin
g m
ode
co
ntroll
er
w
i
th
e
x
te
nde
d Ka
lm
an F
ilter
fo
r
stochastic s
y
stems. IEEE inter
national conf
erenc
e on co
ntrol and autom
ation.
2007: 630-635, .
[6]
KB Park, JJ L
ee. Sli
d
i
ng m
o
de c
ontrol
l
er
with filter
e
d
s
i
gn
al for r
o
b
o
t ma
nip
u
lators
usi
n
g virtua
l
pla
n
t
control
l
er.
J of Mechatro
nics
. 199
7; 7: 277-2
86.
[7]
KW
T
ong, X Z
han
g, Y Z
hang
, Z
Xie, RX Ca
o.
Slidi
ng Mod
e
Varia
b
le Stru
cture Contro
l o
f
Perman
en
t
Magn
et Synchr
ono
us Mach
ine
Based o
n
a N
o
vel R
eac
hin
g
Law
.
J of Proceedings
of the CSEE. 2008;
8: 102-1
06.
[8]
GAO Cun-che
n
, LIU Yu
n-lo
n
g
, LI Yun-
ya
n.
A reac
hin
g
-la
w
m
e
tho
d
for
uncerta
in
discr
ete vari
ab
le-
structure contr
o
l s
y
st
ems. Co
ntrol T
heor
y
&
Appl
icatio
ns. 2
009; 26(
7): 781
-785.
[9]
JK Liu. MAT
L
AB Simulati
on fo
r Slidin
g Mod
e
Contro
l. Beiji
ng
:
T
s
inghua U
n
i
v
ersit
y
Press,
Beiji
n. 200
5.
[10]
Z
Z
H
ANG You-w
e
i, An Introd
ucti
on for W
i
n
e
r and ka
lman
filter theor
y
；
Beiji
ng: Peo
p
l
e
'
s
Educatio
n
Press 198
0.
Evaluation Warning : The document was created with Spire.PDF for Python.