Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
6,
No.
5,
October
2016,
pp.
2239
–
2250
ISSN:
2088-8708
2239
Sliding-Mode
Contr
oller
Based
on
Fractional
Order
Calculus
F
or
a
Class
of
Nonlinear
Systems
Nour
eddine
Bouarr
oudj
*
,
Djamel
Boukhetala
*
,
and
F
ar
es
Boudjema
*
*
LCP
,
Ecole
Nationale
Polytechnique,
10
a
v
.
Hassen
Badi,
BP
.
182,
El-Harrach,
Alger
,
Algeria
Article
Inf
o
Article
history:
Recei
v
ed
Apr
12,
2016
Re
vised
July
18,
2016
Accepted
Aug
7,
2016
K
eyw
ord:
fractional
order
culculus
SMC
FOSMC
in
v
erted
pendulum
ABSTRA
CT
This
paper
presents
a
ne
w
approach
of
fractional
order
sl
iding
mode
controllers
(FOSMC)
for
a
class
of
nonlinear
systems
which
ha
v
e
a
single
input
and
tw
o
outputs
(SIT
O).
Firstly
,
tw
o
fractional
order
sliding
surf
aces
S
1
and
S
2
were
proposed
with
an
intermediate
v
ariable
z
transferred
from
S
2
to
S
1
in
order
to
hierarch
y
the
tw
o
sliding
surf
aces.
Secondly
,
a
control
la
w
w
as
determined
in
order
to
control
the
tw
o
outputs.
A
sliding
control
stability
condition
w
as
obtained
by
using
the
properties
of
the
fractional
order
calculus.
Finally
,
the
ef
fecti
v
eness
and
rob
ustness
of
the
proposed
approach
were
demonstrated
by
comparing
its
performance
with
the
one
of
the
con
v
entional
sliding
mode
controller
(SMC),
which
is
based
on
inte
ger
order
deri
v
ati
v
es.
Simulation
results
were
pro
vided
for
the
case
of
controlling
an
in
v
erted
pendulum
system.
Copyright
c
2016
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Noureddine
Bouarroudj
LCP
,
Ecole
Nationale
Polytechnique,
10
a
v
.
Hassen
Badi,
BP
.
182,
El-Harrach,
Alger
,
Algeria
Email:
autonour@gmail.com
1.
INTR
ODUCTION
Sliding
mode
control
has
lar
gely
pro
v
ed
its
ef
fecti
v
eness
in
numerous
applications,
see,
e.g.,
the
studies
by
Utkin
[1]
and
Slotine
[2].
The
first
step
of
the
SMC
design
is
to
select
a
sliding
surf
ace
that
models
the
desired
closed-
loop
performance
in
state
v
ariable
space.
In
the
second
step,
equi
v
alent
and
hitting
control
la
ws
are
designed
such
that
the
system
state
trajectories
are
forced
to
w
ards
the
sliding
surf
ace
and
slide
along
it
to
the
desired
attitude.
An
adv
antage
of
these
methods
of
control
(SMC)
is
their
rob
ustness
to
parameters
v
ariation
and
bounded
e
xternal
disturbances.
The
rob
ustness
is
attrib
uted
to
the
discontinuous
term
in
the
control
input.
Ho
we
v
er
,
this
discontinuous
term
also
causes
an
undesirable
ef
fect
called
chattering.
Sometimes
this
discontinuous
control
action
can
e
v
en
cause
instabili
ty
of
the
system.
This
ef
fect
can
be
alle
viated
by
,
for
e
xample,
introducing
a
sat
function
and
taking
of
f
the
sgn
function
in
the
hitting
control
la
w
of
the
SMC.
In
the
last
years,
the
control
of
single-input-tw
o
output
non-linear
dynamical
systems
has
risen
s
ome
interest
in
the
control
research
community;
in
general,
the
controller
(input)
is
done
for
the
trajectory
tracking
of
the
tw
o
out-
puts.
PID
controllers
were
applied
in
[3]
to
the
stabilization
and
tracking
control
of
three
types
of
in
v
erted
pendulum.
A
PID+LQR
method
w
as
gi
v
en
in
[4],
in
which
the
LQR
w
as
added
ne
g
ati
v
ely
to
the
PID
in
order
to
ha
v
e
a
resultant
optimal
control.
A
fuzzy
controller
with
estimation
of
scali
ng
f
actors
w
as
studied
in
[5].
An
intelligent
control
system
based
on
an
interv
al
type-2
fuzzy
PD
controller
w
as
studied
in
[6].
The
SMC
design
for
this
kind
of
systems
is
a
v
ery
challenging
task.
Such
de
v
eloped
controllers
include
the
decoupled
SMC
sho
wn
in
[7],
a
Fuzzy
Sliding
Mode
Con-
trol
(FSMC)
that
w
as
tuned
using
ant
colon
y
optimization
[8],
[9]
and
a
decoupled
SM
with
a
fuzzy-neural
netw
ork
controller
[10]
where
the
fuzzy-neural
netw
ork
w
as
used
to
approximate
an
ideal
computational
controller
.
All
these
de
v
eloped
approaches
are
based
on
the
standard
inte
ger
order
calculus.
The
fractional
order
dif
ferentiation
theory
,
which
has
300
years
of
history
and
deals
with
deri
v
ati
v
es
and
inte
grals
of
non-inte
ger
order
,
has
recently
been
redisco
v
ered
by
scientists
and
engineering,
being
applied
in
man
y
fields
and,
among
them,
t
he
field
of
control.
One
of
the
first
applicat
ions
of
fractional
order
deri
v
ati
v
es/inte
grals
to
the
control
of
dynamic
systems
w
as
gi
v
en
by
Oustaloup,
who
de
v
eloped
the
so
called
Commande
Rob
uste
dOrdre
Non
Entier
(CR
ONE)
,which
is
described
in
[11,
12],
along
with
e
xamples
of
application
in
v
arious
fields.
Fractional
order
J
ournal
Homepage:
http://iaesjournal.com/online/inde
x.php/IJECE
Evaluation Warning : The document was created with Spire.PDF for Python.
2240
ISSN:
2088-8708
PID
(FOPID)
controllers
ha
v
e
been
studied
in
[13].
A
fuzzy
FOPID
controller
,
which
uses
the
error
and
its
fractional
order
deri
v
ati
v
e
as
its
input,
and
has
a
fractional
order
inte
grator
in
its
output,
has
been
proposed
in
[14].
Also
in
the
SMC
field,
some
researchers
ha
v
e
introduced
the
fractional
order
calculus
to
design
a
fractional
order
sliding
mode
control
(FOSMC).
The
first
application
using
the
technique
of
frac
tional
order
SMC
returns
to
Calderon
[15,
16]
on
a
DC-DC
po
wer
con
v
erter;
where
the
authors
defined
switching
surf
aces
based
on
the
structures
of
fractional
order
PI
(FOPI)
and
FOPID
controllers.
The
approach
which
uses
the
FOPID
sliding
surf
ace
w
as
also
studied
in
[17,
18].
A
FOSMC
based
on
a
fractional
order
PD
(FOPD)
sliding
surf
ace
w
as
studied
in
[19,
20]
and
a
Fractional
Order
T
erminal
Sliding
Mode
Control
(FO
TSMC)
w
as
proposed
in
[21].
Moti
v
ated
by
the
abo
v
e
discussion,
this
paper
proposes
a
FOSMC
for
a
class
of
single
input
tw
o
outputs
non-linear
systems.
The
control
la
w
is
deri
v
ed
from
these
proposed
fractional
order
sliding
surf
aces
and
its
purpose
is
to
dri
v
e
the
tw
o
outputs
of
the
system
to
their
desired
trajectories.
The
stability
is
guaranteed
by
using
the
L
yapuno
v
theorem.
Simulation
results
illustrate
that
the
proposed
control
design
method
applied
to
the
in
v
erted
pendulum
plant
yield
controller
that
can
rapidly
stabilize
these
system
compared
with
the
con
v
entional
inte
ger
order
controller
.
This
performance
is
guaranteed
through
the
added
e
xtra
parameter
(fractional
order).
The
rest
of
this
article
is
or
g
anized
as
follo
ws.
A
brief
introduction
to
fractional
calculus
is
described
in
Section
2.
System
description
and
control
design
are
de
v
eloped
in
Section
3.
Simulation
results
and
s
ome
conclusions
are
gi
v
en
in
Sections
4
and
5
respecti
v
ely
.
2.
A
BRIEF
INTR
ODUCTION
T
O
FRA
CTION
AL
CALCULUS
The
fractional
dif
fero-inte
gral
operators
are
denoted
by
a
D
t
f
(
t
)
(fractional
calculus),
where
a
and
t
,
are
the
bounds
of
the
operation
and
2
R
is
a
generalization
of
the
standard
inte
gration
and
dif
ferentiation
to
non-inte
ger
order
operators.
The
continuous
dif
fero-inte
gral
operator
is
gi
v
en
by
the
follo
wing:
a
D
t
=
8
<
:
d
dt
f
or
0
1
f
or
=
0
R
t
a
(
d
)
f
or
0
(1)
In
the
literature
W
e
can
find
dif
ferent
definitions
of
the
fractional
dif
fer
-inte
gral
operator
.
But
the
most
commonly
used
definitions
of
fractional
order
deri
v
ati
v
es
are:
The
Riemann-Liouville
(RL)
definition:
a
D
t
f
(
t
)
=
1
(
m
)
d
dt
m
Z
t
a
f
(
)
(
t
)
1
(
m
)
d
(2)
The
Caputos
definition:
a
D
t
f
(
t
)
=
1
(
m
)
Z
t
a
f
m
(
)
(
t
)
1
(
m
)
d
(3)
In
these
e
xpressions,
m
1
<
<
m
,
and
(
:
)
is
the
well-kno
wn
Eulers
g
amma
function:
(
x
)
=
Z
1
0
e
t
t
(
x
1)
dt;
x
>
0
(4)
On
the
other
hand,
Grunw
ald-Letnik
o
v
(GL)
reformulated
the
definition
of
the
fractional
order
dif
fer
-inte
gral
operator
as
follo
ws:
a
D
t
f
(
t
)
=
l
im
h
!
0
1
h
(
t
a
)
=h
X
k
=0
(
1)
k
k
f
(
t
k
h
)
(5)
Since
the
numerical
simulation
of
a
fractional
dif
ferential
equation
is
not
as
simple
as
an
ordinary
dif
ferential
equation,
the
Laplace
transform
method
is
often
used
as
a
tool
for
solving
fractional
order
dif
ferential
equations
that
arise
in
engineering
applications
[22],
[23].
The
Laplace
transforms
of
the
pre
vious
fractional
order
deri
v
ati
v
es
are
gi
v
en
belo
w
.
IJECE
V
ol.
6,
No.
5,
October
2016:
2239
–
2250
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
2241
The
Laplace
transform
of
the
(RL)
definition
is
as
follo
ws
[22],
[24]:
L
f
0
D
t
f
(
t
);
s
g
=
s
F
(
s
)
(
m
1)
X
k
=0
s
k
h
0
D
(
k
1)
t
f
(
t
)
i
t
=0
(6)
The
Laplace
transform
of
the
Caputos
definition
is
gi
v
en
by
[24]:
L
f
0
D
t
f
(
t
);
s
g
=
s
F
(
s
)
(
m
1)
X
k
=0
s
(
k
1)
f
k
(0)
(7)
Where
s
denotes
the
Laplace
operator
.
F
or
zero
initial
conditions,
the
Laplace
transforms
of
the
fractional
order
deri
v
ati
v
es
of
Riemann-Liouville
and
Caputo
are
reduced
to
(8)
[24],
[25].
L
(
0
D
t
f
(
t
))
=
s
F
(
s
)
(8)
In
this
paper
,
the
fractional
order
element
s
is
approximated
by
the
Oustaloups
filter
.
The
Oustaloups
filter
[26]
approximates
G
(
s
)
=
s
;
2
R
(9)
by
a
rational
function
of
the
form
^
G
(
s
)
=
K
N
Y
k
=
N
s
+
w
0
k
s
+
w
k
(10)
where
its
parameters
(zeros,
poles,
and
g
ain)
are
determined
by
the
follo
wing
formulas:
w
0
k
=
w
b
:
w
h
/
w
b
(
k
+
N
+0
:
5(1
))
=
(2
N
+1)
w
k
=
w
b
:
w
h
/
w
b
(
k
+
N
+0
:
5(1+
))
=
(2
N
+1)
K
=
w
h
(11)
In
these
e
xpressions,
(
2
N
+
1
)
is
the
order
of
the
filter
and
wb
and
wh
are
respecti
v
ely
the
lo
w
and
high
transient-frequencies.
In
the
follo
wing,
some
properties
of
the
Caputos
definition
of
fractional
order
deri
v
at
i
v
es
that
will
be
used
in
this
paper
are
gi
v
en.
The
belo
w
desirable
properties
are
v
alid
under
causality
and
when
the
function
is
dif
fer
-inte
grated
taking
as
starting
point
the
point
when
the
function
starts
[24]:
-
Fractional
order
deri
v
ati
v
e
of
the
fractional
order
inte
gration
of
the
function
f
(
t
)
:
a
D
t
(
a
D
t
f
(
t
))
=
f
(
t
)
(12)
-
Fractional
order
inte
gration
of
the
fractional
order
deri
v
ati
v
e
of
the
function
f
(
t
)
:
a
D
t
(
a
D
t
f
(
t
))
=
f
(
t
)
m
1
X
k
=0
f
k
(
a
)
k
!
(
t
a
)
k
(13)
-
The
fractional
order
deri
v
ati
v
e
is
a
linear
operator:
a
D
t
(
f
(
t
)
+
g
(
t
))=
a
D
t
f
(
t
)+
a
D
t
g
(
t
)
(14)
-
The
fractional
order
inte
gration
is
a
linear
operator:
a
D
t
(
f
(
t
)
+
g
(
t
))=
a
D
t
f
(
t
)+
a
D
t
g
(
t
)
(15)
Evaluation Warning : The document was created with Spire.PDF for Python.
2242
ISSN:
2088-8708
3.
SYSTEM
DESCRIPTION
AND
CONTR
OL
DESIGN
3.1.
System
description
Consider
the
single-input
tw
o-output
non-linear
system,
which
can
be
represented
in
the
follo
wing
state
space
form:
_
x
1
=
x
2
_
x
2
=
f
1
(
x
)
+
b
1
(
x
)
u
+
d
1
(
t
)
_
x
3
=
x
4
_
x
4
=
f
2
(
x
)
+
b
2
(
x
)
u
+
d
2
(
t
)
(16)
Where
X
=
[
x
1
;
x
2
;
x
3
;
x
4
]
T
is
the
state
v
ector
,
f
1
(
x
)
,
f
2
(
x
)
and
b
1
(
x
)
,
b
2
(
x
)
are
nonlinear
functions,
u
is
the
control
input,
d
1
(
t
)
,
d
2
(
t
)
are
bounded
e
xternal
disturbances.
Moreo
v
er
assume
that
y
=
[
x
1
;
x
3
]
T
is
the
output
v
ector
.
Assumption
1
:
The
bounded
e
xternal
disturbances
satisfy
the
follo
wing
inequalities:
j
d
1
(
t
)
j
1
(17)
j
d
2
(
t
)
j
2
(18)
D
(
1)
t
d
1
(
t
)
1
(19)
D
(
1)
t
d
2
(
t
)
2
(20)
Where
1
,
2
,
1
and
2
are
kno
wn
positi
v
e
constants.
3.2.
Sliding
Mode
Contr
oller
(SMC)
Design
The
principle
of
the
proposed
methodology
to
design
the
SMC
for
system
(16)
is
the
follo
wing:
-
Decouple
the
global
system
into
tw
o
subsystems.
The
first
one
contains
the
states
x
1
,
x
2
,
and
a
sliding
surf
ace
S
1
=
_
e
1
+
:e
1
is
defined
for
it,
where
e
1
=
x
1
x
1
d
and
1
is
a
positi
v
e
constant.
The
second
subsystem
contains
the
state
s
x
3
,
x
4
,
and
a
sliding
surf
ace
S
2
=
_
e
3
+
:e
3
is
also
defined
for
it,
where
e
3
=
x
3
x
3
d
and
2
is
another
positi
v
e
constant.
Note
that
the
control
ob
j
ecti
v
e
is
to
force
a
motion
of
the
states
of
the
first
subsystem
to
w
ards
the
sliding
surf
ace
S
1
=
0
,
con
v
er
ging
therefore
to
x
1
=
x
1
d
,
x
2
=
_
x
1
d
.
The
second
objecti
v
e
is
to
force
a
motion
of
the
states
of
the
second
subsystem
to
w
ards
the
sliding
surf
ace
S
2
=
0
,
con
v
er
ging
therefore
to
x
3
=
x
3
d
,
x
4
=
_
x
3
d
.
-
The
use
of
a
control
signal
u
=
u
1
calculated
from
the
sliding
surf
ac
e
S
1
causes
the
con
v
er
gence
of
only
the
states
x
1
,
x
2
to
their
desired
v
alues.
And
the
use
of
a
control
signal
u
=
u
2
calculated
from
the
sliding
surf
ace
S
2
causes
the
con
v
er
gence
of
only
the
states
x
3
,
x
4
to
their
desired
v
alues.
In
order
to
achie
v
e
simultaneous
control
of
all
the
states,
we
tak
e
into
account
the
idea
of
[8]
that
consists
in
using
an
intermediate
v
ariable
z
between
the
sliding
surf
aces.
This
v
ariable
represents
the
information
transferred
from
S
2
to
S
1
,
and
saturates
the
error
e
1
to
k
2
.
Thus
the
sliding
surf
ace
S
1
is
modified
to
S
1
=
_
e
1
+
1
:
(
e
1
z
)
(21)
While
the
sliding
surf
ace
S
2
k
eeps
its
standard
form
S
2
=
_
e
3
+
2
:e
3
(22)
-
This
modification
changes
the
control
objecti
v
e
from
making
X
=
X
d
=
[
x
1
d
;
x
2
d
;
x
3
d
;
x
4
d
]
T
to
making
X
=
^
X
d
=
[
x
1
d
+
z
;
x
2
d
;
x
3
d
;
x
4
d
]
T
where
z
=
:
sat(
S
2
=
2
)
:k
2
;
0
<
k
2
<
1
(23)
And
the
definition
of
the
sat
(
:
)
function
is:
sat(
S
2
=
2
)
=
(
S
2
=
2
)
if
j
S
2
=
2
j
<
1
sgn(
S
2
=
2
)
if
j
S
2
=
2
j
1
(24)
Being
2
is
the
boundary
layer
of
the
sliding
surf
ace
S
2
.
Remark
:
If
S
1
=
0
,
then
we
obtai
n
from
equation
(21)
that
x
1
=
x
1
d
+
z
and
x
2
=
_
x
1
d
.
On
the
other
hand,
if
S
2
6
=
0
,
equation
(23)
sho
ws
that
then
we
ha
v
e
z
6
=
0
.
Consequently
,
we
must
pursue
that
the
control
system
IJECE
V
ol.
6,
No.
5,
October
2016:
2239
–
2250
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
2243
decreases
the
v
alue
of
S
2
,
which
implies
that
the
v
alue
of
z
decreases
too.
In
such
case,
if
z
!
0
and
S
1
!
0
too,
we
w
ould
ha
v
e
that
x
1
!
x
1
d
.
In
sum
mary
,
we
can
state
that
when
S
2
!
0
and
S
1
!
0
,
then
x
3
!
x
3
d
,
z
!
0
and
x
1
!
x
1
d
,
and
the
control
objecti
v
es
w
ould
be
achie
v
ed.
K
eeping
the
system
states
(
x
1
;
x
3
)
on
the
sliding
s
u
r
f
aces
S
1
,
S
2
,
8
t
>
0
guarantees
that
the
tracking
errors
v
ector
(
e
1
,
e
3
)
asymptotically
approaches
to
zero.
The
corresponding
sliding
condition
is
[10]:
V
=
1
2
S
2
1
0
_
V
=
S
1
_
S
1
0
(25)
The
general
control
structure
that
satisfies
the
stability
condition
of
the
sliding
motion
can
be
written
as:
u
=
u
eq
+
u
sw
(26)
Where
u
e
q
is
called
the
equi
v
alent
control
la
w
that
is
deri
v
ed
by
setting
_
S
1
=
0
and
u
s
w
is
the
switching
control
la
w
.
T
aking
the
time
deri
v
ati
v
e
of
(21)
gi
v
es:
_
S
1
=
•
e
1
+
1
:
(
_
e
1
_
z
)
=
(
•
x
1
•
x
1
d
)
+
1
:
(
_
e
1
_
z
)
=
(
f
1
(
x
)
+
b
1
(
x
)
u
+
d
1
•
x
1
d
)
+
1
:
(
_
e
1
_
z
)
(27)
Where,
_
z
is
gi
v
en
as
follo
ws:
_
z
=
8
<
:
k
2
2
:
_
S
2
if
S
2
2
1
0
if
S
2
2
1
(28)
And:
_
S
2
=
•
e
3
+
2
:
_
e
3
=
(
•
x
3
•
x
3
d
)
+
2
:
_
e
3
=
(
f
2
(
x
)
+
b
2
(
x
)
u
+
d
2
•
x
3
d
)
+
2
:
_
e
3
(29)
Substituting
equat
ions
(29),
and
(28)
into
(27),
and
setting
_
S
1
=
0
in
this
last
e
qu
a
tion,
the
equi
v
alent
control
is
obtained:
u
eq
=
1
(
b
1
1
k
2
2
b
2
)
[
f
1
•
x
1
d
1
k
2
2
:
(
f
2
•
x
3
d
)
+
(
1
:
_
e
1
1
2
k
2
2
:
_
e
3
)]
(30)
Then
the
global
control
input
u
is
gi
v
en
by
the
e
xpression:
u
=
1
(
b
1
1
k
2
2
b
2
)
[
f
1
•
x
1
d
1
k
2
2
:
(
f
2
•
x
3
d
)
+
(
1
_
e
1
1
2
k
2
2
:
_
e
3
)
+
(
k
1
:
sgn(
S
1
1
))]
(31)
where
k
1
is
a
positi
v
e
constant
and
1
is
the
boundary
layer
of
t
he
sliding
surf
ace
S
1
.
Substituting
(31)
i
nto
(27)
yields
_
S
1
=
d
1
1
k
2
2
:d
2
k
1
sgn(
S
1
1
)
1
:
sgn(
S
1
1
)
1
k
2
2
:
2
:
sgn(
S
1
1
)
k
1
:
sgn(
S
1
1
)
(32)
Then
simply:
S
1
_
S
1
1
:
1
1
k
2
2
:
2
k
1
:
S
1
1
(33)
From
Eq(33)
it
can
be
concluded
that
the
reaching
condition
is
obtained
from
k
1
(
1
1
k
2
2
:
2
)
.
But,
Eq(31)
will
ha
v
e
high-frequenc
y
switching
near
the
sliding
surf
ace
(
S
1
=
0
)
due
to
the
sgn
function
in
v
olv
ed.
Thus,
in
order
to
reduce
the
chattering
phenomenon,
we
replace
sgn(
S
1
=
1
)
by
sat(
S
1
=
1
)
as
follo
ws:
u
=
1
(
b
1
1
k
2
2
b
2
)
[
f
1
•
x
1
d
1
k
2
2
:
(
f
2
•
x
3
d
)
+
(
1
:
_
e
1
1
2
k
2
2
:
_
e
3
)
+
(
k
1
:
s
at(
S
1
1
))]
(34)
Evaluation Warning : The document was created with Spire.PDF for Python.
2244
ISSN:
2088-8708
3.3.
Fractional
Order
Sliding
Mode
Contr
oller
(FOSMC)
Design
In
this
section
we
will
de
v
elop
a
fractional-order
sliding
mode
control
for
trajectory
tracking
of
the
system
of
Eq(16).
F
or
this
purpose,
the
forms
introduced
in
the
pre
vious
section
are
used.
Let
the
tw
o
sliding
surf
aces
to
be
defined
as
follo
ws:
S
1
=
D
t
e
1
+
1
:
(
e
1
z
)
(35)
S
2
=
D
t
e
3
+
2
:e
3
(36)
W
ith
res
p
e
ct
the
propert
y
of
Caputos
deri
v
ati
v
e
D
t
(
f
(
t
))
=
D
(
m
)
t
d
m
dt
m
(
f
(
t
))
,
S
1
and
S
2
can
be
re
written
as
:
S
1
=
D
(
1)
t
_
e
1
+
1
:
(
e
1
z
)
(37)
S
2
=
D
(
1)
t
_
e
3
+
2
:e
3
(38)
T
aking
deri
v
ati
v
e
of
both
sides
of
Eq(37)
with
respect
to
time,
we
ha
v
e.
_
S
1
=
D
(
1)
t
•
e
1
+
1
:
(
_
e
1
_
z
)
=
D
(
1)
t
(
•
x
1
•
x
1
d
)
+
1
:
(
_
e
1
_
z
)
=
D
(
1)
t
(
f
1
(
x
)
+
b
1
(
x
)
u
+
d
1
•
x
1
d
)
+
1
:
(
_
e
1
_
z
)
(39)
Where,
_
z
has
the
same
form
as
in
Eq(28),
and:
_
S
2
=
D
(
1)
t
•
e
3
+
2
:
_
e
3
=
D
(
1)
t
(
•
x
3
•
x
3
d
)
+
2
:
_
e
3
=
D
(
1)
t
(
f
2
(
x
)
+
b
2
(
x
)
u
+
d
2
•
x
3
d
)
+
2
:
_
e
3
(40)
By
setting
_
S
1
=
0
and
respect
the
properties
of
fractional
deri
v
ati
v
e
gi
v
en
in
section
II;
the
equi
v
alent
control
is
obtained
and
it
has
the
flo
wing
formula:
u
eq
=
1
(
b
1
1
k
2
2
b
2
)
[
f
1
•
x
1
d
1
k
2
2
:
(
f
2
•
x
3
d
)
+
D
(1
)
t
(
1
:
_
e
1
1
2
k
2
2
:
_
e
3
)]
(41)
Then
the
global
control
input
u
is
gi
v
en
by:
u
=
1
(
b
1
1
k
2
2
b
2
)
[
f
1
•
x
1
d
1
k
2
2
:
(
f
2
•
x
3
d
)
+
D
(1
)
t
(
1
_
e
1
1
2
k
2
2
:
_
e
3
)
+
D
(1
)
t
(
k
1
:
sgn(
S
1
1
))]
(42)
Where
u
s
w
is
gi
v
en
by
its
proper
formula
in
Eq(42)
to
satisfy
the
e
xistence
condition
of
sliding
mode.
F
or
the
stability
condition,
substituting
Eq(42)
into
Eq(39);
results
in:
_
S
1
=
D
(
1)
t
D
(1
)
t
(
1
:
_
e
1
1
2
k
2
2
:
_
e
3
)
D
(
1)
t
D
(1
)
t
(
k
1
:
sgn(
S
1
1
))
+
(
1
:
_
e
1
1
2
k
2
2
:
_
e
3
)
+
D
(
1)
t
d
1
1
k
2
2
:D
(
1)
t
d
2
(43)
T
aking
into
account
the
property
of
Caputos
deri
v
ati
v
e
a
D
t
(
a
D
t
f
(
t
))
=
f
(
t
)
m
1
P
k
=0
f
k
(
a
)
k
!
(
t
a
)
k
,
where
m
=
1.
This
lets
us
ha
v
e:
_
S
1
=
k
1
:
(sgn(
S
1
1
))
+
k
1
:
(sgn(
S
1
(0)
1
))
+
(
1
:
_
e
1
(0)
1
2
k
2
2
:
_
e
3
(0))
+
D
(
1)
t
d
1
1
k
2
2
:D
(
1)
t
d
2
(44)
If
one
assume
that
k
1
:
(sgn(
S
1
(0)
1
))
+
(
1
_
e
1
(0)
1
2
k
2
2
_
e
3
(0))
=
0
,
Eq
(44)
will
be:
IJECE
V
ol.
6,
No.
5,
October
2016:
2239
–
2250
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
2245
_
S
1
=
k
1
(sgn(
S
1
1
))
+
D
(
1)
t
d
1
1
k
2
2
:D
(
1)
t
d
2
=
k
1
(sgn(
S
1
1
))
+
1
(sgn(
S
1
1
))
1
k
2
2
:
2
(sgn(
S
1
1
))
(45)
Then
simply:
S
1
_
S
1
1
:
1
1
k
2
2
:
2
k
1
:
S
1
1
(46)
From
Eq(46)
the
reaching
condition
can
be
obtained
from
k
1
(
1
1
k
2
2
:
2
)
.
Otherwise,
if
k
1
:
(sgn(
S
1
(0)
1
))
+
(
1
_
e
1
(0)
1
2
k
2
2
_
e
3
(0))
1
,
this
lets
us
ha
v
e:
S
1
_
S
1
1
:
1
1
k
2
2
:
2
k
1
+
:
S
1
1
(47)
F
or
k
1
(
1
1
k
2
2
:
2
+
)
then
reaching
condition
of
Eq(25)
is
also
v
alid.
Eq(48)
represent
the
control
input
with
sat
function
to
a
v
oid
the
problem
of
chattering.
u
=
1
(
b
1
1
k
2
2
b
2
)
[
f
1
•
x
1
d
1
k
2
2
:
(
f
2
•
x
3
d
)
+
D
(1
)
t
(
1
:
_
e
1
1
2
k
2
2
:
_
e
3
)
+
D
(1
)
t
(
k
1
:
s
at(
S
1
1
))]
(48)
Figure
1
summarizes
the
proposed
FOSMC
for
a
single-input
tw
o
output
non-linear
system.
Figure
1.
Scheme
of
proposed
FOSMC
4.
SIMULA
TION
RESUL
TS
In
this
section,
we
shall
demonstrate
that
the
proposed
FOSMC
is
applicable
to
the
problem
of
trajectory
tracking
of
single
input
tw
o
output
(SIT
O)
system
described
in
Eq
(16)
in
order
to
v
erify
the
theoretical
de
v
elopment
by
using
Matlab/simulink
tools.
W
e
choose
as
e
xample,
the
single-in
v
erted
pendulum
system.
The
structure
of
a
single-in
v
erted
pendulum
is
illustrated
in
figure
2
and
its
dynamic
i
s
described
belo
w
(Eq
49):
_
x
1
=
x
2
_
x
2
=
m
t
g
sin
x
1
m
p
L
sin
x
1
cos
x
1
:x
2
2
L
(
4
3
m
t
m
p
cos
2
x
1
)
+
cos
x
1
L
(
4
3
m
t
m
p
cos
2
x
1
)
u
+
d
_
x
3
=
x
4
_
x
4
=
4
3
m
p
Lx
2
2
sin
x
1
+
m
p
g
sin
x
1
cos
x
1
4
3
m
t
m
p
cos
2
x
1
+
4
3
:
(
4
3
m
t
m
p
cos
2
x
1
)
u
+
d
(49)
And
assuming
that
the
control
signal
u
and
the
e
xternal
disturbance
d
are
as
follo
ws:
d
(
t
)
=
0
:
05
sin
(
t
)
max(
abs
(
u
))
10
N
(50)
Where
x
1
=
,
the
angle
of
the
pole
with
respect
to
the
v
ertical
axis,
x
2
=
_
the
angle
v
elocity
of
the
pole
with
respect
to
the
v
ertical
axis;
x
3
=
x
,
the
position
of
the
cart;
x
4
=
_
x
,
the
v
elocity
of
the
cart.
Evaluation Warning : The document was created with Spire.PDF for Python.
2246
ISSN:
2088-8708
Due
to
its
beha
viour
,
the
trajectory
tracking
problem
for
this
system
is
generally
carried
for
x
3
and
sa
v
e
the
v
erticality
of
the
pole
(
x
1
=0);
in
our
study
we
ha
v
e
discussed
the
follo
wing
tw
o
cases:
Case
1
:
the
both
desired
states
x
1
d
and
x
3
d
are
set
to
0.
Case
2
:
the
desired
states
x
1
d
and
x
3
d
are
set
to
0
and
x
3
d
(
t
)
=
0
:
3
sin(
t
25
)
respecti
v
ely
.
Figure
2.
Single-in
v
erted
pendulum
system
F
or
the
simulation,
the
initial
conditions
are
set
to
[
x
1
0
;
x
2
0
;
x
3
0
;
x
4
0
]
=
[0
;
0
;
0
:
5
;
0]
,
and
the
follo
wing
specifications
are
used:
-
F
or
in
v
erted
pendulum
system:
m
p
=
0
:
1
k
g
;
m
c
=
1kg
;
L
=
0
:
5m
;
g
=
9
:
81m
=
s
2
,
m
t
=
m
c
+
m
p
.
-
As
gi
v
en
belo
w
,
the
parameters
of
both
SMC
and
FOSMC
are
equi
v
alent
(b
ut
for
FOSMC
we
ha
v
e
=0.56)
1
=
1
:
05
;
2
=
1
:
05
;
1
=
0
:
24
;
2
=
0
:
95
;
k
1
=
0
:
91
;
k
2
=
0
:
35
The
simulation
results
are
gi
v
en
by
figures
3
to
10.
a)
0
5
10
15
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
time(sec)
θ
(rad)
x
1
with SMC
x
1
d
x
1
with FOSMC
b)
0
5
10
15
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
time (sec)
x(m)
x
3
with SMC
x
3
d
x
3
with FOSMC
Figure
3.
Simulation
results
of
case
1
without
an
y
disturbance,
a)
Angle
e
v
olution
,
b)
displacement
x
It
can
be
seen
from
figures,
that
x
1
and
x
3
con
v
er
ge
respecti
v
ely
to
the
desired
trajectories
x
1
d
and
x
3
d
.
Also,
the
con
v
er
gence
of
these
states
using
the
proposed
FOSMC
is
f
aster
than
that
of
classical
SMC,
this
is
obtained
under
considerable
magnitude
of
x
1
and
control
signal
(
u
)
in
the
transition
state
for
the
first
one.
W
ith
adding
e
xternal
disturbance,
the
system
is
still
stable.
5.
CONCLUSION
In
this
paper
a
sliding
mode
control
scheme
based
on
fractional
order
calculus
w
as
proposed
for
a
class
of
non-linear
systems,
which
is
characterized
by
a
single
input
and
tw
o
outputs
(SIT
O).
The
proposed
approach
used
tw
o
fractional
order
sliding
surf
aces
with
intermediate
v
ariable
between
them;
in
which
the
control
la
w
w
as
calculated
to
control
the
tw
o
system
outputs.
The
L
yapuno
v
theorem
is
used
to
pro
v
e
the
stability
condition.
Finally
the
simulation
IJECE
V
ol.
6,
No.
5,
October
2016:
2239
–
2250
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
2247
a)
0
5
10
15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
time(sec)
z
with SMC
with FOSMC
b)
0
5
10
15
−4
−2
0
2
4
6
8
time (sec)
u (N)
with SMC
with FOSMC
Figure
4.
Simulation
results
of
case
1
without
an
y
disturbance,
a)
intermediate
v
ariable
z,
b)
control
signal
u
a)
0
5
10
15
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
time(sec)
θ
(rad)
x
1
with SMC
x
1
d
x
1
with FOSMC
b)
0
5
10
15
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
time(sec)
x (m)
with SMC
x
3
d
with FOSMC
Figure
5.
Simulation
results
of
case
1
with
adding
e
xternal
disturbance,
a)
Angle
e
v
olution
,
b)
displacement
x
for
in
v
erted
pendulum
system
ha
v
e
sho
wn
that
the
proposed
FOSMC
gi
v
e
the
best
control
specification
compared
with
the
con
v
entional
one
based
on
inte
ger
order
.
REFERENCES
[1]
V
.
I.
Utkin,
”Sliding
modes
and
their
applications
in
v
ariable
structure
systems”.
Mir
,
Mosco
w
.
1978.
[2]
J.
J
.E.
Slotine,
W
.
Li,
”Applied
nonlinear
control”,
Prentice-Hall,
USA
,
1991.
[3]
J-J.
W
ang,
”Simulation
studies
of
in
v
erted
pendulum
based
on
PID
controllers”,
Simulation
Modelling
Practice
and
Theory
,
v
ol.
19,
pp.
440-449,
2011.
[4]
L-B.
Prasad,
B.
T
yagi,
H-O.
Gupta,
”Modelling
Simulation
for
Optimal
Control
of
Nonlinear
In
v
erted
Pendulum
Dynamical
System
using
PID
Controller
LQR”,
Sixth
Asia
Modelling
Symposium
,
pp.
138-143,
2012.
[5]
S-K.
Oh,
W
.
Pedrycz,
S-B.
Rho,
T
-C.
Ahn,
”P
arameter
estimation
of
fuzzy
controller
and
its
application
to
in
v
erted
pendulum”,
Engineering
Applications
of
Artificial
Intelligence
,
v
ol.
17,
pp.
37-60,
2004.
[6]
A-M.
El-Nag
ar
,
M-El-Bardini,
N-M.
EL-Rabaie,
”Intelligent
control
for
nonlinear
in
v
erted
pendulum
based
on
interv
al
type-2
fuzzy
PD
controller”,
Ale
xandria
Engineering
Journal
,
v
ol.
53,
pp.
23-32,
2014.
Evaluation Warning : The document was created with Spire.PDF for Python.
2248
ISSN:
2088-8708
a)
0
5
10
15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
time(sec)
z
with SMC
with FOSMC
b)
0
5
10
15
−4
−2
0
2
4
6
8
time(sec)
u (N)
with SMC
with FOSMC
Figure
6.
Simulation
results
of
case
1
with
adding
e
xternal
disturbance,
a)
intermediate
v
ariable
z,
b)
control
signal
u
a)
0
10
20
30
40
50
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
time(sec)
θ
(rad)
x
1
with SMC
x
1
d
x
1
with FOSMC
b)
0
10
20
30
40
50
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
time(sec)
x (m)
x
3
with SMC
x
3
d
x
3
with FOSMC
Figure
7.
Simulation
results
of
case
2
without
an
y
disturbance,
a)
Angle
e
v
olution
,
b)
displacement
x
[7]
M-J.
Mahmoodabadi,
S-A.
Mostaghim,
A.
Bagheri,
N.
Nariman-zadeh,
”P
areto
optimal
design
of
the
decoupled
sliding
mode
controller
for
an
in
v
erted
pendulum
system
and
its
stability
simulation
via
Ja
v
a
programming”,
Mathematical
and
Computer
Modelling
,
v
ol.
57,
pp.
1070-1082,
2013.
[8]
J-C.
Lo
and
Y
-H.
K
uo,
”Decoupled
Fuzzy
Sliding-Mode
Control”,
IEEE
transactions
on
fuzzy
systems
,
v
ol.
6,
NO.
3,
pp.
426-435,
August
1998.
[9]
Y
-H.
Chang,
C-W
.
Chang,
C-W
.
T
ao,
H-W
.
Lin,
J-S.
T
aur
,
”Fuzzy
sliding-mode
control
for
ball
and
beam
system
with
fuzzy
ant
colon
y
optimization”,
Expert
Systems
with
Applications
,
v
ol.
39
pp.
3624-3633,
2012.
[10]
L-C.
Hung,
H-Y
.
Chung,
”Decoupled
sliding-mode
with
fuzzy-neural
netw
ork
controller
for
nonlinear
systems”,
International
Journal
of
Approximate
Reasoning
,
v
ol.46,
pp.
74-97,
2007.
[11]
A.
Oustaloup,
”Linear
feedback
control
systems
of
fractional
order
between
1
and
2,
”
IEEE
Int.
Symp.
Circ.
Systems
,
Chicago,
IL
(1981).
[12]
A.
Oustaloup,
”The
CR
ONE
control,
”
ECC
91,
v
ol.
1,
Grenoble,
France,
1991.
[13]
I.
Podlubn
y
,
”Fracti
o
na
l-order
systems
and
PID-controllers”,
IEEE
T
rans.
Autom.
Cont
rol
,
v
ol.
44,
No.
1,
pp.208-214,
1999.
[14]
S.
Das,
I.
P
an,
Sh.
Das,
A.
Gupta,
”A
no
v
el
fractional
order
fuzzy
PID
controller
and
its
optimal
time
domain
IJECE
V
ol.
6,
No.
5,
October
2016:
2239
–
2250
Evaluation Warning : The document was created with Spire.PDF for Python.