Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
6, N
o
. 5
,
O
c
tob
e
r
201
6, p
p
. 2
251
~226
1
I
S
SN
: 208
8-8
7
0
8
,
D
O
I
:
10.115
91
/ij
ece.v6
i
5.9
188
2
251
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
/
IJECE
Hover Position of Quadrotor Ba
se
d o
n
PD-like
Fuz
z
y
Line
ar
Programming
Iswanto
1,2
,
O
y
a
s
Wa
hy
un
ggo
ro
1
, Adh
a
Im
am Ca
hy
adi
1
1
Department of
Electrical Eng
i
n
eering
and
Infor
m
ation
Technolo
g
y
, Universitas
Gadjah Mad
a
, Y
o
g
y
ak
arta, Indon
esia
2
Departm
e
nt
of
Ele
c
tri
cal
Eng
i
n
eering
,
Univ
ers
i
tas Muhammadiy
a
h Yog
y
akarta,
Yog
y
ak
arta, Ind
onesia
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Nov 27, 2015
Rev
i
sed
Au
g 5, 201
6
Accepted Aug 23, 2016
The purpose of
this paper
is to pres
ent th
e a
l
titude
contro
l a
l
gorithm
for
quadrotor
to b
e
able
to fly
at a pa
r
ticu
l
ar
altitude. Sev
e
ral pr
evious
research
ers hav
e
conducted st
udies on quadr
otor al
titud
e
b
y
using PID
control but ther
e are problem
s in the
overshoot and oscillat
i
on. T
o
optim
ize
the control, tunn
ing on PID algorithm mu
st
be first conducted to
determine
proportional an
d derivative
co
nstants.
Hence, the pap
e
r pres
ents al
titud
e
control modification b
y
using
PID-
like fuzz
y withou
t tunin
g
. The PID
algorithm
is
a
control
algor
ith
m
for linear s
y
stem
s. W
h
ile, system
to be
controlled
is a non-linear, so that
lin
ear
ization is needed
b
y
using
equilibr
i
um
. The proposed algori
t
hm
is a
m
odification of the PID algorithm
us
ed as
an alt
i
t
ude contro
l which enab
les
q
u
adrotor to be
s
t
able when
hovering.
The
algorithm used is not PI
D algorith
m with tuning using fuzzy
,
but this is a single input single output
(SISO) control PID-like
fuzzy
lin
ear
programming. The result of the r
e
search shows that quadrotor
can hover in a
rapid raise tim
e, stead
y
stat
e and
settling
tim
e wi
thout perform
in
g overshoot
and osci
lla
tion
.
Keyword:
Altitu
d
e
con
t
rol
Fuzzy
l
i
n
ea
r
pr
og
ram
m
i
ng
Ho
ve
r posi
t
i
o
n
PD-li
k
e
Qua
d
rot
o
r
Copyright ©
201
6 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
:
Iswa
nto
,
Depa
rt
em
ent
of El
ect
ri
cal
E
n
gi
nee
r
i
n
g a
n
d
I
n
f
o
rm
at
i
on Te
chn
o
l
o
gy
,
Uni
v
ersitas Ga
dja
h
M
a
da,
Yo
gy
aka
r
t
a
, In
do
nesi
a.
Em
a
il: iswan
t
o.s3
te1
3
@m
ail.
u
g
m
.ac.id
1.
INTRODUCTION
Qua
d
rot
o
r is
a
n
unm
anne
d ai
rcra
ft
that has
four
m
o
tors wi
th
BLDC
type
at each e
n
d a
n
d
has
bee
n
widely use
d
in civilian applications
, m
i
litary, team
search and re
scue
(SAR) and ae
rial phot
ogra
phs
bec
a
use
its
m
a
n
e
u
v
e
rabilit
y, v
e
rtical t
a
k
e
off an
d
land
ing
,
hov
er
positio
n
and
its a
b
ility fly
i
n
g
in d
a
ng
erou
s areas [1
].
Howev
e
r, qu
ad
ro
tor is a
n
on-lin
ear sy
stem
that are
very
difficult to sta
b
l
e
, t
h
ere
f
ore m
a
ny
resea
r
che
r
s ha
ve
co
ndu
cted
st
ud
ies to
stab
ilize q
u
a
dro
t
o
r
b
y
con
t
ro
llin
g th
e fourth
ro
tors u
s
i
n
g
variou
s m
e
th
o
d
s and
alg
o
rith
m
s
. The co
mm
o
n
l
y u
s
ed
co
nv
en
tional co
n
t
ro
ller is
a p
r
op
ortion
a
l
in
teg
r
al
d
e
ri
v
a
t
i
v
e
(PID)
wh
ich
is a
lin
ear co
n
t
ro
l
[2
].
In a
d
di
t
i
on t
o
PID
co
nt
r
o
l
,
t
h
ere a
r
e m
ode
rn c
o
nt
r
o
l
s
use
d
f
o
r
co
nt
r
o
l
o
p
t
i
m
i
zat
i
on of
a qua
d
r
ot
or
suc
h
as by Ra
ffo et al.
[3]
who used
H
∞
c
o
n
t
rol
al
g
o
ri
t
h
m
s
, an
d bl
ac
kst
e
p
p
i
n
g al
g
o
ri
t
h
m
use
d
by
B
e
s
n
a
r
d et
al. [4
] an
d
Zh
en
g
et al [5
] to
stab
ilize th
e p
o
sitio
n
o
f
th
e
q
u
ad
ro
tor. Mod
e
l
Pred
ictiv
e Con
t
ro
l Algo
rithm an
d
l
i
n
ear
qua
d
r
at
i
c
co
nt
r
o
l
i
s
on
e o
f
t
h
e
m
odern
co
nt
r
o
l
s
us
ed t
o
co
nt
r
o
l
qua
d
r
ot
or
by
R
i
nal
d
i
et
al
.
[6]
a
n
d
Alexis et al. [7] respectively
.
In ad
dition, the propose
d
m
e
thod is also
based on arti
ficial
intelligence to
cont
rol
opt
i
m
izat
i
on q
u
a
d
r
o
t
o
r
,
nam
e
l
y
Genet
i
c
Al
g
o
rith
m
[8]
,
A
d
apti
ve Ne
u
r
o
-
F
u
z
z
y
Infe
re
nce
Sy
ste
m
(
A
N
F
I
S
) [9
],
an
d fu
zzy lo
g
i
c
alg
o
r
ith
m
s
[
1
0].
Su
ch m
e
t
hod
s p
r
o
v
i
d
e
opt
i
m
al
resul
t
s
a
n
d
can
be
i
m
pl
em
ent
e
d
because they
do
not c
o
ntain a
lot of m
a
the
m
atical equations.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 5
,
O
c
tob
e
r
20
16
:
225
1
–
22
61
2
252
PID C
o
n
t
ro
l is a lin
ear con
t
ro
l fo
r lin
ear syste
m
s
such as
m
a
gnet
i
c
act
ua
t
o
r [
1
1]
, Po
si
t
i
on C
ont
rol
[1
2]
, T
r
aject
or
y
C
ont
r
o
l
[1
3
]
, B
r
us
hl
ess
D
C
m
o
t
o
rs [
14]
, an
d B
r
us
he
d
DC
M
o
t
o
r
[
1
5]
so
t
h
at
t
h
e
PI
D
cont
roller
has
been wi
dely used
by som
e
researche
r
s
for exam
ple by Ziegler
Ni
ch
ol
s [
16]
, a
s
a
m
o
t
o
r
cont
rol
l
e
rs si
n
ce PID has a s
i
m
p
l
e
st
ruct
ure
wi
t
h
a
m
e
t
hod
of det
e
rm
i
n
i
ng a com
m
on PID pa
ram
e
t
e
r,
t
hus i
t
h
a
s th
e ab
ility in
supp
ressi
n
g
an
in
terferen
ce well. Ho
wev
e
r, it resu
lts in
a larg
e
p
e
rcen
tag
e
of
o
v
e
rsh
oot an
d
th
e con
t
ro
l sign
alin
g u
s
ed
tend
s to b
e
larg
e,
wh
ich
m
a
y cause saturati
on i
n
the actuat
o
r.
Great
gesture c
ont
rol
req
u
i
r
es
bi
g en
ergy
i
n
w
h
i
c
h
several
m
e
t
hod
s pr
op
ose
d
t
o
i
m
prove t
h
e co
nt
r
o
l
of t
h
e ge
s
t
ure by
usi
ng
h
y
b
ri
d
cont
rol
t
h
at
i
s
Part
i
c
l
e
Swar
m
Opt
i
m
i
zati
on (PS
O
)-
base
d
PID co
nt
r
o
l
[
17]
an
d f
u
zzy
l
ogi
c al
go
ri
t
h
m
-
based
PID
co
nt
r
o
l
[1
8]
.
Som
e
researchers suc
h
as Zefang He and L
o
ng Zh
ao [
1
9]
used P
I
D c
o
n
t
ro
l with
Zieg
l
e
r-Nicho
l
s
tu
n
i
ng
u
s
ed
for qu
adro
t
o
r stabilit
y. By u
s
ing
feedb
a
ck
lin
earizatio
n
t
h
eory, a non
lin
ear Qu
adro
to
r is m
o
d
e
led
into a linear one. Ot
her re
se
arche
r
s suc
h
a
s
Hassan Ta
nveer
et
al
. [20]
and B
o
l
a
n
d
i
et
al
. [21]
use
d
Tay
l
or'
s
m
e
t
hod f
o
r q
u
a
dr
ot
o
r
m
odel
l
i
n
eri
zat
i
on so
t
h
at
t
h
e M
I
M
O
qua
dr
ot
o
r
m
odel
con
v
ert
e
d t
o
SIS
O
. T
uni
n
g
PI
D
usi
n
g I
A
E
was
used
by
B
o
l
a
ndi
et
al
. [
21]
t
o
o
p
t
i
m
i
ze t
h
e cont
r
o
l
o
f
q
u
a
dr
ot
o
r
.
Whi
l
e
aut
o
t
u
ni
n
g
m
e
t
h
o
d
was use
d
by
Tan
v
eer Hassa
n
et
al
. [
20]
t
o
opt
i
m
i
ze
PID c
ont
rol. Some researc
h
ers
use
d
the m
e
thods
of
m
o
d
e
rn
con
t
rol an
d
in
tellig
ent co
n
t
ro
l su
ch
as Ch
en
et al. [2
2
]
using
Neu
r
al Network
for tu
n
i
ng
th
e PID and
M
i
an
& Wa
ng
[2
3]
usi
n
g bac
k
st
ep
pi
n
g
fo
r Tu
ni
n
g
t
h
e
PI
D.
Fuzzy
l
i
n
ear p
r
o
g
ram
m
i
ng has been wi
del
y
used by
some researchers
such as Aza
d
eh et al. [24]
usi
n
g i
t
for set
t
i
ng i
n
t
h
e g
r
ee
nh
o
u
se. B
e
si
de
s bei
n
g use
d
f
o
r t
h
e gree
n
h
o
u
s
e, t
h
e m
e
t
hod
was use
d
t
o
co
nt
r
o
l
i
rri
gat
i
o
n by
Lu et
al
[2
5]
.
It
can easi
l
y
be ap
pl
i
e
d t
o
a l
i
n
ear sy
st
em
and ha
s a
po
we
rf
ul
p
r
o
g
r
am
m
i
ng
structure.
Th
e purpo
se
o
f
th
is p
a
p
e
r is to
presen
t an
al
titu
d
e
control algorithm
for quadro
to
r t
o
hov
er stab
ly at
a cert
a
i
n
hei
g
ht
by
usi
ng
fu
zzy
l
i
n
ear pr
o
g
ram
m
i
ng al
gori
t
h
m
t
h
at
i
s
PD. T
h
e sy
st
e
m
at
i
c
s of t
h
i
s
pape
r
co
nsists o
f
q
u
a
d
r
o
t
or m
o
d
e
lin
g
,
altitu
d
e
con
t
ro
l strateg
i
es, fu
zzy con
t
ro
ller alg
o
r
ith
m
s
, analyzin
g
resu
lt an
d
concl
u
si
o
n
.
2.
R
E
SEARC
H M
ETHOD
Qua
d
rot
o
r syste
m
is a non-linear sy
stem
that has
4 rotors
at each end that
can
be m
odele
d
by
using
Eul
e
ri
an a
ngl
e
s
[2
6]
. In t
h
i
s
pape
r, q
u
a
d
r
o
t
o
r i
s
m
odel
e
d
by
usi
n
g eul
e
ri
an an
gel
t
hus t
h
e q
u
ad
rot
o
r s
y
st
em
has si
x
deg
r
ee
s o
f
free
dom
defi
ned
by
t
w
el
ve st
at
es s
h
o
w
n
i
n
Fi
g
u
re
1
[2
7]
. Si
x
o
u
t
of
t
h
e t
w
el
v
e
st
at
es
regu
late th
e attitu
d
e
of
q
u
a
d
r
o
t
or system in
clud
ing
quad
r
o
t
or’s Eu
ler an
g
l
es of
ro
ll, p
itch
,
yaw i.e.
T
2
and angular s
p
eed
of t
h
e
qua
drotor i.e.
T
r
q
p
4
on t
h
ree
orthogona
l
axes
of t
h
e
body.
Six ot
her
states are three posit
i
ons
nam
e
ly
T
z
y
x
1
, and t
h
ree li
near
spee
d of the ce
nter
of m
a
ss of the
qua
d
r
ot
or ass
o
ciated with fixed re
fe
rence
fram
e
(fram
e
earth) i.e.
T
z
y
x
3
. Th
i
s
quad
r
ot
o
r
sy
st
em
m
odel
used 1
2
eq
uat
i
o
ns o
f
n
on l
i
n
ea
r st
at
e i
n
a gl
ob
al
fram
e
work
as
T
x
x
x
12
1
with the state
vari
a
b
l
e
s as
T
x
4
3
2
1
.
Fi
gu
re
1.
Q
u
a
d
rot
o
r m
odel
i
n
g
wi
t
h
Eul
e
ri
a
n
angl
es
Fi
gu
re
1 s
h
o
w
s
t
h
at
t
h
ere
are t
w
o
co
or
di
nat
e
fram
e
s, one
o
f
whi
c
h i
s
t
h
e
ea
rt
h a
n
d
t
h
e
ot
h
e
r f
r
am
e i
s
the body in t
h
e center
of
gravity of
the
quadrotor
[28].
In t
h
e figure it
is seen that the m
ovem
e
nt
of t
h
e
qua
d
r
ot
or
by
6
deg
r
ees o
f
f
r
e
e
dom
i
nvol
vi
n
g
t
h
e m
ovem
e
nt
of t
r
a
n
sl
at
i
o
n base
d o
n
t
h
e
x, y
and z a
x
i
s
. The
ro
tation
o
f
th
e
q
u
a
dro
t
o
r
is roll ro
tatio
n
on
th
e x-ax
is,
p
itch
ro
tation
on
th
e y-ax
is, an
d
yaw ro
tation
on
th
e z-
ax
is. Each
ax
is ro
tatio
n
m
a
trix
can
b
e
written
as:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Ho
ver
Po
sitio
n
o
f
Quad
ro
to
r Ba
sed
o
n
PD-l
ike Fu
zzy Lin
e
a
r
Prog
ra
mmi
n
g (Iswan
to
)
2
253
C
S
S
C
x
R
0
0
0
0
1
,
;
C
S
S
C
y
R
0
0
1
0
0
,
;
1
0
0
0
0
,
C
S
S
C
z
R
(1
)
By u
s
ing
th
e eq
u
a
tion
(1
), ZYX ro
tatio
n matrix
is
d
e
fi
n
e
d as
C
C
C
S
S
C
S
S
S
C
C
C
S
S
S
S
C
S
S
C
S
C
S
C
C
S
S
C
C
R
(2
)
Th
us, t
r
ans
f
or
m
a
t
i
on m
a
t
r
i
x
obt
ai
ne
d i
s
as
f
o
l
l
o
w
s
C
C
S
C
S
C
S
r
q
p
0
0
0
1
(3
)
w
h
er
e
c =
co
s
,
t =
ta
n
an
d s
=
s
i
n
.
Based
o
n
Newt
o
n
'
s seco
nd
law
o
f
m
o
tio
n tran
slatio
n
a
l, th
e fo
llo
wi
n
g
equatio
n
is
ob
tain
ed
mv
v
m
F
(4
)
whe
r
e
T
r
q
p
and
T
z
y
x
v
.
z
y
x
m
r
q
p
z
y
x
m
F
(5
)
In Figure
1, the force acti
n
g
on qu
ad
ro
tor is ob
tain
ed fo
rm
u
l
ated
as
thrust
g
F
F
F
(6
)
Sub
s
titu
tin
g Newton
's secon
d
law
with
th
e fo
rce actin
g on
th
e qu
ad
ro
tor, t
h
e
fo
llowing
eq
u
a
tion
is ob
tain
ed
T
R
mg
z
y
x
m
r
q
p
z
y
x
m
0
0
0
0
(7
)
Sub
s
titu
tin
g the tran
sfo
r
m
a
tio
n
m
a
trix
, th
e
fo
llo
wi
n
g
equ
a
t
i
o
n
is ob
tain
ed
z
y
x
r
q
p
m
c
c
T
mg
c
s
s
s
c
T
s
s
c
s
c
T
z
y
x
m
(8
)
Thus the
equat
i
on
of linear ac
celerati
o
n
i
n
the x
,
y, z is as
fo
llo
ws:
y
r
z
q
s
s
c
s
c
T
m
x
1
(9
)
z
p
x
r
c
s
s
s
c
T
m
y
1
(1
0)
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I
S
SN
:
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-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 5
,
O
c
tob
e
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:
225
1
–
22
61
2
254
x
q
y
p
c
c
T
m
g
z
1
(1
1)
B
a
sed
on
Ne
wt
on'
s sec
o
n
d
l
a
w
of m
o
t
i
o
n
r
o
t
a
tio
n
,
the fo
llowing
eq
u
a
tion
is ob
tain
ed
:
(1
2)
whe
r
e,
is th
e
m
o
men
t
o
f
in
ertia Qu
adro
t
o
r
as shown in
fo
llo
wing
equ
a
tion
z
y
x
0
0
0
0
0
0
(1
3)
wh
ile
T
z
y
x
So t
h
at the
angular accele
r
ation e
quation i
n
t
h
e
x, y, z is as
follows:
r
q
p
x
y
z
x
x
(1
4)
r
p
y
q
y
z
x
y
(1
5)
q
p
r
z
x
y
x
z
(1
6)
An
g
u
l
a
r s
p
ee
d,
p;
q a
n
d
r
can
be
obt
ai
ne
d
fr
o
m
t
h
e l
e
vel
of t
h
e E
u
l
e
r a
n
gl
e
s
usi
n
g
a t
r
a
n
s
f
orm
a
t
i
on m
a
t
r
ix.
W
r
q
p
(1
7)
c
c
s
c
s
c
s
r
q
p
0
0
0
1
(1
8)
So
t
h
at th
e
angu
lar sp
eed
o
f
ro
ll, p
itch an
d
yaw of
the quadrot
o
r
ca
n be de
termined
as
r
q
p
c
s
c
s
c
c
s
c
s
s
c
c
0
0
1
(1
9)
Thu
s
, t
h
e
ro
ll,
p
itch
an
d yaw
are
r
t
c
q
t
s
p
(2
0)
r
s
q
c
(2
1)
r
c
c
q
c
s
(2
2)
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I
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I
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:
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0
8
Hover Position of
Quad
rotor Based on
PD-l
ike
Fuzzy
Linear Programm
i
n
g (Iswanto)
2
255
3.
C
O
N
T
ROL AN
D STRA
TEGY
Altitu
d
e
con
t
rol u
s
es fu
zzy lin
ear prog
ramm
i
n
g, wh
ile
th
e syste
m
wil
l
b
e
co
n
t
ro
lled
b
y
qu
adro
to
r i
n
a no
n-l
i
n
ea
r sy
st
em
. Theref
ore a l
i
n
eari
z
at
i
on m
e
t
hod
fo
r t
h
e sy
st
em
i
s
needed
. No
n-l
i
near m
odel
s
of
q
u
a
dro
t
o
r
is then
lin
earized
at th
e p
o
i
n
t
of
b
a
lan
ce
(e
qu
ilib
ri
u
m
), so
th
at
th
e syste
m
can
b
e
pro
cessed in
a
lin
ear m
o
d
e
l.
In
co
ndu
cting
lin
earization
,
t
h
e po
in
t
o
f
equ
ilib
riu
m
o
f
th
e
qu
adro
to
r m
u
st b
e
d
e
term
in
ed
an
d it
can
b
e
written
as
,
,
,
0
X
f
. So t
h
at the
twelve
non li
near equations
can
be
written as
0
7
1
x
x
x
(2
3)
0
8
2
x
y
x
(2
4)
0
9
3
x
z
x
(2
5)
0
12
11
10
4
5
4
5
4
x
t
c
x
t
s
x
x
x
x
x
x
(2
6)
0
12
11
5
4
4
x
s
x
c
x
x
x
(2
7)
0
12
11
6
5
4
5
4
x
c
c
x
c
s
x
x
x
x
x
(2
8)
0
1
6
4
6
5
4
7
x
x
x
x
x
s
s
c
s
c
T
m
x
x
(2
9)
0
1
6
4
6
5
4
8
x
x
x
x
x
c
s
s
s
c
T
m
y
x
(3
0)
0
1
5
4
9
x
x
c
c
T
m
g
z
x
(3
1)
0
12
11
10
x
x
p
x
x
y
z
x
x
(3
2)
0
12
10
11
x
x
y
q
x
y
z
x
y
(3
3)
0
11
10
12
x
x
r
x
z
x
y
x
z
(3
4)
If it is assu
m
e
d that the equi
librium
point i
s
allocat
ed to several positions in Cartesian coordinates
(x; y; z) and at som
e
positions
of yaw angle
whic
h is defined as
2
x
,
3
x
and
6
x
. Thus, t
h
e value
of
the whole equation
of state
at
this equilibrium
point is
,
,
,
X
, so that it can be
written as
1
x
,
2
x
,
3
x
,
0
4
x
,
0
5
x
,
6
x
,
0
7
x
,
0
8
x
,
0
9
x
,
0
10
x
,
0
11
x
, and
0
12
x
. The
represe
n
tations
of the
state equation an
d
sy
stem
outp
u
t a
r
e s
h
o
w
n as
f
o
llo
ws,
Bu
Ax
x
D
u
Cx
y
, (35)
whe
r
e m
a
trix A an
d B
are
obtaine
d
by
u
s
ing Jac
o
bi
linearization m
e
thods. Matri
x
A and B are
deri
ved
partially to the
n
on-
lin
ear
m
o
del.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
No. 5
,
Octob
e
r
20
16
:
225
1
–
22
61
2
256
,
,
,
|
,
,
,
|
,
,
,
|
,
,
,
|
12
12
1
12
12
1
1
1
X
x
f
X
x
f
X
x
f
X
x
f
A
;
,
,
,
|
,
,
,
|
,
,
,
|
,
,
,
|
4
12
1
12
4
1
1
1
X
u
f
X
u
f
X
u
f
X
u
f
B
The
n
, the A and B m
a
trixes are obtained a
s
6
4
6
6
1
2
2
2
3
2
6
6
6
6
12
12
O
I
O
N
O
I
O
A
;
4
4
4
8
4
12
M
O
B
, with O is
the zero m
a
trix and
I is th
e identity
m
a
trix.
Wh
ile
the N
and M
can be defi
ned
as
0
0
2
2
g
g
N
;
z
z
z
z
y
y
x
x
k
k
k
k
db
db
db
db
m
b
m
b
m
b
m
b
M
0
0
0
0
4
4
The
o
u
tp
ut o
f
t
h
e
qua
d
r
ot
or m
odel ca
n
be
de
f
i
ned
by
the
y
v
ector as
f
o
llo
w
s
T
z
y
x
y
so that the C and
D
m
a
tr
ixies can be
writ
ten as
9
1
3
1
9
3
3
3
12
4
L
O
O
C
and
0
D
with
0
0
0
1
0
0
1
1
1
9
1
L
. T
hus
, the
line
a
rized m
ode
l
of the qu
ad
ro
tor is
u
B
x
A
x
4
12
12
12
x
C
y
12
4
(3
6)
By linearization usi
ng equ
ilibrium
, the whol
e equations of
state
except x,
y, z, and yaw are assum
e
d
to be very sm
all. This
results
in a
ll the i
n
put
s
for all four
m
o
tors a
r
e ass
u
m
e
d to
ha
ve sa
m
e
speed. So t
h
at the
MIMO system can be sim
p
li
fied into
SISO syste
m
by usi
ng t
h
e speed
of altitude change and the altitude
which
have the followi
ng equation
9
3
x
z
x
(3
7)
5
4
1
9
x
x
c
c
T
m
g
z
x
(3
8)
The equation of the altitude syste
m
state of
the
quadrot
or can
be
defi
ned as follows
u
B
x
A
x
x
C
y
(3
9)
whe
r
e the
value of
T
x
x
x
9
3
,
9
3
x
x
x
,
0
0
1
0
A
,
4
0
B
, and
0
1
C
. So that the
trans
f
er f
unctio
n of
z
ca
n be d
e
fine
d by
the f
o
llowi
ng
eq
uat
i
on
2
4
s
G
S
(4
0)
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I
J
ECE
I
S
SN:
208
8-8
7
0
8
Hover Position of
Quad
rotor Based on
PD-l
ike
Fuzzy
Linear Programm
i
n
g (Iswanto)
2
257
4.
FUZ
Z
Y
LOGIC
CONTROLLER
Fuzzy logic algorithm
is
an artific
ial intell
i
g
ence algorithm
s
that ar
e often use
d
by
so
m
e
previo
us
investigat
ors to m
a
ke a
deci
sion [29]
,[30]. The design
of th
e altitude cont
rol
of the
quadrotor to m
a
intain
altitude is direct control as
shown in Fi
gure
2. The fi
gure shows that
there is alti
tude to
cont
rol the
quadrotor
syste
m
. The control desi
gn is
propor
tional cont
rol
which
is a linear control. Altitude
control equation
used i
n
the system
is proportional as
s
een i
n
the followi
ng equation:
p
p
S
K
S
K
G
2
4
(4
1)
whe
r
e
p
K
is a pr
op
o
r
tional c
o
n
s
tant an
d
is m
o
ment of i
n
ertia for each
m
o
torcycle. The stability of
the
sy
stem
from
the eq
uatio
n is determ
ined by
usin
g the r
oot
locus as sh
o
w
n in Fig
u
re
3(
A). T
h
e fi
gu
re
sho
w
s
that there is
one pair
of
poles on t
h
e
im
aginary axis in t
h
e
im
ag
e; that is poles l
o
cated
on the top si
de
of t
h
e
im
aginary axis
value
d
3 and
poles
lo
cated on the
bottom
on the im
aj
inear axis
valu
e
d
3.
The
sy
stem
requi
res
settling ti
m
e
and has the sm
a
llest possi
ble overshoot. Thus it take
s a large dam
p
ing for th
e cont
rol
so that t
h
e
pole at t
h
e zero point
of th
e real
axis of
t
h
e root
l
o
cus will
be
shifted to t
h
e left of the real axis.
Figu
re 2.
B
l
oc
k diag
ram
of
al
titude
co
ntr
o
l o
f
qua
dr
oto
r
Proof: by using the root locus it is
seen that equatio
n (
4
0)
has a p
o
le
value that is not on the left of
the im
aginary axis,
the
n
th
e c
ont
rol is
unsta
ble. T
h
e c
ontr
o
l eq
uatio
n wa
s tested by
usi
ng t
h
e step
s s
h
ow
n in
Figure 3(B
)
. The figure shows
that
the
system
is
not stable
and there is
continuous oscillation.
It proofs that
Deri
vative in the control is necessa
ry to be
com
e
stable. T
h
en
P
D
co
ntr
o
l equatio
n in
the quadrotor syste
m
is as follows:
p
d
p
d
S
K
K
S
K
K
G
2
4
(4
2)
whe
r
e
p
K
is a pro
p
o
r
tio
nal const
a
nt and
d
K
is a derivative co
nsta
nt whic
h value
s
are obtaine
d
by
usin
g the
tuning
respectively. The stabi
lity of the syst
e
m
in equation
(42) is sought
by usi
n
g th
e root
locus. Figure
4(A)
shows that there is a pole lo
ca
ted on t
h
e left
of t
h
e im
aginary axis that
indi
cates the syste
m
will be stabl
e
. T
h
e
equation,
when tested by
using steps, will perform
os
cillation for a short ti
m
e
and then
will be st
able as
shown in
Figure 4(B). It can be is se
en that by using
deri
vatives, t
h
e sy
ste
m
will first perform
oscill
ation,
then t
h
e
oscillations
will be sm
oothe
d and fi
nally it becom
e
s stable.
Figu
re
3.
(
A
)
Diag
ram
of r
o
ot loc
u
s
pr
o
p
o
r
tiona
l co
ntr
o
l
(
B
) Test ste
p
f
o
r
pr
op
o
r
tional
cont
rol
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
No. 5
,
Octob
e
r
20
16
:
225
1
–
22
61
2
258
Figu
re
4.
(
A
)
R
oot l
o
cus
o
f
c
ont
rol t
h
at is
pr
op
o
r
tiona
l
de
rivative.
(B
)
Tes
t
step f
o
r
pr
op
o
r
tional c
o
ntr
o
l
of
Deri
vative
To be o
p
tim
al, a tuning to get
the variable v
a
lues pr
o
p
o
r
tio
nal and d
e
rivat
i
ve is require
d,
so that the
PID-like fuzzy
linear progra
m
m
i
ng is n
ecessary
fo
r the cont
rol. T
h
is co
nt
rol uses SISO fuzzy of m
a
m
d
ani’s
type using three
m
e
m
b
er vari
ables fo
r i
n
put
and
output that is negative
sm
a
ll, zero, and positive sm
all. B
y
usin
g t
h
ree i
n
p
u
t an
d
o
u
tp
ut v
a
riable
s, t
h
e
rule-based is as follows
if (Z is NS) then
(output1 is NS)
if (Z is Z) then
(output1 is Z)
if (Z is PS) t
h
en
(output1 is PS)
Figure 5 shows altitude control
using SISO control wi
th fuzzy
logic algorithm
s
of
m
a
m
d
ani’s
m
e
thod. T
h
is pape
r p
r
ese
n
ts two fuzzy
co
ntr
o
ls i.e fuzzy
contr
o
l as pr
o
p
o
r
tio
nal and
deri
vative w
h
ich ha
v
e
three
varia
b
les of input a
n
d
output set m
e
mber c
o
nsistin
g of ne
gative,
ze
ro
a
n
d po
sitive. The output
of each
of the control
will be summ
e
d
and
connected to thro
tell Quadrotor.
The fir
s
t desig
n
co
n
ducte
d is fuzzy
co
ntr
o
l desig
n
as p
r
op
otional
whic
h
has in
put
with
value er
r
o
r
data range, so that the set variable
s of the in
put o
f
f
u
zzy
cont
rol as p
r
o
p
o
rtio
nal has a r
a
nge
fr
om
-15 to 1
5
sho
w
n in Fig
u
r
e
6 (A
). It is seen
that the m
e
m
b
er set not only uses the
up linear set and do
wn linear
set, but
also
uses a triangles m
e
m
b
e
r
s set. Mem
b
e
r
set is
used
for
linear system
s,
while
the qua
d
r
ot
or sy
stem
s
is
a
non-linear
syste
m
, so t
h
at a lin
earization m
odel is
needed.
Figu
re
5.
B
l
oc
k
diag
ram
of P
D
-like
F
u
zzy
linier
pr
o
g
ram
m
i
ng c
o
ntr
o
l
In addition to
the input,
fuzzy
cont
rol as proportional has out
put
in m
o
tor speeds
pulse
range. The
determ
ination of t
h
e output value is ne
e
d
ed
in o
r
de
r f
o
r
the
cont
rol
to be opti
m
u
m
,
so
tha
t
the ran
g
e
fo
r
m
o
tor
spee
d val
u
e is
- S
p
eed a
n
d +
Spee
d s
h
o
w
n in Fig
u
r
e 6
(B
)
.
Th
e f
i
gu
r
e
shows th
at th
e i
n
put value of the
spee
d
is 1
2
0
0
deri
ve
d
fr
om
the pete
r C
o
r
k
e’s
m
odel [2
4]
.
(A)
(B)
Figu
re
6.
The
s
e
t of
in
put a
n
d
out
put
f
o
r
f
u
zz
y
cont
rol as
p
r
op
o
r
tional
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN:
208
8-8
7
0
8
Hover Position of
Quad
rotor Based on
PD-l
ike
Fuzzy
Linear Programm
i
n
g (Iswanto)
2
259
The sec
o
nd
de
sign c
o
nd
ucted
is in in
put s
e
t desig
n
fo
r f
u
zz
y
contr
o
l as
de
rivative s
h
ow
n
in Fig
u
r
e
7
.
In the
figure it
is seen that
th
e
inp
u
t set
o
f
fu
zzy
co
ntrol
as
deri
vative
has
a ra
nge
f
r
om
dz
to
dz
.
T
h
e v
a
lu
e
will affect the perform
ance
of t
h
e cont
rol
syste
m
.
W
h
en the range value is gr
eat, the
overshoot on
the
qua
drot
or ca
n be sm
oothed
due to the ra
pid
acceleration
va
lu
es can
be rea
d
by the de
riva
tive inputs, so
that it
will generate t
h
e
out
put
val
u
e which
will result in
a decrease in t
h
e
propor
tional
control value.
The design of fuzzy
control out
put
as deri
vatives
is
needed to obtain
op
ti
m
u
m
altitude control of
qua
drot
or. T
h
e
r
e are three set
m
e
m
b
ers
that
is negative small, zero, and
positve sm
all.
This output variable
has ra
n
g
e w
h
i
c
h the
value is
obtaine
d
fr
om
half o
f
th
e pr
op
o
r
tional out
put
s
h
o
w
n
in Figu
re 7.
T
h
e fig
u
re
sho
w
s
that
fuz
z
y
cont
rol
o
u
tp
ut as
deri
vative
s
has
ra
n
g
e
value
fr
om
-50
0
t
o
+
5
0
0
.
(A)
(B)
Figu
re
7.
The
s
e
t of
in
put a
n
d
out
put
f
o
r
f
u
zz
y
cont
rol as
de
rivative
5.
R
E
SU
LTS AN
D ANA
LY
SIS
Altitude control system
fo
r quadrot
or
hover position uses PD-lik
e fuzzy control consi
s
ting
of t
h
ree
p
a
r
t
s: inpu
t,
ou
tpu
t
and
contr
o
l.
The control
was tested by using
peter Corke’s si
m
u
la
tion with has
a
specificatio
n s
h
o
w
n in
Table
1.
The
table s
h
o
w
s t
h
at
the qua
d
r
ot
or
t
o
b
e
use
d
fo
r
the
sim
u
lation
has
4 k
g
weig
ht and
ha
s a
m
o
m
e
nt of inertia x, y
,
and z o
f
0
.
0
8
2
0
kg.m
2
, 0
.
084
5 k
g
.
m
2
an
d
0
.
13
77
kg
.m
2
. W
i
th the
specification
data provided, t
h
e stability of t
h
e
qua
drot
or
syste
m
can tested
by usi
n
g
unit
step.
Unit step is used to test the quadrotor altitude co
ntrol syste
m
for 1 m
e
ter
as shown in Figure 8(A).
B
y
using u
n
it step fo
r fu
zzy
logic co
ntr
o
ller algo
rithm
the alti
tude control can st
abilize the qua
dr
oto
r
,
and it
takes a stea
dy
state at 2.98
th
second, settling ti
m
e
of
1.94 second, an
d
raise tim
e of 1.15 second which is
considered short.
In
perform
i
ng take
off and h
over, there are no
oscillation an
d overshoot
on the
quadrotor.
Havi
ng
tested
by
usi
n
g
unit
step, t
h
e re
sult
s sh
o
w
s that t
h
ere
is no
ove
r
hoot and
osci
llation, the
cont
rol is si
m
u
lated in Peter Corke’s
sim
u
lator. In the simula
tor, firstly the param
e
ter i
s
set to initial
position
of
quadrotor xy (-1,0), the
desired
he
ight of 8
m
e
ter,
and hover
positio
n of the qu
adrotor
xy (-1.0). The
expe
rim
e
nts are per
f
o
rm
ed b
y
inv
o
lvin
g
v
e
rtical win
d
d
i
sturba
nce.
T
h
e si
m
u
lation result shows that the
qua
d
r
ot
or
h
ove
rs at
8 m
e
ter altitude s
h
o
w
n in Fi
gu
re
8(B
)
.
There
are
f
o
u
r
gra
p
hs s
h
o
w
n i
n
fi
gu
re i.e
set
poi
nt
,
disturbance, and Erro
r z.
Altitude controll
er quickly stabilizes
the quadrot
or, and it takes a steady state at
8.
38
4
th
second, settling ti
me of 8.975 seconds and raise
time of 4.02 second
with
PID-Like Fuzzy algorith
m
but it takes
a st
eady state at 13.39
th
second,
settling ti
m
e
of 13.9 seconds
and raise tim
e
of
5.804 second wit
h
PD
L
o
op Sha
p
ing. At 15
th
se
con
d
,t
he
qua
d
r
ot
or is
distu
r
b
e
d by
up
ward v
e
r
tical win
d
f
o
r
2 second
, so
th
at
the quadrot
o
r i
s
shifted
upwa
rd from
the initi
al position. By using PID-
like fuzzy control, it is shifted 1 meter
only and ret
u
rn to its
previous
position.
Table
1. T
h
e
c
h
aracteristic
of quadrotor
No Variable
Value
1 G
9,
81
m
/
s
2
2 M
4
kg
3 I
x
0.
0820 k
g
.m
2
4 I
y
0.
0845 k
g
.m
2
5 I
z
0.
1377
k
g
.m
2
6
B
1.
2953 x 1
0
-5
kg.m
7 D
0.
165
m
8
K
1.
0368 x 1
0
-7
kg.m
2
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
No. 5
,
Octob
e
r
20
16
:
225
1
–
22
61
2
260
Figu
re
8.
(
A
)
T
e
st step
fo
r F
u
z
z
y
cont
rol s
u
c
h
as
P
D
(B
)
Gra
p
h
o
f
q
u
ad
rot
o
r test with
dist
ur
ba
nce
6.
CO
NCL
USI
O
N
Fuzzy linear
programm
ing algorith
m
in PD controller used for a
hover position on quadrotor has
been
presented in this paper.
This algorithm
is not used as
PD tuning
,
but
it is used for
cont
rolling the altitude
of the
quadrotor. Based
on si
m
u
la
tion
re
sult
s, the
r
e a
r
e t
h
r
ee g
r
ap
hs
nam
e
ly
setpoint,
al
titude a
n
d
dist
ur
ba
nce
which show that the
quadrotor can h
over
without overshoot or
oscillation.
When t
h
e quadrot
or is di
sturbed
by vertical wi
nd, the control
can st
abilize it and return i
t
to its previo
us position. Thus, the fuzzy
linear
pr
o
g
ram
m
i
ng a
l
go
rithm
s
can
be a
pplied
o
n
the
qua
d
r
ot
or
w
h
ich
has
a
no
n-
linear sy
stem
.
ACKNOWLE
DGE
M
ENTS
Th
is r
e
sear
ch
was supp
or
ted b
y
PUPT Proj
ect g
r
a
n
t from
DIKTI through Researc
h
Directorate,
Uni
v
ersitas Ga
dja
h
M
a
da wit
h
the co
ntract num
ber:
94
4/
UN
1-
P.
II
I/LT/DIT
-
L
I
T/2
0
1
6
,
awarde
d to A
/
P Dr
.
Oy
as Wahy
un
gg
o
r
o
.
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