Inter
national
J
our
nal
of
Robotics
and
A
utomation
(IJRA)
V
ol.
10,
No.
4,
December
2021,
pp.
308
∼
318
ISSN:
2089-4856,
DOI:
10.11591/ijra.v10i4.pp308-318
❒
308
Adjustment
mechanism
with
sliding
mode
f
or
adapti
v
e
PD
contr
oller
applied
to
unmanned
xed-wing
MA
V
altitude
A.T
.
Espinoza-Frair
e,
A.
S
´
aenz-Esqueda,
F
.
C
´
ortes-Mart
´
ınez
F
acultad
de
Ingenier
´
ıa,
Ciencias
y
Arquitectura,
Uni
v
ersidad
Ju
´
arez
del
Estado
de
Durango,
G
´
omez
P
alacio,
Durango,
M
´
exico
Article
Inf
o
Article
history:
Recei
v
ed
Jul
30,
2020
Re
vised
Jun
10,
2021
Accepted
Jul
23,
2021
K
eyw
ords:
Adapti
v
e
control
Altitude
mode
Unmanned
ABSTRA
CT
This
w
ork
presents
an
adjustment
mechanism
with
the
sliding
modes
technique
to
de-
sign
a
proportional
deri
v
ati
v
e
(PD)
controller
with
adapti
v
e
g
ains.
The
objecti
v
e
and
contrib
ution
are
to
design
a
rob
ust
adjustment
mechanism
in
the
presence
of
unkno
wn
and
not
modeled
perturbations
in
the
system;
this
perturbation
can
be
considered
wind
gusts.
The
rob
ust
adjustment
mechanism
is
designed
with
the
MIT
rule
and
the
gra-
dient
method
with
the
sliding
mode
theory
.
The
adapti
v
e
PD
obtained
is
applied
to
re
gulate
unmanned
x
ed-wing
miniature
aerial
v
ehicle
(MA
V’
s)
altitude.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
A.T
.
Espinoza-Fraire
F
acultad
de
Ingenier
´
ıa,
Ciencias
y
Arquitectura
Uni
v
ersidad
Ju
´
arez
del
Estado
de
Durango
Uni
v
ersidad
1021,
Filadela,
35010
Dgo,
G
´
omez
P
alacio,
Durango,
M
´
exico
Email:
atespinoza@ujed.mx
1.
INTR
ODUCTION
The
de
v
elopment
and
use
of
unmanned
aerial
systems
(U
A
Vs)
ha
v
e
been
increasing
in
the
last
decade
[1],
[2],
and
the
theory
about
adapti
v
e
control
is
fundamental
in
the
de
v
elopment
and
adv
ances
in
this
eld.
And
e
v
en
the
applications
of
the
x
ed-wing
U
A
Vs
are
increasing;
some
applications
are:
forest
re
detection,
in
ci
vil
engineering
(topograph
y
,
analysis
structural
and
others)
[3],
photogrammetry
,
and
military
applications
[4],
car
detection
[5]
or
for
landing
[6].
W
e
can
nd
some
w
orks
in
the
scientic
literature
referents
to
adapti
v
e
control
based
on
the
MIT
rule.
F
or
e
xample,
in
[7]
is
de
v
eloped
a
model
reference
based
on
a
proportional
inte
gral
deri
v
ati
v
e
(PID)
controller
.
Compared
with
a
con
v
entional
or
ordinary
reference
model,
this
is
done
to
get
better
performance
in
the
control
of
the
v
elocity
of
a
DC
motor
.
P
a
w
ar
and
P
arv
at
[8]
is
presented
a
modication
in
the
structure
of
an
model
reference
adapti
v
e
control
(MRA
C)
the
modication
is
based
on
a
PID
controller
as
in
[7].
Still,
the
dif
ference
is
that
in
[8]
the
PID
is
used
between
other
controllers
based
in
MRA
C
and
the
plant,
the
proposed
of
[8]
has
the
objecti
v
e
of
impro
ving
the
transient
response
of
the
plant,
and
it
uses
the
kno
wn
MRA
C
structure
[9].
Whereas
in
[10]
the
direct
model
reference
adapti
v
e
and
an
internal
controller
is
applied
to
doubly
fed
induction
generator
and
in
this
w
ork,
is
proposed
the
adjustment
mechanism
based
on
MIT
rule.
Still
,
in
addition,
the
Perrin
equation
has
been
added
to
this
mechanism
with
an
impro
v
ed
internal
model
controller
lter
design.
Thus,
in
[10]
the
adjustment
mechanism
using
the
Perrin
equation
is
intending
to
a
v
oid
the
selection
of
the
adapti
v
e
g
ain
by
a
heuristic
method.
Priyank
and
Nig
am
[11]
is
presented
the
design
of
a
MRA
C
for
a
second-order
system,
that
is,
is
presented
a
modied
MIT
rule
to
resolv
e
tw
o
problems
that
present
the
MIT
rule,
these
problems
are
that
with
a
suf
ciently
lar
ge
selection
of
the
adaptation
g
ain
or
in
the
magnitude
of
the
reference
signal
the
system
tends
to
the
instability
.
And
then,
to
gi
v
e
a
solution
to
these
problems,
in
[11]
a
normalized
algorithm
with
J
ournal
homepage:
http://ijr
a.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Rob
&
Autom
ISSN:
2089-4856
❒
309
MIT
rule
is
presented
to
de
v
elop
the
control
la
w
.
Riache
et
al.
[12]
is
presented
an
adapti
v
e
rob
ust
controller
applied
to
quadrotor
with
a
serial
robot
manipul
ator
onboard,
the
control
objecti
v
e
in
[12]
is
that
during
the
ight,
mo
v
e
the
robot
arm
and
k
eep
the
desired
trajectory
.
On
the
other
hand,
the
w
orks
[7],
[8],
[10],
[11]
presents
simulations
results
using
the
Matlab
soft
w
are
as
well
as
in
this
w
ork.
W
e
can
nd
other
adapti
v
e
controllers
applied
to
x
ed-wing
unmanned
aerial
v
ehicle
(U
A
V)
as
in
[13]
where
is
presented
the
guidance
system
mak
es
the
airplane
follo
ws
pre-computed
references
using
a
no
v
el
iterati
v
e
model
predicti
v
e
scheme,
which
can
handle
the
nonlinear
optimization
problem
by
successi
v
e
linearizations
(starting
the
algorithm
using
a
rob
ust
l
1
na
vig
ation
la
w
.
On
the
other
hand,
in
[14]
is
presented
an
adapti
v
e
control
to
compensate
the
unkno
wn
parameters
of
an
unmanned
aerial
v
ehicle
with
x
ed-wing
in
normal
condition
ight,
the
control
objecti
v
e
is
to
achie
v
e
a
desired
speed
and
roll
angle,
and
after
that
to
track
desired
path
with
minimum
error
.
An
adapti
v
e
neuro-fuzzy
controller
is
presented
in
[15],
where
is
de
v
eloped
an
autonomous
ight
controller
for
x
ed-wing
U
A
V
based
on
adapti
v
e
neuro-fuzzy
inference
system
(ANFIS).
Three
ANFIS
modules
are
designed
for
controlling
the
altit
ude,
the
heading
angle,
and
the
speed
of
the
U
A
V
.
In
this
w
ay
,
the
U
A
V
position
is
controlled
in
three-dimensional
space:
altitude,
longitude,
and
latitude
position.
The
simulation
results
sho
w
the
capability
of
the
designed
approach
and
its
v
ery
satisf
actory
performance
with
good
stability
and
rob
ustness
ag
ainst
U
A
V
parametric
uncertainties
and
e
xternal
wind
disturbance.
Zhou
et
al.
[16]
is
presented
an
attitude
dynamic
model
of
unmanned
aerial
v
ehicles,
considering
a
strong
coupling
in
the
aerodynamic
model.
Model
uncertainties
and
e
xte
rnal
gust
disturbances
are
considered
during
designing
the
attitude
control
system
for
U
A
Vs
and
feedback
linearization
and
M
RA
C
are
inte
grated
to
design
the
attitude
control
system
for
a
x
ed-wing
U
A
V
.
Qiu
et
al.
[17]
is
presented
the
dynamics
and
attitude
control
of
a
mass-actuated
x
ed-wing
U
A
V
(MFU
A
V)
with
an
internal
slider
.
Based
on
the
deri
v
ed
mathematical
model
of
the
MFU
A
V
,
the
inuence
of
the
slider
parameters
on
the
dynamical
beha
vior
is
ana-
lyzed,
and
the
ideal
installation
position
of
the
slider
is
gi
v
en.
Besides,
it
is
re
v
ealed
that
the
mass-actuated
scheme
has
a
higher
control
ef
cienc
y
for
lo
w-speed
U
A
Vs.
T
o
deal
with
the
coupling,
uncertainty
,
and
dis-
turbances
in
the
dynamics,
an
adapti
v
e
sliding
mode
controller
based
on
fuzzy
system,
radial
basis
function
(RBF)
neural
netw
ork,
and
sliding
mode
control
are
proposed.
P
atel
and
Bhandari
[18]
is
presented
a
neu-
ral
netw
ork-based
nonlinear
adapti
v
e
controller
for
a
x
ed-wing
U
A
V
,
in
[18]
is
used
both
of
ine
and
online
trained
neural
netw
orks.
Multi-layer
perceptron
(MLP)
netw
orks
are
used
for
the
training
of
both
the
of
f-line
and
online
netw
orks.
Ev
en
in
the
scientic
literature,
we
can
nd
some
controllers
for
x
ed-wing
U
A
Vs
that
are
not
adap-
ti
v
e
controllers,
as
in
[19]
is
proposed
a
comprehensi
v
e
approach
combining
backstepping
with
PID
controllers
for
simultaneous
longitudinal
and
lateral-directional
control
of
x
ed-wing
U
A
Vs.
Kayacan
et
al.
[20]
is
pre-
sented
a
learning
control
s
trate
gy
is
preferred
for
the
control
and
guidance
of
a
x
ed-wing
unmanned
aerial
v
ehicle
to
deal
with
lack
of
modeling
and
ight
uncertai
nties.
F
or
learning
the
plant
model
and
changing
w
orking
conditions
online,
a
fuz
zy
neural
netw
ork
(FNN)
is
used
in
parallel
with
a
con
v
entional
proportional
(P)
controller
.
Among
the
learning
algorithms
in
the
literature,
a
deri
v
ati
v
e-free
one,
the
sliding
mode
control
(SMC)
theory-based
learning
algorithm,
is
preferred
as
it
has
been
pro
v
ed
to
be
computationally
ef
cient
in
real-time
applications.
On
the
other
hand,
in
[21]
is
presented
a
m
od
e
l-free
control
(MFC)
that
is
an
algorithm
dedicated
to
systems
with
poor
modeling
kno
wledge.
Indeed,
the
costs
to
deri
v
e
a
reliable
and
representati
v
e
aerodynamic
model
for
U
A
Vs
moti
v
ated
the
use
of
such
a
controller
.
W
e
can
see
that
e
v
ery
application
or
control
theory
applied
to
x
ed-wing
U
A
Vs
is
necessary
to
de
v
elop
an
altitude
control
la
w
.
Then,
in
this
w
ork,
our
control
objecti
v
e
is
to
design
an
altitude
controller
in
the
pres-
ence
of
perturbations
in
unmanned
x
ed-wing
miniature-aerial-v
ehicle
(MA
Vs);
the
perturbations
mentioned
are
the
wind
gusts.
Exists
al
titude
controllers
with
g
ains
denite
x,
b
ut
the
problem
with
such
controllers
is
that
it
w
orks
in
specic
altitudes
(x
ight
points).
On
the
other
side,
adapti
v
e
controllers
e
xist
that
can
w
ork
in
dif
ferent
altitude
points
b
ut
present
some
problems
in
k
eeping
control
objecti
v
es
in
perturbations.
So
in
this
w
ork,
we
ha
v
e
proposed
an
adapti
v
e
controller
that
can
lead
an
unmanned
x
ed-wing
MA
V
to
dif
ferent
altitudes
in
the
presence
of
wind
gusts
(perturbations).
As
is
mentioned
in
[9]
the
problem
to
resolv
e
an
model
reference
adapti
v
e
system
(MRAS)
is
to
determine
the
adjustment
mechanism
to
stabilize
the
system
and
which
achie
v
es
the
error
to
zero.
Then
a
solution
to
this
problem
is
the
de
v
elopment
of
a
proportional-deri
v
ati
v
e
(PD)
controller
with
adapti
v
e
g
ains.
This
adaptation
is
based
on
the
adapti
v
e
scheme
kno
wn
as
MRAS.
Then,
to
achie
v
e
the
control
objecti
v
e,
we
ha
v
e
designed
a
rob
ust
adjustment
mechanism
for
the
adapti
v
e
g
ains
of
a
PD
controller
.
Our
proposal
to
design
it
is
using
the
MIT
rule,
an
approach
to
model-reference
adapti
v
e
control
and
gradient
method
with
sliding
Evaluation Warning : The document was created with Spire.PDF for Python.
310
❒
ISSN:
2089-4856
mode
theory
.
The
obta
ined
rob
ust
adapti
v
e
mechanism
for
the
adapti
v
e
controller
PD
is
going
to
compare
with
the
kno
wn
adapti
v
e
mechanism
de
v
eloped
in
[9],
that
is,
to
demonstrate
the
adv
antages
in
the
error
and
the
control
ef
fort
concerning
de
v
eloped
in
this
w
ork.
The
or
g
anization
of
the
document
is
the
follo
wing:
in
the
section
2.
is
presented
the
longitudinal
model
which
denes
the
x
ed-wing
MA
V
and
in
the
section
3.
is
sho
wn
the
design
of
the
adapti
v
e
mechanism
and
the
PD
controller
.
In
section
4.
is
presented
the
simulation
results
obtained,
and
nally
,
section
5.
presents
the
conclusions
and
the
future
w
ork.
2.
LONGITUDIN
AL
MODEL
T
o
re
gulate
the
altitude
of
the
x
ed-wing
MA
V
is
used
the
aerodynamic
model
which
denes
the
longitudinal
model
of
an
airplane.
Then,
this
aerodynamic
model
has
been
obtained
based
on
t
h
e
second
mo
v
ement
la
w
of
Ne
wton;
some
considerations
are
tak
en
for
the
model
obtention,
that
is,
the
earth
is
considered
as
plane
due
to
the
x
ed-wing
MA
V
is
going
to
y
short
distances,
and
is
not
consider
an
y
e
xible
part
in
the
airplane
for
the
dynamic
model.
Then,
the
longitudinal
model
of
the
airplane
has
been
dened
as
(1),
(2),
(3),
(4)
and
(5).
˙
V
=
1
m
(
−
D
+
T
cos
α
−
mg
sin
γ
)
(1)
˙
γ
=
1
mV
(
L
+
T
sin
α
−
mg
)
sin
γ
)
(2)
˙
θ
=
q
(3)
˙
q
=
M
q
q
+
M
δ
e
δ
e
(4)
˙
h
=
V
sin(
θ
)
(5)
Where
V
is
the
airplane
speed,
α
describes
the
angle
of
attack,
γ
represents
the
ight-path
angle,
and
θ
denotes
the
pitch
angle.
In
addition,
q
is
the
pitch
angular
rate
(concerning
the
y
-axis
of
the
aircraft
body),
T
denotes
the
force
of
engine
thr
u
s
t,
h
is
the
airplane
altitude
[22],
[23]
and
δ
e
represents
the
ele
v
ator
de
viation.
The
aerodynamic
ef
fects
on
the
airplane
are
obtained
by
the
lift
force
L
and
the
drag
force
D
.
The
total
mass
of
the
airplane
is
denoted
by
m
,
g
is
the
gra
vitational
constant,
and
I
y
y
describes
the
component
y
of
the
diagonal
of
the
inertial
matrix.
The
v
alue
of
the
angle
of
attack
is
obtained
by
using
the
follo
wing
relation
α
=
θ
−
γ
[22],
the
Figure1
sho
ws
the
v
ariables
implies
in
the
pure
pitch
motion
to
apply
control
in
altitude.
In
aerodynamics,
M
q
and
M
δ
e
are
the
stability
deri
v
ati
v
es
implicit
in
the
pitch
motion.
The
lift
force
L
,
the
drag
force
D
are
dened
as
(6)
and
(7)
[22],
[23].
L
=
¯
q
S
C
L
(6)
D
=
¯
q
S
C
D
(7)
The
aerodynamic
stability
deri
v
ati
v
es
are
dened
by:
where
¯
q
denotes
aerodynamic
pressure.
S
represents
the
wing
platform
area,
and
¯
c
is
the
mean
aerodynamic
chord.
C
D
and
C
L
are
the
aerodynamic
coef
cients
for
drag
force
and
lift
force,
respecti
v
ely
.
M
q
=
ρS
V
¯
c
2
4
I
y
y
C
m
q
M
δ
e
=
ρV
2
S
¯
c
2
I
y
y
C
m
δ
e
Where:
ρ
:
Air
density
(1.05
kg/m
3
).
S
:
W
ing
area
(0.09
m
2
).
¯
c
:
Standard
mean
chord
(0.14
m
).
b
:
W
ingspan,
(0.914
m
).
I
y
y
:
Moment
of
inertia
in
pitch
(0.17
k
g
·
m
2
).
C
m
q
:
Dimensionless
coef
cient
for
longitudinal
mo
v
ement,
it
is
obtained
e
xperimentally
(-50).
C
m
δ
e
:
Dimensionless
coef
cient
for
ele
v
ator
mo
v
ement,
it
is
obtained
e
xperimentally
(0.25).
Int
J
Rob
&
Autom,
V
ol.
10,
No.
4,
December
2021
:
308
–
318
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Rob
&
Autom
ISSN:
2089-4856
❒
311
L
W
D
V
q
±
e
Figure
1.
Pure
pitching
motion
3.
CONTR
OLLER
DESIGN
T
o
design
the
adapti
v
e
controller
for
altitude,
we
ha
v
e
considered
the
(3),
(4)
and
(5),
this
is
due
to
which
the
(1)
represents
the
v
elocity
of
the
airplane.
Still,
for
the
simulations
of
this
w
ork,
it
is
considered
as
constant,
and
the
(2)
is
the
ight
path
produced
by
the
wind.
In
this
w
ork,
we
are
designing
the
control
la
w
o
v
er
the
solid
(aircraft
body).
F
or
that
reason,
the
control
la
w
is
designed
without
considering
the
wind
equations
which
dene
the
a
irplane
dynamics.
Then,
the
altitude
error
is
dened
as
˜
e
h
=
h
d
−
h
,
where
h
d
is
the
desired
altitude
and
h
is
the
actual
altitude.
The
desired
altitude
is
achie
v
ed
by
controlling
the
pitch
angle.
Thus
we
ha
v
e
dened
an
error
for
this
angle,
gi
v
en
by
˜
e
θ
=
θ
d
−
θ
(
t
)
,
where
θ
d
=
ar
c
tan
(
˜
e
h
/ς
)
is
the
des
ired
pitch
angle,
and
ς
denotes
the
longitude
from
the
center
of
mass
of
the
miniature
aerial
v
ehicle
to
the
nose
of
it.
Consider
the
equations
(3)
and
(4),
δ
e
denes
the
control
input.
Thus,
The
adapti
v
e
control
is
gi
v
en
by
(8).
δ
e
=
ˆ
k
pa
˜
e
θ
+
ˆ
k
v
a
˙
˜
e
θ
(8)
Where
ˆ
k
pa
and
ˆ
k
v
a
are
called
as
the
position
and
v
elocity
g
ains,
respecti
v
ely
,
these
are
the
adapti
v
e
g
ains.
The
g
ains
of
the
PD
control
ha
v
e
implicit
a
subscript
to
indicate
the
algorithm
that
has
been
applied
as
adjustment
mechanism,
a
1
corresponds
to
the
MIT
rule,
a
2
corresponds
to
the
MIT
rule
with
sliding-mode,
a
3
uses
the
MIT
rul
e
with
2-sliding-mode,
and
a
4
represents
the
MIT
rule
with
HOSM.
Therefore,
for
the
design
of
the
MIT
rule,
it
is
introduced
an
error
gi
v
en
by
(9).
e
θ
m
=
θ
m
−
θ
(9)
Where
θ
m
is
the
output
from
the
reference
model,
we
ha
v
e
follo
wed
the
methodology
that
has
been
presented
in
[9]
for
the
MIT
rule,
taking
this
into
account,
the
aerodynamic
model
has
been
transformed
into
the
rep-
resentation
of
a
transference
function
to
de
v
elop
the
deri
v
ati
v
es
of
sensiti
vity;
t
hese
ha
v
e
been
obtained
by
computing
partial
deri
v
ati
v
es
concerning
the
controller
parameters
ˆ
k
pa
and
ˆ
k
v
a
.
Thus,
the
closed-loop
transfer
function
with
the
adapti
v
e
PD
controller
has
been
dened
as
(10).
θ
=
M
δ
e
(
ˆ
k
p
+
ˆ
k
v
s
)
s
2
+
(
M
q
+
M
δ
e
ˆ
k
v
)
s
+
M
δ
e
ˆ
k
p
θ
d
(10)
And
the
model
of
reference
for
the
altitude
dynamics
has
been
dened
as
(11).
θ
m
=
ω
2
n
s
2
+
2
ζ
ω
n
s
+
ω
2
n
θ
d
(11)
Where
ζ
=
3
.
17
and
ω
=
3
.
16
.
Considering
(9),
(10)
and
(11)
and
calculating
the
partial
deri
v
ati
v
es
with
respect
to
ˆ
k
pa
and
ˆ
k
v
a
,
it
is
obtained
as
(12)
and
(13).
∂
e
θ
m
∂
ˆ
k
p
=
M
δ
e
s
2
+
(
M
q
+
M
δ
e
ˆ
k
v
)
s
+
M
δ
e
ˆ
k
p
(
θ
−
θ
d
)
(12)
∂
e
θ
m
∂
ˆ
k
v
=
M
δ
e
s
s
2
+
(
M
q
+
M
δ
e
ˆ
k
v
)
s
+
M
δ
e
ˆ
k
p
(
θ
−
θ
d
)
(13)
Evaluation Warning : The document was created with Spire.PDF for Python.
312
❒
ISSN:
2089-4856
Generally
,
the
e
xpressions
(12)
and
(13)
cannot
be
used
due
to
the
unkno
wn
parameters
ˆ
k
pa
and
ˆ
k
v
a
.
So
that,
an
optimum
case
has
been
assumed
and
it
is
dened
as
(14).
s
2
+
(
M
q
+
M
δ
e
ˆ
k
v
l
)
s
+
M
δ
e
ˆ
k
pl
=
s
2
+
2
ζ
ω
n
s
+
ω
2
n
(14)
thus,
after
these
approximations,
we
ha
v
e
obtained
the
dif
ferential
equations
of
the
adapti
v
e
PD
controller
.
˙
ˆ
k
pa
1
=
−
γ
1
1
s
2
+
2
ζ
ω
n
s
+
ω
2
n
(
θ
−
θ
d
e
θ
m
(15)
˙
ˆ
k
v
a
1
=
−
γ
2
s
s
2
+
2
ζ
ω
n
s
+
ω
2
n
(
θ
−
θ
d
)
e
θ
m
(16)
No
w
,
it
is
proposed
an
MIT
rule
with
second-order
sliding
mode;
this
approach
is
dif
ferent
t
han
the
dened
in
[9].
Thus,
it
is
dened
a
sliding-mode
surf
ace
as
s
1
=
˙
θ
m
−
q
+
k
1
e
θ
m
(we
are
searching
increase
the
stability
of
the
adjustment
mechanism),
where
k
1
is
a
positi
v
e
g
ain.
Then,
the
dif
ferential
equations
of
the
adapti
v
e
controller
,
with
the
methodology
by
sliding-mode,
are
gi
v
en
by
(17)
and
(18).
˙
ˆ
k
pa
2
=
−
γ
1
1
s
2
+
2
ζ
ω
n
s
+
ω
2
n
(
θ
−
θ
d
)
(
β
p
(
s
1
))
(17)
˙
ˆ
k
v
a
2
=
−
γ
2
s
s
2
+
2
ζ
ω
n
s
+
ω
2
n
(
θ
−
θ
d
)
(
β
v
(
s
1
))
(18)
Where
β
p
and
β
v
are
positi
v
e
v
alues.
Due
to
the
chattering
ef
fect
of
the
rst
order
sliding-mode,
let
us
design
an
adjustment
mechanism
with
a
second-order
sliding
mode.
This
second-order
sliding
mode
includes
a
rob
ust
dif
ferentiator
of
rst-order
[24].
This
dif
ferentiator
is
dened
by
(19).
˙
x
0
=
v
0
=
−
λ
0
|
x
0
−
s
1
|
1
/
2
(
x
0
−
s
1
)
+
x
1
˙
x
1
=
−
λ
1
(
x
1
−
v
0
)
(19)
Where
x
0
and
x
1
are
real-time
estimations
of
s
1
and
˙
s
1
,
respecti
v
ely
.
The
v
alues
of
λ
1
and
λ
2
are
positi
v
es
and
constants.
Thus,
the
dif
ferential
equations
of
the
adapti
v
e
PD
controller
with
a
second-
order
sliding
mode
are
dened
by
(20)
and
(21).
˙
ˆ
k
pa
3
=
−
γ
1
1
s
2
+
2
ζ
ω
n
s
+
ω
2
n
(
θ
−
θ
d
)
(
β
p
(
s
1
)
+
β
p
2
(
˙
s
1
))
(20)
˙
ˆ
k
v
a
3
=
−
γ
2
s
s
2
+
2
ζ
ω
n
s
+
ω
2
n
(
θ
−
θ
d
)
(
β
v
(
s
1
)
+
β
v
2
l
(
˙
s
1
))
(21)
Where
β
p
,
β
p
,
β
v
and
β
v
are
positi
v
e
denite
g
ains.
T
o
reduce
or
eliminate
the
chattering
ef
fect
in
the
second-order
sliding
mode,
we
h
a
v
e
des
igned
an
adjustment
mechanism
with
HOSM.
T
o
design
the
adjustment
mechanism,
it
i
s
necessary
a
rob
ust
dif
ferentiator
of
second-order
[24],
which
is
gi
v
en
by
(22).
˙
x
0
=
v
0
=
−
λ
0
|
x
0
−
s
1
|
2
/
3
(
x
0
−
s
1
)
+
x
1
˙
x
1
=
v
1
=
−
λ
1
|
x
1
−
v
0
|
1
/
2
(
x
1
−
v
0
)
+
x
2
(22)
˙
x
2
=
−
λ
2
|
x
2
−
v
1
|
Where
x
0
,
x
1
y
x
2
are
real-time
estimations
of
s
1
,
˙
s
1
and
¨
s
1
.
The
v
alues
of
λ
0
,
λ
1
and
λ
2
are
dened
as
positi
v
e
constants.
Finally
,
the
dif
ferential
equations
of
the
adapti
v
e
PD
controller
with
HOSM
are
dened
by
(23)
and
(24).
Int
J
Rob
&
Autom,
V
ol.
10,
No.
4,
December
2021
:
308
–
318
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Rob
&
Autom
ISSN:
2089-4856
❒
313
˙
ˆ
k
pa
4
=
−
γ
1
1
s
2
+
2
ζ
ω
n
s
+
ω
2
n
(
θ
−
θ
d
)
(
α
pl
[
¨
s
1
l
+
2(
|
˙
s
1
l
|
3
+
|
s
1
l
|
2
)
1
/
6
(
˙
s
1
l
+
|
s
1
l
|
2
/
3
(
s
1
l
))])
(23)
˙
ˆ
k
v
a
4
=
−
γ
2
s
s
2
+
2
ζ
ω
n
s
+
ω
2
n
(
θ
−
θ
d
)
(
α
v
[
¨
s
1
+
2(
|
˙
s
1
|
3
+
|
s
1
|
2
)
1
/
6
(
˙
s
1
+
|
s
1
|
2
/
3
(
s
1
))])
(24)
Where
α
p
and
α
v
are
positi
v
e
and
constant
g
ains.
4.
SIMULA
TION
RESUL
TS
T
o
describe
the
simulations
results
with
the
MIT
-rule
with
sliding
mode
theory
,
we
ha
v
e
analyzed
the
results
with
the
L
2
-norm
[25],
that
is,
to
analyze
the
error
signals
and
the
control
ef
fort
with
the
dif
ferents
adapti
v
e
mechanism
proposed.
Then,
we
ha
v
e
applied
the
L
2
-norm
to
the
error
(25):
L
2
[
e
h
]
=
s
1
T
−
t
0
Z
T
t
0
∥
e
h
∥
2
dt
(25)
The
L
2
-norm
is
also
used
to
obtain
the
ef
fort
of
the
control
la
w
,
and
it
is
dened
as
(26):
L
2
[
δ
e
]
=
s
1
T
−
t
0
Z
T
t
0
∥
δ
e
∥
2
dt
(26)
Thus,
with
the
use
of
the
(25)
and
(26)
are
obtained
the
errors
and
ef
forts,
see
the
T
able
1.
T
able
1.
L
2
-norm
for
the
errors
and
the
ef
forts
of
the
control
la
ws
on
the
altitude
mo
v
ement
Adapti
v
e
mechanism
Altitude
[m]
L
2
[
e
h
]
L
2
[
δ
e
]
MIT
1.2949
0.2876
MIT
-SM
1.2913
0.2689
MIT
-2SM
1.0856
0.2362
MIT
-HOSM
1.0773
0.2519
The
simulations
results
for
the
altitude
control
applying
the
MIT
rule
[9],
are
presented
in
the
Figure
2,
in
the
upper
graphic
of
the
Figure
2
is
presented
the
response
of
the
MIT
rule
and
in
the
lo
wer
graphic
of
the
same
gure,
sho
ws
the
controller
response.
Analyzing
the
results
obtained
in
the
T
able
1
is
appreciated
that
the
PD
controller
with
the
adapti
v
e
mechanism
based
on
the
MIT
rule
has
presented
more
error
than
the
MIT
with
the
sliding
mode
theory
,
that
is,
the
MIT
rule
is
0
.
278%
,
16
.
1635%
and
16
.
8044%
bigger
than
MIT
rule
with
sliding
mode
(MIT
-SM),
the
MIT
rule
with
tw
o
sliding
modes
(MIT
-2SM)
and
the
MIT
rule
with
high
order
sliding
mode
(MIT
-HOSM),
respecti
v
ely
.
Meanwhile,
the
PD
control
ef
fort
with
the
MIT
rule
is
bigger
than
the
other
technique
in
the
st
udy
,
that
is,
with
the
adapti
v
e
mechanism
by
the
MIT
rule,
the
PD
control
ef
fort
is
6
.
5021%
,
17
.
8721%
and
12
.
4131%
bigger
than
the
MIT
-SM,
the
MIT
-2SM
and
the
MIT
-HOSM,
respecti
v
ely
(see
the
T
able
1).
On
the
other
hand,
the
error
of
the
PD
controller
with
the
adapti
v
e
mechanism
based
on
the
MIT
rule
with
the
sliding
mode
is
15
.
9297%
bigger
than
the
MIT
-SM
and
is
16
.
5725%
bigger
than
the
MIT
-HOSM.
The
results
obtained
with
the
adapti
v
e
mechanism
bas
ed
on
MIT
rule
with
sliding
mode
are
presented
in
the
Figure
3,
where
the
upper
graphic
of
the
same
gure
we
can
appreciate
the
con
v
er
gence
to
the
desired
v
alues
in
spite
of
the
noise
applied
in
the
control
system.
In
T
able
1
we
can
see
that
the
PD
controller
with
the
adapti
v
e
mechanism
based
on
the
MIT
rule
with
sliding
mode
applies
a
control
signal
12
.
1607
bigger
than
the
adapti
v
e
mechanism
based
on
the
MIT
rul
e
with
Evaluation Warning : The document was created with Spire.PDF for Python.
314
❒
ISSN:
2089-4856
tw
o
sliding
modes
(MIT),
and
e
v
en
the
adapti
v
e
mechanism
with
the
MIT
rule
with
the
sliding
mode
the
control
ef
fort
is
6
.
3221%
bigger
than
the
MIT
-HOSM.
In
the
lo
wer
graphic
of
the
Figure
3
is
sho
wn
the
control
signal
generated
by
the
PD
with
the
adapti
v
e
mechanism
based
on
the
MIT
rule
with
the
sliding
mode.
Time (s)
0
100
200
300
400
500
600
700
800
900
1000
Altitude (m)
-2
0
2
4
6
Altitude (MIT-Sign)
Reference
Actual altitude
Time (s)
0
100
200
300
400
500
600
700
800
900
1000
u
θ
(deg)
-20
-10
0
10
20
Control signal MIT-Sign
Figure
2.
Adapti
v
e
mechanism
based
on
the
MIT
-rule
with
sign
function
Time (s)
0
100
200
300
400
500
600
700
800
900
1000
Altitude (m)
-2
0
2
4
6
Altitude (MIT-SM)
Reference
Actual altitude
Time (s)
0
100
200
300
400
500
600
700
800
900
1000
u
θ
(deg)
-20
-10
0
10
20
Control signal MIT-SM
Figure
3.
Adapti
v
e
mechanism
based
on
the
MIT
-rule
with
sliding
mode
Figure
4
is
presented
the
results
obtained
by
the
PD
controller
based
on
the
MIT
-2SM,
in
the
upper
graphic
is
appreciated
the
response
of
the
PD
controller
with
the
adapti
v
e
mechanism
based
on
the
MIT
-2SM.
In
T
able
1
we
can
see
that
the
PD
controller
based
on
the
MIT
-2SM
has
presented
an
error
15
.
9297%
bigger
than
the
MIT
-HOSM,
b
ut
the
adapti
v
e
mechanism
based
on
the
MIT
with
tw
o
sliding
modes
has
pre-
sented
a
PD
control
ef
fort
6
.
2327%
smaller
than
the
MIT
-HOSM.
In
the
lo
wer
graphic
of
Figure
4
is
presented
the
response
of
adapti
v
e
mechanism
based
on
the
MIT
rule
with
tw
o
sliding
modes.
Meanwhile,
the
response
of
PD
controller
with
the
adapti
v
e
mechanism
based
on
the
MIT
-HOSM
is
presented
in
Figure
5,
in
the
upper
graphic
of
the
same
gure
is
sho
wn
the
con
v
er
gence
to
the
desired
v
alues
and
in
the
upper
graphic
is
presented
the
controller
response.
Int
J
Rob
&
Autom,
V
ol.
10,
No.
4,
December
2021
:
308
–
318
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Rob
&
Autom
ISSN:
2089-4856
❒
315
The
PD
controller
based
on
the
MIT
-HOSM
has
a
lo
wer
error
in
comparison
with
the
other
adapti
v
e
mechanisms
presented
in
this
w
ork
and
e
v
en
presents
a
smaller
control
action
when
is
compared
with
the
PD
controller
with
the
adapti
v
e
mechanism
based
on
the
MIT
r
ule
and
with
them
based
on
the
MIT
rule
with
sliding
mode.
An
e
xception
occurs
when
it
is
compared
with
the
adapti
v
e
mechanism
based
on
the
MIT
rule
with
tw
o
sliding
modes
(see
T
able
1).
And
nally
,
the
adv
antage
of
using
the
PD
controller
with
the
adapti
v
e
mechanism
based
on
the
MIT
rule
with
high
order
sliding
mode
is
the
reduction
in
the
undesired
chattering
ef
fect
in
the
control
signal,
the
e
v
olution
of
the
chattering
reduce
e
v
en
with
the
perturbation
in
the
system,
this
can
be
appreciated
in
the
Figure
6.
Time (s)
0
100
200
300
400
500
600
700
800
900
1000
Altitude (m)
-2
0
2
4
6
Altitude (MIT-2SM)
Reference
Actual altitude
Time (s)
0
100
200
300
400
500
600
700
800
900
1000
u
θ
(deg)
-20
-10
0
10
20
Control signal MIT-2SM
Figure
4.
Adapti
v
e
mechanism
based
on
the
MIT
-rule
with
tw
o
sliding
mode
Time (s)
0
100
200
300
400
500
600
700
800
900
1000
Altitude (m)
-4
-2
0
2
4
6
Altitude (MIT-HOSM)
Reference
Actual altitude
Time (s)
0
100
200
300
400
500
600
700
800
900
1000
u
θ
(deg)
-20
-10
0
10
20
Control signal MIT-HOSM
Figure
5.
Adapti
v
e
mechanism
based
on
the
MIT
-rule
with
HOSM
Evaluation Warning : The document was created with Spire.PDF for Python.
316
❒
ISSN:
2089-4856
Tiempo (s)
0
100
200
300
400
500
600
700
800
900
1000
u
θ
(deg)
-1
-0.5
0
0.5
1
Control signal MIT-Sign
Time (s)
0
100
200
300
400
500
600
700
800
900
1000
u
θ
(deg)
-1
-0.5
0
0.5
1
Control signal MIT-SM
Time (s)
0
100
200
300
400
500
600
700
800
900
1000
u
θ
(deg)
-1
-0.5
0
0.5
1
Control signal MIT-2SM
Time (s)
0
100
200
300
400
500
600
700
800
900
1000
u
θ
(deg)
-1
-0.5
0
0.5
1
Control signal MIT-HOSM
Figure
6.
Control
signals
zoom
5.
CONCLUSION
The
adapti
v
e
mechanism
based
on
the
MIT
rule
presented
an
error
and
control
ef
fort
bigger
than
the
MIT
rule
wi
th
the
sliding
mode
techniques.
Despite
it,
the
adapti
v
e
controller
with
the
MIT
rule
as
an
adapti
v
e
mechanism
for
the
controlle
r
g
ains
achie
v
es
the
desired
altitude.
The
adapti
v
e
mechanism
based
on
the
MIT
rule
with
high
order
sliding
mode
has
presented
a
better
performance
than
the
other
adapti
v
e
mechanisms
presented
in
this
w
ork,
considering
that
the
altitude
error
is
the
smallest.
Ev
en
this
adapti
v
e
mechanism
for
the
PD
controller
has
presented
less
control
ef
fort
than
the
adapti
v
e
mechanisms
based
on
the
MI
T
rule
and
MIT
rule
with
sliding
mode.
The
PD
controller
with
the
adapti
v
e
mechanism
based
on
the
MIT
rule
with
high
order
sliding
mode
has
presented
a
considerable
reduction
of
the
chattering
ef
fect
.
The
future
w
ork
consists
of
the
implementation
(real-time
ight
tests)
of
this
technique
in
a
miniature
aerial
v
ehicle
to
analyze
the
performance
of
the
PD
controller
with
the
adapti
v
e
mechanisms
proposed
in
this
w
ork.
A
CKNO
WLEDGMENT
The
authors
w
ould
lik
e
to
thank
F
acultad
de
Ingenier
´
ıa,
Ciencias
y
Arquitectura
From
the
Uni
v
ersidad
Ju
´
arez
del
Estado
de
Durango
for
the
support
during
the
de
v
elopment
of
this
w
ork.
Int
J
Rob
&
Autom,
V
ol.
10,
No.
4,
December
2021
:
308
–
318
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Rob
&
Autom
ISSN:
2089-4856
❒
317
REFERENCES
[1]
R.
Beard
and
T
.
Mclai
n,
Small
unmanned
aircraft:
theory
and
practice
,
United
Kingdom:
Princeton
Uni
v
ersity
Press,
2012.
[2]
J.
Guerrero
and
R.
Lozano,
Flight
formation
control
,
USA:
W
ile
y
,
2012,
[3]
T
.
Espinoza,
A.
Saenz,
S.
Del
Rio,
F
.
G
´
omez
and
V
.
L
´
opez,
Aplicaciones
de
los
Drones
en
Ingenier
´
ıa
Ci
vil
,
Me
xico:Alf
aome
g
a,
2021.
[4]
T
.
Espinoza,
A.Dzul,
R.
Lozano
and
P
.
P
arada,
“Backstepping-Sliding
Mode
Controllers
Applied
to
a
Fix
ed-W
ing
U
A
V
,
”
J
ournal
of
Intellig
ent
Robotic
Systems
,
v
ol.
73,
pp.
67-79,
2014,
doi:
10.1007/s10846-
013-9955-y
.
[5]
M.
ElMikaty
and
T
.
Stathaki,
“Car
Detection
in
Aerial
Images
of
Dense
Urban
Areas,
”
in
IEEE
T
r
ansactions
on
Aer
ospace
and
Electr
onic
Systems
,
v
ol
54,
no.
1,
pp.
51-63,
2018,
doi:
10.1109/T
AES.2017.2732832.
[6]
Y
.
Shen,
Z.
Rahman,
D.
Krusienski
and
J.
Li,
“
A
vision-based
automatic
safe
landing-site
detection
sys-
tem,
”
in
IEEE
T
r
ansactions
on
Aer
ospace
and
Electr
onic
Systems
,
v
ol.
49,
no.
1,
pp.
294-311,
2013,
doi:
10.1109/T
AES.2013.6404104.
[7]
A.
A.
Ahmed,
E.
A.
Abdelmageed
and
M.
Mohammed,
“
A
No
v
el
Model
Reference
Adapti
v
e
Controller
Design
for
a
Second
Order
System,
”
2015
International
Confer
ence
on
Computing
,
Contr
ol,
Network-
ing
,
Electr
onics
and
Embedded
Systems
Engineering
(ICCNEEE)
,
2015,
pp.409-413,
doi:
10.1109/ICC-
NEEE.2015.7381402.
[8]
R.
J.
P
a
w
ar
and
B.
J.
P
arv
at,
“Design
and
Implementation
of
MRA
C
and
Modied
MRA
C
technique
for
In
v
erted
Pendulum,
”
2015
International
Confer
ence
on
P
ervasive
Computing
(ICPC)
,
2015,
pp.
1-6,
doi:
10.1109/PER
V
ASIVE.2015.7087168.
[9]
K.
J.
˚
Astr
¨
om,
B.W
ittenmark,
Adapti
v
e
Control
,
2nd
Edition,
Prentice
Hall,
1994.
[10]
N.
Amuthan
and
S.
Singh,
“Direct
Model
Reference
Adapti
v
e
Internal
Model
Controller
using
Per
-
rin
equation
Adjustment
Mechanism
for
DFIG
W
ind
F
arms,
”
2008
IEEE
Re
gion
10
and
the
Thir
d
international
Confer
ence
on
Industrial
and
Information
Systems
,
2008,
pp.
1-6,
doi:
10.1109/ICI-
INFS.2008.4798352.
[11]
J.
Priyank
and
M.
Nig
am,
“Design
of
a
Model
Reference
Adapti
v
e
Controller
Using
Modied
MIT
Rule
for
a
Second
Order
System,
”
Advance
in
Electr
onic
and
Electric
Engineering
,
v
ol.
3,
no.
4,
pp.
477-484,
2013.
[12]
S.
Riache,
M.
Kidouche
and
A.
Rezoug,
“
Adapti
v
e
rob
ust
nonsingular
terminal
sliding
mode
design
controller
for
quadrotor
aerial
manipulator
,
”
TELK
OMNIKA
(T
elecommunication
Computing
Electr
onics
and
Contr
ol)
,
v
ol.
17,
no.
3,
pp.
1501-1512,
2019,
doi:
10.12928/telk
omnika.v17i3.10470.
[13]
S.
Ga
vilan
,
R.
V
azquez
and
S.
Esetban,
“T
rajectory
tracking
for
x
ed-wing
U
A
V
using
model
predic-
ti
v
e
control
and
adapti
v
e
backsttepping,
”
IF
A
C-P
aper
sOnLine
,
v
ol.
48,
no.
9,
pp.
132-137,
2015,
doi:
10.1016/j.if
acol.2015.08.072.
[14]
H.
M.
Nasab
and
N.
Na
v
azani,
“
Adapti
v
e
control
for
trajectory
tracking
of
an
Unmanned
Aerial
V
ehicle,
”
Advanced
Engineering
F
orum
,
v
ol.
17,
pp.
101-110,
2016,
doi:
10.4028/www
.scientic.net/AEF
.17.101.
[15]
A.
Serhan
and
S.
Qin,
“
Autonomous
intelligent
ight
control
of
x
ed-wing
U
A
V
based
on
adapti
v
e
neuro-
fuzzy
inference
system,
”
International
J
ournal
of
Resear
c
h
in
Engineering
and
T
ec
hnolo
gy
,
v
ol.
5,
no.
9,
pp.
92-100,
2016,
doi:
10.15623/IJRET
.2016.0509014.
[16]
W
.
Zhou,
K.
Y
in,
R.
W
ang
and
Y
.
W
ang,
“Design
of
attitude
control
system
for
U
A
V
based
on
feedback
linearization
and
adapti
v
e
control,
”
Mathematical
Pr
oblems
in
Engineering
,
v
ol.
2014,
p.
492680,
2014,
doi:
10.1155/2014/492680.
[17]
X.
Qiu,
M.
Zhang,
W
.
Jing
and
C.
Gao,
“Dynamics
and
adapti
v
e
sliding
mode
control
of
a
mass-actuated
x
ed-wing
U
A
V
,
”
International
J
ournal
of
Aer
onautical
and
Space
Sciences
volume
,
v
ol.
22,
pp.
886–897,
2021,
doi:10.1007/s42405-020-00344-w
.
[18]
N.
P
atel
and
S.
Bhandari,
“Rob
ust
nonlinear
adapti
v
e
control
of
a
x
ed-wing
U
A
V
using
multilayer
per
-
ceptrons,
”
AIAA
Guidance
,
Navigation,
and
Contr
ol
Confer
ence
,
2017,
pp.1-16,
doi:
10.2514/6.2017-
1524.
[19]
D.
Sartori,
F
.
Quagliotti,
M.
Rutherford
and
K.
V
ala
v
anis,
“Implementation
and
testing
of
a
backstepping
controller
autopilot
for
x
ed-wing
U
A
Vs,
”
J
ournal
of
Intellig
ent
Robotic
Systems
volume
,
v
ol.
76,
pp.
505-525,
2014,
doi:
10.1007/s10846-014-0040-y
.
[20]
E.
Kayacan,
A.
Khanesar
,
J.
Herv
as
and
M.
Re
yhanoglu,
“Learning
control
of
x
ed-wing
unmanned
Evaluation Warning : The document was created with Spire.PDF for Python.