Neural Network Control Based on Adaptive Observer for Quadrotor Helicopter

International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.3, July 2012
DOI:10.5121/ijitca.2012.2304 39
Neural Network Control Based on Adaptive
Observer for Quadrotor Helicopter
Hana Boudjedir1
, Omar Bouhali1
and Nassim Rizoug2
1
LAJ Lab, Automatic department, Jijel University, Algeria
hana_boudjedir@yahoo.fr, bouhali_omar@yahoo.fr
2
Mecatronic Lab, ESTACA School, Laval, France
nrizoug@yahoo.fr
ABSTRACT
A neural network control scheme with an adaptive observer is proposed in this paper to Quadrotor
helicopter stabilization. The unknown part in Quadrotor dynamical model was estimated on line by a
Single Hidden Layer network. To solve the non measurable states problem a new adaptive observer was
proposed. The main purpose here is to reduce the measurement noise amplification caused by conventional
high gain observer by introducing some changes in observer’s original structure that can minimize the
variance and the amplitude of the noisy signal without increasing tracking error. The stability analysis of
the overall closed-loop system/ observer is performed using the Lyapunov direct method. Simulation results
are given to highlight the performances of the proposed scheme.
KEYWORDS
Adaptive observer; high gain observer; Neural Network control; Quadrotor; Lyapunov stability.
1. INTRODUCTION
As their application potential both in the military and industrial sector strongly increases,
miniature unmanned aerial vehicles (UAV) constantly gain in interest among the research
community [1]. Quadrotor Helicopter is considered as one of the most popular UAV platform.
The main reasons for all this attention have stemmed from its simple construction and its large
payload as compared with the conventional helicopter [2].
The Quadrotor is an under actuated system from where it has six degrees of freedom controlled
only by four control inputs. To solve the Quadrotor UAV tracking control problem many
techniques have been proposed [2-7] where the control objective is to control three desired
Cartesian positions and a desired yaw angle.
In [1-3] Backstepping and sliding mode techniques have been used to control the Quadrotor. In
[4] the H∞ robust control law was proposed and a PID and LQR controls have been applied in
[5]. Unfortunately these techniques are limited if the dynamic model is completely or partially
unknown and/or if the states variables are unavailable.

40
The unknown nonlinear functions problem in the control law can be solved by using adaptive
control technique based on universals approximators as artificial neural network [6]. Several
Neural Networks adaptive schemes have been proposed for Quadrotor control [6-10].
To avoid the state variables measurement assumptions, state observers can be introduced in the
control scheme such as in [9] where a neuro-sliding mode observer was proposed in order to
reduce the noise measurement amplification. In [10] other observer was used to estimate the
velocity, the corrective term here is the superposition of a observation errors proportional term
and a feed-forward neural networks output used to estimate unknown functions.
High-gain observers have evolved over the past two decades as an important tool for the design of
output feedback control of nonlinear systems [11]. However this observer has two drawbacks, the
first one is the peak phenomena and the second one is its high sensitivity to noise measurements.
Several methods have been proposed in the literature to reduce the last phenomena as in [11]
were two observation gains values have been used. The biggest value was used in the transient
response and the smallest observation gain was used in permanent regime. In [12] and [13] an
adaptation laws is utilized to compute the optimal observation gain. However those methods
present some limitations in particular when the observation gain has to be too large as in the case
of quadrotor dynamic.
In the present paper a new robust adaptive scheme that does not require any prior information on
model dynamics and state measurement is proposed. The SHL NN in the control loop is used to
estimate the unknown part in Quadrotor’s dynamic model. States variables reproduction is made
by a proposed adaptive observer. The main purpose of the new observer is to reduce noise
measurements amplification by the minimization of the variance and the amplitude of the noisy
signal without introducing an increase in the tracking error. The learning law is derived based on
the Lyapunov stability theory.
This paper is organized as follows. In Section 2, the dynamical model of the Quadrotor will be
presented. Problem formulation will be posed in section 3. In Sections 4 and 5, neural adaptive
controller based on high gain observer and the proposed observer will be developed respectively.
Simulation results are given in section 6 to show the effectiveness and feasibility of the proposed
observer at noise measurement presence. The conclusion is given in section 7.
2. QUADROTOR DYNAMICS
Such its name indicates Quadrotor helicopter is composed of four propellers in cross
configuration where each rotors pair situated on the same branch; turn in opposite direction of the
other pair to avoid the device rotation on her (Figure. 1).

41
3
M
1
M
3
F
2
F
1
F
4
M
2
M
4
F
Z
Y X
E
z
y
x
B
m g
o
O
Figure 1. Model of conventional Quadrotor.
Flights mode in quadrotor are defined according to the direction and the velocity of each rotor.
Vertical ascending (descending) flight is created by increasing (decreasing) thrust forces.
Applying speed difference between front and rear rotors we will have pitching motion which is
defined as a rotation motion around Y axis coupled with a translation motion along X axis. The
same analogy is applied to obtain rolling motion, but by changing the side motors speed this time
and as result we will have a rotation motion around X axis coupled with a translation motion
along Y axis. The last flight mode is yaw motion; this one is obtained while increasing
(decreasing) speed of motors (1.3) compared to (2.4) motors speed. Unlike pitch and roll motions,
yaw rotation is the result of reactive torques produced by rotors rotation.
By using the formalism of Newton-Euler [2, 3, 6, 9] the dynamic model of a Quadrotor can be
expressed as:
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )



















−
−
=
−
−
=
−
+
=
−
−
+
=
−
Ω
+
−
+
=
−
Ω
−
−
+
=
g
z
k
t
U
C
C
m
z
y
k
t
U
C
S
C
S
S
m
y
x
k
t
U
S
S
C
S
C
m
x
I
k
I
I
I
I
t
U
I
k
I
t
J
I
I
I
I
t
U
I
k
I
t
J
I
I
I
I
t
U
ftz
fty
ftx
z
frz
z
y
x
z
y
fry
y
r
r
y
x
z
y
x
frx
x
r
r
x
z
y
x


























1
1
1
2
2
2
1
1
1





























(1)
Where : m: Quadrotor mass, kp : Thrust factor, kd : drag factor, ωi : angular rotor speed, J=diag(Ix,
Iy, Iz): inertia matrix, Kft=diag(kftx, kfty, kftz), drag translation matrix, Kfr=diag(kfrx, kfry, kfrz): friction
aerodynamic coefficients, [ ]T
z
y
x
=
 : position vector, [ ]T



 = represents the angles of
roll, pitch and yaw and ( )
∑
=
+
−
=
Ω
4
1
1
1
i
i
i
r  .

42
The control inputs according to the angular velocities of the four rotors are given by:
( )
( )
( )
( ) 

























−
−
−
−
=














2
4
2
3
2
2
2
1
1
0
0
0
0







d
d
d
d
p
p
p
p
p
p
p
p
k
k
k
k
lk
lk
lk
lk
k
k
k
k
t
U
t
U
t
U
t
U
(2)
2.2. Virtual control
In this section, three virtual control inputs will be defined to ensure that the Quadrotor follows a
specified trajectory. Those virtual controls are [9]:
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )







=
−
=
+
=
t
U
C
C
t
U
t
U
C
S
C
S
S
t
U
t
U
S
S
C
S
C
t
U
z
y
x
1
1
1












(3)
The physical interpretation of theses virtual control, means that the control of translation motion
depends on three common inputs are: θ, φ and U1(t). This requires that the rolling and pitching
motions must take a desired trajectory to guarantee the control task of translation motion. Using
equation (3) the desired trajectories in rolling and pitching are defined as follow [9]:
( ) ( )
( ) ( ) ( )
( ) ( )
( )
















 +
=










+
+
−
=
t
U
t
U
S
t
U
C
t
U
t
U
t
U
C
t
U
S
t
U
z
y
d
x
d
d
z
y
x
d
y
d
x
d






arctan
arcsin
2
2
2
(4)
with φd, θd and ψd are the desired trajectories in roll, pitch and yaw respectively.
3. PROBLEM FORMULATION
After the use of the virtual control defined in eq. (3), the Quadrotor helicopter dynamical model
can be described by the following differential equation system:
( )
( ) ( ) ( )
( ) ( ) ( ) ( )







+
=
+
=
t
u
X
g
X
f
y
t
u
X
g
X
f
y
p
p
p
r
p
r
p

1
1
1
1
1
(5)
Where the state space form is represented by:
( ) ( ) ( )









+
=
+
=
=
=
Σ
i
i
i
i
i
i
r
i
i
i
i
b
x
y
t
u
X
g
X
f
x
x
x
i
1
,
2
1



(6)

43
with: ( ) ( )
[ ]T
r
p
p
p
r p
y
y
y
y
y
y
X
1
1
1
1
1 ,
,
1
−
−
= 



 : state space vector, [ ]T
p
u
u
U 
1
= : the
input vector, [ ]T
p
y
y
Y 
1
= : output vector which is assumed available for measurement, fi(X),
gi(X) are smooth nonlinear unknown functions and bi is the measurement noise.
Assumption.1: The desired output trajectory p
i
ydi :
1
, = and its first ri derivatives are smooth and
bounded.
Assumption.2: The gain gi(X) is bounded, positive definite and slowly time varying.
Tracking error ei and filtered errors si are defined by following equation:
( ) ( )





>
Φ
=






+
=
−
=
−
0
,
i
1
i
i
i
r
i
i E
e
dt
d
s
t
i
y
t
di
y
i
e
i


(7)
From filtered error expression we can conclude that the convergence of tracking error and its
derivatives to zero is guaranteed when si converge to zero [14]. For that raison, our control
objective will be based on the synthesis of control law that allows the convergence to zero of the
filtered error.
The filtered errors time derivative of can be expressed by:
( ) ( ) ( ) p
i
E
e
t
u
X
g
X
f
v
s i
i
r
i
i
i
i
i
i
i
:
1
,
0
)
(
=
Φ
+
=
−
−
=
 (8)
With:
[ ]
[ ]
( )
[ ]
( ) ( )
( )
( ) ( )















−
−
−
=
+
=
−
=
=
=
=
Φ
=
Φ
−
−
=
−
−
−
∑
j
r
i
i
ij
r
j
j
i
ij
r
d
i
i
T
r
i
i
i
i
r
i
i
i
r
i
i
i
i
i
i
i
i
j
i
r
r
e
y
v
r
j
p
i
e
e
e
E







!
1
!
!
1
1
:
1
,
:
1
,
,
0
,
1
1
1
1
1
,
1
0
1
,
1
i




(9)
The following equation expresses the ideal control that can guarantee the closed loop system
performances:
( ) ( ) ( ) p
i
t
u
t
u
t
u pd
i
eq
i
i :
1
, =
+
=
∗
∗
∗
(10)
With:
( ) ( ) ( )
( )
( )





>
Φ
=
=
−
=
∗
∗
−
0
,
1
i
i
i
i
i
i
pd
i
i
i
i
eq
i
k
E
k
s
k
t
u
X
f
v
X
g
t
u
(11)
By substituting (10) in (8), we will have:

44
p
i
s
s
s
k
g
s i
i
i
i
i
i :
1
0 =
≤
⇒
−
= 
 (12)
Which implying that si→0 when t→∞, and therefore ( )
0
→
j
i
e p
i
r
j i :
1
,
1
:
1 =
−
= when
t→∞[14].
Unfortunately the application of this control law is impossible if the dynamic model and the state
variables are unknown as this case. From this fact, six SHL NN will be implemented in control
loop in order to approximate online the equivalent controller defined in (11). The non measured
states problem will be solved by using a proposed observer.
4. SYNTHESIS OF NEURAL ADAPTIVE CONTROL WITH A HIGH GAIN
OBSERVER
The proposed adaptive control is given by the following expression:
( ) ( ) ( ) ( )
( )
( ) ( )
p
i
s
sign
w
t
u
E
k
s
k
t
u
t
u
t
u
t
u
t
u
i
i
r
i
i
i
i
i
i
pd
i
pd
i
r
i
eq
i
i
:
1
ˆ
ˆ
ˆ
ˆ =







=
Φ
=
=
+
+
=
(13)
Where:
 pd
i
u : is the PD term, the ideal PD control is given by (11), However the state vector X is not
available to measure so
∗
pd
i
u is replaced by pd
i
u ,
 r
i
u : is the robust term used for errors approximation compensation,
 eq
i
u : is the neural adaptive control term which is an approximate of ideal equivalent control
defined in (11).

The vector i
Ê is the estimated of Ei defined in eq. (9) and ℜ
∈
i
ŵ is the estimated of i
w , which
will be defined later.
4. 1. Adaptive control Design
According to approximation theorem [15] a SHL network presented in Figure.2 can approximate
the unknown implicit ideal equivalent controller defined in (11) as follows:
( ) ( ) p
i
V
W
t
u i
i
T
i
T
i
eq
i :
1
=
+
=
∗


 (14)

45
i
1

Wi
Vi
ad
i
u
i
2

nei





Figure 2: Network used in the control schemes.
where : nei input neurons number, nci hidden neurons number, nsi=1 output neurons number,
[ ]T
i
i
di
di
di
i e
e
y
y
y 



=
 input vector and σ() is sigmoid activation function, i
i ns
nc
i
W ×
ℜ
∈ and
i
i nc
ne
i
V ×
ℜ
∈ are the ideal weight and i
 is the reconstruction error.
Since the optimal weights and the input vector are unknown, it is necessary to estimate them by
an adaptation mechanism so that the output feedback control law can be realized. i
Ŵ and i
V
ˆ are
the estimate of Wi and Vi. Thus, the adaptive control approximating the ideal SHLNN output
defined in (14) is given by :
( ) p
i
V
W
u T
i
T
i
eq
i :
1
ˆ
ˆ
ˆ =
= 
 (15)
with : [ ]T
i
i
di
di
di
i e
e
y
y
y 


 ˆ
ˆ
ˆ =
 is i
 estimated.
Assumption.3: m
i
F
i W
W ,
≤ , m
i
F
i V
V ,
≤ . with : Wi,m et Vi,m are unknown positive constants.
The identification error of the equivalent control is :
( ) ( ) ( )
( ) ( ) i
i
T
i
T
i
T
i
i
T
i
T
i
eq
i
eq
i
eq
i
w
V
V
W
V
W
t
u
t
u
t
u
+
′
+
=
−
=
∗




 ˆ
~
ˆ
ˆ
ˆ
ˆ
ˆ
~
~
(16)
Where, wi presents the estimation errors:
( ) ( )
{ } ( )
( ) 







+
′
+
+
−
=
i
i
T
i
i
T
i
T
i
i
T
i
T
i
i
T
i
T
i
T
i
i
V
V
W
V
O
W
V
V
W
w









ˆ
~
ˆ
ˆ
~
ˆ
~
ˆ
2
(17)
Where : i
i
i W
W
W ˆ
~
−
= and i
i
i V
V
V ˆ
~
−
= are parameter estimation errors.
Assumption.4 : i
i w
w ≤ . with : i
w is an unknown positive constant.
To achieve the goal of controlling the weights adaptation laws are defined by:

46
( )
( )
( )
( )





−
′
=
−
=
i
i
i
T
i
T
i
i
i
Vi
i
i
i
i
i
T
i
Wi
i
V
V
W
s
F
V
W
s
V
F
W
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ









(18)
with : FWi>0, FVi>0 are the adaptive gains and κi is a positive constant. The estimated of i
w is
given by:
0
,
ˆ
ˆ >
= i
i
i
i s
w 

 (19)
4. 2. High gain Observer Design
The first time derivative of Ei is giving by :
( )





=
+
Φ
−
=
i
i
i
i
i
i
i
i
i
i
E
C
e
s
B
E
B
A
E 

0
(20)
With:
















=
















=
















=
0
0
0
1
,
1
0
0
0
,
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0











T
i
i
i C
B
A
For this equation system; the conventional high gain observer used has the following form:
( )





=
+
Φ
−
=
i
i
i
i
i
i
i
i
i
i
i
E
C
e
e
K
E
B
A
E
ˆ
ˆ
~
ˆ
ˆ
0 

(21)
With:
[ ]T
r
i
r
i
i
i
i
i
i
i
K 1
1
,
2
,
1
,
−
−
= 



  (22)
And : δi>>1 and 0
,
,
, 1
,
2
,
1
, >
−
i
r
i
i
i 

  are design parameter. The notation i
Ê is the estimate of Ei,
and i
i
i e
e
e ˆ
~ −
= .
As we already noted in first section this observer is characterized by its sensibility to noise
measurement. For that raison a new adaptive observer is proposed in this paper.
5. PROPOSED OBSERVER DESIGN
The proposed adaptive observer is given by the following equation:

47
( ) ( ) ( )





=
−
+
Φ
−
=
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
E
C
e
e
K
t
e
K
t
E
B
A
E
ˆ
ˆ
~
ˆ
ˆ
0 



(23)
Where:
( ) ( )
i
f
i
i
i
i
i
i
i e 





 −
+
−
= ,
2
,
1
,
~
 (24)
With: 0
,
, ,
2
,
1 >
i
i
i 

 , ( ) i
i 
 =
0 , i
f
i 
 <
< ,
1 , and, [ [
1
;
0
∈
i
 .
In this case the filtered error dynamic can express by the following equation:
( )
1
,
~
i
i
i
i
i
i
i w
s
k
s
k
g
s −
−
−
=
 (25)
Where :
( ) ( )








−
+
′
+
=
r
i
i
i
T
i
i
T
i
T
i
i
T
i
T
i
i
u
w
V
V
W
V
W
w




 ˆ
~
ˆ
ˆ
ˆ
ˆ
ˆ
~
1
, (26)
Assumption.5 : 3
,
2
,
1
,
1
,
~
~
i
i
i
i
i
i c
V
c
W
c
w +
+
≤ . With : 1
,
i
c , 2
,
i
c , 3
,
i
c are unknown positive
constants.
Assumption.6 : 2
,
1
, i
i
i
i s
e 
 +
≤ ,with : 0
, 2
,
1
, >
i
i 
 .
The observer error dynamic is given by :
( ) ( ) ( ) i
i
i
i
i
i
i
i
i
i
i
i
i e
K
t
s
B
E
B
E
A
t
E 1
~
~
~
0 −
+
+
Φ
−
= 

 

(27)
Where :
E
E
E ˆ
~
−
= ,
( )
( )
( ) 















−
−
−
−
=
−
−
−
−
0
0
0
1
0
0
0
1
0
0
0
1
2
1
,
3
2
,
2
,
1
,









t
t
t
A
i
i
i
i
r
i
r
i
r
i
r
i
i
i
i
i







The matrix i
A is stable, then it exists 0
>
= T
i
i P
P and 0
>
= T
i
i Q
Q , as : [14]
p
i
Q
P
A
A
P i
i
T
i
i
i :
1
=
−
=
+ (28)
5. 1. Proof
Consider the following Lyapunov function candidate:

48
( )
( ) ( )



















+
+
+
=
=
+
=
∑
∑
=
−
−
−
=
p
i
i
i
i
V
T
i
i
W
T
i
i
i
P
i
i
i
T
i
w
V
F
V
tr
W
F
W
tr
s
g
L
E
P
E
L
L
L
L
i
i
1
2
1
1
2
1
2
1
1
2
1
~
1
~
~
~
~
2
1
~
~
2
1

(29)
After introducing the following equations (13), (18), (19), (25) and (26) and by using the 5th
and
the 6th
assumption the first time derivative of Lyapunov function can express by:
( ) ( ) ( )
∑
=














+






−
−






−
−
−
−






−
−
≤
p
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
c
V
c
W
c
s
c
k
E
c
Q
t
L
1
10
,
2
7
,
2
6
,
2
5
,
2
9
,
min
~
2
~
2
1
~
2




 (30)
Where :
i
i
i
i
i
i
i
i
i g
B
P
k
B
P
c Φ
+
Φ
= 0
4
, ,
( )







 +
+
Φ
=
i
i
i
i
i
i
i
i
i
i
i
i
k
K
t
P
g
B
P
k
c


 1
,
5
,
2
1
,
( )
2
1
,
6
,
i
i
i
i
i
i
g
B
P
c
c
+
Φ
= ,
( )
2
2
,
7
,
i
i
i
i
i
i
g
B
P
c
c
+
Φ
= ,
( ) ( )
( )
2
2
,
3
,
8
,
i
i
i
i
i
i
i
i
i
i
i
K
t
P
g
B
P
c
c



+
+
Φ
= ,
1
7
,
6
,
5
,
4
,
9
, +
+
+
+
= i
i
i
i
i
i c
c
c
k
c
c , ( )
2
2
2
8
,
10
, ˆ
2
i
i
i
i
i
i V
W
s
c
c +
+
+
=

.
Let's suppose that: ci,5<1, μi(t)>ci,9/λmin(Qi) and κi>2Max(ci,6, ci,7). So the Lyapunov function is
decreasing if i
s , i
E
~ , i
W
~ and i
V
~ are outsiders of the compact sets i
s
Ω ,
i
E
~
Ω ,
i
W
~
Ω and
i
V
~
Ω
respectively, where:
( )
( ) ( )



























−
≤
=
Ω










−
≤
=
Ω










−
≤
=
Ω










−
≤
=
Ω
7
,
10
,
~
6
,
10
,
~
9
,
min
10
,
~
5
,
10
,
2
2
~
/
~
,
2
2
~
/
~
2
2
~
/
~
,
1
/
i
i
i
i
i
V
i
i
i
i
i
W
i
i
i
i
i
i
E
i
i
i
i
i
s
c
c
V
V
c
c
W
W
c
Q
t
c
E
E
c
k
c
s
s
i
i
i
i




(31)
6. SIMULATION RESULTS
The simulation experiments are performed to compare the ordinary high gain observe and the
proposed observer.
The simulation parameters are given in Tables I and II. The 3rd
Figure show simulation results
obtained by using the proposed and the conventional approach at the presence of noise

49
measurement and with 50% parametric variation from time t = 10 sec.
Table 1: Quadrotor Parameters.
Symbol Value Symbol Value
m
l
kd
kp
Jr
Ix
Iy
0.486 (Kg)
0.25 (m)
3.23e-7 (NMs2
)
2.98e-5 (Ns2
)
2.83e-5 (KgM2
)
3.82e-3 (KgM2
)
3.82e-3 (KgM2
)
Iz
gft
kftx , kfty
kftz
kfrφ , kfrθ
kfrψ
7.65e-3 (KgM2
)
9.81 (Ms2
)
5.56e-4 (Ns/M)
6.35e-4 (Ns/M)
5.56e-4 (Ns/r)
6.35e-4 (Ns/r)
Table 2: Controller and Observer Parameters.
Controller Proposed Observer
Symbol Value Symbol Value
λη
λξ
Kξ
Kη
λx, λy,
λz
λφ, λθ
, λψ
FWx,y,z
FWθ,φ,ψ
FVx,y,z
FVθ,φ,ψ
κx, κy,
κz
κφ, κθ
,κψ
daig(10, 10, 10)
diag (1, 1, 10)
daig(1.5, 1.5, 20)
diag (0.2, 0.2, 0.2)
0.01
0.01
0.01
0.01
3 I3.3
I3.3
5 I5.5
I5.5
0.1
0.1
0.1
0.1
μx,y,z,f
μθ,φ,ψ,f
ςx,y,z,1
ςθ,φ,ψ,1
ςx,y,2
ςθ,φ,z,2
ςψ,2
δx,y,φ,θ
δz
δψ
ϑx,y,z,1
ϑx,y,z,2
ϑ θ,φ,ψ,1
ϑ θ,φ,2
ϑψ,2
βx,y,z
β θ,φ,ψ


 ,
,
,
, z
y
x


2
2
10
5
0.1
0.2
0.3
20
30
40
10
40
5
20
30
0.5
0.8
0.01
0.05
The conventional high gain observer produces large measurement noise amplification has created
degradations in system’s pursuit and by consequence an important control signal as is presented
as in Figure.3. a which can cause instability if the measurement noise is so important.
Unlike the results achieved by the proposed observer where a significant minimization of
unfavorable effect was obtained for the system output and control signals as is shown in the
Figure. 3. b where control signal amplitude is ten times less than the one gotten by using ordinary
observer.
desired output Real output estimated output Signal control

50
0 10 20 30 40 50
-0.2
-0.15
-0.1
-0.05
0
0.05
(3. a. 1)
0 1
0 2
0 3
0 4
0 5
0
-0
.1
5
-0
.1
-0
.0
5
0
(3. b. 1)
0 10 20 30 40 50
-4
-2
0
2
4
(3. a. 2)
0 1
0 2
0 3
0 4
0 5
0
-0
.5
0
0
.5
(3. a. 3)
0 1
0 2
0 3
0 4
0 5
0
-0
.1
0
0
.1
0
.2
0
.3
(3. a. 4)
0 10 20 30 40 50
-4
-2
0
2
4
(3. a. 5)
0 1
0 2
0 3
0 4
0 5
0
-0
.5
0
0
.5
(3. a. 6)
0 1
0 2
0 3
0 4
0 5
0
-2
-1
0
1
2
(3. b. 2)
0 1
0 2
0 3
0 4
0 5
0
-0
.2
0
0
.2
(3. b. 3)
0 1
0 2
0 3
0 4
0 5
0
-0
.1
0
0
.1
0
.2
0
.3
(3. b. 4)
t(s)
t(s)
t(s) t(s)
t(s)
t(s)
t(s)
t(s)
t(s)
t(s)
φ
φ
Uφ
Uφ
Uθ
θ
θ

 



51
0 10 20 30 40 50
-5
0
5
(3. b. 5)
0 1
0 2
0 3
0 4
0 5
0
-0
.5
0
0
.5
(3. b. 6)
0 1
0 2
0 3
0 4
0 5
0
0
0
.5
1
(3. a. 7)
0 1
0 2
0 3
0 4
0 5
0
-0
.2
0
0
.2
0
.4
0
.6
(3. a. 8)
0 1
0 2
0 3
0 4
0 5
0
-0
.1
-0
.0
5
0
0
.0
5
0
.1
(3. a. 9)
0 1
0 2
0 3
0 4
0 5
0
-0
.4
-0
.2
0
0
.2
0
.4
0
.6
(3. a. 10)
0 1
0 2
0 3
0 4
0 5
0
-0
.5
0
0
.5
(3. a. 11)
0 10 20 30 40 50
0
0.5
1
(3. b. 7)
0 1
0 2
0 3
0 4
0 5
0
0
0
.1
0
.2
0
.3
0
.4
(3. b. 8)
0 1
0 2
0 3
0 4
0 5
0
-0
.0
1
-0
.0
0
5
0
0
.0
0
5
0
.0
1
(3. b. 9)
t(s)
t(s)
t(s)
t(s)
t(s) t(s)
t(s) t(s)
t(s)
t(s)
Uθ
Uψ Uψ
Ux
ψ ψ



 

x


52
0 10 20 30 40 50
-0.2
0
0.2
0.4
0.6
(3. b. 10)
0 1
0 2
0 3
0 4
0 5
0
-0
.5
0
0
.5
(3. b. 11)
0 1
0 2
0 3
0 4
0 5
0
-0
.4
-0
.2
0
0
.2
0
.4
(3. a. 12)
0 10 20 30 40 50
-0.2
0
0.2
0.4
(3. a. 13)
0 1
0 2
0 3
0 4
0 5
0
0
0
.2
0
.4
0
.6
(3. a. 14)
0 1
0 2
0 3
0 4
0 5
0
4
6
8
1
0
(3. a. 15)
-0.5
0
0.5
-0.5
0
0.5
0
10
20
(3. a. 16)
0 1
0 2
0 3
0 4
0 5
0
-0
.4
-0
.2
0
0
.2
0
.4
(3. b. 12)
0 1
0 2
0 3
0 4
0 5
0
-0
.2
0
0
.2
(3. b. 13)
0 1
0 2
0 3
0 4
0 5
0
0
0
.2
0
.4
0
.6
(3. b. 14)
t(s)
t(s)
t(s)
t(s) t(s)
t(s) t(s)
t(s) t(s)
Ux
Uz
Uy
Uy
x
y
z
x

y
 y

z
 z


53
0 1
0 2
0 3
0 4
0 5
0
4
6
8
1
0
(3. b. 15)
-0.5
0
0.5
-0.5
0
0.5
0
10
20
(3. b. 16)
Figure. 3: Simulation results
7. CONCLUSION
A new adaptive observer was developed for an under actuated Quadrotor UAV. The control
scheme is based on the use of SHL neural networks for each Quadrotor subsystem in order to
estimate on line the unknown nonlinear dynamical modal in Quadrotor helicopter. The proposed
observer is based on conventional high gain observer structure where new parameters are added
to the last one in order to reduce the amplification of measurement noise generated by ordinary
observer. The proposed observer have proved its capacity where a very important minimization in
the amplitude and the variance of the noisy signal is gotten without having degradations in the
performances as it was shown by the numeric simulation results. This method does not require
any prior knowledge about dynamic model and states variables. The uniformly ultimately
boundedness of the tracking error and all signals in the overall closed-loop system is proved using
Lyapunov's direct method. The effectiveness of the proposed approach was validated by
simulation of a Quadrotor control at noise measurement presence.
REFERENCES
[1] P. Adigbli : Nonlinear Attitude and Position Control of a Micro Quadrotor using Sliding Mode and
Backstepping Techniques. 3rd US-European Competition, MAV07 & EMAV07, 17-21, France,
2007
[2] S.Bouabdallah and al : Backstepping and sliding mode Techniques Applied to an Indoor Micro
Quadrotor. In Proc 2005 IEEE ICRA, 2005
[3] T. Madani and A.Benallegue: Backstepping Control with Exact 2-Sliding Mode Estimation for a
Quadrotor Unmanned Aerial Vehicle. Proc of IEEE/RSJ International CIRS, 2007.
[4] A.Mokhtari, A.Benallegue and B.Daachi: Robust Feedback Linearization and GH∞ Controller for a
Quadrotor Unmanned aerial vehicle. Journal of Electrical Engineering, vol.57, No.1, pp 20.27, 2006.
[5] S.Bouaddallah and al : PID vs. LQ control Techniques Applied to an Indoor Micro Quadrotor. In
Proc IEEE ICIRS, Japan, 2004.
[6] A.Das, F.Lewis and K.Subbarao : Backstepping Approach for Controlling a Quadrotor Using
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PSC, pp. 454-458, 2008.
[8] J.C.Raimundez & al : Adaptive Tracking control for a Quad-rotor. ENOC-2008, saint Pertersburg,
Russia July 2008.
[9] O.Bouhali and H.Boudjedir : Neural Network control with neuro-sliding mode observer applied to
Quadrotor Helicopter. In Proc INISTA’11, pp 24-28, Istanbul, Turkey 2011.
[10] T.Dieks and S.Jagannathan: Neural Networks Out Feedback of a Quadrotor UAV. Proc IEEE, 47th
CDC, pp 3633-3639, Mexi, Dec 2008.
[11] H.Khalil : High-Gain Observers in Nonlinear Feedback Control. Proc IEEE, IC on Control,
Automation and Systems, 2008
t(s)
Uz
x
y
z

54
[12] M.Farza and al : Dynamic High Gain Observer Design. Proc IEEE, 6th International Multi-
Conference on systems, signals and Device, 2009.
[13] E.Bullinger and F.Allgower : An Adaptive High-Gain Observer for Nonlinear Systems. Proc IEEE,
the 36th CDC USA, 1997.
[14] J.J.Slotine : Applied nonlinear control. Prentice Hall, 1991.
[15] S.Jagannathan : Neural networks control of nonlinear Discrete-time systems. CRC press, Taylor &
Francic Group, 2006.

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