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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 AdaptiveObserver for Quadrotor Helicopter
Hana Boudjedir1 , Omar Bouhali1 and Nassim Rizoug2
1 LAJ Lab, Automatic department, Jijel University, [email protected], [email protected]
2Mecatronic Lab, ESTACA School, Laval, [email protected]
ABSTRACT
A neural network control scheme with an adaptive observer is proposed in this paper to Quadrotorhelicopter stabilization. The unknown part in Quadrotor dynamical model was estimated on line by aSingle Hidden Layer network. To solve the non measurable states problem a new adaptive observer wasproposed. The main purpose here is to reduce the measurement noise amplification caused by conventionalhigh gain observer by introducing some changes in observer’s original structure that can minimize thevariance and the amplitude of the noisy signal without increasing tracking error. The stability analysis ofthe overall closed-loop system/ observer is performed using the Lyapunov direct method. Simulation resultsare 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 researchcommunity [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 largepayload as compared with the conventional helicopter [2].
The Quadrotor is an under actuated system from where it has six degrees of freedom controlledonly by four control inputs. To solve the Quadrotor UAV tracking control problem manytechniques have been proposed [2-7] where the control objective is to control three desiredCartesian 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 partiallyunknown and/or if the states variables are unavailable.
International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.3, July 2012
40
The unknown nonlinear functions problem in the control law can be solved by using adaptivecontrol technique based on universals approximators as artificial neural network [6]. SeveralNeural Networks adaptive schemes have been proposed for Quadrotor control [6-10].To avoid the state variables measurement assumptions, state observers can be introduced in thecontrol scheme such as in [9] where a neuro-sliding mode observer was proposed in order toreduce the noise measurement amplification. In [10] other observer was used to estimate thevelocity, the corrective term here is the superposition of a observation errors proportional termand 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 ofoutput feedback control of nonlinear systems [11]. However this observer has two drawbacks, thefirst 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 transientresponse and the smallest observation gain was used in permanent regime. In [12] and [13] anadaptation laws is utilized to compute the optimal observation gain. However those methodspresent some limitations in particular when the observation gain has to be too large as in the caseof quadrotor dynamic.
In the present paper a new robust adaptive scheme that does not require any prior information onmodel dynamics and state measurement is proposed. The SHL NN in the control loop is used toestimate the unknown part in Quadrotor’s dynamic model. States variables reproduction is madeby a proposed adaptive observer. The main purpose of the new observer is to reduce noisemeasurements amplification by the minimization of the variance and the amplitude of the noisysignal without introducing an increase in the tracking error. The learning law is derived based onthe Lyapunov stability theory.
This paper is organized as follows. In Section 2, the dynamical model of the Quadrotor will bepresented. Problem formulation will be posed in section 3. In Sections 4 and 5, neural adaptivecontroller 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 proposedobserver 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 crossconfiguration where each rotors pair situated on the same branch; turn in opposite direction of theother pair to avoid the device rotation on her (Figure. 1).
International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.3, July 2012
41
3M
1M3F
2F
1F
4M
2M
4F
Z
Y X
E
zy
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 isdefined as a rotation motion around Y axis coupled with a translation motion along X axis. Thesame analogy is applied to obtain rolling motion, but by changing the side motors speed this timeand as result we will have a rotation motion around X axis coupled with a translation motionalong 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 beexpressed as:
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( )( )( ) ( )( )( ) ( )( )
−−=
−−=
−+=
−−
+=
−Ω+−+=
−Ω−−
+=
gzktUCCm
z
yktUCSCSSm
y
xktUSSCSCm
x
I
k
I
II
I
tU
I
k
I
tJ
I
II
I
tU
I
k
I
tJ
I
II
I
tU
ftz
fty
ftx
z
frz
z
yx
z
y
fry
y
rr
y
xz
y
x
frx
x
rr
x
zy
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): frictionaerodynamic coefficients, [ ]Tzyx= : position vector, [ ]T = represents the angles of
roll, pitch and yaw and ( )∑=
+−=Ω4
1
11i
ii
r .
International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.3, July 2012
42
The control inputs according to the angular velocities of the four rotors are given by:
( )( )( )( )
−−−
−=
24
23
22
211
00
00
dddd
pp
pp
pppp
kkkk
lklk
lklk
kkkk
tU
tU
tU
tU
(2)
2.2. Virtual control
In this section, three virtual control inputs will be defined to ensure that the Quadrotor follows aspecified trajectory. Those virtual controls are [9]:
( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )
=
−=
+=
tUCCtU
tUCSCSStU
tUSSCSCtU
z
y
x
1
1
1
(3)
The physical interpretation of theses virtual control, means that the control of translation motiondepends on three common inputs are: θ, φ and U1(t). This requires that the rolling and pitchingmotions must take a desired trajectory to guarantee the control task of translation motion. Usingequation (3) the desired trajectories in rolling and pitching are defined as follow [9]:
( ) ( )( ) ( ) ( )
( ) ( )( )
+=
++
−=
tU
tUStUC
tUtUtU
CtUStU
z
ydxdd
zyx
dydxd
arctan
arcsin222
(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 modelcan be described by the following differential equation system:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
+=
+=
tuXgXfy
tuXgXfy
ppprp
r
p
11111
(5)
Where the state space form is represented by:
( ) ( ) ( )
+=
+=
=
=Σ
iii
iiiri
ii
i
bxy
tuXgXfx
xx
i
1,
21
(6)
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43
with: ( ) ( )[ ]Trppp
r pyyyyyyX 11111 ,,1
−−= : state space vector, [ ]TpuuU 1= : the
input vector, [ ]TpyyY 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 piydi :1, = and its first ri derivatives are smooth andbounded.
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
iii
r
ii Eedt
ds
tiytdiyie
i
(7)
From filtered error expression we can conclude that the convergence of tracking error and itsderivatives to zero is guaranteed when si converge to zero [14]. For that raison, our controlobjective will be based on the synthesis of control law that allows the convergence to zero of thefiltered error.
The filtered errors time derivative of can be expressed by:
( ) ( ) ( ) piEetuXgXfvs iir
iiiiiii :1,0)( =Φ+=−−= (8)
With:
[ ]
[ ]( )[ ]
( ) ( )
( )( ) ( )
−−−=
+=
−===
=Φ
=Φ
−
−
=
−
−
−
∑jr
i
iij
r
j
jiij
rdi
i
Triiii
riii
rii
i
i
i
i
i
i
jir
r
eyv
rjpieeeE
!1!
!1
1:1,:1,
,0
,1
1
1
1
1,10
1,1i
(9)
The following equation expresses the ideal control that can guarantee the closed loop systemperformances:
( ) ( ) ( ) pitututu pdi
eqii :1, =+=
∗∗∗ (10)
With:
( ) ( ) ( )( )( )
>Φ==
−=∗
∗ −
0,
1
iiiiiipdi
iiieqi
kEksktu
XfvXgtu(11)
By substituting (10) in (8), we will have:
International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.3, July 2012
44
pissskgs iiiiii :10 =≤⇒−= (12)
Which implying that si→0 when t→∞, and therefore ( ) 0→jie pirj i :1,1:1 =−= when
t→∞[14].
Unfortunately the application of this control law is impossible if the dynamic model and the statevariables are unknown as this case. From this fact, six SHL NN will be implemented in controlloop in order to approximate online the equivalent controller defined in (11). The non measuredstates 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:
( ) ( ) ( ) ( )( )
( ) ( )pi
ssignwtu
Eksktu
tutututu
iiri
iiiiipdi
pdi
ri
eqii
:1
ˆˆ
ˆˆ =
=
Φ==
++=
(13)
Where:
pdiu : is the PD term, the ideal PD control is given by (11), However the state vector X is not
available to measure so∗pd
iu is replaced by pdiu ,
riu : is the robust term used for errors approximation compensation,
eqiu : is the neural adaptive control term which is an approximate of ideal equivalent control
defined in (11).The vector iE is the estimated of Ei defined in eq. (9) and ℜ∈iw is the estimated of iw , whichwill be defined later.
4. 1. Adaptive control Design
According to approximation theorem [15] a SHL network presented in Figure.2 can approximatethe unknown implicit ideal equivalent controller defined in (11) as follows:
( ) ( ) piVWtu iiT
iT
ieqi :1=+=
∗ (14)
International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.3, July 2012
45
i1
WiVi
adiu
i2
nei
Figure 2: Network used in the control schemes.
where : nei input neurons number, nci hidden neurons number, nsi=1 output neurons number,
[ ]Tiidididii eeyyy = input vector and σ() is sigmoid activation function, ii nsnciW ×ℜ∈ and
ii ncneiV ×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. iW and iV arethe estimate of Wi and Vi. Thus, the adaptive control approximating the ideal SHLNN outputdefined in (14) is given by :
( ) piVWu Ti
Ti
eqi :1ˆˆˆ == (15)
with : [ ]Tiidididii eeyyy ˆˆˆ = is i estimated.
Assumption.3: miFi WW ,≤ , miFi VV ,≤ . with : Wi,m et Vi,m are unknown positive constants.
The identification error of the equivalent control is :
( ) ( ) ( )( ) ( ) ii
Ti
Ti
Tii
Ti
Ti
eqi
eqi
eqi
wVVWVW
tututu
+′+=
−=∗
ˆ~ˆˆˆˆˆ~
~(16)
Where, wi presents the estimation errors:
( ) ( ) ( )( )
+′+
+−=
iiT
iiT
iT
i
iT
iT
iiT
iT
iT
ii
VVW
VOWVVWw
ˆ~ˆˆ~ˆ~ˆ 2
(17)
Where : iii WWW ˆ~ −= and iii VVV ˆ~ −= are parameter estimation errors.
Assumption.4 : ii ww ≤ . with : iw is an unknown positive constant.
To achieve the goal of controlling the weights adaptation laws are defined by:
International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.3, July 2012
46
( )( )( )( )
−′=
−=
iiiT
iT
iiiVii
iiiiT
iWii
VVWsFV
WsVFW
ˆˆˆˆˆˆˆ
ˆˆˆˆˆ
(18)
with : FWi>0, FVi>0 are the adaptive gains and κi is a positive constant. The estimated of iw is
given by:
0,ˆˆ >= iiii sw (19)
4. 2. High gain Observer Design
The first time derivative of Ei is giving by :
( )
=
+Φ−=
iii
iiiiiii
ECe
sBEBAE 0 (20)
With:
=
=
=
0
0
0
1
,
1
0
0
0
,
0000
1000
0100
0010
Tiii CBA
For this equation system; the conventional high gain observer used has the following form:
( )
=
+Φ−=
iii
iiiiiiii
ECe
eKEBAE
ˆˆ
~ˆˆ0
(21)
With:
[ ]Tririiiii
i
iK 1
1,2,1,−
−= (22)
And : δi>>1 and 0,,, 1,2,1, >−iriii are design parameter. The notation iE is the estimate of Ei,
and iii eee ˆ~ −= .
As we already noted in first section this observer is characterized by its sensibility to noisemeasurement. 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:
International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.3, July 2012
47
( ) ( ) ( )
=
−+Φ−=
iii
iiiiiiiiiiii
ECe
eKteKtEBAE
ˆˆ
~ˆˆ0
(23)
Where:
( ) ( )ifiiiiiii e −+−= ,2,1,~ (24)
With: 0,, ,2,1 >iii , ( ) ii =0 , ifi << ,1 , and, [ [1;0∈i .
In this case the filtered error dynamic can express by the following equation:
( )1,~
iiiiiii wskskgs −−−= (25)
Where :
( ) ( )
−+
′+=
rii
iT
iiT
iT
iiT
iT
ii
uw
VVWVWw
ˆ~ˆˆˆˆˆ~
1, (26)
Assumption.5 : 3,2,1,1,
~~iiiiii cVcWcw ++≤ . With : 1,ic , 2,ic , 3,ic are unknown positive
constants.
Assumption.6 : 2,1, iiii se +≤ ,with : 0, 2,1, >ii .
The observer error dynamic is given by :
( ) ( ) ( ) iiiiiiiiiiiii eKtsBEBEAtE 1~~~
0 −++Φ−= (27)
Where :
EEE ˆ~ −= ,
( )
( )( )
−−
−−
=
−−
−−
000
100
010
001
21,
32,
2,
1,
t
t
t
A
i
i
i
i
riri
riri
ii
i
i
The matrix iA is stable, then it exists 0>= Tii PP and 0>= T
ii QQ , as : [14]
piQPAAP iiTiii :1=−=+ (28)
5. 1. Proof
Consider the following Lyapunov function candidate:
International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.3, July 2012
48
( )
( ) ( )
+++=
=
+=
∑
∑
=
−−−
=p
ii
iiV
TiiW
Tiii
P
iii
Ti
wVFVtrWFWtrsgL
EPEL
LLL
ii
1
211212
11
21
~1~~~~
2
1
~~
2
1
(29)
After introducing the following equations (13), (18), (19), (25) and (26) and by using the 5th andthe 6th assumption the first time derivative of Lyapunov function can express by:
( ) ( ) ( )∑
=
+
−−
−−
−−
−−
≤p
iiii
iii
i
iiiiiii
cVcWc
sckEcQt
L1
10,
2
7,
2
6,
2
5,
2
9,min
~
2
~
2
1~
2
(30)
Where :
iiiiiiiii gBPkBPc Φ+Φ= 04, ,( )
++Φ=
i
iiiiiiiiiii k
KtPgBPkc
1,5, 2
1,
( )2
1,6,
iiiiii
gBPcc
+Φ= ,
( )2
2,7,
iiiiii
gBPcc
+Φ= ,
( ) ( )( )2
2,3,8,
iiiiiiiiiii
KtPgBPcc
++Φ= ,
17,6,5,4,9, ++++= iiiiii ccckcc , ( )2228,10, ˆ
2 iiii
ii VWscc +++= .
Let's suppose that: ci,5<1, μi(t)>ci,9/λmin(Qi) and κi>2Max(ci,6, ci,7). So the Lyapunov function isdecreasing if is ,
iE~ ,
iW~ and
iV~ are outsiders of the compact sets
isΩ ,iE
~Ω ,iW
~Ω andiV
~Ω
respectively, where:
( )
( ) ( )
−≤=Ω
−≤=Ω
−≤=Ω
−≤=Ω
7,
10,~
6,
10,~
9,min
10,~
5,
10,
2
2~/
~
,2
2~/
~
2
2~/
~
,1
/
ii
iiiV
ii
iiiW
iii
iiiE
ii
iiis
c
cVV
c
cWW
cQt
cEE
ck
css
i
i
i
i
(31)
6. SIMULATION RESULTS
The simulation experiments are performed to compare the ordinary high gain observe and theproposed observer.
The simulation parameters are given in Tables I and II. The 3rd Figure show simulation resultsobtained by using the proposed and the conventional approach at the presence of noise
International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.3, July 2012
49
measurement and with 50% parametric variation from time t = 10 sec.
Table 1: Quadrotor Parameters.
Symbol Value Symbol Valuemlkd
kp
JrIx
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 , kfrθkfrψ
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 ObserverSymbol Value Symbol Value
ληλξKξKη
λx, λy,λz
λφ, λθ, λψ
FWx,y,z
FWθ,φ,ψFVx,y,z
FVθ,φ,ψκx, 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)
0.010.010.010.013 I3.3
I3.3
5 I5.5
I5.5
0.10.10.10.1
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,θδz
δψϑx,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
,,,, zyx
22
105
0.10.20.320304010405
20300.50.8
0.010.05
The conventional high gain observer produces large measurement noise amplification has createddegradations in system’s pursuit and by consequence an important control signal as is presentedas 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 ofunfavorable effect was obtained for the system output and control signals as is shown in theFigure. 3. b where control signal amplitude is ten times less than the one gotten by using ordinaryobserver.
desired output Real output estimated output Signal control
International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.3, July 2012
50
0 10 20 30 40 50-0.2
-0.15
-0.1
-0.05
0
0.05
(3. a. 1)
0 10 20 30 40 50-0.15
-0.1
-0.05
0
(3. b. 1)
0 10 20 30 40 50-4
-2
0
2
4
(3. a. 2)
0 10 20 30 40 50
-0.5
0
0.5
(3. a. 3)
0 10 20 30 40 50
-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 10 20 30 40 50
-0.5
0
0.5
(3. a. 6)
0 10 20 30 40 50
-2
-1
0
1
2
(3. b. 2)
0 10 20 30 40 50
-0.2
0
0.2
(3. b. 3)
0 10 20 30 40 50-0.1
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
UφUφ
Uθ
θ θ
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0 10 20 30 40 50
-5
0
5
(3. b. 5)
0 10 20 30 40 50
-0.5
0
0.5
(3. b. 6)
0 10 20 30 40 500
0.5
1
(3. a. 7)
0 10 20 30 40 50
-0.2
0
0.2
0.4
0.6
(3. a. 8)
0 10 20 30 40 50-0.1
-0.05
0
0.05
0.1
(3. a. 9)
0 10 20 30 40 50-0.4
-0.2
0
0.2
0.4
0.6
(3. a. 10)
0 10 20 30 40 50-0.5
0
0.5
(3. a. 11)
0 10 20 30 40 500
0.5
1
(3. b. 7)
0 10 20 30 40 50
0
0.1
0.2
0.3
0.4
(3. b. 8)
0 10 20 30 40 50-0.01
-0.005
0
0.005
0.01
(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
International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.3, July 2012
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0 10 20 30 40 50
-0.2
0
0.2
0.4
0.6
(3. b. 10)
0 10 20 30 40 50-0.5
0
0.5
(3. b. 11)
0 10 20 30 40 50-0.4
-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 10 20 30 40 500
0.2
0.4
0.6
(3. a. 14)
0 10 20 30 40 50
4
6
8
10
(3. a. 15)
-0.50
0.5
-0.50
0.50
10
20
(3. a. 16)
0 10 20 30 40 50-0.4
-0.2
0
0.2
0.4
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.3, July 2012
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Figure. 3: Simulation results
7. CONCLUSION
A new adaptive observer was developed for an under actuated Quadrotor UAV. The controlscheme is based on the use of SHL neural networks for each Quadrotor subsystem in order toestimate on line the unknown nonlinear dynamical modal in Quadrotor helicopter. The proposedobserver is based on conventional high gain observer structure where new parameters are addedto the last one in order to reduce the amplification of measurement noise generated by ordinaryobserver. The proposed observer have proved its capacity where a very important minimization inthe amplitude and the variance of the noisy signal is gotten without having degradations in theperformances as it was shown by the numeric simulation results. This method does not requireany prior knowledge about dynamic model and states variables. The uniformly ultimatelyboundedness of the tracking error and all signals in the overall closed-loop system is proved usingLyapunov's direct method. The effectiveness of the proposed approach was validated bysimulation of a Quadrotor control at noise measurement presence.
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