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 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.

mailto:[email protected]




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).


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 .


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)


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:


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)


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:


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:


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:


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


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


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


51

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


52

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

(3. b. 12)

0 10 20 30 40 50-0.2

0

0.2

(3. b. 13)

0 10 20 30 40 500

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

xy

z

x

y y

z z


53

0 10 20 30 40 50

4

6

8

10

(3. b. 15)

-0.50

0.5

-0.50

0.50

10

20

(3. b. 16)

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.

REFERENCES

[1] P. Adigbli : Nonlinear Attitude and Position Control of a Micro Quadrotor using Sliding Mode andBackstepping 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 MicroQuadrotor. In Proc 2005 IEEE ICRA, 2005

[3] T. Madani and A.Benallegue: Backstepping Control with Exact 2-Sliding Mode Estimation for aQuadrotor 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 aQuadrotor 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. InProc IEEE ICIRS, Japan, 2004.

[6] A.Das, F.Lewis and K.Subbarao : Backstepping Approach for Controlling a Quadrotor UsingLagrange Form Dynamics. J Intell Robot syst, pp 127-151, 2009.

[7] C. Nicol, & al: Robust Neural Network of a Quadrotor Helicopter, in proc IEEE Fuzzy informationPSC, 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 toQuadrotor 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, 47thCDC, 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

xy

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.

neural network control based on adaptive observer for quadrotor helicopter

Documents