موفق


 
 

     
Al-Khwarizmi  
Engineering   

Journal  
Al-Khwarizmi Engineering Journal, Vol. 7, No. 4, PP 27-40 (2011)  

  

 
Control on a 2-D Wing Flutter Using an Adaptive 

Nonlinear Neural Controller 
 
Mauwafak A. Tawfik*     Mohammed I. Mohsin*      Hayder S. Abd Al-Amir** 

   * Department of Mechanical Engineering /University of Technology 
** Department of Mechanical Engineering /Institute of Technology  

Email: has_04@yahoo.com 

   
(Received 23 February 2011; Accepted 6 July 2011) 

 
 
Abstract 
 

An adaptive nonlinear neural controller to reduce the nonlinear flutter in 2-D wing is proposed in the paper. The 
nonlinearities in the system come from the quasi steady aerodynamic model and torsional spring in pitch direction. 
Time domain simulations are used to examine the dynamic aero elastic instabilities of the system (e.g. the onset of 
flutter and limit cycle oscillation, LCO). The structure of the controller consists of two models :the modified Elman 
neural network (MENN) and the feed forward multi-layer Perceptron (MLP). The MENN model is trained with off-line 
and on-line stages to guarantee that the outputs of the model accurately represent the plunge and pitch motion of the 
wing and this neural model acts as the identifier. The feed forward neural controller is trained off-line and adaptive 
weights are implemented on-line to find the flap angles, which controls the plunge and pitch motion of the wing. The 
general back propagation algorithm is used to learn the feed forward neural controller and the neural identifier. The 
simulation results show the effectiveness of the proposed control algorithm; this is demonstrated by the minimized 
tracking error to zero approximation with very acceptable settling time even with the existence of bounded external 
disturbances. 

 
Keywords: Adaptive Nonlinear Control, Flutter, Nonlinear system, Neural Network. 
 
 
1. Introduction 
 

The performance of aircraft is often limited by 
adverse aero elastic interactions such as flutter. 
Flutter is defined as a dynamic instability of a 
flight vehicle associated with the interaction of 
aerodynamic, elastic, and inertial forces. If flutter 
can be controlled at cruise speeds, lighter wings 
can be designed and consequently more efficient 
airplanes. It is therefore, in the aircraft designer’s 
best interest to design innovative ways in which 
flutter can be controlled without making the 
resulting structure too heavy The nonlinear flutter 
models behavior is close to the systems in nature. 
However, the control on the flutter resulting from 
nonlinear models is not an easy task and need 
more development. In reality nonlinearities may 
be presented in various forms [1,2] .Recently, 
many researches in this field proposed different  

 

 
nonlinear flutter controllers, where the linear 
controller   could   not   effectively   suppress   the 
flutter [3]. Palaniappan, et al. [4] developed a 
feedback algorithm for the control of nonlinear 
flutter. The actuators are jets in the walls through 
which there is a small mass flow, either by way of 
blowing or suction. Afkhami and Alighanbair [5] 
presented nonlinear controller to control flutter. 
Integral-input-to-state stability concept is utilized 
for the construction of a feedback controller. 
Haiwei and Jinglong [6] proposed the robust 
flutter analysis of a nonlinear 2-D wing section 
with structural and aerodynamic uncertain using 
μ-method. The parametric uncertainty was 
adopted to describe the uncertainties in structure 
and aerodynamics. The nonlinear system was 
linearized at equilibrium point and μ analysis is 
performed for a set of values of flow velocities to 
generate the lower and upper bounds of robust 
flutter speed. For a typical section with a 

mailto:has_04@yahoo.com


Mauwafak A. Tawfik                              Al-Khwarizmi Engineering Journal Vol. 7, No. 4, PP 27-40(2011)   
 

28 
 

structural nonlinearity, Zeng and Singh [7] have 
derived a nonlinear adaptive control based on the 
model reference adaptive control theory. The 
system design was based on an output feedback 
method that eliminates the requirement for full 
state reconstruction. However, the derived control 
system exhibits somewhat larger flap deflections 
in simulations than others. 

The neural networks were introduced in aero 
elastic field as nonlinear controller or flutter 
prediction device. Melin and Castillo[8] combined 
adaptive model-based control using neural 
networks with the method for modeling using 
fuzzy logic, and fractal theory to obtain a new 
hybrid neuro-fuzzy-fractal method for the control 
of nonlinear dynamic aircraft. The adaptive 
controller can be used to control chaotic and 
unstable behavior in aircraft systems. Chen, et al. 
[9] presented an approach using artificial neural 
networks (ANN) algorithm for predicting the 
flutter derivatives of rectangular section models 
without wind tunnel tests. The database of flutter 
derivatives was identified from a back-
propagation (BP) ANN model that is built using 
experimental dynamic responses of rectangular 
section models in smooth flow as the input/output 
data. These limited sets of database are employed 
as input/output data to establish a prediction ANN 
frame model to further predict the flutter 
derivatives for other rectangular section models 
without conducting wind tunnel tests. The results 
presented indicate that this ANN prediction 
scheme works reasonably well.  

The contribution of the present work is the 
utilization of a relatively simple approximation 
neural network to identify the posture of the non 
linear 2-D wing system and to design an adaptive 
nonlinear neural controller.  

 
 

2. Method of Analysis 
2.1. Two Dimensional Aero elastic Wing 
Model 
 

The 2-D aero elastic wing section is shown in 
Figure (1). The governing equations of motion are 
provided in [5] and are given as:  

Lhkhcbmxhm ha −=+++
.....

hα  

MkchbmxI a =+++ αααα α )(
.

a

....
              …(1) 

where h is the plunge displacement and α is the 
pitch angle. In Eq. (1), m is the mass of the wing; 
b is the semichord of the wing; I is the moment of 

inertia; xα is the nondimensionalized distance of 
the center of mass from the elastic axis; kh is  
plunge stiffness coefficient; )(ααk is nonlinear 
pitch stiffness ch and cα are plunge and pitch 
damping coefficients, respectively; L and M are 
the aerodynamic lift and moment. In the present 
work, the nonlinear quasi steady aerodynamic 
model including stall effect [10] and flap angle 
effect [5] is as follow 

22
2

11
23

3
2

      

      )(

βρ

βρααρ

β

βα

L

LeffeffL

bCU

bCUcbCUL

+

+−=
 

22
22

11
223

3
22

       

      )(

βρ

βρααρ

β

βα

m

meffeffm

CbU

CbUcCbUM

+

+−=

                                                                       … ( 2 ) 
where ρ is the air density, U is the flow 

velocity, and CLα and Cmα are the lift and moment 
coefficients. CLβ and Cmβ are the lift and moment 
coefficients per flap angle. β1 and β2 are flap 
angles. c3 is a nonlinear parameter associated with 
stall model. αeff is the effective angle of attack 
defined by 

 /Uα-d)b.(/Uhα
..

eff 50++=α                 …(3) 

     where d is the non dimensional distance from 
the mid chord to the elastic axis. Parameter c3 is 
defined as follow for NACA 0012 airfoil (a 
symmetric wing section by National Advisory 
Committee for Aeronautics), 

   c3 = 0.00034189(180/π)3/CLα                       …(4) 

This aerodynamic model is valid for                   
αeff ∈  (−11, 11) degree. 
The function kα(α) is considered as a polynomial 
given by [10] 

   kα(α) = kα1 + kα2α + kα3α2,                            … (5) 

where kαj , j = 1, 2, 3  are constants. 
 

 
Fig.1. Two-Dimensional Aero elastic Model. 



Mauwafak A. Tawfik                              Al-Khwarizmi Engineering Journal Vol. 7, No. 4, PP 27-40(2011)   
 

29 
 

0 5 10 15 20 25 30 35 40 45 50

-0.1

-0.05

0

0.05

0.1

time (s ec)

al
fa

 (
ra

d)

U=9.8 m/sec

0 5 10 15 20 25 30 35 40 45 50
-4

-3

-2

-1

0

1

2

3

4
x 10

-3

time (s ec)

pl
un

ge
(m

)

U=9.8 m/sec

0 5 10 15 20 25 30 35 40 45 50
-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

time (s ec)

al
fa

 (
ra

d)

U=9.9 m/sec

0 5 10 15 20 25 30 35 40 45 50
-3

-2

-1

0

1

2

3
x 10

-3

time (s ec)

pl
un

ge
(m

)

U=9.9 m/sec

2.2.  Nonlinear Flutter analysis 
 

The proposed nonlinear controller must work 
in the unstable region .Time domain simulation is 
used to examine the dynamic aeroelastic 
instabilities of the system (e.g. the onset of flutter 
and limit cycle oscillation (LCO)) [11].The 
simulation is  performed by solving eqs (1-5) and 
using Runge Kutta method for different velocities 
and initial conditions.It was found that LCO 
appears at U=9.9m/sec and never appear at speed 
less than what ever the initial conditions. 
Therefore the flutter speed is 9.9m/sec and the 
proposed nonlinear controller must give a good 

performance at speed higher than that value 
(unstable region).  

Figure (2) shows the plunge h and pitch angle 
α responses at speed 9.8m/sec while the responses 
at speed 9.9m/sec are shown in the figure (3) The 
time responses of the plunge h and pitch angle α 
corresponding to the uncontrolled system at 
U=30m/sec, and 40m/sec are shown in figure 
(4).Clearly, the system responses exhibit LCO 
behaviour. It is clear that the amplitudes of LCO 
increase with the velocity. The function of 
controller is to reduce amplitudes of LCO at short 
settling time in order to eliminate the risk of 
damage of wing structure.   
 

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 

Fig.2. Time history of Pitch Angle and  
Plunging At 9.8 M/S Speed and Initial 

Conditions ( radh 1.0)0(;0)0( == α ). 

Fig.3. Time History of Pitch Angle and 
Plunging at 9.9 M/S Speed and Initial 

Conditions () radh 1.0)0(;0)0( == α . 
 



Mauwafak A. Tawfik                              Al-Khwarizmi Engineering Journal Vol. 7, No. 4, PP 27-40(2011)   
 

30 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 

2.3. Adaptive Nonlinear Neural Control 
Methodology 
 

The approach to control wing motion depends 
on the available information about the system and 
the control objectives. The 2-D wing system is 
considered as modified Elman recurrent neural 
networks model. The first step in the procedure of 
the control structure is the identification of 
nonlinear dynamics of 2-D wing system from the 
input-output data. Then a feed forward neural 
controller is designed using feed forward multi-
layer perceptron neural network to find flap 
angles that control the plunge and pitching wing 
motion. 

The proposed structure of the adaptive 
nonlinear neural controller can be given in the 
form of block diagram as shown in figure (5). It 
consists of: 

1- Neural Network Identifier of 2-D wing system . 
2- Feed forward Neural Controller. 

In the following sections, each part of the 
proposed controller will be explained in details. 
 
 
2.4. Nonlinear 2-D Wing System Neural 
Network Identifier 
 

The modified Elman recurrent neural network 
model is applied to construct the 2-D wing system 
neural network identifier as shown in figure (5). 
The nodes of input, context, hidden and output 
layers are highlighted. The network uses two 
configuration models, series-parallel and parallel 
identification structures, which are trained using 
dynamic back-propagation algorithm. 
 
 
 
 
 
 
 

Fig.4. Time History of Pitch Angle and Plunging at 30m/s and 40m/sec Speed With Initial Conditions 
( radh 1.0)0(;0)0( == α ). 

0 5 10 15 20 25 30 35 40 45 50
-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

time (sec)

al
fa

 (
ra

d)
U=30 m/sec

0 5 10 15 20 25 30 35 40 45 50
-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

time (sec)

al
fa

 (
ra

d)

U=40 m/sec

0 5 10 15 20 25 30 35 40 45 50
-0.03

-0.02

-0.01

0

0.01

0.02

0.03

time (sec)

pl
un

ge
(m

)

U=40 m/sec

0 5 10 15 20 25 30 35 40 45 50
-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

time (sec)

pl
un

ge
(m

)

U=30 m/sec  

  



Mauwafak A. Tawfik                              Al-Khwarizmi Engineering Journal Vol. 7, No. 4, PP 27-40(2011)   
 

31 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Fig.5. The Proposed Structure of the Adaptive Nonlinear Neural  
Controller for the 2-D Wing . 

 
Fig.6. The Modified Elman Recurrent Neural Networks Acts as the Plunging  

and Pitch Motion of the Wing. 

o
cγ  



Mauwafak A. Tawfik                              Al-Khwarizmi Engineering Journal Vol. 7, No. 4, PP 27-40(2011)   
 

32 
 

The structure shown in figure (6) is based on 
the following equations: 

}),(),({)( VbbiaskVCkGVHFk oγγ =                 …(6) 

)),(()( WbbiaskWkO γ=                                     …(7) 

where VH,VC and W are weight matrices, Vb  
and Wb  are weight vectors and F is a non-linear 
vector function. The multi-layered modified 
Elman neural network, shown in figure (7), is 
composed of many interconnected processing 
units called neurons or nodes. The network 
weights are denoted as follows: 

VH :Weight matrix of the hidden layers. 
VC : Weight matrix of the context layers. 
Vb : Weight vector of the hidden layers. 
W : Weight matrix of the output layer. 
Wb : Weight vector of the output layer. 
L : Denotes linear node. 
H : Denotes nonlinear node with sigmoidal 
function. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Fig.7. The Multi-Layer Perceptron Neural 
 Networks of the Feed forward Neural Controller. 

 
 

In order to improve the ability of network 
memory, self-connections, with fixed value λ , are 
introduced into the context units of the network to 
give these units a certain amount of inertia [13]. 
The introduction of self-connections in the context 
units increases the possibility of modelling high-
order systems by Elman network.  
The output of the context unit in the modified 
Elman network is given by: 

)1()1()( −+−= kkk c
o
c

o
c ργλγγ                     …(8) 

where )(kocγ  and )(kcγ are the outputs of the 
context and hidden units respectively. λ  is the 
feedback gain of the self-connections and ρ is the 

connection weight from the hidden units (jth) to 
the context units (cth) at the context layer.  The 
value of λ  and ρ  are selected randomly between 
(0 and 1) [13] . To explain these calculations, 
consider the general jth neuron in the hidden layer. 
The inputs to this neuron consist of an i– 
dimensional vector, where i is the number of the 
input nodes. Each of the inputs has VH and VC 
weights associated with it. 
Vb is the weight vector for the bias input set 
equal to -1 to prevent the neurons quiescent. The 
first calculation within the neuron consists of 
calculating the weighted sum jnet  of the inputs as 
[13 and 14]: 

j

C

c

o
cjc

nh

i
ijij VbbiasVCGVHnet ×+×+×= ∑∑

== 11

γ         …(9) 

Where j.is the number of the hidden nodes, c is 
the number of the context nodes and G is the 
input vector. The outputs of the identifier are the 
modelling plunge and pitch motion and are 
defined as:   T

mmm hq ),( α=  
where 

mh : plunge motion of the wing identifier.  

mα Pitch motion of the wing identifier. 
The learning algorithm will be used to adjust 

the weights of dynamical recurrent neural 
network. Dynamic back propagation algorithm is 
used to train the Elman network. The sum of the 
square of the differences between the desired 
outputs Thq ),( α= and neural network identifier 
outputs T

mmm hq ),( α= is given by equation (10). 

))()((
2
1

1

22∑
=

−+−=
np

i
mmhhE αα                            …(10) 

where np is the number of patterns. 
 

The connection matrix between hidden layer 
and output layer is kjW   

kj
kj W

E
kW

∂
∂

−=+∆ η)1(                                          …(11) 

where η  is the learning rate. 

kj

k

k

k

k

m

mkj W
net

net
o

o
kq

kq
E

W
E

∂
∂

∂
∂

∂
+∂

+∂
∂

=
∂
∂ )1(

)1(
              …(12) 

kjkj ekW ××=+∆ γη)1(                                      …(13)         

)1()()1( +∆+=+ kWkWkW kjkjkj                            …(14) 

The connection matrix between the input layer 
and hidden layer is 

jiVH   



Mauwafak A. Tawfik                              Al-Khwarizmi Engineering Journal Vol. 7, No. 4, PP 27-40(2011)   
 

33 
 

ji
ji VH

E
kVH

∂
∂

−=+∆ η)1(                                   …(15) 

ji

j

j

j

j

k

k

k

k

m

mji VH
net

net
net

net
o

o
kq

kq
E

VH
E

∂

∂

∂

∂

∂
∂

∂
∂

∂
+∂

+∂
∂

=
∂

∂ γ

γ
)1(

)1(
  …(16)  

∑
=

×′×=+∆
K

k
kjkijji WeGnetfkVH

1

)()1( η               …(17)  

)1()()1( +∆+=+ kVHkVHkVH jijiji                     …(18) 

The connection matrix between context layer 
and hidden layer is jiVC   

jc
jc VC

E
kVC

∂
∂

−=+∆ η)1(                                      …(19) 

jc

c

c

j

j

k

k

k

k

m

mjc VC
net

net
net

net
o

o
kq

kq
E

VC
E

∂
∂

∂

∂

∂
∂

∂
∂

∂
+∂

+∂
∂

=
∂

∂ γ

γ
)1(

)1(
    

                      …(20) 

∑
=

×′×=+∆
K

k
kjk

o
cjjc WenetfkVC

1

)()1( γη               …(21) 

)1()()1( +∆+=+ kVCkVCkVC jcjcjc                     …(22) 

 
 
2.5.  Feed Forward Neural Controller 
 

The Feed Forward Neural Controller (FFNC) 
is essential to stabilize the tracking error of the 
wing system when the response of the wing is 
drifted from the desired condition during transient 
state and kept the steady-state tracking error at 
zero. The controller generates flap angles )(1 kβ  
and )(2 kβ  control action that minimizes the 
cumulative error between the desired condition 
and the output response of the wing. The FFNC is 
supposed to learn the adaptive inverse model of 
the wing with off-line and on-line stages to 
calculate wing's reference input flap angle and 
keep the wing stable without flutter state in the 
presence of any disturbances or dynamics 
parameters changing. 

To achieve FFNC, a multi-layer Perceptron 
model is used as shown in figure (7). The network 
notations are as follows: 

Vffc : W eight ma tr ix of t he hidd en la yer s. 
bffcV : Weight vector of the hidden layers. 

Wffc : Weight matrix of the output layer. 
ffcWb : W eight vector  of the output la yer .  

To explain these calculations, consider the 
general ath neuron in the hidden layer shown in 
figure (7.) The inputs to this neuron consist of an 
n–dimensional vector, where n is the number of 
the input nodes. Each input has an associated 

weight of Vffc . The first calculation within the 
neuron is to calculate the weighted sum of the 
inputs, anetc  as [15, 16 and 17]: 

a

nhc

a
nana VbffcbiasZVffcnetc ×+×= ∑

=1

                 …(23) 

where nhc is the number of the hidden nodes. 
Next, the output of the neuron 

ah is calculated as 
the continuous sigmoid function of the 

anetc  as: 

)( aa netcHc =γ                                                …(24) 

1
1

2
)( −

+
=

− anetca e
netcH                                …(25) 

Once the outputs of the hidden layer have 
been calculated, they are passed to the output 
layer. 

In the output layer, two linear neurons are 
used to calculate the weighted sum netco of its 
inputs, which are the output of the hidden layer as: 

b

nhc

a
abab WbffcbiascWffcnetco ×+×= ∑

=1

γ                …(26) 

where 
baWffc  are the weights between the hidden 

neuron acγ  and the output neurons. Then the sum 
( bnetco ) will be passed through a linear activation 
function of slope 1; another slope can be used to 
scale the output, as: 

)( bb netcoLOc =                                                …(27) 

The outputs of the feedforward neural 
network controller represent flap angles, )(1 kβ  
and )(2 kβ .  
        The training of the feedforward neural 
controller is performed off-line as shown in figure 
(8). And adaptive weights are adapted on-line. It 
depends on the posture neural network identifier 
to find the wing Jacobian through the neural 
identifier model. This approach is currently 
considered as one of the better approaches that 
can be followed to overcome the lack of initial 
knowledge. 
 
 
 
 
 
 
 
 
 
 



Mauwafak A. Tawfik                              Al-Khwarizmi Engineering Journal Vol. 7, No. 4, PP 27-40(2011)   
 

34 
 

 
 

 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 

 The dynamic back propagation algorithm is 
employed to realize the training the weights of the 
feedforward neural controller. The sum of the 
square of the differences between the desired 
posture Trrr hq ),( α= and neural network posture 

T
mmm hq ),( α= is: 

))()((
2
1

1

22∑
=

−+−=
npc

i
mrmr hhEc αα

                          …(28) 

where npc is the number of patterns. 
 

To achieve equation (28) a modified Elman 
neural network will be used as posture identifier. 
This task is carried out using an identification 
technique based on series-parallel and parallel 
configuration with two stages to learn the posture 
identifier. The first stage is an off-line 
identification, while the second stage is an on-line 
modification of the weights of the obtained wing 
neural identifier. The on-line modifications are 
necessary to keep tracking any possible variation 
in the dynamic parameters of the 2-D wing 
system. Back Propagation Algorithm (BPA) is 
used to adjust the weights of the posture neural 
identifier to learn dynamic of the 2-D wing 
system by applying a simple gradient decent rule. 

The connection matrix between the hidden 
layer and the output layer is 

baWcont   

ba
ba Wffc

Ec
kWffc

∂
∂

−=+∆ η)1(                                   …(29) 

ba

b

b

b

b

b

b

m

mba Wffc
netc

netc
oc

oc
k

k
kq

kq
Ec

Wffc
Ec

∂
∂

∂
∂

∂
∂

∂
+∂

+∂
∂

=
∂

∂ )(
)(

)1(
)1(

β
β

    

                                 …(30) 

ab
b

b

b

m

mba

cnetcf
oc

k
k

kq
kq
Ec

Wffc
Ec

γ
β

β
)(

)(
)(

)1(
)1(

′
∂

∂
∂

+∂
+∂

∂
=

∂
∂    …(31) 

)1(

))()((
2
1

)1(

22

+∂

−+−∂
=

+∂
∂

∑
kq

hh

kq
Ec

m

mrmr

m

αα
             …(32) 

)(
)1(

k
kq

Jacobian
b

m

β∂
+∂

=                                   …(33) 

)(
)(

)(
)1(

)(
)1(

k
net

net
net

net
ko

ko
kq

k
kq

b

j

j

j

j

k

k

k

k

m

b

m

β

γ

γβ ∂

∂

∂

∂

∂
∂

∂
∂

∂
+∂

=
∂

+∂            …(34) 

For linear activation function in the outputs layer: 

)()(
)1(

k
net

net
net

k
kq

b

j

j

j

j

k

b

m

β
γ

γβ ∂
∂

∂
∂

∂
∂

=
∂

+∂                    …(35) 

For nonlinear activation function in the hidden 
layer: 

)(
)(

)(
)1(

1 k
net

netfW
k

kq

b

j
j

K

k
kj

b

m

ββ ∂
∂

′=
∂

+∂
∑

=

                  …(36) 

∑ ∑
= =

′=
∂

+∂ nh

j

K

k
kjjbj

b

m WVHnetf
k

kq
1 1

)(
)(

)1(
β

                   …(37)   

Substituting equations (32 and 37) into equation 
(31), )1( +∆ kWffcba  becomes: 

)))1((

))1((()()1(

2

1
1

jm

jm

nh

j
jbjaba

Wke

WkehVHnetfckWffc

++

+′×=+∆ ∑
=

α

ηγ              

                                                                       …(38)   

)1()()1( +∆+=+ kWffckWffckWffc bababa                  …(39) 

The connection matrix between the input layer 
and the hidden layer is 

a nVffc  

a n
a n Vffc

Ec
kVffc

∂
∂

−=+∆ η)1(                     …(40) 

 

Fig.8. The Structure of the Feed forward Neural Controller 
for Wing Model. 



Mauwafak A. Tawfik                              Al-Khwarizmi Engineering Journal Vol. 7, No. 4, PP 27-40(2011)   
 

35 
 

×
∂
∂

×
∂

∂
×

∂
∂

=
∂

∂

b

b

b

b

ba n netc
oc

oc
k

k
Ec

Vffc
Ec )(

)(
β

β

an

a

a

a

a

b

Vffc
netc

netc
c

c
netc

∂
∂

×
∂
∂

×
∂

∂ γ
γ

                                …(41) 

×
∂

∂
∂

+∂
+∂

∂
=

∂
∂

b

b

b

m

ma n oc
k

k
kq

kq
Ec

Vffc
Ec )(

)(
)1(

)1(
β

β

an

a

a

a

a

b

b

b

Vffc
netc

netc
c

c
netc

netc
oc

∂
∂

×
∂
∂

×
∂

∂
×

∂
∂ γ

γ
                       …(42) 

na

B

b
ba

b

m

ma n

ZnetcfWffc
k

kq
kq
Ec

Vffc
Ec

×′××
∂

+∂
+∂

∂
=

∂
∂

∑
=

)(
)(

)1(
)1( 1β

   …(43) 

Substituting equations (32 and 37) into equation 
(43), )1( +∆ kVffca n becomes: 

)))1(())1((()(

)()1(

21
1 1

1

jmjm

nh

j

I

i
jij

B

b
baanan

WkeWkehVHnetf

WffcnetcfZkVffc

+++′

′=+∆

∑ ∑

∑

= =

=

α

η
   

                                  …(44) 

The B and I are equal to two because there are 
two outputs in the feedforward neural controller.  

)1()()1( +∆+=+ kVffckVffckVffct ananan               …(45) 
 

Once the feedforward neural controller has 
learned, it generates the flap control action to keep 
the output of the wing at reference value and to 
overcome any external disturbances during 
motion.  
 
 
3. Results and Discussion 
 

The proposed controller is verified with 
computer simulation using C++ program. The 
dynamics model of 2-D wing system described in 
section 2 is used. The simulation is carried out by 
tracking a desired plunging and pitch angle during 
flutter or limit cycle oscillation condition. The 
parameter values of a two Degree Of Freedom (2-
DOF) airfoil system (typical section model) are 
taken from [5] see table (1). 

The fist stage of operation is to set the 
position (plunging motion) and orientation about 
the elastic axis (pitch angle) neural network 
identifier. This task is performed using series-
parallel and parallel identification technique 
configuration with modified Elman recurrent 
neural networks model. The identification scheme 
of the nonlinear MIMO (2-DOF) airfoil system 
are needed to input-output training data pattern to 
provide enough information about dynamics (2-
DOF) wing model to be modelled. This can be 
achieved by injecting a sufficiently rich input 
signal to excite all process modes of interest while 
also ensuring that the training patterns adequately 

covers the specified operating region. A hybrid 
excitation signal has been used for the 2-D wing 
model. Figure (9) show the input signals )(1 kβ  and 

)(2 kβ ,.  
 
 Table 1, 
 System Parameters [5]  
b  0.135 m 
d -0.45 
m  12.387 kg 
xa  0.25 
I 0.065 kg.m2 
kh  2844.4 N/m 
kα(α)  12.77+53.47α+1003α2 N/rad  [11] 
ch 27.43 
cα 0.036 
ρ 1.225 kg/m3 
CLα  6.28 
Cmα -0.635 
CLβ1  3.358 
C Lβ2  3.458 
C mβ1  -0.635 
C mβ2  -0.735 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

The training set is generated by feeding a 
Pseudo Random Binary Sequence (PRBS) signals, 
to the model and measuring its corresponding 
outputs, position (plunging motion) and 
orientation (pitch angle), with a sampling time of 
0.01 second. This value was found adequate for 
the stability and convergence of solution. Back 
propagation learning algorithm is used with the 
modified Elman recurrent neural network of the 
structure 5-7-7-2. The number of nodes in the 
input, hidden, context and output layers are 5, 7, 7 
 and 2 respectively as shown in figure (6).  

Fig.9. The PRBS Input Flap Angles Signals Used 
 to Excite the Wing Model.  



Mauwafak A. Tawfik                              Al-Khwarizmi Engineering Journal Vol. 7, No. 4, PP 27-40(2011)   
 

36 
 

0 20 40 60 80 100 120 140 160 180 200
-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Sampling Time 0.01 Sec

pl
un

gi
ng

 (
m

)

 

 
h
hm

A training set of 200 patterns has been used 
with a learning rate of 0.1 and variable speed 
inputs U=[25, 30, 35, 40 and 45] m/sec. After 
5439 epochs, the identifier outputs of the neural 
network, position (plunge motion) and orientation 
about (pitch angle), are approximated to the actual 
outputs as shown in figure (10). The objective 
cost function MSE is less than 16 × 10-5 as shown 
in figure (11). 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 

Parallel configuration is used to guarantee the 
similarity between the outputs of the neural 
network identifier and the actual outputs of the 
plunging motion and pitch angle of the wing. At 
3859 epochs the same training set patterns has 
been achieved with an MSE less than 1.9×10-6. 

The testing set is generated by difference 
feeding a PRBS signals as shown in figure (12), 
and it is applied to the system.  

Figure (13) compare the time response of the 
parallel mode output with the actual plant output, 
and there is excellent identification. 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 

Fig.10. The Response of the Neural Network 
 Identifier with the Actual 2-D Wing Model Output 

 for the Learning Set. 

Fig.13. The Response of the Neural Network 
Identifier with the Actual 2-D Wing Model 

Output for the Testing Set. 

Fig.12. The PRBS Input Flap Angles Signals Used  
for Testing 2-D Wing Identifier Model. 

Fig.11. The MSE of the Cost function. 

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.001

0 600 1200 1800 2400 3000 3600 4200 4800 5400

Epochs

M
S

E

 



Mauwafak A. Tawfik                              Al-Khwarizmi Engineering Journal Vol. 7, No. 4, PP 27-40(2011)   
 

37 
 

 
The final stage of the proposed controller is 

feedforward neural controller. It uses multi-layer 
perceptron neural network 6-13-2 as shown in 
figure (7). The desired conditions has been 
learned by the feedforward neural controller with 
off-line and on-line adaptation stages using back 
propagation algorithm as shown in figure (8) to 
find the suitable control action.  

The controller performance is simulated at 
three values of the flight speed (30, 35, 45 m/sec) 
in unstable region and at different initial 
conditions of plunging motion and pitch angle. 
Figure (14) shows the closed loop responses for 
the controlled 2-D wing system. The controller 
reach the requirements and the closed-loop 
simulation obtained is stable .The over shoot and 
settling time increase slightly with the increasing 
of the velocity. Also the oscillation during the 
transient period appears with the increasing of the 
velocities, but its amplitude is small and converge 
of desired condition is quick with settling time 
0.4sec at high velocity U= 40m/sec. 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 

 
In figure (15) the responses of the controller 

effort (flap angles) at U=30m/sec and 40m/sec are 
shown. It is clear that the control action β1 has 
large values in comparing with another control 
action β2 .Also the control actions never exceed 
± 0.3 rad. The values of flap angles are limited 
from -0.3 to 0.3 rad. to make the controller works 
in logical limits. 

 

 
 
 

The values of the initial conditions are 
varying to make the nonlinear stable or unstable, 
so the present controller performance is tested at 
different initial conditions as shown in figure (16). 
When the initial values of plunging and pitch 
angle increase the over shoot and the oscillation 
increase during the transient period. The plunging 
initial value has large effect on the responses than 
pitch angle initial value. But the present controller 
can give acceptable performance and reaches to 
desired condition at very short settling time about 
0.2sec. This result compares very well with 
reference [5]. 

 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-10

-8

-6

-4

-2

0

2

4
x 10

-3

time (sec)

pl
un

gi
ng

 (
m

)

 

 
U=30m/sec
U=35m/sec
U=40m/sec

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

time (sec)

al
fa

 (
ra

d)

 

 
U=30m/sec
U=35m/sec
U=40m/sec

Fig.14. System Responses with Controller at  
Different Speed and Initial Conditions 

 ( radh 1.0)0(;0)0( == α ). 

Fig.15. Control Efforts at Different Speed 
 and Initial Conditions ( r a dh 1.0)0(;0)0( == α ). 



Mauwafak A. Tawfik                              Al-Khwarizmi Engineering Journal Vol. 7, No. 4, PP 27-40(2011)   
 

38 
 

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

time (sec )

al
fa

 (r
ad

)

U=40m/s ec

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-8

-6

-4

-2

0

2

4
x  10

-3

time (sec )

pl
un

gi
ng

 (
m

)

U=40m/sec

 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
Finally, to investigate the present controller 

performance at difficult conditions, the 
disturbance action is introduced on the wing 
model at high velocity U= 40m/sec. In figure (17) 
the responses of controller with sinusoid 
disturbance has been shown. The amplitude of 
disturbance is taken as 300% the maximum values 
of control actions and frequency 10 Hz. In spite of 
the existence of bounded disturbances the 
adaptive learning and robustness of neural 
controller show small effect of these disturbances. 
The controller achieves the desired condition with 
very small fluctuation. For plunging and pitch 
motions the fluctuation was 0.00055 m and 
0.0068 rad. respectively. These values are less 
than the maximum fluctuation for the same wing 
section [18] (about 0.048b for plunging and 0.024 
rad. for pitch motion). 
 
 
4. Conclusions 

In the current work, it has been shown that the 
proposed controller has the capability to generate  
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
Fig.17. System Responses with Sinusoid 

Disturbanceat initial Conditions 
( radh 1.0)0(;0)0( == α ). 

 
smooth and suitable flap commands, 1β  and 2β  
without sharp spikes. Moreover, it has the 
capability of effectively eradicating the tracking 
errors for the 2-D wing model. The principal 
conclusions may be summarized as follows: 
- Excellent identification is achieved when 
comparing the time response of the parallel mode 
output with the actual plant output. 
- The controller reach the requirements and the 
closed-loop simulation obtained is stable. The 
over-shoot and settling time increase slightly with 
the velocity. The amplitude of oscillation during 
the transient period is small and converge to 
desired condition. 
- The controller worked in logical limits, the 
control action never exceeds ± 0.3 rad. 
- The plunging initial value has large effect on the 
response than pitch angle initial value. 
- Simulation results show that the proposed 
controller is robust and effective in comparison 
with the controller in [5] in terms of fast response 
with minimum settling times and minimum 

Fig.16. System Responses with Controller  
at Different Initial Conditions. 



Mauwafak A. Tawfik                              Al-Khwarizmi Engineering Journal Vol. 7, No. 4, PP 27-40(2011)   
 

39 
 

tracking error despite the presence of bounded 
external disturbances. 
 
 
5. References 
 
[1] Shams Sh., Sadr M.H.,and  Haddadpour H.     

" Nonlinear aeroelastic response of slender 
wings based on Wagner function", Thin-
Walled Structures ,Vol 46 (2008) pp.1192– 
1203 

[2] Woolston DS, Runyan HL,and Andrews RE. 
"An investigation of certain types of 
structural nonlinearities on wing and control 
surface flutter" J Aeronaut Sci, Vol 
24,(1957),pp.57–63. 

[3] Frampton, K.D., and Clark, R.L. 
"Experiments on control of limit cycle 
oscillations in a typical section" , AIAA 
Paper 99-1466’. Proc.40th Structures, 
Structural Dynamics and Material Conf., St. 
Louis, MO, (1999), pp. 2195–2055 

[4] Palaniappan K, Sahu  P.and  Alonso J.J             
" Design of Adjoint Based Laws for wing 
flutter control",American Institute of 
Aeronautics and Astronautics Paper AIAA-
(2009)-148. 

[5] Afkhami S. and . Alighanbari H. "Nonlinear 
control design of an airfoil with active flutter 
suppression in the presence of disturbance", 
IET Control Theory Appl., Vol. 1, No. 6, 
November (2007) 

[6] Haiwei,  Y. and  Jinglong H. " Robust flutter 
analysis of a nonlinear aeroelastic system  
with parametric uncertainties", Aerospace 
Science and Technology , Vol 13 ,(2009) 
pp.139–149 

[7] Zeng, Y., and Singh, S.N "Output feedback 
variable structure adaptive control of an 
aeroelastic system", J. Guidance Control 
Dyn.,Vol 21, (1998), pp. 830–837 

[8] Melin P.and Castillo O." Adaptive intelligent 
control of aircraft systems with a hybrid 

approach combining neural networks, fuzzy 
logic and fractal theory", Applied Soft 
Computing vol3, (2003) pp.353–362 

[9] Chen C., Wu J.and Chen J. " Prediction of 
flutter derivatives by artificial neural 
networks", Journal of Wind Engineering and 
Industrial Aerodynamics ,Vol. 96 ,(2008), 
pp.1925–1937 

[10] Demenkov M. "Form Local to Global 
Stabilizility of Aeroelastic oscillation",18th 
IEEE International Conference on Control 
Apllictions  ,Saint Petersburg ,Russia, 2009.  

[11] Abbas1 L K, Chen1 Q. and Milanese A.        
" Non-linear aeroelastic investigations of 
store(s)-induced limit cycle oscillations", J. 
Aerospace Engineering Vol. 222 Part G 
(2008). 

[12] Ogata K.Modern Control Engineering, 
Prentice-Hall International.Inc(1970) 

[13] Pham D. T. and Xing L.  " Neural Networks 
for Identification, Prediction and Control", 
Springer, (1995).  

[14] S. Omatu, M. Khalid, and R. Yusof "Neuro-
Control and its Applications" London: 
Springer-Velag, 1995. 

[15] Zurada J. M., Introduction to Artificial 
Neural Systems. Jaico Publishing House, 
Pws Pub Co. (1992). 

[16] K. S. Narendra and K. parthasarathy, 
“Identification and control of dynamical 
systems using neural networks,” IEEE Trans. 
Neural Networks, vol. 1,pp. 4-27, 1990. 

[17] K. S. Narendra and K. parthasarathy,  
“Gradient methods for the optimization of 
dynamical systems containing neural 
networks,” IEEE Trans. Neural Networks, 
vol. 2 No. 2, pp. 252-262, 1991. 

[18] Abu-Tabikh,M.I." Modeling of Steady and 
Unsteady Tturblunent Boundary Layer 
Separation Using Vortex Hybrid Method ", 
Ph.D thesis, Department of Mechanical 
Engineering, U.O.T, Baghdad,1997.

 
 
 
 
 
 
 
 
 
 
 
 
 



 )2011( 27 - 40 ، صفحة4 ، العدد7مجلة الخوارزمي الھندسیة المجلد                       موفق علي توفیق                                       
 

 

40 
 

  
  مسیطرثنائي األبعاد باستخدام جناح  السیطرة على رفرفة 

  متكیفعصبي ال خطي  
  

  **األمیر حیدر صباح عبد            *     محسن إدریسمحمد . د                   *  موفق علي توفیق. د     
  الجامعة التكنولوجیة  /قسم ھندسة المكائن والمعدات * 

  بغداد  -معھد التكنولوجیا /قسم الھندسة المیكانیكیة **
  
  
  

  الخالصة
  

ت الالخطی ة ف ي منظوم ة    الم ؤثرا  تنش ا  .ثن ائي األبع اد   جن اح  مسیطر عصبي الخطي متكیف للسیطرة عل ى الرفرف ة لنم وذج    اقتراح البحث تم في ھذا 
غی ر مس تقر م ن خ الل فح ص       ت م تحدی د المنطق ة الت ي یك ون فیھ ا النظ ام        .الت أرجح  شبھ المستقر والنابض أاللتوائي باتج اه  الجناح من النموذج االیرودینامي

من نموذجین ھما الشبكة العصبیة  تتكون ھیكلیة المسیطر. رفرفة والتذبذب الدوري المحدداستجابتھ مع الزمن حیث تم إیجاد السرعة التي تبدأ عندھا ظاھرة ال
الخ ط  مرحل ة  و ل ق الخ ط المغ ھم ا مرحل ة   ن م رحلتی ف ي  )  MENN( النم وذج  لقد تم تأھیل. )MLP(و بیرسبتون متعدد الطبقات ) MENN(المحسنة أللمن

ت م   .حركة التأرجحیة والحركة العمودیة لتكوین النموذج العص بي المع رف   مخرج منظومة الجناح وھومع  العصبي خرج النموذجم المفتوح لضمان تطابق
الخ ط المفت وح إلیج اد زاوی ة الخافق ة المطلوب ة الت ي         من خالل الخط المغلق ثم تم تحدیث األوزان لھذا المسیطر م ن خ الل   تأھیل المسیطر العصبي األمامي

 نتائج المحاكاة لھذا المسیطر العصبي الالخطيكانت . االنتشار الخلفي لتأھیل النموذجینتم استخدام خوارزمیة .والحركة العمودیة التأرجحتسیطر على حركة 
  .وضاء خارجیةمع وجود ض لى صفر وبزمن استقرار مناسبمن خالل تقلیل الرفرفة إ فعالة