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