Intelligent Control and Automation, 2011, 2, 176-181
doi:10.4236/ica.2011.23021 Published Online August 2011 (http://www.SciRP.org/journal/ica)
Copyright © 2011 SciRes. ICA
Identification and Adaptive Control of Dynamic Nonlinear
Systems Using Sigmoid Diagonal Recurrent Neural
Network
Tarek Aboueldahab1, Mahumod Fakhreldin2
1Cairo Metro Company, Ministry of Transport, Cairo, Egypt
2Computers and Systems Department, Electronic Research Institute, Cairo, Egypt
E-mail: heshoaboda@hotmail.com, mafakhr@mcit.gov.eg
Received April 28, 2011; revised May 12, 2011; accepted May 19, 2011
Abstract
The goal of this paper is to introduce a new neural network architecture called Sigmoid Diagonal Recurrent
Neural Network (SDRNN) to be used in the adaptive control of nonlinear dynamical systems. This is done
by adding a sigmoid weight victor in the hidden layer neurons to adapt of the shape of the sigmoid function
making their outputs not restricted to the sigmoid function output. Also, we introduce a dynamic back
propagation learning algorithm to train the new proposed network parameters. The simulation results showed
that the (SDRNN) is more efficient and accurate than the DRNN in both the identification and adaptive con-
trol of nonlinear dynamical systems.
Keywords: Sigmoid Diagonal Recurrent Neural Networks, Dynamic Back Propagation, Dynamic Nonlinear
Systems, Adaptive Control
1. Introduction
The remarkable learning capability of neural networks is
leading to their wide application in identification and
adaptive control of nonlinear dynamical systems [1,2-5,6]
and the tracking accuracy depends on neural networks
structure, which should be chosen properly [7-13].
Feedforward Neural Network (FNN) [4] is a static
mapping and can not reflect the dynamics of the nonlin-
ear systems without using Tapped Delay Lines (TDL)
[7,9]. Fully connected Recurrent Neural Network (RNN)
[9,14,15] contains interlink between neurons to reflect
this dynamics but it suffers both structure complexity
and the poor performance accuracy [9,15]. Based on Lo-
cally Recurrent Globally Feedforward network architec-
tures (LRGF) many researchers focused in (DRNN)
which doesn't contain interlink between hidden layer
neurons leading to the network structure complexity re-
duction [15,8-10].
However, in all these architectures, the hidden layer
neurons output restricted to the sigmoid function output
which represents a major disadvantage in the network
behavior and significantly reduces its performance accu-
racy. Therefore, a new architecture called Sigmoid Di-
agonal Recurrent Neural Network (SDRNN) based on
the hidden layer sigmoid weight and its associated dy-
namic back propagation learning algorithm is proposed.
Simulation results show that SDRNN is more suited than
the (DRNN) for identification and adaptive control of
nonlinear dynamical systems [9].
This paper is organized as follows: Section II presents
some background concerning application of neural net-
works in adaptive control, Section III introduce the new
architecture and its associated dynamical learning algo-
rithm for adaptation of the sigmoid weight. Simulation
results are shown in Section IV, and finally, conclusion
and future work is shown in Section V.
2. Neural Network in Nonlinear System
Identification and Control
In the identification stage of the adaptive control of
nonlinear dynamical system, a neural network identifier
model for the system to be controlled is developed. Then,
this identifier is used to represent the system while train-
ing the neural network controller weights in the control
T. ABOUELDAHAB ET AL.177
stage [6,8,10,12].
2.1. Identification
If a set of data (measurements) can be carried out on a
nonlinear dynamic system, an identifier could be derived
whose dynamic behavior should be as close as possible
to this system. The identifier model is selected based on
whether all the system states or only its output are meas-
ured [1,3,4,9].
The state space representation of a nonlinear system is
given by the following equation:
 

 
1, and ,
x
kxkukyk xkuk (1)
where: the input to the system,

m
uk R
n
x
kR
the states of the system,

p
yk R
:n
R
is the outputs of the
system, the nonlinear mappings functions
and are dynamic and
smooth [4,9].
:
nm n
RR Rp
,
R
Usually, all the system states are not measured for
process representation, so the Input/Output representa-
tion which is also called Nonlinear Autoregressive Mov-
ing Average (NARMA) representation given by the fol-
lowing equation is used instead [4,6,9]
 
 
,1,, 1
,,1
yk dykykyk n
ukuk m
 

(2)
where d is the relative degree (or equivalent delay) of the
system and it is assumed that both the order of the sys-
tem n and relative degree are specified while the nonlin-
ear function is unknown. ()
In general, any discrete time dynamical nonlinear sys-
tem can be represented using NARMA—Output Error
Model representation shown in Figure 1 and has the fol-
lowing forms:
 


ˆ,1,,
1,,1,,
1,
mmm
m
ykd ykyk
yknukuk
uk mWk
 
 

(3)
It is crucial to note that, the present identifier output

m
y
k represented by the nonlinear mapping function
()
is a dynamic mapping function because it depends
on its past outputs. Thus, the feed forward neural net-
work can not be used to represent this mapping function
and the recurrent neural network which is a dynamic non
linear mapping is used instead. [9].
2.2. Control
According to both inverse function theorem and implicit
()
ym(k)
u(k)
Z
-1 Z-1 Z-1
Z-1 Z-1 Z-1
Figure 1. NARMA—output error model.
function theorem [4,5,9], if the state vector
kof the
nonlinear system given by equation (1) is accessible,
then the system output
y
kcan make exact tracking of
a general output
ky
using a control low given by
the following equation
 
,ukxkyk d
 (4)
If the nonlinear dynamical system given by equations
(1-2) is observable, then the control law can also
be represented in terms of its past values:

uk
1, 1,ukuk m
 as well as previous dynamic
system outputs

,, 1yk n
yk and
ykd
.
If
rk
k
represents the information that is needed at
the instant to implement the control law (i.e.,
rk ykd

) so the above equation can be re-
written as follows [1,3-5,9]:
 

,1,, 1,
(),(1),,1,
ukykykyk n
uk ukukmrk



,
(5)
As this control law is represented by the nonlinear
dynamical mapping function

, so a neural network
controller model represented by the nonlinear dynamical
mapping function
can be used to achieve the con-
trol law and can be written as:

 
,1,, 1,,
1,,1,,
ukykykyk nuk
ukukmrk Wk



,
(6)
and
Wk
is the set of parameters of the neural network
controller.
As the neural network controller output
ukdepends
on its past outputs, thus, the recurrent neural network
controller is used to evaluate the controller parameters [9].
The structure of the closed loop nonlinear predictive
controller consisting of the nonlinear system, the nonlin-
ear identifier and the nonlinear controller is shown in
Figure 2.
3. The Sigmoid Diagonal Recurrent Neural
Network(SDRNN)
In this section, our proposed architecture and its associ-
Copyright © 2011 SciRes. ICA
T. ABOUELDAHAB ET AL.
178
Figure 2. The structure of the closed loop nonlinear predic-
tive controller.
ated modified dynamic back propagation learning algo-
rithm to learn the new added sigmoid weight vector are
presented
Define the following to obtain the mathematical model
of the proposed neural network architecture:
i,h,o are the number of neurons in input, hidden,
and output layers respectively.
n nn
I
W is the input weight matrix connecting between
input layer and the hidden layer, ,
S
W
D
W
th
are the sig-
moid weight vector and the diagonal weight vector of the
hidden layer, and is the output weight matrix con-
necting between hidden layer and the output layer
O
W
Assume, at sample k, the input to the input neu-
ron in the input layer is
i

I
k, so the output of the neu-
ral network can be calculated as follows:
The net input to the sigmoid neuron in the hidden
layer can be calculated as follows
th
j
 
1
1
ni
DI
jjj iji
i
H
inkWHkWIk
 
(7)
And it’s output can be calculated as


s
j
j
js
j
fHinkW
Hk W
(8)
The output of the neuron in the output layer can
be written as follows
th
m
 
1
h
no
mjm
j
YkWHk

j
(9)
For the standard Diagonal Recurrent Neural Network,
the sigmoid weight vector is set to be one and
for the normal Feed Forward Neural Networks the di-
agonal vector

S
Wk
D
Wk is set to be zero.
Learning Algorithm
Given the structure of the network described by Equa-
tions (7)-(9) and applying the gradient descent method [9,
16] to update the network weights the partial derivatives
of the neural network predictor output with re-
spect to network weights are given by

m
Yk

mO
jm
Yk
H
jk
W
(10a)

O
jm
m
Yk
SS
jJ
Wk
WW
 (10b)

mD
j
Yk
W
O
jm j
WP
k (10c)

mI
ij
Yk
W
O
jm ij
WQ
k (10d)
where:



SS
jj jj
SS
jj
fHin
1
()
h
n
j
k
k WfHinkW
kWW

 
()1 1
Dj
j
jj j P
PkH kWk

()
j i
k I
(11a)

1
() ()
h
nD
ijj ij
j
QkkW QkC
 
(11b)
and
 



1
00,
s
j
kW

j,
00
s
jj
j
jij
fHinkW
Hin k
PQ



From Equation (9), the partial derivatives of the neural
network predictor output represented by the
nonlinear mapping function with respect
to output weight matrix

m
Yk
Ik
O

,W
j
m
W is:

j
O
jm
k
m
Y
H
k
W
S
which lead to (10a).
Also, the partial derivative with respect to sigmoid
weight vector
j
W is

j
mO
jm
ss
jj
H
k
W
WW


Yk
And from Equations (7) and (8),
Copyright © 2011 SciRes. ICA
T. ABOUELDAHAB ET AL.179




() 1
1
s
jj
j
ss s
jj j
s
jj
s
s
jj
fHinkW
Hk
WW W
fHinkWWW








 



1s
jj jj
j
ss ss
jj jj
fHink WfHink W
Hk
WW WW
 
 

s
which lead to (10b).
From Equation (9), the partial derivatives with respect
to diagonal weight vector
D
j
W is


j
mOO
jmjm j
DD
jj
Hk
Yk WW
WW
 

Pk
and from Equation (8)
 
1
s
j
j
j
Ds D
jj j
fHinkW
Hk
WW W







1s
jj
jj
Ds D
j
jj j
fHinkW
H
kHin
Hin k
WW W


 

k
and from Equation (7)
  
1
1
jj
D
jj
DD
jj
Hin kHk
Hk W
WW



which leads to (10c), (11a).
Also, the partial derivative with respect to input
weight matrix
I
ij
W is
()
() ()
j
OO
mjmjm ij
II
ij ij
Hk
Yk WW
WW
 

Qk
and from Equation (8)
 
1
s
j
j
j
Is I
ij jij
fHinkW
Hk
WW W







1s
jj
j
Is I
j
ij jij
fHinkWj
H
k
Hin kWW W

 
Hink
and from Equation (7)
  
1
j
D
ij
II
ij ij
Hin kHk
Ik W
WW



j
which leads to (10d), (11b).
The full proof of this lemma is given in details in ref-
erence number [9].
4. Results and Discussion
In this section, extensive experimentation is carried out
in an attempt to demonstrate the performance of the
SDRNN architecture and compare it to the DRNN archi-
tecture in nonlinear system identification and in adaptive
control of nonlinear dynamical systems. As a measure to
test the performance, we use the Mean Square Error
(MSE) criteria between the actual nonlinear system out-
put and the neural network output [8-10]. It is worth
mentioning that in our proposed network, there is no
need to use momentum term or learning rate adaptation
[8, 10] because the training is done using the adaptation
of the sigmoid function shape leading to reduction of the
learning algorithm complexity [9].
Example 1: (N o nl i ne a r s y stem identificati o n )
A benchmark problem is employed, the identification
of a dynamical system. The example is taken from [7,9],
where the nonlinear system to be identified is governed
by the following difference equation

  

  
22
22
1
121 2
11 2
11 2
Yk
Yk YkYkukYk
YkYk
uk
Yk Yk

1




(12)
As it can be seen, the current output of the plant
1Yk
depends on three previous outputs
,1,Yk 2Yk Yk
and two previous inputs
1k,uk u
. A NARMA-Output Error Model given
by Equation (3) is considered for identification of this
nonlinear dynamical system. The neural network identi-
fier output
1
m
Yk
is the approximation of the
nonlinear system output
1Yk. Thus the input to the
neural network identifier are the three previous identifier
outputs i.e. (

,Yk
2,1Yk
mm
Yk m) and the two
previous inputs
,1ukuk
and the size of the neu-
ral network identifier is 5-8-1’ (5 input units, 8 hid-
den units, and one output unit).
In order to comply with previous results reported in
the literature, a new training data containing 200 batches
of 900 patterns is generated. For each data batch, the
input
uk is an independent and identically distrib-
uted uniform in the range between
while the
testing data set is composed of 1000 samples with a sig-
nal described by
1, 1

 

sin π25 250
1 250500
()1 500750
0.3sin π250.1sin π32
0.6sin π10 7501000
kk
k
uk k
kk
kk


 

 
(13)
As a measure to test the identification performance,
Copyright © 2011 SciRes. ICA
T. ABOUELDAHAB ET AL.
180
the MSE using the standard DRNN is 0.0017 while using
our proposed SDRNN, it is reduced to be 1.1049e-004.
The nonlinear system output, the SDRNN identifier out-
put and the DRNN identifier output are shown in Figure
3.
The solution of the problem without using the sigmoid
weight vector as similar to the neural network identifier
schemes in [2, 8-10], the system performance is very
poor because the restriction to the output hidden layer
neurons. While using our proposed architecture, this re-
striction is avoided due to the adaptation of the sigmoid
function shape.
Example 2: (Rigid non-minimum phase model)
[7,12,13]
The nonlinear system is given by the following dif-
ference equation
 

 
2
115 2
11
Yk
Ykuk uk
Yk


(14)
And the reference model is giving by the following
dynamical difference equation [7,10,12]:
 

sin 2π25sin2π10
and
0.6 1
rr
rk kk
YkYk rk

 
(15)
It can be seen that the current system output
1Yk
depends on its previous output and two previous

Yk
Figure 3. The nonlinear system output, the SDRNN identi-
fier output and the DRNN identifier output.
inputs
uk and
1uk
. Thus, the NARMA-Output
Error Model given by Equation (3) is considered for sys-
tem identification and the neural network identifier in-
puts are the previous identifier output and the
two previous inputs and the size of the neural network
identifier is 3-6-1’ (3 input units, 6 hidden units, and one
output unit).

m
Yk
Consequently, in the nonlinear adaptive control phase,
the current neural network controller output
uk de-
pends on it's previous output , reference model
input
1uk
rk and reference output beside the
previous neural network identifier output

Yk
r
m
Yk. A
NARMA-Output Error Model given by equation (3) is
considered for the neural network controller and the size
of this controller is 4-7-1’ (4 input units, 7 hidden units,
and one output unit).
After 10000 iterations for training identifier weights
with uniformly distributed random signal, both identifier
and controller start closed-loop control. The MSE for
100 samples using DRNN is 0.3127 while using our
proposed SDRNN it enhances to be 0.0141.
The final diagonal and sigmoid weights in both the
standard network controller and the proposed controller
are shown in Table 1.
The reference model output, the SDRNN controller
output and the DRNN controller output are shown in
Figure 4.
From Ta ble 1 and Figure 4, it is obvious that the val-
ues of the diagonal weights in the DRNN controller is
located within a wide range between 16.4877 (neuron
number 4) and -45.6087 (neuron number 7) while in the
SDRNN it is located is located in a narrow range be-
tween 0.2047 (neuron number 5) and -2.7992 (neuron
number 6). This great difference is due to the existence
of the sigmoid weight vector adapting the shape of the
sigmoid function enabling the neural network controller
output to get the appropriate values that can efficiently
reduce the MSE between the actual reference t model
and the nonlinear system output.
Table 1. Diagonal and sigmoid weights of neural network
controllers.
Diagonal Weight Sigmoid
Weight
Hidden
Layer
Neuron
Number Standard NetworkProposed Network Proposed
Network
1
2
3
4
5
6
7
–6.0076
–10.7195
–20.0180
16.4877
–19.3239
–15.1893
0.1645
–45.6087
-1.9830
-0.9809
-0.3553
0.2047
-2.7992
-0.0002
0.6095
0.0007
0.0005
0.1742
1.2101
-0.1767 0.7825
Copyright © 2011 SciRes. ICA
T. ABOUELDAHAB ET AL.
Copyright © 2011 SciRes. ICA
181
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