 Open Access
 Total Downloads : 448
 Authors : Shravani Chandaka
 Paper ID : IJERTV1IS9211
 Volume & Issue : Volume 01, Issue 09 (November 2012)
 Published (First Online): 29112012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
The Sugeno Fuzzy Model Identification Method In Parameter Variation of Systems
Shravani Chandaka
Department of ECE,Chaitanya Institute of Technology, Visakhapatnam,India
Abstract – This paper presents the sugeno model identification method by which a great number of systems whose parameters vary dramatically with working states can be identified via Fuzzy Neural Networks (FNN). The suggested method could overcome the drawbacks of traditional linear system identification methods which are only effective under certain narrow working states and provide a global dynamic description based on which further control of such systems may be carried out. The simulation results of a secondorder parameter varying system demonstrate the effectiveness of the method.
system. Finally, the conclusions of the paper are given in section 5.

TS FUZZY MODEL
Parameter varying systems which possess m working state characteristic variables, q inputs and single output can be described by the TS fuzzy model consisting of R rules where the ith rule can be represented as:
Keywords – Parameter Varying Systems, TS Fuzzy Model, Fuzzy
Rule i: if z
is Ai ,k1 , z
is Ai ,k2 ,,and z
is Ai ,km
Neural Networks (FNN), Identification.
1 1 2 2 m m
1 1 2 2
then y i a i x

a i x
ai x
(1)
q
q
1 INTRODUCTION
Controlled systems whose parameters vary dramatically with working states, namely parameter varying systems, are widely encountered in practical industrial situations. Although traditional linear system identification methods have been well established in the last twenty years, it can only be used under a certain narrow range of working conditions.
i 1,2,, R. k j 1,2,,rj .
Where R is the number of rules in the TS fuzzy model.
z j j 1,2,,m is the jth characteristic variable, which
i
reflects the working state of the systems and can be selected as input, output or other variables affecting the parameters of system dynamics. xl l 1,2,,q is the lth model input. y
is the output of the ith rule. For the ith rule, Ai ,k j is the k th
l
j
j
Moreover, traditional controllers based on such models j j
cannot cope with the changes in process dynamic effectively.
fuzzy subset of z
. ai is the coefficient of the consequent. r
Therefore, developing a global dynamic model and establishing the corresponding control schemes for the parameter varying systems are deeply expected.
Takagi and Sugeno [1,2,3] proposed a new type of fuzzy model (TS model) which has been widely used in many disciplines. Describing complex systems is one of the most important applications since the mathematical expression of the model is convenient to design controllers. Recently, the authors [4] suggested an identification method of the TS
is the fuzzy partition number of z j . For simplicity of induction, we let rj r and r is determined by both the complexity and the accuracy of the model.
10 20 m0
Once a set of working state variables z , z ,, z and
model input variables x10 , x20 ,, xq 0 are available, then the
R
R
output of the TS model under such working states can be calculated by the weightedaverage of each yi :
fuzzy model for nonlinear systems via Fuzzy Neural
Networks (FNN). It has been proved effective in describing the systems. In this paper, the TS fuzzy model is generalized
y i yi i1
i i1
(2)
to the parameter varying systems, and an identification method based on FNN is presented. The simulation results of a secondorder system verify the effectiveness of the method.
where yi is determined by the consequent equation of the ith
rule. The truthvalue i of the ith rule can be calculated as:
m
This paper is organized as follows: Section 2 gives mathematical expression of the TS fuzzy model and point out the reason why the model is fit for parameter varying systems. Identification method for the model via FNN is suggested in
section 3. The structure and the training algorithm for the
i Ai ,k j z
j
j 0
j1
Furthermore, the equation (2) can be rewritten as:
y a x a x
R R R
i i i i i
1 1 q q
(3)
(4)
i1
i1
i1
networks are also given in details. Section 4 gives the simulation results of a secondorder parameter varying
From (4), one can see that the TS fuzzy model can be expressed as an ordinary linear equation under certain working states since the truthvalue i is only determined by
the working state variables. As i varies with working state,
I D w
z w
TS fuzzy model is a coefficientvarying linear equation. For
:
j ,k
g j ,k j c j ,k
(6)
all possible varying ranges of working states, the TS fuzzy
O D I D
j ,k
j ,k
model reflects the relationships between model parameters and working states. Therefore, the global dynamic
Signature in layer (F) is the fuzzy minimum node and the inputoutput relationships for the ith rule can be written as:
characteristics of the parameter varying systems can be
( F )
m E
represented.
Ii
:
min
j 1 and k i , j
Oj ,k
(7)
O F I F


FUZZY NEURAL NETWORKS TS FUZZY MODEL IDENTIFICATION METHOD
i i
j 1
m l
i i,l
1r
1

Structure of the FNN
i, j int
l 0
r m j
1
(8)
According to (1~3), the structure of FNN presented here consists of a premise, consequent and fuzzy inference. For systems which posses m working state characteristic variables, q inputs and a single output, the FNN used for the
i,0 1. i 1,2,, R. j 1,2,,m. k 1,2,,r
j ,k
j ,k
where I and O are input and output of the nodes which correspond to the kth fuzzy subset of z j in layer
TS model identification is shown in Fig. 1. The circles and
the squares in the figure represent the units of the networks.
respectively;
I
i
and O
are input and output of the nodes
i
The notations between the units denote the connection weights. The units without any notation just deliver the
which correspond to the ith rule in layer respectively; the
c
central point and gradient of the kth fuzzy subset for z j are
signals from input to output.
determined by both wg
j ,k
and w
j ,k
; i, j represents the

Normalization of the working state variables
Layers (A)~(B) of the FNN are used to normalize the working state variables in case of saturation of the premise
1 2
m
nodes. Assuming P samples z p , z p ,, z p p 1,2,, P are
connective relationship between the ith rule and the kth fuzzy subset of z j .
The membership functions of the working state variables are determined by activation functions of the nodes in layer (E). In this paper, the following activation functions are
available for training the networks, the jth wrking state variable of the pth sample can be normalized as:
taken:
e x
1 e x
, k 1
z p w z p w
(5)
f x e x 2
k
, k 2,3,,r 1
(9)
j s j j
p
t j
p
x
where z j
is the normalized working state variable of z j ;
1 1 e
, k r
w and w
s j t j
are the coefficients and biases of
which realize fuzzy partition as shown in Fig. 2.
normalization respectively:
w
2
s
j maxz p minz p
3) Consequent and fuzzy inference
Layers (G)~(J), which are used to implement the linear
j j equations of the TS fuzzy model, are consequent parts of the
maxz p minz p
j
j
FNN. As for the ith rule of the consequent, inputoutput
q
t
j
w relation realized can be written as:
2 O J w x
(10)
j 1,2,,m. p 1,2,, P
i
j 1
a j ,i j
2) Premise
where w
a j ,i
is the coefficient of xi
in rule i .
The premise parts of the FNN include Layers (C)~(F) which are used for fuzzy partition and truthvalue calculations. Signature in layer (D) ,which is the sum
node, realizes the following operations for the kth fuzzy
Layers (K)~(M) realize the fuzzy inference as shown in (2).


Learning algorithm
a
Two kinds of parameters need to be learnt by the FNN.
subset of z j :
One is wg
j ,k
and w
j ,k
, which determine central points
and gradients of the membership functions in the premise.
O F
E
The other is w
a j ,i
, which determines local linear
i j
min O
j ,k
m
1 and k i , j
(15)
relationships of the consequent.
m
O
E
m
E E
p p p p p p
j ,k
IOj ,k

Ol ,k
Assuming P samples z1 , z2 ,, zm , x1 , x2 ,, xq
j 1 and k i , j
l 1 and l j
p 1,2,, P
are available for training the FNN and the
p
where:
1 ,
O E O E
corresponding teacher signal is t
. Once the pth sample is
IO E O E
j ,k
l ,k
(16)
put on the networks, the actual output y p of the networks can
j ,k
l ,k
0 ,
O E O E
j ,k
l ,k
i
O
be obtained. Thus, the learning error function of the sample can be defined as:
Therefore, O F p
E p j ,k
can be calculated by
E p
1 t p y p 2
(11)
O F p
i
IO E p O E p
m
2 O E p j ,k
l ,k
Under this definition, the total error function of all the samples can be written as:
j ,k
l 1 and l j
1 , O E p O E p
(17)
P 1 P
j ,k
l ,k
E E p
t p y p 2
(12)
0 ,
O E p O E p
2 j ,k
l ,k
p1
p1
Moreover,
O E p
I E p
can be obtained from (10) as
According to the GradientDescent learning algorithm, one can obtain:
follows:
j ,k
j ,k
p
E p E
y p
J p
O E p O E p 1
, k = 1
Ol
E p j ,1
j ,1
O
w
P 1 y p
O J p
w
j ,k 2 I E p O E p
, k = 2,3,,r – 1
(18)
a l ,i
l a l ,i
(13)
I E p
j ,k
j ,k
j ,k
O E p 1 O E p
, k r
P R
j ,r
j ,r
t p y p O J p
l
O F p x p
k
p1
k 1
l
From (14), (17) and (18), E
w
g j ,k
can be obtained.
E P E
y p

Using the same method mentioned above, E
w
c j ,k
can
w p1 y p
w
also be represented by
g j ,k
g j ,k
p
p
E P E
y p
P E
R y p
O F p
w
c
y w
O
i
c j ,k
p1
j ,k
i
p1 y p
i 1 and k i , j O F p
E p j ,k
P E p
R y p
O F p
i
O E p
I E p
p1 y p
O F p
O E p

j ,k


j ,k
i1 and k
i , j i
j ,k
I E p
w
j ,k
P
g j ,k
E p j ,k
E p
E p
O
I

j ,k
t p y p
P
p1
I j ,k
wc
j ,k
(19)
R

O
F p y p
O
F p
F p
O
O


i
t p y p p1
i 1 and k i , j
i l
R
l 1
E p
j ,k
R R
O F p
O F p y p
O F p i
O E p
i l
O E p

j ,k


z p
i1 and k i , j
l1
j ,k
I E p j
j ,k
(14)
O E p
In order to solve
F p E p
O O
i j ,k
in (14) , the following

j ,k 1

I E p
equivalent transition for (7) is needed:
j ,k
Therefore, the final tuning equations of the premise and consequent parameters of the FNN can be written as:
a j ,i a j ,i a j ,i
w n 1 w n E w
wg n 1 wg n E wg
(20)
(21)
shown in Fig 4 and Fig. 5. As shown in Fig. 4, where the solid line and the dotted line denote the expected output of the system and the actual output of the networks respectively,
j ,k
j ,k
j ,k
most of the samples have good performance to describe the
c c
c
w n 1 w n E
w
(22)
actual outputs of the system.
j ,k
j ,k
j ,k
where n is the training times; and are learning rates. In this paper, we use the adaptive backpropagation algorithm suggested by the authors [5].


SIMULATION EXAMPLE
Considering the following secondorder parameter varying system:
Finally, we use another ten groups of z to verify the performance of the resulted FNN TS fuzzy model and the results are shown in Fig. 6. The same conclusion can be drawn from it. Therefore, the suggested TS model identification method is strongly effective to obtain the global dynamic model of parameter varying systems.

CONCLUSIONS
y s 1
(23)
This paper generalizes the TS model to the parameter
u s 1 Ts 2
where the time constant T is affected by a working state variable z z 0.3,0.9 . Suppose the relationship between
them is:
varying systems and presents the corresponding identification method via FNN. The proposed method can effectively realize the identification of parameter varying systems whereas the traditinal linear system identification methods
T 20 20 z 0.3
(24)
can not. Furthermore, control of such systems based on the
Once the sample time T0 is given, the discrete time description of the system could be obtained:
0
y k 2 T 2 T T y k 1 T 2 y k 2
(25)
0 0
T 2 u k 1T T 2
In this paper, the sample time T0 is taken as 5 seconds. Curves 1, 2 and 3 in Fig. 3 show the unit step response of the system at z 0.3, z 0.6 and z 0.9 respectively and one can see that variations between the different z s are very large.
Using the suggested FNN TS model identification method, we select z as a working state variable for the input of premise in the FNN and take u k 1 , y k 1 and y k 2
as input variables for the TS model. The aim of the identification is to obtain the global model which is suitable for all the possible working states of the system. First, ten states are selected randomly, and 310 groups of training data are obtained by exerting 5order M sequels which have the range of 1 on the system. All of the weights in the consequent of the FNN are selected between 0.1 and 0.1 randomly, and
the fuzzy partition number r is selected to 7 as shown in Fig.
2.
In order to fasten the convergence rate of the networks, the following parameters are used as the initial value of the adaptive BP algorithm shown in [5]:
0
1 s
1 0.9, 1 0.4 , 1.4 , 0.6, E 0.5
The final convergence conditions are taken as:

The number of the samples which have satisfied
wellestablished TS fuzzy model can be carried out and this further research field creates for us. The simulation results of a secondorder parameter varying system have fully verified the effectiveness of this method.
It should be noted that a more effective way is provided to establish fuzzy control rules for the multiworkingstates situations. Based on the model, performance of fuzzy controller will be greatly improved under such situations.
7 REFERENCE

T. Takagi and M. Sugeno, Fuzzy identification of system and its applications to modeling and control, IEEE Trans. on System Man and Cybernetics, vol. SMC15, no. 1, 1985, pp. 116132.

M. Sugeno and G. T. Kang, Structure identification of fuzzy model, Fuzzy Sets and Systems, vol. 28, 1988, pp. 15 33.

M. Sugeno and K. Tanaka, Successive identification of a fuzzy model and its applications to prediction of a complex system, Fuzzy Sets and Systems, vol. 42, 1992, pp. 315334.

Xie Keming and Zhang Jianwei. A linear fuzzy model identification method based on fuzzy neural networks, in Proceedings of the 2nd Worldwide Chinese Intelligence Control and Intelligence Automation Conference, 1997.

Xie Keming and Zhang Jianwei, An Adaptive Backpropagation Algorithm Based on Error Rate of Change, submitted to Journal of Taiyuan University of Technology.
t p y p t p 0.05 has exceeded 95 percent of the total samples.


Training times has exceeded the maximum times specified as 10000.
After training the FNN 868 times, the networks converged by satisfying condition 1) and the final simulation results are
Fig. 1. Structure of FNN Fig. 4. Results of training
Fig. 2. Fuzzy subsets of working state variables
Fig. 3. Unit step response under different working state
Fig. 5. Membership functions after training
Fig. 6. Results of verification