Rough Sets Current Trends Conference, RSCTC'98, Warsaw, Poland, June 22 26, 1998
The Takagi-Sugeno Fuzzy Model Identification Method
of Parameter Varying Systems*
Xie Keming**
T. Y. Lin***
Zhang Jianwei**
and
** Department of Automation, Taiyuan University of Technology, Taiyuan, 030024, P. R. China
*** Department of Mathematics and Computer Science, San Jose State University, San Jose, CA 95192, U. S. A.
Abstract - This paper presents the TS 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 global dynamic description
based on which further control of such systems may be carried
out. Simulation results of a second-order parameter varying
system demonstrate the effectiveness of the method.
Keywords - Parameter Varying Systems, TS Fuzzy Model, Fuzzy
Neural Networks (FNN), Identification.
system. Finally, the conclusions of the paper are given in
section 5.
2. 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 i-th rule can be represented as:
i ,k
i ,k
i ,k
Rule i: if z1 is A1 , z 2 is A2 , , and z m is Am
1
2
m
then y  a 1 x 1  a 2 x 2  a q x q
i
i
i
(1)
i  1,2 , , R. k j  1,2 , , r j .
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.
Moreover, traditional controllers based on such models can
not cope with the changes on process dynamic effectively.
Therefore, developing 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 mathematics expression of the
model is convenient to design controllers. Recently, the
authors [4] suggested an identification method of the TS
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
to the parameter varying systems, and an identification
method based on FNN is presented. Simulation results of a
second-order system verify the effectiveness of the method.
This paper is organized as follows: Section 2 gives
mathematics 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
networks are also given in details. Section 4 gives the
simulation results of a second-order parameter varying
i
where R is the number of rules in the TS fuzzy model.
z j  j  1,2,, m is the j-th characteristic variable, which
reflects the working state of the systems and can be selected
as input, output or other variables affecting the parameters of
i
system dynamics. xl  l  1,2 ,, q  is the l-th model input. y
i ,k j
is the output of the i-th rule. For the i-th rule, A j
is the k j -th
i
l
fuzzy subset of z j . a is the coefficient of the consequent. r j
is the fuzzy partition number of z j . For simplicity of
induction, we let r j  r and r is determined by both the
complexity and the accuracy of the model.
Once a set of working state variables z10 , z20 ,, zm0 and




model input variables x10 , x20 ,, xq 0 are available, then the
output of the TS model under such working states can be
i
calculated by the weighted-average of each y :
R
y   y
i
R
i
i 1

i
(2)
i 1
i
where y is determined by consequent equation of the i-th
i
rule. The truth-value  of the i-th rule can be calculated as:
m
   Aj
i
j 1
i ,k j
z 
j0
(3)
Furthermore, equation (2) can be rewritten as:
R
 R i i
 R i
i i
y     a1 x1    a q xq   
(4)
 i 1
 i 1
i 1
From (4), one can see that the TS fuzzy model can be
expressed as an ordinary linear equation under certain
working states since the truth-value  is only determined by
i
the working state variables. As  varies with working state,
TS fuzzy model is a coefficient-varying linear equation. For
all possible varying ranges of working states, the TS fuzzy
model reflects the relationships between model parameters
and working states. Therefore, the global dynamic
characteristics of the parameter varying systems can be
represented.
i
node, realizes the following operations for the k-th fuzzy
subset of z j :
  D
z j  wc
 I j , k  wg
j ,k
j ,k
:   D 
(6)
 D
 O j , k  I j , k
 
 
Signature ‘  ’ in layer (F) is the fuzzy minimum node and the
input-output relationships for the i-th rule can be written as:
m
 (F)
 E
 I i  min O j , k
(7)
:
j 1 and k    i , j 
O  F   I  F 
 i
i
j 1


 m l 
 1
 i      i , l   1 r
1
  i , j   int l  0
 m j 


r
(8)




3. FUZZY NEURAL NETWORKS TS FUZZY MODEL
IDENTIFICATION METHOD

A. Structure of the FNN
According to (1~3), the structure of FNN presented here
consists of premise, consequent and fuzzy inference. For
systems which posses m working state characteristic
variables, q inputs and single output, the FNN used for the
TS model identification is shown in Fig. 1. The circles and
the squares in the figure represent the units of the networks.
The notations between the units denote the connection
weights. The units without any notation just deliver the
signals from input to output.
  i ,0  1. i  1,2,, R. j  1,2,, m. k  1,2,, r
 
correspond to the k-th fuzzy subset of z j in layer  
 
 
determined by both w g
s
j
and

 
t

are the coefficients and biases of
j
2

t
j
max z


3) Consequent and fuzzy inference
   min z 
max z   min z 
w  
2
s
;   i , j  represents the
which realize fuzzy partition as shown in Fig. 2.
normalization respectively:
w 
j ,k
2
p
w 
 
and wc
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
taken:
e  x 1  e  x , k  1

x
f k  x   e
, k  2 ,3, , r  1
(9)

x
1 1  e
, k r
where z j is the normalized working state variable of z j ;
w 
j ,k
connective relationship between the i-th rule and the k-th
fuzzy subset of z j .

p
 
respectively; I i and Oi are input and output of the nodes
which correspond to the i-th rule in layer   respectively; the
central point and gradient of the k-th fuzzy subset for z j are
Layers (A)~(B) of the FNN are used to normalize the
working state variables in case of saturation of the premise
p
p
p
nodes. Assuming P samples z1 , z2 ,, zm  p  1,2,, P are
available for training the networks, the j-th working state
variable of the p-th sample can be normalized as:
 p

p
(5)
z j  ws  z j  wt 
j
j
 
 
where I j , k and O j , k are input and output of the nodes which
1) Normalization of the working state variables


p
j
p
j
p
j
p
j
Layers (G)~(J), which are used to implement the linear
equations of the TS fuzzy model, are consequent parts of the
FNN. As for the i-th rule of the consequent, input-output
relation realized can be written as:
j
j  1,2 , , m. p  1,2 , , P
 J
Oi
 
2) Premise
where wa
j ,i
q
 
  wa
j 1
j ,i
xj
(10)
is the coefficient of x i in rule i .
Layers (K)~(M) realize the fuzzy inference as shown in (2).
The premise parts of the FNN include Layers (C)~(F)
which are used for fuzzy partition and truth-value
calculations. Signature ‘  ’ in layer (D) ,which is the sum
B. Learning algorithm
2
Two kinds of parameters need to be learnt by the FNN.
 
One is w g
 
and wa
j ,k
j ,k
 F
, which determine central points
Assuming
samples
P
 p  1,2,, P
z
p
1
p
2
p
m
p
1
p
2
, z ,, z , x , x ,, x
p
q


 F p
Oi
 E p
 E p

 E p

 E p
I j , k
P
I j , k
 
 wg
follows:

p
p
 E p

O j , k
 E p
I j , k
I j , k
 E p
I j , k
also be represented by
 p
p
P 
E
E
y
  p 
 wc j ,k p1  y  wc j ,k

 
 
O j , k

 E p

E p
I j , k
p 1
l 1
 F p
 F p

j ,k
can be obtained.
 
j ,k
can




 p
F p 
p

P 
R
E
y
O
  p 
   F  p  i E  p 
O j , k 
p 1  y
i 1 and k   i , j   Oi






l
 
 wg
Using the same method mentioned above, E  wc

 O
(18)

From (14), (17) and (18), E
I j , k
 
 wc
  t  y

(17)


E p
R

can be obtained from (10) as

O j , k







  Oi F  p  y p
i  1 and k    i , j  
R
 E p
 Ol , k
 E p
  E p
O
 O j ,1  1 , k = 1
 j ,1
 E p
 E p
  2 I j , k  O j , k
, k = 2 ,3, , r - 1
  E p
 E p
, k r
 O j ,r  1  O j ,r
 E p
P
  t  y
p 1
j ,k
 E p
Moreover, O j , k
 
O j , k
E p
j ,k
l 1 and l  j
 1 , O  E  p  O  E  p
j ,k
l ,k

 E p
 E p
 0 , O j , k  Ol , k


 F p
p

R
 E
Oi
y






 F p
 E p
p
O j , k
p  1  y
i  1 and k    i , j  
 Oi

m
 E p

P
 I O

O j , k
(16)
can be calculated by
 F p
 
 
O j , k

(15)



1 , O  E   O  E 
j ,k
l ,k

 E
 E
 0 , O j , k  Ol , k

 E
 E p
Therefore, Oi
p

 E
I O j , k  Ol , k
corresponding teacher signal is t . Once the p-th sample is
p
put on the networks, the actual output y of the networks can
be obtained. Thus, the learning error function of the sample
can be defined as:
1 p
p
p 2
E  t  y 
(11)
2
Under this definition, the total error function of all the
samples can be written as:
P
1 P p
p
p 2
(12)
E  E   t  y
2 p 1
p 1
According to the Gradient-Descent learning algorithm, one
can obtain:
 p
 J p 
p
p 

O l
E
E
y

  p 

 J p
 wa l ,i P 1  y Ol
 wa l ,i 
(13)
P 
R



 J p
p
p
   t  y   Ol
Ok F  p   x lp 



p 1 
k 1


p
P 

E
E
y

  p 
p  1  y

 wg
 wg
j ,k

j ,k 
 

where:
are available for training the FNN and the

O j ,k
j 1 and k    i , j 
m


 O j ,Ek   I O j ,Ek  Ol,Ek 
j 1 and k    i , j  
l 1 and l  j
j ,i
relationships of the consequent.
 E
min
m
and gradients of the membership functions in the premise.
The other is wa , which determines local linear
 
m

Oi
Oi
 E p
O j , k
p
p
j ,k







  Oi F  p  y p
i 1 and k    i , j  




R
E p

p
zj 


O j , k
E p
I j , k
(19)

 O
R
l 1
F p
l

 E p 
O j ,k 
F p
Oi

  1 

(14) Therefore, the final tuning equations of the premise and
in (14) , the following consequent parameters of the FNN can be written as:
equivalent transition for (7) is needed:
 F p
In order to solve Oi
 E p
O j , k
3
w 
a
w 
g
w 
c
j ,i
j ,k
j ,k
 n  1   wa   n    E   wa 
j ,i
j ,i
(20)
 n  1  w g
(21)
 
 n  1   wc 
j ,k
j ,k
 
 n     E  w g
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,
most of the samples have good performance to describe the
actual outputs of the 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.
j ,k
 n    E   wc 
j ,k
(22)
where n is the training times;  and  are learning rates. In
this paper, we use the adaptive back-propagation algorithm
suggested by the authors [5].
4. SIMULATION EXAMPLE
Considering the following second-order parameter varying
system:
y  s
1
(23)

2
u s  1  Ts
where the time constant T is affected by a working state
variable z  z   0.3,0.9  . Suppose the relationship between
them is:
T  20  20   z  0.3
(24)
Once the sample time T0 is given, the discrete time
description of the system could be obtained:
 

5. CONCLUSIONS
This paper generalizes the TS model to the parameter
varying systems and presents the corresponding identification
method via FNN. The proposed method can effectively
realize the identification of parameter varying systems
whereas the traditional linear system identification methods
can not. Furthermore, control of such systems based on the
well-established TS fuzzy model can be carried out and this
further research fields creates for us. Simulation results of a
second-order 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 multi-working-states
situations. Based on the model, performance of fuzzy
controller will be greatly improved under such situations.
y k   2  T  T  T0  y k  1  T  y k  2
2
 T0  u k  1
2
2
 T  T 
2
(25)
0
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 5-order 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]:
  1  0.9 ,   1  0.4 ,  0  1.4 ,  1  0.6 , E s  0.5
The final convergence conditions are taken as:
1) the number of the samples which have satisfied
p
p
p
t y
t  0.05 has exceeded 95 percent of the total
samples.
2) 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

6. ACKNOWLEDGMENTS
The authors gratefully acknowledge the financial support
of the Visiting Scholar Foundation of Shanxi Province, P. R.
China
7 REFERENCE
[1] T. Takagi and M. Sugeno, “Fuzzy identification of
system and its applications to modeling and control”, IEEE
Trans. on System Man and Cybernetics, vol. SMC-15, no. 1,
1985, pp. 116-132.
[2] M. Sugeno and G. T. Kang, “Structure identification of
fuzzy model”, Fuzzy Sets and Systems, vol. 28, 1988, pp. 1533.
[3] 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. 315-334.
[4] 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.
[5] Xie Keming and Zhang Jianwei, “An Adaptive
Backpropagation Algorithm Based on Error Rate of Change”,
submitted to Journal of Taiyuan University of Technology.

4
Fig. 1. Structure of FNN
Fig. 4. Results of training
Fig. 2. Fuzzy subsets of working state
variables
Fig. 5. Membership functions after training
Fig. 6. Results of verification
Fig. 3. Unit step response under different
working state
5