*
Supported by the Visiting
Scholar
Foundation
of Shanxi Province P. R. China
1
A Fuzzy Neural Network Based on the TS Model with Dynamic Consequent Parameters and its
Application to Steam Temperature Control System in Power Plants*
Keming Xie
Department of Automation
Taiyuan University of Technology
Taiyuan, Shanxi, 030024, P.R.Chi
na
kmxie@tyut.edu.cn
T. Y. Lin
Department of Mathematics
and Computer Science
San Jose State University
San Jose , CA 95192, USA
tylin@cs.sjsu.edu
Jianfeng Nan
Department of Automation
Taiyuan University of Technology
Taiyuan, Shanxi, 030024, P.R.China
ABSTRACT
This paper
presents
a new fuzzy

neural network based
on the Takagi

Sugeno(TS) model with dynamic
consequent parameters. In the first step, this network
adopts the least

square method for rough

tuning the
consequent parameters; this is an off

line
processing. It
then in the second step employs the error
back

propagation to fine

tune the consequent
parameters, which is on

line. The fusion of fuzzy logic
and neural network enabling us to captures the physical
meaning in the model. In summary, the app
roach is
a
semantic oriented approximation of non

linear maps;
the optimization of the parameters is fast and efficient.
The network is applied to the cascade control system of
superheated steam temperature in power plants. The
approach is simulated in MAT
LAB. The simulation
shows the method is effective; fast in response,
minimal in overshoot, and robust.
Key words
Fuzzy neural

network, TS model, Cascade
control, Superheated steam temperature
1. Introduction
There are two characteristics in the fuzzy
inference rule
model [1] presented by Takagi and Sugeno (short for TS
model). The first one is that all rules in the model are
expressible by linear equations. This fact allows us to
express the global output of the model in a succinct
mathematical express
ion. So the classical linear control
method can be easily employed to design the non

linear
controller. The second one is that the partitioning of the
antecedent of the inference rules depends on whether
there is a local linear relation between the input a
nd
output. This makes it easy to use
linear model
of region
step

down
to describe complex global dynamic
characteristics when there is a major change in
operation
.
By combining fuzzy systems with neural networks and
making full use of the complimentary na
ture of two
approaches, fuzzy neural networks are applied to
intelligent control. The essential idea is that the
mechanisms of fuzzy systems are transformed to the
corresponding structures of neural networks [2].
Inference methods of fuzzy systems are tran
slated to
neural networks. By using the learning capability of
neural networks, auto

adjustments of antecedent and
consequent parameters can be achieved. The major
advantage of this method is that when the target
information is not adequate, the informatio
n of past
experience can be used to structure the neural network.
Using the capability of learning by examples in neural
networks, the fuzzy relationships between the input and
the output can be captured, revised and summarized.
Least square method is fun
damental in the classical
identification theory and is widely used. Both the
one

time

completing algorithm
and recursive
algorithm
can easily be realized in engineering. Its obvious
advantage is strong robust.
The
back propagation (BP)
learning algorithm c
an effectively revise the weights and
thresholds
of
hidden nodes
.
The feed forward neural
networks (FFNF) presented by the authors in reference [3]
focused on studying the problems of the topological
structure of a network. The present paper will stress th
e
algorithms of training networks and will present a fuzzy
TS
neural network with a dynamic consequent parameters
(DFNN)
. In this paper, the least square method combined
with the BP is used to train the networks. The new method
is effective; it not only ov
ercomes the drawbacks but also
takes the advantage of the merits of the two methods. First,
this network
employs the least square method
for
rough

tun
ing
the consequent parameters, which is off

line.
Then, it employs the back

propagation method
for
fine

tu
ning the consequent parameters, which is on

line.
This
method
has
captured
physical meaning in models
and achieved an excellent fusion between fuzzy logic and
neural network
.
T
his method is a powerful semantic
o
riented approximation of non

linear maps; the
optimization of parameters is fast and efficient
2. Topological structure of DFNN
The rules of TS
fuzzy
model can be expressed as follows
R
i
: if x
1
(k) is A
1
, x
2
(k) is A
2
, x
3
(k) is A
3
i =1, 2,
…
, R
(1)
where
R
is the number of rules in the TS fuzzy model
,
*
Supported by
the Visiting
Scholar
Foundation
of Shanxi Province P. R. China
2
x
1
(k), x
2
(k), and x
3
(k)
are three input variables,
is the
output of the i

th rule.
and
are fuzzy
subset of
x
1
(k), x
2
(k), and x
3
(k)
respectively, whose parameters of
membership function are called antecedent parameters,
the coefficients and the constants in equation (1) are
called consequent parameters
.
The number of fuzzy
granulations,
x
1
(k), x
2
(k),
and x
3
(k),
is determined by
jointly the complexity
and precision
of the model.
Suppose a group of input vector (
x
1
(k), x
2
(k), x
3
(k)
) is
given, then the global output y in the TS fuzzy model can
be obtained by the weighted average of the output
as
follows
(2)
where
is determined by the conclusive equation of the
i

th rule,
is
the weight of the firing strength layer
to
the i

th rule of the inpu
t vector, which is calculated by the
equation
(3)
(3)
where
represents the fuzzy
minimizing
.
In order to realize the smooth connection of a local linear
input

output re
lation in a fuzzy subs
pace
, TS fuzzy model
uses
the fuzzy
logic
inference based on the fuzzy
granulation of the input space [3]. So the ability of
describing nonlinear characteristics in the model depends
mainly on the granulation method and the precision
in the
input space.
The structure of DFNN network is shown in Fig.1 below,
which consists of 5 layers.
(a) Input layer: Input layer transforms the input vector
s
to
the next layer,
and
the i

th neur
on
is relative
to the i

th
element of the vector
s
, i=1,
2,…, n, where n is the
dimension of the input vector
s
.
(b) Fuzzy layer: the function of the fuzzy layer is similar
to the one of fuzzy logic controller (FLC). Because every
node in the previous layer responds to
nodes, the
number o
f nodes in the fuzzy layer is
and every
node has an action of membership function. In this paper,
the Gaussian membership function is employed. There is
a physical meaning to every node which represents a
fuzzy subset
that is
a ling
uistic variable such as NL, NM,
NS, NZ, PZ, PS, PM, PL and so on. The antecedent
parameters consist of the mean value and deviation in the
membership function.
is the number of the fuzzy
partition of the i

th input node in (a) layer
.
(c)
Firing strength
layer: Every rule adaptation
grade
is
calculated in this layer. The number of nodes is
respondent to the total number N
w
of rules. A neur
on
node has a function of the fuzzy logic and
computing
. If
i
(
t) represents the adaptation
grade
of the i

th rule, one
has
i
(t) = min {
mj
(t), …
kl
(t)}
(4)
where i=1,
2,
…,N
w
; m,
k=1,
2,
…,
n, m
k; j=1,
2,
…
,
N
m
;
l
=1,
2,
…,N
k
;
N
w
=
.
(d) Normalized layer: Normali
zing calculation is carried
out in this layer
:
(5)
(e) Linear combination layer:
u
i
(t) = a
i
x
1
(t) + b
i
x
2
(t) + c
i
x
1
(t)
is the consequence of every node, which is determined by
the input vector and
consequent parameters a
i
, b
i
, and c
i
*
Supported by
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3
w
hich
are evaluated by the learning mechanism. The
neuron in this layer has only one node which acts as a
linear weighted sum. The output is control function:
(6)
3. DFNN learning algorithm
In this paper, the control tactics is as follows: data
collected by the
cascade
control system is used to rough
tune the parameters of a network
off

line
. A group of data
is collected on line and
learn to make the
consequent parameters satisfy the demands on the index
of the performance
.
3.1 Rough tuning
In simulating the cascade control system,
Collect
p
group
of data (e,
ce,
T
×
e,
u)
,
that are teacher signals
to train
network, where e, ce,
T
×
e
and u are the
error
, the change
rate of the
error
, the
error
integral, and control
a
ction. The
matrix equation is
Ax=B,
where
x
is the vector composed
by all consequen
t
parameters.
Where
the dimension of
x
is
n
×
1,the dimension of
A
and
B
are
P
×
n and
P
×
1
re
spectively.
The
least square method is employed to
minimize 
Ax

B

, then the least square estimation :
(7)
where
，
is the pseudo inverse of
A
(
must
be non

singular)
.
Because there is a large computation in
the equation above and it becomes an ill

conditioned one
when
is singular, a recurrence formul
ae is
employed to calculate the least square solution of
.
Suppose
represents the i

th row vector of the matrix
A
and
represents the i

th element in the matrix
B
,
one
has [3]
,
,
(8)
where
is matrix covariance, the least square solution
x* is x
p
. The initial values of the consequent parameters
can be determined in advance according to
experie
nce
values of the controllers.
The initial values of the matrix of covariance
can
be determined as follows
S
o
= r I
(9)
where r is a larger positive number,
I
is a
n
×
n
unit
ma
trix
.
After
p
group of data being trained, the rough
tune values of the consequent parameters can be
obtained
and put its inquiry
library
3.2
. F
ine
tuning
The integral square

error criterion is adopted as
follows:
(10)
and
x
1
(k)= e(k), x
2
(k)= Te(k), x
3
(k)=(e(k)

e(k

1))/T (11)
(13)
(14)
where T is sampling period,
k
is the sampling
moment
l
is the number of learning iteration
x
1
(k), x
2
(k),
and
x
3
(k)
are the error, the error integral and the error differential
signal
s
respectively
a
i
(l), b
i
(l) and c
i
(l)
are coefficients
of the rule consequence
.
Particularly, the teacher signal
s
her
e
are
different from
that in rough tuning. The former are the mapping
relation between the
error
, the change rate of error, the
error
integral and the expected output of the closed
loop
control
system. One often cannot determine the
expected control tactic
s to the given the deviation, the
change rate of deviation, and the deviation integral, but
he can presents the expected output response curve. So
in the paper
(x
1
, x
2
, x
3
, r)
is selected as the teaching
signal, where r is the expected output of the
cascad
e
control
system in order to make the characteristics of
the
DFNN controller
network better. In this process
there is a error propagation from y to u. For the sake of
simplification in simulation, one doesn’t consider any
particular mathematical model inst
ead of the
approximate expression below
(15)
where
and
are the con
se
quence linear
function value and rule grade.
In order to prevent the
denominator from being zer
o when
u(k)=u(k

1)
one
take
(16)
The alternate is feasible because the equation (16) is
equality. Evaluated criterion can be given as function
(10)
.
If
J
is less than 0.05 directly apply the model without
any learning
. Otherwise a learning is carried out. In
general, the learning
times are
3
～
5
，
which
are
related
to the
sampling
period and the learning
rate
.
Because
the learning process is always controlled within one
*
Supported by
the Visiting
Scholar
Foundation
of Shanxi Province P. R. China
4
sampling
period, the time of the learning will
have
an
upper value. That is, although the maximal time of the
learning has been
reached.
J
is still larger than 0.05. At
this moment, the parameters are improved to some
extent and the controller can satisfy the demand on the
quickness of manufactory. The learning of the
antecedent parameters in which the intransitive error
algorithm
was used is discussed in detail in reference
[3].
4. S
uperheater steam temperature
cascade control system
The superheated steam temperature is an important
index in operation of monoblock unit in the power
plants. It has an important thing with the heat
efficiency of a monoblock unit and it will heavily
affect the safe and economic operation in the power
plants. Generally, the superheated steam temperature is
controlled at 540+(5/

10)
.
That is, 530~545
is
suitable and
reasonable
. Because the superheated
steam temperature object has a large inertia and a
larger delay time, so how to control superheated steam
temperature efficiently is a point for attention. At
present the typical control system pattern of the
superheated steam temperature is the cascade c
ontrol
system which employs the desuperheating water as the
manipulated variable.
The
cascade
control system of the steam temperature is
shown in Fig.2, where
the main
controller
employs
the
DFNN algorithm and vice controller adopts the PI
algorithm. The
steam temperature object is separated
into two
fields
.
is the prior field and
the inertia field
. And
(17)
(18)
5
.
Simulation
This cascade control system with DFNN algorithm is
simulated in MATLAB. Fig.3 shows the comparison of
two step responses, in which the response 1 and
response 2 represent DFNN algorithm (as main
controller) and
PID algorithm (as main controller)
respectively. It is seen from the Fig.3 that the control
performance is improved obviously when DFNN
instead of PID is employed. The former has a zero of
overshoot
and the later 0.0954. Fig.4 shows
comparison of the abil
ities to reject disturbance. In
simulation the disturbance
and
when they are entered in 500 second. It can
be seen from Fig. 4 that DFNN has a better ability to
reject disturbance than the traditional PID alg
orithm.
Fig.3 T
he comparison of step responses
for DFNN and
PID
Fig.4 The comparison of abilities of reject disturbances
for DFNN and PID
*
Supported by
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5
6. Conclusion
The DFNN algorithm presented in this paper
combines the classical least square method with BP
algorithm, not only employs the strong robust of least
square and the clear conception and the precision of BP
algorithm but also
overcomes their drawbacks. The
drawbacks existed in the network is a slower
calculation. Although the DFNN can be employed to
improve the control quality in the serial steam
temperature cascade control system. The simulation
shows the method is effective,
fast in response,
minimal overshoot, and robust.
References
[1].Takagi, M.Sugeno. Fuzzy Identification of Systems and Its
Applications to Modeling and Control. IEEE Transactions on
Systems, Man, and Cybernetics, Vol.SMC

15, No.1(1985):
116~132
[2] Xie Ke
ming, Zhang Jianwei. A Linear Fuzzy Model
Identification Method Based on Fuzzy Neural Networks.
Proceedings of 2
nd
World Congress on Intelligent Control and
Intelligent Automation Conferences (CWCICIA’97), vol.1:
316~320
[3] Xie keming, Nan jianfeng A Fast
Fuzzy

Neural Feedback
Network and its Application in Modeling. Proceedings of
ICAIE’98, 1998, 499~502
[4] Zhang Yuduo, Wang Manjia. Thermotechnical Automatic
Control System. Beijing: Press of Hydroelectric, 1987: 201

203
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