326
IEEE
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AND
CYBERNETICS—PART
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26,
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2,
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1996
that
their differences weighs more than their similarities. Of course,
it
depends on the reference class of properties to be compared ....
ACKNOWLEDGM
ENT
The
author thank J.
Logan
and D.
Elias
for their comments, and
the anonymous referees.
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,
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[33] , "Is probability theory sufficient for dealing
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North-Holland,
1986, pp. 103-116.
Learning
Fuzzy Inference
Systems
Using
an Adaptive Membership Function Scheme
A.
Lotfi
and
A.
C.
Tsoi
Abstract—An
adaptive membership function scheme for general ad-
ditive
fuzzy
systems
is proposed in this paper. The proposed scheme
can
adapt
a proper membership function for any nonlinear input-
output mapping, based upon a minimum number of rules and an
initial
approximate membership
function.
This
parameter
adjustment procedure
is
performed by computing the error between the actual and the desired
decision
surface.
Using
the proposed adaptive scheme for fuzzy system,
the number of rules can be
minimized.
Nonlinear
function approximation
and
truck backer-upper control system are employed to
demonstrate
the
viability
of the proposed method.
I.
INTRODUCTION
Fuzzy
inference
systems
have found many applications in recent
years. The simplicity of the design procedure of such
systems
is
a
dominant attraction in various industrial as
well
as household
products. In most cases, the design of a fuzzy inference system is
related to the ways in
which
an expert or a
skilled
human operator
would
operate
in
that
special domain.
Among
the various successful
applications of fuzzy inference
systems
we can mention are the
application
of fuzzy theory in the subway system in the city of Sendai,
Japan [13]; the detection of load and control of the washing cycle of
a
washing machine, the automatic focusing of the video camera and
nuclear reactor control [1].
Despite the brisk and stimulating promotion of fuzzy theory [14]
from
academic research to production
line,
there
is
still
a lack of
a
fuzzy system theory for the study of fuzzy inference systems.
However,
some
attempts
have recently been made [4]. Techniques
which
have been successfully applied in particular domain may not be
applied
to problems arising from
another
domain. Therefore a general
design
method is required. To move one
step
in this direction, an
Manuscript
received December 12. 1993: revised
March
30, 1995.
The
authors
are
with
the Department of
Electrical
and Computer Engineer-
ing.
University
of
Queensland,
Brisbane 4072,
Australia
(lotfia@elec.uq.oz.au;
act@elec.uq.oz.au).
Publisher
Item Identifier S 1083-4419(96)02299-6.
1083-M19/96S05.00 © 1996
IEEE
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327
Adaptive
Membership Function
Scheme
(AMFS)
for Fuzzy Inference
Systems
(FIS) is proposed in this paper.
The
first
attempt
to provide a general theory for the
"realization"
1
of
a
fuzzy inference system was proposed by Jang [2] who introduced
Generalized
Neural Networks based fuzzy inference systems. The
network was able to adjust its
parameters
such
that
the error between
the desired and the actual decision surface was gradually reduced. The
fuzzy
inference method used was based on a specific type of fuzzy
inference system introduced earlier by
Takagi
and Sugeno [11]. The
consequent premise of each rule is assumed to be a crisp value
rather
than a fuzzy value. Our proposed scheme is to employ the same type
of
network except
that
we
will
use a general additive fuzzy system
which
has already proposed by
Pacini
and
Kosko
[10].
Using
the proposed adaptive scheme for fuzzy systems, fewer rules
are required to correspond the expert exemplar or expert knowledge
to the fuzzy system.
The
structure of this paper is as
follows:
first the fuzzy inference
systems
will
be introduced,
followed
by the description of
an
adaptive
membership function scheme and rule minimization.
Application
to
nonlinear function approximation and the control system for backing-
up trucks are employed to illustrate the proposed scheme. Pertinent
conclusions
will
be drawn from the implications of the proposed
method.
II.
FUZZY
INFERENCE
SYSTEMS
A
crucial
step
in the design of
FIS
2
is determination of appropriate
knowledge-base parameters. A knowledge
base
consists of
three
major sub-systems
which
can be varied in the design of a FIS [5].
These
three
major sub-systems are:
•
A
data-base
containing membership function of
linguistic
values
for
both the antecedent and the consequent
•
The fuzzy reasoning mechanism
•
The number of rules used in the fuzzy rule-base
A
general inference mechanism,
which
is often used in the repre-
sentation of human reasoning, can be represented as
follows:
P
1
: If
Xis.4'
then
Yis£\
else
P'
: If
Xis.4'
then
Yis£'.
else
P"
:
Q:
If
Xis.4"
X
is
A'
then Y is B", else
A
YisB'
The
antecedent vector A' = [m, x
2
. •••, .v
m
\ is an m vector
with
elements
which
are linguistic variables in the universe of U —
[Ui,
U
2
,
• •
•. Cm]. The consequent vector Y = [y
t
. 1/2.
• •
•. 2/*]
is
a A- vector
with
elements
which
are linguistic variables
in
the universe of V = [V\. V
2
.
• •
•, 14]. Vectors A' =
[A\,A
2
,---,A
,
3
,---.A
l
„
l
]
andB' = [B\. B
2
.
• •
•. B),
• •
•,
B<
k
]
are vectors of
linguistic
values (linguistic labels) referring to the fuzzy
variables X and Y, respectively. Vector A' is the crisp observation
vector and B' is the crisp conclusion vector.
Fuzzy
sets
correspond
to each fuzzy variable can be shown as
follows:
j
= 1, 2,
• •
•. m i = 1, 2,
1
»
w
(1)
1
The
verb "realization" is used here to denote the explicit construction of
an
implementation of a fuzzy inference system.
2
In
control engineering literature, this may be referred to as a
Fuzzy
Logic
Controller
(FLC).
B)={»
B
i(v)/v)
j
= 1. 2. •
veVj
-.A-
«
= 1.2,
(2)
To
use the
AMFS,
it is desirable
that
the membership functions
employed
have a continues first derivative. We have investigated
[7] different piecewise continuous membership functions (triangular,
trapezoidal,
Cauchy and Gaussian)
with
the Gaussian MF showing
the
best
performance. Therefore, the membership function /( for
antecedent and consequent premises in the fuzzy values A) and B)
are defined as
follows:
.
2l fa'
U
—
IJij
(3)
j
= 1.
• •
•, k i = 1.
• •
•,
11
(4)
where a,j. pij. f3ij, o,j, pij, ffij are unknown constant parameters.
As
will
be shown subsequently,
these
parameters
can be adjusted on-
line
using a gradient decent algorithm. We further assume
that
the
universe of antecedent and consequent i.e., U and V are
limited
to
a
specific domain interval, i.e.,
Uj=\Uj
V+], j = l,
•••.m:
Vj=[v
3
r
v;\. j =
i.---,k.
(5)
Vector
X contains in linguistic variables
which
are connected
together by a
"liaison"
operator
AND.
The consequent vector Y
comprises of
A-
linguistic variables. It is reasonable to assume
that
there
is no relationship between this linguistic variable yj and the
other linguistic variables yt, I ^ j. Therefore, such an inference
might
be decomposed into A- inferences
with
antecedent vector X
and
consequent linguistic variable jy, j = 1. 2. •••.
A-
separately.
Without
loss
of
generality we can assume
that
the consequent premise
is
just one variable, i.e.,
A-
= 1, (&i
}
=
<Ti,
pij = pi. /3i
}
= $i).
For
making an inference "Y is £?'" from a set of rules P and
observation Q, different methods of reasoning under different fuzzy
implication
concepts have been studied e.g., [8], Since the output of
the decision engine should be a crisp value, numerous methods for
defuzzification
also have been proposed [5].
Among
these
methods,
the centroid method has been shown to be more effective.
Pacini
and
Kosko
[10] have proven
that
if correlation product inference
determines the output fuzzy values, the global centroid F can be
computed from
local
consequent premise centroids. i.e.,
^widli
F
=
1=1
£«*/•
(6)
where wi, d, and J; are rule
firing
weights,
local
centroid, and
area of consequent premise, respectively. Based on our definition for
membership functions we have:
d(v) =
1
/.(«)
r
v cxp <
—
V
—
<Ti
1&'
dv
--
j
v
exp
t -
Pi
•}Pi
1
dv.
(7)
(8)
(9)
328
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III.
ADAPTIVE MEMBERSHIP FUNCTION SCHEME
There
are different approaches for extracting
fuzzy
if-then rules
automatically,
based on a
fuzzy
model of the system [6] or a
numerical-fuzzy
approach [12].
Still,
there
is an unknown question
regarding the assignment of a membership function to each
fuzzy
value.
It is obvious
that
altering: a) the membership function of
linguistic
values, b)
fuzzy
reasoning mechanisms, or c) the number
of
rules,
will
affect the overall input-output mapping.
Altering
the
membership function has a dominant effect on the two other factors
[7].
It can be said
that
for a
fixed
number of rules and different
fuzzy
reasoning methods, changing the membership function can achieve
the same input-output mapping. Alternatively, for a
fixed
fuzzy
reasoning method we can achieve the same input-output mapping
with
different number of rules and different membership functions.
Consider
the generalized neural network based
fuzzy
inference
system introduced by Jang [2].
This
contains a multilayer feedforward
network in
which
each node performs a particular function (node
function).
There are two types of nodes:
1) the nodes
which
have
fixed
parameters
(they are
called
circle
node functions by Jang)
2) the nodes
which
depend on a set of
parameters
specific to the
node (Jang
called
this type of nodes a square node function)
The
performance of the node and consequently the performance of
the system changes
with
altering
these
parameters.
Our
proposed algorithm
uses
a neural network
which
contains four
layers,
with
2
circle
and 2 square layers. The first and third layers
(containing
only
square nodes) represent the membership functions
given
in
fuzzy
values A) and B' (in this case j = 1), respectively.
The
other two layers contain
only
circle
nodes. The node function
of
the second layer is a simple multiplication and the node function
of
the fourth layer is the actual output of the system governed by
(6). Fig. 1 shows the structure of the adaptive network based
fuzzy
inference system.
The
cost function for
minimizing
the error arising
from
all the
square nodes in the first and the third layers, is defined as
follows:
P
=I
p
=
YJJ" ~F
v
f (10)
p=\
where E" is the square of the difference between the actual F" and
the desired T
p
output of the system for the pth training data. We
assume the number of exemplars in the training data set is P. The
parameters
in the first and the third layer/membership functions in
the antecedent and the consequent premises are defined as 0^ =
[<r,j. pij, /3ij], 6j =
[<T,,
p,, ,#,]. To update the parameters, we
can
use a
steepest
descent gradient method to minimize the cost
function
E. The values
AQij
and A0, at (r + l)th instant, where
AQij(t)
=
e,j(t)
- G,j(r - 1), and A0,(«) is defined in a similar
fashion.
It is given as a function of the values at the «th instant as
follows:
AQij(t
+ 1) = -
nVE
tJ
+
cxAQiiit)
(11)
A0,(f
+ 1) = - </V£, + aA0,(t) (12)
where V Eij, V£
;
, and n, are gradients of the
parameters
and the
learning
rate,
which
can be expressed as
follows:
/'
p=i
Fig.
1. Adaptive network based
fuzzy
inference system.
VEi =
J3
v
-£f
(13)
where
V£f
J
=
V£?
=
P=I
dE"
dE" dE
p
dcr.j
' dp,j'
dPij
dE"
OE" dE
p
[
da, ' dpi ' d0i
and
'/
=
Y
t
^E
lJ
f
+
Y.^E,f
(14)
(15)
The
constant
parameter
a is the
momentum
of the gradient descent
and
the constant k is the
step
size of the gradient descent. The
gradients defined in (14) are analytically available (see Appendix)
making
the presented network realizable.
IV.
RULE MINIMIZATION
Since
acquiring the expert knowledge of a
skilled
domain specialist
in
the
form
of
fuzzy
value for each
fuzzy
rule is an arduous
step
in the design procedures,
there
should be some methods to
determine the proper meaning of related
fuzzy
values.
AMFS
gives
this opportunity to the controller designer. As
long
as the
parameters
of
the membership function of
fuzzy
values are changing, we can
obtain
the same decision surface
with
different rules.
Based
on our empirical results
(these
results are shown in Sections
V
and VI in sequel), for a system without very "spiky" convex
nonlinearity
(not necessary smooth)
3
the minimum and maximum
number of rules can be expressed as
follows:
2
m
< n < 3" (16)
Therefore,
for a system
with
two inputs, four to nine rules are
sufficient.
We can
start
with
a minimum number of rules, and in the
case of deficiency, increase it toward the maximum number of rules.
3
In
general, it is
difficult
to describe exactly what this means in practice.
One
way in
which
we can understand this is explained later in Sections V
and VI.
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(a)
(C
(b) ]
>o
Fig.
2. Initial membership functions
with
4 rules for (a) the first input, (b)
Fig.
4. The membership functions after 250 epochs of training
with
4 rules
the second input, and (c) consequent. for (a) the first input, (b) the second input, and (c) consequent.
2S0
Fig.
3. Error between the
target
and the trained surface for 250 epochs of
training
with
momentum a = 0.95 and
step
size n = 0.01.
There are two types of expert information for justifying member-
ship function of fuzzy values in FIS.
1) Expert Knowledge:
This
is the simplest situation, when some
linguistic
rules or desired decision surface is accessible. Adjust-
ing
membership function of fuzzy values using
AMFS
would
commence
with
initial
membership functions
with
a minimum
number of rules. Rules
will
be added as required in the design
process. The
final
membership function of fuzzy values, is
obtained when the actual decision surface converges to the
desired one.
2) Expert Exemplar:
This
is a specific case of (1) and it is more
applicable
to real systems. A set of input-output pairs of desired
system response (expert training) is available. The procedure
of
training the FIS has been explained earlier. Naturally, the
more training exemplars we have the
better
performance of the
overall
system
will
be.
V.
APPLICATION
TO
NONLINEAR FUNCTION
APPROXIMATION
To
demonstrate
the
viability
of the proposed method, we use
AMFS
in
this section to approximate a nonlinear function
with
a set of fuzzy
rules. The
target
nonlinear function
which
we are going to train our
Fig.
5. Diagram of simulated truck and loading zone.
fuzzy
system is taken as
follows:
r*=[3e«
/10
-l],»h(^-)
+
^t
4
+
e
•T2/101
(xi
+4)7r
(17)
30 ' ' 10
The
fuzzy inference system contains 4 rules (i = 4) and 2 inputs
for
antecedent
(j — 2). The universe Ui and Da are both [-10 10].
Based
on some understanding from the desired surface, we can
assign
initial
values for the membership function
parameters.
With
an appropriate combination of the
step
size and the momentum, the
network converges.
The
initial
membership functions for
antecedent
and consequent
are depicted in Fig. 2. The
AMFS
has been employed to reduce
the error between the desired nonlinear function and fuzzy inference
system. The percentage of error is shown in
Fig.
3 for 250 epochs
of
training when the momentum a = 0 and
step
size of gradient
descent K = 0.01. The membership functions after 250 epochs of
training
are shown in Fig. 4.
VI.
APPLICATION
TO
TRUCK BACKER-UPPER CONTROL
In
the real
world,
backing a truck to a loading zone is a
difficult
problem
except for a
skilled
truck driver. If we
elicit
the
skilled
driver experience in a fuzzy if-then rule format, and can be assured
that
the fuzzy controller is working
with
the
same
set of rules,
we
would
obtain the
same
trajectory. For truck backing-up control.
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0 -SO ~ 0 -00 0-80
(a) (b) (c)
Fig.
6. Control surface of fuzzy controller after 300 epochs training
with:
(a) 35 rules, (b) 9 rules, and (c) 4 rules.
Nguyen
and
Widrow
[9] use expert exemplars to train an
artificial
neural network based controller.
Kong
and
Kosko
[3] proposed a
fuzzy
logic
controller
with
35 expert rules, and they compared their
results obtained from FLC
with
results achieved by using a neural
network controller. FLC has been shown to give more appropriate
tracking
results. A neural network controller only
uses
numerical
data, whereas FLC employs linguistic rules concluded from expert
drivers
explicitly.
To combine the above two methods, Wang and
Mendel
[12] utilized numerical-fuzzy approach
with
almost the
same
rules as [3], but
with
different membership functions for fuzzy values.
The
simulated truck (which is allowed to move backward only),
and
the loading zone are depicted in Fig. 5. The truck in our
simulation
is the cab
part
of [9] and the
same
truck for [3] and
[12] except for the size of the yard, the definition of steering and
the azimuth angle. Since our study is performed in simulation, the
dynamics of the truck backing-up system is required. We used the
following
approximate kinematics [12].
x(t + 1) = x(t) +
(;{cos
[cj>(t)
+ 0(t)] + sin [0(t)] sin
[<£(<)]}
</(r + 1) = «,(r) + v{sm
[<£(*)
+
<?(<)]
- sin [0(f)] cos
[J>(t)]}
| 2 sin [0(f)]"
4>(t
+ l)=<fr(t)
"4
(18)
where x, </, and
<j>
are
rear
center of truck coordinate and azimuth
angle of truck in yard, respectively. They can be considered as
state
variables of the system
which
indicate position and direction of the
truck in yard at any instant of time. 8 is the steering angle to direct
the truck to the loading zone Xj and (//. Constant
parameters
v and
/ are truck speed and length of the truck, respectively. The control
goal
is to
steer
the truck from any
initial
position to prespecified
loading
dock
with
a right azimuth angle (e>/ = 90) and coincided
rear
position.
The steering angle 0 is the control action
which
is provided
by
the designed fuzzy controller. Since we presuppose
adequate
clearance between the truck and the loading dock,
state
variable y can
be abandoned for the reason
that
it becomes a dependent variable.
Therefore,
the inputs to the controller are x and
4>.
The range of
variables for simulated truck and controller are as
follows:
x
<E
\x~
#e [r
x+]=[0
100]
^+]=[-90
270]
<?+]=[-30
30].
The
truck speed v = 5 and the length of the truck t = 4. The
maximum
width of the yard is y = 100. Desired loading dock
position
is .17 = 50 and yj = 100. Positive
attitude
of azimuth
angle
4>
is clockwise
with
respect to the horizontal
line.
Steering
angle 0 is positive when the steering wheels
rotate
counterclockwise.
We
start
with
fuzzy value and rules specified in [3]. There are 35
rules
with
seven,
five,
and seven fuzzy values for azimuth angle
<j>,
coordinate x, and steering angle 0, respectively. The control surface
is
depicted in Fig. 6(a).
(a) (b)
Fig.
7. Truck trajectory of fuzzy controller for both expert knowledge and
expert exemplars
with:
(a) 9 rules, and (b) 4 rules.
In
the next step,
AMFS
is used for controllers
with
9 and 4 rules.
Final
decision surface after 300 epochs of training is illustrated in
Fig.
6(b) and (c) for 9 and 4 rules, respectively. Truck trajectory of
the fuzzy controller
with
9 and 4 rules after training from 4 different
initial
position (XQ, (po) is shown in Fig. 7.
The
afore-mentioned procedure has been repeated for 20
sets
of
expert exemplars from randomly selected
initial
points in the (x, y)
plane. The results obtained from 9 and 4 rules case studies are almost
the
same
as before. We initiate
with
the
same
membership function
for
fuzzy values as before. Truck trajectory of the fuzzy controller
from
four
initial
state
with
9 and 4 rules trained through 20 expert
exemplars are depicted in Fig. 7. In both
cases
for 4 and 9 rules the
trajectory is
followed
perfectly after training.
VII.
CONCLUSIONS
Through
illustrated examples, in this paper it has been shown
that
changing
the membership functions of fuzzy values can affect the
overall
input-output mapping of
FIS.
This
change might be directed to
applying
the minimum number of rules and consequently
simplifying
the controller design. The proposed scheme can be used to achieve
any continuous nonlinear surfaces, but the gradient descent method is
not capable of convergence for very sharp nonlinearities. It has been
shown
the control surface
with
fewer rules is more smooth, and this
smoothness can be thought of as being more robust and fault-tolerant.
APPENDIX
In
this appendix, the gradient of learning
parameters
for Gaussian
membership functions is derived.
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