Neuro-Fuzzy Control
Adriano Joaquim de Oliveira Cruz
NCE/UFRJ
adriano@nce.ufrj.br
Neuro-Fuzzy Systems
 Usual
neural networks that simulate
fuzzy systems
 Introducing
*@2001 Adriano Cruz
fuzziness into neurons
*NCE e IM - UFRJ
Neuro-Fuzzy ‹#›
ANFIS architecture
 Adaptive
Neuro Fuzzy Inference
System
 Neural system that implements a
Sugeno Fuzzy model.
*@2001 Adriano Cruz
*NCE e IM - UFRJ
Neuro-Fuzzy ‹#›
Sugeno Fuzzy Model





A typical fuzzy rule in a Sugeno fuzzy model
has the form
If x is A and y is B then z = f(x,y)
A and B are fuzzy sets in the antecedent.
z=f(x,y) is a crisp function in the consequent.
Usually z is a polynomial in the input
variables x and y.
When z is a first-order polynomial the system
is called a first-order Sugeno fuzzy model.
*@2001 Adriano Cruz
*NCE e IM - UFRJ
Neuro-Fuzzy ‹#›
Sugeno Fuzzy Model
z1=p1x+q1y+r1
m
m
A1
B1
y
x
m
w1
w1 z1  w2 z 2
z
w1  w2
m
B2
A2
w2
x
y
z2=p2x+q2y+r2
*@2001 Adriano Cruz
*NCE e IM - UFRJ
Neuro-Fuzzy ‹#›
Sugeno First Order Example
 If
x is small then y = 0.1x + 6.4
 If x is median then y = -0.5x + 4
 If x is large then y = x – 2
Reference: J.-S. R. Jang, C.-T. Sun and E. Mizutani, Neuro-Fuzzy and Soft
Computing
*@2001 Adriano Cruz
*NCE e IM - UFRJ
Neuro-Fuzzy ‹#›
Comparing Fuzzy and Crisp
small
medium
(b) Overall I/O Curve for Crisp Rules
8
large
1
6
0.8
0.6
Y
Membership Grades
(a) Antecedent MFs for Crisp Rules
0.4
2
0.2
0
-10
0
-10
-5
small
medium
0
5
10
X
(d) Overall I/O Curve for Fuzzy Rules
8
large
0.6
0.4
*@2001 Adriano Cruz
4
2
0.2
0
-10
-5
6
0.8
Y
Membership Grades
0
5
10
X
(c) Antecedent MFs for Fuzzy Rules
1
4
-5
0
X
5
10
0
-10
*NCE e IM - UFRJ
-5
0
X
5
10
Neuro-Fuzzy ‹#›
Sugeno Second Order Example




If x is small and y is small then z = -x + y +1
If x is small and y is large then z = -y + 3
If x is large and y is small then z = -x + 3
If x is large and y is large then z = x + y + 2
Reference: J.-S. R. Jang, C.-T. Sun and E. Mizutani, Neuro-Fuzzy and Soft
Computing
*@2001 Adriano Cruz
*NCE e IM - UFRJ
Neuro-Fuzzy ‹#›
Membership Grades
Membership Functions
Small
0.8
0.6
0.4
0.2
0
-5
Membership Grades
Large
1
-4
-3
-2
-1
0
X
1
2
Small
1
3
4
5
4
5
Large
0.8
0.6
0.4
0.2
0
-5
-4
*@2001 Adriano Cruz
-3
-2
-1
0
Y
1
*NCE e IM - UFRJ
2
3
Neuro-Fuzzy ‹#›
Output Surface
*@2001 Adriano Cruz
*NCE e IM - UFRJ
Neuro-Fuzzy ‹#›
ANFIS Architecture
Layer 1
Layer 2 Layer 3
A1
x
A2
B1
y
B2
x
w1
O1,2
P
N
P
Layer 4 Layer 5
N
w2
y
w1
w1 f1
f
S
w2
w2 f 2
x
y

Output of the ith node in the l layer is
denoted as Ol,i
*@2001 Adriano Cruz
*NCE e IM - UFRJ
Neuro-Fuzzy ‹#›
ANFIS Layer 1





Layer 1: Node function is
O1,i  m Ai ( x), for i  1,2 or
O1,i  m Bi  2 ( y ) for i  3,4
x and y are inputs.
Ai and Bi are labels (e.g. small, large).
m(x) can be any parameterised membership
function.
These nodes are adaptive and the
parameters are called premise parameters.
*@2001 Adriano Cruz
*NCE e IM - UFRJ
Neuro-Fuzzy ‹#›
ANFIS Layer 2

Every node output in this layer is defined as:
O2,i  wi  m Ai ( x) T m Bi ( y ), for i  1,2



T is T-norm operator.
In general, any T-norm that perform fuzzy
AND can be used, for instance minimum and
product.
These are fixed nodes.
*@2001 Adriano Cruz
*NCE e IM - UFRJ
Neuro-Fuzzy ‹#›
ANFIS Layer 3

The ith node calculates the ratio of the ith
rule’s firing strength to the sum of all rules’
firing strength
wi
O3,i  wi 
for i  1,2
w1  w2


Outputs of this layer are called normalized
firing strengths.
These are fixed nodes.
*@2001 Adriano Cruz
*NCE e IM - UFRJ
Neuro-Fuzzy ‹#›
ANFIS Layer 4

Every ith node in this layer is an adaptive
node with the function
O4,i  wi f i  wi ( pi x  qi y  ri ) for i  1,2


Outputs of this layer are called normalized
firing strengths.
pi, qi and ri are the parameter set of this node
and they are called consequent parameters.
*@2001 Adriano Cruz
*NCE e IM - UFRJ
Neuro-Fuzzy ‹#›
ANFIS Layer 5
 The
single node in this layer calculates
the overall output as a summation of all
incoming signals.
 wi f i
O5,1   wi f i  i
 wi
i
i
*@2001 Adriano Cruz
*NCE e IM - UFRJ
Neuro-Fuzzy ‹#›
ANFIS Layer 5
 Every
ith node in this layer is an
adaptive node with the function
O4,i  wi f i  wi ( pi x  qi y  ri ) for i  1,2
 Outputs
of this layer are called
normalized firing strengths.
*@2001 Adriano Cruz
*NCE e IM - UFRJ
Neuro-Fuzzy ‹#›
Alternative Structures
 The
structure is not unique.
 For
instance layers 3 and 4 can be
combined or weight normalisation can
be performed at the last layer.
*@2001 Adriano Cruz
*NCE e IM - UFRJ
Neuro-Fuzzy ‹#›
Alternative Structure cont.
Layer 1
Layer 2 Layer 3
A1
x
A2
B1
y
B2
*@2001 Adriano Cruz
w1
O1,2
x
y
P
Layer 4 Layer 5
w1 f1
S
w2 f1
P
w2
x
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y
S
 wi f i
f
/
 wi
Neuro-Fuzzy ‹#›
Training Algorithm
 The
f
f
function f can be written as
w1
w2

f1 
f2
w1  w2
w1  w2
 w1 ( p1 x  q1 y  r1 )  w2 ( p2 x  q2 y  r2 )
 There
is a hybrid learning algorithm
based on the least-squares method and
gradient descent.
*@2001 Adriano Cruz
*NCE e IM - UFRJ
Neuro-Fuzzy ‹#›
Example





Modeling the function
sin( x) sin( y)
z
xy
Input range [-10,+10]x[-10,+10]
121 training data pairs
16 rules, with four membership functions
assigned to each input.
Fitting parameters = 72; 24 premise and 48
consequent parameters.
*@2001 Adriano Cruz
*NCE e IM - UFRJ
Neuro-Fuzzy ‹#›
Initial and Final MFs
Degree of membership
1
0.8
0.6
0.4
0.2
0
-10
Degree of membership
Initial MFs on Y
-5
0
5
input1
Final MFs on X
10
1
0.8
0.6
0.4
0.2
0
-10
-5
*@2001 Adriano Cruz
0
input1
5
10
1
0.8
0.6
0.4
0.2
0
-10
Degree of membership
Degree of membership
Initial MFs on X
-5
0
5
input2
Final MFs on Y
10
1
0.8
0.6
0.4
0.2
0
-10
*NCE e IM - UFRJ
-5
0
input2
5
10
Neuro-Fuzzy ‹#›
Training Data
Training data
ANFIS Output
1
1
0.5
0.5
0
0
10
10
10
0
-10 -10
Y
Y
X
0
-10 -10
X
step size curve
0.2
0.35
0.3
0.15
step size
root mean squared error
error curve
0.1
0.05
0
10
0
0
0.25
0.2
0.15
0
50
epoch number
*@2001 Adriano Cruz
100
0.1
0
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50
epoch number
100
Neuro-Fuzzy ‹#›