A First Course in FUZZY and NEURAL CONTROL

178 CHAPTER 5. NEURAL NETWORKS FOR CONTROL

Therefore,

4 wjk =

XN

q=1

4 qwjk

= −η

XN

q=1

∂Eq

∂wjk

=

XN

q=1

−ηδqjxqk

The delta rule leads to the search for a weight vectorw∗such that the gradient
ofE atw∗is zero. For some simple single-layer neural networks,w∗is the
unique absolute minimum ofE(w).

Example 5.3In this example, we implement the logical function AND by
minimizing the errorE. Consider the training set T consisting of binary-
inputs/bipolar-targets:

T={(xq,yq),q=1, 2 , 3 , 4 }

withxq∈{ 0 , 1 }^2 andyq∈{− 1 , 1 }. Specifically,

x^1 =

°

x^11 ,x^12

¢

=(1,1) y^1 =1

x^2 =

°

x^21 ,x^22

¢

=(1,0) y^2 =− 1

x^3 =

°

x^31 ,x^32

¢

=(0,1) y^3 =− 1

x^4 =

°

x^41 ,x^42

¢

=(0,0) y^4 =− 1

An appropriate neural network architecture for this problem is the following,
with linear activation functionf(x)=x.

We have

E=

X^4

q=1

°

xq 1 w 1 +xq 2 w 2 −w 0 −yi

¢ 2

The weight vector(w∗ 0 ,w∗ 1 ,w 2 ∗)that minimizesEis the solution of the system
of equations
∂E
∂wj

=0,j=0, 1 , 2