184 CHAPTER 5. NEURAL NETWORKS FOR CONTROL
Figure 5.9. Two-layer feedforward perceptronanalyze a single feedforward and backpropagation step for the initialized network
with activation functionf(z)=1+^1 e−z.
The weight matrices for the specified network are
d =0. 9x =
1
x 1
x 2
=
1
1
3
W =
∑
w 10 w 11 w 12
w 20 w 21 w 22∏
=
∑
3 − 12
12 0
∏
u =
u 10
u 11
u 12
=
− 3
1
− 2
The output of the hidden node is given byai=wjïxandhj=f(aj)=
1
1+e−aj,fromwhichwecalculate
a 1 = (3)(1) + (−1)(1) + (2)(3) = 8
a 2 = (1)(1) + (2)(1) + (0)(3) = 3
h =[h 0 ,h 1 ,h 2 ]T=[1, 0. 9997 , 0 .9526]TThe network output is therefore
y 1 = f(u∑h)=f((−3)(1) + (1)(0.9997)−(2)(0.9526))
= f(− 3 .9055) = 0. 0197The error signals are computed as follows:
δy 1 =(y 1 −d)(y 1 )(1−y 1 )
=(0. 0197 − 0 .9)(0.0197)(1− 0 .0197) =− 0. 0170
δh 1 =(h 1 )(1−h 1 )(δy 1 )(u 11 )
=(0.9997)(1− 0 .9997)(− 0 .0197)(1) =− 5. 098 ◊ 10 −^6
δh 2 =(h 2 )(1−h 2 )(δy 1 )(u 12 )
=(0.9526)(1− 0 .9526)(− 0 .0170)(−2) = 8. 056 ◊ 10 −^4