A First Course in FUZZY and NEURAL CONTROL

224 CHAPTER 6. NEURAL CONTROL

nonlinear neural network model at each sampling time.
Assume that a neural network input-output model of the system to be
controlled is described as a function of past outputsy(t−i)and past inputs
u(t−d−i)in the form
y(t)=g(x(t))

where the vector

x(t)=[y(t−1),...,y(t−n),u(t−d),...,u(t−d−m)]T

defines the state of the system. At timet=τ, linearize the function

g=g(x 1 ,...,xn+m+1)

around the current statex(τ)to obtain the approximate model

yò(t)=−a 1 yò(t−1)−...−anyò(t−n)+b 0 uò(t−d)+...+òu(t−d−m)

where

yò(t−i)=y(t−i)−y(τ−i)

òu(t−i)=u(t−i)−u(τ−i)

ai = −

∂g(x(t))

∂xi

Ø

Ø

Ø

Ø

t=τ

for 1 ≤i≤n

bi = −

∂g(x(t))

∂xn+i+1

Ø

ØØ

Ø

t=τ

for 0 ≤i≤m

For a multi-layer perceptron (MLP) network withnxinputs, one hidden
layer ofnhtanhunits, and a linear output

y(t)=

Xnh

j=1

Wjtanh

√nx

X

k=1

wkjxk(t)+w 0 j

!

+W 0

the derivative of the output with respect to inputxi(t)is calculated in accor-
dance with

∂g(x(t))

∂xi(t)

=

Xnh

j=1

Wjwji

√

1 −tanh^2

√nx

X

k=1

wjkxk(t)+wj 0

!!

where
(x 1 ,...,xnx)=(y 1 ,...,yn,u 0 ,u 1 ,...,um)

The approximate model can also be expressed as

y(t)=−

Xn

i=1

aiy(t−i)+

Xm

i=0

biu(t−i)+

√

y(τ)+

Xn

i=1

aiy(τ−i)−

Xm

i=0

biu(τ−i)

!