The Development of Neural Network Based System Identification ...

4.3 SYSTEM IDENTIFICATION WITH NEURAL NETWORK 97
ŷ1
yˆn
Output
Layer
ŷ1
yˆn
1
Bias Input
1
2
h
Hidden
Layer
1
Bias Input
1
h
MLP
connection
Linear
connection
1
1
Bias Input
Bias Input
y(t-1)
∆T
y(t-2)
∆T
y(t-ny)
u(t-1)
∆T
u(t-2)
∆T
u(t-nu)
Input Layer
y(t-1)
∆T
y(t-2)
∆T
y(t-ny)
u(t-1)
∆T
u(t-2)
∆T
u(t-nu)
Input Layer
Input 1 Input 2
(a)
Input 1 Input 2
(b)
ˆ
1( 1)
y t yˆ t1
n
Output
Layer
Self
Connection
1
Hidden
Layer
Bias Input
1
h
x 1
Context Units
Input
Layer
1
Bias Input
x k
y(t)
u(t)
(c)
Figure 4.9 The model structure using different ANN networks architecture (a) The NNARX based
model structure using MLP network (b) The NNARX based model structure using HMLP network;
and (c) Prediction/Forecasting using modified Elman network with reduced weight connection from
context units to hidden units.
ϕ (t) = [ ϕ 1 ϕ 2 · · ·
]
ϕ m
[
= y(t − 1) y(t − 2) · · · y(t − n y )
]
u(t − 1) u(t − 2) · · · u(t − n u )
(4.26)
where w hj is the weights matrix between the input layer and the hidden layer and W ih
is the weights matrix between the hidden layer and the output layer. The functions
f j (∗) and F i (∗) are non-linear activation function for neurons in each hidden and output
layers. The symbol H denotes the number of neurons in the hidden layer while b1 and
b2 are the bias elements for the input layer and output layer. The number of inputs