The Development of Neural Network Based System Identification ...

6.3 PRINCIPLE OF INSTANTANEOUS LINEARISATION 163
Similarly, the linear model extracted from the linearisation of HMLP network with
tangent hyperbolic and linear activation function in hidden and output units is given
by:
⎡ ⎛
⎞⎤
∂ŷ i (k)
H
∂ϕ j (k) = ∑
m∑
W 2 ih W 1 hj
⎣1 − tanh 2 ⎝ W 1 hj ϕ j (t) + B1 h
⎠⎦ +
h=1
j=1
m∑
W 3 ij
with h = 1, 2, 3 · · · H and i = 1, 2, 3 · · · n (6.12)
j=1
For the HMLP network with linear activation function in both hidden and output units,
the linearisation term is expressed as:
∂ŷ i (k)
H
∂ϕ j (k) = ∑
W 2 ih W 1 hj +
h=1
m∑
W 3 ij
j=1
with h = 1, 2, 3 · · · H and i = 1, 2, 3 · · · n (6.13)
Finally, the linear model extracted from the modified Elman network with tangent
hyperbolic and linear activation functions for hidden and output processing units is
formulated such as:
⎡ ⎛
∂ŷ i (k)
H
∂ϕ j (k) = ∑
m∑
W 2 ih W 1 hj
⎣1 − tanh 2 ⎝ W 1 hj ϕ j (t) + B1 h +
h=1
j=1
⎞⎤
m∑
W 3 k x k (k) ⎠⎦
with h = 1, 2, 3 · · · H and i = 1, 2, 3 · · · n (6.14)
k=1
For linear activation function in hidden and output layers, the linear model for Elman
network is given as:
∂ŷ i (k)
H
∂ϕ j (k) = ∑
W 2 ih W 1 hj
h=1
with h = 1, 2, 3 · · · H and i = 1, 2, 3 · · · n (6.15)