The Development of Neural Network Based System Identification ...

162 CHAPTER 6 NEURAL NETWORK BASED PREDICTIVE CONTROL SYSTEM
where,
a ny = −
∂ŷ(k)
∂y(k − n y ) ∣ ; b nu = − ∂ŷ(k)
ϕ(k)=ϕ(τ)
∂u(k − n u ) ∣ (6.7)
ϕ(k)=ϕ(τ)
and,
ỹ(k − n y ) = y(k − n y ) − y(τ − n y );
ũ(k − n u ) = u(k − n u ) − u(τ − n u ) (6.8)
The constant n y and n u are the size of the past output and input measurements. The
approximate model in Equation (6.6) can be alternatively expressed similar to Equation
(4.18), where the coefficients a ny and b nu are collected in the polynomial A(q −1 ) and
B(q −1 ) as follows:
A(q −1 ) = 1 + a 1 q −1 + · · · + a ny q −ny ;
B(q −1 ) = b 0 + b 1 q −1 + · · · + b nu q −nu (6.9)
The instantaneous linearisation of the NN model can be derived by taking the partial
derivative of the NN model prediction with respect to each system input [Norgaard,
2000, Lawrynczuk, 2007b,a]. Applying the chain rules to Equation (4.36) with respect to
regression vector, the linearisation model for the MLP network with tangent hyperbolic
and linear activation function in hidden and output units yields:
⎡ ⎛
⎞⎤
∂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
with h = 1, 2, 3 · · · H and i = 1, 2, 3 · · · n (6.10)
For the case of linear units in both hidden and output layers, the linearisation is given
by:
∂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 (6.11)