The Development of Neural Network Based System Identification ...

4.3 SYSTEM IDENTIFICATION WITH NEURAL NETWORK 95
are related through a linear differential equation. Considering single input and single
output (SISO) case, the input-output relationship of a linear dynamic system described
in this research is given as follows:
y(t) + a 1 y(t − 1) + · · · + a ny y(t − n y ) = b 1 u(t) + b 2 u(t − 2)
+ · · · + b nu u(t − n u ) + v(t) (4.17)
where n y and n u are the sizes of past output and input observations and v(t) is the
system disturbance. Then the general polynomial equation (4.17) can be rewritten in
terms of the time shift operator q −1 as:
A(q −1 )y(t) = B(q −1 )u(t) + v(t) (4.18)
where A(q −1 ) and B(q −1 ) are polynomials in the time shift operator:
A(q −1 ) = 1 + a 1 q −1 + · · · + a ny q −ny
B(q −1 ) = 1 + b 1 q −1 + · · · + b nu q −nu (4.19)
The model in (4.17) or (4.18) describes the dynamic relationship between the input and
the output signals. Similar to definition in Section 4.2.1, the input-output relationship
(coefficient of ARX model) and the lagged input output data is express in terms of
parameter vector θ and time regression vector ϕ(t) as:
θ = [ a 1 a 2 · · · a ny b 1 b 2 · · · b nu
] T
(4.20)
ϕ(t) = [y(t − 1) · · · y(t − n y ) u(t − 1) · · · u(t − n u )] T (4.21)
The equation (4.17) then can be rewritten for k-step ahead prediction as:
ŷ (t + k |t, θ ) = ϕ T (t + k)θ + v(t) (4.22)
This model describe the observed output variable y(t) as an unknown linear combination
of the component of the observed time regression vector ϕ with system disturbance