compared with the values obtained in the experiment in order to determine whether it is
a good estimation of the system response.
The black box model structure which considers input and output signals can be
expressed as the linear equation shown in 4.1 where e(t) is the noise error term.
y(t)+aly(t--1)+...+an y(t-na) = blu(t- 1)+...+bnbu(t-nb)+e(t) (4.1)
Then this equation can be expressed in terms of the initial output signal as shown
in 4.2.
y(t) = -aly(t 1) -... -ay(t na) + blu(t- 1) +...+ bu(t --nb) +e(t) (4.2)
This is typically referred to as an ARX model, which defines the autoregressive
part to be the output terms in 4.2, and the input terms in 4.2 as the extra input.
So the initial output values as well as the input and output terms on the right hand
side of 4.2 are collected in matrix form for each time interval. This makes it possible
to solve for the regression coefficients since the initial output and the input values are
known.
The initial output values for each time interval can be expressed as in 4.3 in terms
of the input and output values as well .
I --11 --r U Ut
nb al
/ l /l1 _/ 1 ; 1 1 an (4.3)
/n-" yiI" _-" '/" 9 n
y \ na U1 p