Then the regression coefficients are obtained by using the matrix equation 4.4 and
solving for the matrix of coefficients X as shown in 4.5.
B =AX (4.4)
A"-B =X (4.5)
A transformation as shown in 4.6 is then applied to equation 4.1 in order to obtain
a transfer function as shown in 4.7. In this transfer function the B term contains all the
input coefficients from equation 4.6 and the A terms consists of all the coefficients in
the output terms.
y(t) at) + t) +... + az y(t) = blzlu(t) +... + b,,z u(t) + e(t) (4.6)
yu 1 =BA1 (4.7)
A Tustin transformation is then done using Matlab in order to create a continuous
time version of the discrete time system. This is done using a standard bilinear
transformation such as shown in 4.8.
z = 1 + 2(sT/2) + 2(sT/2)2 + 2(sT/2)3 +... (4.8)
The ARX model approximation is just one of several types of model structures
which can be used for system identification. An ARMAX model structure can similarly
be used but was not used in this project because an initial simple estimation was
desired. The ARMAX model considers the basic properties that the ARX model uses
but also includes a moving average term, which considers the noise in its coefficient
calculations.