User`s guide

3 Linear Model Identification

Polynomial Model Structure

You can estimate the following types of linear p olynomial model structures:

Aqyt

ut nk

()()

()

()=−

()

∑

The polynomials A, B

, C, D,andF

contain the time-shift operator q. u

is the

ith input, nu is the total number of inputs, and nk

is the ith input delay that

characterizes the delay response time. The variance of the white noise e(t)

is assumed to be

. For more information about the time-shift operator, see

“Understanding the Time-Shift Opera tor q” on page 3-43.

Note This form is completely eq uivalent to the Z-transform form: q

corresponds to z.

To estimate polynomial models, you must specify the model order as a set of

integers that represent the number of coefﬁcients for each polynomial y ou

include in your selected structure—na for A, nb for B , nc for C, nd for D ,

and nf for F . You must also specify the number of samples nk corresponding

to the input delay— dead time—given by the number of samples before the

output responds to the input.

The number of coefﬁcients in denominator polyn omials is equal to the number

of poles, and the number of coefﬁcients in the numerator polynomials is equal

to the number of zeros plus 1. When the dynamics f rom u(t) to y(t) contain a

delay of nk samples, then the ﬁrst nk coefﬁcients of B are zero.

For more i nfo rmation about the family of transfer-function models, see the

corresponding section in System Identiﬁcation: Theory for the User,Second

Edition, by Lennart Ljung, Prentice Hall PTR, 1999.

3-42