User`s guide

Identifying Input-Output Polynomial Models
Understanding the Time-Shift Operator q
The general polynomial equation is written in terms of the time-shift operator
q. To understand this time-shift operator, cons ider the following discrete-time
difference equation:
yt a yt T a yt T
but T but T
() ( ) ( )
() ( )
+−+−=
−+ −
12
12
2
2
where y(t) is the output, u(t) is the input, and T is the sampling interval. q
-1
is a time-shift operator that compactly represents such difference equations
using
qu t u t T() ( )=−
:
yt a q yt a q yt
qut bqut
Aq
() () ()
() ()
(
++=
+
−−
−−
1
1
2
2
1
2
2
b
or
1
))() ()()yt Bqut=
In this case,
Aq aq aq()=+ +
−−
1
1
1
2
2
and
Bq bq bq()=+
−−
1
1
2
2
.
Note This q description is completely e quivalent to the Z-transform form: q
corresponds to z.
Definition of a Discrete-Time Polynomial Model
These model structures are subsets of the following ge ne ral p olyno m ia l
equation:
Aqyt
Bq
Fq
ut nk
Cq
Dq
et
i
i
ii
i
nu
()()
()
()
()
()
()=−
()
+
=
∑
1
The model structure s differ by how many of these polynomials are included
in the structure. Thus, different model structures provide varying levels of
flexibility for modeling the dynamics and noise characteristics. For m ore
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