User`s guide

Identifying Input-Output Polynomial Models
For a system with nu inp uts and ny outputs, A(q)isanny-by- ny matrix. A(q)
can be represented as a polynomial in the shift operator q
-1
:
Aq I Aq A q
ny na
na
()=+ ++
−−
1
1
…
For more information about the time-shift operator, see “Understanding the
Time-Shift Operator q” on page 3-43.
A(q) can also be represented as a matrix:
Aq
aq aq a q
aq aq a q
aqa
ny
ny
ny n
()
() () ()
() () ()
()
=
11 12 1
21 22 2
1
…
…
…………
yynynyq
qa
2
()
()
…
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
where the matrix element a
kj
is a polynomial in the shift operator q
-1
:
aq aq a q
kj kj kj
kj
na
na
kj
kj
()=+ ++
−
−
δ
11
…
δ
kj
represents the Kronecker delta, w hich equals 1 for k=j and equals 0
for k≠j. This polynomial describes h ow the old values of the jth output are
affected by the kth output. The ith row of A(q) represents the contribution of
the past output values for predict the current value of the ith output.
B(q)isanny-by-ny matrix. B(q) can be represented as a polynomial in the
shift operator q
-1
:
Bq B Bq B q
nb
nb
()=+ ++
−−
01
1
…
3-47