User's Manual

Chapter 1 Introduction
Xmath Model Reduction Module 1-12 ni.com
and also:
Reλ
i
(A
22
)<0 and .
Usually, we expect that,
in the sense that the intuitive argument hinges on this, but it is not necessary.
Then a singular perturbation is obtained by replacing by zero; this
means that:
Accordingly,
(1-2)
Equation 1-2 may be an approximation for Equation 1-1. This means that:
• The transfer-function matrices may be similar.
• If Equation 1-2 is excited by some u(·), with initial condition x
1
(t
o
), and
if Equation 1-1 is excited by the same u(·) with initial condition given
by,
•x
1
(t
o
) and x
2
(t
o
) = –A
–1
22
A
21
x
1
(t
o
)–A
22
–1
B
2u
(t
o
),
then x
1
(·) and y(·) computed from Equation 1-1 and from Equation 1-2
should be similar.
• If Equation 1-1 and Equation 1-2 are excited with the same u(·), have
the same x
1
(t
o
) and Equation 1-1 has arbitrary x
2
, then x
1
(·) and y(·)
computed from Equation 1-1 and Equation 1-2 should be similar after
a possible initial transient.
As far as the transfer-function matrices are concerned, it can be verified that
they are actually equal at DC.
Reλ
i
A
11
A
12
A
22
1–
A
21
–()0<
Reλ
i
A
22
()Reλ
i
A
11
A
12
A
22
1–
A
21
–()«
x
·
2
A
21
x
1
A
22
x
2
B
2
u++ 0=
or
x
2
A–
22
1–
A
21
x
1
A
22
1–
B
2
u–=
x
·
1
A
11
A
12
A
22
1–
A
21
=()x
1
B
1
A
12
A
22
1–
B
2
–()u+=
yC
1
C
2
A
22
1–
A
21
–()x
1
DC
2
A
22
1–
B
2
–()u+=