Chapter 1 Introduction
Xmath Model Reduction Module 1-12 ni.com
and also:
Reλi(A22)<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 Equation1-2 is excited by some u(·), with initial condition x1(to), and
if Equation 1-1 is excited by the same u(·) with initial condition given
by,
•x
1(to) and x2(to) = –A–122A21x1(t o)–A
22–1B2u(to),
then x1(·) and y(·) computed from Equation1-1 and from Equation 1-2
should be similar.
• If Equation1-1 and Equation 1-2 are excited with the same u(·), have
the same x1(to) and Equation 1-1 has arbitrary x2, then x1(·) and y(·)
computed from Equation1-1 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.
ReλiA11 A12A22
1–A21
–()0<
ReλiA22
()ReλiA11 A12 A22
1–A21
–()«
x
·2
A21x1A22x2B2u++ 0=or x2A–22
1–A21x1A22
1–B2u–=
x
·1A11 A12A22
1–A21
=()x1B1A12A22
1–B2
–()u+=
yC
1C2A22
1–A21
–()x1DC
2A22
1–B2
–()u+=