National Instruments NI MATRIXx Xmath User Manual
Page 19
Chapter 1
Introduction
1-12
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and also:
Re
λ
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.
Re
λ
i
A
11
A
12
A
22
1
–
A
21
–
(
) 0
<
Re
λ
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
+
=
y
C
1
C
2
A
22
1
–
A
21
–
(
)x
1
D C
2
A
22
1
–
B
2
–
(
)u
+
=