User's Manual

Chapter 1 Introduction
© National Instruments Corporation 1-13 Xmath Model Reduction Module
Similar considerations govern the discrete-time problem, where,
can be approximated by:
mreduce( ) can carry out singular perturbation. For further discussion,
refer to Chapter 2, Additive Error Reduction. If Equation 1-1 is balanced,
singular perturbation is provably attractive.
Spectral Factorization
Let W(s) be a stable transfer-function matrix, and suppose a system S with
transfer-function matrix W(s) is excited by zero mean unit intensity white
noise. Then the output of S is a stationary process with a spectrum Φ(s)
related to W(s) by:
(1-3)
Evidently,
so that Φ( jω) is nonnegative hermitian for all ω; when W( jω) is a scalar, so
is Φ( jω) with Φ( jω)=|W( jω)|
2
.
In the matrix case, Φ is singular for some ω only if W does not have full
rank there, and in the scalar case only if W has a zero there.
Spectral factorization, as shown in Example 1-1, seeks a W(jω), given
Φ(jω). In the rational case, a W(jω) exists if and only if Φ(jω) is
x
1
k 1+()
x
2
k 1+()
A
11
A
12
A
21
A
22
x
1
k()
x
2
k()
B
1
B
2
uk()+=
yk()
C
1
C
2
x
1
k()
x
2
k()
Du k()+=
x
1
k 1+()A
11
A
12
IA
22
–()
1–
A
21
+[]x
1
k()+=
B
1
A
12
IA
22
–()
1–
B
2
+[]uk()
y
k
C
1
C
2
IA
22
–()
1–
A
21
+[]x
1
k()+=
DC
2
IA
22
–()
1–
B
2
+[]uk()
Φ s() Ws()W′ s–()=
Φ jω() Wjω()W
*
jω()=