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 Equation1-1 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 Example1-1, seeks a W(jω), given
Φ(jω). In the rational case, a W(jω) exists if and only if Φ(jω) is
x1k1+()
x2k1+()
A11 A12
A21 A22
x1k()
x2k()
B1
B2
uk()+=
yk() C1C2
x1k()
x2k() Du k()+=
x1k1+()A11 A12 IA
22
–()
1–A21
+[]x1k()+=
B1A12 IA
22
–()
1–B2
+[]uk()
ykC1C2IA
22
–()
1–A21
+[]x1k()+=
DC
2IA
22
–()
1–B2
+[]uk()
Φs() Ws()W′s–()=
Φjω() Wjω()W*jω()=