National Instruments NI MATRIXx Xmath - Manual (Page 20)

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(j
ω), 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
u k ( )
+ =
y k ( ) C
1
C
2
x
1
k ( )
x
2
k ( )
Du k ( )
+ =
x
1
k 1 +
( ) A
11
A
12
I A
22
–
( ) 1 – A
21
+
[ ]x
1
k ( ) +
=
B
1
A
12
I A
22
–
( )
1 – B
2
+
[ ]u k ( )
y
k
C
1
C
2
I A
22
–
( ) 1 – A
21
+
[ ]x
1
k ( ) +
=
D C
2
I A
22
–
( ) 1 – B
2
+
[ ]u k ( )
Φ s ( ) W s ( )W′ s
–
( )
=
Φ jω ( ) W j ω ( )W
*
j ω ( )
=