National Instruments NI MATRIXx Xmath, Техническое описание

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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
( j
ω
) is nonnegative hermitian for all
ω
; when W ( j
ω
) is a scalar, so
is
Φ
( j
ω
) with
Φ
( j
ω
) = | W ( j
ω
)|
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
( j
ω
). In the rational case, a W ( j
ω
) exists if and only if
Φ
( j
ω
) 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
j
ω ( )
W j
ω ( )
W
*
j
ω ( )
=