National Instruments NI MATRIXx Xmath User Manual
Page 36
Chapter 2
Additive Error Reduction
© National Instruments Corporation
2-13
Next, Schur decompositions of W
c
W
o
are formed with the eigenvalues of
W
c
W
o
in ascending and descending order. These eigenvalues are the square
of the Hankel singular values of
Sys
, and if
Sys
is nonminimal, some can
be zero.
The matrices V
A
, V
D
are orthogonal and S
asc
, S
des
are upper triangular. Next,
submatrices are obtained as follows:
and then a singular value decomposition is found:
From these quantities, the transformation matrices used for calculating
SysR
are defined:
and the reduced order system is:
An error bound is available. In the continuous-time case it is as follows. Let
G( j
ω) and G
R
( j
ω) be the transfer function matrices of
Sys
and
SysR
,
respectively.
For the continuous case:
V
′
A
W
c
W
o
V
A
S
asc
=
V
′
D
W
c
W
o
V
D
S
des
=
V
lbig
V
A
0
I
nsr
=
V
rbig
V
D
I
nsr
0
=
U
ebig
S
ebig
V
ebig
V
′
lbig
V
rbig
=
S
lbig
V
lbig
U
ebig
S
ebig
1 2
⁄
=
S
rbig
V
rbig
V
ebig
S
ebig
1 2
⁄
=
A
R
S
lbig
′
AS
rbig
=
A
R
CS
rbig
=
B
R
S
lbig
′
B
=
D
G j
ω
( ) G
R
j
ω
( )
–
∞
2
σ
nsr 1
+
σ
nsr 2
+
...
σ
ns
+
+
+
(
)
≤