Onepass algorithm, Onepass algorithm -18 – National Instruments NI MATRIXx Xmath User Manual
Page 41
Chapter 2
Additive Error Reduction
2-18
ni.com
being approximated by a stable G
r
(s) with the actual error (as opposed to
just the error bound) satisfying:
Note
G
r
is optimal, that is, there is no other G
r
achieving a lower bound.
Onepass Algorithm
The first steps of the algorithm are to obtain the Hankel singular values of
G(s) (by using
hankelsv( )
) and identify their multiplicities. (Stability of
G(s) is checked in this process.) If the user has specified
nsr
and this does
not coincide with one of 0,n
1
,n
2
, ... an error message is obtained; generally,
all the
σ
i
are different, so the occurrence of error messages will be rare.
The next step of the algorithm is to calculate the sum G(s) = G
r
(s) + G
u
(s),
following [SCL90]. (A separate function
ophred( )
is called for this
purpose.) The controllability and observability grammians P and Q are
found in the usual way.
AP + PA
′ = –BB′
QA + A
′Q = –C′C
and then a singular value decomposition is obtained of the
matrix
:
There are precisely n
i
– n
i – 1
zero singular values, this being the multiplicity
of
σ
n
i
. Next, the following definitions are made:
G s
( ) G
r
s
( )
–
∞
σ
ns
=
QP
σ
n
i
2
I
–
U
1
U
2
S
B
0
0 0
V
1
′
V
2
′
QP
σ
n
i
2
I
–
=
A
11
A
12
A
21
A
22
U
1
′
U
2
′
=
σ
n
i
2
A
′ QAP
+
(
) V
1
V
2
(
)
B
1
B
2
U
1
′
U
2
′
QB
=
C
1
C
2
[
]
CP V
1
V
2
[
]
=