Multipass algorithm, Multipass algorithm -20 – National Instruments NI MATRIXx Xmath User Manual
Page 43
Chapter 2
Additive Error Reduction
2-20
ni.com
to choose the D matrix of G
r
(s), by splitting
between G
r
(s) and G
u
(s).
This is done by using a separate function
ophiter( )
. Suppose G
u
(s) is
the unstable output of
stable( )
, and let K(s) = G
u
(–s). By applying the
multipass Hankel reduction algorithm, described further below, K(s) is
reduced to the constant K
0
(the approximation), which satisfies,
that is, if it is larger than,
then one chooses:
This ensures satisfaction of the error bound for G – G
r
given previously,
because:
Multipass Algorithm
We now explain the multipass algorithm. For simplicity in first explaining
the idea, suppose that the Hankel singular values at every stage or pass are
distinct.
1.
Find a stable order ns – 1 approximation G
n – 1
(s) of G(s) with:
(This can be achieved by the algorithm already given, and there is no
unstable part of the approximation.)
D˜
K s
( ) K
0
–
∞
σ
1
K
( ) ... σ σ
n
s
n
i
–
K
( )
+
+
≤
σ
≤
n
i
1
+
G
( ) ... σ
n
s
G
( )
+
+
G
u
s
–
( ) K
0
–
∞
σ
k
G
( )
k
n
i
1
+
=
n
s
∑
≤
G
r
G˜
r
K
0
+
=
G
u
G˜
u
K
0
+
=
G G
r
–
∞
G G˜
r
G˜
u
–
–
G˜
u
K
0
–
(
)
+
∞
=
G G˜
r
G˜
u
–
–
∞
=
K K
0
–
∞
+
σ
n
i
G
( ) σ
n
i
1
+
G
( ) ... σ
n
s
G
( )
+
+
+
≤
G j
ω
( ) G
ns 1
–
j
ω
–
∞
σ
ns
G
( )
=