Bst( ), Bst( ) -3 – National Instruments NI MATRIXx Xmath User Manual
Page 49
Chapter 3
Multiplicative Error Reduction
© National Instruments Corporation
3-3
bandwidth at the expense of being larger outside this bandwidth, which
would be preferable.
Second, the previously used multiplicative error is
. In the
algorithms that follow, the error
appears. It is easy to
check that:
and
This means that if either bound is small, so is the other, with the bounds
approximately equal.
Two algorithms for multiplicative reduction are presented:
bst( )
,
a mnemonic for balanced stochastic truncation, and
mulhank( )
.
Roughly, they relate to one another in the same way that
redschur( )
and
ophank( )
relate, that is, one focuses on balanced realization
truncation and the other on Hankel norm approximation. Some of the
similarities and differences are as follows:
•
When the errors are small, the error bound formula for
bst( )
is
about one half of that for
bst( )
.
•
With
bst( )
, the actual multiplicative error as a function of frequency
goes to zero as
ω→∞ (or, after using an optional transformation given
in the algorithm description, to zero as
ω→ 0); with
mulhank( )
, the
error tends to be more uniform as a function of frequency.
•
bst( )
can handle nonsquare reduction, while
mulhank( )
cannot.
•
Both algorithms are restricted to stable G(s); both preserve right half
plane zeros, that is, these zeros are copied into the reduced order
object; both have difficulties with j
ω-axis zeros of G(s), but these
difficulties are not insuperable.
bst( )
[SysR,HSV] = bst(Sys,{nsr,left,right,bound,method})
The
bst( )
function calculates a balanced stochastic truncation of
Sys
for
the multiplicative case.
G Gˆ
–
(
)Gˆ
1
–
δ
G Gˆ
–
(
)Gˆ
1
–
=
δ jω
( )
∞
Δ jω
( )
∞
1
Δ jω
( )
∞
–
-------------------------------
≤
Δ jω
( )
∞
δ jω
( )
∞
1
δ jω
( )
∞
–
-------------------------------
≤