National Instruments Xmath bst( )

Chapter 3 Multiplicative Error Reduction

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( ),

amnemonic 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 jω-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

ˆ1–

δGG

–()G

ˆ1–

δjω()

∞

Δjω()

∞

1Δjω()

∞

–

-------------------------------

≤

Δjω()

∞

δjω()

∞

1δjω()

∞

–

-------------------------------

≤