National Instruments Xmath Truncation of Balanced Realizations

Chapter 2 Additive Error Reduction

Xmath Model Reduction Module 2-2 ni.com

Truncation of Balanced Realizations

A group of functions can be used to achieve a reduction through truncation

of a balanced realization. This means that if the original system is

(2-1)

and the realization is internally balanced, then a truncation is provided by

The functions in question are:

• balmoore( )

•balance( ) (refer to the Xmath Help)

• truncate( )

• redschur( )

One only can speak of internally balanced realizations for systems which

are stable; if the aim is to reduce a transfer function matrix G(s) which

contains unstable poles, one must additively decompose it into a stable part

and unstable part, reduce the stable part, and then add the unstable part back

in. The function stable( ), described in Chapter 5, Utilities, can be used

to decompose G(s). Thus:

G(s) = Gs(s) + Gu(s)(Gs(s) stable, Gu(s) unstable)

Gsr(s) = found by algorithm (reduction of Gs(s))

Gr(s) = Gsr(s) + Gu(s) (reduction of G(s))

·1

·2

A11 A12

A21 A22

u+=

yC1C2xD

·1A11x1B1u+=

1x1Du+=