National Instruments NI MATRIXx Xmath - Manual (Page 47)

© National Instruments Corporation 3-1 Xmath Model Reduction Module
3
Multiplicative Error Reduction
This chapter describes multiplicative error reduction presenting
two reasons to consider multiplicative rather than additive error reduction,
one general and one specific.
Selecting Multiplicative Error Reduction
The general reason to use multiplicative error reduction is that many
specifications are given using decibels; ±1 db corresponds to a
multiplicative error of about 12%. Specifications regarding phase shift also
can be regarded as multiplicative error statements: ±0.05 radians of phase
shift is like 5% multiplicative error also.
The more specific reason arises in considering the problem of plant
approximation, with a high order (possibly very high order) plant being
initially prescribed, with no controller having been designed, and with a
requirement to provide a simpler model of the plant, possibly to allow
controller design. Consider the arrangement of Figure 3-1, controller C(s)
designed for G(s)j with G(s) = (I +
Δ)G(s).
Figure 3-1. Controller C(s) Designed for Multiplicative Error Reduction
The full order plant is G = (I +
Δ)G, and the reduced order model is .
Since , this means that
Δ is the multiplicative error.
Another way one could measure the multiplicative error would be as
. In the matrix plant case, interchange of the order of the
product gives two more possibilities again.
The following multiplicative robustness result can be found in [Vid85].
C Gˆ
Δ
Gˆ
G Gˆ –
( )Gˆ 1 – Δ
=
G Gˆ –
( )Gˆ 1 –