National Instruments NI MATRIXx Xmath - Manual (Page 27)

Chapter 2 Additive Error Reduction
Xmath Model Reduction Module 2-4 ni.com
proper. So, even if all zeros are unstable, the maximum phase shift when
ω
moves from 0 to
∞ is (2n – 3)π/2. It follows that if G(jω) remains large in
magnitude at frequencies when the phase shift has moved past (2n – 3)
π/2,
approximation of G by G
r
will necessarily be poor. Put another way, good
approximation may depend somehow on removing roughly cancelling
pole-zeros pairs; when there are no left half plane zeros, there can be no
rough cancellation, and so approximation is unsatisfactory.
As a working rule of thumb, if there are p right half plane zeros in the
passband of a strictly proper G(s), reduction to a G
r
(s) of order less than
p + 1 is likely to involve substantial errors. For non-strictly proper G(s),
having p right half plane zeros means that reduction to a G
r
(s) of order less
than p is likely to involve substantial errors.
An all-pass function exemplifies the problem: there are n stable poles and
n unstable zeros. Since all singular values are 1, the error bound formula
indicates for a reduction to order n – 1 (when it is not just a bound, but
exact) a maximum error of 2.
Another situation where poor approximation can arise is when a highly
oscillatory system is to be replaced by a system with a real pole.
Reduction Through Balanced Realization Truncation
This section briefly describes functions that reduce( ) , balance( ) ,
and truncate( ) to achieve reduction.
• balmoore( )
— Computes an internally balanced realization of a
system and optionally truncates the realization to form an
approximation.
• balance( )
— Computes an internally balanced realization of a
system.
• truncate( )
— This function truncates a system. It allows
examination of a sequence of different reduced order models formed
from the one balanced realization.
• redschur( )
— These functions in theory function almost the same
as the two features of balmoore( ) . That is, they produce a
state-variable realization of a reduced order model, such that the
transfer function matrix of the model could have resulted by truncating
a balanced realization of the original full order transfer function
matrix. However, the initially given realization of the original transfer
function matrix is never actually balanced, which can be a numerically
hazardous step. Moreover, the state-variable realization of the reduced