User's Manual

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