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 Gr 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 Gr(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 Gr(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 tran sfer
function matrix is never actually balanced, which can be a numerically
hazardous step. Moreover, the state-variable realization of the reduced