Chapter 2
Additive Error Reduction
2-4
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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
(
j
ω
) 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
