User's Manual

Chapter 3 Multiplicative Error Reduction
© National Instruments Corporation 3-5 Xmath Model Reduction Module
These cases are secured with the keywords right and left, respectively.
If the wrong option is requested for a nonsquare G(s), an error message will
result.
The algorithm has the property that right half plane zeros of G(s) remain as
right half plane zeros of G
r
(s). This means that if G(s) has order nsr with n
+
zeros in Re[s] > 0, G
r
(s) must have degree at least n
+
, else, given that it has
n
+
zeros in Re[s] > 0 it would not be proper, [Gre88].
The conceptual basis of the algorithm can best be grasped by considering
the case of scalar G(s) of degree n. Then one can form a minimum phase,
stable W(s) with |W(jω)|
2
= |G(jω)|
2
and then an all-pass function (the phase
function) W
–1
(–s) G(s). This all pass function has a mixture of stable and
unstable poles, and it encodes the phase of G(jω). Its stable part has n
Hankel singular values σ
i
with σ
i
≤ 1, and the number of σ
i
equal to 1 is the
same as the number of zeros of G(s) in Re[s] > 0. State-variable realizations
of W,G and the stable part of W
–1
(–s)G(s) can be connected in a nice way,
and when the stable part of W
–1
(–s)G(s) has a balanced realization, we say
that the realization of G is stochastically balanced. Truncating the balanced
realization of the stable part of W
–1
(–s)G(s) induces a corresponding
truncation in the realization of G(s), and the truncated realization defines an
approximation of G. Further, a good approximation of a transfer function
encoding the phase of G somehow ensures a good approximation, albeit in
a multiplicative sense, of G itself.
Algorithm with the Keywords right and left
The following description of the algorithm with the keyword right is
based on ideas of [GrA86] developed in [SaC88]. The procedure is almost
the same when
left is specified, except the transpose of G(s) is used; the
algorithm finds an approximation in the same manner as for
right, but
transposes the approximation to yield the desired G
r
(s).
1. The algorithm checks
• That the system is state-space, continuous, and stable
• That a correct option has been specified if the plant is nonsquare
• That D is nonsingular; if the plant is nonsquare, DD´ must be
nonsingular