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 Gr(s). This means that if G(s) has order nsr with n+
zeros in Re[s] > 0, Gr(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 σi with σi ≤ 1, and the number of σi 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 Gr(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