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Algorithm with the Keywords right and left
© National Instruments Corporation
These cases are secured with the keywords
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
zeros in Re[s] > 0, G
(s) must have degree at least n
r
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(j )|
–1
function) W
(–s) G(s). This all pass function has a mixture of stable and
unstable poles, and it encodes the phase of G(j ). Its stable part has n
Hankel singular values
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
and when the stable part of W
that the realization of G is stochastically balanced. Truncating the balanced
realization of the stable part of W
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.
The following description of the algorithm with the keyword
based on ideas of [GrA86] developed in [SaC88]. The procedure is almost
is specified, except the transpose of G(s) is used; the
the same when
left
algorithm finds an approximation in the same manner as for
transposes the approximation to yield the desired G
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
Chapter 3
right
(s). This means that if G(s) has order nsr with n
r
2
2
and then an all-pass function (the phase
= |G(j )|
with
1, and the number of
i
i
–1
(–s)G(s) can be connected in a nice way,
–1
(–s)G(s) has a balanced realization, we say
–1
(–s)G(s) induces a corresponding
3-5
Multiplicative Error Reduction
and
, respectively.
left
, else, given that it has
+
equal to 1 is the
i
is
right
, but
right
(s).
r
Xmath Model Reduction Module
+
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