Algorithm with the keywords right and left, Algorithm with the keywords right and left -5 – National Instruments NI MATRIXx Xmath User Manual

Page 51

Chapter 3

Multiplicative Error Reduction

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

(s). This means that if G(s) has order nsr with n

zeros in Re[s] > 0, G

(s) must have degree at least n

, else, given that it has

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

ω)|

= |G(j

ω)|

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(j

ω). Its stable part has n

Hankel singular values

with

≤ 1, and the number of σ

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

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

(s).

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