Right and left, Right and left -15 – National Instruments NI MATRIXx Xmath User Manual
Page 61
Chapter 3
Multiplicative Error Reduction
© National Instruments Corporation
3-15
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)|
2
= |G(j
ω)|
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(j
ω). 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. The algorithm computes an additive Hankel norm reduction of
the stable part of W
–1
(–s)G(s) to cause a degree reduction equal to the
multiplicity of the smallest
σ
i
. The matrices defining the reduced order
object are then combined in a new way to define a multiplicative
approximation to G(s); as it turns out, there is a close connection between
additive reduction of the stable part of W
–1
(–s)G(s) and multiplicative
reduction of G(s). The reduction procedure then can be repeated on the new
phase function of the just found approximation to obtain a further reduction
again in G(s).
right and left
A description of the algorithm for the keyword
right
follows. It is based
on ideas of [Glo86] in part developed in [GrA86] and further developed
in [SaC88]. The procedure is almost the same when
{left}
is specified,
except the transpose of G(s) is used; the following algorithm finds an
approximation, then transposes it to yield the desired G
r
(s).
1.
The algorithm checks that G(s) is square, stable, and that the transfer
function is nonsingular at infinity.
2.
With G(s) = D + C(sI–A)
–1
B square and stable, with D nonsingular
[
rank(d)
must equal number of rows in d] and G(j
ω) nonsingular for
all finite
ω, this step determines a state variable realization of a
minimum phase stable W(s) such that,
W´(–s)W(s) = G(s)G´(–s)
with:
W(s) = D
w
+ C
w
(sI–A
w
)
–1
B
w
The various state variable matrices in W(s) are obtained as follows. The
controllability grammian P associated with G(s) is first found from
AP + PA´ + BB´ = 0, then:
A
w
= AB
w
= PC´+BD´D
w
= D´
The algorithm checks to see if there is a zero or singularity of G(s)
close to the j
ω-axis. The zeros are determined by calculating the