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