Chapter 3
Multiplicative Error Reduction
3-16
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eigenvalues of A – B/D * C with the aid of
schur( )
. If any real part
of the
eigenvalues
is less than
eps
, a warning is displayed.
Next, a stabilizing solution
Q
is found for the following Riccati
equation:
The function
singriccati( )
is used; failure of the nonsingularity
condition of
G
(
j
ω
) will normally result in an error message. To obtain
the best numerical results,
singriccati( )
is invoked with the
keyword
method="schur"
.
The matrix
C
w
is given by
.
Notice that
Q
satisfies
, so that
P
and
Q
are
the controllability and observability grammians of
This strictly proper, stable transfer function matrix is the strictly
proper, stable part (under additive decomposition) of
θ
(
s
)=
W
–T
(
–s
)
G
(
s
), which obeys the matrix all pass property
θ
(
s
)
θ
'
(
–s
)
=I
. It is the
phase matrix
associated with
G
(
s
).
3.
The Hankel singular values
ν
i
of
are
computed, by calling
hankelsv( )
. The value of
nsr
is obtained if
not prespecified, either by prompting the user or by the error bound
formula ([GrA89], [Gre88], [Glo86]).
(3-3)
(with
ν
i
≥ ν
i
+ 1
≥ ⋅⋅⋅
being assumed). If
ν
k
=
ν
k
+ 1
= ... =
ν
k
+
r
for some
k
, (that is,
ν
k
has multiplicity greater than unity), then
ν
k
appears once
only in the previous error bound formula. In other words, the number
of terms in the product is equal to the number of distinct
ν
i
less than
ν
nsr
. There are restrictions on
nsr
.
nsr
cannot exceed the dimension
of a minimal realization of
G
(
s
); although
ν
i
≥
i
+ 1
⋅⋅⋅
,
nsr
must obey
n
nsr
>
n
nsr
+1
; and while 1
≥ ν
i
for all
i
, it is necessary that 1>
ν
nsr
+ 1
. (The
number of
ν
i
equal to 1 is the number of right half plane zeros of
G
(
s
).
They must be retained in
G
r
(
s
), so the order of
G
r
(
s
),
nsr
, must at least
be equal to the number of
ν
i
equal to 1.) The software checks all these
conditions. The minimum order permitted is the number of Hankel
QA A
′
Q
C B
′
w
Q
–
(
)′
DD
′
(
)
1
–
C B
w
′
Q
–
(
)
+
+
0
=
C
w
D
1
–
C B
′
w
Q
–
(
)
=
QA A
′
Q C
′
w
C
w
+
+
0
=
F s
( )
C
w
sI A
–
(
)
1
–
B
=
F s
( )
C
w
sI A
–
(
)
1
–
B
=
v
nsr
1
+
G
1
–
G G
r
–
(
)
∞
1
v
j
+
(
)
1
–
j
nsr
1
+
=
ns
∏
≤
≤
