Chapter 2
Additive Error Reduction
©
National Instruments Corporation
2-15
Algorithm
The algorithm does the following. The system
Sys
and the reduced order
system
SysR
are stable; the system
SysU
has all its poles in
Re
[
s
] > 0. If
the transfer function matrices are
G
(
s
),
G
r
(
s
) and
G
u
(
s
) then:
•
G
r
(
s
) is a stable approximation of
G
(
s
).
•
G
r
(
s
) +
G
u
(
s
) is a more accurate, but not stable, approximation of
G
(
s
),
and optimal in a certain sense.
Of course, the algorithm works with state-space descriptions; that of
G
(
s
)
can be minimal, while that of
G
r
(
s
) cannot be.
These statements are explained in the
section. If
onepass
is
specified, reduction is calculated in one pass. If
onepass
is not called or is
set to 0
{onepass=0}
, reduction is calculated in (number of states of
Sys - nsr
) passes. There seems to be no general rule to suggest which
setting produces the more accurate approximation
G
r
. Therefore, if
accuracy of approximation for a given order is critical, both should be tried.
As noted previously, if an approximation involving an unstable system is
desired, the default
{onepass=1}
is specified.
Behaviors
The following explanation deals first with the keyword
{onepass}
.
Suppose that
σ
1
,
σ
2
,...,
σ
ns
are the Hankel Singular values of
S
, which has
transfer function matrix
G
(
s
). Suppose that the singular values are ordered
so that:
Thus, there are
n
1
equal values, followed by
n
2
–
n
1
equal values, followed
by
n
3
–
n
2
equal values, and so forth.
The order
nsr
of
G
r
(
s
) cannot be arbitrary when there are equal Hankel
singular values. In fact, the orders shown in Table 2-1 for the strictly stable
G
r
(all poles in
Re
[
s
]<0) and strictly unstable
G
u
(all poles
Re
[
s
]>0) are
possible (and there are no other possibilities).
σ
1
σ
2
...
σ
n
1
=
=
=
σ
n
1
1
+
...
>
σ
n
1
1
+
...
σ
n
2
σ
n
2
1
+
...
>
=
=
σ
n
m
1
–
1
+
σ
n
m
1
–
2
+
σ
n
m
=
σ
ns
(
)
0
≥
=
=
>
