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Chapter 4
Frequency-Weighted Error Reduction
4-20
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Additional Background
A discussion of the stability robustness measure can be found in [AnM89]
and [LAL90]. The idea can be understood with reference to the transfer
functions E(s) and E
and [LAL90]. The idea can be understood with reference to the transfer
functions E(s) and E
r
(s) used in discussing
type="right perf"
. It is
possible to argue (through block diagram manipulation) that
•
C(s) stabilizes P(s) when E(s) stabilizes (as a series compensator) with
unity negative feedback
unity negative feedback
.
•
E
r
(s) also will stabilize [P(s)I], and then C
r
(s) will stabilize P(s),
provided
(4-14)
Accordingly, it makes sense to try to reduce E by frequency-weighted
balanced truncation. When this is done, the controllability grammian for
E(s) remains unaltered, while the observability grammian is altered. (Hence
Equation 4-5, at least with Q
balanced truncation. When this is done, the controllability grammian for
E(s) remains unaltered, while the observability grammian is altered. (Hence
Equation 4-5, at least with Q
yy
= I, and Equation 4-12 are the same while
Equation 4-6 and Equation 4-13 are quite different.) The calculations
leading to Equation 4-13 are set out in [LAL90].
leading to Equation 4-13 are set out in [LAL90].
The argument for
type="left perf"
is dual. Another insight into
Equation 4-14 is provided by relations set out in [NJB84]. There, it is
established (in a somewhat broader context) that
established (in a somewhat broader context) that
The left matrix is the weighting matrix in Equation 4-14; the right matrix is
the numerator of C(j
the numerator of C(j
ω) stacked on the denominator, or alternatively
E(j
ω) +
This formula then suggests the desirability of retaining the weight in the
approximation of E(j
approximation of E(j
ω) by E
r
( j
ω).
Pˆ s
( )
P s
( ) I
=
C j
ωI A
–
K
E
C
+
(
)
1
–
B I C j
ωI A
–
K
E
C
+
(
)
1
–
K
E
–
E j
ω
( ) E
e
j
ω
( )
–
[
]
∞
1
<
C j
ωI A
–
K
E
C
+
(
)
1
–
B
I C j
ωI A
–
K
E
C
+
(
)
1
–
K
E
–
{
}
K
r
sI A BK
R
+
–
(
)
1
–
K
E
I C j
ωI A
–
BK
R
+
(
)
1
–
K
E
+
×
I
=
0
I