is equivalent to the closed-loop convex speci cation
0( ) ( )
;P
j
!
K
j
!
1 +
(9.18)
0( ) ( )
2 for
bw
!
!
:
P
j
!
K
j
!
In (9.17) we might interpret
as the command input to tracking error transfer
;S
function, so that the speci cation (9.17), and hence (9.18), limits the logarithmic
sensitivity of the command input to tracking error transfer function with respect to
changes in 0.
P
9.2
MAMS Log Sensitivity
It is possible to generalize Bode's results to the MAMS case. We consider the
MAMS 2-DOF control system (see section 2.3.4), with I/O transfer matrix
= ( + 0 ~) 1
;
0
T
I
P
K
P
K
:
y
r
If the plant is perturbed so that 0 becomes 0 + 0, we have
P
P
P
+ = ( + ( 0 + 0) ~) 1
;
( 0 + 0)
T
T
I
P
P
K
P
P
K
y
r
Retaining only rst order terms in 0 we have
P
( + 0 ~) 1
( +
) 1
( +
) 1
;
0
0 ~ ;
0 ~
0 ~ ; 0
T
'
I
P
K
P
K
;
I
P
K
P
K
I
P
K
P
K
y
r
y
y
y
r
= ( + 0 ~) 1
1
;
0( + ~ 0);
I
P
K
P
I
K
P
K
y
y
r
=
0( + ~ 0) 1
;
(9.19)
S
P
I
K
P
K
y
r
where = ( + 0 ~) 1 is the (output-referred) sensitivity matrix of the MAMS
;
S
I
P
K
y
2-DOF control system.
Now suppose that we can express the change in 0 as
P
0 =
frac
0
0
P
P
P
:
We can interpret frac
0 as an output-referred fractional perturbation of 0, as shown
P
P
in gure 9.4. Then from (9.19) we have
frac
0
0( + ~ 0) 1
frac
;
=
0
T
'
S
P
P
I
K
P
K
S
P
T
y
r
so that
frac
T
'
T
T
where frac
frac
0
(9.20)
T
'
S
P
:
This is analogous to (9.2): it states that the output-referred fractional change in
the I/O transfer matrix is, to rst order, the sensitivity matrix times the
T
S
output-referred fractional change in 0.
P
9.2 MAMS LOG SENSITIVITY
203
+
q
r
p
y
u
0
P
+
frac
0
P
An output-referred fractional perturbation of the transfer ma-
Figure
9.4
trix 0. In the SASS case, 0
log 0.
P
P
P
The design speci cation,
max( frac( )) 0 01 for
bw max( frac
)) 0 20
(9.21)
<
0 (
T
j
!
:
!
!
P
j
!
:
which limits the rst order fractional change in to 1% over the bandwidth bw,
T
!
despite variations in 0 of 20%, is therefore equivalent to the closed-loop convex
P
speci cation
max( ( )) 0 05 for
bw
(9.22)
S
j
!
:
!
!
:
We remind the reader that the inequality in (9.21) holds only to rst order in
frac
0 : its precise meaning is
P
frac
lim max(
( )) 0 01
T
j
!
:
0
bw
max( frac
)) 0 20 for ! ! :
P
!
0 (
:
P
j
!
0
9.2.1
Cruz and Perkins’ Comparison Sensitivity
Cruz and Perkins gave another generalization of Bode's log sensitivity to the MAMS
2-DOF control system using the concept of comparison sensitivity, in which the
perturbed closed-loop system is compared to an equivalent open-loop system.
The open-loop equivalent system consists of 0+ 0 driven by the unperturbed
P
P
actuator signal, as shown at the top of gure 9.5. For 0 = 0, the open-loop
P
equivalent system is identical to the closed-loop system shown at the bottom of
gure 9.5. For 0 nonzero, however, the two systems di er. By comparing the rst
P
order changes in these two systems, we can directly see the e ect of the feedback
on the perturbation 0.
P
The transfer matrix of the open-loop equivalent system is
ole = ( 0 + 0)( + ~ 0) 1
;
T
P
P
I
K
P
K
y
r
so that
ole = 0( + ~ 0) 1
;
T
P
I
K
P
K
:
y
r
204