11.4
Sensitivity Specifications
11.4.1
A Log Sensitivity Specification
We consider the plant perturbation
std
0 ( ) =
std
0 ( )
P
s
P
s
, a gain variation in std
0 (see section 9.1.3). Figure 11.13 shows the level curves of
i.e.
P
the maximum logarithmic sensitivity of the magnitude of the I/O transfer function
13, over the frequency range 0
1, to these gain changes, ,
H
!
i.e.
sup @
log
) = max
( )
(11.12)
0
1
13(
=0
jH
j
!
j
0
1 j<S j! j
!
@
!
where
( ) = 1
(a)
(b)
(c)
13 ( ) +
13 ( ) + (1
) 13 ( )
S
j
!
;
H
j
!
H
j
!
;
;
H
j
!
:
As expected, the level curves in gure 11.13 bound convex subsets of slice.
H
2
0:8
0:7
0:6
0:5
0:4
1:5
0:3
0:2
0:1
1
0:5
0:4
0
0:5
0:6
;0:5
0:7
0:8
;1
;1
;0:5
0
0:5
1
1:5
2
Level curves of the logarithmic sensitivity of the magnitude
Figure
11.13
of the I/O transfer function
, over the frequency range 0
1, to
H
!
13
gain changes in the plant
, given by (11.12).
std
P
0
When the function (11.12) takes on the value 0.3, the maximum rst order
change in 13( ) , over 0
1, with a 25% plant gain change is exp(0 075),
jH
j
!
j
!
:
or 0 65dB. In gure 11.14 the actual maximum change in 13( ) is shown for
:
jH
j
!
j
points on the 0 3 contour of the function (11.12).
:
11.4 SENSITIVITY SPECIFICATIONS
261
2
q
0 85dB
:
0 63dB
:
q
1:5
0 58dB
:
q
0 55dB
:
q
1
0 53dB
:
q
0:5
0 63dB
:
q
0 51dB
:
0 54dB
q
0
0 35dB
:
q
:
q
0 39dB
:
q
;0:5
0 57dB
:
q
;1
;1
;0:5
0
0:5
1
1:5
2
To rst order, the peak change in 13( ) for 0
1
Figure
11.14
jH
j
!
j
!
along the 0.3 contour in gure 11.13, for a 25% gain change in std
0 , will be
P
0 65dB. The 0.3 contour from gure 11.13 is shown, together with the actual
:
peak change in 13( ) for 0
1 for several points on the contour.
jH
j
!
j
!
11.4.2
A Step Response Sensitivity Specification
In section 9.3 we considered the sensitivity of the I/O step response at t = 1 to
plant gain changes, i.e., Pstd
0 = Pstd
0 :
s (1) = @s(1)
@
:
=0
Figure 11.15 shows the subset of slice for which
H
s (1) 0:75:
j
j
This speci cation is equivalent to
1 Z 1 (1 T(j!))T(j!)
;
2
j!
ej! d! 0:75
(11.13)
;1
where
T(j!) = H(a)
13 (j!) + H(b)
13 (j!) + (1
)H(c)
13 (j!):
;
;
As we showed in section 9.3, and as is clear from gure 11.15, the step response
sensitivity speci cation (11.13) is not convex.
262