1:4
1:2
1
0:8
0:6
pert( )
s
t
0:4
approx( )
s
t
0:2
( )
s
t
0
;0:2
0
1
2
3
4
5
6
7
8
9
10
t
When std
0 is replaced by 0 8 std
0 , the step response changes
Figure
9.6
P
:
P
from to pert. The rst order approximation of pert is given by approx( ) =
s
s
s
s
t
( ) 0 2 ( ).
s
t
;
:
s
t
1:6
1:4
1:2
1
() 0:8
t
s
0:6
0:4
0:2
0
;0:2
0
1
2
3
4
5
6
7
8
9
10
t
The sensitivity of the step response to plant gain changes is
Figure
9.7
shown for the controller (a). The rst order approximation of the step
K
response falls in the shaded envelope when std
0 is replaced by
std
0 , for
P
P
0 8
1 2.
:
:
9.3 GENERAL DIFFERENTIAL SENSITIVITY
207
9.3.2
Some Convex Approximations
In many cases there are useful convex approximations to speci cations that limit
general di erential sensitivities of the closed-loop system.
Consider the speci cation
( ) 0 75
for
0
(9.27)
js
t
j
:
t
which limits the sensitivity of the step response to gain variations in 0. This
P
speci cation is equivalent to
pk step 0 75
kS
T
k
:
which is not closed-loop convex. We will describe two convex approximations for
the nonconvex speci cation (9.27).
Suppose
min( )
( ) max( ) for
0
(9.28)
s
t
s
t
s
t
t
is a design speci cation (see gure 8.5). A weak approximation of the sensitivity
speci cation (9.26) (along with the step response speci cation (9.28)) is that a
typical (and therefore xed) step response satis es the speci cation:
typ
75
kS
T
kpk step
0:
where typ is the transfer function that has unit step response
T
) +
)
typ( ) = min(
max(
s
t
s
t
s
t
2
:
A stronger approximation of (9.27) (along with the step response speci ca-
tion (9.28)) requires that the sensitivity speci cation be met for every step response
that satis es (9.28):
max
min( )
( ) max( ) for
0
0 75
f
kS
v
k
j
s
t
v
t
s
t
t
g
:
:
1
This is an inner approximation of (9.27), meaning that it is tighter than (9.27).
208