9.1.3
Example: Gain Variation
Our rst example concerns a change in gain, i.e.
0( ) =
0( )
(9.10)
P
s
P
s
so that
log 0( )
P
j
!
'
:
Hence from (9.8{9.9) we have
log ( )
( )
(9.11)
jT
j
!
j
'
<S
j
!
( )
( )
(9.12)
6
T
j
!
'
=S
j
!
:
It follows, for example, that the closed-loop convex speci cation
( 0) = 0
<S
j
!
guarantees that the magnitude of the I/O transfer function at the frequency 0 is
!
rst order insensitive to variations in .
To give a speci c example that compares the rst order deviation in ( ) to
jT
j
!
j
the real deviation, we take the standard example plant and controller (a) described
K
in section 2.4, and consider the e ect on ( ) of a gain perturbation of = 25%,
jT
j
!
j
which is about 2dB. Figure 9.1 shows:
std
(a)
( ) =
0 ( )
( )
P
j
!
K
j
!
jT
j
!
j
1 + std
0 ( ) (a)( )
P
j
!
K
j
!
which is the nominal magnitude of the I/O transfer function
pert
std
(a)
( ) = 1 25 0 ( )
( )
:
P
j
!
K
j
!
jT
j
!
j
1 + 1 25 std
0 ( ) (a)( )
:
P
j
!
K
j
!
which is the actual magnitude with the 25% gain increase in std
0
and
P
approx( ) = ( ) exp 0 25
1
jT
j
!
j
jT
j
!
j
:
<
1 + std
0 ( ) (a)( )
(9.13)
P
j
!
K
j
!
which is the magnitude of the perturbed I/O transfer function predicted by the rst
order perturbation formula (9.11). For this example, the rst order prediction gives
a good approximation of the perturbed magnitude of the I/O transfer function,
even for this 2dB gain change in std
0 .
P
9.1 BODE’S LOG SENSITIVITIES
199
10
5
0
;5
dB ;10
( )
p
ert
jT
j
!
j
;15
( )
appro
x
jT
j
!
j
;20
( )
jT
j
!
j
;25
;30
0:1
1
10
!
When
is replaced by 1 25
, the I/O transfer function's
std
std
Figure
9.1
P
:
P
0
0
magnitude changes from
to pert . approx is a rst order approxima-
jT
j
jT
j
jT
j
tion of
computed from (9.13). In this example the eect of a plant
p
ert
jT
j
gain change as large as 25% is well approximated using the dierential sen-
sitivity.
9.1.4
Example: Phase Variation
In this example we study the e ects on of a phase variation in 0, i.e.,
T
P
0( ) + 0( ) =
( )
)
j
!
0(
P
j
!
P
j
!
e
P
j
!
:
In this case we have, from (9.8{9.9),
log ( )
( ) ( )
(9.14)
jT
j
!
j
'
;
!
=S
j
!
( )
( ) ( )
(9.15)
6
T
s
'
!
<S
j
!
:
To guarantee that ( 0) is, for example, rst order insensitive to phase variations
jT
j
!
j
in 0( 0), we have the speci cation
P
j
!
( 0) = 0
=S
j
!
which is closed-loop a ne.
We now consider a speci c example that compares the actual e ect of a phase
variation in 0 to the e ect predicted by the rst order perturbational analy-
P
sis (9.14). As above, our plant is the standard example described in section 2.4,
together with the same controller (a). The speci c perturbed std
0 is
K
P
std
5
0 ( ) +
std
0 ( ) = 1 ; s
P
s
P
s
2 5 +
s
s
200