T
0 + 0
ole
p
y
P
P
q
p
u
y
+
K
0
r
0
P
;
q
r
pert
T
pert
q
pert
u
+
0
r
p
y
K
0 + 0
P
P
;
In the open-loop equivalent system, shown at top, the actuator
Figure
9.5
signal drives 0+ 0, so there is no feedback around the perturbation 0.
u
P
P
P
The benet of feedback can be seen by comparing the rst order changes in
the transfer matrices from to ole
p and pert
p , respectively (see (9.23)).
r
y
y
Comparing this to the rst order change in the I/O transfer matrix, given in (9.19),
we have
ole
(9.23)
T
'
S
T
:
This simple equation shows that the rst order variation in the I/O transfer
matrix is equal to the sensitivity transfer matrix times the rst order variation in
the open-loop equivalent system. It follows that the speci cation (9.22) can be
interpreted as limiting the sensitivity of the I/O transfer matrix to be no more than
5% of the sensitivity of I/O transfer matrix of the open-loop equivalent system.
9.3
General Differential Sensitivity
The general expression for the rst order change in the closed-loop transfer matrix
due to a change in the plant transfer matrix is
H
+
(
) 1
+
(
) 1
(9.24)
;
;
H
'
P
P
K
I
;
P
K
P
P
K
I
;
P
K
P
z
w
z
u
y
u
y
w
z
u
y
u
y
w
+
(
) 1
(
) 1
;
;
P
K
I
;
P
K
P
K
I
;
P
K
P
:
z
u
y
u
y
u
y
u
y
w
9.3 GENERAL DIFFERENTIAL SENSITIVITY
205
The last term shows that
(which is a complicated object with four indices)
@
H
=@
P
y
u
has components that are given by the product of two closed-loop transfer functions.
It is usually the case that design speci cations that limit the size of
are
@
H
=@
P
y
u
not closed-loop convex, since, roughly speaking, a product can be made small by
making either of its terms small.
9.3.1
An Example
Using the standard example plant and controller (a), described in section 2.4, we
K
consider the sensitivity of the I/O step response with respect to gain variations in
std
0 . Since
std
0 =
std
0 , the sensitivity
P
P
P
( ) = ( )
@
s
t
s
t
=0
@
is simply the unit step response of the transfer function
std
0
(a)
P
K
(1 + std
(9.25)
0
(a))2 = ST:
P
K
This transfer function is the product of two closed-loop transfer functions, which is
consistent with our general comments above.
In gure 9.6 the actual e ect of a 20% gain reduction in std
0 on the step response
P
is compared to the step response predicted by the rst order perturbational analysis,
approx( ) = ( ) 0 2 ( )
s
t
s
t
;
:
s
t
with the controller (a). The step response sensitivity with this controller is shown
K
in gure 9.7. For plant gain changes between 20%, the rst order approximation
to the step response falls in the shaded envelope ( ) 0 2 ( ).
s
t
:
s
t
We now consider the speci cation
(1) 0 75
(9.26)
js
j
:
which limits the sensitivity of the step response at time = 1 to gain variations in
t
std
0 . We will show that this speci cation is not convex.
P
The controller (a) yields a closed-loop transfer matrix (a) with (a)(1) =
K
H
s
0 697, so (a) satis es the speci cation (9.26). The controller (b) yields a closed-
:
H
K
loop transfer matrix (b) with (b)(1) = 0 702, so (b) also satis es the speci ca-
H
s
:
H
tion (9.26). However, the average of these two transfer matrices, ( (a) + (b)) 2,
H
H
=
has a step response sensitivity at = 1 of 0 786, so ( (a) + (b)) 2 does not satisfy
t
:
H
H
=
the speci cation (9.26). Therefore the speci cation (9.26) is not convex.
206