Feedback and Sensitivity
The ability of feedback to make a system less sensitive to changes in the plant is discussed
in essentially every book on feedback and control see Mayr May70] for a history of
this idea. An early discussion (in the context of feedback ampliers) can be found in
Black Bla34], in which we nd:
...by building an amplier whose gain is deliberately made, say 40dB higher
than necessary, and then feeding the output back on the input in such a
way as to throw away the excess gain, it has been found possible to eect
extraordinary improvement in constancy of amplication ... By employing
this feedback principle, ampliers have been built and used whose gain varied
less than 0.01dB with a change in plate voltage from 240V to 260V whereas]
for an amplier of conventional design and comparable size this change would
have been 0.7dB.
For a later discussion see Horowitz Hor63, ch3]. A concise discussion appears in chapter
1,
, of Callier and Desoer CD82a].
On
the
A
dvantages
of
F
e
e
db
ack
Differential Sensitivity
Bode Bod45] was the rst to systematically study the eect of small (dierential) changes
in closed-loop transfer functions due to small (dierential) changes in the plant. On page
33 of Bod45] we nd (with our corresponding notation substituted),
The variation in the nal gain characteristic ] in dB, per dB change in the
T
gain of 0], is reduced in the ratio ].
P
S
A recent exposition of dierential sensitivity can be found in chapter 3 of Lunze Lun89].
Comparison Sensitivity
The notion of comparison sensitivity was introduced by Cruz and Perkins in CP64]
see also the book edited by Cruz Cru73]. The idea of an open-loop equivalent system,
however, is older. In NGK57, 1.7], it is called the
of
x
e
quivalent
c
asc
ade
c
on
gur
ation
the control system. Recent discussions of comparison sensitivity can be found in Callier
and Desoer CD82a, ch1] and Anderson and Moore AM90, 5.3].
x
Sensitivity Specifications that Limit Control System Performance
The idea that sensitivity or robustness specications can limit the achievable control system
performance is explicitly expressed in,
, Newton, Gould, and Kaiser NGK57, p23]:
e.g.
Control systems often employ mechanical, hydraulic, or pneumatic elements
which have less reproducible behavior than high quality electric circuit ele-
ments. This practical problem often causes the control designer to stop short
of an optimum design because he knows full well that the parameters of the
physical system may deviate considerably from the data on which he bases
his design.
A more recent paper that raised this issue, in the context of regulators designed by state-
space methods, is Doyle and Stein DS81].
