Some Comments on Reduction of
Parameter Variations by Use of
Feedback
Hongcai Zhang
Richard S. Marleau
Senior Member, IEEE
Summary: T ~ vformulae
o
regarding reduction of output variation ofan elementaql
feedback system due to variations in system
parameters are derived and discussed. Certain confusing and often erroneously treated
aspects are detailed and illustrated.
If AG(s)isassumedsmallcomparedto
1 + G(s)inthedenominator,thenthedisturbance in the output is
Obviously,theresult
of equation (4) is
a b e t t e ra p p r o x i m a t i o nt h a nt h a to f
equation (2). Equation (2) is essentiallyof
no valueandshouldbeconsideredanincorrect approximation.
Conclusion
Both authors, one from The People’s Republic of China and the other from the United
Realizing how readilythis error was made
States, have often encountered student diffiin quality textbooks suggests that special care
which is sometimes accepted as the measure
culty in understanding how feedback reduces of the system response error.
should be taken in presenting this topic to
the incremental output variations caused by
persons learning feedback concepts for the
Starting again with equation (2), but subincremental variations in system parameters.
stituting 6G(s)for AG(s) in the denominator, fist time. This reaffirms the continuing care
There exist two different approximation solu- we have
thatmust be takenindealingwithsmall
tions in texthooks [ 1 2,3] in use today.This
difference was eliminatedin later editions of
[2,3]. Rather than start out with the concept
of systemsensitivity, the ultimate tool for
such investigations, this presentation follows
[l G(s)] G(s) AG(s) - G(s) [l + G(s) + ~G(s)]
the leadof the textbooks used in introductory
1 + G(s) + 6G(s)
feedback control-system courses.
Givenaunityfeedbacksystemwith
- [I + G(s)]AG(s) - G(s) 6G(s)
(3)
forward-path gain G(s), input R(s) and out[l + G(s) f 6G(s)] [ 1 f G(s)]
put C(s), let us suppose that G(s)is subject to
an incremental change AG(s), this increment
being of smallamplitudecomparedto
that of G(s).
differences in large values, especially when
Assuming 6G(s) = AG(s),and thatboth
For the closed-loop, feedback system sub- these terms are small compared to G(S), we
approximations are involved.
ject to variations AG(s) in G(s), the differhave
ence between the output of the system with
References
the perturbed parameter G(s) + AG(s) and
[
I
]
K.
Ogata,
“Moderrz
Control Engineering,”
that from the system with nominal parameter
1
Englewood
Cliffs,
NJ: Prentice-Hall,
AC
=
G(s) is
[I + G ( S ) ] ’ ~ ( ~ )
Inc.,1970.
[2] R. C. Dorf, “ModernControl Systems,”
G(s) + AG(s)
AC,(S) =
Addison-WesleyPublishingCompany,
1 + G(s) + AG(s)
Inc., 1967; Second Edition, 1974; Thirdmiwhich does not agree with equation (2).
tion,1980.
The
result
of
equation
(2)
can
be
obtained
- 1 + G(s) *R(s)
(1)
[3] B.C. Kuo, “AutomaticControlSystems,”
fromequation (4)by assuming G ( s ) = 0
Englewood Cliffs, NJ: Prentice-Hall,
andG(s) 4 G( s ) . T h e d i f f e r e n c e i n
Inc., 1962; Second Edition, 1967; Third Ediequations (2) and (4) results from erroneous
tion,1975.
ReceivedMay 31, 1983;revisedSeptember23,
subtraction to obtain the small difference in
1983,andNovember23,1983.Accepted
in re[4] J . B . Cruz, Jr., “FeedbackSystems,” New
vised form by Associate Editor G . H. Hostetter.
two “large numbers.”
York: McGraw-Hill Book Company, 1972.
~
+
a]
0272-1708/84/0joo-0317$01.03 0 1984 IEEE
may I984
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