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Section 4.3 Sensitivity of Control Systems to Parameter Variations 239
4.3 SENSITIVITY OF CONTROL SYSTEMS TO PARAMETER VARIATIONS
A process, represented by the transfer function G(s), whatever its nature, is subject
to a changing environment, aging, ignorance of the exact values of the process para-
meters, and other natural factors that affect a control process. In the open-loop sys-
tem, all these errors and changes result in a changing and inaccurate output.
However, a closed-loop system senses the change in the output due to the process
changes and attempts to correct the output. The sensitivity of a control system to pa-
rameter variations is of prime importance. A primary advantage of a closed-loop
feedback control system is its ability to reduce the system's sensitivity [1^4,18].
For the closed-loop case, if G c(s)G(s) » 1 for all complex frequencies of inter-
est, we can use Equation (4.2) to obtain (letting T d(s) = 0 and N(s) = 0)
Y(s) ^ R(s).
The output is approximately equal to the input. However, the condition G c($)G(s) >>> 1
may cause the system response to be highly oscillatory and even unstable. But the fact
that increasing the magnitude of the loop gain reduces the effect of G(s) on the output
is an exceedingly useful result. Therefore, the first advantage of a feedback system is
that the effect of the variation of the parameters of the process, G(s), is reduced.
Suppose the process (or plant) G(s) undergoes a change such that the true plant
model is G(s) + AG(s). The change in the plant may be due to a changing external
environment or natural aging, or it may just represent the uncertainty in certain
plant parameters. We consider the effect on the tracking error E(s) due to AG(s).
Relying on the principle of superposition, we can let T d(s) = N(s) = 0 and consid-
er only the reference input R(s), From Equation (4.3), it follows that
1 + G c(s){G{s) + AG{s))
Then the change in the tracking error is
, -G c(s) AG(s)
AE( =
{S) {S)
(1 + G c(s)G(s) + G c(s) AG(s))(l + G c(s)G(s)) '
Since we usually find that G c(s)G(s) » G c(s) AG(s), we have
-G c(s) AG(s) n/ x
AE(s) « ^-R(s).
W 2
(1 + L(s))
We see that the change in the tracking error is reduced by the factor 1 + L(s),
which is generally greater than 1 over the range of frequencies of interest.
For large L(s), we have 1 + L(s) ~ L(s), and we can approximate the change
in the tracking error by
1 AG(J)
A£W x m (49)
~W) m -
Larger magnitude L{s) translates into smaller changes in the tracking error (that is,
reduced sensitivity to changes in AG(s) in the process). Also, larger L{s) implies