Page 267 - Modern Control Systems

Section 4.3 Sensitivity of Control Systems to Parameter Variations 241

Very often, the transfer function of the system T(s) is a fraction of the form [1]
N(s, a)
T(s, a) = (4.15)
D(s,ay
where a is a parameter that may be subject to variation due to the environment.
Then we may obtain the sensitivity to a by rewriting Equation (4.11) as
d In T d In TV dlnD
iJ/v — Or, (4.16)
din a din ex d In a
«0 «0
is the nominal value of the parameter.
where a 0
An important advantage of feedback control systems is the ability to reduce the effect
of the variation of parameters of a control system by adding a feedback loop. To obtain
highly accurate open-loop systems, the components of the open-loop, G(s), must be
selected carefully in order to meet the exact specifications. However, a closed-loop
system allows G(s) to be less accurately specified, because the sensitivity to changes or
errors in G(s) is reduced by the loop gain L(s).This benefit of closed-loop systems is a
profound advantage for the electronic amplifiers of the communication industry. A
simple example will illustrate the value of feedback for reducing sensitivity.

EXAMPLE 4.1 Feedback amplifier
An amplifier used in many applications has a gain -K a, as shown in Figure 4.4(a).
The output voltage is
VQ = -K av in. (4.17)
We often add feedback using a potentiometer R p, as shown in Figure 4.4(b). The
transfer function of the amplifier without feedback is
T = -K a, (4.18)
and the sensitivity to changes in the amplifier gain is
T
S K. = 1. (4.19)
The block diagram model of the amplifier with feedback is shown in Figure 4.5,
where

)8 (4.20)

+ o o + ^inO— f

°~x nr-
FIGURE 4.4
(a) Open-loop
amplifier.
(b) Amplifier with
feedback. (a) (b)