Page 279 - Modern Control Systems

Section 4.7 The Cost of Feedback 253

or 10%. By contrast, the steady-state error of the closed-loop system, with
AK/K = 0.1, is e c(oo) = 1/91 if the gain decreases. Thus, the change is

Ae (00) (4 55)
< = w ~ h -

and the relative change is

i , \ / = 0.0011, (4.56)
KOI
or 0.11%.This is a significant improvement, since the closed-loop relative change is
two orders of magnitude lower than that of the open-loop system.

4.7 THE COST OF FEEDBACK

Adding feedback to a control system results in the advantages outlined in the previ-
ous sections. Naturally, however, these advantages have an attendant cost. The first
cost of feedback is an increased number of components and complexity in the sys-
tem. To add the feedback, it is necessary to consider several feedback components;
the measurement component (sensor) is the key one. The sensor is often the most
expensive component in a control system. Furthermore, the sensor introduces noise
and inaccuracies into the system.
The second cost of feedback is the loss of gain. For example, in a single-loop sys-
tem, the open-loop gain is G c(s)G(s) and is reduced to G c(s)G(s)/(l + G c(s)G(s))
in a unity negative feedback system. The closed-loop gain is smaller by a factor of
1/(1 + G c(s)G(s)), which is exactly the factor that reduces the sensitivity of the sys-
tem to parameter variations and disturbances. Usually, we have extra open-loop
gain available, and we are more than willing to trade it for increased control of the
system response.
We should note that it is the gain of the input-output transmittance that is
reduced. The control system does retain the substantial power gain of a power
amplifier and actuator, which is fully utilized in the closed-loop system.
The final cost of feedback is the introduction of the possibility of instability.
Whereas the open-loop system is stable, the closed-loop system may not be always
stable. The question of the stability of a closed-loop system is deferred until Chapter 6,
where it can be treated more completely.
The addition of feedback to dynamic systems causes more challenges for the
designer. However, for most cases, the advantages far outweigh the disadvantages,
and a feedback system is desirable. Therefore, it is necessary to consider the addi-
tional complexity and the problem of stability when designing a control system.
Clearly, we want the output of the system, Y(s), to equal the input, R(s). How-
ever, upon reflection, we might ask, Why not simply set the transfer function
G(s) = Y(s)/R(s) equal to 1? (See Figure 4.2, assuming T d(s) = 0.) The answer to
this question becomes apparent once we recall that the process (or plant) G(s)
was necessary to provide the desired output; that is, the transfer function G(s) rep-
resents a real process and possesses dynamics that cannot be neglected. If we set