Page 273 - Modern Control Systems

Section 4.5 Control of the Transient Response 247

which is approximately
Y(s) - -N(s), (4.35)
for large loop gain L(s) — G c(s)G(s). This is consistent with the earlier discussion
that smaller loop gain leads to measurement noise attentuation. Clearly, the design-
er must shape the loop gain appropriately.
The equivalency of sensitivity, S^, and the response of the closed-loop system
tracking error to a reference input can be illustrated by considering Figure 4.3. The
sensitivity of the system to G(s) is

SG = = ( 4 3 6 )
1 + G c(s)G(s) 1 + L(sY '
The effect of the reference on the tracking error (with T d(s) = 0 and N(s) = 0) is
E(s) 1 1
(4.37)
R(s) 1 + G c(s)G(s) 1 + L(s)'
In both cases, we find that the undesired effects can be alleviated by increasing the
loop gain. Feedback in control systems primarily reduces the sensitivity of the system
to parameter variations and the effect of disturbance inputs. Note that the measures
taken to reduce the effects of parameter variations or disturbances are equivalent,
and fortunately, they reduce simultaneously. As a final illustration, consider the
effect of the noise on the tracking error:
E(s) = G c(s)G(s) = L(s)
{ }
T d(s) 1 + G c(s)G(s) 1 + L(sY '
We find that the undesired effects of measurement noise can be alleviated by de-
creasing the loop gain. Keeping in mind the relationship

S(s) + C(s) = 1,
the trade-off in the design process is evident.

4.5 CONTROL OF THE TRANSIENT RESPONSE

One of the most important characteristics of control systems is their transient re-
sponse. The transient response is the response of a system as a function of time. Be-
cause the purpose of control systems is to provide a desired response, the transient
response of control systems often must be adjusted until it is satisfactory. If an open-
loop control system does not provide a satisfactory response, then the process, G(s),
must be replaced with a more suitable process. By contrast, a closed-loop system can
often be adjusted to yield the desired response by adjusting the feedback loop para-
meters. It is often possible to alter the response of an open-loop system by inserting a
suitable cascade controller, G c(s), preceding the process, G(s), as shown in Figure 4.12.
Then it is necessary to design the cascade transfer function, G c(s)G(s), so that the re-
sulting transfer function provides the desired transient response.