Page 348 - Modern Control Systems

322 Chapter 5 The Performance of Feedback Control Systems
represented in terms of the poles and zeros of its transfer function T(s). On the other
hand, system performance is often analyzed by examining time-domain responses,
particularly when dealing with control systems.
The capable system designer will envision the effects on the step and impulse
responses of adding, deleting, or moving poles and zeros of T(s) in the s-plane. Like-
wise, the designer should visualize the necessary changes for the poles and zeros of
T(s), in order to effect desired changes in the model's step and impulse responses.
An experienced designer is aware of the effects of zero locations on system
response. The poles of T(s) determine the particular response modes that will be
present, and the zeros of T(s) establish the relative weightings of the individual
mode functions. For example, moving a zero closer to a specific pole will reduce
the relative contribution of the mode function corresponding to the pole.
A computer program can be developed to allow a user to specify arbitrary sets
of poles and zeros for the transfer function of a linear system. Then the computer
will evaluate and plot the system's impulse and step responses individually. It will
also display them in reduced form along with the pole-zero plot.
Once the program has been run for a set of poles and zeros, the user can modify the
locations of one or more of them. Plots may then be presented showing the old and new
poles and zeros in the complex plane and the old and new impulse and step responses.

5.6 THE STEADY-STATE ERROR OF FEEDBACK CONTROL SYSTEMS

One of the fundamental reasons for using feedback, despite its cost and increased
complexity, is the attendant improvement in the reduction of the steady-state error
of the system. As illustrated in Section 4.6, the steady-state error of a stable closed-
loop system is usually several orders of magnitude smaller than the error of an
open-loop system. The system actuating signal, which is a measure of the system
error, is denoted as E a(s). Consider the closed-loop feedback system shown in
Figure 5.18. According to the discussions in Chapter 4, we know from Equation (4.3)
that with N(s) - 0, T d(s) = 0, the tracking error is

E{S)
= 1 + ^ , ) 0 ( , ) ^
Using the final value theorem and computing the steady-state tracking error yields
1
lime(r) = <? ss = ^s——————R(s). (5.23)
r->co v-K) I + G c(s)G{s)
It is useful to determine the steady-state error of the system for the three standard
test inputs for the unity feedback system. Later in this section we will consider
steady-state tracking errors for non-unity feedback systems.

Step Input. The steady-state error for a step input of magnitude A is therefore
s(A/s) A
= lim
e w
.v-ol + G c(s)G(s) 1 + lim G c(s)G(s)'
s—»0