Page 332 - Modern Control Systems

306 Chapter 5 The Performance of Feedback Control Systems

determine initially whether the system is stable; we can achieve this goal by using
the techniques of ensuing chapters. If the system is stable, the response to a specific
input signal will provide several measures of the performance. However, because
the actual input signal of the system is usually unknown, a standard test input signal
is normally chosen. This approach is quite useful because there is a reasonable cor-
relation between the response of a system to a standard test input and the system's
ability to perform under normal operating conditions. Furthermore, using a stan-
dard input allows the designer to compare several competing designs. Many control
systems experience input signals that are very similar to the standard test signals.
The standard test input signals commonly used are the step input, the ramp input,
and the parabolic input. These inputs are shown in Figure 5.2. The equations repre-
senting these test signals are given in Table 5.1, where the Laplace transform can be
obtained by using Table 2.3 and a more complete list of Laplace transform pairs can
be found at the MCS website. The ramp signal is the integral of the step input, and the
parabola is simply the integral of the ramp input. A unit impulse function is also use-
ful for test signal purposes. The unit impulse is based on a rectangular function

0, otherwise,

where e > 0. As e approaches zero, the function {t) approaches the unit impulse
f €
function 5(f), which has the following properties:
oo
8(t - a)g(t) dt = g{a). (5.1)
/
oo

/(0

FIGURE 5.2
Test input signals:
(a) step, (b) ramp, (a) (b) (c)
and (c) parabolic.

Table 5.1 Test Signal Inputs
Test Signal r{t) R{s)
Step r(t) = A,t > 0 R{s) = Ajs
= 0, r < 0
Ramp r(t) = At, t > 0 R{s)- = Ajs 2
= 0, t < 0
2
Parabolic r(t) = At , t > 0 R(s) = lAjs 71
= 0, l < 0