Page 336 - Modern Control Systems

310

FIGURE 5.6
Response of a
second-order
system for an
impulse function
input.

As £ decreases, the closed-loop roots approach the imaginary axis, and the response
becomes increasingly oscillatory. The response as a function of £ and time is also
shown in Figure 5.5(b) for a step input.
The Laplace transform of the unit impulse is R(s) = 1, and therefore the output
for an impulse is

Y(s) = (5.10)
2
.v + 2£w„s + o)f t
which is T(s) = Y(s)/R(s), the transfer function of the closed-loop system. The
transient response for an impulse function input is then

0)n r .
X0 = je sm.(a) n(3t), (5.11)

which is the derivative of the response to a step input. The impulse response of the
second-order system is shown in Figure 5.6 for several values of the damping ratio £.
The designer is able to select several alternative performance measures from the
transient response of the system for either a step or impulse input.
Standard performance measures are usually defined in terms of the step response
of a system as shown in Figure 5.7. The swiftness of the response is measured by the
rise time T r and the peak time T p. For underdamped systems with an overshoot, the
0-100% rise time is a useful index. If the system is overdamped, then the peak time
is not defined, and the 10-90% rise time 7 f| is normally used. The similarity with
which the actual response matches the step input is measured by the percent over-
shoot and settling time T s. The percent overshoot is defined as

J
P.O. = ' r X 100% (5.12)