0195136047.pdf

16.2 FEEDBACK CONTROL SYSTEMS 793

function, whereas the natural frequencyωnof a system provides a measure of the settling time
(see Figure 16.2.6). The settling time is no greater thants= 3 /ξ ωnfor the response to a step input
to reach 95% of its steady-state value. For a 1% tolerance band, the settling time is no greater than
ts= 5 /ξ ωn. Figure 16.2.10 depicts the percent maximum overshoot as a function of the damping
ratioξfor a linear second-order system. For two different systems having the same damping ratio
ξ, the one with the larger natural frequency will have the smaller settling time in responding to
input commands or load disturbances.
The particular form of the forced solution of Equation (16.2.26) depends upon the type of
forcing function used. If a step input of magnituder 0 is applied, thenR(s)=r 0 /s, so that the
complete transformed solution becomes

C(s)=

(r

0

s

) ω 2

n

s^2 + 2 ξωns+ω^2 n

(16.2.27)

Note that the steady-state solution is generated by the pole associated withR(s), while the transient
terms resulting from a partial-fraction expansion are associated with the poles of the denominator.
In order to meet the requirements for the steady-state performance, as well as dynamic
performance of the feedback control system of Figure 16.2.7, it becomes necessary to provide
independent control of both performances. Practical methods that have gained widespread accep-
tance are

• Error-rate control


• Output-rate control


• Integral-error (or reset) control.

ERROR-RATECONTROL

A system is said to possess error-rate damping when the generation of the output in some way
depends upon the rate of change of the actuating signal. For the system of Figure 16.2.7, if the

0.1

10

20

30

40

50

60

70

80

90

100

0.2 0.3 0.4 0.5 0.6 0.7

Percent maximum overshoot

Damping ratio ξ

0 0.8 0.9 1.0

Figure 16.2.10Plot of percent maxi-

mum overshoot as a function of the

damping ratio for linear second-order

system.