Page 276 - Modern Control Systems

250 Chapter 4 Feedback Control System Characteristics

; i ' i ; ; i ; 1 1 1
r
1.0 ! ! i i 1 i 1 !
0.9
CI osed-loop
0.8
j i l l .
I ' ! !
0.7 — —- i 1 j i ^ -._^p]_
FIGURE 4.15 1 t J
The response of the 0.6
open-loop and 0.5 r — r - —
closed-loop speed ! O p e n - l o o p | |
control system 0.4 i
when r = 10 and ! ! JS j (without feedback)
KiK aK t = 100. The 0.3 4 - -
'"' • / \ • " ~ \ i
time to reach 98% -Kp-f-; —
of the final value for 0.2 i i . ! - I
the open-loop and 0.1 —
closed-loop system ~A~\ I 1 T i 1"! • 1
is 40 seconds and / i • !
0.4 seconds, 0 1 2 3 5 6 7 8 9 10 11 12 13 14 15
respectively. Time (s)

motor and its associated torque signal must be larger for the closed-loop than for
the open-loop operation. Therefore, a higher-power motor will be required to
avoid saturation of the motor. The responses of the closed-loop system and the
open-loop system are shown in Figure 4.15. Note the rapid response of the
closed-loop system.
While we are considering this speed control system, it will be worthwhile to de-
termine the sensitivity of the open- and closed-loop systems. As before, the sensitivity
of the open-loop system to a variation in the motor constant or the potentiometer
is unity. The sensitivity of the closed-loop system to a variation is
constant k 2 in K m
[s + (iAi)]
oT QT QG
s + (K aK tK x + lVn"

Using the typical values given in the previous paragraph, we have
(s + 0.10)
Si
s + 10
We find that the sensitivity is a function of s and must be evaluated for various values of
frequency. This type of frequency analysis is straightforward but will be deferred until a
later chapter. However, it is clearly seen that at a specific low frequency—for example,
s = jo) = ;'l—the magnitude of the sensitivity is approximately |s£J = 0.1.

4.6 STEADY-STATE ERROR

A feedback control system is valuable because it provides the engineer with the
ability to adjust the transient response. In addition, as we have seen, the sensitivity
of the system and the effect of disturbances can be reduced significantly. However,
as a further requirement, we must examine and compare the final steady-state error