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258 Chapter 4 Feedback Control System Characteristics

TAs)
Controller Rover
K(s + l)(s + 3) •i 1 Y(s)
• (s +
s 2 + 4s + 5 ID > l)(s + 3) position
(a)

FIGURE 4.21 Rover
Control system for + ^) •i 1 Y{s)
the rover, (a) Open- R(s) H V fc ' K }V > . {s +1)(.9 + 3)
loop (without position
feedback). T
(b) Closed-loop
with feedback. (b)
and the transfer function for the closed-loop system is

Y(s) K
T c(s) = 2 (4.63)
R(s) s + 4s + 3 + K
Then, for K = 2,
T(s) = T 0(s) = T c(s) = 2
s + 4s + 5
Hence, we can compare the sensitivity of the open-loop and closed-loop systems for
the same transfer function.
The sensitivity for the open-loop system is

,7 _ dT 0 K
SK h (4.64)
~ dKT 0~
and the sensitivity for the closed-loop system is
K s 2 + 4s + 3
dT c
Olf' — 2 (4.65)
dK T c s + 4s + 3 + K
To examine the effect of the sensitivity at low frequencies, we let s = jco to obtain
2
(3 - (o ) + j4w
SP = 2 (4.66)
(3 + K - o) ) + j4w
For K = 2, the sensitivity at low frequencies, w < 0.1, is s£'| — 0.6.
|
A frequency plot of the magnitude of the sensitivity is shown in Figure 4.22.
Note that the sensitivity for low frequencies is
|S£| < 0.8, for a) < 1.
The effect of the disturbance can be determined by setting R(s) = 0 and letting
T d(s) = 1/s. Then, for the open-loop system, we have the steady-state value
1
y(oo) = lim s (4.67)
s-i) I (S + 1)(5 + 3)15 3