Page 271 - Modern Control Systems

Section 4.4 Disturbance Signals in a Feedback Control System 245
T,,{s)
Tj{s)

*• a)(s)

•H(s)

(a) (b)

FIGURE 4.10 Closed-loop system, (a) Signal-flow graph model, (b) Block diagram model.

Therefore, if G x(s)H(s) is made sufficiently large, the effect of the disturbance can
be decreased by closed-loop feedback. Note that

KgK m Kb
G^His)
R„
since K a ^> K b. Thus, we strive to obtain a large amplifier gain, K a, and keep
< 2 CI. The error for the system shown in Figure 4.10 is
R a
E(s) = R(s) - (o(s),

and R(s) = (o^s), the desired speed. For calculation ease, we let R(s) = 0 and ex-
amine a>(s).
To determine the output for the speed control system of Figure 4.9, we must
consider the load disturbance when the input R(s) = 0. This is written as
-1/(/5 + b)
(o(s) = Us)
1 + (K tK aK m/R a)[\/(Js + b)] + (K mK h/R a)[\/{Js + b)]
- 1
•Us)- (4.30)
Js + b + (KJR (l)(K tK a + K b)
The steady-state output is obtained by utilizing the final-value theorem, and we have
- 1
limco(/) = limfaufc)) = , .„ ._ W r , ., D\ (431)
^oo s ^ }) b + (KjR a)(K tK a + K h)

when the amplifier gain K a is sufficiently high, we have
w(co) ^-D = o, c(oo). (4.32)
-R n
K aK niK t
The ratio of closed-loop to open-loop steady-state speed output due to an undesired
p steady-
disturbance is
w c(oo) RJb + K mK b
(4.33)
w 0(oo) KaK mK,
and is usually less than 0.02.
This advantage of a feedback speed control system can also be illustrated by
considering the speed-torque curves for the closed-loop system, which are shown in