Page 274 - Modern Control Systems

248 Chapter 4 Feedback Control System Characteristics

FIGURE 4.12 Controller Process
Cascade controller G c(s) C(i)
system (without /e(.v) O Controller o O Yls) «(v) G c(s) • G(s) Y(s)
feedback). Process
(a) Signal-flow graph.
(b) Block diagram. (a) (b)

To make this concept more comprehensible, consider a specific control system,
which may be operated in an open- or closed-loop manner. A speed control system,
as shown in Figure 4.13, is often used in industrial processes to move materials and
products. Several important speed control systems are used in steel mills for rolling
the steel sheets and moving the steel through the mill [19]. The transfer function of
the open-loop system (without feedback) was obtained in Equation (2.70). For
(o(s)/V a(s), we have

(4.39)
K }
V a(s) T,S + r
where
K„ RJ
K,= and T\ =
RcP + K hK m R ab + K hK n
In the case of a steel mill, the inertia of the rolls is quite large, and a large armature-
controlled motor is required. If the steel rolls are subjected to a step command for a
speed change of

(4.40)

the output response is
<o(s) = G(s)V a(s). (4.41)
The transient speed change is then

/Tl
u)(t) = #i(A 2 E)(l - e~' )- (4.42)
If this transient response is too slow, we must choose another motor with a dif-
ferent time constant T h if possible. However, because T\ is dominated by the load
inertia,/, it may not be possible to achieve much alteration of the transient response.

if = Constant field current

FIGURE 4.13
Load
Open-loop speed Lc
control system
(without feedback).