Modern Control Engineering

Section 2–3 / Automatic Control Systems 23

is called the differential gap. A differential gap is indicated in Figure 2–7(b). Such a dif-

ferential gap causes the controller output u(t)to maintain its present value until the ac-

tuating error signal has moved slightly beyond the zero value. In some cases, the differential

gap is a result of unintentional friction and lost motion; however, quite often it is inten-

tionally provided in order to prevent too-frequent operation of the on–off mechanism.

Consider the liquid-level control system shown in Figure 2–8(a), where the electromag-

netic valve shown in Figure 2–8(b) is used for controlling the inflow rate. This valve is either

open or closed. With this two-position control, the water inflow rate is either a positive con-

stant or zero. As shown in Figure 2–9, the output signal continuously moves between the

two limits required to cause the actuating element to move from one fixed position to the

other. Notice that the output curve follows one of two exponential curves, one correspon-

ding to the filling curve and the other to the emptying curve. Such output oscillation be-

tween two limits is a typical response characteristic of a system under two-position control.

(a) (b)

U 1

U 2

e u U^1

U 2

e u

Differential gap

+– +

Figure 2–7
(a) Block diagram of
an on–off controller;
(b) block diagram of
an on–off controller
with differential gap.

115 V

Float

R

C h

(a) (b)

qi

Movable iron core

Magnetic coil

Figure 2–8
(a) Liquid-level
control system;
(b) electromagnetic
valve.

h(t)

(^0) t
Differential
gap
Figure 2–9
Levelh(t)-versus-t
curve for the system
shown in Figure 2–8(a).