Modern Control Engineering

Problems 157

B–4–10.Consider the liquid-level control system shown in
Figure 4–52. The inlet valve is controlled by a hydraulic
integral controller. Assume that the steady-state inflow rate
is and steady-state outflow rate is also the steady-state
head is steady-state pilot valve displacement is
and steady-state valve position is We assume that the set
point corresponds to the steady-state head The set
point is fixed. Assume also that the disturbance inflow rate
qd, which is a small quantity, is applied to the water tank at
t=0. This disturbance causes the head to change from to
This change results in a change in the outflow rate
byqo. Through the hydraulic controller, the change in head
causes a change in the inflow rate from to (The
integral controller tends to keep the head constant as much
as possible in the presence of disturbances.) We assume that
all changes are of small quantities.

Q

–

Q +qi.

–

H

–

+h.

H

–

H

–

R.

– Y

–

.

X

–

H =0,

–

,

Q

–

Q ,

–

We assume that the velocity of the power piston (valve)

is proportional to pilot-valve displacement x,or

whereK 1 is a positive constant. We also assume that the

change in the inflow rate qiis negatively proportional to the

change in the valve opening y,or

whereKvis a positive constant.

Assuming the following numerical values for the system,

C=2m^2 , R=0.5secm^2 , Kv=1m^2 sec

a=0.25m, b=0.75m, K 1 =4 sec–1

obtain the transfer function H(s)/Qd(s).

qi=-Kv y

dy

dt

=K 1 x

C(Capacitance)

R

(Resistance)

a b

h

Y+y qd

Q+qi

H+h

Q+qo

x

Figure 4–52

Liquid-level control system.