Example Problems and Solutions 141
A–4–2. Consider the liquid-level system shown in Figure 4–28. In the system, and are steady-state
inflow rates and and are steady-state heads. The quantities qi1,qi2,h 1 ,h 2 ,q 1 , and qoare con-
sidered small. Obtain a state-space representation for the system when h 1 andh 2 are the outputs
andqi1andqi2are the inputs.
Solution.The equations for the system are
(4–32)
(4–33)
(4–34)
(4–35)
Elimination of q 1 from Equation (4–32) using Equation (4–33) results in
(4–36)
Eliminatingq 1 andqofrom Equation (4–34) by using Equations (4–33) and (4–35) gives
(4–37)
Define state variables x 1 andx 2 by
x 1 =h 1
x 2 =h 2
the input variables u 1 andu 2 by
u 1 =qi1
u 2 =qi2
and the output variables y 1 andy 2 by
y 1 =h 1 =x 1
y 2 =h 2 =x 2
Then Equations (4–36) and (4–37) can be written as
x# 2 =
1
R 1 C 2
x 1 - a
1
R 1 C 2
+
1
R 2 C 2
bx 2 +
1
C 2
u 2
x# 1 =-
1
R 1 C 1
x 1 +
1
R 1 C 1
x 2 +
1
C 1
u 1
dh 2
dt
=
1
C 2
a
h 1 - h 2
R 1
+qi2-
h 2
R 2
b
dh 1
dt
=
1
C 1
aqi1-
h 1 - h 2
R 1
b
h 2
R 2
=qo
C 2 dh 2 =Aq 1 +qi 2 - qoBdt
h 1 - h 2
R 1
=q 1
C 1 dh 1 =Aqi 1 - q 1 Bdt
H– 1 H– 2
Q– 1 Q– 2
C 1 C 2
R 1 R 2
Q 1 +q 1
Q 1 +qi 1 Q 2 +qi 2
Q 1 +Q 2 +qo
H 1 +h 1 H 2 +h 2
Figure 4–28
Liquid-level system.