Modern Control Engineering

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.