Modern Control Engineering

104 Chapter 4 / Mathematical Modeling of Fluid Systems and Thermal Systems

Q+q

Tank 1 Tank 2

H 1 +h 1

R 1 H 2 +h 2 R 2

Q+q 2

C 1 Q+q^1 C 2

Q:

H 1 :

H 2 :

Steady-state flow rate

Steady-state liquid level of tank 1

Steady-state liquid level of tank 2

Figure 4–2

Liquid-level system

with interaction.

Ifqiis considered the input and hthe output, the transfer function of the system is

If, however,qois taken as the output, the input being the same, then the transfer

function is

where we have used the relationship

Liquid-Level Systems with Interaction. Consider the system shown in Figure

4–2. In this system, the two tanks interact. Thus the transfer function of the system is not

the product of two first-order transfer functions.

In the following, we shall assume only small variations of the variables from the

steady-state values. Using the symbols as defined in Figure 4–2, we can obtain the

following equations for this system:

(4–3)

(4–4)

(4–5)

(4–6)

Ifqis considered the input and q 2 the output, the transfer function of the system is

(4–7)

Q 2 (s)

Q(s)

=

1

R 1 C 1 R 2 C 2 s^2 +AR 1 C 1 +R 2 C 2 +R 2 C 1 Bs+ 1

C 2

dh 2

dt

=q 1 - q 2

h 2

R 2

=q 2

C 1

dh 1

dt

=q-q 1

h 1 - h 2

R 1

=q 1

Qo(s)=

1

R

H(s)

Qo(s)

Qi(s)

=

1

RCs+ 1

H(s)

Qi(s)

=

R

RCs+ 1

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