Modern Control Engineering

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

whereX(s)=l[x]and If qi, the change in flow through the pneumatic

actuating valve, is proportional to x, the change in the valve-stem displacement, then

whereQi(s)=lCqiDandKqis a constant. The transfer function between qiand

becomes

whereKvis a constant.

The standard control pressure for this kind of a pneumatic actuating valve is between

3 and 15 psig. The valve-stem displacement is limited by the allowable stroke of the

diaphragm and is only a few inches. If a longer stroke is needed, a piston–spring

combination may be employed.

In pneumatic actuating valves, the static-friction force must be limited to a low value

so that excessive hysteresis does not result. Because of the compressibility of air, the

control action may not be positive; that is, an error may exist in the valve-stem position.

The use of a valve positioner results in improvements in the performance of a pneu-

matic actuating valve.

Basic Principle for Obtaining Derivative Control Action. We shall now present

methods for obtaining derivative control action. We shall again place the emphasis on

the principle and not on the details of the actual mechanisms.

The basic principle for generating a desired control action is to insert the inverse of

the desired transfer function in the feedback path. For the system shown in Figure 4–12,

the closed-loop transfer function is

If@G(s)H(s)@1, then C(s)/R(s)can be modified to

Thus, if proportional-plus-derivative control action is desired, we insert an element

having the transfer function 1/(Ts+1)in the feedback path.

C(s)

R(s)

=

1

H(s)

C(s)

R(s)

=

G(s)

1 +G(s)H(s)

Qi(s)

Pc(s)

=Kc Kq=Kv

pc

Qi(s)

X(s)

=Kq

Pc(s)=lCpcD.

R(s) C(s)

G(s)

H(s)

+–

Figure 4–12

Control system.

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