Modern Control Engineering

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

+–

(a)

(b)

e

a

b

X+x

Ps

Ri

Rd

CC

Pc+pc

(Ri Rd)

E(s) X(s) Pc(s)

b K

a+b

a

a+b

A

ks

1

RdCs+ 1

1

RiCs+ 1

• 
  

+

Figure 4–16

(a) Pneumatic

proportional-plus-

integral-plus-

derivative controller;

(b) block diagram of

the controller.

whereKis a constant,Ais the area of the bellows, and ksis the equivalent spring constant

of the combined bellows. If which is usually the

case, the transfer function can be simplified to

where

Obtaining Pneumatic Proportional-Plus-Integral-Plus-Derivative Control

Action. A combination of the pneumatic controllers shown in Figures 4–14(a) and

4–15(a) yields a proportional-plus-integral-plus-derivative controller, or a PID con-

troller. Figure 4–16(a) shows a schematic diagram of such a controller. Figure 4–16(b)

shows a block diagram of this controller under the assumption of small variations in the

variables.

Kp=

bks

aA

, Ti=RC

Pc(s)

E(s)

=Kpa 1 +

1

Ti s

b

@KaARCsC(a+b)ks(RCs+1)D@1,

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