Modern Control Engineering

Section 4–3 / Pneumatic Systems 121

(a) (b)

(c)

(d)

e

a

b

X+x

Pc+pc

Ps

R

C

pc

K

K

x

e

t

t

t

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

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

a

a+b

b

a+b

b

a+b

A

ks

a

a+b

A

ks

a

a+b

A

ks

1

RCs+ 1

1

RCs+ 1

III

+– ++

+–

+

• 
  

Figure 4–15
(a) Pneumatic
proportional-plus-
integral controller;
(b) step change in e
and the corre-
sponding changes in
xandpcplotted
versust; (c) block
diagram of the
controller;
(d) simplified block
diagram.

Note that the integral control action in the controller takes the form of slowly

canceling the feedback that the proportional control originally provided.

A block diagram of this controller under the assumption of small variations in the

variables is shown in Figure 4–15(c). A simplification of this block diagram yields

Figure 4–15(d). The transfer function of this controller is

Pc(s)

E(s)

=

b

a+b

K

1 +

Ka

a+b

A

ks

a 1 -

1

RCs+ 1

b