Modern Control Engineering

Section 4–4 / Hydraulic Systems 135

e

a

b

x

y

R R

k 1 k 2

Area=A

z

Figure 4–24
Schematic diagram
of a hydraulic
proportional-plus-
integral-plus-
derivative controller.

where

A block diagram for this system is shown in Figure 4–23(b). From the block diagram the

transfer function Y(s)/E(s)can be obtained as

Under normal operation we have Hence

where

Thus the controller shown in Figure 4–23(a) is a proportional-plus-derivative controller

(PD controller).

Obtaining Hydraulic Proportional-Plus-Integral-Plus-Derivative Control Action.

Figure 4–24 shows a schematic diagram of a hydraulic proportional-plus-integral-plus-

derivative controller. It is a combination of the proportional-plus-integral controller

and proportional-plus derivative controller.

If the two dashpots are identical except the piston shafts, the transfer function

Z(s)/Y(s)can be obtained as follows:

(For the derivation of this transfer function, refer to Problem A–4–9.)

Z(s)

Y(s)

=

T 1 s

T 1 T 2 s^2 +AT 1 +2T 2 Bs+ 1

Kp=

b

a

, T=

RA^2 r

k

Y(s)

E(s)

=Kp(1+Ts)

@aKC(a+b)s(Ts+1)D@1.

Y(s)

E(s)

=

b

a+b

K

s

1 +

a

a+b

K

s

1

Ts+ 1

T=

RA^2 r

k