Modern Control Engineering

Section 4–4 / Hydraulic Systems 133

+

• 
  

(a)(b)

Spring Area=A

constant=k

Density

of oil =r

Oil

under

pressure

Resistance=R

e

x

a

b

y

z

E(s) b X(s) Y(s)

a+b

a

a+b

K

s

Ts

Ts+ 1

Z(s)

or

Taking the Laplace transforms of both sides of this last equation, assuming zero initial

conditions, we obtain

The transfer function of this system thus becomes

Let us define RA^2 rk=T.(Note that RA^2 rkhas the dimension of time.) Then

Clearly, the dashpot is a differentiating element. Figure 4–21(c) shows a block diagram

representation for this system.

Obtaining Hydraulic Proportional-Plus-Integral Control Action. Figure 4–22(a)

shows a schematic diagram of a hydraulic proportional-plus-integral controller. A block

diagram of this controller is shown in Figure 4–22(b). The transfer function Y(s)/E(s)

is given by

Y(s)

E(s)

=

b

a+b

K

s

1 +

Ka

a+b

T

Ts+ 1

Z(s)

Y(s)

=

Ts

Ts+ 1

=

1

1 +

1

Ts

Z(s)

Y(s)

=

s

s+

k

RA^2 r

sY(s)=sZ(s)+

k

RA^2 r

Z(s)

dy

dt

=

dz

dt

+

kz

RA^2 r

Figure 4–22
(a) Schematic diagram of a hydraulic proportional-plus-integral controller; (b) block diagram of the controller.