Modern Control Engineering

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

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(a) (b)

e

a

b

x

z y

R

k

q

P 2 P 1

Area = A

Density of oil = r

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

Z(s)

b

a+b

a

a+b

K

s

1

Ts+ 1

In such a controller, under normal operation with the

result that

where

Thus the controller shown in Figure 4–22(a) is a proportional-plus-integral controller

(PI controller).

Obtaining Hydraulic Proportional-Plus-Derivative Control Action. Figure 4–23(a)

shows a schematic diagram of a hydraulic proportional-plus-derivative controller. The

cylinders are fixed in space and the pistons can move. For this system, notice that

Hence

or

Z(s)

Y(s)

=

1

Ts+ 1

y=z+

A

k

qR=z+

RA^2 r

k

dz

dt

q dt=rA dz

q =

P 2 - P 1

R

k(y-z)=AAP 2 - P 1 B

Kp=

b

a

, Ti=T=

RA^2 r

k

Y(s)

E(s)

=Kpa 1 +

1

Ti s

b

@KaTC(a+b)(Ts+1)D@1,

Figure 4–23

(a) Schematic diagram of a hydraulic proportional-plus-derivative controller; (b) block diagram of the controller.

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