стр. 51 |

Air

Figure 6-1.11 Phosphate pellets drier.

Mois, %

(a)

Mois, %

c,%TO

H

Sensor/transmitter

(b)

Controller

c set + m Mois, %

e

GC

% CO

% TO

% TO -

c,%TO

H

Sensor/transmitter

(c)

Figure 6-1.12 Developing the block diagram of the drier control system.

135

CONTROL LOOP STABILITY

Conveyor

Controller

belt

m

c set + Mois, %

e Speed

GCB

GC

% CO

% TO rpm

% TO -

c,%TO

H

Sensor/transmitter

(d)

Conveyor

Drier

Controller

belt

m

c set + e Mois, %

Speed

GCB G1

GC

rpm

% CO

% TO

% TO -

c,%TO

H

Sensor/transmitter

(e)

HV IMois

Btu / lb %

G3 G4

Conveyor

Controller Drier

belt

Mois, %

m

c set + Speed

e GCB

GC G1

% CO

% TO rpm

% TO -

c,%TO

H

Sensor/transmitter

(f)

Figure 6-1.12 Continued.

0.016 50 1.0

GV = G1 = H=

3s + 1 30 s + 1 10 s + 1

The time constants are in seconds. The gain of 1.0 in H is obtained by

136 BLOCK DIAGRAMS AND STABILITY

(100 - 0)%TO 100 %TO %TO

= = 1.0

(150 - 50)в€ћF 100в€ћF в€ћF

To study the stability of any control system, control theory says that we need only

to look at the characteristic equation of the system. For block diagrams such as the

one shown in Fig. 6-1.7, the characteristic equation is given by

1 + GCGVG1H = 0 (6-2.1)

That is, the characteristic equation is given by one (1) plus the multiplication of all

the transfer functions in the loop, all of that equal to zero (0). Thus

(0.016)(50)(1)GC

1+ =0 (6-2.2)

(3s + 1)(30 s + 1)(10 s + 1)

or

900 s 3 + 420 s 2 + 43 s + (1 + 0.8GC ) = 0 (6-2.3)

Note that the transfer functions of the disturbances are not part of the characteris-

tic equation, and therefore they do not affect the stability of the loop.

Let us п¬Ѓrst look at the stability when a P controller is used; for this controller

GC = KC. The characteristic equation is then

900 s 3 + 420 s 2 + 43 s + (1 + 0.8GC ) = 0 (6-2.4)

This equation is a polynomial of third order; therefore, there are three roots in this

polynomial. As we may remember, these roots can be either real, imaginary, or

complex. Control theory and mathematics says that for any system to be stable, the

real part of all the roots must be negative; Fig. 6-2.1 shows the stability region. Note

from Eq. (6-2.4) that the locations of the roots depend on the value of KC , which is

the same thing as saying that the stability of the loop depends on the tuning of the

стр. 51 |