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Usually we have to compromise among many conflicting and damaging specifications and have to adjust the system parameters to achieve a suitable performance although it is not possible to obtain all desired optimum specifications… Read TextPreview

- Subject: Technology
- Type: Research Paper
- Level: Masters
- Pages: 6 (1500 words)
- Downloads: 0
- Author: deontaebechtela

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Although the prime objective of a control system is to have optimum performance yet without any adjustment it is rare indeed. Usually we have to compromise among many conflicting and damaging specifications and have to adjust the system parameters to achieve a suitable performance although it is not possible to obtain all desired optimum specifications. The control system adjustment or stabilization and alteration is carried out by introducing a compensating block within the control loop that will not allow the steady state error to exceed to a certain limit. This compensating block is called compensator or filter and physically it may be an electric network or mechanical equipment containing levers, springs, dashpots, etc. The compensator may be placed cascade with forward transfer function Gx(s) or in feed back path as shown in fig.1.1. The selection of location for inserting the compensator largely depends on the control system, the necessary physical modifications and results desired (D’Azzo, 1995, pp. 319-320). The available techniques to achieve improvement in the response of feedback control system are time domain, frequency domain, root locus and disturbance rejection technique. In next section we will examine cascade and feed back path compensators using frequency domain technique and a comparison is made relative to merits of each design.

design.

Cascade Compensator

Let we consider a system as shown in fig. 1.2, then forward transfer function in

terms of frequency domain with a compensator be,

G(j) = Gc(j) Gx(j)

or, Gc(j) = .(1)

Now for lag compensators, the equation (1) in terms of frequency domain becomes

(D'Azzo, 1995, pp. 333),

Lag Compensator Gc(j) = .(2)

Also for Lead Compensator Gc(j) = . .(3)

Fig. 1.1 Block Diagrams of Control Systems with Feedback Compensators

Fig. 1.2 Feedback System with Compensators

The polar plots of equations (2) and (3) can be plotted as shown in fig. 1.3. From

graph we observe that the cascade lag compensator results in a clockwise rotation of

G(j) and can produce a large increase in gain (Km) with a small decrease in m where

as cascade lead compensator results in the counter clockwise rotation of G(j) and

produce a large increase in m with small increase in Km where both Km and m

determine the transit response of a feedback system.

Fig. 1.3 Polar graphs of Cascade Compensator

Also from equation (1), Gc(j) = Gc(j)

Solving for ', we have (Dorf, 1989, p. 266), (D'Azzo, 1995, pp. 370-371).

= tan-1T - tan-1T

= tan-1 . (3)

or T + T + = 0 . (4)

Where for lag compensator > 1 and < 1 for lead compensator. Differentiating

equation (3) w.r.t. and setting it equal to '0, we have,

m = . (5)

Substituting the value in (3) we have the max. phase shift,

() max = sin-1 . (6)

From equation (6) we can calculate typical shift values as given in table (a) which are

very useful in design application of lead compensator as we will see in design examples.

0

0.1

0.2

0.3

0.4

() max

90

54.9

41.8

32.6

19.5

Table. a

Design Example of Phase - lag Compensator Network

Let we consider an uncompensated transfer function give as,

GH(j) =

GH(j) = (where Kv = K/20)

Let we compensate this network to Kv = 20 with phase margin of 45.

If we observe the uncompensated Bode diagram as given in fig. 1.4 we see it has as

Fig. 1.4 Bode Diagram

shown in solid lines then we see that uncompensated system has a phase margin of 20.

Allowing 5 for an increase in phase -lag compensator, we relocate the frequency,

(where = -130) which is to be our crossover ...Download file to see next pagesRead More

design.

Cascade Compensator

Let we consider a system as shown in fig. 1.2, then forward transfer function in

terms of frequency domain with a compensator be,

G(j) = Gc(j) Gx(j)

or, Gc(j) = .(1)

Now for lag compensators, the equation (1) in terms of frequency domain becomes

(D'Azzo, 1995, pp. 333),

Lag Compensator Gc(j) = .(2)

Also for Lead Compensator Gc(j) = . .(3)

Fig. 1.1 Block Diagrams of Control Systems with Feedback Compensators

Fig. 1.2 Feedback System with Compensators

The polar plots of equations (2) and (3) can be plotted as shown in fig. 1.3. From

graph we observe that the cascade lag compensator results in a clockwise rotation of

G(j) and can produce a large increase in gain (Km) with a small decrease in m where

as cascade lead compensator results in the counter clockwise rotation of G(j) and

produce a large increase in m with small increase in Km where both Km and m

determine the transit response of a feedback system.

Fig. 1.3 Polar graphs of Cascade Compensator

Also from equation (1), Gc(j) = Gc(j)

Solving for ', we have (Dorf, 1989, p. 266), (D'Azzo, 1995, pp. 370-371).

= tan-1T - tan-1T

= tan-1 . (3)

or T + T + = 0 . (4)

Where for lag compensator > 1 and < 1 for lead compensator. Differentiating

equation (3) w.r.t. and setting it equal to '0, we have,

m = . (5)

Substituting the value in (3) we have the max. phase shift,

() max = sin-1 . (6)

From equation (6) we can calculate typical shift values as given in table (a) which are

very useful in design application of lead compensator as we will see in design examples.

0

0.1

0.2

0.3

0.4

() max

90

54.9

41.8

32.6

19.5

Table. a

Design Example of Phase - lag Compensator Network

Let we consider an uncompensated transfer function give as,

GH(j) =

GH(j) = (where Kv = K/20)

Let we compensate this network to Kv = 20 with phase margin of 45.

If we observe the uncompensated Bode diagram as given in fig. 1.4 we see it has as

Fig. 1.4 Bode Diagram

shown in solid lines then we see that uncompensated system has a phase margin of 20.

Allowing 5 for an increase in phase -lag compensator, we relocate the frequency,

(where = -130) which is to be our crossover ...Download file to see next pagesRead More

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