Design of Lead and Lag Compensator

7/31/2019 Design of Lead and Lag Compensator

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(GNP/Control Systems-2012)

Design of Lead and Lag compensator

Lead Compensator: Phase lead provided by the lead network is used to improve the phasemargin (and hence to improve stability)

Transfer function of a lead compensator is1

1)(

s

ssG

C

with 1 .

See the example in slide-5 of lecture-17. Design a lead compensator for a unity feedback system

with)2(

)(

ss

KsG . Design specifications are Kv = 10, Phase margin = 60.

Kv is higher means system is more accurate. Phase margin is higher means system is more

stable.

Letm

is the phase lead of the compensator,u

is the phase margin of the uncompensated system

andd

is the desired phase margin (the given specification) after compensator.

Now you can start to do the problem by following the steps given below. Note that lag and lead

design can be done analytically without drawing Bode plots.

1. First find the amplifier K to satisfy the accuracy condition.Since Kv = K/2, K=10

2. With this K, draw Bode plots and find the phase margin of the uncompensated system.(Alternatively you can find the phase margin analytically).

Phase margin of the uncompensated system( u ) is 25.2

(See slide-6 of lecture-17).

3. Find the phase lead (m

) of the compensator.

eudm

wheree

is the extra phase to account for the shifting of gain cross over frequency.

(When we add a lead network, gain cross over frequency shifts to right resulting in lesser

phase margin). You can guess it as 155e

In this problem we take 8e

.m

=43. (See slide-7 of lecture-17)

4. Find the parameter using the formulaIn this example 17.0

5. Find the new gain cross-over frequency where the uncompensated system has a gain ofm

m

sin1

sin1

)1

log(20


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In this example the uncompensated system has a gain cross over frequency of

4.25rad/sec. When we are adding a lead compensator the gain cross over will be higherthan 4.25. We place the lead network in such a way that the frequency at which the

maximum phase lead occurs (m

)at the new cross over frequency. We know the gain of

the lead network at this frequency is

In this example

Find the frequency at which the uncompensated system has a gain of -7.7dB. In this

example it is found as 6.8 rad/sec

6. Equate the frequency found in step 5 tom

Hence

The lead compensator transfer function is

7. Check the phase margin of the compensated system. If the phase margin is less redesignit.

Lag Compensator: Gain reduction of the lag network is used to improve the phase

margin (and hence to improve stability)

Transfer function of a lag compensator is1

1)(

s

ssG

C

with 1 .

See the example in slide-2 of lecture-18. Design a lag compensator for a unity feedback system

with)4)(1(

)(

sss

KsG . Design specifications are Kv = 5, Phase margin = 43.

1. First find the amplifier K to satisfy the accuracy condition.Since Kv = K/4, K=20

2. With this K, draw Bode plots of the uncompensated plant.See slide-3 of lecture-18

)

1

log(20

dB7.7)1

log(20

8.61

m

36.0

106.0136.0

s

s


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3. Find the new gain cross over frequency (c

) where the uncompensated plant has a phase

margin ofed

whered

is the desired phase margin ande

is the extra phase to

compensate for the phase lag of the compensator. 155e

In this problemed

=43+12=55

That means the frequency at which the phase of the uncompensated has phase of -125

is 0.52rad/sec. 52.0

c

If you are doing it analytically (without drawing Bode plots) you can skip step-2.Remember there is no need to find the phase margin of the uncompensated system in the

design of lag networks.

4. Find the gain of the uncompensated plant at the new cross over frequency and equate it toand find

In this problem, the gain at 0.52 rad/sec (the new gain cross over frequency) is 20 dB.

10

5. Place the upper corner frequency one decade to one octave below the new gain cross overfrequency.

cc

5.0

11.0

You know the corner frequencies of the lag network are

1(upper corner frequency) and

1(lower corner frequency)

In this problem we select

Hence the transfer function of the lag compensator is

6. Check the phase margin of the compensated system. If the phase margin is less redesignit.

log20

20log20

13.01

177

17.7

s

s

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Design of Lead and Lag Compensator