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CONTROL SYSTEMS

Compensators

Dr. S. SUMATHI

Associate Professor

RNSIT

Bengaluru

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COMPENSATORS

Compensators are subsystems introduced into the system to meet design

specifications

The desired behavior of the system is specified in terms of transient

measures and the steady state error.

Compensation may Cascaded or Feedback

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COMPENSATORS

Compensators are of three types

1. Lead Compensator

2. Lag Compensator

3. Lead -Lag Compensator

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Lead Compensator

Lead compensators have a transfer function of the form

Pole is located to the left of zero (zero near origin)

Lead Compensator

improves transient response

increases stability margin

increases system error constant by limited range

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Lag Compensator

Lag compensators have a transfer function of the form

Pole is located to the right of zero (pole near origin)

Lag Compensator

improves steady state behavior

preserves transient response

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Lag-Lead Compensator

Lag-Lead compensators have a transfer function of the form

Lag-Lead Compensator is a combination lag compensator

and lead compensator connected in series

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Lead Compensator

It has a zero at s=-1/ꞇ and pole at s=-1/αꞇ with zero

closer to origin

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Lead compensator can be realized by an electrical network

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The DC attenuation caused by α can be compensated

by an amplifier connected in cascade. Therefore the

sinusoidal transfer function is

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Since α < 1, the output leads the input under steady state condition

hence the name lead compensator.

The frequency ωm at which maximum phase shift occurs is given by

the solution

ωm is the geometric mean of the two corner frequencies

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The maximum phase lead φm is given by

Therefore

Cross multiplying and solving

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The magnitude of Gc(jω) at ω=ωm is

The magnitude of Gc(jω) at ω=ωm in decibels is

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Design of Lead compensator

Specifications: System error constant and Phase margin

STEP 1: Determine the open loop gain K required to satisfy

the specified error constant

STEP 2: Using this value of K draw the magnitude and

phase Bode plots.

Determine the phase margin, (φpm1) and

gain crossover frequency (ωg1)

If phase margin is not satisfactory design

lead compensator

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STEP3: Determine the phase lead required

where φs is the specified phase margin

STEP 4: let φm=φl and determine the α parameter using

If φm required is more than 60 deg it is recommended to use

two identical compensators contributing a maximum lead of

φl /2

ϵ-margin of safety

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STEP 5: Calculate the DB gain

Locate the frequency at which the uncompensated system

has a gain of

This frequency ωg2=ωm of the compensated system.

We know that

STEP6: Compute the two corner frequencies of the

compensator

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The transfer function of the compensator

STEP 7: Draw the magnitude and phase plot of the

compensated system and check the phase margin.

If the phase margin is still low increase ϵ and go to step 3.

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EFFECT OF PHASE LEAD COMPENSATOR

It adds a pole and a zero (with zero to the right of pole) to the

forward path transfer function.

•It adds more damping to the system.

•The rise time and settling time reduces

• Improves the phase margin and gain margin of the closed loop system

•The band width of the closed loop system increases ( corresponding to

faster system)

•The steady state error is not affected

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Design a lead compensator for a unity feedback system with an

open loop transfer function

for the specifications of Kv=10 and φm= 36 deg.

SOLUTION

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Therefore the open loop transfer function

The sinusoidal transfer function is

The gain cross over frequency is 3.3 rad/sec Phase margin is 17 deg

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The magnitude contribution corresponding to ωm is

From the bode plot we find ωg2 corresponding to -3.75 dB as 4 rad/sec

Therefore

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Therefore the compensator transfer function

The open loop Transfer function of lead compensated system is

At ω=4

Therefore phase margin is 38o

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Lag compensator

Since β>1 the steady state output lags behind the input

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The lag compensator can be realized by the following electrical

network

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The frequency (ωm) at which maximum phase lag φm

can be obtained by

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Lag compensator Design Lag network acts like a low pass filter attenuating high frequencies by

The phase lag is undesirable at the gain crossover frequency of the

compensated system.

Therefore the attenuation characteristic of the lag network is

used for compensation purpose.

STEP 1: Determine the open loop gain necessary to satisfy the specified

error constant.

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STEP 2: With this value of K draw the magnitude and phase Bode

plot and determine the phase margin φpm1 and the GCF ωg1 of the

uncompensated system.

If the phase margin is not satisfactory go to step 3.

STEP 3: Find the frequency ωg2 at which the system makes a phase

margin of

Where φpm2 is measured above -180o line φs is the specified phase

margin and ϵ =5o to 15o.

Lag compensation is possible only if there exists a range of frequencies

in which the system has a phase angle less negative than -180o+φ.

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STEP4: Measure the gain at ωg2 and equate it to required high frequency

attenuation 20 log β. And calculate β.

STEP 5: choose the upper cut of frequency ω2=1/ꞇ of the lag network

one octave to one decade below ωg2.

so that phase lag contribution at this frequency is negligible.

STEP 6: With β and ꞇ determined the transfer function of lag compensator

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STEP 7: Draw the magnitude and phase bode plot of the

compensated system and check the resulting margin

STEP 8: If the specifications are not met redesign the compensator

by choosing another value of ꞇ

Example

Compensate the system with the open loop transfer function

to meet the following specifications.

Velocity error constant

Damping ratio

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SOLUTION

The given time domain specifications are converted in to frequency domain

specifications

We know

The open loop transfer function of an uncompensated system may be written

as

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The bode plot of uncompensated system shows that GCF=2.25.rad/sec

and phase margin= -4deg

The phase margin(desired)

The frequency corresponding to this phase angle

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Placing the upper corner frequency of the compensator

two octave below, we have

The log magnitude at ωc2 =20 dB

Therefore

Therefore the lower corner frequency

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The transfer function of the lag network is

The open loop transfer function of compensated system

If we draw the bode plot , the phase margin of

compensated system can be found to be 42 deg.

The step response of the uncompensated system (can be

found out using SIMULINK)shows peak overshoot as 32%

and settling time as 2 sec

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Comparison of Phase lag and phase lead network

Phase Lead network Phase Lag network

Band width Increases, High frequency

gain increases Band width decreases

Dynamic response becomes faster Dynamic response slows down

Susceptible to high frequency noise High frequency noise suppressed

No significant decreases in steady

state error Steady state error is reduced

Application : When fast dynamic

response is required

Application : When low steady state

error is required

Cannot be applied when phase angle

of uncompensated system is

decreasing rapidly near Gain cross

over frequency

Cannot be applied when

uncompensated system phase angle in

low frequency region is not sufficient to

provide requisite phase margin

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Lag-Lead Compensator

Lag-Lead compensators have a transfer function of the form

Lag-Lead Compensator is a combination lag compensator

and lead compensator connected in series

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The equation Gc(s) can be realized by a single electric

lead lag network as shown below

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Lead compensation is applied to system to achieve desired

error constant and any phase margin, even though the

uncompensated system may be absolutely unstable.

Lead compensation results in increased bandwidth and faster

response.

For higher order systems and systems with large error

constants, large leads are required resulting in excessively

large bandwidth which is undesirable from noise point of view.

For such systems lag compensation is desired, provided the

uncompensated system is absolutely stable.

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Design procedure for lag-lead compensator

STEP1: For the specified error constant determine the open

loop gain necessary

STEP 2: With this value of K draw the magnitude and phase

Bode plot and determine the phase margin and the GCF of

the uncompensated system.

STEP 3: First design the lag section to provide only partial

compensation of phase margin. Choose GCF such that it

is higher than the GCF if the system is fully lag compensated.

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STEP 4: determine the value of β, such that the high

frequency attenuation provided by the Lag network is equal

to the magnitude of the uncompensated system at this

frequency.

STEP 5: calculate the value of ꞇ1such that the upper cut off

frequency of lag network is below the gain crossover

frequency

STEP 6: calculate the lower cut-off frequency ω1=1/βꞇ

So the lag network design is over. Find the transfer function,

Draw the magnitude and phase bode plot of the lag

compensated system and determine the GCF and phase

margin,

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STEP 7: For the lead section independent value of α

cannot be chosen. So select α=1/β and calculate the

maximum lead provided by the lead section by

STEP 8: Choose the compensated cross over

frequency to coinside with ωm(corresponding to

φm).

This is the frequency where lag section has a gain

of -10 log β db.

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Calculate ꞇ2 and αꞇ2 and write the lead compensator

transfer function.

STEP 9: Combine the transfer function of the lag and lead

sections to get the lag-lead compensator transfer function.

Draw the bode plot and determine the phase margin.

Step 10: If the specifications are not met , redesign the

system by modifying the values of β and ꞇ .

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Lead lag compensator design

The open loop transfer function of a unity feedback system is

given by

Compensate the system to meet the following specifications

Velocity error constant Kv >= 25 s-1

Phase margin >= 60o

Solution

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The sinusoidal transfer function is

The bode plot of uncompensated system shows that the

GCF=6.5 rad/sec, and phase margin is -16o

Therefore the uncompensated system is unstable for the

given value of K. It is stable only for limited valve of K.

So lead lag network is used for compensation.

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First lag compensator is used to partially compensate phase

margin.

Let it provide 36o so that the new crossover frequency is

2rad/sec

The magnitude at 2 rad/sec is 18 dB therefore

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The upper cut off frequency of the lag network is two octaves

below this GCF

The lower cut off frequency

Therefore the T. F of lag compensator

The lag compensated network

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For the design of lead section

Therefore

To utilize this phase lead ,at ωm the magnitude must be

From the lag compensated system we can find that the

magnitude -9 db occurs at frequency ωm =3.8 rad/sec

Therefore

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The lead compensator T.f is