compensator design using bode plot

7/24/2019 Compensator Design Using Bode Plot

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1 Compensator Design Using Bode Plot

In this lecture we would revisit the continuous time design techniques using frequency domain

since these can be directly applied to design for digital control system by transferring the loop

transfer function in -plane to -plane.

1.1 Phase lead compensator

If we look at the frequency response of a simple PD controller, it is evident that the magnitude of

the compensator continuously grows with the increase in frequency.

The above feature is undesirable because it amplifies high frequency noise that is typically

present in any real system.

In lead compensator, a first order pole is added to the denominator of the PD controller at

frequencies well higher than the corner frequency of the PD controller.

typical lead compensator has the following transfer function.

where,

is the ratio between the pole !ero break point "corner# frequencies.

$agnitude of the lead compensator is . nd the phase contributed by the

lead compensator is given by

Thus a significant amount of phase is still provided with much less amplitude at high

frequencies.

The frequency response of a typical lead compensator is shown in %igure & where the magnitude


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varies from to and ma'imum phase is always less than ()* "around

+)* in general#.

%igure & %requency response of a lead compensator

It can be shown that the frequency where the phase is ma'imum is given by

The ma'imum phase corresponds to


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The magnitude of

Example 1: onsider the following system

Design a cascade lead compensator so that the phase margin "P$# is at least /* and steady state

error for a unit ramp input is 0 ).& .

The lead compensator is

where,

1teady state error for unit ramp input is

P$ of the closed loop system should be /*. 2et the gain crossover frequency of the

uncompensated system with 3 be 4g .


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Phase angle at 4g5 6.& is -() - tan -&6.& 5 - &+7* . Thus the P$ of the uncompensated system

with 3 is &8*.

If it was possible to add a phase without altering the magnitude, the additional phase leadrequired to maintain P$5 /* is /* - &8* 5 79* at 4 g5 6.& rad:sec.

;owever, maintaining same low frequency gain and adding a compensator would increase thecrossover frequency. s a result of this, the actual phase margin will deviate from the designed

one. Thus it is safe to add a safety margin of < to the required phase lead so that if it devaites

also, still the phase requirement is met. In general < is chosen between /* to &/*.

1o the additional phase requirement is 79* = &)* 5 69* , The lead part of the compensator willprovide this additional phase at 4ma'.

Thus

The only parameter left to be designed is >. To find >, one should locate the frequency at which

the uncompensated system has a logarithmic magnitude of .


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1elect this frequency as the new gain crossover frequency since the compensator provides a gain

of at 4ma'. Thus

In this case 4ma'5 4g new 5 .& . Thus

The lead compensator is thus

?ith this compensator actual phase margin of the system becomes (.+* which meets the designcriteria.

The

corresponding@ode plot is shown in %igure 7

Figure 2:@ode plot of the compensated system for A'ample &
http://nptel.ac.in/courses/108103008/module5/lec6/3.html#m5l6f2http://nptel.ac.in/courses/108103008/module5/lec6/3.html#m5l6f2


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Example 2:

Bow let us consider that the system as described in the previous e'ample is subCect to a sampleddata control system with sampling time T 5 ).7 sec. Thus

The bi-linear transformation


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will transfer into w -plane, as

please try the simplificationE

?e need first design a phase lead compensator so that P$ of the compensated system is at least

/)* with 3v5 7 . The compensator in w -plane is

Design steps are as follows.

F 3 has to be found out from the 3vrequirement.

F $ake 4ma'5 4gnew.

F ompute the gain crossover frequency 4gand phase margin of the uncompensated

system after introducing 3 in the system.

F t 4gcheck the additional:required phase lead, add safety margin, find out .

alculate G from the required

F 1ince the lead part of the compensator provides a gain of , find out the

frequency where the logarithmic magnitude is . This will be the new

gain crossover frequency where the ma'imum phase lead should occur.

F alculate > from the relation


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Bow,

Hsing $T2@ command marginJJ, phase margin of the system with 3 5 7 is computed as

6&.+* with 4g5 &.7+ rad:sec, as shown in %igure 6.

Figure 3:@ode plot of the uncompensated system for A'ample 7

Thus the required phase lead is /)* - 6&.+* 5 &8.* . fter adding a safety margin of &&.+* ,

becomes 6)* . ;ence

%rom the frequency response of the system it can be found out that at 4 5 &.9/ rad:sec, the

magnitude of the system is . Thus 4ma'5 4gnew 5 &.9/ rad:sec. This gives
http://nptel.ac.in/courses/108103008/module5/lec6/5.html#m5l6f3http://nptel.ac.in/courses/108103008/module5/lec6/5.html#m5l6f3http://nptel.ac.in/courses/108103008/module5/lec6/5.html#m5l6f3


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Kr,

Thus the controller in w-plane is

The @ode plot of the compensated system is shown in %igure .

Figure 4:@ode plot of the compensated system for A'ample 7

Le-transforming the above controller into ! -plane using the relation , we get the

controller in ! -plane, as


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End

1 Lag Compensator Design

In the previous lecture we discussed lead compensator design. In this lecture we would see howto design a phase lag compensator

1.1 Phase lag compensator

The essential feature of a lag compensator is to provide an increased low frequency gain, thusdecreasing the steady state error, without changing the transient response significantly.

%or frequency response design it is convenient to use the following transfer function of a lagcompensator.

?here,

The above e'pression is only the lag part of the compensator. The overall compensator is

Typical obCective of lag compensator design is to provide an additional gain of G in the low

frequency region and to leave the system with sufficient phase margin.

The frequency response of a lag compensator, with G5 and >56, is shown in %igure & where the

magnitude varies from d@ to ) d@.


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%igure & %requency response of a lag compensator

1ince the lag compensator provides the ma'imum lag near the two corner frequencies, tomaintain the P$ of the system, !ero of the compensator should be chosen such that 4 5 &: > is

much lower than the gain crossover frequency of the uncompensated system.

In general, > is designed such that &: > is at least one decade below the gain crossover frequency

of the uncompensated system. %ollowing e'ample will be comprehensive to understand the

design procedure.

Example 1: onsider the following system

Design a lag compensator so that the phase margin "P$# is at least /)* and steady state error to a

unit step input is .


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The overall compensator is

where,

?hen

1teady state error for unit step input is

Thus,

Bow let us modify the system transfer function by introducing 3 with the original system. Thus

the modified system becomes

P$ of the closed loop system should be /)*. 2et the gain crossover frequency of the

uncompensated system with 3 be 4g .


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Lequired P$ is /)*. 1ince the P$ is achieved only by selecting 3, it might be deviated from this

value when the other parameters are also designed. Thus we put a safety margin of /* to the P$

which makes the required P$ to be //*.

To make 4g5 7.8 rad:sec, the gain crossover frequency of the modified system, magnitude at 4 g

should be &. Thus

Putting the value of 4gin the last equation, we get 3 5 /.&. Thus,

The only parameter left to be designed is >.

1ince the desired P$ is already achieved with gain 3, ?e should place 4 5 &: > such that it does

not much effect the P$ of the modified system with 3. If we place &: > one decade below thegain crossover frequency, then

or,



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?ith this compensator actual phase margin of the system becomes /7.9*, as shown in %igure 7,

which meets the design criteria.

Figure 2:@ode plot of the compensated system for A'ample &

Example 2:

Bow let us consider that the system as described in the previous e'ample is subCect to a sampleddata control system with sampling time T 5 ).& sec. ?e would use $T2@ to derive the plant

transfer function w -plane.

Hse the below commands.
http://nptel.ac.in/courses/108103008/module5/lec7/3.html#m5l7f2http://nptel.ac.in/courses/108103008/module5/lec7/3.html#m5l7f2


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>> s=t!"s"#$

>> gc=1%!!s&1#'!(.)'s&1##$

>> g*=c2+!gc,(.1,"*oh"#$

Mou would get

The bi-linear transformation

will transfer into w-plane. Hse the below commands

>> aug=-(.1,1$

>> g/ss = 0ilin!ss!g*#,1,"ust",aug#

>> g/=t!g/ss#

to find out the transfer function in w-plane, as

The @ode plot of the uncompensated system is shown in %igure 6.


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Figure 3:@ode plot of the uncompensated system for A'ample 7

?e need to design a phase lag compensator so that P$ of the compensated system is at least /)*

and steady state error to a unit step input is ).&. The compensator in w -plane is

where,

1ince , for ).& steady state error.

Bow let us modify the system transfer function by introducing 3 to the original system. Thus themodified system becomes


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P$ of the closed loop system should be /)*. 2et the gain crossover frequency of theuncompensated system with 3 be 4g. Then,

Lequired P$ is /)*. 2et us put a safety margin of /*. Thus the P$ of the system modified with3 should be //*.

@y solving the above, 4g5 7. rad:sec. Thus the magnitude at 4gshould be &.

Putting the value of 4gin the last equation, we get 3 5 .&6 .

Thus,

If we place &: > one decade below the gain crossover frequency, then


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or,

Thus the controller in w -plane is

Le-transforming the above controller into ! -plane using the relation , we get


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Lag -lead Compensator

?hen a single lead or lag compensator cannot guarantee the specified design criteria, a lag-lead

compensator is used.

In lag-lead compensator the lag part precedes the lead part. continuous time lag-leadcompensator is given by

where,

The corner frequencies are , , , . The frequency response is shown in %igure

&.


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%igure & %requency response of a lag-lead compensator

In a nutshell,

If it is not specied hich t!pe of compensator has to "e designed# onesho$ld rst chec% the P& and B' of the $ncompensated s!stem ithad($sta"le gain ).

F If the @? is smaller than the acceptable @? one may go for lead compensator. If the

@? is large, lead compensator may not be useful since it provides high frequency

amplification.

F Kne may go for a lag compensator when @? is large provided the open loop system is

stable.

F If the lag compensator results in a too low @? "slow speed of response#, a lag-lead

compensator may be used.

1.1 Laglea+ compensator +esign

Example 1onsider the following system with transfer function

Design a lag-lead compensator "s# such that the phase margin of the compensated system is at

least /* at gain crossover frequency around &) rad:sec and the velocity error constant 3vis 6).

The lag-lead compensator is given by

where,

?hen


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Thus 3 5 6) . @ode plot of the modified system 3N"s# is shown in %igure 7. The gain crossover

frequency and phase margin of 3N"s# are found out to be (.99 rad:sec and -&9.7* respectively.

%igure 7 @ode plot of the uncompensated system for A'ample &

1ince the P$ of the uncompensated system with 3 is negative, we need a lead compensator tocompensate for the negative P$ and achieve the desired phase margin.

;owever, we know that introduction of a lead compensator will eventually increase the gain

crossover frequency to maintain the low frequency gain.

Thus the gain crossover frequency of the system cascaded with a lead compensator is likely to be

much above the specified one, since the gain crossover frequency of the uncompensated systemwith 3 is already (.99 rad:sec.

Thus a lag-lead compensator is required to compensate for both.

?e design the lead part first.


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%rom %igure 7, it is seen that at &) rad:sec the phase angle of the system is -&(8*.

1ince the new 4gshould be &) rad:sec, the required additional phase at 4g, to maintain the

specified P$, is / - "&8) - &(8# 5 +6* . ?ith safety margin 7*,

nd

which gives . ;owever, introducing this compensator will actually increase the gain

crossover frequency where the phase characteristic will be different than the designed one. This

can be seen from %igure 6.


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Figure 3:%requency response of the system in A'ample & with only a lead compensator

The gain crossover frequency is increased to 76.7 rad:sec. t &) rad:sec, the phase angle is -&6*

and gain is &7.+ d@. To make this as the actual gain crossover frequency, lag part should providean attenuation of -&7.+ d@ at high frequencies.

t high frequencies the magnitude of the lag compensator part is . Thus ,

which gives . Bow, should be placed much below the new gain crossover

frequency to retain the desired P$. 2et be ).7/. Thus


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The frequency response of the system after introducing the above compensator is shown in

%igure , which shows that the desired performance criteria are met.

Figure 4: %requency response of the system in A'ample & with a lag-lead compensator

compensator design using bode plot

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