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 beca use it amplifies high frequency noise that is typically present in any real system. In lead compensator, a first order pole is added to the den ominator of the PD controller at frequencies well higher than the corner frequenc y of the PD controller. typical lead compensator has the following transfer function.
where,
is the ratio between the pole !ero break point "corner# frequencies.
$agnitude of the lead compensator is lead compensator is given by
. nd the phase contributed by the
Thus a significant amount of phase is still provided with much less amplitude at high frequencies. The frequency response of a typical lead compensator is shown in %igure & where the magnitude
varies from +)* in general#.
to
and ma'imum phase is always less than ()* "around
%igure & %requency response of a lead compensator It can be shown that the frequency where the phase is ma'imum is given by
The ma'imum phase corresponds to
The magnitude of
Example 1: onsider the following system
Design a cascade lead compensator so that the phase margin "P$# is at least /* and stead y 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 .
Phase angle at 4g 5 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 lead required to maintain P$5 /* is /* - &8* 5 79* at 4 g 5 6.& rad:sec. ;owever, maintaining same low frequency gain and adding a compensator would increase the crossover frequency. s a result of this, the actual phase margin will deviate from the designed one. Thus it is safe to add a safety margin of < to the required phase lead so that if it devaites also, still the phase requirement is met. In general < is chosen between /* to &/*. 1o the additional phase requirement is 79* = &)* 5 69* , The lead part of the compensator will provide this additional phase at 4ma' . Thus
The only parameter left to be designed is >. To find >, one should locate the frequency at which
the uncompensated system has a logarithmic magnitude of
.
1elect this frequency as the new gain crossover frequency since the compensator provides a gain
of
at 4ma'. Thus
In this case 4ma' 5 4g new 5 .& . Thus
The lead compensator is thus
?ith this compensator actual phase margin of the system becomes (.+* which meets the design criteria. Figure 2: @ode plot of the compensated system for A'ample &
@ode plot is shown in %igure 7
The corresponding
Example 2:
Bow let us consider that the system as described in the previous e'ample is subCect to a sampled data control system with sampling time T 5 ).7 sec. Thus
The bi-linear transformation
will transfer
into w -plane, as
please try the simplificationE
?e need first design a phase lead compensator so that P$ of the compensated system is at least /)* with 3 v 5 7 . The compensator in w -plane is
Design steps are as follows. •
F 3 has to be found out from the 3 v requirement. F $ake 4ma' 5 4gnew. F ompute the gain crossover frequency 4g and phase margin of the uncompensated system after introducing 3 in the system. F t 4g check 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
Bow,
Hsing $T2@ command marginJJ, phase margin of the system with 3 5 7 is computed as 6&.+* with 4g 5 &.7+ rad:sec, as shown in %igure 6.
Figure 3: @ode plot of the uncompensated system for A'ample 7
Thus the required phase lead is /)* - 6&.+* 5 &8.* . fter adding a safety margin of &&.+* , becomes 6)* . ;ence
%rom the frequency response of the system it can be found out that at 4 5 &.9/ rad:sec, the
magnitude of the system is
. Thus 4ma' 5 4gnew 5 &.9/ rad:sec. This gives
Kr,
Thus the controller in w-plane is
The @ode plot of the compensated system is shown in %igure .
Figure 4: @ode plot of the compensated system for A'ample 7
Le-transforming the above controller into ! -plane using the relation controller in ! -plane, as
, we get the
End………
1 Lag Compensator Design In the previous lecture we discussed lead co mpensator design. In this lecture we would see ho w to design a phase lag compensator
1.1 Phase lag compensator The essential feature of a lag compensator is to provide an increased low frequency gain, thus decreasing the steady state error, without changing the transient response significantly. %or frequency response design it is convenient to use the following transfer function of a lag compensator.
?here,
The above e'pression is only the lag part of the compensator. The overall compensator is
Typical obCective of lag compensator design is to provide an additional gain of G in the low frequency region and to leave the system with sufficient phase margin. The frequency response of a lag compensator, with G5 and >56, is shown in %igure & where the magnitude varies from
d@ to ) d@.
%igure & %requency response of a lag compensator
1ince the lag compensator provides the ma'imum lag near the two corner frequencies, to maintain the P$ 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$# is at least /)* and steady state error to a unit step input is
.
The overall compensator is
where,
?hen 1teady state error for unit step input is
Thus, Bow let us modify the system transfer function by introducing 3 with the original system. Thus the modified system becomes
P$ of the closed loop system should be /)*. 2et the gain crossover frequency of the uncompensated system with 3 be 4g .
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 4g 5 7.8 rad:sec, the gain crossover frequency of the modified system, magnitude at 4g should be &. Thus
Putting the value of 4g in 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 the gain crossover frequency, then
or, The overall compensator is
?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 sampled data control system with sampling time T 5 ).& sec. ?e would use $T2@ to derive the plant transfer function w -plane. Hse the below commands.
>> s=t!"s"#$ >> 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.
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 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 . Then,
Lequired P$ is /)*. 2et us put a safety margin of /*. Thus the P$ of the system modified with 3 should be //*.
@y solving the above, 4g 5 7. rad:sec. Thus the magnitude at 4g should be &.
Putting the value of 4g in the last equation, we get 3 5 .&6 . Thus,
If we place &: > one decade below the gain crossover frequency, then
or, Thus the controller in w -plane is
Le-transforming the above controller into ! -plane using the relation
, we get
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-lead compensator is given by
where,
The corner frequencies are &.
,
,
,
. The frequency response is shown in %igure
%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# one sho$ld rst chec% the P& and B' of the $ncompensated s!stem ith ad($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 provide s high frequency amplification. F Kne may go for a lag compensator when @? is large provided the ope n 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 1 onsider 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 3 v is 6). The lag-lead compensator is given by
where,
?hen
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 to compensate 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 system with 3 is already (.99 rad:sec. Thus a lag-lead compensator is required to co mpensate for both. ?e design the lead part first.
%rom %igure 7, it is seen that at &) rad:sec the phase angle of the system is -&(8*. 1ince the new 4g should be &) rad:sec, the required additional phase at 4g, to maintain the specified P$, is / - "&8) - &(8# 5 +6* . ?ith safety margin 7*,
nd
which gives . ;owever, introducing this compensator will actually increase the gain crossover frequency where the phase characteristic will be different than the designed one. This can be seen from %igure 6.
Figure 3: %requency response of the system in A'ample & with only a lead compensator
The gain crossover frequency is increased to 76.7 rad:sec. t &) rad:sec, the phase angle is -&6* and gain is &7.+ d@. To make this as the actual gain crossover frequenc y, lag part should provide an attenuation of -&7.+ d@ at high frequencies.
t high frequencies the magnitude of the lag compensator part is
which gives
. Bow,
. Thus ,
should be placed much below the new gain crossover
frequency to retain the desired P$. 2et
be ).7/. Thus
The overall compensator is
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
