EXPERIMENT NO.
- 8
OBJECT: Design lead-lag compensator for the given system
There are three types of compensators — lag, lead and lag-lead compensators. These are most
commonly used.
Lag Compensator
The Lag Compensator is an electrical network which produces a sinusoidal output having the
phase lag when a sinusoidal input is applied. The lag compensator circuit in the ‘s’ domain is
shown in the following figure.
Here, the capacitor is in series with the resistor R2 and the output is measured across this
combination.
𝑉𝑜 (𝑠) 1 𝑠+1�
The transfer function of this lag compensator is 𝑉𝑖 (𝑠)
= 𝛼 (𝑠+1� 𝜏 )
𝛼𝜏
𝑅1 +𝑅2
Where, 𝜏 = 𝑅2 𝐶 𝛼= 𝑅2
From the above equation, α is always greater than one.
From the transfer function, we can conclude that the lag compensator has one pole at s=−1/ατ
and one zero at s=−1/τ. This means, the pole will be nearer to origin in the pole-zero
configuration of the lag compensator.
𝑉𝑜 (𝑗𝜔) 1 𝑗𝜔+1�
Substitute, s=jω in the transfer function, 𝑉 (𝑗𝜔)
= 𝛼 (𝑗𝜔+1� 𝜏 )
𝑖 𝛼𝜏
Phase angle 𝜙 = 𝑡𝑎𝑛−1 𝜔𝑡 − 𝑡𝑎𝑛−1 𝛼𝜔𝑡
We know that, the phase of the output sinusoidal signal is equal to the sum of the phase angles
of input sinusoidal signal and the transfer function.
So, in order to produce the phase lag at the output of this compensator, the phase angle of the
transfer function should be negative. This will happen when α >1.
Lead Compensator
The lead compensator is an electrical network which produces a sinusoidal output having phase
lead when a sinusoidal input is applied. The lead compensator circuit in the ‘s’ domain is shown
in the following figure.
Here, the capacitor is parallel to the resistor R1 and the output is measured across resistor R2.
𝑉𝑜 (𝑠) 𝑠𝜏+1
The transfer function of this lead compensator is - 𝑉 (𝑠)
= 𝛽(𝛽𝑠𝜏+1)
𝑖
𝑅2
Where, 𝜏 = 𝑅1 𝐶 𝛽=
𝑅1 +𝑅2
From the transfer function, we can conclude that the lead compensator has pole at s=−1/β and
zero at s=−1/βτ.
𝑉 (𝑗𝜔) 𝑗𝜔𝜏+1
Substitute, s=jω in the transfer function 𝑉𝑜(𝑗𝜔) = 𝛽(𝑗𝜔𝛽𝜏+1)
𝑖
Phase angle 𝜙 = 𝑡𝑎𝑛−1 𝜔𝑡 − 𝑡𝑎𝑛−1 𝛽𝜔𝑡
We know that, the phase of the output sinusoidal signal is equal to the sum of the phase angles
of input sinusoidal signal and the transfer function.
So, in order to produce the phase lead at the output of this compensator, the phase angle of the
transfer function should be positive. This will happen when 0<β<1. Therefore, zero will be
nearer to origin in pole-zero configuration of the lead compensator.
Lag-Lead Compensator
Lag-Lead compensator is an electrical network which produces phase lag at one frequency
region and phase lead at other frequency region. It is a combination of both the lag and the lead
compensators. The lag-lead compensator circuit in the ‘s’ domain is shown in the following
figure.
This circuit looks like both the compensators are cascaded. So, the transfer function of this
circuit will be the product of transfer functions of the lead and the lag compensators.
𝑉𝑜 (𝑠) 𝑠𝜏1 + 1 1 𝑠 + 1�𝜏2
= 𝛽( ) ( )
𝑉𝑖 (𝑠) 𝛽𝑠𝜏1 + 1 𝛼 𝑠 + 1�𝛼𝜏
2
𝑉𝑜 (𝑠) 𝑠+1�𝜏1 𝑠+1�𝜏
αβ=1, 𝑉 (𝑠)
= 𝐾( 1 )( 1 2 )
𝑖 𝑠+ �𝛽𝜏 𝑠+ �𝛼𝜏2
1
Where, 𝜏1 = 𝑅1 𝐶1 𝜏2 = 𝑅2 𝐶2
Lead compensation essentially yields an appreciable improvement in transient response and a
small change in steady-state accuracy. It may accentuate high-frequency noise effects. Lag
compensation on the other hand, yields an appreciable improvement in steady-state accuracy at
the expense of increasing the transient-response time. Lag compensation will suppress the
effects of high-frequency noise singles. Lead-Lag compensation combines the characteristics of
both lead compensation and lag compensation. The use of a lead or lag compensator raises the
order of the system by (unless cancellation occurs between the zero of the compensator and a
pole of the uncompensated open-loop transfer function). The use of a lead-lag compensator
raises the order of the system by (unless cancellation occurs between zeros of the lead-lag
compensator and poles of the uncompensated open-loop transfer function), which means that
the system becomes more complex and it is more difficult to control the transient response
behavior
Exercise 1: Lead Compensator
In order to examine the transient-response characteristics of the designed system, we shall
obtain the unit-step and unit-ramp response curves of the compensated and uncompensated
system with MATLAB. Note that the closed-loop transfer functions of the uncompensated and
compensated system are given, respectively,
𝐶(𝑠) 4
= 2
𝑅(𝑠) 𝑠 + 2𝑠 + 4
𝐶(𝑠) 166.8𝑠 + 735.588
= 3
𝑅(𝑠) 𝑠 + 20.4𝑠 2 + 203.6𝑠 + 735.588
Solution: The MATLAB code for obtaining the unit step responses of both compensated and
uncompensated systems is
*****Unit-Step Responses*****
num=[0 0 4];
den=[1 2 4];
numc=[0 0 166.8 735.588];
denc=[1 20.4 203.6 735.588];
t=0:0.2:6;
[c1,x1,t]=step(num,den,t);
[c2,x2,t]=step(numc,denc,t);
plot(t,c1,'*',t,c2,'+')
grid
title('Unit-Step Responses of Compensated and Uncompensated Systems')
xlabel('t Sec')
ylabel('Outputs')
text(0.4,1.31,'Compensated System')
text(1.55,0.88,'Uncompensated System')
Unit-Step Responses of Compensated and Uncompensated Systems
1.4
Compensated System
1.2
Uncompensated System
0.8
Outputs
0.6
0.4
0.2
0
0 1 2 3 4 5 6
t Sec
The MATLAB code for obtaining the ramp responses of both compensated and uncompensated
systems is
*****Unit-Ramp Responses*****
num1=[0 0 0 4];
den1=[1 2 4 0];
num1c=[0 0 0 166.8 735.588];
den1c=[1 20.4 203.6 735.588 0];
t=0:0.05:5;
[y1,z1,t]=step(num1,den1,t);
[y2,z2,t]=step(num1c,den1c,t);
plot(t,y1,'*',t,y2,'+',t,t,'--')
grid
title('Unit-Ramp Responses of Compensated and Uncompensated Systems')
xlabel('t Sec')
ylabel('Outputs')
text(0.89,3.7,'Compensated System')
text(2.25,1.1,'Uncompensated System'
Unit-Ramp Responses of Compensated and Uncompensated Systems
5
4.5
Compensated System
3.5
3
Outputs
2.5
1.5
Uncompensated System
1
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
t Sec
Observations & conclusion:
1) Lead compensation -achieves the desired result through the merits of its phase–lead
contribution, whereas lag compensation accomplishes the result through the merits of its
attenuation property at high frequencies.
a. Lead compensation is commonly used for improving stability margins.
b. Lead compensation yields a higher gain crossover frequency than is possible with
lag compensation. The higher gain crossover frequency means larger band width. A
large band width means reduction in the settling time. The band width of a system
with lead compensation is always greater than that with lag compensation.
Therefore, if a large band width or fast response is desired, lead compensation
should be employed. If, however noise signals are present, then a large band width
may not be desirable since it makes the system more susceptible to noise signals
because of increase in the high frequency gain.
c. Lead compensation requires an additional increase in gain to offset the attenuation
inherent in the lead network. This means that lead compensation will require a larger
gain than that required by lag compensation. A larger gain, in most cases, implies
larger space, greater weight, and higher cost.
2. Lag compensation -reduces the system gain at higher frequencies without reducing the system
gain at lower frequencies. Since the system band width is reduced, the system has a slower
speed to respond. Because of the reduced high frequency gain, the total system gain can be
increased, and thereby low frequency gain can be increased and the steady state can be
improved. Also, any high frequency noises involved in the system can be attenuated.
3. If both fast responses and good static accuracy are desired, a lag –lead compensator may be
employed. By use of the lag–lead compensator, the low–frequency gain can be increased (which
means an improvement in steady- state accuracy), while at the same time the system band width
and stability margins can be increased. Although a large number of practical compensation tasks
can be accomplished with lead, lag –lead compensators; for complicated system simple
compensation by use of these compensators may not yield satisfactory results. Then, different
compensators having different pole-zero configurations must be employed.
