Control system compensator lag lead

Control System
Lag, Lead, Lag-Lead Compensator
Dr. Nilesh Bhaskarrao Bahadure
nbahadure@gmail.com
https://www.sites.google.com/site/nileshbbahadure/home
June 30, 2021
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Control System June 30, 2021 1 / 30

Overview I
1 Necessary of Compensation
2 Methods of Compensation
3 Phase Lead Compensation
Effect of Phase Lead Compensation
Advantages of Phase Lead Compensation
Disadvantages of Phase Lead Compensation
4 Phase Lag Compensation
Effect of Phase Lag Compensation
Advantages of Phase Lag Compensation
Disadvantages of Phase Lag Compensation
5 Phase Lag Lead Compensation
6 Comparison between lead and lag compensators
7 Thank You

Necessary of Compensation
1 In order to obtain the desired performance of the system, we use
compensating networks. Compensating networks are applied to the
system in the form of feed forward path gain adjustment.

2 Compensate a unstable system to make it stable.

3 A compensating network is used to minimize overshoot.

4 These compensating networks increase the steady state accuracy of
the system. An important point to be noted here is that the increase
in the steady state accuracy brings instability to the system.

4 These compensating networks increase the steady state accuracy of
the system. An important point to be noted here is that the increase
in the steady state accuracy brings instability to the system.
5 Compensating networks also introduces poles and zeros in the system
thereby causes changes in the transfer function of the system. Due to
this, performance specifications of the system change.

Methods of Compensation
1 Series compensation
Series Compensation
Connecting compensating circuit between error detector and plants known
as series compensation.
Figure : series compensator

1 Series compensation
2 Feedback compensation
Series Compensation
Connecting compensating circuit between error detector and plants known
as series compensation.
Figure : series compensator

Feedback Compensation
When a compensator used in a feedback manner called feedback
compensation.
Figure : Feedback compensator

Phase Lead Compensation

A system which has one pole and one dominating zero is known as lead
network.

network. If we want to add a dominating zero for compensation in control
system then we have to select lead compensation network.

The lead compensator is an electrical network which produces a sinusoidal
output having phase lead when a sinusoidal input is applied.

output having phase lead when a sinusoidal input is applied.The lead
compensator circuit in the ’s’ domain is shown in the figure 3

output having phase lead when a sinusoidal input is applied.The lead
compensator circuit in the ’s’ domain is shown in the figure 3
Figure : Lead Compensator

Phase Lead Compensation...

Figure : Lead Compensator

From the circuit we get,

I1 = C
d
dt
(ei − eo)

I1 = C
d
dt
(ei − eo)
I2 =
ei − eo
R1

I1 = C
d
dt
(ei − eo)
I2 =
ei − eo
R1
I = I1 + I2
Also,

I1 = C
d
dt
(ei − eo)
I2 =
ei − eo
R1
I = I1 + I2
Also,
I =
eo
R2
Equating above equations we get,

I1 = C
d
dt
(ei − eo)
I2 =
ei − eo
R1
I = I1 + I2
Also,
I =
eo
R2
Equating above equations we get,
eo
R2
= C
d
dt
(ei − eo) +
ei − eo
R1

Now let us determine the transfer function for the given network and the
transfer function can be determined by finding the ratio of the output
voltage to the input voltage. So taking Laplace transform of both side of
above equations,

above equations,
Eo(s)
R2
=
1
R1
[Ei (s) − Eo(s)] + Cs[Ei (s) − Eo(s)]

above equations,
Eo(s)
R2
=
1
R1
[Ei (s) − Eo(s)] + Cs[Ei (s) − Eo(s)]
Eo(s)
Ei (s)
=
1+sCR1
R1
R1+R2+sR1R2C
R2R1

above equations,
Eo(s)
R2
=
1
R1
[Ei (s) − Eo(s)] + Cs[Ei (s) − Eo(s)]
Eo(s)
Ei (s)
=
1+sCR1
R1
R1+R2+sR1R2C
R2R1
Eo(s)
Ei (s)
=
R2
R1 + R2
[
1 + sCR1
1 + sR1R2C
R1+R2
]

above equations,
Eo(s)
R2
=
1
R1
[Ei (s) − Eo(s)] + Cs[Ei (s) − Eo(s)]
Eo(s)
Ei (s)
=
1+sCR1
R1
R1+R2+sR1R2C
R2R1
Eo(s)
Ei (s)
=
R2
R1 + R2
[
1 + sCR1
1 + sR1R2C
R1+R2
]
On substituting the α = R1+R2
R2
and T = R1R2
R1+R2

above equations,
Eo(s)
R2
=
1
R1
[Ei (s) − Eo(s)] + Cs[Ei (s) − Eo(s)]
Eo(s)
Ei (s)
=
1+sCR1
R1
R1+R2+sR1R2C
R2R1
Eo(s)
Ei (s)
=
R2
R1 + R2
[
1 + sCR1
1 + sR1R2C
R1+R2
]
R2
and T = R1R2
R1+R2
Where, T and α are respectively the time constant and attenuation
constant, we have

above equations,
Eo(s)
R2
=
1
R1
[Ei (s) − Eo(s)] + Cs[Ei (s) − Eo(s)]
Eo(s)
Ei (s)
=
1+sCR1
R1
R1+R2+sR1R2C
R2R1
Eo(s)
Ei (s)
=
R2
R1 + R2
[
1 + sCR1
1 + sR1R2C
R1+R2
]
R2
and T = R1R2
R1+R2
Where, T and α are respectively the time constant and attenuation
constant, we have
Transfer Function =
Eo(s)
Ei (s)
=
1
α
[
1 + sT
1 + αsT
]

The above network can be visualized as an amplifier with a gain of 1
α . Let
us draw the pole zero plot for the above transfer function.

The above network can be visualized as an amplifier with a gain of 1
α . Let
us draw the pole zero plot for the above transfer function.
Figure : Pole Zero Plot

Clearly we have −1
T (which is a zero of the transfer function) is closer to
origin than the −1
αT - (which is the pole of the transfer function).

αT - (which is the pole of the transfer function). Thus we
can say in the lead compensator zero is more dominating than the pole
and because of this lead network introduces positive phase angle to the
system when connected in series.
Let us substitute s = jω in the above transfer function and also we have
α < 1. On finding the phase angle function for the transfer function we
have

have
Transfer Function =
Eo(jω)
Ei (jω)
=
1
α
[
1 + jωT
1 + jαωT
]

have
Transfer Function =
Eo(jω)
Ei (jω)
=
1
α
[
1 + jωT
1 + jαωT
]
θ(ω) = tan−1
(ωT) − tan−1
(αωT)

1 The slope of the magnitude plot reduces at the gain crossover
frequency so that relative stability improves and error decrease due to
error is directly proportional to the slope.

2 Phase margin increases.

2 Phase margin increases.
3 Response become faster.

1 Due to the presence of phase lead network the speed of the system
increases because it shifts gain crossover frequency to a higher value.

1 Due to the presence of phase lead network the speed of the system
increases because it shifts gain crossover frequency to a higher value.
2 Due to the presence of phase lead compensation maximum overshoot
of the system decreases.

Disadvantages of Phase Lead Compensation
1 Steady state error is not improved.

Phase Lag Compensation

A system which has one zero and one dominating pole ( the pole which is
closer to origin that all other poles is known as dominating pole) is known
as lag network.

A system which has one zero and one dominating pole ( the pole which is
closer to origin that all other poles is known as dominating pole) is known
as lag network. If we want to add a dominating pole for compensation in
control system then, we have to select a lag compensation network. The
basic requirement of the phase lag network is that all poles and zeros of
the transfer function of the network must lie in (-)ve real axis interlacing
each other with a pole located or on the nearest to the origin.

Phase Lag Compensation...

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 figure 6.

Figure : Phase Lag Compensator

Here, the capacitor is in series with the resistor R2 and the output is
measured across this combination.

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The transfer function of this lag compensator is

We will have the output at the series combination of the resistor R2 and
the capacitor C. From the above circuit diagram, we get

ei = iR1 + iR2 +
1
C
Z
idt

ei = iR1 + iR2 +
1
C
Z
idt
eo = iR2 +
1
C
Z
idt

ei = iR1 + iR2 +
1
C
Z
idt
eo = iR2 +
1
C
Z
idt
voltage to the input voltage.

ei = iR1 + iR2 +
1
C
Z
idt
eo = iR2 +
1
C
Z
idt
voltage to the input voltage.
Taking Laplace transform of above two equation we get,
Ei (s) = R1I(s) + R2I(s) +
1
Cs
I(s)
Eo(s) = R2I(s) +
1
Cs
I(s)

Transfer Function =
Eo(s)
Ei (s)
=
R2 + 1
Cs
R1 + R2 + 1
Cs

Transfer Function =
Eo(s)
Ei (s)
=
R2 + 1
Cs
R1 + R2 + 1
Cs
G(s) =
1 + R2Cs
1 + (R1 + R2)Cs

Transfer Function =
Eo(s)
Ei (s)
=
R2 + 1
Cs
R1 + R2 + 1
Cs
G(s) =
1 + R2Cs
1 + (R1 + R2)Cs
On substituting the T = R2C and β = R1+R2
R1
, Where, T and β are
respectively the time constant and DC gain

Transfer Function =
Eo(s)
Ei (s)
=
R2 + 1
Cs
R1 + R2 + 1
Cs
G(s) =
1 + R2Cs
1 + (R1 + R2)Cs
R1
Transfer Function = G(s) =
1 + Ts
1 + βTs

Transfer Function =
Eo(s)
Ei (s)
=
R2 + 1
Cs
R1 + R2 + 1
Cs
G(s) =
1 + R2Cs
1 + (R1 + R2)Cs
R1
Transfer Function = G(s) =
1 + Ts
1 + βTs
The above network provides a high frequency gain of 1
β . Let us draw the
pole zero plot for the above transfer function.

Figure : Pole Zero Plot

T (which is a zero of the transfer function) is far to
βT (which is the pole of the transfer function). Thus we
can say in the lag compensator pole is more dominating than the zero and
because of this lag network introduces negative phase angle to the system
when connected in series.

β < 1. On finding the phase angle function for the transfer function we
have

β < 1. On finding the phase angle function for the transfer function we
have
θ(ω) = tan−1
(ωT) − tan−1
(βωT)

1 Gain crossover frequency increases.

2 Bandwidth decreases.

3 Phase margin will be increase.

3 Phase margin will be increase.
4 Response will be slower before due to decreasing bandwidth, the rise
time and the settling time become larger.

1 Phase lag network allows low frequencies and high frequencies are
attenuated.

1 Phase lag network allows low frequencies and high frequencies are
attenuated.
2 Due to the presence of phase lag compensation the steady state
accuracy increases.

Disadvantages of Phase Lag Compensation
1 Due to the presence of phase lag compensation the speed of the
system decreases.

Phase Lag Lead Compensation

With single lag or lead compensation may not satisfied design
specifications.

specifications. For an unstable uncompensated system, lead compensation
provides fast response but does not provide enough phase margin whereas
lag compensation stabilize the system but does not provide enough
bandwidth.

bandwidth. So we need multiple compensators in cascade.

bandwidth. So we need multiple compensators in cascade.
Given below is the circuit diagram for the phase lag- lead compensation
network.

Phase Lag Lead Compensation...

Figure : Phase Lag Lead Compensator

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)
Vi (s)
=
Eo(s)
Ei (s)
= β

sτ1 + 1
βsτ1 + 1

1
α
s + 1
τ2
s + 1
ατ2
!

Vo(s)
Vi (s)
=
Eo(s)
Ei (s)
= β

sτ1 + 1
βsτ1 + 1

1
α
s + 1
τ2
s + 1
ατ2
!
We know αβ = 1.

Vo(s)
Vi (s)
=
Eo(s)
Ei (s)
= β

sτ1 + 1
βsτ1 + 1

1
α
s + 1
τ2
s + 1
ατ2
!
We know αβ = 1.
⇒
Vo(s)
Vi (s)
=
Eo(s)
Ei (s)
=
s + 1
τ1
s + 1
βτ1
!
s + 1
τ2
s + 1
ατ2
!

Vo(s)
Vi (s)
=
Eo(s)
Ei (s)
= β

sτ1 + 1
βsτ1 + 1

1
α
s + 1
τ2
s + 1
ατ2
!
We know αβ = 1.
⇒
Vo(s)
Vi (s)
=
Eo(s)
Ei (s)
=
s + 1
τ1
s + 1
βτ1
!
s + 1
τ2
s + 1
ατ2
!
Where,

Vo(s)
Vi (s)
=
Eo(s)
Ei (s)
= β

sτ1 + 1
βsτ1 + 1

1
α
s + 1
τ2
s + 1
ατ2
!
We know αβ = 1.
⇒
Vo(s)
Vi (s)
=
Eo(s)
Ei (s)
=
s + 1
τ1
s + 1
βτ1
!
s + 1
τ2
s + 1
ατ2
!
Where,
τ1 = R1C1

Vo(s)
Vi (s)
=
Eo(s)
Ei (s)
= β

sτ1 + 1
βsτ1 + 1

1
α
s + 1
τ2
s + 1
ατ2
!
We know αβ = 1.
⇒
Vo(s)
Vi (s)
=
Eo(s)
Ei (s)
=
s + 1
τ1
s + 1
βτ1
!
s + 1
τ2
s + 1
ατ2
!
Where,
τ1 = R1C1
τ2 = R2C2

Comparison between lead and lag compensators

Lead compensator

Lead compensator Lag compensator

High pass

High pass Low pass

High pass Low pass
Approximates derivative plus propor-
tional control

High pass Low pass
tional control
Approximates integral plus propor-
tional control

High pass Low pass
tional control
tional control
Contributes phase lead

High pass Low pass
tional control
tional control
Contributes phase lead Attenuation at high frequencies

High pass Low pass
tional control
tional control
Increases the gain crossover fre-
quency

High pass Low pass
tional control
tional control
quency
Moves the gain-crossover frequency
lower

High pass Low pass
tional control
tional control
quency
lower
Increases bandwidth

High pass Low pass
tional control
tional control
quency
lower
Increases bandwidth Reduces bandwidth

Thank you
Please send your feedback at nbahadure@gmail.com

Thank you
Please send your feedback at nbahadure@gmail.com
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Control system compensator lag lead

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