Lag Compensation - Dr. Imtiaz Hussain
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Transcript Lag Compensation - Dr. Imtiaz Hussain
Modern Control Systems (MCS)
Lecture-16-17-18
Lag Compensation
Dr. Imtiaz Hussain
Assistant Professor
email: [email protected]
URL :http://imtiazhussainkalwar.weebly.com/
Lecture Outline
Lag Compensation
β’ Lag compensation is used to improve the steady state error
of the system.
β’ Generally Lag compensators are represented by following
transfer function
πΊπ π =
ππ +1
πΎπ π½
π½ππ +1
β’ Or
πΊπ π = πΎπ
1
π
1
π +
π½π
π +
,
,
β’ Where πΎ πis gain of lag compensator.
(Ξ² > 1)
(Ξ² > 1)
Lag Compensation
πΊπ π =
π +10
3
π +1
(π½ = 10)
,
Bode Diagram
Pole-Zero Map
30
Magnitude (dB)
1
0.5
20
15
10
5
0
Phase (deg)
0
-0.5
-1
-10
25
-8
-6
-4
Real Axis
-2
0
-30
-60
-2
10
0
10
Frequency (rad/sec)
2
10
Lag Compensation
β’ Consider the problem of finding a suitable compensation network
for the case where the system exhibits satisfactory transientresponse characteristics but unsatisfactory steady-state
characteristics.
β’ Compensation in this case essentially consists of increasing the
open loop gain without appreciably changing the transientresponse characteristics.
β’ This means that the root locus in the neighborhood of the
dominant closed-loop poles should not be changed appreciably, but
the open-loop gain should be increased as much as needed.
Lag Compensation
β’ To avoid an appreciable change in the root loci, the angle
contribution of the lag network should be limited to a small
amount, say less than 5°.
β’ To assure this, we place the pole and zero of the lag network
relatively close together and near the origin of the s plane.
β’ Then the closed-loop poles of the compensated system will be
shifted only slightly from their original locations. Hence, the
transient-response characteristics will be changed only slightly.
Lag Compensation
β’ Consider a lag compensator Gc(s), where
1
πΊπ π = πΎπ
π +π
1
π +π½π
,
(Ξ² > 1)
β’ If we place the zero and pole of the lag compensator very close to
each other, then at s=s1 (where s1is one of the dominant closed
loop poles then the magnitudes π 1 +
1
π
and π 1 +
equal, or
1
π +
π β
πΎ
πΊπ (π 1 ) = πΎπ
π
1
π +
π½π
1
π½π
are almost
Lag Compensation
β’ To make the angle contribution of the lag portion of the
compensator small, we require
1
π +
π
β5° < πππππ
1
π +
π½π
< 0°
β’ This implies that if gain πΎπ of the lag compensator is set equal to 1,
the alteration in the transient-response characteristics will be very
small, despite the fact that the overall gain of the open-loop
transfer function is increased by a factor of π½, where π½>1.
Lag Compensation
β’ If the pole and zero are placed very close to the origin, then the
value of π½ can be made large.
β’ A large value of π½ may be used, provided physical realization of the
lag compensator is possible.
β’ It is noted that the value of T must be large, but its exact value is
not critical.
β’ However, it should not be too large in order to avoid difficulties in
realizing the phase-lag compensator by physical components.
Lag Compensation
β’ An increase in the gain means an increase in the static error
constants.
β’ If the open loop transfer function of the uncompensated system is
G(s), then the static velocity error constant Kv of the uncompensated
system is
πΎπ£ = lim π πΊ(π )
π β0
β’ Then for the compensated system with the open-loop transfer
function Gc(s)G(s) the static velocity error constant πΎπ£ becomes
πΎπ£ = lim π πΊπ π πΊ(π ) = πΎπ£ limπΊπ π
π β0
π β0
1
π +
π =πΎ πΎπ½
πΎπ£ = πΎπ£ limπΎπ
π£ π
1
π β0
π +
π½π
Lag Compensation
β’ The main negative effect of the lag compensation is that
the compensator zero that will be generated near the
origin creates a closed-loop pole near the origin.
β’ This closed loop pole and compensator zero will generate a
long tail of small amplitude in the step response, thus
increasing the settling time.
Electronic Lag Compensator
β’ The configuration of the electronic lag compensator using
operational amplifiers is the same as that for the lead
compensator.
1
π
+
πΈπ (π ) π
4 πΆ1
π
1 πΆ1
=
πΈπ (π ) π
3 πΆ2 π + 1
π
2 πΆ2
π = π
1 πΆ1
π½π = π
2 πΆ2
π
4 πΆ1
πΎπ =
π
3 πΆ2
π
2 πΆ2 > π
1 πΆ1
Electronic Lag Compensator
β’ Pole-zero Configuration of Lag
Compensator
π
2 πΆ2 > π
1 πΆ1
Electrical Lag Compensator
β’ Following figure shows lag compensator realized by
electrical network.
π
1
π
2
πΆ
πΈ2 (π )
π
2 πΆπ + 1
=
πΈ1 (π )
π
1 + π
2 πΆπ + 1
Electrical Lag Compensator
πΈ2 (π )
π
2 πΆπ + 1
=
πΈ1 (π )
π
1 + π
2 πΆπ + 1
π = π
2 πΆ
π
1 + π
2
π½=
>1
π
2
β’ Then the transfer function becomes
πΈ2 (π )
ππ + 1
=
πΈ1 (π )
π½ππ + 1
Electrical Lag Compensator
πΈ2 (π )
ππ + 1
=
πΈ1 (π )
π½ππ + 1
β’ If an RC circuit is used as a lag compensator, then it is
usually necessary to add an amplifier with an adjustable
gain πΎπ π½ so that the transfer function of compensator is
πΈ2 (π )
ππ + 1
= πΎπ π½
πΈ1 (π )
π½ππ + 1
1
π +
πΈ2 (π )
π
= πΎπ
1
πΈ1 (π )
π +
π½π
Mechanical Lag Compensator (Home Work)
Design Procedure
β’ The procedure for designing lag compensators by the rootlocus method may be stated as follows.
β’ We will assume that the uncompensated system meets the
transient-response specifications by simple gain adjustment.
β’ If this is not the case then we need to design a lag-lead
compensator which we will discuss in next few classes.
Design Procedure
β’ Step-1
β Draw the root-locus plot for the uncompensated system whose
open-loop transfer function is G(s).
β Based on the transient-response specifications, locate the
dominant closed-loop poles on the root locus.
Design Procedure
β’ Step-2
β Assume the transfer function of the lag compensator to be
given by following equation
πΊπ π =
ππ +1
πΎπ π½
=πΎπ
π½ππ +1
1
π +
π
1
π +
π½π
β Then the open-loop transfer function of the compensated
system becomes Gc(s)G(s).
Design Procedure
β’ Step-3
β Evaluate the particular static error constant specified in the
problem.
β Determine the amount of increase in the static error constant
necessary to satisfy the specifications.
Design Procedure
β’ Step-4
β Determine the pole and zero of the lag compensator that
produce the necessary increase in the particular static
error constant without appreciably altering the original
root loci.
β The ratio of the value of gain required in the specifications
and the gain found in the uncompensated system is the
required ratio between the distance of the zero from the
origin and that of the pole from the origin.
Design Procedure
β’ Step-5
β Draw a new root-locus plot for the compensated system.
β Locate the desired dominant closed-loop poles on the root locus.
β (If the angle contribution of the lag network is very smallβthat
is, a few degreesβthen the original and new root loci are almost
identical.
β Otherwise, there will be a slight discrepancy between them.
β Then locate, on the new root locus, the desired dominant closedloop poles based on the transient-response specifications.
Design Procedure
β’ Step-6
β Adjust gain of the compensator from the magnitude condition so
that the dominant closed-loop poles lie at the desired location.
β πΎπ will be approximately 1.
Example-1
β’ Consider the system shown in following figure.
β’ The damping ratio of the dominant closed-loop poles is
0.491. The undamped natural frequency of the dominant
closed-loop poles is 0.673 rad/sec. The static velocity error
constant is 0.53 secβ1.
β’ It is desired to increase the static velocity error constant Kv to
about 5 secβ1 without appreciably changing the location of
the dominant closed-loop poles.
Example-1 (Step-1)
β’ The dominant closed-loop poles of given system are
s = -0.3307 ± j0.5864
Example-1 (Step-2)
β’ According to given conditions we need to add following
compensator to fulfill the requirement.
πΊπ π =
ππ +1
πΎπ π½
=πΎπ
π½ππ +1
1
π +
π
1
π +
π½π
Example-1 (Step-3)
β’ The static velocity error constant of the plant (πΎπ£ ) is
1.06
πΎπ£ = limπ πΊ(π ) = lim π
π β0
π β0
π π +1 π +2
= 0.53π β1
β’ The desired static velocity error constant ( πΎπ£ ) of the
compensated system is 5π β1 .
πΎπ£ = lim π πΊπ π πΊ(π ) = πΎπ£ limπΊπ π
π β0
π β0
1
π +
π =πΎ πΎπ½
πΎπ£ = πΎπ£ limπΎπ
π£ π
1
π β0
π +
π½π
Example-1 (Step-3)
1
π =πΎ πΎπ½
πΎπ£ = πΎπ£ limπΎπ
π£ π
1
π β0
π +
π½π
π +
πΎπ£ = πΎπ£ πΎπ π½
5 = 0.53π½
π½ = 10
Example-1 (Step-4)
β’ Place the pole and zero of the lag compensator
πΊπ π =πΎπ
β’ Since π½ = 10, therefore
πΊπ π =πΎπ
1
π +
π
1
π +
π½π
1
π
0.1
π +
π
π +
Example-1 (Step-4)
Solution-1
β’ Place the zero and pole of the lag compensator at s=β0.05
and s=β0.005, respectively.
β’ The transfer function of the lag compensator becomes
πΊπ π
π +0.05
=πΎπ
π +0.005
β’ Open loop transfer function is given as
πΊπ π
π +0.05
1.06
πΊ(π )=πΎπ
π +0.005 π (π +1)(π +2)
πΊπ π
πΎ(π +0.05)
πΊ(π )=
π (π +0.005)(π +1)(π +2)
π€βπππ πΎ = 1.06πΎπ
Example-1 (Step-5)
Solution-1
β’ Root locus of uncompensated and compensated systems.
β’ New Closed Loop poles
are
π = β0.31 ± π0.55
Example-1 (Step-5)
Solution-1
β’ Root locus of uncompensated and compensated systems.
Example-1 (Step-6)
Solution-1
β’ The open-loop gain K is determined from the magnitude
condition.
πΎ(π + 0.05)
π (π + 0.005)(π + 1)(π + 2)
=1
π =β0.31+π0.55
πΎ = 1.0235
β’ Then the compensator gain πΎπ is determined as
πΎ = 1.06πΎπ
πΎ
πΎπ =
= 0.9656
1.06
Example-1 (Step-6)
β’ Then the compensator transfer function is given as
π + 0.05
πΊπ π = 0.9656
π + 0.005
Solution-1
Example-1 (Final Design Check)
Solution-1
β’ The compensated system has following open loop transfer
function.
πΊπ π
1.0235(π +0.05)
πΊ(π )=
π (π +0.005)(π +1)(π +2)
β’ Static velocity error constant is calculated as
πΎπ£ = lim π πΊπ π πΊ(π )
π β0
1.0235 π + 0.05
πΎπ£ = lim π [
]
π β0 π π + 0.005 π + 1 π + 2
1.0235 0.05
πΎπ£ =
= 5.12π β1
0.005 1 2
Example-1 (Step-4)
Solution-2
β’ Place the zero and pole of the lag compensator at s=β0.01
and s=β0.001, respectively.
β’ The transfer function of the lag compensator becomes
πΊπ π
π +0.01
=πΎπ
π +0.001
β’ Open loop transfer function is given as
πΊπ π
π +0.01
1.06
πΊ(π )=πΎπ
π +0.001 π (π +1)(π +2)
πΊπ π
πΎ(π +0.01)
πΊ(π )=
π (π +0.001)(π +1)(π +2)
π€βπππ πΎ = 1.06πΎπ
Example-1 (Step-5)
Solution-2
β’ Root locus of uncompensated and compensated systems.
Root Locus
6
β’ New Closed Loop poles
are
Compensated System
4
Imaginary Axis
π = β0.33 ± π0.55
Actual System
2
0
-2
-4
-6
-8
-6
-4
-2
Real Axis
0
2
4
Example-2
β’ Design a lag compensator for following unity
feedback system such that the static velocity error
constant is 50 sec-1 without appreciably changing the
closed loop poles, which are at π = β2 ± π 6.
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END OF LECTURE-16-17-18