LINEAR CONTROL SYSTEMS
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LINEAR CONTROL LINEAR CONTROL SYSTEMS SYSTEMS Ali Karimpour Assistant Professor Ferdowsi University of Mashhad
description
LINEAR CONTROL SYSTEMS. Ali Karimpour Assistant Professor Ferdowsi University of Mashhad. Topics to be covered include : Lag controller design. Lecture 28. Controller design in the frequency domain. Phase (deg) Magnitude (db). A phase-lag controller. 0. - PowerPoint PPT Presentation
Transcript of LINEAR CONTROL SYSTEMS
LINEAR CONTROLLINEAR CONTROL SYSTEMS SYSTEMS
Ali Karimpour
Assistant Professor
Ferdowsi University of Mashhad
Lecture 28
Ali Karimpour Dec 2010
2
Lecture 28
Controller design in the frequency domain
Topics to be covered include:
Lag controller design
Lecture 28
Ali Karimpour Dec 2010
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A phase-lag controller
1
1)(
s
sasG
1aLet
(rad/sec)Frequency
10101010 3210
0
10
20
30
20
10
90
0
90
0
10
20
30
40
50
Mag
nitu
de (
dB)
100
101
102
103
0
45
90
Pha
se (
deg)
Bode Diagram
Frequency (rad/sec)
Pha
se (
deg)
Mag
nitu
de (
db)
/1 a/1
alog20
1
1sin
a
am m
Lecture 28
Ali Karimpour Dec 2010
4
0
0
20 log a
a
11
Design fundamental of a lag controller
Bode Diagram
Frequency (rad/sec)
100
101
102
103
-180
-135
-90
Pha
se (
deg)
-60
-40
-20
0
20
40
60
Mag
nitu
de (
dB)
PM=25°
How can the lag controller help us?
PM=40° Is it different from a Proportional controller?
What is the effect of a lag controller on BW?
What is the effect of a lag controller on noise effect?
Consider a minimum phase system.
What is the effect of a lag controller on BW? Speed of system?
Analysis Design
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Design procedure of a phase-lag controller in the frequency domain
Step 1: Consider with as a phase-lag controller. 1
1)(
s
saksGc
1a
Step 2: Try to fix k according to the performance request, otherwise let k=1
Step 3: Sketch the Bode plot of the system (with the fixed k ) without controller.
Step 5: Find the required gain by lag controller and derive the parameter a. a
?
10
1 newcross
a
Note: If the plant has another gain k, let 1
1)(
s
sasGc
Step 7: Check the designed controller.
0
0
20 log a
a
11
Step 4: According to desired PM (GM) choose the new gain crossover frequency (Phase crossover frequency). (reduce it a little).
0)log(20|gain Required db a
Step 6: Put the right corner of the controller sufficiently far from crossover frequency.
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Example 1: Design a lag controller for the following system such that the phase margin be 45° and the ramp error constant be 100. Find the Mp of overall system.
)25( ss
k )(sC)(sGc
+
-
)(sR
Step 1: Consider with as a phase-lag controller. 1
1)(
s
saksGc
1a
Note: If the plant has another gain k, let 1
1)(
s
sasGc
11
1)(
as
sasGc
Step 2: Try to fix k according to the performance request, otherwise let k=1
100)25(
)(lim0
ss
kssGk c
sv 2500k
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Example 1: Design a lag controller for the following system such that the phase margin be 45° and the ramp error constant be 100. Find the Mp of overall system.
)25( ss
k )(sC)(sGc
+
-
)(sRStep 3: Sketch the Bode plot of the system (with the
fixed k ) without controller.
)25(
2500)(
sssG
)125/(
100
ssBode Diagram
Frequency (rad/sec)
100
101
102
103
-180
-135
-90
Pha
se (
deg)
-60
-40
-20
0
20
40
60
Mag
nitu
de (
dB)
PM=25°
25newcroos 20new
cross
Step 4: According to desired PM (GM) choose the new gain crossover frequency (Phase crossover frequency). (reduce it a little).
Lecture 28
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Example 1: Design a lag controller for the following system such that the phase margin be 45° and the ramp error constant be 100. Find the Mp of overall system.
)25( ss
k )(sC)(sGc
+
-
)(sR
Bode Diagram
Frequency (rad/sec)
100
101
102
103
-180
-135
-90
Pha
se (
deg)
-60
-40
-20
0
20
40
60
Mag
nitu
de (
dB)0)log(209 a
35.010 20
9
a
10
1 newcross
a
2 41.1
1
1)(
s
sasGc
141.1
15.0
s
s
9
Step 5: Find the required gain by lag controller and derive the parameter a.
0)log(20|gain Required db a
Step 6: Put the right corner of the controller sufficiently far from crossover frequency.
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Example 1: Design a lag controller for the following system such that the phase margin be 45° and the ramp error constant be 100. Find the Mp of overall system.
)25( ss
k )(sC)(sGc
+
-
)(sRStep 7: Check the designed controller.
Bode Diagram
Frequency (rad/sec)
100
101
102
103
-180
-135
-90
Pha
se (
deg)
-60
-40
-20
0
20
40
60
Mag
nitu
de (
dB)
281354.0
1354.0sin
mm
)( jG
2/1,71.0/1 a
dba 9log20
19.1/1 a
141.1
135.041.1
1
1)(
s
s
s
sasGc
0
0
20 log a
a
11
)()( jGjGc
45PM
Lecture 28
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Example 1: Design a lag controller for the following system such that the phase margin be 45° and the ramp error constant be 100. Find the Mp of overall system.
)25( ss
k )(sC)(sGc
+
-
)(sRStep 7: Check the designed controller.
-100
-50
0
50
100
Mag
nitu
de (
dB)
10-2
10-1
100
101
102
103
-180
-135
-90
Pha
se (
deg)
Bode Diagram
Frequency (rad/sec)
)( jG
)()( jGjGc
45PM
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Example 1: Design a lag controller for the following system such that the phase margin be 45° and the ramp error constant be 100. Find the Mp of overall system.
)25( ss
k )(sC)(sGc
+
-
)(sRFinding Mp
-360 -315 -270 -225 -180 -135 -90 -45 0-60
-40
-20
0
20
40
60
80
6 dB 3 dB
1 dB 0.5 dB
0.25 dB 0 dB
-1 dB
-3 dB
-6 dB
-12 dB
-20 dB
-40 dB
-60 dB
Nichols Chart
Open-Loop Phase (deg)
Ope
n-Lo
op G
ain
(dB
)
)( jG
)()( jGjGc
Mp=3 db
Without controller M p=6 db
After applying
controller
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Example 2: Design a lag controller for the following system such that the phase margin be 45° and the ramp error constant be 100.
)20)(10( sss
k )(sC)(sGc
+
-
)(sR
Step 1: Consider with as a phase-lag controller. 1
1)(
s
saksGc
1a
Note: If the plant has another gain k, let 1
1)(
s
sasGc
11
1)(
as
sasGc
Step 2: Try to fix k according to the performance request, otherwise let k=1
100)20)(10(
)(lim0
sss
kssGk c
sv 20000k
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-100
-50
0
50
100
Mag
nitu
de (
dB)
10-1
100
101
102
103
-270
-225
-180
-135
-90
Pha
se (
deg)
Bode Diagram
Frequency (rad/sec)
Example 2: Design a lag controller for the following system such that the phase margin be 45° and the ramp error constant be 100.
Step 3: Sketch the Bode plot of the system (with the fixed k ) without controller.
)20)(10(
20000)(
ssssG
)120/)(110/(
100
sss
PM= -22°5.5newcross 4new
cross
)20)(10( sss
k )(sC)(sGc
+
-
)(sR
5.5
Step 4: According to desired PM (GM) choose the new gain crossover frequency (Phase crossover frequency). (reduce it a little).
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-100
-50
0
50
100
Mag
nitu
de (
dB)
10-1
100
101
102
103
-270
-225
-180
-135
-90
Pha
se (
deg)
Bode Diagram
Frequency (rad/sec)
Example 2: Design a lag controller for the following system such that the phase margin be 45° and the ramp error constant be 100.
0)log(2030 a
0316.010 20
30
a
10
1 newcross
a
4.0 79
1
1)(
s
sasGc
179
15.2
s
s
30
)20)(10( sss
k )(sC)(sGc
+
-
)(sR Step 5: Find the required gain by lag controller and derive the parameter a.
0)log(20|gain Required db a
Step 6: Put the right corner of the controller sufficiently far from crossover frequency.
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Example 2: Design a lag controller for the following system such that the phase margin be 45° and the ramp error constant be 100.
)20)(10( sss
k )(sC)(sGc
+
-
)(sRStep 7: Check the designed controller.
-150
-100
-50
0
50
100
150
200
Mag
nitu
de (
dB)
10-4
10-3
10-2
10-1
100
101
102
103
-270
-225
-180
-135
-90
Pha
se (
deg)
Bode Diagram
Frequency (rad/sec)
)( jG
)()( jGjGc
50PM
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16-270 -225 -180 -135 -90-150
-100
-50
0
50
100
150Nichols Chart
Open-Loop Phase (deg)
Ope
n-Lo
op G
ain
(dB
)
Example 2: Design a lag controller for the following system such that the phase margin be 45° and the ramp error constant be 100.
)20)(10( sss
k )(sC)(sGc
+
-
)(sRStep 7: Check the designed controller.
)( jG
)()( jGjGc
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Example 3: Design a lag controller for the following system such that the gain margin be 10 db and the ramp error constant be 100.
)20)(10( sss
k )(sC)(sGc
+
-
)(sR
Step 1: Consider with as a phase-lag controller. 1
1)(
s
saksGc
1a
Note: If the plant has another gain k, let 1
1)(
s
sasGc
11
1)(
as
sasGc
Step 2: Try to fix k according to the performance request, otherwise let k=1
100)20)(10(
)(lim0
sss
kssGk c
sv 20000k
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-100
-50
0
50
100
Mag
nitu
de (
dB)
10-1
100
101
102
103
-270
-225
-180
-135
-90
Pha
se (
deg)
Bode Diagram
Frequency (rad/sec)
Step 3: Sketch the Bode plot of the system (with the fixed k ) without controller.
)20)(10(
20000)(
ssssG
)120/)(110/(
100
sss
Step 4: According to desired PM (GM) choose the new gain crossover frequency (Phase crossover frequency). (reduce it a little).
GM= -11 db
15newcross 15new
cross
)20)(10( sss
k )(sC)(sGc
+
-
)(sR
Example 3: Design a lag controller for the following system such that the gain margin be 10 db and the ramp error constant be 100.
Lecture 28
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19
-100
-50
0
50
100
Mag
nitu
de (
dB)
10-1
100
101
102
103
-270
-225
-180
-135
-90
Pha
se (
deg)
Bode Diagram
Frequency (rad/sec)
Step 5: Find the required gain by lag controller and derive the parameter a.
0)log(20|gain Required db a
0)log(2025 a
056.010 20
25
a Step 6: Put the right corner of the
controller sufficiently far from crossover frequency.
10
101
a1 9.17
1
1)(
s
sasGc
19.17
1
s
s
15
)20)(10( sss
k )(sC)(sGc
+
-
)(sR
Example 3: Design a lag controller for the following system such that the gain margin be 10 db and the ramp error constant be 100.
Lecture 28
Ali Karimpour Dec 2010
-100
-50
0
50
100
Mag
nitu
de (
dB)
10-1
100
101
102
103
-270
-225
-180
-135
-90
Pha
se (
deg)
Bode Diagram
Frequency (rad/sec) 20
Step 7: Check the designed controller.
631056.0
1056.0sin
mm
)( jG
1/1,06.0/1 a
dba 25log20
25.0/1 a
19.17
19.17056.0
1
1)(
s
s
s
sasGc
0
0
20 log a
a
11
)()( jGjGc dbGM 10
Example 3: Design a lag controller for the following system such that the gain margin be 10 db and the ramp error constant be 100.
)20)(10( sss
k )(sC)(sGc
+
-
)(sR
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Example 4: Design a lag controller for the following system such that the phase margin be 45° and the open loop bandwidth be 10 rad/sec
2
1
s
)(sC)(sGc
+
-
)(sR
It is not possible explain why?
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Exercises
1 Derive the gain margin of the compensated system in example 1.
2 Derive the phase margin of the compensated system in example 3.
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Exercises
3 Following is the open loop transfer function of a system. a) What is the velocity error constant. (answer 80)b) Design lag controller such that PM=45°. (answer Gc(s)=(0.5s+1)/(1.56s+1))c) Design lag controller such that PM=45° and the velocity error constant be 200. (answer Gc(s)=(0.49s+1)/(3.75s+1))
10-1
100
101
102
103
-180
-135
-90
Pha
se (
deg)
Bode Diagram
Frequency (rad/sec)
-60
-40
-20
0
20
40
60
80
Mag
nitu
de (
dB)
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Exercises
-150
-100
-50
0
50
Mag
nitu
de (
dB)
10-1
100
101
102
103
-270
-225
-180
-135
-90
Pha
se (
deg)
Bode Diagram
Frequency (rad/sec)
4 Following is the open loop transfer function of a system. a) Design a controller that the GM of system be 50 db.b) Design lag controller such that PM=45° and the velocity error constant be 30.
