linear control systems
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 PresentationTRANSCRIPT
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
3
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
Lecture 28
Ali Karimpour Dec 2010
5
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.
Lecture 28
Ali Karimpour Dec 2010
6
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
Lecture 28
Ali Karimpour Dec 2010
7
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
Ali Karimpour Dec 2010
8
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.
Lecture 28
Ali Karimpour Dec 2010
9
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
Ali Karimpour Dec 2010
10
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
Lecture 28
Ali Karimpour Dec 2010
11
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
Lecture 28
Ali Karimpour Dec 2010
12
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
Lecture 28
Ali Karimpour Dec 2010
13
-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).
Lecture 28
Ali Karimpour Dec 2010
14
-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.
Lecture 28
Ali Karimpour Dec 2010
15
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
Lecture 28
Ali Karimpour Dec 2010
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
Lecture 28
Ali Karimpour Dec 2010
17
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
Lecture 28
Ali Karimpour Dec 2010
18
-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
Ali Karimpour Dec 2010
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
Lecture 28
Ali Karimpour Dec 2010
21
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?
Lecture 28
Ali Karimpour Dec 2010
22
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.
Lecture 28
Ali Karimpour Dec 2010
23
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)
Lecture 28
Ali Karimpour Dec 2010
24
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.