2008/09 MECH466 : Automatic Control 1
MECH466: Automatic Control
MECH466: Automatic Control
Dr. Ryozo Nagamune
Dr. Ryozo Nagamune
Department of Mechanical Engineering
Department of Mechanical Engineering
University of British Columbia
University of British Columbia
Lecture 20
Lecture 20
Frequency response shaping
Frequency response shaping
2008/09 MECH466 : Automatic Control 2
Course roadmap
Course roadmap
Laplace transform
Laplace transform
Transfer function
Transfer function
Models for systems
Models for systems
•
•
electrical
electrical
•
•
mechanical
mechanical
•
•
electromechanical
electromechanical
Linearization
Linearization
Modeling
Modeling
Analysis
Analysis
Design
Design
Time response
Time response
•
•
Transient
Transient
•
•
Steady state
Steady state
Frequency response
Frequency response
•
•
Bode plot
Bode plot
Stability
Stability
•
•
Routh
Routh
-
-
Hurwitz
Hurwitz
•
•
Nyquist
Nyquist
Design specs
Design specs
Root locus
Root locus
Frequency domain
Frequency domain
PID & Lead
PID & Lead
-
-
lag
lag
Design examples
Design examples
Matlab
Matlab
simulations & laboratories
simulations & laboratories
2008/09 MECH466 : Automatic Control 3
Typical shaping goal (review)
Typical shaping goal (review)
Steady
Steady
-
-
state accuracy
state accuracy
Sensitivity
Sensitivity
Disturbance rejection
Disturbance rejection
Noise
Noise
reduction
reduction
Transient
Transient
Response speed
Response speed
Transient
Transient
Overshoot
Overshoot
Relative stability
Relative stability
Relative stability
Relative stability
Noise
Noise
reduction
reduction
2008/09 MECH466 : Automatic Control 4
Frequency response shaping
Frequency response shaping
(Loop shaping)
(Loop shaping)
Reshape
Reshape
Bode plot of
Bode plot of
G(j
G(j
ω
ω
)
)
into a
into a
“
“
desired
desired
”
”
shape of
shape of
by a series connection of
by a series connection of
appropriate
appropriate
C(s
C(s
).
).
G(s
G(s)
)
C(s
C(s)
)
Stable plant
Stable plant
Controller
2008/09 MECH466 : Automatic Control 5
An advantage of Bode plot (review)
An advantage of Bode plot (review)
Bode plot of a series connection G
Bode plot of a series connection G
1
1
(s)G
(s)G
2
2
(s) is
(s) is
the addition of each Bode plot of G
the addition of each Bode plot of G
1
1
and G
and G
2
2
.
.
Gain
Gain
Phase
Phase
We use this property to design
We use this property to design
C(s
C(s
) so that
) so that
G(s)C(s
G(s)C(s
) has a
) has a
“
“
desired
desired
”
”
shape of Bode plot.
shape of Bode plot.
2008/09 MECH466 : Automatic Control 6
Simple controllers
Simple controllers
We use simple controllers for shaping.
We use simple controllers for shaping.
Gain
Gain
Lead and lag compensators
Lead and lag compensators
G(s
G(s)
)
C(s
C(s)
)
Stable plant
Stable plant
Controller
Controller
Bode plot of a gain (review)
Bode plot of a gain (review)
dB
dB
deg
deg
Effect of gain
Effect of gain
C(s
C(s
) on
) on
L(j
L(j
ω
ω
)
)
10-2 10-1 100 101 102 103 -100
-50 0 50 100
10-2 10-1 100 101 102 103 -180
-160 -140 -120 -100
In case of K>1,
In case of K>1,
Gain increases
Gain increases
uniformly, but phase
uniformly, but phase
does not change.
does not change.
Typically,
Typically,
(Steady state) L(0)
(Steady state) L(0)
(Speed)
(Speed)
ωg
ω
g
(Stability &
(Stability &
overshoot) PM
2008/09 MECH466 : Automatic Control 9
Bode plots of lead and lag
Bode plots of lead and lag
C(s
C(s
)
)
10-2 10-1 100 101 102 103 -20
-15 -10 -5 0
10-2 10-1 100 101 102 103 -60
-40 -20 0
10-2 10-1 100 101 102 103
0 5 10 15 20
10-2 10-1 100 101 102 103
0 20 40 60
Lead
Lead
compensator
compensator
Lag
Lag
compensator
compensator
PHASE LEAD
PHASE LEAD
PHASE LAG
PHASE LAG
MEMORIZE THESE
MEMORIZE THESE
SHAPES!!!
SHAPES!!!
2008/09 MECH466 : Automatic Control 10
Straight
Straight
-
-
line approximations
line approximations
dB
dB
+20
+20
deg
deg
+45
+45
dB
dB
--
20
20
deg
deg
--
45
45
Lead (z<p)
Lead (z<p)
Lag (p<z)
Lag (p<z)
dB
dB
+20
+20
deg
deg
+45
+45
dB
dB
--
20
20
deg
deg
--
45
45
--
45
45
+45
+45
2008/09 MECH466 : Automatic Control 11
Guideline of lead/lag design
Guideline of lead/lag design
Gain+lag
Gain+lag
Lead
Lead
Lag
Lag
Lead
Lead
2008/09 MECH466 : Automatic Control 12
Effect of a lag
Effect of a lag
C(s
C(s
) on
) on
L(j
L(j
ω
ω
)
)
Destabilizing effect
Destabilizing effect
•
•
Decreasing
Decreasing
ωg
ω
g
Select z much (at least 1
Select z much (at least 1
decade) less than
decade) less than
ω
ω
g
g
10-2 10-1 100 101 102 103 -20
-15 -10 -5 0
10-2 10-1 100 101 102 103 -60
2008/09 MECH466 : Automatic Control 13
10
-210
-110
010
110
210
3-100
-50
0
50
100
10
-210
-110
010
110
210
3-180
-160
-140
-120
-100
Lag+gain
Lag+gain
C(s
C(s
) design
) design
PM: 28 deg at
PM: 28 deg at
ω
ωg
g
=47
=47
rad/s
rad/s
PM
PM
PM: 27 deg at
PM: 27 deg at
ω
ωg
g
=47
=47
rad/s
rad/s
2008/09 MECH466 : Automatic Control 14
Guideline of lead/lag design
Guideline of lead/lag design
Gain+lag
Gain+lag
Lead
Lead
Lag
Lag
Lead
Lead
10-2 10-1 100 101 102 103 0
5 10 15 20
10-2 10-1 100 101 102 103 0
20 40 60
Effect of a lead
Effect of a lead
C(s
C(s
) on
) on
L(j
L(j
ω
ω
)
)
Stabilizing effect
Stabilizing effect
Increasing
Increasing
ωg
ω
g
Select
Select z&p
z&p
around
around
ω
ω
g
g
10
110
210
3-60
-40
-20
0
20
10
110
210
3-180
-160
-140
-120
-100
Example of a lead
Example of a lead
C(s
C(s
) design
) design
PM: 28 deg at
PM: 28 deg at
ω
ωg
g
=47
=47
rad/s
rad/s
PM
PM
PM: 47 deg at
PM: 47 deg at
ω
2008/09 MECH466 : Automatic Control 17
10-2 10-1 100 101 102 103 0
5 10 15 20
10-2 10-1 100 101 102 103 -60
-40 -20 0 20 40
Lead
Lead
-
-
lag compensator
lag compensator
2008/09 MECH466 : Automatic Control 18
10
-210
-110
010
110
210
3-100
-50
0
50
100
10
-210
-110
010
110
210
3-180
-160
-140
-120
-100
Example of a lead
Example of a lead
-
-
lag
lag
C(s
C(s
) design
) design
PM: 28 deg at
PM: 28 deg at
ω
ωg
g
=47
=47
rad/s
rad/s
PM
PM
PM: 47 deg at
PM: 47 deg at
ω
ωg
g
=60
=60
rad/s
rad/s
2008/09 MECH466 : Automatic Control 19
Step responses
Step responses
0 0.1 0.2 0.3 0.4 0.5
0 0.5 1 1.5
Uncompensated (
Uncompensated (
C(s
C(s
)=1)
)=1)
Lead
Lead
-
-
lag compensated
lag compensated
Less overshoot is due to larger PM.
Less overshoot is due to larger PM.
Faster response is due to larger
Faster response is due to larger
w
wg
g
.
.
2008/09 MECH466 : Automatic Control 20
Ramp responses
Ramp responses
0.48 0.485 0.49 0.495 0.5 0.48
0.485 0.49 0.495 0.5
Uncompensated (
Uncompensated (
C(s
C(s
)=1)
)=1)
Kv
Kv
=100
=100
Lead
Lead
-
-
lag compensated
lag compensated
Kv
Kv
=1000
=1000
Ramp reference
Ramp reference
Smaller steady
2008/09 MECH466 : Automatic Control 21
An example
An example
Consider a system
Consider a system
Analysis for
Analysis for
C(s
C(s
)=1
)=1
Stable
Stable
PM at least 12 deg
PM at least 12 deg
GM at least 3.5 dB
GM at least 3.5 dB
G(s
G(s)
)
C(s
C(s)
)
Plant
Plant
Controller
Controller
These values are too
These values are too
small for good
small for good
transient response!
transient response!
10-2 10-1 100 101 102
-100 -50 0 50
10-2 10-1 100 101 102
-250 -200 -150 -100
2008/09 MECH466 : Automatic Control 22
Gain compensation
Gain compensation
PM is specified to be 50 deg.
PM is specified to be 50 deg.
In this example, to
In this example, to
increase PM
increase PM
by gain
by gain
compensation, we need to lower the gain curve.
compensation, we need to lower the gain curve.
10-2 10-1 100 101 102 -100
-50 0 50
10-2 10-1 100 101 102 -250
-200 -150 -100
Bode plot for
Bode plot for
C(s
C(s
)=0.286
)=0.286
Uncompensated (
Uncompensated (
C(s
C(s
)=1)
)=1)
Gain compensated
Gain compensated
Low freq. gain
Low freq. gain
decreases.
decreases.
0
5
10
15
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Step responses
Step responses
K=0.455 (PM=35deg)
K=0.455 (PM=35deg)
K=0.286 (PM=50deg)
K=0.286 (PM=50deg)
K=0.158 (PM=65deg)
2008/09 MECH466 : Automatic Control 25
Phase
Phase
-
-
lag compensator (review)
lag compensator (review)
dB
dB
deg
deg
--
45
45
--
20
20
10-2 10-1 100 101 102 103 -20
-15 -10 -5 0
10-2 10-1 100 101 102 103 -60
-40 -20 0
2008/09 MECH466 : Automatic Control 26
Phase
Phase
-
-
lag
lag
C(s
C(s
) design
) design
1.
1.
To satisfy low frequency requirement, adjust
To satisfy low frequency requirement, adjust
DC gain of OL system by a constant gain K.
DC gain of OL system by a constant gain K.
Analysis for
Analysis for
C(s
C(s
)=1
)=1
Stable
Stable
PM at least 12 deg
PM at least 12 deg
GM at least 3.5 dB
GM at least 3.5 dB
G(s
G(s)
)
C(s
C(s)
)
Plant
Plant
Controller
Controller
We try to design phase
We try to design phase
-
-
lag
lag
C(s
C(s
) which gives
) which gives
•
•
PM 50deg
PM 50deg
•
•
Low frequency gain same as the original plant.
Low frequency gain same as the original plant.
2008/09 MECH466 : Automatic Control 27
After Step 1
After Step 1
10-2 10-1 100 101 102
-100 -50 0 50
10-2 10-1 100 101 102
-250 -200 -150 -100
OK
OK
2008/09 MECH466 : Automatic Control 28
Phase
Phase
-
-
lag
lag
C(s
C(s
) design
) design
2.
2.
Find the frequency
Find the frequency
ω
ω
g
g
(which will become gain
(which will become gain
crossover frequency after compensation)
crossover frequency after compensation)
where
where
In this example,
In this example,
2008/09 MECH466 : Automatic Control 29
After Step 2
After Step 2
10-2 10-1 100 101 102
-100 -50 0 50
10-2 10-1 100 101 102
-250 -200 -150 -100
PM=55
PM=55
2008/09 MECH466 : Automatic Control 30
Phase
Phase
-
-
lag
lag
C(s
C(s
) design
) design
3.
3.
Set z and p as
Set z and p as
dB
dB
--
20
20
deg
deg
--
45
45
For small phase lag at
For small phase lag at
ω
ω
g
g
For setting new gain crossover at
For setting new gain crossover at
ω
ω
g
g
10-2 10-1 100 101 102 -100
-50 0 50
10-2 10-1 100 101 102 -250
-200 -150 -100
After Step 3
After Step 3
PM=50
PM=50
0
5
10
15
0
0.5
1
1.5
2
Step responses
Step responses
C(s
C(s
)=0.286 (PM=50deg,
)=0.286 (PM=50deg,
ω
ω
gg=0.5)
=0.5)
C(s
C(s
)=
)=
C
C
LagLag(s
(s
) (PM=52.3deg,
) (PM=52.3deg,
ω
ω
gg=0.4)
=0.4)
Small overshoot is due to larger PM.
Small overshoot is due to larger PM.
Slower response is due to smaller
Slower response is due to smaller
w
wg
g
.
.
C(s
2008/09 MECH466 : Automatic Control 33
67
67.5
68
68.5
69
69.5
70
64
65
66
67
68
69
70
Ramp responses
Ramp responses
Smaller steady
Smaller steady
-
-
state error is due to larger
state error is due to larger
Kv
Kv
.
.
C(s
C(s
)=0.286 (
)=0.286 (
Kv
Kv
=0.572)
=0.572)
Ramp reference
Ramp reference
C(s
C(s
)=
)=
C
C
LagLag(s
(s
) (
) (
Kv
Kv
=2)
=2)
C(s
C(s
)=1
)=1
(
(
Kv
Kv
=2)
=2)
2008/09 MECH466 : Automatic Control 34
Summary and exercises
Summary and exercises
Gain controller design in Bode plot
Gain controller design in Bode plot
Gain changes uniformly over frequencies.
Gain changes uniformly over frequencies.
Phase does not change.
Phase does not change.
Lag compensator design in Bode plot
Lag compensator design in Bode plot
Lag compensator can be used for
Lag compensator can be used for
•
•
Improving PM by maintaining low freq. gain, or
Improving PM by maintaining low freq. gain, or
•
•
Improving low freq. gain by maintaining PM
Improving low freq. gain by maintaining PM
Low freq. gain
Low freq. gain
determines steady state error,
determines steady state error,
disturbance rejection, while
disturbance rejection, while
PM
PM
does overshoot.
does overshoot.
Read Sections 11.1, 11.2, and 11.3.
Read Sections 11.1, 11.2, and 11.3.
2008/09 MECH466 : Automatic Control 35
Lab report hand
Lab report hand
-
-
in policy
in policy
The last lab report must be handed
The last lab report must be handed
-
-
in in person
in in person
to TA Mr. Roland Lang (ICICS 065), or the
to TA Mr. Roland Lang (ICICS 065), or the
instructor (Kaiser 3104), by
instructor (Kaiser 3104), by
5pm April 9
5pm April 9
(Thursday).
(Thursday).