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Transcript feedback control
Feedback Control System
Dr.-Ing. Erwin Sitompul
Textbook and Syllabus
Textbook:
Gene F. Franklin, J. David Powell, Abbas
Emami-Naeini, “Feedback Control of Dynamic
Systems”, 6th Edition, Pearson International
Edition.
Syllabus:
1.
2.
3.
4.
5.
6.
IDR 192,000
Introduction
Dynamic Models
Dynamic Response
A First Analysis of Feedback
The Root-Locus Design Method
The Frequency-Response Design Method
USD 112.50
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Grade Policy
Final Grade = 10% Homework + 20% Quizzes +
30% Midterm Exam + 40% Final Exam +
Extra Points
Homeworks will be given in fairly regular basis. The average
of homework grades contributes 10% of final grade.
Homeworks are to be written on A4 papers, otherwise they
will not be graded.
Homeworks must be submitted on time. If you submit late,
< 10 min.
No penalty
10 – 60 min. –20 points
> 60 min.
–40 points
There will be 3 quizzes. Only the best 2 will be counted.
The average of quiz grades contributes 20% of final grade.
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Grade Policy
Midterm and final exam schedule will be announced in time.
Make up of quizzes and exams will be held one week after
the schedule of the respective quizzes and exams, at the
latest.
The score of a make up quiz or exam can be multiplied by 0.9
(the maximum score for a make up is 90).
Extra points will be given every time you solve a problem in
front of the class. You will earn 1 or 2 points.
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Chapter 1
INTRODUCTION
Feedback Control System
Dr.-Ing. Erwin Sitompul
Introduction
• Control is a series of actions directed for making a system
variable adheres to a reference value (can be either
constant or variable).
• The reference value when performing control is the desired
output variable.
• Process, as it is used and understood by control engineers,
means the component to be controlled.
Fundamental structures of control are classified based on the
information used along the control process:
1. Open-loop control / Feedforward control
2. Closed-loop control / Feedback control
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Process
Reference
Measurement
Disturbance
noise
Performance
Input
Measurement
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Open-loop vs. Feedback Control
The difference:
In open-loop control, the system does not measure the
actual output and there is no correction to make the actual
output to be conformed with the reference value.
In feedback control, the system includes a sensor to
measure the actual output and uses its feedback to
influence the control process.
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Examples
Open-loop control
Feedback control
Example: an electric toaster,
a standard gas stove.
Example: automated filling
up system, magic jar, etc.
• The controller is constructed
based on knowledge or
experience.
• The process output is not
used in control computation.
• The output is fed back for
control computation.
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Plus-Minus of Open-loop Control
+ Generally simpler than closed-loop control
+ Does not require sensor to measure the output
+ Does not, of itself, introduce stability problem
– Has lower performance to match the desired output
compared to closed-loop control
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Plus-Minus of Feedback Control
–
–
–
–
More complex than open-loop control
May have steady-state error
Depends on the accuracy of the sensor
May have stability problem
+ Process controlled by well designed feedback control can
respond to unforeseen events, such as: disturbance,
change of process due to aging, wear, etc.
+ Eliminates the need of human to adjust the control variable
reduce human workload
+ Gives much better performance than what is possibly given
by open loop control: ability to meet transient response
objectives and steady-state error objectives
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Chapter 2
DYNAMIC MODELS
Feedback Control System
Dr.-Ing. Erwin Sitompul
Dynamic Models
A Simple System: Cruise Control Model
Write the equations of motion for the speed and forward
motion of the car shown below, assuming that the engine
imparts a force u, and results the car velocity v, as shown.
Using the Laplace transform, find the transfer function
between the input u and the output v.
u
(Force)
x (Position)
v (Velocity)
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Dynamic Models
Applying the Newton’s Law for
translational motion yields:
u bv ma
u bx mx
u bv mv
b
u
v v
m
m
MATLAB (Matrix Laboratory) is
the standard software used in
control engineering:
V (s b m) U m
V (s)
1m
U (s) s b m
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In the end of this course, you are
expected to be able to know how to
use MATLAB for basic applications.
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Dynamic Models
m 1000 kg
b 50 Ns/m
u 500 N
In MATLAB windows:
Response of the car velocity v
to a step-shaped force u:
10
9
8
7
Amplitude
With the parameters:
6
5
4
3
2
1
0
0
V (s)
1m
U (s) s b m
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20
40
60
80
100
120
Time (sec)
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Dynamic Models
A Two-Mass System: Suspension Model
m1 : mass of the wheel
m2 : mass of the car
x,y : displacements from equilibrium
r : distance to road surface
Equation for m1:
ks ( x y) b( x y) kw ( x r ) m1x
Equation for m2:
ks ( y x) b( y x) m2 y
Rearranging:
ks
kw
kw
b
x ( x y) ( x y)
x
r
m1
m1
m1
m1
ks
b
y
( y x)
( y x) 0
m2
m2
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Dynamic Models
Using the Laplace transform:
L
x (t )
X (s)
d
L
x (t )
sX ( s )
dt
to transfer from time domain to frequency domain yields:
b
s X ( s) s X ( s) Y ( s)
m1
ks
kw
kw
X ( s ) Y ( s ) X ( s ) R( s )
m1
m1
m1
2
ks
b
s Y ( s)
s Y ( s ) X ( s )
Y ( s ) X ( s ) 0
m2
m2
2
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Dynamic Models
Eliminating X(s) yields a transfer function:
kw b
ks
s
m1m2
b
Y ( s)
R( s )
b
kw ks
b 3 ks k kw 2 kw b
4
s
s
s
s
m1m2
m1 m2
m1 m2 m1
m1m2
Y (s)
output
F (s)
transfer function
R( s)
input
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Dynamic Models
Bridged Tee Circuit
v1
v Ri
V1 Vi V1 Vo
sC1V1 0
R1
R2
v L di
dt
i C dv
dt
V (s) R I (s) V (s) sL I (s) I (s) sC V (s)
Resistor
Inductor Capacitor
Vo V1
sC2 (Vo Vi ) 0
R2
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Dynamic Models
RL Circuit
v1
V1 Vi V1 V1 Vo
0
1
1
s
V1 Vo Vo 0
V1 Vo ( s 1)
s
1
Further calculation and
eliminating V1,
1 Vo
V1 2 Vi
s s
1 Vo
Vo s 1 2 Vi
s s
Vo 2s 3 Vi
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Vo
1
Vi 2s 3
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Chapter 3
DYNAMIC RESPONSE
Feedback Control System
Dr.-Ing. Erwin Sitompul
Review of Laplace Transform
Time domain
Problem
f (t )
Frequency domain
L
F (s)
difficult
operations
easy
operations
1
L
g (t )
G ( s)
Solution
L f (t ) F (s)
f (t )e st dt
s j
1
L
F ( s)
f (t )
0 j
c
1
2 j
c
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st
F ( s )e ds
j
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Properties of Laplace Transform
1. Superposition
L f1 (t ) f2 (t ) F1 (s) F2 (s)
2. Time delay
L f (t ) u(t ) es F (s)
3. Time scaling
1 s
L f (at ) F
a a
4. Shift in Frequency
L e at f (t ) u (t ) F ( s a)
5. Differentiation in Time
L f (t ) s F ( s ) f (0 )
L f (t ) s 2 F ( s ) s f (0 ) f (0 )
L f n (t ) s n F ( s ) s n 1 f (0 )
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s f n 2 (0 ) f n 1 (0 )
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Properties of Laplace Transform
6. Integration in Time
t
1
L f (t )dt F ( s )
0
s
7. Differentiation in Frequency
dF ( s )
L t f (t )
ds
8. Convolution
F1 ( s ) F2 ( s ) L f1 (t ) f 2 (t )
F1 ( s ) F2 ( s ) 2 j L f1 (t ) f 2 (t )
f1 (t ) f 2 (t )
f () f (t )d
1
2
0
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Table of Laplace Transform
(t )
1
unit impulse
t
1(t )
1
unit step
t
r (t )
1
1
unit ramp
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t
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Laplace Transform
Example:
2t
Obtain the Laplace transform of f (t ) (t ) 2 1(t ) 3e , t 0.
F ( s ) L (t ) L 2 1(t ) L 3e 2t
L (t ) 2 L 1(t ) 3 L e 2t
1
1
1 2 3
s
s2
s2 s 4
s ( s 2)
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Laplace Transform
Example:
Find the Laplace transform of the function shown below.
g (t )
4
0
1
2
3
4
t
g (t ) 4 1(t 2) 4 1(t 3)
G ( s) L g (t )
e2 s
e3s
4
4
s
s
4 2 s 3s
(e e )
s
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Inverse Laplace Transform
The steps are:
1. Decompose F(s) into simple terms using partial-fraction
expansion.
2. Find the inverse of each term by using the table of Laplace
transform.
Example:
Find y(t) for Y ( s)
( s 2)( s 4)
.
s( s 1)( s 3)
c3
c1
c2
Y (s)
s s 1 s 3
c1 ( s 1)( s 3) c2 s( s 3) c3 s( s 1)
s( s 1)( s 3)
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Inverse Laplace Transform
(c1 c2 c3 )s 2 (4c1 3c2 c3 )s 3c1
1s 2 6s 8
Y ( s)
s( s 1)( s 3)
s(s 1)(s 3)
Comparing the c1 c2 c3 1
coefficients
4c1 3c2 c3 6
3c1 8
8
3
1
c1 , c2 , c3
3
2
6
81 3 1 1 1
Y ( s)
3 s 2 s 1 6 s 3
y (t ) L1 Y ( s )
8 1 1 3 1 1 1 1 1
L L
L
3
s 2
s 1 6
s 3
8
3 t
1 3t
1(t ) e 1(t ) e 1(t )
3
2
6
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Initial and Final Value Theorem
lim y (t ) lim s Y ( s )
s
t 0
lim y (t ) lim s Y ( s )
t
Only applicable to stable
system, i.e. a system with
convergent step response
s 0
Example:
Find the final value of the system corresponding to
3( s 2)
Y ( s)
s( s 2 2s 10)
y () lim y (t ) lim s Y ( s )
t
s 0
3( s 2)
3 2
y() lim s
0.6
2
s 0
s( s 2s 10)
10
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Initial and Final Value Theorem
Example:
Find the final value of the system corresponding to
3
Y (s)
s( s 2)
3
3
y () lim y (t ) lim s Y ( s) lim s
t
s 0
s 0
s( s 2) 2
WRONG
Since
3
3 2 3 2
Y ( s)
s( s 2)
s
s2
y(t ) L1 Y (s) 3 2 1(t ) 3 2 e2t 1(t ) NOT convergent
NO limit value
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Initial and Final Value Theorem
Example:
Find the final value of
2
Y ( s) 2
s 4
2
y () lim y (t ) lim s Y ( s) lim s 2
0
t
s 0
s 0
s 4
WRONG
Since
2
Y (s) 2
y (t ) sin 2t
s 4
Erwin Sitompul
periodic signal
NOT convergent
NO limit value
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Homework 1
2.6
3.4 (b)
3.5 (c)
3.6 (e)
Deadline: 10.05.2011, 7:30 am.
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