Post on 03-Jun-2018
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Repetitive Control Theory
Basics of Control Theory
Open Loop Versus Closed Loop
1 Open Loop systems are always stable unless plant behavior
is unstable where as closed loop system, if properly
designed can compensate for instability of plant for
example of balancing of Inverted Pendulum in closed loop.
2 Closed loop can improve transient response and
compensate for disturbances, nonlinearity of the plant etc.
3 Open loop control is simple where as closed loop is
complex as compared with open loop as controller design
requires additional hardware and has to be design for a
stable operation.
Goals for Design of a Controller in a Closed Loop
Control
Fast Transient Response
Zero steady state error
Should be stable over complete range of operation.
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Example of Stability by Perturbation
+5
-5V +15
-
-
+5
10
10
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+5V-5V +15V-15V
-5V
+5V
10K
10K
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Analog Controller Versus Digital Controllers
Implementation of analog controllers is quite old and well
known from time of steam engine in the recent history.
Analog Controllers have wide band-width
Easy Implementations.
Hardware size is proportional to number of functions to be
implemented.
Can be adopted for linear as well as non-linear plant
behavior.
Can not log/save Data
Fault conditions will be lost on power failure.
Advantages of Digital Controller
Some of the advantages of digital controller over
analog controller are listed below.
Flexibility :Any changes in methodology or control
parameters dont require changes in hardware.
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Advanced Control :Due to implementation of digital
control, It has been possible to implement more
advanced control schemes such as Dead-Beat control,
Disturbance observer, Repetitive control, Kalman Filter
Park and Clark transforms etc. to achieve better
performance. Can be designed for linear as well as non
linear plant behavior.
Communication & Control : System cancommunicate with several other systems as well as
with master controller to implement factory
automation. This helps in setting control parameters
remotely.
Reduction in Hardware size of controller :
As one DSP chip can be used for generating
several PID loops and signal monitoring & contro
hence require minimal system
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Data Logging and Fault recording:Inevent of fau
condition all the data and fault conditions can be
recovered for further action
Multiplexing and Multitasking :With general
purpose digital hardware, many more signals can be
sampled by just adding a multiplexers which increases
the flexibility of the control and one DSP chip can
perform several operations such as computation of
control law, signal sampling, fault recording,communication etc. Fault recording could be
summarized as one of the best facility that digital
control can offer.
Disadvantages of Digital Controller
Sampling Effect :Unless sampled at higher rates ca
result into aliasing of signal and there by loosin
important data
Time Delay Effect:Sampling introduces delays. Dela
introduces frequency dependent phase lag with n
change in amplitude and Phase lag reduces closed-loo
stability
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Quantization and Delay Effects : A-to-D conversio
causes quantization errors. This reduces accuracy o
measurement of input/output signals. Effect of this ca
be reduced by higher order A/D converters.
Effect of Time Delay on Step Response and Stability
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Some Popular Techniques Used In Digital Control
Dead-Beat Control
Predictive Control
Digital PID
Observer based Sensor-less Control
Repetitive Control Etc
Closed-loop system
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But for AC Reference Input, PI loop gain has finite
gain and can not track the AC input hence to make
the controller track exactly to the given AC reference,
resonant controller scheme was introduced.
Gain
1H
10H
100Hz
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but then with with the advent of microprocessors dq
transformation technique was developed. This uses
conventional PI loop and AC quantities are
transformed in to DC to get zero steady state error.
Principal of Repetitive Control
Repetitive control intends to track/reject
arbitrary periodic signals/disturbance of afixed period
Tracking/Disturbance rejection of periodicsignals appear in many applications
Hard disk/CD drives
Electric power supply Robotic motions
Steppers in IC productions
And many others
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History: The First Example
Magnet power supply for a protonsynchrotron (Nakano and others) for
Ring Magnet
Control Objective:
Control the power supply curve(periodically) to thefollowing shape:
Precision requirement: order of 0.1V!
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Internal Model Principle of a Control System
According to internal model principal if Laplace transform of
Vin(s) and Gc(s) should have same poles to exactly track the
reference.
If input signal or the disturbance is periodic then it can berepresented by Fourier series as
So the minimal system Gc
will consist of following poles.
GC(S)Vin(s)
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1)........(1..........
)20
2n2(s
2)0
(n1)(n
sscG
)2.(..........!7
6)'(
!5
4)'(
!3
2)'(1
)'(
)'sin(
sss
s
s
This polynomial has roots at s = +/- 1, +/- 2, and so on.
So
)3.().........23
2'1)(
22
2'1)(
21
2'1(
)'(
)'sin( sss
s
s
Expanding right-hand side we get
...].
23
2'
22
2'
21
2'......'...)
23
1
22
1
22
1
21
1('...)
22
1
21
1(1[
'
)'sin( 42 sss
ss
s
s
Comparing coefficients of equn 2 & 4 we get following.
6
2...........
24
1
23
1
22
11
and similarly
120
2........
25
1.
24
1
24
123
123
122
1
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In any given function we may replace a variable with another
variable with out changing its value.
Let be s = j*sin equn 4 then we get
)5.().........23
21)(
22
21)(
21
21(
)(
)sin( sss
sj
sj
)6........(
1
)2
21(
)sin(
n n
s
sj
sj
)7........(
1
)22
1(sinh
n n
s
s
s
)8.........(
1 )1(
1
sinh
2
2
n
n
ss
s
)9.........(
)1(
11
sinh1
2
2
n
n
sss
)10.........(
)1(
11
sinh2
12
2
0
n
T
n
sss
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)11.........(
)1(
11
sinh2
12
20
n
T
n
sss
Again substitute S -> S/0then
Then
)12)........((.
)1(
11
)sinh(
2
1
202
2
0
scG
n
sss
n
T
But
So
)13)........((
)sinh(
2
2
scGsT
T
To calculate the transfer function of the above equn first we have to
substitute s = j and T = 2/ 0and then we get
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)( 0
0
2
1
j
e
j
Te
and
)sin()0
2
1(
0
0
T
j
e
j
Te
So the minimal system for Gc(s) is given as below.
This can be derived by using following block.
)( 0
2
1
j
e
0
2
j
e
1,0
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or
Verror(s) = Gc(s)
Periodic Waveform Generator
Verror(s) Verror(s)
T T 2T 3T0
Periodic Waveform Generator
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z-N
Q(z-1
)
ZK
+ +
error
Repetitive Control Block
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+ -
+10VRepetitive
Control
+ +
error
e-sT
Repetitive Control Block
+ +
e-sT
+
-
+10V
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+ +
e-sT
+
-
+10V +5V
5V
+5V
0 V
+ +
e-sT
+
-
+10V +7.5V
7.5V
+2.5V
+5V
+ +
e-sT
+
-
+10V +8.75V
8.75V
+1.25V
+7.5V