Repetitive Control

8/12/2019 Repetitive Control

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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

Repetitive Control

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