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 ControlFast Transient ResponseZero steady state errorShould be stable over complete range of operation.

Example of Stability by Perturbation

+5V-5V +15V

-15V

-5V +5

V

10K

10K

-5V

+5V

10K

10K

+5V-5V +15V-15V

Analog Controller Versus Digital ControllersImplementation 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 don’t require changes in hardware.

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 can communicate 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 & control hence require minimal system

Data Logging and Fault recording: Inevent of fault condition all the data and fault conditions can be recovered for further actionMultiplexing 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 can result into aliasing of signal and there by loosing important data

Time Delay Effect : Sampling introduces delays. Delay introduces frequency dependent phase lag with no change in amplitude and Phase lag reduces closed-loop stability

Quantization and Delay Effects : A-to-D conversion causes quantization errors. This reduces accuracy of measurement of input/output signals. Effect of this can be reduced by higher order A/D converters.Effect of Time Delay on Step Response and Stability

Some Popular Techniques Used In Digital Control

Dead-Beat Control Predictive Control Digital PID

Observer based Sensor-less Control Repetitive Control Etc

If we assume that GH >> 1, then the overall transfer function simplifies to

Closed-loop system

if GH>>1

If reference input is a DC quantity then using PI loop will give zero steady state error.

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

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1Hz

10Hz

100Hz

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 a fixed period

Tracking/Disturbance rejection of periodic signals appear in many applications

Hard disk/CD drives Electric power supply Robotic motions Steppers in IC productionsAnd many others

History: The First Example

Magnet power supply for a proton synchrotron (Nakano and others) for Ring Magnet

Control Objective:

Control the power supply curve (periodically) to the following shape:Precision requirement: order of 0.1V!

Internal Model Principle of a Control System

GC(S)Vin(s)

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 be represented by Fourier series as

So the minimal system Gc will consist of following poles.

This polynomial has roots at s = +/- 1, +/- 2, and so on.So

Expanding right-hand side we get

Comparing coefficients of equn 2 & 4 we get following.

and similarly

In any given function we may replace a variable with another variable with out changing its value.

Let be s’ = j*s in equn 4 then we get

Again substitute S -> S/ω0 thenThen

But

So

To calculate the transfer function of the above equn first we have to

substitute s = jω and T = 2π/ ω0 and then we get

and

So the minimal system for Gc(s) is given as below.

This can be derived by using following block.

)( 0

21

je

0

2j

e

1,0

Verror(s) = Gc(s)

or

Periodic Waveform Generator

Verror(s) Verror(s)

T

= Gc(s)

Plant

T

= Gc(s)

Plant

2T

= Gc(s)

Plant

3T

= Gc(s)

Plant

0

= Gc(s)

Plant

Periodic Waveform Generator

z-N

Q(z-1)

ZK

+ +error

Repetitive Control Block

+ -

+10V RepetitiveControl

+ +

error

e-sT

Repetitive Control Block

+ +

e-sT

+

-

+10V

+ +

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