DCS Chapter2

8/14/2019 DCS Chapter2

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CHAPTER 2

SIGNAL PROCESSING INDIGITAL CONTROL


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CONFIGURATION OF THE BASIC

DIGITAL CONTROL SCHEME.

A/D D/AComputer

Clock

Final

controlelement Plant

S/HAnti-aliasing

filterSensor

Digital

set -point

Disturbance Controlled

output


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The analog feedback signal that comes from thesensor is usually of low frequency but often includehigh frequency noise.


DIGITAL CONTROL SCHEME

Sensor


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Anti-aliasing filter is a low pass filter that filters outhigh frequency noise from the analog signal that

comes from the sensor.



Anti-aliasing Filter


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A/D conversion systemconverts the analog signal todigital signalafter anti-aliasing process.

A/D conversion system consists of A/D converterpreceded by sample and hold (S/H) device.

A/D converter converts a voltage or current amplitudeat its input into a binary code. However, theconversion is not instantaneous.

This input signal variationduring the conversion timeof the A/D converter can lead to erroneousresults.

Thus, high performance A/D conversion systemsinclude a S/H device which keeps the input to the A/D

converter constantduring the conversion time.



A/D Conversion System


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Digital computer processes the sequence ofnumbers by an algorithm and produces a newsequence of numbers.

Since data conversions and computations takestime, there is always computational delay thatdegrades the control system performance.

This can be minimized by the proper choice ofhardware and the proper design of software for thecontrol algorithm.



Computer


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Real-time clock in the computer synchronizes all

the events of A/D conversion-computation-D/A

conversion.



Clock


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D/A conversion system converts the sequence ofnumbers in numerical code into a piecewise

continuous-time signal.



D/A Conversion System


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The output of the D/A converter is fed to the plantthrough the actuator (final control element)to controlits dynamics.



Plant & Final Control Element


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This basic control scheme shows a single feedback

loop and assumes a uniform sampling operation;

only one sampling rate exist in the system and thesampling period is constant.

A digital control system having multiple loops may

have multiple-rate sampling;different samplingperiods in different feedback paths.




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IMPLEMENTATION PROBLEMS IN

DIGITAL CONTROL

MAIN IMPLEMENTATION

PROBLEMS IN DIGITAL CONTROL

QUANTIZATION

EFFECTS

SAMPLING

EFFECTS


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DIGITAL CONTROL

Quantization Effects

The conversion of signals from analog into digital

form and vice versa is performed by electronicdevices (A/D and D/A converters) of finiteresolution.

The analog signalgets tied to these finite number ofquantization levels in the process of conversion todigitalform.

As a result, a valuable part of information about thesignal is lost.


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DIGITAL CONTROL

Quantization Effects

In addition, any computer employed as a real-time

controller performs all the necessary calculationswith limited precision.

As a result, a truncation error occurs after each

arithmetic operation has been performed .


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DIGITAL CONTROL

Sampling Effects

Process of sampling and reconstructionaffects the

amount of information available to the computerand degrades the control system performance.

In sampling theorem, sampling period should be

chosen such that

where is the strict bandwidthof the

signal being sampled.

mT /m


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DIGITAL CONTROL

Sampling Effects

This condition ensures that

a) there is no lossof information due to sampling

b) the continuous time signal can completely

recoveredfrom its sample by using an ideal lowpass filter.


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DIGITAL CONTROL

Sampling Effects

However, there are two problems associated with

the use of this theorem in practical control systems:

a) Real signals are not band-limitedand hence

strict bandwidth limits are not defined.

b) The ideal low-pass filter, needed for the

distortionless reconstruction of continuous-time

signals from its samples, is not physically

realizable. Practical devices, such as the D/A

converter, introduces distortions.

Thus, sampling process degrades the control system

performance.


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PRINCIPLES OF SIGNAL CONVERSION

SIGNAL CONVERSION

ANALOG TO DIGITALSIGNAL CONVERSION DIGITAL TO ANALOGSIGNAL CONVERSION


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ANALOG TO DIGITAL SIGNAL CONVERSION

Analog signal Discrete-time signal

Digital signal Digital word

Sampling

Quantization

Coding


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An analog signalcannot be stored in digital computers.

Therefore it must be converted to a form that will beaccepted by digital computers.

.


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The sampling intervalcorresponds to a sampling rate

of of 100 samples/sec. The choice of the sampling rateis important because it

determines how accurately the discrete-time signalcan represent the original signal.



Sampling


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The time axis of the discrete-time signalis labeled

by samplenumberand index k (k=0, 1, 2, )

ANALOG TO DIGITAL SIGNAL CONVERSIONSampling


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The sample values can be represented by a sequence of numbersys,...}4.0,4.1,8.2,4.2,7.1{sy


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In general, kkyys 0)},({

)(kywhere denotes the kth number in the sequence.This is a one-sided sequence, whereys=0 for k


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ANALOG TO DIGITAL SIGNAL CONVERSIONQuantization

This signal is quantified to four quantization levels.


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ANALOG TO DIGITAL SIGNAL CONVERSIONCoding

. The final step in converting an analog signal to a

form acceptable by digital computers is coding.

The encoder maps each quantized sample value

into a digital word.


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.Sampling, quantization and coding operation is performed

by an A/D converter.

Sampler Quantizer Encoder

Continuous-timecontinuous-amplitude

signal

Digital

words

Discrete-time

continuous-amplitude

signal

Discrete-time

discrete-amplitude

signal


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D/A conversion is a process of producing an analogsignal from a digital signal.

Thus it is the reverse ofA/D conversionprocess.

D/A conversionis performed by D/A converter.

In a D/A converter:

a) the decoder generate output samples fromthe binary-form digital signals produced by the

machineb) the zero- order holdconverts these samples to

analog form

DIGITAL TO ANALOG SIGNAL CONVERSION



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DIGITAL TO ANALOG SIGNAL CONVERSION

DecoderZero-order

hold

Digital

words

Discrete-

time signal

Analog

signal

The operations performed by a D/A converter

Sampled sequence Analog output


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Unit Sample Sequence

BASIC DISCRETE-TIME SIGNALS

Delayed unit sampled sequence

1 for k= n

0 otherwise )( nk


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An arbitrary sequence can be represented by sum of scaled,

delayed unit sample sequence.

The sequence in this figure can be represented by:


Unit Sample Sequence

...)2()2()1()1()()0()( krkrkrkr

0

)()(n

nknr


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Unit Step Sequence

0

1)(k

for k 0

otherwise


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Unit Ramp Sequence

0,0

0,)(

k

kkkr


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Sinusoidal Sequence)()cos()( kkAkr

where is the frequency in radians and is the phase.


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TIME-DOMAIN MODELS FOR

DISCRETE-TIME SYSTEMS

A discrete time system is defined mathematically as

a transformation or an operator that maps input

sequence r(k)into an outputsequencey(k).

Time-domain model is one of mathematicalrepresentations for linear time-invariant discrete-

time systems.

The time-domain models that will be discussed in

this lecture are:

a) Difference Equation Models

b) Impulse Response Models


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A linear time-invariant discrete time systemcomposed

of n dynamic elements can be analyzed by using asingle nth-order difference equationas its model.

Difference Equation Models


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The integral is approximated by

Subtracting equation (2.1) from equation (2.2),




1

00 )()()(

k

m

kT

mTTrdrkTc

TkT k

m

mTTrdrTkTc0

0

)()()(

)()()( kTTrkTcTkTc

(2.1)

(2.2)

(2.3)


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In general, the nth order linear difference equationrelating the outputy(k)to input r(k)is given by:

The coefficients aiand bjare real constants; k, m, andnare integers with mn.


)(...)1()( 1 kyankyanky n)(...)1()( 10 krbmkrbmkrb m (2.1)


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For m=n, the nth order linear difference equation

relating the outputy(k)to input r(k)is given by:




)(...)1()( 1 kyankyanky n

)(...)1()( 10 krbnkrbnkrb n (2.2)


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Shifting the origin from k=n to k=0, the equivalentdifference equation modelis obtained:




)(...)1()( 1 nkyakyaky n )(...)1()( 10 nkrbkrbkrb n (2.3)

The initial conditions of this model are {y(-1), y(-2), ,

y(-n)}.Given {y(-1), , y(-n)}, the initial conditions {y(0), ,

y(n-1)} of the model (2.2) can be determined by

successively substitutingk=-n,-n+1, , -2, -1,inEqn. (2.2).


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In impulse response model, the system is assumed

to be linear, time invariant and initially relaxed.

A linear time-invariant initially relaxed system can be

characterized by its impulse response.



Impulse Response Models


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Letg(k) be the response of initially relaxed linear time-

invariantdiscrete-time system to an impulse (k).

Due to time-invariance property, the response to

(k-n) will beg(k-n).

By linearity property, the response to an input signal

r(k):




...)2()2()1()1()()0()( kgrkgrkgrky

0

0);()(j

kjkgjr(2.4)


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Equation (2.4) is usually called the convolution sum.

For a causal systemthe equation becomes:

(2.5)




k

j

kjkgjrky0

0);()()(


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Another important observation concerns the symmetry

of the situation. Let k-j=m in Eqn. (2.5), then

(2.6)

Reversing the order of summation,

(2.7)




0

)()()(km

mgmkrky

k

m

mkrmgky

0

)()()(


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If the weighting sequence is known, the discrete timesystems response to any input can be evaluated byusing convolution summation.

Convolution summation, c(k)is defined as:.



Weighting Sequence of Linear Discrete

Systems

k

m

mkrmhkc0

)()()(


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Consider the input and weighting sequence of the system:


Systems


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By convolution process:

*Note that k=3


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The discrete-time system response to input r(m):




Systems


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Examples

Suppose a signal varies between 0

and 10 Volts (which is termeddynamic range) and it is required thatthe signal must be represented in the

digital computer to the nearest 5 mV,that is, the resolution is expected tobe 5 mV. Determine how many bitsthe ADC must have to achieve thisobjective.


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Examples

Suppose a radar search antenna at the KLIA

rotates at 10 rev/min, and data points

corresponding to the position of flight MH 1001

are plotted on the controllers screen once per

antenna revolution. MH 1001 is traveling directlytowards the airport at 600 km/hr. A feedback

control system is established through the

controller who gives course corrections to the

pilot. The controller wishes to do so at 10 km oftravel of the aircraft, and his instructions consist

of course headings in integral degree values.


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Examples a) What is the sampling rate, in seconds, of the range

signal plotted on the radar screen?

b) What is the sampling rate, in seconds, of the

controllers instructions?

c) Identify the following signals as continuous, discrete,

or digital:

i) the aircrafts range from the airport.

ii) the range data as plotted in the radar screen.

iii) the controllers instructions to the pilot.

iv) the pilots actions on the aircraft control surfaces.d) Is this a continuous, sampled data, or digital control

system?


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Quiz#1

With a sampling frequency of 200 kHz,what is the system output (DAC) for the

input signals having the following

frequencies a) 157.6 kHz?

b) 261 kHz?

c) 54.9 kHz? d) 500.7 kHz?

e) 184.2 kHz?

f 100.1 kHz?


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Quiz In a particular temperature control system

application, it is desired to measure thetemperature in the range -20 to 50C withan accuracy of 1C. Suppose a

temperature transducer with a gain of 0.02V/C and accuracy of 0.5C is used inthis system, determine the number of ADCbits for this system. Also determine the

mean-square-error quantization power.

DCS Chapter2

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