discrete control system(intro)

8/13/2019 Discrete Control System(Intro)

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INTRODUCTION TODISCRETE CONTROL

SYSTEM


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UPON COMPLETION OF THIS CHAPTER STUDENTSSHOULD BE ABLE T0:

DESCRIBE THE ADVANTAGES AND DISADVANTAGES OF

DIGITAL CONTROL

DIFFERENTIATE BETWEEN ANALOGUE AND DIGITAL SYSTEM

EXPLAIN THE SAMPLING PROCESS AND SIGNAL COVERSION

FROM ANALOG TO DIGITAL SIGNAL


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INTRODUCTION

The use of a digital computer as a compensator (controller) device has

grown during the past three decades as the price and reliability of digitalcomputers have improved dramatically .

The cost of microcontrollers and digital signal processors has reduced

and you can buy an 8-bit micro controller for a few RM.

A digital control system uses digital signals and a digital computer tocontrol a process.

The measurement data are converter from analog form to digital form by

means of the analog-to-digital converter shown in Figure 1.1.

Figure 1.1: A block diagram of a computer control system


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Digital control systems are used in many applications: for

machine tools, metalworking processes, chemical processes,

aircraft control, and automobile traffic control and others.

Digital Control Problem

Usually trying to control a continuous time system, but using a

digital controller to control it.

Need to be able to convert from continuous time signal to discrete

time signal (A/D converter) this is just sampling Need to be able to convert from discrete time to continuous time

signal (D/A converter) may ways to do this


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Flexibility.With analogue circuitry it would be difficult to suddenly change a

resistor to modify the integral gain while the plant is operating! So it

allows operators to, modify, to tune, to experiment with the controller.

With a good user interface any parameter inside that piece of control

software can be adjusted.

Improved measurement sensitivity

the use of digitally coded signals, digital sensors and transducers, and

microprocessors; reduced sensitivity to signal noise; and the

capability to easily reconfigure the control algorithm in software.


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Both algorithmic and decision-making control in one

package

Usually Smaller/lighter

Usually needs Less power

Often More precise Can Re-program

It's easy for parameters to be passed around in a program, far easier than

hardware signals.

More complex control can be realised

It allows algorithms such as vector control of machines involving

flux modelling and reference frame transformation to be carried

out.


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Computation is performed serially,

Instructions take a finite time, albeit in the region of nano-

seconds, but if sever hundred instructions are required to carry

out a task then this time can add up. This can limit the

complexity, but each year faster processors are launched and this

problem becomes less significant for all but the cheapest micros.

New theory involved

There is generally a resistance by industry to adopt new ideas,

although this is becoming less of a problem.


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Sampled Data Systems

The basic difference between these controllers is that the digital

system operates on discrete signals (or samples of the sensed

signal) rather than on continuous signals.

A typical sampled data control system is shown in Figure 1.2.

Figure 1.2


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Sampled Data Systems

The digital computer performs the compensation function within

the system.

The A/D converter converts the error signal, which is a continuous

signal, into digital form so that it can be processed by the computer.

At the computer output the D/A converter converts the digital output

of the computer into a form which can be used to drive the plant.

A sampler is basically a switch that closes every T seconds, as shown

in Figure 1.3.

Figure 1.3


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The ideal sampling process can be considered as the

multiplication of a pulse train with a continuous signal, i.e.

Figure 1.4


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where P(t) is the delta pulse train and expressed as:

Taking the Laplace transform gives:


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A D/A converter converts the sampled signal into a continuous signal y(t).

The D/A can be approximated by a zero-order hold (ZOH) circuit as shown in

Figure 1.5.

This circuit remembers the last information until a new sample is obtained, i.e. the

zero-order hold takes the value r(nT) and holds it constant for and the value r(nT)

is used during the sampling period.

*( )r t

Figure 1.5: sampler and zero-order hold


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The transfer function of a zero-order hold is given by

A sampler and zero-order hold can accurately follow the input

signal if the sampling time T is small compared to the

transient changes in the signal.

Where H(t) is the step function, and taking the Laplace transform yields

Figure 1.6: waveforms after the sampler and zero-order hold


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Example 1.1

Figure 1.7 shows an ideal sampler followed by a zero-order hold. Assuming the

input signal r(t) is as shown in the figure. Show the waveforms after the

sampler and also after the zero-order hold.

Solution

Figure 1.7


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Systems The z-transform can be defined through the following equation:

Its used in sampled data systems just as the Laplace transformation

is

used in continuous-time systems.

The z-transform of the function r(t) is Z[r(t)]=R(z) which is given by:

The response of a sampled data system can be determined easily by

finding the z-transform of the output and then calculating the inversez-transform, just like the Laplace transform techniques used in

continuous-time systems.


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EXPONENTIAL FUNCTION


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SINE FUNCTION

so that

FromPreviously we already knowthe z-transform of an

exponential function is:

)

2cos

2sin

jx jx

jx jx

From

e e x

e e x


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COSINE FUNCTION

Previously we already know

the z-transform of an

exponential function is:

2cos

2sin

jx jx

jx jx

e e x

e e x


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THE Z TRANSFORM OF COMMON DISCRETE TIME FUNCTIONS


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THE Z TRANSFORM OF COMMON DISCRETE TIME FUNCTIONS (cont..)


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Determine G(z) for the system with transfer function:

Example 1.2

Solution:

G(s) can be express as a sum of its partial fractions:

The inverse Laplace transform of this equation is :


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In z-transforms we can write this equation as:


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Properties of z-Transforms

1. Linearity property

Suppose that the z-transform of f (nT ) is F(z) and the z-transform of g(nT ) is

G(z). Then

and for any scalar a

2. Left-shifting property

Suppose that the z-transform of f (nT ) is F(z) and let y(nT ) = f (nT+mT ).

Then

If the initial conditions are all zero, i.e. f (iT ) =0, i = 0,1,2,..., m1, then,


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3. Right-shifting property

Suppose that the z-transform of f (nT ) is F(z) and let y(nT ) = f (nTmT).

Then

If f(nT ) = 0 for k < 0, then the theorem simplifies to

4. Attenuation property

Suppose that the z-transform of f (nT ) is F(z). Then,

This result states that if a function is multiplied by the exponential eanT

then in the z-transform of this function z is replaced by zeaT

.


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5. Initial value theorem

Suppose that the z-transform of f (nT ) is F(z). Then the initial value of the

time response is given by

6. Final value theorem

Suppose that the z-transform of f (nT ) is F(z). Then the final value of the

time response is given by

Note that this theorem is valid if the poles of (1 z 1)F(z) are inside the

unit circle or at z =1.


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Example 1.3

The z-transform of a unit ramp function r(nT ) is

Find the z-transform of the function 5r(nT ).

Solution

Using the linearity property of z-transforms,

Example 1.4

The z-transform of trigonometric function r(nT) =sinnwT is

find the z-transform of the function y(nT) = e 2T sin nW T .


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Example 1.5

discrete control system(intro)

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