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C H A P T E R 1

BKC3863 Advanced Process Control

BKF 3863ADVANCED PROCESS CONTROL

Semester I 2011/2012

C H A P T E R 1

BKC3863 Advanced Process Control 2

CHAPTER 1

Digital sampling and z-Transforms

Digital computer and its components, Sampling

continuous signals, Reconstruction of continuous

signals from their discrete-time values, Analog to digital

converters, Digital to analog converters, multiplexer,

Discrete time model of a first-order and second-order

processes, z – Transform analysis for digital control

C H A P

T E R 1


Control Diagram of a Typical ControlLoop (Blending Process)

Controller

F1

T 1

T

F

F 2

T 2

TC

Actuator

System

TT

Sensor

System C H A P

T E R 1


DIGITAL COMPUTER



C H A P T E R 1


BASIC CONCEPTS OF COMPUTERHARDWARE

– Programs and data are stored in the same memory:primary memory.

– The computer can only perform one instruction at a time.

CPU

(Central Processing Unit)

Input

Units

Output

Units

Primary Memory

C H A P T E R 1


BASIC CONCEPTS OF COMPUTERHARDWARE

• Input/Output (I/O): Refers to the process ofgetting information into and out of thecomputer.

– Input: Those parts of the computerreceiving information to programs.

– Output: Those parts of the computer that

provide results of computation to theperson using the computer.

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SOURCES OF DATA FOR THE COMPUTER

• Two types of data stored within a computer:

– Original data or information: Data beingintroduced to a computing system for the first time.

• Computers can deal directly with printed text,pictures, sound, and other common types ofinformation.

– Previously stored data or information: Data thathas already been processed by a computer and isbeing stored for later use.

• These are forms of binary data useful only to

the computer.

• Examples: Floppy disks, DVD disks, and musicCDs.

C H A P

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INPUT DEVICES

• Two categories of input hardware:

– Those that deal with original data.

– Those that handle previously stored data.

• Input hardware: Those that deal with original data.

– Keyboard

– Mouse

– Voice recognition hardware

– Scanner

– Digital camera

• Digitizing: The process of taking a visual image, or

audio recording and converting it to a binary form forthe computer.

– Used as data for programs to display, play ormanipulate the digitized data.



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INPUT DEVICES• Connecting Hardware to the computer:

– Hardware needs access through some general input/output

connection.

• Port: The pathway for data to go into and out of the

computer from external devices such as keyboards.

– There are many standard ports as well as custom

electronic ports designed for special purposes.

– Ports follow standards that define their use.

» SCSI, USB: Multiple peripheral devices (chain).

» RS-232, IDE: Individual peripheral devices.

• Peripheral device: A piece of hardware like a printer or

disk drive, that is outside the main computer.

C H A P T E R 1


INPUT DEVICES• Connecting Hardware to the computer:

(continued)

– Hardware needs software on the computer thatcan service the device.

• Device driver: Software addition to theoperating system that will allow the computerto communicate with a particular device.

• Common Basic Technologies for StoringBinary Information:

– Electronic

– Magnetic

– Optical

C H A P

T E R 1


INPUT DEVICES

• Electronic Circuits

– Most expensive of the three forms forstoring binary information.

– A flip-flop circuit has either one electronic

status or the other. It is said to flip-flop fromone to the other.

– Electronic circuits come in two forms:

• Permanent

• Non-permanent

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INPUT DEVICES

• Magnetic Technology

– Two parts to most of the magnetic forms of informationstorage:

• The medium that stores the magnetic information.

– Example: Floppy disk. Tiny spots on the disk are

magnetized to represent 0s and 1s.• The device that can “read” that information from the

medium.

– The drive spins the disk.

– It has a magnetic sensing arm that moves overthe disk.

– Performs nondestructive reading.



C H A P T E R 1


INPUT DEVICES

• Optical

– Uses lasers to “read” the binary informationfrom the medium, usually a disc.

• Millions of tiny holes are “burned” intothe surface of the disc.

• The holes are interpreted as 1s. Theabsence of holes are interpreted as 0s. C

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INPUT DEVICES

• Secondary Memory Input Devices

– These input devices are used by a computer tostore information and then to retrieve thatinformation as needed.

• External to the computer.

• Commonly consists of floppy disks, hard diskdrives, or CD-ROMs.

– Secondary memory uses binary.

• The usual measurement is the byte.

– A byte consists of 8 binary digits (bits). Thebyte is a standard unit.

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INPUT DEVICES

• Capacity - The amount of informationthat can be stored on the medium.

Unit Description Approximate Size

1 bit 1 binary digit1 nibble 4 bits

1 byte 8 bits 1 character

1 kilobyte 1,024 bytes ≈1/2 page, double spaced

1 megabyte 1,048,576 bytes ≈500,000 pages

1 million bytes

1 gigabyte 1,073,741,824 bytes ≈5 million pages

1 billion bytes

1 terabyte 1 trill ion bytes ≈5 billion pages

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INPUT DEVICES

• Type of Access

• Sequential - Obtained by proceeding

through the storage medium from the

beginning until the designated area is

reached (as in magnetic tape).

• Random Access - Direct access (as in

floppy and hard disks).



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PRIMARY MEMORY

• Primary storage or memory: Is where the data and

program that are currently in operation or being

accessed are stored during use.

– Consists of electronic circuits: Extremely fast and

expensive.

– Two types:

• RAM (non-permanent)

– Programs and data can be stored here forthe computer’s use.

– Volatile: All information will be lost once the

computer shuts down.

• ROM (permanent)

– Contents do not change.

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THE CENTRAL PROCESSING UNIT

• The Central Processing Unit ( CPU)

– Often referred to as the “brain” of the computer.

– Responsible for controlling all activities of the computer system.

– The three major components of the CPU are:

1. Arithmetic Unit (Computations performed)

Accumulator (Results of computations kept here)

2. Control Unit (Has two locations where numbers are kept)

Instruction Register (Instruction placed here for analysis)

Program Counter (Which instruction will be performed next?)

3. Instruction Decoding Unit (Decodes the instruction)

– Motherboard: The place where most of the electronics including the

CPU are mounted.

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OUTPUT DEVICES

• Output units store and display information (calculated

results and other messages) for us to see and use.

– Floppy disk drives and Hard disk drives.

– Display monitors: Hi-resolution monitors come in two types:

• Cathode ray tube (CRT) - Streams of electrons make

phosphors glow on a large vacuum tube.

• Liquid crystal display (LCD) - A flat panel display that

uses crystals to let varying amounts of different colored

light to pass through it.

– Developed primarily for portable computers.

C H A P T E R 1


OUTPUT DEVICES

• Audio Output Devices

– Windows machines need special audio card for audio output.

– Macintosh has audio playback built in.

– Audio output is useful for:

• Music

– CD player is a computer.

– Most personal computers have CD players that can

access both music CDs and CD-ROMs.

• Voice synthesis (becoming more human sounding.)

• Multimedia

• Specialized tasks (i.e.: elevator’s floor announcements)



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OUTPUT DEVICES

• Optical Disks: CD-ROM and DVD

– CD-ROM (Compact Disk - Read Only Memory)

• By its definition, CD-ROM is Read Only.

• Special CD drives “burn” information into blank

CDs.

– Burn: A laser is used to “burn” craters into

the surface to represent a binary 1.

– Two main types of CDs:» CD-R (Compact Disk - Recordable)

» CD-WR (Compact Disk - ReWritable)

• It takes longer to write to a CD-R than a hard

drive.

• Special software is needed to record.

C H A P T E R 1


Additional Hardware for ProcessControl Computer

Hardware Floating Point Processor – Performs with

very high speed floating-point arithmetic operations

Real Time Clock – Every digital computer used for

process control must have a real time clock.

Uninterrupted Power Supply (UPS) – In the event of

power failure this device will run the computer Watchdog Timer – If the control program “hung-up”

in a never ending loop then this device will alert the

operator and control engineer that the computer lost

control of the process

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Components and Signals of aTypical Control Loop

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DIGITAL COMPUTER CONTROL



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MULTIPLEXER

It quite often happens, in the design of large-scale digital systems,

that a single line is required to carry two or more different digital

signals. Of course, only one signal at a time can be placed on the one

line. What is required is a device that will allow us to select, at

different instants, the signal we wish to place on this common line.

Such a circuit is referred to as a Multiplexer .

A multiplexer performs the function of selecting the input on any one

of 'n' input lines and feeding this input to one output line.

C H A P T E R 1


MULTIPLEXERMultiplexers are used as one method of

reducing the number of integrated circuit

packages required by a particular circuit

design. This in turn reduces the cost of

the system.

Assume that we have four lines, C 0, C 1,

C 2 and C 3, which are to be multiplexed

on a single line, Output (f). The four input

lines are also known as the Data Inputs.Since there are four inputs, we will need

two additional inputs to the multiplexer,

known as the Select Inputs, to select

which of the C inputs is to appear at the

output. Lines A and B are called Select

Inputs

C H A P T E R 1


Sampler

Green Line – Continuous Signal

Blue Dots – Discrete Signal

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Sampler

Sampler is a switch, which closes every T seconds and

remains closed for an infinitesimally short period of time.

As the sampling period tends zero, the sampled

representation comes closer to the continuous signal but

requires an explosively large number of sampled values.

On the other hand, as the sampling period increases,

fewer sampled values are required, but the sampled

representation of a continuous signal deteriorates, and

the reconstruction of the original signal becomes poor or

sample.



C H A P T E R 1


Sampler

C H A P T E R 1


Sampler

t ___

0 0

0.632

0.865

0.9500.982

0.993

τ

2τ

3τ

4τ

5τ

Response of first-order system to a step of magnitude, M

C H A P T E R 1


Sampling of oscillating signal

Sampling an oscillating signal more than 2 times per

cycle of oscillation; otherwise its impossible to

reconstruct the original signals from its sampled values.

C H A P T E R 1


Hold Element



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First-order Hold Element

C H A P T E R 1


Comparison of Zero and First-order Hold

Elements

Slowly varying Signals

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Comparison of Zero and First-order HoldElements

Rapidly Changing Signals

C H A P T E R 1


Analog-to-Digital Converter

03 07 10 14 09 02 00 04



C H A P T E R 1


Analog-to-Digital Converter Resolution:

Suppose a binary number with N bits is to represent ananalog value ranging from 0 to A

There are 2N possible numbers (including zero).

Resolution = A / (2N – 1)

For example, consider a voltage range of 0 to 10V and12 bit converter. The 12 bits define 4096 integernumbers, which in turn defines 4095 voltage intervalsbetween 0 and 10.

ADC used for process control allow 20,000 to 100,000conversions per second

C H A P T E R 1


Digital-to-analog converter

Digital-to-analog converter (DAC) function in the reverse manner to ADC.

The 12 bits define 4096 integer numbers,which in turn defines 4095 voltageintervals between 0 and 10.

Then the integer number 516 causes an

analog output of(516/4095) X 10 = 1.26V

C H A P T E R 1


Discrete-Time Model of a Digital PIDController

⎥⎦

⎤⎢⎣

⎡++= ∫

t

D

I

cdt

d dt t t K t c

0

)(1

)()(ε

τ ε τ

ε

The continuous analog action of a PID controller is given by

C H A P T E R 1



)(nT K cε

Then the discrete control action produced by proportional

mode is

Then the discrete control action produced by integral

mode is based on the integration of errors over a time period.

Since the values of the errors are available on a discrete-time

Basis, the integral can be approximated by a numerical

Integration.

∑∫ =

≅n

k

t

kT T dt t 00

)()( ε ε



C H A P T E R 1



∑=

n

k I

c kT T K

0

)(ε τ

C H A P T E R 1



C H A P T E R 1



C H A P T E R 1


Discrete-time Model of a First-orderProcess



C H A P T E R 1


Discrete-time Model of a First-orderProcess

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Discrete-time Model of a Second-orderProcess

C H A P T E R 1


Discrete-time Model of a Second-orderProcess

n pnnnnnn m K y y yT

y y yT

=+−++− +++ )(2)2(1122

2 τ ξ

τ

[OR]

n pnnn mT

K yT T

yT

y2

2

2

2

12 12)1(2τ τ

ξ τ τ

ξ +⎟⎟ ⎠

⎞⎜⎜⎝

⎛ +−−−= ++

C H A P T E R 1


Z-Transforms

The z-transform is the most general concept for the

transformation of discrete-time series.

The Laplace transform is the more general concept for the

transformation of continuous time processes.

For example, the Laplace transform allows you to transform a

differential equation, and its corresponding initial and

boundary value problems, into a space in which the equation

can be solved by ordinary algebra.

The switching of spaces to transform calculus problems into

algebraic operations on transforms is called operational

calculus. The Laplace and z transforms are the most

important methods for this purpose.



C H A P T E R 1


Definition of Z-TransformsConsider a continuous function (signal) y(t) sampled at uniform

intervals of period T. Let the sequence of sampled values be y(o),

y(T), y(2T),…..

C H A P T E R 1


C H A

P T E R 1


Z- Transform of Basic Functions

Unit Step Function:

11

1

1111

1

321

z-

z

...... z z z Z[u(t)]

=−

=

+⋅+⋅+⋅+=

−

−−−

[ ]

onvergencefor c z e

z-e

z

z -e

λ

z e, λ λ z ee Z

-aT

-aT

-aT

-aT

n

nn

n

anT -at

1

1

1

1

1

1

1

1

00

<

=

=

−=

===

−

−

−∞

=

−∞

=

− ∑∑Exponential Function:

C H A

P T E R 1


Z- Transform of Basic Functions

Ramp Function:

Trigonometric Functions:

[ ]

[ ]12zcoswTz

zcoswTzcoswtZ

12zcoswTz

zsinwTsinwtZ

2

2

2

+−

−=

+−=

[ ]

1

1

321

132

1321

1

0

)1(2

320

−

−

−−−

−−−

−−−−

−=

++++=

−++=

++++==

z aTz

...(aT)z (aT)z )aT(z

S z (aT)z aTz

S ...z (aT)z (aT)z )aT(z S at Z

L



C H A P T E R 1


C H A P T E R 1


Properties of Z- Transforms

(z) y )-z ( y(t)t

-

⎥⎦⎤

⎢⎣⎡

→=

∞→ˆ1

1limlim 1

(z) f a(z) f a f a f a Z ∧

+∧

=+⎥⎦⎤

⎢⎣⎡

22112211

1. Linearity

2. Final Value Theorem

3. Initial Value Theorem:

(z) y y(t)t

⎥⎦⎤

⎢⎣⎡

∞→=

→ˆlim

0lim

C H A

P T E R 1


Inversion of z-Transforms

{ }),....3(),2(),(),0()(1

T yT yT y y z y z =⎥⎦⎤

⎢⎣⎡ ∧

−

Using the inversion of z-transform the values of a function at the sampling

instant can be calculated. The inverse of z-transform can be symbolized as

follows.

The inverse Z-transform yields the values of a function at the sampling

instants only and not the continuous function itself.

The inverse Z-transform does not even help us to determine the sampling

Period T for the computed sampled values: y(0), y(T), y(2T),…

The inverse transform of a function does not necessarily yield a

unique Continuous function y(t).

)( z y∧

C H A

P T E R 1


Inversion of z-Transforms

Two methods used to determine inverse z-transform:

1. Partial Fraction Expansion

2. Long division of two polynomials

1.Partial fraction expansion

λ1, λ2,… λn are low-order polynomials in z-1 compute c1,c2,…cn.

Invert each part separately.

)(...

)()()(

)()(ˆ

11

2

2

1

1

1

1

1

−−−−

−

+++== z

c

z

c

z

c

z P

z Q z y

n

n

λ λ λ



C H A P T E R 1


11

311

1

1

11

1

2

1

2

1

1

11

1

21

1

2

31

21

1

21)(ˆ

2

1

1

2

1

31

311)31)(1(

34134)(ˆ

1

1

−−

=−

−

=−

−

−−−−

−

−−

−

−

+

−

−=∴

=−

=

−=−

=

−+

−=

−−=

+−=

+−=

−

−

z y

z

z c

z

z c

z

c

z

c

z z

z

z z

z

z z

z z y

z

z

Partial Fraction Expansion

C H A P T E R 1


Partial Fraction Expansion

C H A

P T E R 1


Long Division Method

C H A

P T E R 1





C H A P T E R 1



4321

321

2.003.119.036.01

42.156.167.24)(ˆ

−−−−

−−−

+−+−

−++=

z z z z

z z z z y

Tutorial: Find the inverse z-transform using long division method

apc chapter 1

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