Fundamentals of Automation
8th semester, 3+1 c.
(Introduction to Automation)
Lecturer: dr. Tibor Csáki
dr. Ildikó Makó
Machine Tool Department
University of Miskolc
Topics
Fundamental ideas in automation. Signals, signal theory. Digital and analogue signals,
systems.
Theoretical fundamentals of digital automation. Boole functions and their realization.
Functional elements of digital systems in automation of machinery.
Linear control theory. Operational blocks, group of blocks, block diagram. Methods for
analysis of control loops. (Laplace transformation, differential equations, frequency response
methods, rootlocus method.)
Compensation of control loops. Basic control loops in machinery. Joined positioning systems
in robotics.
Principle of numerical control. Structure of CNCs.
Planning and simulation of positioning systems. Software structure of CNC controllers.
Realtime operating systems. Solving of control and monitoring tasks.
Cell controller and FMS controller, their functions.
CIM structure. Communication tasks in CIM.
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1. Preface
What is the automation ?
In the ancient times people worked by hand. They made every tasks, every works without any
help. Later they began to do some simple (and later more complicated ) machines, eg. water
wheels for lifting water from channels, mills (water and wind mills ) for milling corns, etc.
They began to use animals to give their force, their power to get work machines, vehicles, etc.
In the XIXth century the machines were able to do many tasks. Steam engines gave the
mechanical energy to machines, but the man had to control every machine.
What does it mean control ? Control consists of some activities:
• observe the phenomena (speed of machine, pressure of steam, temperature of
water, etc.),
• compute (decide) the needed activity (growing or reducing the amount of fuel ),
• set the appropriate device (modify the setting of fuel valves ).
Generally, control has 3 parts: measuring, computing, setting. During evolution machines
became more powerful, more quick, more precise, thus human control doesn't fit their tasks. It
became necessary to control machines by other machines. This control is the automation. This
is the short story of automation.
We can study and we can plan automated systems. The first task is the analysis, and the
second is the synthesis. In the life, in the industry, in the machinery there are a lot of
automated systems. We are dealing with such a systems, in which automation works by
machines, automatic devices.
Topics of this semester is the following:
1. Signals. What is a signal ? Properties of signals. Digital and analogue signals.
Information of signals. Quantification and sampling of signals.
2. Digital automation. Boole algebra. Boole functions. Logic elements. Logic networks.
3. Combinatorial and sequential networks. Methods for planning networks.
4. Digital controllers.
5. Linear continuous control theory. Components of analogue control systems. Open
loop and closedloop characteristics. Operational blocks, groups of blocks, block
diagram.
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6. Method for analyzing control loops. Transient and sinusoidal response. Solving of
differential equations. Laplace transformation, frequency response methods. Bode
diagram. Rootlocus method.
7. Compensation of control loops. Stability . Quality.
8. Basic control loops in machinery. Positioning systems, planning and simulation.
Joined positioning systems in robotics.
9. Numerical control. Tasks of NC. Geometric systems. Machine tasks. Programming of
NC.
10. Structure of CNC. Hardware and software structure of CNC controller. Realtime
operating systems.
11. Monitoring.
12. Systems. CIM, FMS, cell. CIM structure. Tasks in FMS. Cell controller.
13. Communication in CIM. LAN , RS 232, protocol, block, message. DNC.
1.1. Signals.
A signal is a physical quantity or a change in a physical quantity, which can be measured.
There are a lot of signals.(In a certain respect every phenomena are signals.) E.g.:
 sounds of drums in Africa
 intensity of light
 electrical voltage, current
 state of a switch
 etc.
Information: it stops an uncertainty. (You've learned about informatics in 6th semester, thus
you know many things.) Signal has a physical property (state variable), and an intellectual
property ( it contains information ).
Let's see a figure:
Operational program
Reports
Controller
Command signals
Measuring signals
Product
Energy
Material
Machine
• The machine's task, function: to do some operations (machine functions)
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• The controller's task, function: to control the process, to course the machine's tasks.
• The controller gives command signals, and it receives measuring signals to supervise
of carrying out of commands.
• The machine needs materials and energy, and it produces a product.
• The controller is in connection with man. It receives operational program from
worker, and sends information about it's operating conditions.
• There is a transformation between operational program and product. If the program is
good, we can get a good product (in case of good machine and good control
equipment.)
• The sort of signals have large importance in respect of security of information
transfer.
1.1.1. Classifying of signals.
Signal
Analogue Digital
The natural signals These signals are made
are analogous. by men.
continuous quantified
continual
continual
t
f(t)
t
f(t)
sampled sampled
t
f(t)
t
f(t)
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• Measuring always has an uncertainty because noise, and this causes inexact results.
• Information can be handled in noisy circumstances, too
Description needs 2 steps:
1. Quantification: we select a unit ( increment, delta). This is the smallest quantity which we
want to distinguish.
2. Sampling: we select a unit in time. T
s
.
How can we select these units ?
Delta is in conjunction with accuracy and noise. If delta is too small, we'll measure the noise.
If delta is too coarse, we can't measure the signal with appropriate accuracy.
T
s
depends from claim (demand) and from properties of process. What is the speed of change
of signals ? It is obvious, that we must sample the signal more frequently than it's period time.
Every signal has a Fourierspectrum. The signal has a limit frequency. This is such a
frequency, that the frequencies being greater then limit frequency have no significant
amplitude in spectrum.
Shannon's law:
ssh
T/1ff2
=
<
⋅
It must be fulfilled.
There is a special digital signal: the binary signal. The binary signal has a quantification, in
which the quantified signal has 2 values.
The physical appearance of signals may be very different.
t
x
0
1
Binary signals have a very great importance. Digital signals are generally a combination of
some binary signals. The binary signal is available not only for description of numerical
values 0 and 1, but it can explain information about a state of a device ( eg. a relay can or
can't conduct current, etc.).
A system can have more states, these can be described by more binary signals. A machine
also can have more states, which we want to control, and these state changes can be caused by
binary signals.
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1.1.2. Application of signals.
1. Computing system.
Input
peripheral
Computer
Output
peripheral
Digital Digital
Input peripheral: inputs the data, command.
Computer: processes the input and the stored data.
Output peripheral: displays the result.
2. Measuring system.
Sensor
Measuring
signal
processor
Display
Digital Digital
A/D
Analogue
Sensor : measures the states of process (e.g. force, stress, pressure, voltage, torque,
acceleration, vibration, temperature, etc.).
A/D converter: converts analogue signals to digital signals. Measuring signal processor:
processes measured signals, computes characteristic values, makes transformations, stores
data, etc.
Display: displays measured and computed, processed values.
3. Digital control system.
Program
carrier
Man
Control
equipment
Analogue, digital
Machine
Digital
Digital Analogue, digital
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Program carrier: stores the program for control equipment (eg. floppy, CD, pen drive, etc.).
Control equipment: controls the operation of machine. It gives and receives A and D signals
to/from machine, commands and programs from man.
Machine: executes commands of controller.
4. Binary logical controller.
Control
equipment
Input
signals
Output
signals
The control equipment makes output signals from input signals. The transformation between
input and output signals can be described by means of functions. We'll use binary signals,
thus signals have 2 states, 2 values: 0 or 1. We'll use binary algebra ( Boole algebra), and our
functions will be binary ( Boole ) functions.
Let it be:
[ ]
1
0
xeach,x...,,x,xX
i
T
n21
==
input signals, input vector.
[ ]
1
0
yeach,y...,,y,yY
i
T
m21
==
output signals, output vector.
( )
XFY =
in every t time.
We can describe:
( )
( )
.x,...,x,xfy
.
.
.
,x,...,x,xfy
n21mm
n2111
=
=
where f
i
is Boole function of ( Boole or binary ) variables x
1
,...,x
n
.
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This simple binary controller is named as combinatorial network. Any combination of output
corresponds to a combination of input signals, variables. Typical examples are the coder and
decoder circuits.
There are another kind of control equipments, in which the dependency of output isn't so
simple as earlier. The states of output variables depend not only from input variables but the
earlier life of the control equipment.
We can say that the equipment remembers own previous states, it knows the order of
preceding input variables. These are so called sequential networks. It stores the previous state,
thus it has a memory function.
x
1
x
2
x
n
y
1
y
2
y
m
m
1
m
p
F
Memory
G
[ ]
( )
( )
( )
( )
( )
( )
t
t
t
1t
T
p21
M,XGY
M,XFM
m,...,m,mM
=
=
=
+
2. Discrete controllers
2.1. Information
Discrete controllers serve as information transformers, they make output information from
input information. The form of information under processing may be:
 number
 character
 code
 etc.
Let's see the numbers often used in control equipments.
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2.1.1. Numbers
Any number can be expressed in different form. The usual description of a number is the
following:
∑ ⋅=
=0i
i
i
RaN
where N is the number being treated,
a
i
are the figures, the range of figures is
0 ≤ a
i
< R
R is the radix of applied number system.
The most "popular" systems are:
 decimal numbers. R = 10, a
i
= 0,1,2,3,4,5,6,7,8,9
 binary numbers. R = 2, a
i
= 0,1
 octal numbers. R = 8, a
i
= 0,1,2,3,4,5,6,7
 hexadecimal numbers. R = 16, a
i
= 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
Conversions between different forms of numbers:
We can convert any number to decimal number using the equation given above. Man can
compute with decimal numbers, this form is the most common for people.
If we want to convert a number to another, we have to divide the number with the radix of
new number, and the residues will be the figures of the number. The figures are read in
backward direction, that is the first residue is the least significant figure, and the last residue
is the most significant figure.
To convert a decimal number into a binary number is an easy task.
1
2
4
9
19
10011038
=
After this conversion we can get the octal and the hexadecimal numbers simply grouping of
binary figures. To compute the octal form of a binary number we have to make groups with 3
binary figures from right to left ( from least significant to most significant ) and each group
corresponds to an octal figure, where the value of an octal figure is computed by making the
weighted sum of binary figures. The least significant binary figure in the triplet has a
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weighting factor 1 = 2
0
, the figure in the middle of the triplet has a factor 2 = 2
1
, and the most
significant figure ( which is in the lefthand side ) has a factor 4 = 2
2
.
536110.011.101
=
To get the hexadecimal numbers we have to make groups with 4 binary figures, and the
figures have weighting factors 1, 2, 4, 8 respectively.
1101.1011.0011=DB3
These numbers were integer ones, because this is the most simple and most often used type of
numbers in control, especially in the discrete, binary control. There are a lot of standards to
give the real numbers ( for example in computer technics ), but these numbers are out of our
mind.
2.2.2. Codes
In machinery the form of information is generally code, we use coded information in
information gathering, transferring, processing, etc.
There are a lot of different code systems. Some often used code systems are:
 pure binary code. The binary number can be treated as a pure binary code of the
decimal numbers. This code contains weighting factors, these factors are power of 2,
and the value of code is the sum of the weighted components.
 ASCII code. This code is an international standard code of ISO, and it can describe
characters ( numbers, letters, control characters). This code contains 8 bits ( binary
figures) to give the information. We can describe one word of this code as 2
hexadecimal numbers. In the standard the most significant bit is a parity bit, it doesn't
contain information. In the extended code the most significant bit is a normal bit, and
the codes above 127 are graphic codes in the IBM code set, e.g.
 Grey code. This code has the property, that only one bit changes its value, if the
number representing by the code word is incremented or decremented. This code serves
to code numbers. Because of above mentioned properties this code is very useful to
code measuring signals, as digital encoders in machine tools, etc.
 Johnson code. This is a redundant code, because it works as a Moebius strip. The
content of a 4bit Johnson counter changes in the following way:
0000⇒ 0001⇒ 0011⇒ 0111⇒ 1111⇒ 1110⇒ 1100⇒ 1000⇒ 0000
As you can see, this code contains only 8 different words of 2
4
= 16 4bit words.
There are different codes, which can find if some error occurred during information transfer
or processing. The most simple type of these codes is the parity checking code. The standard
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ASCII code contains a parity bit, the 8th bit. It's value depends on the number of 1's in the
code word. In an even parity code the number of 1's must be an even number in every code
word. Eg.:
The code of the character 0 is 30
H
, it contains 2 1's, so the value of parity bit is 0.
The code of the character 1 is 31
H
, it contains 3 1's, which is an odd number, so the parity bit
must be 1, the final code word is B1
H
.
 There are codes, which can correct some errors occurred during information transfer or
processing. Such codes are used in mainframe computer memories for example.
2.2. Logical algebra.
Logical algebra is based on work of an English philosopher Boole.
There are a lot of systems in the nature and in the technics which can have 2 stable states. A
logical statement can be true or false, and it is always or true, or false, exclusively. Similarly a
circuit is opened or closed, but it isn't closed and opened at the same time. It is obvious to
intend values 0 and 1 for false and true. 0 means false, 1 means true (e.g.).
By means of logical algebra we can describe equations, logical functions, these may be
simplified, and we can reduce networks.
2.2.1. Basic operations in logical algebra:
There are 3 basic operations:
OR
AND
NOT
operations.
OR operation (logical addition, disjunction, union).
Its symbol is: +
x=a+b+c+...+n
It means: x equals to 0 if and only if a=0, b=0, ... , n=0.
If any of a,b,c,...,n equals to 1, then x=1.
In other words:
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x=1, if a=1, or b=1 ( or a=1 and b=1 ), or c=1,...e.t.c.
AND operation (logical multiplication, conjunction, intersection).
Its symbol is .
n...cbax ⋅⋅⋅⋅=
It means x equals to 1 if and only if a=1 and b=1 and ...and n=1.
If any one of a,b,c,...,n equals to 0, then x=0.
NOT operation (negation, inversion).
Its symbol is
a
.
ax =
It means: x=1 if and only if a=0, and x=0 if and only if a=1.
Some basic relations:
1. Addition
a0a
=
+
11a
=
+
aaa
=
+
2. Multiplication
00a
=
⋅
a1a
=
⋅
aaa
=
⋅
3. Negation
1aa =+
0aa =⋅
aa =
4. Commutative rules
abba
+
=
+
abba
⋅
=
⋅
5. Associative rules
(
)
(
)
cbacba
+
+
=
+
+
(
)
(
)
cbacba ⋅⋅=⋅⋅
6. Distributive rules
(
)
cabacba ⋅+⋅=+⋅
(
)
(
)
cabacba +⋅+=⋅+
7. DeMorgan rules
n...ban...ba ⋅⋅⋅=+++
n...ban...ba +++=⋅⋅⋅
Truth table:
a
b
ba +
ba +
a
b
ba ⋅
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0
0
0
1
1
1
1
0
1
1
0
1
0
0
1
0
1
0
0
1
0
1
1
1
0
0
0
0
thus we verified 1st De Morgan thesis.
2nd De Morgan thesis:
a
b
ba ⋅
ba ⋅
a
b
ba +
0
0
0
1
1
1
1
0
1
0
1
1
0
1
1
0
0
1
0
1
1
1
1
1
0
0
0
0
Thus we verified 2nd De Morgan thesis, too.
2.2.2. Boole functions.
Function: if we give a rule to order elements of a set, we define a function. If the rule contains
Boole operations, then the function is a Boole function.
Let's see some Boole functions and their's mode of definitions, descriptions.
The most simple Boole functions are those, in which only one independent variable can be
found.
Let the independent variable be x.
x may be 0 or 1, y=f(x).
There are 4 functions,
)x(f0y
1
0
==
,
)x(fxy
1
1
==
,
)x(fxy
1
2
==
,
)x(f1y
1
3
==
.
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Generally: there are N=2
n
Nabors (different variations of independent variables) where n=
number of independent variables, and there are
n
2N
22p ==
If n=1, then N=2, , the above described 4 functions.
42p
2
==
If n=2, then N=2
2
=4, , that is there are 16 functions having 2 independent
variables.
162p
4
==
Let the 2 variables be a and b .
4 nabors are:
a
b
=
2
0
N
0
0
=
2
1
N
0
1
=
2
2
N
1
0
=
2
3
N
1
1
Functions:
2
3
N
2
2
N
2
1
N
2
0
N
f
0
0
0
0
0
f
0
=0
f
1
0
0
0
1
baf
1
⋅=
f
2
0
0
1
0
baf
2
⋅=
f
3
0
0
1
1
af
2
=
f
4
0
1
0
0
baf
4
⋅=
f
5
0
1
0
1
bf
5
=
f
6
0
1
1
0
babaf
6
⋅+⋅=
f
7
0
1
1
1
baf
7
+=
f
8
1
0
0
0
baf
8
⋅=
f
9
1
0
0
1
babaf
9
⋅+⋅=
f
10
1
0
1
0
bf
10
=
f
11
1
0
1
1
baf
2
+=
f
12
1
1
0
0
af
12
=
f
13
1
1
0
1
baf
13
+=
f
14
1
1
1
0
baf
14
+=
f
15
1
1
1
1
f
15
=1
These are 16 2variable Boole functions.
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2.2.3. Boole function systems.
We can select some functions such a way, that all the other functions can be derived from
these.
Some such systems:
1. NOTANDOR
system.
This system is most closed to human thinking. We can give functions as anded statements
being ored together.
For example :
bababaf ⋅+⋅+⋅=
2. NOTOR
system (
NOR
)
This system contains only NOT and OR statements, and the AND statement is described by
means of De Morgan rules:
babaf +=⋅=
3. NOTAND
system (
NAND
)
This system contains only NOT and AND statements, and the OR statement is described by
means of De Morgan rules:
babaf ⋅=+=
In the following sections we'll use NOTANDOR system in theoretical treatments. In the
practice Boole functions are generally realized by means of integrated circuits (ICs), and the
most common ICs are based on NAND system. We can convert NOA system to NAND
system easily, such we'll treat the NOA system.
We can describe a function in many equivalent forms. Among these forms are regular and
irregular ones. If in a regular form all groups of variables contain all of the variables being
negated or not, then this form is called as complete form.
We'll treat the next 2 complete regular forms:
minterm
and
maxterm
forms.
Definition:
Minterm
is such an nvariable Boole function, that the value of function equals to 1 only at 1
nabor of variables. Consequently, there are 2
n
different minterms, that is the number of
minterms = the number of nabors.
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Maxterm
is such an nvariable Boole function, that the value of function equals to 0 only at 1
nabor of variables. Consequently, the number of maxterms = the number of minterms.
In the NOTORAND (NOA) system the minterms are described as logical multiplication of
variables, and the maxterms are realized as logical addition of all the variables (or negated
values of variables, of coarse).
We introduce 2 symbols:
m
indicates a minterm. It's superscript tell us the number of variables, and it's subscript
contains the number of that nabor, at which the value of minterm equals to 1.
Similarly,
M
indicate a maxterm. The meaning of superscript is the same as above, and the
subscript contains the number of that nabor, at which the value of maxterm equals to 0.
Let's see an example:
Number of minterms (and maxterms, of coarse) in a 2variable case is:
42NN
2
Mm
===
These are:
a
b
m
M
N
0
0
0
bam
2
0
⋅=
baM
2
0
+=
N
1
0
1
bam
2
1
⋅=
baM
2
1
+=
N
2
1
0
bam
2
2
⋅=
baM
2
0
+=
N
3
1
1
bam
2
3
⋅=
baM
2
0
+=
In the literature there are other numberings of subscripts, too.
The connection between minterm and maxterm is:
( )
n
i12
n
i
n
Mm
−−
=
, and
( )
n
i12
n
i
n
mM
−−
=
A logical function can be described as logical addition of minterms having significance in the
function. Similarly, a function is logical multiplication of such maxterms, at which the
function has 0 value.
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Let's give some 2variable Boole functions in minterm and maxterm forms:
(
)
(
)
(
)
(
)
babababaMMMMf0f0f
3210000
+⋅+⋅+⋅+=⋅⋅⋅===
after using the axioms of Boole algebra we get:
(
)
(
)
(
)
(
)
0aa0a0abbabba =⋅=+⋅+=⋅+⋅⋅+=
(
)
(
)
(
)
( )
( )
( )
babaaaababbaba
bababaMMMfbamfbaf
3211011
⋅=⋅+⋅=⋅+=⋅+⋅+=
+⋅+⋅+=⋅⋅=⋅==⋅=
(
)
(
)
babababa
aabbbabaMfmmmfbaf
014321114
+=⋅+⋅+⋅=
+⋅++⋅=+===++=+=
babababa1fmmmmf1f
032101515
⋅+⋅+⋅+⋅==+++==
2.2.4. Simplification of logical functions.
A logical function generally contains too much terms, thus it is unsuitable for realization. If
we want to realize a function, then we must simplify it. We can simplify a function in a
heuristic or a systematic way.
Intuitive (heuristic )
simplification method:
we apply the rules of logic algebra for simplifying a function. If we have good luck, we'll find
the most simple form of a given function. In a complicated function having much variables
there is a hard work to find the most simple form. It is better to simplify a function
systematically.
There are a lot of systematic simplification methods. One of these based on Veitch diagram
and called as VeitchKarnaugh method.
Veitchdiagram is a table consisting of cells. Each cell corresponds to a minterm (maxterm).
Simplification based on the fact, that the neighbouring cells can be grouped together, and the
group can be described fewer variables. We can find most simple solution because graphic
features of Veitchdiagram.
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Minterm table:
a
m
0
b
m
1
m
3
m
2
a
b
ba ⋅
ba ⋅
ba
⋅
ba ⋅ a
b
0 1
1
1
We write
1
s to those cells, which contain existing minterms, and
0
s to those cells, which
contain nonexistent minterms.
In practice some conditions are not defined. Cells containing minterms describing notdefined
conditions are filled with x, which means that we can treat these minterms or existing or
nonexistent ones.
In the example we can create 2 groups:
a
b
0 1
1
1
First group contains 2 minterms, m
2
and m
3
The common part of these minterms is variable a.
Second group contains 2 minterms, too. These are m
1
and m
2
, and their's common part is
variable b.
Thus the function can be simplified as:
babababammmf
321
+=⋅+⋅+⋅=++=
There are some rules to create groups
1. Each cell has n neighbouring cells, where n is the number of variables.
2. A group can contain as many cells, that the number of cells be power of 2 ( 1,2,4,8,...)
3. The cells of groups must be logical neighbouring cells. In a large table (which has
more then 4 variables) the logical neighbours aren't physical neighbours.
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Let's see some bigger Veitchtables.
A minterm table for 4 variables:
a
b
c
d
d
cba ⋅⋅⋅
d
cba ⋅⋅⋅
d
cba ⋅⋅⋅
d
cba ⋅⋅⋅
d
cba ⋅⋅⋅
d
cba ⋅⋅⋅
d
cba ⋅⋅⋅
d
cba ⋅⋅⋅
d
cba ⋅⋅⋅
d
cba ⋅⋅⋅
d
cba
⋅
⋅
⋅
d
cba ⋅⋅⋅
d
cba ⋅⋅⋅
d
cba ⋅⋅⋅
d
cba ⋅⋅⋅
d
cba ⋅⋅⋅
We can create a lot of tables, in which the variables are placed in different manner. The
numbering of minterms depends on the order of variables. Each variable has a weighting
factor. This factor is power of 2, and increases from right to left.
E.g.:
e
d
c
b
a
2
4
2
3
2
2
2
1
2
0
16
8
4
2
1
These factors determine the number of minterm. True variables have a multiplier 1, negated
variables have a multiplier 0. Thus the minterm
edcbam ⋅⋅⋅⋅=
has an index
2216180412110
=
⋅
+⋅+⋅
+
⋅+⋅
,
edcbam
22
⋅⋅⋅⋅=
A minterm table for 4 variables is:
Possibilities of grouping ( according to rules).
a
b
c
d
0 1
732
45
14151110
6
121398
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A table for 5 variables is:
a
c
d
0 1
732
45
14151110
6
121398
b
e
20 21 17
2322
24252928
16
19 18
30 31 27 26
Neighbours, grouping.
2.2.5. Realization of logic functions.
Realization of logic functions means, that we must construct a network, in which there are
logic gates, circuits, elements, and connections amongst them, and, finally, we must build this
network.
We'll talk about these elements.
You've learned about basic circuits in electronics, microelectronics. In this subject we apply
some of the elements.
Basic gates.
NOT
gate, inverter.
Symbol:
x
y
1
Function:
xy =
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AND
gate.
Symbol:
&
a
b
y
Function:
bay
⋅=
OR
gate.
Symbol:
1≥
a
b
y
Function:
bay
+=
The above mentioned elements realize the basic logical operations, functions.
By means of these we can realize any Boole function.
E.g.:
babay ⋅+⋅=
(exclusive or function)
a
1
b
1
&
&
1≥
y
In practice we use integrated circuits (ICs) to realize logic functions. The standard circuits are
the NAND and NOR gates, circuits (and inverters ).
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NAND
gate:
Symbol:
a
y
&
b
Function:
bay ⋅=
An example:
&
y
&
x
1
&
x
2
x
4
x
3
432143214321
xxxxxxxxxxxxz ⋅+⋅=⋅+⋅=⋅⋅⋅=
NAND
gate is an inverter, an or gate and an and gate alone.
xy =
y
&
x
2121
xxxxy +=⋅=
&
y
&
x
1
&
x
2
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2121
xxxxy ⋅=⋅=
x
1
&
y
&
x
2
NOR
gate
Symbol:
a
y
b
1≥
Function:
bay +=
It is the dual pair of the NAND gate.
Realization of basic elements:
xy =
y
1≥
2121
xxxxy ⋅=+=
y
x
1
x
2
1≥
1≥
1≥
2121
xxxxy +=+=
x
1
y
x
2
1≥
1≥
There are those gates ( integrated circuits ), which have more than 2 inputs.
In a package generally there are more gates.
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E.g.:
SN 7400 contains 4x2 NAND gates
SN 7402 contains 4x2 NOR gates
SN 7404 contains 6 inverters
SN 7410 contains 3x3 NAND gates
SN 7420 contains 4x2 NAND gates, etc.
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2.3. Sequential networks.
Another type of discrete control systems is characterized by the fact, that its output is defined
in a time t
2
by its input and its previous state ( in time t
1
). The interval t
2
t
1
is the system
reaction time. The above system is called yet as:
 finite inner? state automaton
 multiple beat network
The sequential network makes a connection among 3 finite state (event) systems:
 input state
 inner state
 output state
How can we give these connections ?
Let it be:
X
= [x
1
,x
2
,...,x
n
] the input vector
Y
= [y
1
,y
2
,...,y
m
] the output vector
M
= [ m
1
,m
2
,...,m
p
] the vector of inner state
M
(t
2
)=
F
(
X
(t
1
),
M
(t
1
))
Y
(t
2
)=
G
(
X
(t
2
),
M
(t
2
))
We can describe these functions as usual Boole functions.
Another method is to give a table (so called coded transition and output table). In this table
we give the input and inner (memory ) variables in time t
1
, and the answer of the network,
that is the values of inner and output variables in time t
2
.
x
1
x
2
m
1
(t
1
)
m
2
(t
1
)
m
1
(t
2
)
m
2
(t
2
)
We can arrange the elements of table in form a matrix, as:
Nabors of inner
variables
Nabors of input variables
Nabors of inner variables
in time t
2
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E.g.
Let it be 1 input variable x, and 2 inner variables, m
1
and m
2
.
t
1
t
2
x
m
1
(t
1
)
m
2
(t
1
)
m
1
(t
2
)
m
2
(t
2
)
0
0
0
0
0
stable state
1
0
0
0
1
unstable state
1
0
1
0
1
stable state
0
0
1
1
1
unstable state
0
1
1
1
1
stable state
1
1
1
1
0
unstable state
1
1
0
1
0
stable state
0
1
0
0
0
unstable state
0 0
0 1
0 0
11 0 1
0 0 1 0
1 011
0 1
1 0
1 1
m
1
m
2
0
1
x
We can represent this network on a time diagram, too.
The above example in a time diagram is :
x
m
1
m
2
Finally we can describe the sequential network by means of transition (and output ) graph.
00
11
10
01
1
1
00
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The nodes of graph are the inner states, and the branches are the input states.
2.3.1. Realization of sequential networks
.
There are 2 kinds of real sequential networks :
 asynchronous
 synchronous
networks.
In the asynchronous networks the state transitions are determined by the elements of networks
only. The main functional unit is the RS flop (memory, trigger ) in these networks. RS flop
has 2 inputs:
S is the set input, R is the reset input.
It's Boole equation is:
( )
( )
t1t
12
QRSQ,tmRStm ⋅+=⋅+=
+
that is m=1 if S=1 (and R not = 1 ), or if m was 1 and R=0.
With other words:
m=1 if set (S=1 ), or ( and until ) if it was set and not reset.
We can create an RS flop from NAND ( or NOR ) gates as:
Q
&
&
S
&
R
Q
&
The output of RS flop is often marked ( named ) as Q.
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In the synchronous networks the state transitions are determined by a special signal, named
clock signal. The clock signal ( the edge of clock signal ) enables the state transitions.
The main functional unit is the
JK flop
.
It's state transition table is:
J
0
0
1
1
K
0
1
0
1
Q
t
0
0
0
1
1
1
1
0
1
0
K
J
=JQ
t
+KQ
t
Q
t
0
1
0
Q
t+1
=
0
1
0
1
1
J
K
C
Q
Q
C is the clock input.
Time diagram:
C
J
K
Q
t
t
t
t
There is an advanced kind of synchronous elements, the masterslave flop. It has 2 stages, the
master and the slave. The master stage takes a sample from the input at the positive edge of
the clock signal, and the slave stage makes the output at the negative edge of the clock signal.
Such a way the input change is isolated from output change.
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Let's see some simple sequential units and ICs.
E.g. SN 74LS73
J
K
C
Q
Q
R
D latch, D flop.
We can create it from a JK flop as:
C
D
Q
t
t
t
J
K
C
Q
Q
1
T
D
It stores the state of the D input until the next clock signal. It is an important element in the
digital technique.
E.g. SN 7474
S
C
D
Q
Q
R
T
It has static (asynchronous ) inputs for setting and resetting the required state, a data input D
and a clock input C.
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T flop.
We can create it from a JK flop as:
T
Q
t
t
t
C
J
K
C
Q
Q
T
T
If the state of the input T is High, the output changes its state at every clock pulse, thus the
output frequency
TQ
f5,0f ⋅=
We can use the T flop as a frequency divider.
2.3.2. Higher level functional units.
 combinatorial
 coder/decoder
 multiplexer
 comparator
 arithmeticlogicalunit ALU
 sequential
 register
 counter
Coder makes a code from events. Event means that among inputs not more then 1 is true.
E.g.
x
2
x
1
x
0
y
1
y
0
0
0
0
0
0
0
0
1
0
1
0
1
0
1
0
1
0
0
1
1
y
1
,y
0
contain the coded information about which event was occurred (which x is 1).
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Decoder makes event code from code.
Only one output of the decoder is active at a time.
E.g. SN 74138
G
1
G
2
G
3
A
B
C
Q
0
Q
1
Q
2
Q
3
Q
7
Q
6
Q
5
Q
4
Enable
Inputs
Code
(address)
inputs
Outputs
&
Multiplexer circuit (organizer circuit ).
Principle:
x
1
x
n
.
.
.
y
u
0
. . . u
n
The information of one of the inputs is connected to the output depending from contents of u
control (address ) lines. We can select, we can address an input and connect to the output.
E.g. SN 74151
Demultiplexer circuit makes the inverse activity of the multiplexer: one input is connected to
selected output.
A
B
C
D
0
D
1
D
2
D
3
D
7
D
6
D
5
D
4
E
Q
W=Q
Enable
input
Data
inputs
Address
inputs
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E.g. SN 74154
Q
0
Q
1
Q
2
Q
3
Q
7
Q
6
Q
5
Q
4
Q
8
Q
9
Q
10
Q
11
Q
12
Q
13
Q
14
Q
15
A
B
C
D
G
1
G
2
&
Comparator circuit compares two set of signals as binary numbers.
E.g. SN 7485
A
1
A
2
A
3
Cascade
inputs
A
0
>
=
<
B
1
B
2
B
3
B
0
=
>
=
<
Via cascade inputs it can be used to compare more bit wide binary numbers.
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Binary adder adds 2 onebit wide binary number and the carry of the previous adder.
entals of Automation
34
BAC
BABAS
⋅=
⋅+⋅=
ALU can execute some arithmetic and logical instructions on (e.g.) 4bit binary numbers.
(E.g. SN 74181)
Registers.
Tasks of registers are:
1. Store of data in this case register works as a little, quick memory.
2. Shifting in this case register moves the data.
Outputs of registers may be:
 totem pole. This name derived from Indian totem pole made from wood. In this type of
output transistors are built on one another. This is a quick output, but it mustn't be
connected outputs together.
 open collector. We can construct wiredor connections, bus systems, etc., using this
type of output. It has nonsymmetrical properties.
 tristate. It has symmetrical output and as we can command the output to a high
impedance state, the outputs may be connected to a bus line, e.g.
Functions of registers are:
 parallel setting, loading via parallel input lines
 parallel reading via parallel output lines
 shifting via serial input and output lines
Control lines of registers may be:
A
B
C
S
C
i+1
T.CsákiI. Makó: Fundam
C
B
A
S
C
0
0
0
0
0
0
0
1
1
0
0
1
0
1
0
0
1
1
0
1
1
0
0
1
0
1
0
1
0
1
1
1
0
0
1
1
1
1
1
1
J JC CK K
S R
Q Q Q Q
& &
LOAD
Shift
Clock
Serial input
Parallel
load
Reset
V
TS output
1
S R
 clock signal
 Right/Left shift direction
 Load
 Output enable
We can construct a ring register, if the output of a register is connected to the input ( serial ).
E.g. SN 74194 4bit register
S
1
AS
DS
Parallel
inputs
S
0
C
D
R
B
Q
A
Q
B
Q
C
Q
D
Mode
control
Serial
data
C
Clock
A
Reset
Parallel
inputs
S
1
S
0
C
Function
0
0
X
Store
1
0
Left (DS as serial data)
0
1
Right (AS as serial
data)
1
1
Parallel load
Latch register: it temporarily stores data.
It may be used as synchronizer circuit, too.
Counters.
Functions of counters are:
 counting of pulses
 storing of data
The counter may be unidirectional or reversible.
Its content differs from register's content. The content of a register is a binary vector (event
code, other code, etc. ), while the content of a counter is always a number whether in coded
form or not.
Code of counters are:
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 pure binary
 BCD  binary coded decimal
 special (Johnson, Gray, etc.)
Mode of operation may be:
 synchronous : input pulse operates all the cells parallel.
 asynchronous : input pulse operates the input cell, information propagate seriously. (simple,
but slow.)
Binary counter
C
t
R
E
Q
0
Q
1
Q
2
t
t
t
t
t
1 2 3 4 5
1
0
1
5T
J
C
K
R
Q
J
C
K
R
Q
J
C
K
R
Q
Számlálandó
pulzusok
Reset
E
Q
0
Q
1
Q
2
BCD counter
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Q
0
J
C
K
Q
0
Számlálandó
pulzusok
E
J
C
K
Q
2
J
C
K
Q
1
J
C
K
Q
3
&
Q
1
&
Q
2
&
Q
3
Q
3
C
t
Q
0
Q
1
Q
2
t
t
t
t
1 2 3 4 5
Q
3
6 7 8 9
10
E.g. SN 74192 BCD, SN 74193 binary
Contents of a counter in case of counting up are:
CD
R
Parallel
outputs
CU
C
D
L
B
Q
0
Q
1
Q
2
Q
3
Count up
Load
CC
Carry
Borrow
A
Reset
Parallel
inputs
CW
Count down
7
6
5
4
3
2
1
0
1

2
t
Contents of a counter in case of counting down are:
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
t
2.4. Memories.
The next main types of functional units are memories. Memory is such a device, which can
store data for further usage. How do memory work ?
Let's imagine a table ( or cupboard ) with drawers. Each drawer has a number (identity
number). The content of a drawer is the data, and the number of the drawer is the address. If
we want to store something, we can put it into a drawer, and if we want to use this thing, we
can get it from the drawer signed the number of order (serial number).
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1.
Book
2
Pen
3
Paper
Organization of a memory.
Address
(RAM)
Data
input
Data
output
Organization of memories:
Measure of storing places may be:
1 bit bitorganization mem.
4 bits wordorg. mem.
8 bits byteorg. mem.
9 bits byte with parity
16 bits word
32 bits word
39 bits word with error code
.
.
.
Types of memories.
1. RAM random access memory. Data may be stored and taken out in any kind of order, and
the writing and the reading of data is quick.
SRAM static random access memory
DRAM dynamic random access memory (data must be refreshed frequently)
CRAM CMOS random access memory (it has a stand by state, in which it can store
data with very low power consumption.)
2. ROM read only memory. Data are stored in an unchanging way, the user can only read
them.
3. PROM programmable read only memory. The user can erase and program it many times,
but the writing is significantly slower then the reading .
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EPROM erasable read only memory. UV light can erase its content, and a higher voltage
can write it.
EAROM electrically erasable read only memory. High voltage (2025 V can erase it).
Examples:
8316 ROM Capacity : 2 kbytes.
321
11
CSCSCSEn
k220482N
⋅⋅=
===
ROM
D
0
D
1
D
2
D
3
D
4
D
5
D
6
D
7
CS
1
A
0
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
A
9
A
10
CS
2
CS
3
8708 EPROM Capacity : 1 kbytes.
D
0
D
1
D
2
D
3
D
4
D
5
D
6
D
7
PR
A
0
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
A
9
CS
25V
1ms
Quartz window for UVlight
erasing
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8111 SRAM Capacity : 256 bytes
RAM
CS
1
A
0
A
1
A
2
A
3
A
4
A
5
A
6
A
7
CS
2
R/W
D
0
D
1
D
2
D
3
D
4
D
5
D
6
D
7
2.5. Digital controllers.
As we mentioned earlier there is a type of control equipments in which binary signals are
measured and used as input signals, and binary signals are used as output signals to control
the operation of a machine, a device, an equipment , etc. We can represent them as follows:
Input
Logical
network
Output
.
.
.
.
.
.
.
.
.
.
If we make a control device, we must plan and realize a network, which can make output
signals from input signals.
There are two methods for this:
 parallel
 serial
methods.
Principle of the parallel logic:
( )
n1ii
x,...,xfy =
Each of such functions is realized separately, these subnetworks ( NAND gates, flops, etc.)
compute Boole functions simultaneously, each y
i
is ready independently from each other.
A Boole function may be computed by
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 Boolealgebraic operations
 taking out from a table
In the second method the table is stored in a ROM (PROM). The memory assigns a data for
each address. The address is the state of input lines, and the data is the output.
We can create a sequential networks in this way, too. E.g. we use a separate memory for
flops, R and S inputs are connected to 11 address lines, and its output is connected to the
appropriate line of main ROM.
E.g.
x
0
.
.
.
x
n
y
0
.
.
.
y
m
A
0
.
.
.
A
n
D
0
.
.
.
D
m
A
0
.
.
.
A
t
Q
0
.
.
.
Q
p
Input ROM Output
ROM
Principle of the serial logic:
We can determine the value of y
i
s in another way, too. We compute first the value of y
1
, then
we compute the value of y
2
, etc., one after the other.
Really, this is a special computer computing Boole equations. Its name is PLC, programmable
logical controller.
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The block scheme of a simple PLC is:
PU SC PC
MEM
I/O U
OR
START
STOP
IR MUX
MUX
(melyik input)
(input v. output)
LU A
DMUX MUXMR
C
C C
C C
C C C
C
OM
OC
OC
A
S
What can do this simple PLC ?
It can read a variable to A (Accumulator).
It can store a variable from A to output register.
It can negate the value of A.
It can logically add A and selected (input or output
or inner) variable.
It can logically multiply A and selected (input or output or inner ) variable.
By means of this activities this PLC can solve any logical equations, thus it can control binary
processes, equipment.
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3. Linear Control Theory
3.1. The Control System and Its Components.
The control system consists of a controlled equipment (system) and a controlling system. The
controlled system is an independent, existing equipment, machine, which is the object of the
control. The controlling system is a set of each equipments by which the control is done.
During control there are several changes in the system. To achieve the desired changes we
need some time, thus a control system is a dynamic one.
More exactly :
dynamic system
is a system in which the value of output signal depends on
the value of input signal as well as the earlier values of input (and output) signals.
The
block diagram
is the flow map linking all the variables (signals) to each other through
the operational blocks. The operational block is a symbol. It represents a (mathematical or
physical or other ) transformation between input and output signals.
For example: we can treat our car as a control system. The setting of the accelerator pedal is
the input signal and the velocity ( or the displacement ) of the car is the output signal. There is
a ( quite difficult) connection between the accelerator setting and the velocity, and this can be
described by a differential equation ( approximately, of course, because the very exact
description is a very difficult thing. )
The task of a control (system) is to achieve, that let a selected signal be a constant, desired
value, or that let the selected signal follow the input signal ( the reference signal) in different
technological processes.
The equipment, in which the material and energy changes are carried out, is the controlled
equipment. We influence the operation of the controlled equipment by means of control. The
dynamic properties of the controlled equipment are expressed by the controlled section.
The controller is a set of equipments doing the control process (e.g. measuring, sensing the
values of variables, decision, intervening in process ).
The quantity under control is the controlled variable, signal.
The developing of its desired value is disturbed by the unwanted disturbing signal(s). The
control want to eliminate the effect of the disturbing signal by influencing the modified
signal. This modified signal is the input signal of the controlled section.
To solve a control task we must first select the controlled signal, the signal we want to
control. After this we select the modified signal, which can influence the controlled signal and
which is easily modifyable for us. The other signals acting on the controlled section are the
disturbing signals.
3.1.1. Grouping of control systems
( from some points of view.)
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Point of view
Version
Mode of operation
Continuouscontinuous
(common)
in value in time
Discretecontinuous
(relay)
discontinuous
a. two state
b. three state
Continuousinterrupted
(sampled)
Discontinuousinterrupted
(digital)
Function between input and output signals of
members of control system
Linear
a. with constant parameters
b. with variable (timedependent)
parameters
Nonlinear
a. with constant parameters
b. with variable parameters
The reference signal
Constant (valuekeeper control loop)
Variable (follower)
(Time function)
(program control)
The nature (property) of signal
Deterministic
Stochastic
The number of controlled variables
One
More
etc
Control is
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1. continuouscontinuous, if the signals are explained at every time ( each signal is continuous
function of time) and each member of control loop has a continuous static characteristic.
2. discretecontinuous, if each signal exists at every time , but at least one signal has only
discrete values. (e.g. electric iron with 2state temperature control.)
3. continuoussampled, if the transfer of signals from one point to another happens at discrete
times ( e.g. at every 10 milliseconds ).
4. digital, if the values of sampled signals are discrete ones.
A control loop is linear, if each member of loop is linear. A control loop member is linear, if
the static characteristic between its input and output signal is linear, a straight line.
It means, that if there are 2 input signals, x
1
and x
2
, and x
1
gives an output signal y
1
, and x
2
gives an output signal y
2
, then x
1
+x
2
give an output signal with value y
1
+y
2
, and
give an output signal with value
2211
xAxA ⋅+⋅
2211
yAyA
⋅
+
⋅
. The behaviour of a linear
control system is described by linear differential equations. If the parameters of the equations
are constant, then we can say that it is a constantparameter system, else it is a variable
parameter system.
In most cases some members of the control loop have nonlinear properties. There are a lot of
nonlinearities, but we'll deal with linear systems first of all. A nonlinear system can be
approximated with an equivalent linear system. The error done by approximation depends on
nonlinearity, etc..
The goal of a control may be to maintain a constant controlled value ( if the reference signal
is constant ) or to follow a variable reference signal. In the first case (valuekeeper control
loop ) we want to achieve a constant controlled variable in spite of disturbing signals. In this
case the most important property of control loop is the ratio of disturbance suppress. In the
second case the reference signal, i.e. the input signal of the control loop can change its value
in time, this change has generally a maximum speed and a maximum value.
The task of the control loop is to follow the reference signal as close as it possible. The most
important property of the control loop is the ability of following the reference. In practice the
control loop has to follow the reference and has to maintain the desired value at the same
time. The distinction is made by the more important goal.
There is a special follower control loop named as servo loop, servo system. In a servo system
the controlled variable is a position of a mechanical element, e.g. a distance, a rotating shaft
turn. In machinery, and in machine tools and robotics, the servo systems have a very great
importance because the motion is a very important and significant function in them.
We'll deal with positioning systems later in detail.
The outer signals ( disturbing signals ) acting to the control loop are generally unknown for
us, we don't know their time functions. We can examine the control loops supposing a typical
signal ( step, impulse, etc.). In some cases it is a better way to examine the system with
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statistical ( or stochastic) methods. Some properties of outer signals can be discussed by
means of probability theory.
In the single cases the control loop has only one controlled variable, but there are some tasks
in which we must impose more signal's value of control loop. The control task is determined
if the number of modified signals equal to the number of the controlled signals.
3.1.2. General Requirements to Control Loops
Substantial task of the control loop is
a. to maintain the constant value of the controlled signal in a valuekeeper control loop.
b. to follow the reference signal as close as it possible in a follower control loop.
The most essential requirement is to make such a control loop, that let the control loop reach a
steady state, i.e. let the control loop be stable. A stable control loop would be ideal, if the
controlled signal followed the reference signal exactly.
In this case the control loop would have no error or in ready state or in transitional state. But
an ideal control loop doesn't exist because of timestoring (energystoring) members between
reference and controlled signals.
As we try to approximate the ideal control, the control loop became more and more difficult
and precious. We have to achieve a reasonable compromise between the cost and the quality
of control.
The accuracy of the control is measured by the control difference. The control difference is
the difference between the desired and the existing value of the controlled signal.
The better is the control the smaller is this difference (generally ).
The quality of control is characterized by sensitivity, too. The sensitivity gives the measure,
that if a parameter changes in the loop, how the change of a characteristic function will be.
The better is a control loop the least is influenced the function.
The goodness of a control is determined by transients. The qualitative characteristics of a
control loop can be measured by drawn the controlled signal answering to a step input signal.
The qualitative characteristics are as follows :
t
T
c
2 ∆
x
c
x
c
x
c max
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a. maximum overshoot of the controlled signal
(
)
( )
[ ]
%100
x
xx
c
cmaxc
⋅
∞
∞
−
=σ
where is the steadystate value of controlled signal.
( )
∞
c
x
b. control time (settling time ) is that time, after which the controlled signal doesn't differ
from its steadystate value with more than +d %. It means, that:
( ) ( )
( )
c
c
cc
Ttif
x
xtx
≥∆<
∞
∞−
∆ is the dynamic accuracy of the control, its value generally is 5 %.
c. Rise time: 10% to 90%
d. Damped frequency of oscillations.
We can describe the number of swingings during T
c
control time, too.
The better is a control loop, the smaller is the overshoot, the smaller is the T
c
control time,
the smaller is the value of d, and the smaller is the number of swingings.
There are a lot of cases, in which the input signal is a velocity step or an acceleration step. In
these cases not the controlled signal but the error signal is the most important and valuable
signal, and thus the qualitative characteristics refer to the error signal.
The introduction of the qualitative characteristics make it possible to compare some control
loops realizing the same control tasks. Of coarse, the technological demands determine, that
what is the better control loop for the given point of view.
There are other qualitative characteristics also in the control theory, the integral criteria. The
goal is to decrease the value of the given criterion. Some criteria are:
a.
( ) ( )
[ ]
dttxxI
0
ccl
∫
−∞=
∞
this is the linear control area. The better is the control, the smaller is this area. But this
criterion is applicable only in aperiodic systems, because in a swinging system it gives a good
result, as the positive and negative areas compensate each other, however the control loop is
not so good.
b.
( ) ( )
dttxxI
0
cca
∫
−∞=
∞
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this integral counts all the areas with positive sign.
Its analytical determination has a lot of trouble.
c.
( ) ( )
[ ]
dttxxI
2
0
ccn
∫
−∞=
∞
the squared control area. It can be determined analytically easily. Because squaring the big
errors are weighted with greater significance.
d. Timeweighted integrals.
( ) ( )
[ ]
dtttxxI
m
0
cclm
⋅
∫
−∞=
∞
( ) ( )
dtttxxI
m
0
ccam
⋅
∫
−∞=
∞
( ) ( )
[ ]
dtttxxI
m2
0
ccn
⋅
∫
−∞=
∞
These integrals punish the later differences, and thus the control time have to decrease to
fulfil the requirements.
Summary
.
The goal of control is to maintain the required value of the controlled variable. The basic,
fundamental operations of the control are the sensing, the decision and the intervention.
According to this, the components of the control loop are the sensors, reference signal makers
and difference makers (sensing and decision ), the signal formers and the amplifiers, and the
intervening members.
Each control loop has to fulfil the requirements of the technological process. These
requirements are done in static and dynamic state.
3.1.3. Blocks, block diagrams
.
The block diagram shows the connection between the signals in a control loop. A block has
an input signal and an output signal. The connection can be described by a transfer coefficient
and a transfer function. The transfer coefficient characterizes the block ( the member of
control loop ) in the steady state. After transients the output signal has a constant value, if the
input signal has a constant value, too. The rate of output signal / input signal is the transfer
coefficient, it has generally a dimension, too.
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There are functions, which can describe the transient properties of output signal. These are the
different transfer functions.
We'll deal with these functions later in this semester, now it is enough to know that they exist.
We can describe the resultant functions of a network if we know the rules to transform the
block diagrams.
In a linear system there are some simple rules for this task.
We can determine the resultant of two serially linked block by simply multiplying their
transfer coefficients or transfer functions.
x
o
x
12
x
i
A
1
A
2
=
x
o
x
i
21
AAR ⋅
=
The resultant of two parallel linked block is the sum of their coefficients or transfer functions.
x
o
x
i
A
1
A
2
=
x
o
x
i
R=A
1
+A
2
x
i
x
o1
x
o2
Finally the resultant of a block with a feedback block is:
A
1
A
v
x
o
x
i
if the feedback is negative.
Let us see some examples for these.(...)
We'll use the theorem of superposition.
3.2. Methods for treating of linear control loops.
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3.2.1 The differential equations method.
The different technical ( physical and chemical ) processes can be described by similar
differential equations. Electrical and magnetic phenomena, the flow of gases and fluids,
heating and cooling processes, mechanical movements etc. can be described by differential
equations. As a control loop can control different technical, technological processes , and they
are built up from members with different physical properties, their operation can be described
by differential equations, too.
The behaviour of control loops can be determined by means of differential equations. The
describing and solving of differential equations is often a very hard task, thus they had
developed some methods to simplify the treat. First let us see the method of differential
equations.
Any linear system can be described as follows:
i0
i
1
1m
i
1m
1m
m
i
m
m
o0
o
1
1n
o
1n
1n
n
on
n
xb
dt
dx
b
dt
xd
b
dt
xd
b
xa
dt
dx
a
dt
xd
a
dt
xd
a
++++=
=++++
−
−
−
−
−
−
K
K
where x
o
is the output signal of the system, x
i
is the input signal of the system, a
0
,a
1
,...,a
n
,
b
0
,b
1
,...,b
m
constants are the parameters of the system.
We can describe it as:
( )
( )
∑ ⋅=∑ ⋅
==
m
kk
k
i
k
k
n
ii
i
o
i
i
00
dt
xd
b
dt
xd
a
In physically realizable systems m<= n ( according to the law of causeresult ).
If i
0
=0 and j
0
=0, then the system is
proportional
, if i
0
=0 and j
0
> 0 then the system is
differentiating
, and if i
0
> 0 and j
0
=0 then the system is
integrating
one.
In the control theory we introduce time constants instead of coefficients, thus we can describe
the differential equation as:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+++Τ⋅=++++
−
−
−
− i
i
1
m
i
m
m
mo
o
1
1n
o
1n
1n
1n
n
o
n
n
n
x
dt
dx
T...
dt
xd
Ax
dt
dx
T...
dt
xd
T
dt
xd
T
or
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∑
+⋅=
∑
+
==
k
i
k
m
1k
k
ki
i
o
i
n
1i
i
io
dt
xd
TxA
dt
xd
Tx
In the equation T
i
and T
j
are time constants ( their dimensions are sec.s ), and
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( )
( )
∞
∞
==
i
o
0
0
x
x
a
b
A
is the transfer coefficient, which gives
the ratio of the output signal and the input signal in steady state. The transfer coefficient can
have a physical dimension (if the dimensions of the input and output signal are not the
same.)
This interpretation of the transfer coefficient is true if a
0
and b
0
are not zeros, or i
0
=j
0
=0. If for
example a
0
=0, (i
0
=1), then the transfer coefficient is
1
0
a
b
A =
, and the interpretations of time constants changes.
To solve an nth order differential equation we need n initial conditions.
By solving the differential equation of the control system we get the output signal in time as
the response to the supposed input signal.
The inhomogeneous differential equation can be solved as the sum of the general solution of
the homogeneous differential equation and a particular solution of the equation.
( ) ( ) ( )
txtxtx
opoho
+=
If the input signal is unchanging or periodic, the xop is a particular solution in steady state.
The solution of coarse depends on the input signal. The x
oh
homogeneous solution gives the
transient output signal of the system.
As it is known from mathematics , the transient solution can be got by means of characteristic
equation of the differential equation. If we search the transient solution in the from :
( )
∑
⋅=
=
⋅
n
1i
tp
ioh
i
eCtx
and we put it into the homogeneous equation, we get the following connection:
( )
0eCapa...papa
tp
n
1i
i01
1n
1n
n
n
i
=⋅
∑
⋅+⋅++⋅+⋅
⋅
=
−
−
This connection is true if and only if at p=p
i
there is :
( )
0apa...papapK
01
1n
1n
n
n
=+⋅++⋅+⋅=
−
−
This is the characteristic equation, and p=p
i
(i=1,2,...,n) are the n roots of this characteristic
equation.
The unexcited system can be described as:
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( )
∑
⋅=⋅++⋅+⋅=
=
⋅⋅⋅⋅
n
1i
tp
i
tp
n
tp
2
tp
1oh
in21
eCeC...eCeCtx
if only simple roots exist.
If there are multiple roots, too, the solution is :
( )
( )
∑
⋅+⋅⋅++⋅+=
+=
⋅
⋅
−
n
1qi
tp
i
tp
1q
q21oh
i
q
eCetC...tCCtx
here p
q
is a multiple ( qtimes) root, q ≤ n.
The roots of the characteristic equation are or real or conjugated complex values. In case of
conjugated complex roots (
ii1iiii
jp,jp
ω
⋅
−
α
−
=
ω
⋅
+
α−=
+
) the suitable part of the
solution is:
)tsin(eC)tcos(eC
eeCeeC)t(x
i
t"
ii
t'
i
tjt
1i
tjt
ioh
ii
iiii
⋅ω⋅⋅+⋅ω⋅⋅=
=⋅⋅+⋅⋅=
⋅α−⋅α−
⋅ω⋅−⋅α−
+
⋅ω⋅⋅α−
where
1ii
'
i
CCC
+
+=
1ii
"
i
CjCjC
+
⋅−⋅=
The C
i
constants can be determined from the initial conditions after writing the
inhomogeneous equation.
A system is stable, if their transients are damped, i.e. the transient solution approaches to zero
if time go to infinite.
It does, if each root of the characteristic equation has a negative real part.
The p
i
roots depend only on the parameters of the system, and they are independent from the
input signal.
A particular solution of the inhomogeneous equation can be determined by means if the
method of varying the constants, for example. There are a lot of cases in which simple
physical meditations can help us to determine the constants.
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An example:
There is an electrical quadropole ( it may be a member of a control loop ):
R
1
R
1
C
i
u
i
u
o
We connect a voltage u
i
=U
i
to its input at time t=0.
What will the uo output voltage be ?
The equations describing this network are:
∫
+⋅=
0
2o
dti
C
1
Riu
( )
oi
1
uu
R
1
i −⋅=
We can arrange them as
( )
( )
∫
−
⋅
+−⋅=
0
oi
1
oi
1
2
o
dtuu
RC
1
uu
R
R
u
∫
⋅
⋅
+⋅=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+⋅+
∫
⋅
⋅
0
i
1
i
1
2
1
2
o
0
o
1
dtu
RC
1
u
R
R
R
R
1udtu
RC
1
Let us differentiate it
i
1
i
1
2
1
21o
o
1
u
RC
1
dt
du
R
R
R
RR
dt
du
u
RC
1
⋅
⋅
+⋅=
+
⋅+⋅
⋅
( )
dt
du
RCuu
dt
du
RRC
i
2io
o
21
⋅⋅+=+⋅+⋅
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( )
CRRT
211
⋅
+=
CRT
22
⋅=
i
i
2o
o
1
u
dt
du
Tu
dt
du
T +⋅=+⋅
The characteristic equation is:
01pT
1
=+⋅
Its root is
1
T
1
p −=
The general solution of the homogeneous equation is
( )
1
T
t
1oh
eCtu
−
⋅=
One particular solution of the inhomogeneous equation is
u
op
(t)=U
i
because after charging the capacitor the entire input voltage is on the output.
The full solution is :
( ) ( ) ( )
i
T
t
1opoho
UeCtututu
1
+⋅=+=
−
The initial condition is :
( )
i
21
2
o
U
RR
R
0u ⋅
+
=
because the time t=0 the capacitance can be approximated as a shortcircuit.
Thus
i
21
2
i1
U
RR
R
UC ⋅
+
=+
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21
2
i1
RR
R
UC
+
⋅−=
The solution is :
( )
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅
+
−⋅=
−
1
T
t
21
1
io
e
RR
R
1Utu
We can represent it as:
u
i
t
t
u
o
T
1
21
2
i
RR
R
U
+
⋅
Summary
:
The method of differential equations is suitable for treating the dynamic properties of a
control loop or a part of a control loop. It has some disadvantages : It is a hard task to solve a
differential equation in particular with difficult exciting functions. The constants can be
determined only after the full search of the solution of the inhomogeneous equation.
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3.2.2. Typical reference signals for examining the control loops.
To solve an inhomogeneous differential equation it is a common way to give a typical
examining input signal to the system. The solution can be got easily if the input signal is very
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