Jan 1, 1993 - system design using fuzzy logic controllers, and to design and ..... Characteristics . ..... along with experimental data to test the robustness of the controllers. Chapter ...... is used to construct the change of error sequence Ae(kT) = e(kT) - e((k-1)T), which ... r(kt) defines a range for y(kT) and y(t) at steady-state.
DEPARTMENT COIJ.EGE
OF OF
ENGINEERING
DOMINION
OLD
NORFOLK,
DESIGN
ELECq'RICAL
& TECHNOLOGY
23529
IMPLEMENTATION
OF FUZZY
By Osama
A. Abihana,
Principal
Final For
Graduate
Investigator:
Research
Oscar
Assistant
R. Gonzalez
Report the
period
Prepared
January
Aeronautics
Langley
1, 1993
for
National
Research
Hampton,
and
Space
Administration
Center
VA
23681-0001
Under P.O. Capt. U.S.
#L20278D Gregory
W. Walker,
of Contact
by the
Old Dominion P.O. Box 6369 Norfolk,
O July
Point
Army
Submitted
1993
Virginia
ENGINEERING
UNIVERSITY
VIRGINIA
AND
& COMPUTER
University
Research
23508-0369
Foundation
LOGIC
CONTROLLERS
ACKNOWLEDGEMENTS This is a thesis entitled
"Design
1992 to January Center
being submitted
and Implementation 1, 1993.
through purchase
U.S. Army,
NASA
Langley
in lieu of a final report of Fuzzy Logic
for the research
Controllers"
This work was supported
by the NASA
order
of contact
no. L20278D,
Research
point
Center,
Mail Stop 286.
ii
for the period Langley
project July 27,
Research
Capt. Gregory
W. Walker,
ABSTRACT The main overview
of fuzzy
system fuzzy
objectives
design
sets and fuzzy
using
logic
controller
fundamental
concepts
fuzzy operations to understand
design
of fuzzy
the concepts
a four-step
and robustness procedure
our
design
actuator Research
methods,
fuzzy
based
and
logic
variables,
control
control
systems.
implementation
Fuzzy
of a fuzzy
logic
Research
Third, Finally,
controller
of the Free Center.
Second,
we The
the capabilities
This is followed experience.
term sets,
procedure.
showing
the
it is important
values,
design
a
sets and basic
methods.
analysis.
Langley
and implement
systems,
for stability
the direction
for control
we first present
linguistic
system
from our design
techniques
at NASA
for control
via four examples,
to be used to control Vehicle
logic.
and defuzzification
logic
is illustrated of fuzzy
In this thesis
In addition,
a self-contained
a methodology
and to design
sets and fuzzy
of linguistic
that we developed
two Lyapunov
develop
for a real system.
are defined.
procedure
logic,
are to present
fuzzy logic controllers,
fuzzy rule base, inference introduce
of our research
Flight
by a tuning we present we present for
a linear
Rotorcraft
TABLE
OF CONTENTS
LIST OF TABLES
..............................................................................................
vi
LIST OF FIGURES
...........................................................................................
vii
INTRODUCTION 1.1 Introduction
............................................................................................ ................................................................................................
1 1
1.2 1.3
Overview of Fuzzy Controls .................................................................... Thesis Structure ........................................................................................
2 3
FUZZY
SETS AND FUZZY LOGIC IN CONTROL SYSTEMS ................... 2.1 Introduction .....................................................................................
6 6
2.2
Sets ........................................................................................ Definition of Fuzzy Sets ...................................................... Fuzzy Set Operations ......................................................... Linguistic Variables ......................................................... Logic ................................................................................... Fuzzy Rule Base ................................................................. InferenceMethods ..............................................................
8 8 11 20 23 23 25
Fuzzy Systems in Control Systems .............................................. 2.4.1 Defuzzification .................................................................... 2.4.2 Defuzzification Method .....................................................
29 29 31
2.4.3 Center of Gravity Method .................................................. Example ......................................................................................... Conclusions ....................................................................................
32 35 41
LOGIC CONTROLLERS (FLCs) ....................................................... 3.1 Introduction ....................................................................................
42 42
2.3
2.4
2.5 2.6 FUZZY
3.2
Fuzzy 2.2.1 2.2.2 2.2.3 Fuzzy 2.3.1 2.3.2
Fuzzy Logic Control System Design ........................................... 44 3.2.1 Step One. Acquire Plant Information ................................ 47 3.2.2 Step Two. Select Term Sets For the Linguistic Variable.48 3.2.2.1 Example ................................................................. 52 iv
3.3 3.4 3.5 3.6 3.7
3.2.3 Step Three. Form the Fuzzy Rule Base ........................... 3.2.4 Step Four. Tune the Fuzzy Controller ............................... Design Example One ..................................................................... Design Example Two .................................................................... Design Example Three .................................... _............................ Design Example Four ................................................................... Conclusions ...................................................................................
Appendices
3A, 3B, 3C ...........................................................................
56 59 60 76 87 91 93 94
OVERVIEW OF STABILITY ANALYSIS OF FLC's ................................. 4.1 Introduction .................................................................................
109 109
4.2
Stability Analysis Using Lyapunov's Direct Method ............... 4.2.1 Example ............................................................................. 4.2.2 Conclusions .....................................................................
110 115 120
4.3
Another Approach Using Lyapunov's Direct Method .............. 4.3.1 StabilityCriterion ............................................................. 4.3.2 Example ............................................................................ 4.3.3 Conclusions ....................................................................
120 121 123 126
Appendix
128
ACTUATOR RESEARCH 5.1
4A ................................................................................
DESIGN FOR THE FREE FLIGHT ROTORCRAFT VEHICLE ................................................................................... 131 Introduction ............................................................................... 131
5.2
System 5.2.1 5.2.2 5.2.3
5.3 5.4 5.5
Design Procedure ....................................................................... Testing Results ........................................................................... Conclusions .................................................................................
Appendix
Description and Specifications ..................................... Physical Characteristics ................................................... Mechanical Characteristics .............................................. Electrical Characteristics ..................................................
................................................................................................
CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH 6.1 Conclusions ................................................................................. 6.2
Suggestions
For Further
Research .............................................
BIBLIOGRAPHY ........................................................................................ APPENDIX A .................................................................................................
V
133 133 134 134 135 138 142 148 .... 154 156 158 158 162
LIST
OF TABLES
Table 2.1.
Sample
Table
Range of signals ............................................................................
48
Table 3.2.
Example
57
Table
3.3.
Rule base for the example ...........................................................
63
Table
3.4.
Results
of varying
the command
74
Table
3.5.
Results
of varying
the pole locations
Table
3.6.
Results
of varying
the dc gain by 20% .......................................
75
Table
3.7.
Results
of varying
the command
86
Table
3.8.
Results
of varying
the pole locations
Table
3.9.
Results
of varying
the dc gain by 20% .......................................
Table
4.1.
Partitioned
3.1.
look up table ....................................................................
of a rule base .................................................................
input ........................................ by 20% ............................
input ........................................ by 20% ............................
state space .................................................................
vi
37
74
86 87 124
LIST OF FIGURES
Figure
2.1.
The membership
Figure
2.2.
Membership
functions
for two valued
Figure
2.3.
Membership
functions
Figure
2.4.
Membership
functions
Figure
2.5.
Membership functions for the complement of WARM HOT ................................................................................................
Figure
2.6.
Membership
functions
for WARM
union NOT WARM ........... 21
Figure
2.7.
Membership
functions
for WARM
intersection
Figure
2.8.
Max-min
inference
method ..........................................................
28
Figure
2.9.
Max-dot
inference
method ..........................................................
29
Figure 2.10.
Graphical
functions
representation
Figure
2.11.
Figure
3.1.
A basic block diagram
Figure
3.2.
Block diagram
Figure
3.3.
Example
Figure
3.4.
Graphical
WARM
and HOT ............................ logic ...............................
12
for WARM
union HOT ..........................
14
for WARM
intersection
15
of control
HOT .................
rules ...................................
for fuzzy controlled
of the filter in Figure
representation
functions
system ....................
3.1 .....................................
..............................................
of two rules with overlapping vii
and 16
NOT WARM...22
Determination of the control input by means of center gravity method ..............................................................................
of membership
10
39 of 40 45 46 51
sets ..... 53
Figure
Figure
3.5.
3.6.
Graphical representation of two roles with non sets ............................................................................................. Membership
functions
of fuzzy
sets error,
control input ....................................................
w
change
overlapping 54 of error, and
._............................
65
Figure
3.7.
Stages of fuzzy logic controllers
.................................................
68
Figure
3.8.
Step response before tuning the controUer, control included .....................................................................................
input 69
Figure
3.9.
Step response
after the first tuning,
Figure
3.10.
Step response
after the second
Figure
3.11.
Step response
to different
Figure
3.12.
Step response
to variation
in pole locations
..............................
72
Figure
3.13.
Step response
to variation
in dc gain ....... . .................................
73
Figure
3.14.
Step response
before tuning,
Figure
3.15.
Step response
to different
Figure
3.16.
Step response
to variation
in pole locations
Figure
3.17.
Step response
to variation
in dc gain .........................................
Figure
3.18.
Ramp response .............................................................................
82
Figure
3.19.
Step response
to disturbance
at the input ...................................
83
Figure
3.20.
Step response
to disturbance
at the output .................................
84
Figure
3.21.
Step response
to disturbance
at the input and output ................
85
Figure
3.22.
Step response
to variation
control
tuning,
command
control
command
control
oo_
....... 70
input included.71
inputs ...............................
input included
.................
inputs ...............................
in pole locations
VIII
input included
..............................
................................
71
78 79 80 81
89
Figure
3.23.
Step response
to variation
Figure
3.24.
Step response
of unstable
Figure
3.25.
Fuzzy sets for example
one, before tuning...._ ...........................
96
Figure
3.26.
Fuzzy
one, after first tuning ...........................
97
Figure
3.27.
Fuzzy sets for example
one, after second
98
Figure
3.28.
Fuzzy sets for example
two ......................................................
101
Figure
3.29.
Fuzzy
four .....................................................
102
Figure
4.1
No P-region .................................................................................
117
Figure
4.2
CriticalP-region
..........................................................................
118
Figure
4.3
P-region
appears .........................................................................
19
Figure
4.4
Step response
Figure
5.1.
Closed loop step response
before tuning ...................................
140
Figure
5.2.
Closed loop step response
after tuning ......................................
143
Figure
5.3.
Closed
due to horizontal
144
Figure
5.4.
Closed loop step response
due to vertical
Figure
5.5.
Closed loop step response
of different
controllers
Figure
5.6.
Closed loop step response
of different
controllers
Figure
5.7.
Fuzzy
sets for the linear actuator
before tuning ........................
150
Figure
5.8.
Fuzzy
sets for the linear actuator
after first tuning ..................
151
sets for example
sets for example
in dc gain ......................................... plant ..................................................
tuning ......................
of example ...........................................................
loop step response
ix
loading ................
loading .................... ....................
90 92
127
145 146
under load. 147
Figure
5.9.
Fuzzy
sets for the linear actuator after second
X
tuning ............ 152
CHAPTER
ONE
INTRODUCTION
1.1 Introduction Fuzzy logic processes
operator
FLCs
are fuzzy
of a process. variables.
and fuzzy
are used
in control
logic.
Fuzzy
linguistic
variables,
linguistic
methods,
and defuzzification logic
control
illustrated via four examples control
controllers.
systems. Third,
values, methods. design
showing
completeness
operations
to understand
Second,
we
are defined.
In
the concepts
rule bases,
present
four
procedure
is
of fuzzy
to tune fuzzy
two
of
inference
a basic
and robustness
by a procedure
of the sets
The design
the capabilities
the human
of fuzzy
we introduce
procedure.
for
the data is
description
concepts
term sets, fuzzy
This is followed for
fuzzy
design
can model
on a linguistic
it is important
system
that
system
or where
the fundamental
sets and basic
systems,
model
systems
are based
We first present
for control
step fuzzy
expert
They
addition,
logic
(FLCs)
that do not admit a mathematical
imprecise.
process
controllers
logic
Lyapunov-based
2 techniques
for
stability
analysis.
research,
since these techniques
designs.
Finally,
controller
we
1.2 Overview Since
of Fuzzy
fuzzy
laboratory
speed
engine
setting
area
be applied
and implementation
in NASA
of fuzzy
for
Langley
future
to our control of a fuzzy logic
the direction
set theory
was developed (Mamdani, steam
of the Free
Research
Center.
by Lofti
for control
applications.
by Mamdani
1974).
Zadeh
to control
The purpose
by using
heat
on the engine.
Because
of the successes
conducted
by Mamdani
logic
was generated.
Since
many others have been developed
(Zadeh, In 1974 a small
was to regulate
pressure
experiments control
currently
that it is well suited
and boiler
and the throttle
important
Controls
controller
steam
is an
to be used to control
Vehicle
the development
it was found
the first
engine
Research
cannot
the design
for a linear actuator
Flight Rotorcraft
1965),
present
This
applied
to the boiler
and co-workers,
an interest
that first application
of fuzzy
and tested to prove the worthiness
of these in fuzzy set theory of those
controllers. Fuzzy control to processes
where
generated other
a lot of enthusiasm
control
techniques
were
because
it can be applied
not efficiem
or simply
3 failed
to do the
mathematical
job.
model
Moreover,
but rather
fuzzy
a linguistic
logic
controllers
description
require
of the process.
no The
Q
control
task
IF..THEN
is achieved
rules that capture
a control
process
would,
One drawback fine
tune
the
difficult
procedure
to achieve
The tuning
1.3 Thesis
sets
and
Those
rules
when
includes
results.
consuming.
This
In this
response reshaping
when
of
applied
to
input to the process.
is the lack of systematic best
the desired
control
a series
techniques
to
process
is
introduce
a
tuning
thesis
tuning
we fuzzy
of the membership
controllers. functions
to
response.
Structure
The theory
of the chapters
of fuzzy sets and fuzzy logic
comparison understand
of fuzzy
expertise.
to attain
and time
The organization
of fuzzy
human
of fuzzy control
procedure
the desired
the use
in turn, give the desired
controller
sometimes
achieve
through
controllers
between
and appreciate
relate to fuzzy
control
logic as it directly
is presented
two-valued
logic
the theory
are defined
in the thesis follows.
in Chapter and
fuzzy
of fuzzy
Two. logic
sets.
and illustrated
relates to the design This
includes
a
to help
the reader
Moreover,
terms that
graphically.
An example
4 is presented to help in understanding
the application
of fuzzy
sets and fuzzy
logic. t
Chapter
Three
is an
controllers.
In this
chapter,
controllers
is presented
logic
control
invariant,
system
controller
techniques
with experimental Chapter controllers. briefly.
In this chapter, However,
problems.
these
This implies
analyzing
the stability
A real-world in Chapter
Five.
fuzzy
logic
of fuzzy
logic
single-output,
Second-order
type 0 and
phase
plant.
The
and the way to approach
them
fuzzy
controllers
data to test the robustness is an
of
In the examples,
for single-input,
for tuning
Four
design
the design
by examples.
plants.
examples
the
for
as a non-minimum
for these
Moreover,
actuator
is designed
as well
to
a procedure
followed
and continuous
considered,
introduction
overview
of
two approaches approaches
are also
linear,
problem The design
time
1 systems
are
design
of a
is explained.
presented
along
of the controllers. the
stability
to stability
are all aimed
analysis analysis
of
fuzzy
are discussed
at specific
that until now there has been no general of fuzzy
a fuzzy
classes method
of for
logic controllers.
is solved
and its controller
of a fuzzy
logic
to be used on the Free Flight Rotorcraft
controller Research
design
is presented
to control Vehicle
a linear (FFRRV)
5 is included with the specifications by testing
the controller
The presented
conclusion in Chapter
Fuzzy Chapters
requirements.
under various
load conditions
of the
and
thesis
topics
This is followed
to show its robusmess. for
further
research
are
Six.
sets and the rule base used
Two,
Three,
with the C code listings loop systems
and system
in the design
and Five are included that were written
in appendices
before
of the controllers
and after tuning
for the simulation
at the end of each chapter.
in
along
of the closed-
CHAPTER FUZZY
SETS
AND
TWO
FUZZY
LOGIC
IN CONTROL
SYSTEMS
2.1 Introduction The theory University
of
generalization solution
of fuzzy sets was developed California
of real-world
applicability,
mathematics chapter, control
system
concepts
the objects
making
Prade,
it more fuzzy
with uncertain
data.
of fuzzy
Kandel
applicable
In addition
a
in the logic
to its real
area of research
and Lee,
sets and fuzzy
represents
sets and fuzzy
is now an important
1980;
It
of the
logic
1979).
in
In this
that are used
in
are presented.
to introduce
sets which
1965).
In particular,
set theory
and
design
In order
theory,
decisions
fuzzy
(Zadeh,
set theory,
problems.
(Dubois
the main
conventional
Berkeley
of conventional
may be used to make world
at
in 1965 by Lofii Zadeh
and
appreciate
fuzzy
are based on two-valued
of the universal
set belong 6
sets,
logic.
consider
first
In conventional
to or do not belong
the set
to specific
7 sets. This is due to the fact that two-valued we assign
an object
odd-even,
black-white,
performed
on processes
the classification categories etc.
etc.
This
that are precise
of numbers
for example,
might
consider
by another
A historical sophism
tall,
or dictates
can
and well def'med. However,
warm,
another
be
many
are all relative.
warm by an individual
easily
Such a process
might
near,
tall,
For instance
consider
could
is
engineering
hot, fast, turbulent,
person
that
0-1, good-bad,
of classification
as odd or even.
or what is considered
very warm
type
imposes
for example,
that the terms in the above example
what one person height
to one of two categories,
are ill-defined,
Notice
logic
medium
be classified
as
individual.
example
of fuzzy notation
and can best illustrate
comes
the classification
from an ancient
dilemma
(Pedrycz,
Greek 1989),
"...one seed does not constitute a pile nor two nor three .. from the other side everybody agrees that a 100 million seeds constitute a pile. What therefore is the appropriate limit? Can we say that 325,647 constitute a pile but 325,648 do?" From the above discussion to a set.
seeds don't
we see the need to assign
degree
of belonging
The concept
Zadeh
did just that and is introduced
of a fuzzy
in the next
section.
to an object set formulated
some by
8 2.2 Fuzzy 2.2.1
Sets
Definition Fuzzy
grade
set theory
of belonging
complete certain
object
generalizes
(1).
by A.
in degrees
set warm
with
degrees
belongs
example,
the
universe
of discourse
real interval
defining
to the
of 1.0 while
with
a grade
set warm
X.
The
In this example temperature
by a membership
the value
For example,
set of all possible
Formally,
from
a fuzzy function
of a set to allow the
full exclusion
with
a grade
temperatures universe
of warm belongs
of 70 degrees
and the temperature of 0.0.
the closed
to
From
a universal
of 50
the
above
set or a
can be discrete interval
or
on the real
Fahrenheit.
set A on a universe RA(X) which
notion
of 80 degrees
of discourse
X=[40 °, 100°], in degrees
the fuzzy
forms
of a
the link of x to the
the temperature
of 0.50,
(0) up to
of the membership
set A, R^(x), the stronger
a grade
to the set warm
line, denoting
varies
concept
we can say that a temperature
belongs
continuous.
the original
The higher
x to the fuzzy
described
expressed
Sets
to the set, which
membership
category
the
of Fuzzy
of discourse
X is characterized
elements
of X into the closed
maps
[0,1] as follows: PA: X --_ [0, 1],
9 where
pA(X) expresses
simplicity simply
if there
the degree that x belongs
is no confusion,
denoted by A.
to some
the membership
category
function
A fuzzy set A can be represented
A.
For
pA will be
by an ordered pair
A = {(x,IaA(X))l x e X}. Hence, a fuzzy setcan be viewed by plottingx vcrsus laA(X). In addition,for discrete X, a tablecan alsobe used to representa fuzzy set.In conventional set theory,there is also a membership function called the characteristic function which can only take two values 0 or I, dcnoting exclusion or inclusion, respectively, of an objectin a set. For cxarnplc,considerthe universeof discourseX of temperaturesin [40°,100°].Wc can definetwo fuzzy setsWARM
and HOT
on X which are
characterized by theirrespectivemembership functions.Figure 2.1 shows the membership functionsfor WARM functionsare leftout,such as VERY
and HOT. HOT.
Additionalmembership
Given thatx=65 ° wc determine
the degree x belongs to cach fuzzy set by findingthe point at which x intersects thc membership function,thatis,findingthe value of WARM(x) and HOT(x). WARM
In thiscase a temperatureof 65° belongs to the fuzzy set
with a dcgrcc of 0.75 and to HOT
with a degree of 0.25. In
conventionalsets,the characteristic functionleadsto sharpboundaries as
I0
Warm
Hot
10
85
Figure
2.1.
The
membership
functions
WARM
and
HOT.
11 seen in Figure used.
2.2.
Triangular
In this example
and trapezoidal
a triangular
membership
membership
functions
function
are commonly
was used
ql,
in fuzzy control membership
Other membership
applications.
functions
and
monotonically
functions
include
increasing
and
gaussian decreasing
functions.
2.2.2 Fuzzy
Set Operations
All conventional
set operations
In fact, when
a fuzzy
set operation
conventional
set result
is obtained
In this
section
we present
three
have been is performed (Zadeh,
basic
generalized
to fuzzy
on a conventional
1965;
operations
Kandel including
sets.
set, the
and Lee,
1979).
some
of their
properties. In set theory the operations denoted
by Ac'_B, AuB
defined
as follows:
of union,
, and NOT
A, respectively.
AUB = {xeXlxeA
AQB
= {x6X
intersection,
IxeA
and complement The
operations
are are
oz xeB},
(2.1)
and
(2.2)
x6B},
12
_)
Figure
2.2.
_)
Membership
_)
function
1_)
for
two
valued
logic.
13 and
-- {x6Xl
Operations
in fuzzy
functions above
(Zadeh,
operations
sets are defined
1965).
in terms
In particular,
in (2.1),
(2.2),
(2.3)
xfA}.
of their
for fuzzy
sets
fuzzy
membership
A and B on X the
and (2.3) become
(AUB)
(x)
= max(A(x),B(x))
(AnB)
(x)
--min(A(x),B
(x))
xeX,
(2.4)
xeX,
(2.5)
and
A(x)
where operator
the union
operator
corresponds
corresponds set operations
= 1 - A(x)
corresponds
to the AND
to the NOT
function.
to the OR function,
function, Notice
in (2.4) - (2.6) is a new fuzzy
in (2.4) - (2.6) are illustrated
in Figures
(2.6)
xeX,
and
the complement
that the result set.
the intersection operator
of the three
The three fuzzy
2.3 - 2.5 for the fuzzy
fuzzy
operations
sets WARM
14
i
10
0S
40
Figure
2.3.
The
60
membership
80
function
I00
of
WARM
union
HOT.
15
10 / /
05 / /
\ /
40
Figure
2.4.
The membership intersection
function HOT.
of
WARM
16
10
05
411
Figure
2.5.
60
I_
The membership functions of WARM and HOT.
II_
of
the
complements
17 and HOT
defined
2.3 - 2.5 consider
in Section
2.2.1.
the following
To aid in the interpretations
subset
of Figures
of temperatures
T = {60 °, 65 °, 70 °, 75 °, 80°}. The
degrees
WARM
of the membership
of these
temperatures
to the fuzzy
sets
and HOT are given by WARM
(T) = {I,0.75,0.5,0.25,0}
and HOT
(T) = { 0, 0.25,0.5,0.75,I}.
The fuzzy operationsin (2.4)- (2.6)must be satisfied for allx _ X.
In
particular, they must be satisfied for x _ T. The degree of membership of temperaturesin T to the fuzzy setsthatresultfrom (2.4)- (2.6)isgiven by (WARM
u HOT)(T) =
{max(l,0),max(0.75,0.25),max(0.5,0.5), max(0.25,0.75),max(0, I)} = { 1, 0.75, (WARM
n HOT)(T)
{min(1,0), = { 0, 0.25, (NOT and
0.5, 0.75,
1 }; -
min(0.75,0.25), 0.5, 0.25,
WARM)(T)
min(0.5,0.5),
min(0.25,
0};
= {0, 0.25,
0.5, 0.75,
1 };
0.75),
min(0,
1)}
18 (NOT HOT) (T) --- {1, 0.75, 0.5, 0.25, 0}. For completeness, are stated (1989).
below. First,
functions
some fuzzy
For further
two
are identical
fuzzy
set properties
information
sets are equal
for all x _ X.
in the form
see Zadeh, if and
(1965)
only
De Morgan's
if their
Laws
of theorems or Pedrycz, membership
are given
(ANa) (x) = _(x)U _(x)
by
(2.7)
and
(AUB)
De Morgan's (2.6).
(x)
laws can be easily
A(X)
proven
using
(2.8)
.
the basic
operations
in (2.4)-
The first, becomes 1 - Max{A(x),
In order to verify
B(x)}
the equality,
< B(x) need to be tested.
= Min{ 1 - A(x),
the two possible
1 - A(x);
if A(x) < B(x) we have, 1 - B(x) = 1 - B(x). The distributive
cases:
If A(x) > B(x) we have, 1 - A(x)--
while
NB(x)
laws are given by
1 - B(x)}. A(x) > B(x) and A(x)
19
A _
(BUC)
=
(A/_B) U
(2.9)
(AnC)
and
°
A U
The properties
(B_C)
of absorption
= (AUB)
n
(AUC)
and idempotency
(2.10)
.
also hold:
(AnB)
U A : A,
(2.11)
(AUB)
n A = A,
(2.12)
AUA
= A,
(2.13)
A.
(2.14)
and
AnA:
However,
the following
laws are not satisfied,
AUX,
X
(2.15)
¢' O.
(2.16)
and
ANA
This is expected of two values.
since fuzzy To illustrate
sets do not impose
that an object
why the laws are not satisfied
take on one
consider
the fuzzy
20 set WARM. WARM fuzzy
The membership functions of WARM
n NOT
WARM
are given
set to be universal,
membership
function
that is, be equal
to define
the concept
logic
2.2.3
Linguistic
function
of linguistic
and
In order
for a
of discourse,
its
a fuzzy set is empty
is zero on X. of the above statements
variables
which
we need
is the comer
stone
of
Variables variable
rather than with
is a variable
numbers.
variables
are considered
statement
such as the temperature
variable
WARM
control.
A linguistic words
to its universe
we move on to the application
fuzzy
2.6 and 2.7.
should be unity on X. In addition,
if and only if its membership Before
in Figures
u NOT
temperature
linguistic
variable
linguistic
values.
with
variables.
values
Since each linguistic
may
is referred
in a
that the linguistic
The linguistic
X
with
the measured
For example,
of temperatures.
of discourse
The set of linguistic variable.
is represented
applications,
value hot.
of discourse
universe
value
is hot, we are saying
has the linguistic
set in the universe
of the linguistic
In control
to be linguistic
is a fuzzy
whose
value hot
In general, take
on
a
several
to as the term set
value is a fuzzy set on X, the
21
1.J J
/
/
/
05
Figure
2.6.
The membership intersection
NOT
function WARM.
for
WARM
22
10
05
Figure
2.7.
The membership WARM.
function
for
WARM
union
NOT
23 term
set
represents
a fuzzy
functions
of the linguistic
linguistic
values
values
are made
of X,
where
to overlap.
the
membership
An example
of some
are:
PB: positive
big,
PS: positive
small,
Z:
partitioning
zero,
NS: negative
small,
NB: negative
big.
and
A finer
granularity
2.3 Fuzzy 2.3.1
can be obtained
expert
Fuzzy
Rule
rule base
Lee (1990) rule base
linguistic
values.
Base sets, a fuzzy
can be provided
or learned
to derive
more
Logic
In order to apply fuzzy This
by considering
a fuzzy
by an artificial
to the system, neural
rule base for control
and Kosko
(1992).
are of the form
rule base needs
for example,
network. system
The fuzzy
(Hill, Horstkotte,
to be specified. by a human
Some of the methods applications
control
are presented
or production
and Teichrow,
1990):
used in
rules in the
24 IF premise where
the premise
THEN consequence,
is a set of conditions
(2.17)
to be specified
and the consequence Q
is a set of actions to be taken.
The premise
relations
represented
example,
let L1, L2, and L 3 be three linguistic
x3, respectively. respectively, addition,
by linguistic
Let
where
variables
x 1, x 2, and
B,2}, and {C1,..., Cn3}, respectively. Ri: IF L I is A i AND Additional easily
linguistic
taken into account.
the cartesian represented
x 3 be
variables
variables
defined
samples
of
are fuzzy values.
For
on Xl, x2, and
L l, L 2, and
and x 3 is to be determined.
L3, In
and L3 be given by {A1,..., A_I}, {BI,..., Then the i th fuzzy L2 is B i THEN in the premise
The premise
of _
rule is of the form
L3 is Ci
(2.18)
and consequence
is a fuzzy relation
product XI×X 2. This fuzzy relation
can be
defined
on
is a fuzzy set and it can be
by Premise
where
and their linguistic
x I and x 2 are known,
let the term sets for L1, _,
and the consequence
= {((xl,x2),
Premise(xl,x2))
[ (xl,x2)e
X1xX2},
(2.19)
we can take Premise(x
In this case, conjunctively
it is assumed with
the
1, x2) = min (Ai(xl), that the fuzzy
AND
operation.
Bi(x2)).
sets in the premise It is also
possible
(2.20) are combined to use,
for
25 example, the OR operation to combine the fuzzy sets. depends
on the inference
method
chosen.
This
The
is discussed
operation
further
in the
III
next section. An example
of a rule to control
a linear actuator
is
R1: IF E is PB AND AE is PS THEN Here E, AE, and control PB linguistic
values
2.3.2 Inference
be taken taken
input
input are the linguistic
of applying
rule premise
from the premise.
the
degree
to the rule's
is called performing
and PB, PS, and
variable.
of membership
conclusion
an inference.
In the example to be a fuzzy
computed
to determine
One is inferring in the previous
for a
the action the action
section,
to
to be
each fuzzy
implication
Ai. xBi---_C
is a fuzzy set. This fuzzy implication
membership
variables
in the term sets of each linguistic
rule R can be considered
which
is PB.
Methods
The process production
control
(2.21)
k,
is a fuzzy set in X_xX2xX 3 with
function R(x l, x 2, x 1) = Ll(xl)*L2(x2)*La(x3),
(2.22)
26 where the most commonly
used operations
for * are product
and union (Lee,
1990). I,
The two main are the max-min Horstkotte, methods
are
inference
methods
inference
method
and Teichrow,
1990).
referred
and the max-dot In Kosko
respectively.
the value
to be assigned
to the output
(max-min)
to the degree
of membership
In either
to form the final output
inference
membership
very similar
results.
However,
the max-dot
method
is preferred
method.
is either
the output
can be computed
and observing
to which
and we can find the degree or the OR (max)
methods
of fulfillment
operations.
inference
concept
is that
or clipped or
together
both methods
faster
give
implementation computationally
are illustrated
in Figures
In Figures
2.8 and 2.9
1990).
Temperature
From the appropriate
and Pressure rules will fire
of each rule by applying
The output
(Hill,
All the clipped
to computer
the samples
sets they belong.
two
are then combined
it is much
and Teichrow,
method
(max-dot)
In reality,
it comes
because
by taking
the basic
scaled
fuzzy logic
correlation-product
for the premise.
function.
when
these
and
method,
Both inference
2.8 and 2.9 (Hill, Horstkotte,
(min)
(1992)
sets for all the rules that set this output
than the max-min
inference
to as correlation-minimum
encoding,
scaled
that are used in applying
fuzzy
the AND
set is then scaled
or
27 Figures
clipped by the result of the min or max operation. both
the
AND
(min)
the
and
OR
(max)
operations
2.8 and 2.9 show on both
inference
Q
methods. The following membership
function
comments
apply to the max-dot
of the i th fuzzy
implication
Ri.(xl,x2,x3) = rain(Ai(xl),
is given
function
of the consequence
W i = min(Ai(xl),
If there
are
sets _.
A method is needed
base which
will be called
n fuzzy
to determine
the output fuzzy
o = R1.UR2U...
A better
approach
(Kosko,
1992)
method.
rules,
(2.23)
a combined set.
scaled then
The
by
Bi(x2))xC i,
which is the membership Bi(x2)).
inference
by the weight
there
fuzzy
are
n fuzzy
set for the rule
One approach
is to let
URn.
(2.24)
is to add the membership
functions
as
follows
n
"o -- _
wxci"
(2.25)
t-1.
The latter
approach
in the rule base fuzzy
set is X 3.
is preferred,
grows.
Note
in particular, that the universe
as the number of discourse
of fuzzy
rules
of the output
28
rule : if Temperatureis Low or PressureIs Low then throttleis I_edlum
Throttle
P is Low
T is Low
ls I_1
rule ' if Tenl)erature is Low and Pressureis Low then throttmeis radium
Throttle
P is Low Figure
2.8.
Max-Min
Inference
is l_d
Method.
29
rule : if Temperatureis Low or PressureIs Low then throttleis Medium
T i5 Low
P is Low
rule : if Te_erature is Lowand PressureIs Low then throttleis medlum
,/xx T is Low Figure
P is Low 2.9.
Max-Dot
Throttleis Med Inference
Method.
30 2.4 Fuzzy 2.4.1
Systems
in Control
Defuzzification The previous
systems. engine. linguistic one
A fuzzy
system
The fuzzy
system
values,
method
consists
system
variable
applications, first.
process variable
an inference
represented
fuzzy
(MISO)
section,
with
variables
considering
with an appropriate the degree
fuzzy
set.
of measured by
by
can be handled
we may
each
term set.
of membership
sets in the corresponding
of the previous
of fuzzy
rules fire and together
the output
in f'lnding
inputs,
single-output
which
variable
and
inputs
is accomplished
to the fuzzy
with the example
More
the crisp vector
This
consists
set of fuzzy set O.
determine
components
rule base
multi-input,
to be a linguistic
The fuzzification
Continuing
fuzzy
several
the main
of a fuzzy
maps a given
chosen,
to be fuzzified
measured
described
logic is used to determine
In control
measured
has
into an output
Fuzzy
the inference
each
section
at a time by forming
systems.
needs
Systems
term
of sets.
find that
x_ is A_, x_ is A2, X2 is B5, and x 2 is B 6. The fuzzy
sets for which
the degree
inputs
to the fuzzy
system.
which
corresponds,
for example,
of membership
The output
of the fuzzy
to the degree
is non-zero system
of membership
are used as is a fuzzy
set
of the control
31 inputs.
In control
converted
applications,
into a crisp munerical
and it is described logic
systems
controllers
value.
This process
in the next section. is given
2.4.2 Defuzzification
the output fuzzy
described
requires
a nonfuzzy
stage.
above
is a fuzzy
value.
that best
methods
for performing
procedure membership
This
method
performance
function
establishes
defuzzification. method
used (Hill,
and
is discussed
than
the latter
because
using
one of the inference
set of controls.
However,
a method function.
is needed There
the max-procedure below. method
It yields, (Lee,
to pick
are several that are used
defuzzification in general, 1990).
not consider
and Teichrow,
a process
for a defuzzification
Two of the methods
it does
Horstkotte,
the need
value,
the membership
of gravity
is hardly
obtained
at a crisp output
represents
The former
steady-state
of fuzzy
Method
a value
method.
description
in the next chapter.
In order to arrive
are the center
to be
is called defuzzification
A more detailed
The output of the fuzzy controller methods
set needs
the
1990).
better
The maxshape
of the
32 2.4.3
Center
Of Gravity
Method
The center of gravity method picks the output value corresponding
to
el,
the center of gravity of the output membership an output.
In other words,
of the summed the IF_THEN
area, which
in this method is contributed
the action is given by the center by the inputs
(the premise
system
is the control
input
u.
set is denoted by O then the control input u, determined
the centroid control
input,
part of
rule).
The output of the fuzzy fuzzy
function as the crisp value for
to the axis corresponding
to the universe
If the output by projecting
of discourse
of the
is given by
44J
f
xO(z)dz
--
,
fo(_)
(2 26)
dx
whenever
foc
_r) d'r ,
0,
(2.27)
33 These
integrals
membership numerical more
function
than
output fuzzy
in our applications
of the integrals
makes
the max method.
the integrals
As described
defined
since
is non zero only over a f'lnite range
evaluation
complex
evaluate
are well
the center
Fommately,
the output fuzzy of values.
of gravity
The
method
it is not necessary
to
at all. in the previous
section,
the membership
function
of the
set is given by
n
n
(2.28)
where
C corresponds
of the
control
sections). directly using
to one of the fuzzy values
input
It is possible related
(Kosko,
linguistic
variable
in the term set {C1,
(denoted
to show that the centroid
to the centroids
of C i.
The control
by
...,
Cn3
I_,3 in the previous
of the output input
fuzzy
set is
can be evaluated
1992)
n
E
u--
w:t c:t I i
i--I n
}
(2.29)
34 where rule
n is the number that
has
corresponding
of fuzzy
fired,
wi =
to the centroid
min(Ai(xl),
with respect
is simply
the value
of the control
is easily
computed.
In addition,
then
Therefore,
I = _ for the control
all
fired, and for the ith fuzzy
Bi(x2)),
ci is the
of C, and _ is the area under
sets are symmetric
same
rules that have
to a vertical
line passing
control
C i. If the fuzzy
through
input on the axis of symmetry. if the area under
i and
it can
input is simply
be
each fuzzy
canceled
input
from
it, then ci This value
set Ci is the the
equation.
given by
wl Ci
u -
(2.30)
n
If the fuzzy sets Ci are unimodal in triangular fuzzy
or trapezoidal
set is the same,
with peak belief values
membership
then the control
functions)
over its centroid
and the area under
input is also given
(as each
by
n
ciO (c±) u = i=I rl
_. O(c±)
(2.31)
35 2.5 Example Suppose
we have a system
inputs
are error and derror
input
of the plant.
normalized
(the change-of-error)
The fuzzy
with the universe
for inputs
and output
of discourse
-1 and a change-of-error are the following
control
R 3 :
IF error THEN
and the output
In addition,
are all
the term sets
a crisp output,
In addition,
suppose
consider
an error of
that the rules that get
control
derror
derror
is PS
change
is Z
input is Z;
is NS AND control
change
input is NS;
R 2 : IF error is Z AND THEN
[-6, 6].
is the control
ones:
R 1 : IF error is Z AND THEN
inputs
two
in this case.
how to obtain of 1.75.
The
and one output.
and the output
sets for both
are identical
In order to illustrate
fired
with two inputs
input
derror
change
is NS
change
is Z
is PS;
and R4 : IF error is NS AND derror THEN Figures
control
input is PB.
2.10 and 2.11 illustrate
the fuzzification
of the controller
inputs,
the
36 max-dot inference method, and the center of gravity Figure
defuzzification
method.
2.10 show the fuzzy sets Z and NS for error, PS, Z, and NS for error t
change,
and the fuzzy
additional
lines
inference
method.
illustrate
0.5, w 2 = 0.25, resulting
(2.25).
solved
membership
given
Equation
the
respectively.
function
output
used
input.
in the
Figure
contributed can
by
for the control membership
input
by w_ =
2.11 shows the
be obtained
the
rules.
using
The
Equation
corresponds
function.
The
max-dot
for the four rules are given
functions
value
scaling
to the
It can be easily
(2.30):.
[0.5x(-2)+O.25xO+OxO+O.25x4] (0.5+0.25+0+0.25)
the centroids
(2.32)
=0.0,
of the NS, Z, PS, and PB membership
functions
are
by ci = -2, c2 = 0, c3 = 2, and c4 = 4, respectively. These
calculation, input.
crisp
of the combined
I=
where
factors
membership
The resulting
using
to fred
The scaling
output
centroid
how
w 3 -- 0, and w4 = 0.25,
scaled
combined
sets NS, Z, PS, and PB for the control
These
calculations each
can be implemented
error and error
results
such a way that given
change
can then be stored an error and error
on a computer.
will give
a corresponding
in the form change
After
of a look-up
we can then
each
control table
in
look up the
37 control input.
An example of a look up-table is given in Table 2.1.
Error
Error change -4 I -3
-2 ]-1
]
0 ]
1
2 I
3 I
4
-4
5
4
4
3
3
2
1
1
-1
-3
5
4
3
2
2
1
0
0
-2
-2
4
3
3
2
1
1
0
-1
-3
-I
4
3
2
1
1
0
1
-2
-3
0
3
3
2
1
0
-1
-2
-3
-3
1
3
2
1
0
-1
-1
-2
-3
-4
2
3
1
0
-1
-1
-2
-3
-4
-4
3
2
0
0
-1
-2
-2
-3
-4
-5
4
1
-1
-1
-1
-2
-3
-4
-4
-5
Table 2.1. Sample look-up table. The following
procedure shows how a control input is determined from the
look up table. • Suppose
the set point
= 1.
• Output
of system
at t_ = 4.
• Output
of system
at t2 = 2.
• Erroratt_=4• Erroratt
2=2-1
1=3. = 1.
38 • Error change = 1 - 3 -- -2. From
the look-up
table we can find the control
t2 to be 1. For errors and error changes interpolation
will be necessary
the simplicity use
of determining
of a computer
implement
makes
in real time.
input to the process
that do not appear
to find the control the control the
in the table, linear
input. This example
input to a process.
calculation
process
at time
simple
shows
Moreover, and
easy
the to
39
tm
\ |
O Rule one
2
2
4
-2
,/
\ 0
-2
-4
-2
2
-!
I
-2
I
-4
2
-2
0
-2
2.10.
1
Graphical
2
0
_lt
!
_
7',,
-1
Figure
2
I
I
I Iide
1
-4
2\ 1
I
-4
0
f_,r
representation
of
control
rules.
4O
I_oership
function
1.0 Contributed
by rule 1
_.75 Corrtr ibuted by rule 2 0.50
/
-4
-2
0
Cor_ibut_ byrole4
2
4
6
Universe of discourse
Figure
2.11.
Determination of the control input means of center of gravity method.
by
41 2.6 Conclusions In this chapter,
we presented
an overview
of fuzzy sets and fuzzy logic Q
as they
apply
operations
to control
on fuzzy
introduced defuzzification
and
the
logic
It is expected researchers
design.
analytically
concepts
of
Moreover,
and a rule
we
graphically. base,
inference
illustrated
Fuzzy
logic
methods,
the was and
were illustrated.
In conclusion, fuzzy
sets
systems
this chapter
served
and how they can be applied to be a useful new to the field.
research
as an introduction to design
guide, together
fuzzy
to fuzzy logic
sets,
controllers.
with the references,
to
CHAPTER FUZZY
LOGIC
3
CONTROLLERS
3.1 Introduction A
fuzzy
logic
controller
is
generalization
of the expert systems
applications.
The main difference
expert
systems
system,
is in the way
uncertainty
expert
system
linguistic
variables
logic controller knowledge
engine
Initially
fuzzy
fuzzy
expert
a probabilistic
which
systems
is embedded approximate
approach.
interpretation
operator
is a (AI)
and AI
In an AI expert A fuzzy
in the way humans
In fact, one
a human
operator
system
uncertainty.
uncertainty
and the defuzzifier
to a given
handle
sets.
is that it models
of the human
inference operator
and fuzzy
between
using
to handle
expert
widely used in artificial intelligence
they
is handled
attempts
a fuzzy
of a control in the fuzzy the response
do, using of a fuzzy
process.
The
rule base.
The
of the human
set of inputs. logic
applications
characterized
mathematical
models
controllers by
slow
of the process,
were time
applied constants
but reasonably
42
in
process
and
controlled
control
lacking
the
by a human
43 operator. More recently, fuzzy logic controllers have been applied to more typical
electrical
engineering
control
problems
such as motor
control
(Li and
Q
Lau,
1989),
aircraft
(Chiu
mentioned faster
these
offered
and proportional proportional
(MRAC)
or in many The
includes
goal
new
cases better
guidelines
of fuzzy
up with
logic
model techniques
1992).
for the tuning
controllers.
single-output to multivariable
Furthermore,
fuzzy
logic
integral
reference
all of
controllers
derivative adaptive
to prove
control
is to present
examples
All of the
to single-input,
between
of
can be used with
have been applied
than classical
roll control
applications.
(PI), proportional
control
1992),
controllers
(Heisemer,
(PD),
of this chapter
will be followed
robustness
integral
classical
general
logic
comparisons
derivative
and other
more
In addition
such as a turbo fan engine
(FLCs),
This
systems.
(Kosko,
and many
fuzzy logic controllers
applications
(PID),
1989)
problem
show that fuzzy
constant
processes,
problems
pendulum
and Chand,
applications
time
(SISO)
good
the inverted
control
that FLCs
are as
techniques.
our design of fuzzy
to illustrate
procedure logic
which
controllers.
the capabilities
and
44 3.2 Fuzzy
Logic
Control
System
A basic block diagram
Design
for fuzzy logic
control
is given in Figure 3.1 Q
where
r(kT) is a sequence
sequence output
of tracking of the plant.
control
systems
of command
errors,
that is useful
The controller
to construct
the change
is a rough diagram
consists
implementation (Langari
of four blocks
signal,
Ae(kT)
= e(kT)
in Figure
1992). The fuzzy
is a
and y(t) is the of fuzzy
to other
control
First, a filter is used
of the rate of change
in Figure
- y(kT)
implementation
and comparison
of the filter is shown
illustrated
= r(kT)
of two main blocks.
of error sequence
and Berenji,
e(kT) input
sampled-data
for analysis
first order approximation
also possible consists
u(t) is the control
This is a typical
techniques.
requests,
- e((k-1)T), of error.
3.2.
A block
Other filters
logic controller
3.7; their
which
design
are
(FLC)
is described
next.
A Design
Procedure
Suppose an appropriate typically
that the control sampling
used to design
period fuzzy
configuration
in Figure
T is chosen. logic
controllers.
3.1 is chosen
The following
main
and that steps
are
45
= 5) derror=5; if (derror -0.
X(k)---0 for the system
if there exists a Lyapunov
function
is negative
- V(x) < 0 with AV(x) = 0 only if x = 0.
4. V(x) approaches
stable
- V(x)
positive
is asymptotically
definite
matrix
stable
in the large
P such that
ATi P Ai -P
