Sergei_Astapov_Fuzzy_Control_lecture_slides.pdf

fuzzy control

1. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Fuzzy Logic: Principles and Applications
ISS0023 Intelligent Control Systems
Sergei Astapov
Laboratory for Proactive Technologies
Department of Computer Control
Tallinn University of Technology, Estonia

2. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Self Introduction
Engineer at the Laboratory for Proactive Technologies
(ProLab)
PhD student at the Department of Computer Control
Research topics
Band-limited signal analysis
Signal processing and data mining algorithms
Classication and decision-making algorithms
Room: U02-305
E-mail: [email protected]

3. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Lecture Overview
1. Concepts of Fuzzy Logic
1.1 Introduction
1.2 Fuzzy Sets
1.3 Fuzzy Set Operations
1.4 Membership Functions
1.5 Fuzzy Rules
2. Fuzzy Inference Systems
2.1 Introduction
2.2 Mamdani FIS
2.3 Mamdani Inference
2.4 Defuzzication
2.5 Mamdani FIS Editor
2.6 Takagi-Sugeno FIS
2.7 Sugeno FIS Editor
3. Applications
3.1 Fuzzy PID Controller
3.2 Fuzzy %Anything%
3.3 Adaptive FIS
3.4 Adaptive Control
3.5 Clustering and Classication
3.6 Discussion

4. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Outline
1. Concepts of Fuzzy Logic
1.1 Introduction
1.2 Fuzzy Sets
1.3 Fuzzy Set Operations
1.4 Membership Functions
1.5 Fuzzy Rules
2. Fuzzy Inference Systems
2.1 Introduction
2.2 Mamdani FIS
2.3 Mamdani Inference
2.4 Defuzzication
2.5 Mamdani FIS Editor
2.6 Takagi-Sugeno FIS
2.7 Sugeno FIS Editor
3. Applications
3.1 Fuzzy PID Controller
3.2 Fuzzy %Anything%
3.3 Adaptive FIS
3.4 Adaptive Control
3.5 Clustering and Classication
3.6 Discussion

5. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Why Go Fuzzy?
Fuzzy logic models human expertise and knowledge in some
task or application
Consider conventional binary logic
Variables may take values of TRUE or FALSE (0 or 1)
Try then to answer a simple question with binary logic
What do you consider warm temperature?
How to answer?
You could try to give a value or interval of warm temperature
But then when does the temperature become cold or hot?

6. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
The Fuzzy Way of Thinking t (°C)
μ(t)
10 20 30 400-10-20-30-40
1
hotwarm
warm hot
t (°C)
μ(t)
10 20 30 400-10-20-30-40
1
coolchillycoldfreezing
freezing cold chilly cool
Binary logic
Fuzzy logic

7. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
The Concept of Fuzzy Logic
Fuzzy logic variables have a range of truthfulness from 0 to 1
Fuzzy logic operates withlinguistic variables, like
temperature instead oft(

C)
Each variable has a specic number oflinguistic values, like
hot or cold
Fuzzy inference is performed usinglinguistic rules, e.g.
IF temperature is cold THEN dress warm
The linguistic values and their truth degree are quantied
usingmembership functions(MF)

8. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
A Little Bit of History
1964 : Lot A. Zadeh, UC Berkeley, introduced the paper on fuzzy sets
Idea of grade of membership
Imperfection and noise in the real world
Sharp criticism from academic community
19651975 : Zadeh continued to broaden the foundation of fuzzy set theory
Fuzzy multistage decision-making
Fuzzy similarity relations
Fuzzy restrictions, linguistic hedges
1970s : Research was mainly centered in Japan
1974 : E. H. Mamdani, UK, developed the rst fuzzy logic controller
1977 : Dubois applied fuzzy sets in a comprehensive study of trac conditions
19761987 : Industrial application of fuzzy logic in Japan and Europe
1987Present : Widespread application

9. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Outline
1. Concepts of Fuzzy Logic
1.1 Introduction
1.2 Fuzzy Sets
1.3 Fuzzy Set Operations
1.4 Membership Functions
1.5 Fuzzy Rules
2. Fuzzy Inference Systems
2.1 Introduction
2.2 Mamdani FIS
2.3 Mamdani Inference
2.4 Defuzzication
2.5 Mamdani FIS Editor
2.6 Takagi-Sugeno FIS
2.7 Sugeno FIS Editor
3. Applications
3.1 Fuzzy PID Controller
3.2 Fuzzy %Anything%
3.3 Adaptive FIS
3.4 Adaptive Control
3.5 Clustering and Classication
3.6 Discussion

10. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Conventional and Fuzzy Sets
LetXbe a space of objects andxbe a generic element ofX. A
classical setA,AX, is dened as a collection of elements
x2X, such that each elementxcan either belong or not belong
to the setA:
The classical set thus can be characterized asA=fxjx2Xg:
By dening acharacteristic functionfor eachx, we can represent
the classical setAby a set of ordered pairs(x;0)or(x;1), which
indicatex =2Aorx2Arespectively.
In a fuzzy set the characteristic function is allowed to have values
of membership between 0 and 1.

11. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Fuzzy Set Denition
Denition 1 (Fuzzy set)
IfXis a collection of objectsx, then afuzzy setAinXis
dened as a set of ordered pairs:
A=f(x; A(x))jx2Xg; (1)
whereA(x)is called themembership function(MF) for the
fuzzy setA.
In fuzzy set theory classical sets are referred to ascrispsets and
the values as crisp values.
Xis usually referred to as theuniverse of discourse. It represents
the range of values the fuzzy variables may take.
Universes of discourse may be either discrete or continuous.

12. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Linguistic Variables and Values
Fuzzy sets usually carry names appealing in our daily linguistic usage
The universe is called alinguistic variableand its sets are called
linguistic values
The universe of discourseXis partitioned into several fuzzy sets,
with MFs coveringXin a more or less uniform manner
Example 1
Consider the universeXof linguistic variable temperature. The
universe may be dened dierently, depending on the application. We
may set it from the lowest to the highest temperature a typical human
being can live in, e.g.[50;50]

C.
We partition the universe into 6 fuzzy sets: freezing, cold, chilly,
cool, warm, hot. These sets are characterized by MFs
f reezing(x),cold(x),chilly(x),cool(x),warm(x),hot(x).

13. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Relevant Properties of Fuzzy Sets
Denition 2 (Support)
Thesupportof a fuzzy setAis the set of all pointsx2X, such that
A(x)>0:
support(A) =fxjA(x)>0g: (2)
Denition 3 (Core)
Thecoreof a fuzzy setAis the set of all pointsx2X, such that
A(x) = 1:
core(A) =fxjA(x) = 1g: (3)
Denition 4 (Crossover points)
Acrossover pointof a fuzzy setAis a pointx2X, at which
A(x) = 0:5:
crossover(A) =fxjA(x) = 0:5g: (4)

14. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Relevant Properties of Fuzzy Sets Continued
Denition 5 (Normality)
A fuzzy setAisnormalif its core is nonempty, i.e. we can always
nd a pointx2X, such thatA(x) = 1.
Denition 6 (Fuzzy singleton)
A fuzzy set, the support of which is a single point inXwith
A(x) = 1is called afuzzy singleton.
Denition 7 (Symmetry)
A fuzzy setAissymmetricif its MF is symmetric around a
certain pointx=c, namely,A(c+x) =A(cx),8x2X.

15. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Relevant Properties of Fuzzy Sets Continued
Denition 8 (Open left, open right, closed sets)
A fuzzy setAis:
open leftiflimx!1A(x) = 1,limx!+1A(x) = 0;
open rightiflimx!1A(x) = 0,limx!+1A(x) = 1;
andclosediflimx!1A(x) = limx!+1A(x) = 0.warm hot
x
μ(x)
10 20 30 400-10-20-30-40
1
0.5
freezing cold chilly cool
core
crossover points
support
all are normal
symmetric
open left open right
temperature is 25 °C
singleton

16. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Outline
1. Concepts of Fuzzy Logic
1.1 Introduction
1.2 Fuzzy Sets
1.3 Fuzzy Set Operations
1.4 Membership Functions
1.5 Fuzzy Rules
2. Fuzzy Inference Systems
2.1 Introduction
2.2 Mamdani FIS
2.3 Mamdani Inference
2.4 Defuzzication
2.5 Mamdani FIS Editor
2.6 Takagi-Sugeno FIS
2.7 Sugeno FIS Editor
3. Applications
3.1 Fuzzy PID Controller
3.2 Fuzzy %Anything%
3.3 Adaptive FIS
3.4 Adaptive Control
3.5 Clustering and Classication
3.6 Discussion

17. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Containment
Denition 9 (Containment or subset)
Fuzzy setAiscontainedin fuzzy setB(orAis asubsetofB),
iA(x)B(x)for allx:
AB()A(x)B(x): (5)x
μ(x)
1
A
B

18. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Complement
Denition 10 (Complement or negation)
Thecomplementof a fuzzy setA, denoted byAor:A, or
NOTAis dened as

A
(x) = 1A(x): (6)x
μ(x)
1
A
x
μ(x)
1
NOT A

19. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Union
Denition 11 (Union or disjunction)
Theunionof two fuzzy setsAandBis a fuzzy setC, written as
C=A[BorC=AORB, the MF of which is related to those
ofAandBby
C(x) = max (A(x); B(x)) =A(x)_B(x): (7)x
μ(x)
1
A B
x
μ(x)
1
A OR B

20. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Intersection
Denition 12 (Intersection or conjunction)
Theintersectionof two fuzzy setsAandBis a fuzzy setC,
written asC=A\BorC=AANDB, the MF of which is
related to those ofAandBby
C(x) = min (A(x); B(x)) =A(x)^B(x): (8)x
μ(x)
1
A B
x
μ(x)
1
A AND B

21. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Cartesian Product and Co-product
Denition 13 (Cartesian product and co-product)
LetAandBbe fuzzy sets inXandY, respectively. The
Cartesian productofAandB, denoted byAB, is a fuzzy set
in the product spaceXYwith the membership function
AB(x; y) = min (A(x); B(y)): (9)
Similarly, the Cartesian co-productA+Bis a fuzzy set with the
membership function
A+B(x; y) = max (A(x); B(y)): (10)

22. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Outline
1. Concepts of Fuzzy Logic
1.1 Introduction
1.2 Fuzzy Sets
1.3 Fuzzy Set Operations
1.4 Membership Functions
1.5 Fuzzy Rules
2. Fuzzy Inference Systems
2.1 Introduction
2.2 Mamdani FIS
2.3 Mamdani Inference
2.4 Defuzzication
2.5 Mamdani FIS Editor
2.6 Takagi-Sugeno FIS
2.7 Sugeno FIS Editor
3. Applications
3.1 Fuzzy PID Controller
3.2 Fuzzy %Anything%
3.3 Adaptive FIS
3.4 Adaptive Control
3.5 Clustering and Classication
3.6 Discussion

23. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Preface
A fuzzy set is completely characterized by its MF
As the universe most often consists of real values,XR, it
is convenient to dene MFs as continuous functions
For a single linguistic variable the MFs are one-dimensional
Combining the universes of dierent linguistic variables, MFs
of higher dimensions may be derived
Here the most commonly applied MF types are presented

24. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Straight-Line MF:Triangular MF
Denition 14 (Triangular MF)
Atriangular MFis specied by three parametersfa; b; cgas follows:
triangle (x;a; b; c) =
8
>
>
>
<
>
>
>
:
0; xa:
xa
ba
; axb:
cx
cb
; bxc:
0; cx:
(11)
It may also be described byminandmaxas
triangle (x;a; b; c) = max

min

xa
ba
;
cx
cb

;0

:(12)
The parametersaandclocate the feet of the triangle andb its
peak.

25. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Straight-Line MF: Trapezoidal MF
Denition 15 (Trapezoidal MF)
Atrapezoidal MFis specied by four parametersfa; b; c; dgas follows:
trapezoid (x;a; b; c; d) =
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
0; xa:
xa
ba
; axb:
1; bxc:
dx
dc
; cxd:
0; dx:
(13)
An alternative expression usingminandmaxis
trapezoid (x;a; b; c; d) = max

min

xa
ba
;1;
dx
dc

;0

:(14)
The parametersaanddlocate the feet of the trapezoid andbandc
its shoulders.

26. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Smooth MF: Gaussian and Bell MF
Denition 16 (Gaussian MF)
AGaussian MFis specied by two parametersfc; gas follows:
gaussian (x;c; ) =e

1
2(
xc
)
2
: (15)
The parametercrepresents the MF center anddetermines the
MF width.
Denition 17 (Generalized bell MF)
Ageneralized bell MFis specied by three parametersfa; b; cg
as follows:
bell (x;a; b; c) =
1
1 +

xc
a


2b
; (16)
wherebis usually positive (ifb <0;then the MF becomes an
upside-down bell).

27. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
MATLAB MF Examples 0 20 40 60 80 100
0
0.2
0.4
0.6
0.8
1
Membership Grades
(a) Triangular MF: trimf(x,[20,60,80])
0 20 40 60 80 100
0
0,2
0,4
0,6
0,8
1
Membership Grades
(b) Trapezoidal MF: trapmf(x,[10,20,60,95])
0 20 40 60 80 100
0
0.2
0.4
0.6
0.8
1
Membership Grades
(c) Gaussian MF: gaussmf(x,[20,50])
0 20 40 60 80 100
0
0.2
0.4
0.6
0.8
1
Membership Grades
(d) Generalized Bell MF: gbellmf(x,[20,4,50])

28. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Changing the Parameters of Bell MF -10 -5 0 5 10
0
0.2
0.4
0.6
0.8
1
(a) Changing 'a'
-10 -5 0 5 10
0
0.2
0.4
0.6
0.8
1
(b) Changing 'b'
-10 -5 0 5 10
0
0,2
0,4
0,6
0,8
1
(c) Changing 'c'
-10 -5 0 5 10
0
0.2
0.4
0.6
0.8
1
(d) Changing 'a' and 'b'

29. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Straight-line and Smooth MFs: Analysis
What are the advantages and drawbacks of straight-line and
smooth MFs?
Straight-line MFs
Simple formulas: computational eciency
Zero points strictly dened:
Good, when boundary strictness is needed
Bad, when fuzzy sets cannot be adequately characterized by
sudden drops to zero membership
Limitations due to linearity
Simple for manual tuning, unsuited for automated tuning
Smooth MFs:
Non-linear: higher exibility
Best for automated tuning (adaptive systems)
Less straight-forward: more problems during initial design

30. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Open Membership Functions
Denition 18 (Sigmoidal MF)
Asigmoidal MFis specied by two parametersfa; cgas follows:
sig (x;a; c) =
1
1 +e
a(xc)
; (17)
whereacontrols the slope of the crossover pointc.
An open triangular MF is obtained by specifyinginfas a left or right
foot parameter, e.g.trimf(x,[3,7,inf])-5 0 5 10 15
0
0.2
0.4
0.6
0.8
1
Membership Grades
(a) Sigmoidal MF: sigmf(x,[1,5])
-5 0 5 10 15
0
0.2
0.4
0.6
0.8
1
Membership Grades
(b) Triangular MF: trimf(x,[3,7,inf])

31. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Asymmetric Membership Functions
There are numerous ways to get asymmetric smooth MFs. One way
is taking the dierencejy1y2jand producty1y2of sigmoid MFs:-10 -5 0 5 10
0
0.2
0.4
0.6
0.8
1
y1
y2
(a) y1 = sig(x;1,-5); y2 = sig(x;2,5)
-10 -5 0 5 10
0
0.2
0.4
0.6
0.8
1
(b) |y1 - y2|
-10 -5 0 5 10
0
0.2
0.4
0.6
0.8
1
y1 y3
(c) y1 = sig(x;1,-5); y3 = sig(x;-2,5)
-10 -5 0 5 10
0
0.2
0.4
0.6
0.8
1
(d) y1*y3

32. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Asymmetric MF: Left-Right MF
Denition 19 (Left-right MF)
Aleft-right MFis specied by three parametersf; ; cgas
LR (x;; ; c) =
(
FL

cx


; xc;
FR

xc


; xc;
(18)
whereFL(x)andFR(x)are monotonically decreasing functions
dened on[0;1)withFL(0) =FR(0) = 1and
limx!1FL(x) = limx!1FR(x) = 0.

33. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Asymmetric MF: Left-Right MF Example
Example 2
LetFL(x) =
p
max (0;1x
2
),FR=e
jxj
3
. Then applying (18)
we can generate dierent curves, e.g. (a)lr_mf(x,60,10,65);
and (b)lr_mf(x,10,40,25);0 20 40 60 80 100
0
0.2
0.4
0.6
0.8
1
Membership Grades
(a)
0 20 40 60 80 100
0
0.2
0.4
0.6
0.8
1
Membership Grades
(b)

34. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Asymmetric MF: Two-Sided Gaussian MF
Denition 20 (Two-sided Gaussian MF)
Atwo-sided Gaussian MFis dened by four parametersfc1; 1; c2; 2g
as
gaussian2 (x;c1; 1; c2; 2) =
8
>
>
<
>
>
:
exp
h

1
2

xc1
1
i
; xc1;
1; c 1< xc2;
exp
h

1
2

xc2
2
i
; c2x;
(19)
wherec1; 1are the parameters of the left-most curve andc2; 2are the
parameters of the right-most curve.
The two-sided Gaussian is essentially a mixture of two Gaussian
functions dened by (15). It is computed in MATLAB using the
gauss2mffunction.

35. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Remarks on Membership Functions
The presented MFs are only the most common ones
For a full glossary of available MFs refer to the MATLAB
Fuzzy Toolbox manual and other sources
Be creative! Nobody forbids you from inventing your own MFs
Non-normality and other properties of MFs can be achieved
by mathematical manipulations on existing MFs or by dening
one's own MFs
Two-dimensional MFs are not discussed here, for further study
please refer to literature

36. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Outline
1. Concepts of Fuzzy Logic
1.1 Introduction
1.2 Fuzzy Sets
1.3 Fuzzy Set Operations
1.4 Membership Functions
1.5 Fuzzy Rules
2. Fuzzy Inference Systems
2.1 Introduction
2.2 Mamdani FIS
2.3 Mamdani Inference
2.4 Defuzzication
2.5 Mamdani FIS Editor
2.6 Takagi-Sugeno FIS
2.7 Sugeno FIS Editor
3. Applications
3.1 Fuzzy PID Controller
3.2 Fuzzy %Anything%
3.3 Adaptive FIS
3.4 Adaptive Control
3.5 Clustering and Classication
3.6 Discussion

37. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Fuzzy IF-THEN Rules
Denition 21 (Fuzzy if-then rule)
Afuzzy if-then rule, also known as afuzzy rule,fuzzy implication, or
fuzzy conditional statement, assumes the form
IFxisATHENyisB; (20)
whereAandBare linguistic values dened by fuzzy sets on universes of
discourseXandY, respectively. The expressionxisAis called the
antecedentorpremise, whileyisBis called theconsequenceor
conclusion.
Expression (20), which is abbreviated asA!B, can be dened as a binary
fuzzy relationRon the product spaceXY:R=A!B.Rcan be viewed
as a fuzzy set of two-dimensional MF
R(x; y) =f(A(x); B(y));
where the functionfis called the fuzzy implication function, that transforms
the membership degrees ofxinAandyinBinto those of(x; y)inA!B.

38. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Multiple Input Multiple Output Rules
Let premise linguistic variablesxi,i= 1;2; : : : ; nand consequence
linguistic variablesyj,j= 1;2; : : : ; mtake on values of their universes of
discourseXiandYj, respectively. Letxibe characterized by a set of
linguistic values
Ai=

A
k
i:k= 1;2; : : : ; Ni

;
andyjbe characterized by a set of linguistic values
Bj=

B
l
j:l= 1;2; : : : ; Mi

:
Then a MIMO rule with number of inputsnand number of outputsm
can be written as
IFx1isA
p
1
ANDx2isA
q
2
AND: : :ANDxnisA
r
n
THENy1isB
s
1ANDy2isB
u
2AND: : :ANDymisB
v
m:
(21)

39. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Linguistic Operators
A large number of operators may be applied to linguistic
terms in fuzzy rules
Negation, e.g. notwarm
Connectives:and,or,either,neither, etc.
Hedges:too,very,more or less,quite,extremely, etc.
For example more or lesswarmbut not toowarm
Here onlynot,and,oroperators are discussed as they are
most common and sucient in the majority of applications
In practice we will use only theandoperator

40. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Outline
1. Concepts of Fuzzy Logic
1.1 Introduction
1.2 Fuzzy Sets
1.3 Fuzzy Set Operations
1.4 Membership Functions
1.5 Fuzzy Rules
2. Fuzzy Inference Systems
2.1 Introduction
2.2 Mamdani FIS
2.3 Mamdani Inference
2.4 Defuzzication
2.5 Mamdani FIS Editor
2.6 Takagi-Sugeno FIS
2.7 Sugeno FIS Editor
3. Applications
3.1 Fuzzy PID Controller
3.2 Fuzzy %Anything%
3.3 Adaptive FIS
3.4 Adaptive Control
3.5 Clustering and Classication
3.6 Discussion

41. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Fuzzy Inference Systems
Afuzzy inference system(FIS) or, as it is also known in dierent
application areas,fuzzy expert system,fuzzy model,fuzzy
associative memoryandfuzzy logic controller(FLC), is a
computing framework based on the concepts of fuzzy theory, fuzzy
if-then rules and fuzzy reasoning.
FIS have many application areas
Automatic control and robotics
Classication and clustering
Pattern recognition
Decision analysis and expert systems

42. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Generic FIS Structure Defuzzification
Fuzzification
Inference
mechanism
Rule-base
...
...
x
1
x
2
x
n
y
1
y
2
y
n
Crisp
inputs
Crisp
outputs
Fuzzified
inputs
Fuzzified
conclusions
Fuzzication: transformation of crisp values to fuzzy sets
Rule-base: contains a selection of fuzzy rules
Inference mechanism: performs a certain inference procedure
upon the rules and derives a conclusion
Defuzzication: transformation of output fuzzy sets to crisp
values

43. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
What Will Be Discussed Defuzzification
Fuzzification
Inference
mechanism
Rule-base
Reference input
r(t)
Process
Inputs
u(t)
Outputs
y(t)
Fuzzy logic controller
We investigate two most common FIS types:
Mamdani and Takagi-Sugeno fuzzy models
An example of a fuzzy control system is provided along the
coarse of investigation

44. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Controlled Process: Inverted Pendulum θ
F
x
r(t)
Inverted
pendulum
u(t) y(t)
Fuzzy logic
controllerd
dt
Σ
-
+
r(t) — reference θ angle
u(t) — force (N)
y(t) — θ angle (rad)
e(t) = r(t) − y(t)
e(t)

45. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Outline
1. Concepts of Fuzzy Logic
1.1 Introduction
1.2 Fuzzy Sets
1.3 Fuzzy Set Operations
1.4 Membership Functions
1.5 Fuzzy Rules
2. Fuzzy Inference Systems
2.1 Introduction
2.2 Mamdani FIS
2.3 Mamdani Inference
2.4 Defuzzication
2.5 Mamdani FIS Editor
2.6 Takagi-Sugeno FIS
2.7 Sugeno FIS Editor
3. Applications
3.1 Fuzzy PID Controller
3.2 Fuzzy %Anything%
3.3 Adaptive FIS
3.4 Adaptive Control
3.5 Clustering and Classication
3.6 Discussion

46. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Mamdani Type FIS
Was proposed as the rst attempt to control a steam engine
and boiler combination by a set of linguistic control rules
obtained from experienced human operators
The most straight-forward cognitive approach to transferring
knowledge into fuzzy models
Design steps
Choose controller inputs and outputs (linguistic variables)
Assign linguistic values to every variable
Derive control rules for every possible scenario
Choose proper MF for every linguistic value
Specify the parameters of the inference mechanism
Test, observe behavior, tune

47. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
FIS Linguistic Variables and Values
For our inverted pendulum example we choose the following
inputs and outputs:
error describese(t) =r(t)y(t)
change-in-error describes
d
dt
e(t)
force describesu(t)
The linguistic variables take on the following values:
negative large or neglarge, represented by -2
negative small or negsmall, represented by -1
zero, represented by 0
positive small or possmall, represented by 1
positive large or poslarge, represented by 2

48. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Fuzzy Rules
Recall the general MIMO rule structure (15). Substituting mathematical
characters with our assigned linguistic labels and values, we get rules of
the following structure:
(a) IF error is neglarge AND change-in-error is neglarge THEN force is poslarge
(b) IF error is zero AND change-in-error is possmall THEN force is negsmall
(c) IF error is poslarge AND change-in-error is negsmall THEN force is negsmallF F F
(a) (b) (c)

49. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Fuzzy Rule-Base
The number of rules for a MISO FIS is at most
Q
n
i=1
Ni, whereNi
is the number of linguistic values for thei-th linguistic premise
variable. (All possible combinations of premise linguistic values.)
In our case the number of rules is equal to55 = 25.
Continuing the logic of the previous three rule cases, we can derive
the rule-base, presented as a table.
force change-in-error
-2-1012
error
-222210
-12210-1
0 210-1-2
1 10-1-2-2
2 0-1-2-2-2

50. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Membership Functions e(t) (rad)π/40-π/4-π/2
zeronegsmallneglarge possmall poslarge
π/2
de(t)/dt (rad/s)
zeronegsmallneglarge possmall poslarge
u(t) (N)10 200-10-20
zeronegsmallneglarge possmall poslarge
-30 30
π/80-π/8-π/4 π/4

51. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Fuzzication
Singleton fuzzication: apply a fuzzy singleton
fuz
Ai
(x)to the
premise variable universe, perform intersection.
This method is applied when measurement noise is not accounted for the
crisp input values are certain. In Gaussian fuzzication a Gaussian is used as
a fuzzication function, which accounts for inconsistency in the input signal.e(t) (rad)π/40-π/4-π/2
zeronegsmallneglarge possmall poslarge
π/2
de(t)/dt (rad/s)
zeronegsmallneglarge possmall poslarge
π/80-π/8-π/4 π/4
e(t) = -9π/20
de(t)/dt = 9π/80
Crisp inpute(t) =9=20:

\neglarge
(e) = min

neglarge(e);
fuz
1
(e)

=
min(0:75;1) = 0:75;

\negsmall
(e) = min

negsmall(e);
fuz
1
(e)

=
min(0:25;1) = 0:25;all other zero.
Crisp input_e(t) = 9=80:

dzero( _e) = min

zero( _e);
fuz
2
( _e)

=
min(0:125;1) = 0:125;

\possmall
( _e) = min

possmall( _e);
fuz
2
( _e)

=
min(0:875;1) = 0:875;all other zero.

52. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Outline
1. Concepts of Fuzzy Logic
1.1 Introduction
1.2 Fuzzy Sets
1.3 Fuzzy Set Operations
1.4 Membership Functions
1.5 Fuzzy Rules
2. Fuzzy Inference Systems
2.1 Introduction
2.2 Mamdani FIS
2.3 Mamdani Inference
2.4 Defuzzication
2.5 Mamdani FIS Editor
2.6 Takagi-Sugeno FIS
2.7 Sugeno FIS Editor
3. Applications
3.1 Fuzzy PID Controller
3.2 Fuzzy %Anything%
3.3 Adaptive FIS
3.4 Adaptive Control
3.5 Clustering and Classication
3.6 Discussion

53. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Mamdani Inference Mechanism Steps
Calculate thering strengthfor each rule in the rule-base
Determine whichrules are onusing the ring strengths
Determineimplied fuzzy sets performfuzzy implication
Determineoverall implied fuzzy set performfuzzy
aggregation*
*Performed in case of applying specic types of defuzzication.
If defuzzication uses implied fuzzy sets, the step is not performed.

54. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Firing Strength of a Premise
The ring strength of a rule is the degree of certainty that the rule premise
holds for the given inputs. Its calculation depends on the linguistic operators
used in the structure of a premise.
For any linguistic variablesx1andx2the typical operators are the following:
Fuzzy complement (NOT):
Dened in (6) as
^
A
k
1
(x1) = 1^
A
k
1
(x1)
Fuzzy union (OR):
Dened in (7) as maximum^
A
k
1
[
^
A
l
2
(x1; x2) = max

^
A
k
1
(x1); ^
A
l
2
(x2)

Alternative: algebraic sum
^
A
k
1
[
^
A
l
2
(x1; x2) =^
A
k
1
(x1) +^
A
l
2
(x2)^
A
k
1
(x1)^
A
l
2
(x2)
Fuzzy intersection (AND):
Dened in (8) as minimum^
A
k
1
\
^
A
l
2
(x1; x2) = min

^
A
k
1
(x1); ^
A
l
2
(x2)

Alternative: algebraic product^
A
k
1
\
^
A
l
2
(x1; x2) =^
A
k
1
(x1)^
A
l
2
(x2)

55. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Firing Strength: More Complex Premises
In premises with more complex logic, the ring strength is calculated by
partitioning the premise into simpler terms.
Example 3
The premise
IFx1is
^
A
2
1ANDx2is
^
A
1
2ANDx3is NOT
^
A
5
3ORx4is
^
A
3
4
yields the ring strength
premise(x1; x2; x3; x4) =
max
h
min

^
A
2
1
(x1); ^
A
1
2
(x2);1^
A
5
3
(x3)

; ^
A
3
4
(x4)
i
:
Also there exists an option to use a rule certainty weight. This way, for
thei-th rule, the ring strength is multiplied by the weightwi, which
species how certain we are in this specic rule compared to other rules.
Keep in mind that there are more alternatives to AND and OR
operations, you can also specify your custom ones.

56. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Which Rules Are On
The rule is considered being on if its premise is non-zero:
premise(x1; x2; : : : ; xn)>0
An optional step, that reduces the number of computations
Alternatively, perform fuzzy implication over the whole
rule-base, but you will be doing a large number of operations
over zero values
Example 4
Consider a FIS with 3 inputs and 10 MFs per input. The number of rules
is then at most10
3
= 1000. With the universes partitioned by so many
rules, the number of on rules at any given time will be quite small. If
for example10rules are on, then mark those rules and perform later
steps with10sets of parameters, instead of using the whole rule-base and
performing100times more computations, mainly with zeros.

57. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Implied Fuzzy Sets: Fuzzy Implication
The implied fuzzy set of an outputyjfor a rulei, which has a
consequentB
k
j
, and a premise degree of membership equal to

premise(i)(x1; x2; : : : ; xn), is characterized by
^
B
k
j
(yj) = min


premise(i)(x1; x2; : : : ; xn);
B
k
j
(yj)

:
Alternatively the algebraic product can be dened as the
implication operation:
^
B
k
j
(yj) =
premise(i)(x1; x2; : : : ; xn)
B
k
j
(yj):
An implied fuzzy set is computed for every rule that is on.

58. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Overall Implied Fuzzy Set: Fuzzy Aggregation
The overall implied fuzzy set
^
Bjof an outputyj, which
incorporates the implied fuzzy sets
n
^
B
k
j
;
^
B
l
j
; : : : ;
^
B
p
j
o
is
characterized by
^
Bj
(yj) = max

^
B
k
j
(yj); ^
B
l
j
(yj); : : : ; ^
B
p
j
(yj)

:
Alternatively the algebraic sum can be dened as the aggregation
operation:
^
Bj
(yj) =^
B
k
j
(yj) +^
B
l
j
(yj) + +^
B
p
j
(yj)
^
B
k
j
(yj)^
B
l
j
(yj): : : ^
B
p
j
(yj):

59. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Mamdani Inference: Example e(t) (rad)π/40-π/4-π/2
zeronegsmallneglarge possmall poslarge
π/2
de(t)/dt (rad/s)
zeronegsmallneglarge possmall poslarge
π/80-π/8-π/4 π/4
e(t) = -9π/20
de(t)/dt = 9π/80
u(t) (N)10 200-10-20
zeronegsmallneglarge possmall poslarge
-30 30
u(t) (N)10 200-10-20
zeronegsmallneglarge possmall poslarge
-30 30
u(t) (N)10 200-10-20
zeronegsmallneglarge possmall poslarge
-30 30
Apply implication (min)
Apply AND (min)
u(t) (N)10 200-10 30
Apply aggregation (max)

60. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Mamdani Inference: Example Computations
From the fuzzication stage we have established that we have four fuzzy values:

\neglarge
(e) = 0:75;
\negsmall
(e) = 0:25;
dzero( _e) = 0:125;

\possmall
( _e) = 0:875. Thus the rules that are on are:
IF error is neglarge AND change-in-error is zero THEN force is poslarge
IF error is neglarge AND change-in-error is possmall THEN force is possmall
IF error is negsmall AND change-in-error is zero THEN force is possmall
IF error is negsmall AND change-in-error is possmall THEN force is zero
Compute the ring strengths of the four rules usingminfor the AND operator:

premise(1)(e;_e) = min(
\neglarge
(e);
dzero( _e)) = min(0:75;0:125) = 0:125;

premise(2)(e;_e) = min(
\neglarge
(e);
\possmall
( _e)) = min(0:75;0:875) = 0:75;

premise(3)(e;_e) = min(
\negsmall
(e);
dzero( _e)) = min(0:25;0:125) = 0:125;

premise(4)(e;_e) = min(
\negsmall
(e);
\possmall
( _e)) = min(0:25;0:875) = 0:25.

61. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Mamdani Inference: Example Computations Continued
Implied fuzzy sets are derived from the rule premises usingminas:

\poslarge(1)
(u) = min


premise(1)(e;_e); poslarge(u)

=
= min(0:125;1) = 0:125;

\possmall(2)
(u) = min


premise(2)(e;_e); possmall(u)

=
= min(0:75;1) = 0:75;

\possmall(3)
(u) = min


premise(3)(e;_e); possmall(u)

=
= min(0:125;1) = 0:125;

dzero(4)(u) = min


premise(4)(e;_e); zero(u)

= min(0:25;1) = 0:25.
The overall implied fuzzy set is obtained by fuzzy aggregation usingmaxas:

\overall
(u) =
max


\poslarge(1)
(u);
\possmall(2)
(u);
\possmall(3)
(u);
dzero(4)(u)

.

62. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Mamdani Inference: Principle Justication
The choice of linguistic operator functions, fuzzy inference and
fuzzy aggregation operations is based on the assertions that:
We can be no more certain in our premises than we are certain in
our data.
We can be no more certain in our conclusions than we are certain
in our premises.u(t) (N)10 200-10-20
zeronegsmallneglarge possmall poslarge
-30 30 u(t) (N)10 200-10 30
Aggregation (max)Inference (product)

63. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Outline
1. Concepts of Fuzzy Logic
1.1 Introduction
1.2 Fuzzy Sets
1.3 Fuzzy Set Operations
1.4 Membership Functions
1.5 Fuzzy Rules
2. Fuzzy Inference Systems
2.1 Introduction
2.2 Mamdani FIS
2.3 Mamdani Inference
2.4 Defuzzication
2.5 Mamdani FIS Editor
2.6 Takagi-Sugeno FIS
2.7 Sugeno FIS Editor
3. Applications
3.1 Fuzzy PID Controller
3.2 Fuzzy %Anything%
3.3 Adaptive FIS
3.4 Adaptive Control
3.5 Clustering and Classication
3.6 Discussion

64. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Defuzzication
The result of fuzzy inference is the implied fuzzy set (or sets). For
the systems, where a crisp value is required from the FIS, the
operation called defuzzication is applied to the implied sets.
A number of defuzzication strategies exist, and it is not hard to
invent more, suiting your specic application.
Each provides a means to choose a crisp outputy
crisp
j
based on
either the implied fuzzy sets or the overall implied fuzzy set.
Reviewed defuzzication methods:
Center of gravity (COG)
Center-average
Maximum criterion: mean of maximum (MOM), smallest of
maximum (SOM), largest of maximum (LOM)
Center of area (COA)

65. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Defuzzication on IFS: Center of Gravity
Denition 22 (Center of gravity)
InCenter of gravity(COG) defuzzication the outputy
crisp
j
is
computed using the center of area and area of each implied fuzzy set:
y
crisp
j
=
P
R
i=1
b
j
i

Yj
^
B
i
j
(yj) dyj
P
R
i=1

Yj
^
B
i
j
(yj) dyj
; (22)
whereRis the number or rules,b
j
i
is the center of area of the MF ofB
p
j
associated with the implied fuzzy set
^
B
i
j
for thei-th rule and

Yj
^
B
i
j
(yj) dyj
denotes the area under^
B
i
j
(yj).

66. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Defuzzication on IFS: Center of Gravity
The COG is easy to compute if you have simple areas under
implied fuzzy set MFs, e.g. triangles with tops chopped o while
using triangle MFs andminfor implication.
Notice though, that for this method to be reliable the fuzzy system
must be dened such that
R
X
i=1

Yj
^
B
i
j
(yj) dyj6= 0
for allxi. This is achieved if for every possible combination of
inputs the consequent fuzzy sets all have nonzero area.
Also areas must be computable, thus we cannot useopenMFs for
output fuzzy sets.

67. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Defuzzication on IFS: Center-Average
Denition 23 (Center-average)
InCenter-averagedefuzzication the outputy
crisp
j
is computed using
the centers of each of the output MFs and the maximum certainty of
each of the implied fuzzy sets:
y
crisp
j
=
P
R
i=1
b
j
i
sup
yj
n
^
B
i
j
(yj)
o
P
R
i=1
sup
yj
n
^
B
i
j
(yj)
o; (23)
wheresup
yj
denotes the supremum (i.e. the least upper bound) of the
implied fuzzy set^
B
i
j
(yj).

68. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Defuzzication on IFS: Center-Average
The center-average is easy to compute if implied fuzzy set MFs
have a single maximum, e.g. reduced triangles while using triangle
MFs and product for implication in this case
sup
yj
n
^
B
i
j
(yj)
o
= max

^
B
i
j
(yj)

:
Notice though, that the fuzzy system must be dened such that
R
X
i=1
sup
yj
n
^
B
i
j
(yj)
o
6= 0
for allxi. This is achieved as in the case of COG.

69. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Defuzzication on IFS: Center-Average
If usingnormalMFs for output fuzzy sets, then for many inference
strategies we have
sup
yj
n
^
B
i
j
(yj)
o
=
premise(i)(x1; x2; : : : ; xn);
which is the ring strength of rulei. The formula for
defuzzication is then given by
y
crisp
j
=
P
R
i=1
b
j
i

premise(i)(x1; x2; : : : ; xn)
P
R
i=1

premise(i)(x1; x2; : : : ; xn)
; (24)
where
P
R
i=1

premise(i)(x1; x2; : : : ; xn)6= 0;8ximust be ensured.
The shape of the output MFs does not matter, as bounds of
supremum subsets can be dened using singletons.

70. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Defuzzication on The Overall IFS: Maximum Criterion
For the MOM, SOM and LOM defuzzication the crisp output is
chosen as a point on the output universeYj, for which the overall
implied fuzzy set
^
Bjreaches its maximum:
y
crisp
j
2
(
arg sup
Yj
n
^
Bj
(yj)
o
)
:
MOM, SOM and LOM dier in the strategy of choosing the crisp
value from this subset.u(t) (N)10 200-10 30
supremum
MOM
LOM
SOM

71. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Defuzzication on The Overall IFS: MOM
Denition 24 (Mean of maximum)
Dene a fuzzy set
^
B

j
Yjwith a MF dened as
^
B

j
(yj) =
(
1; ^
Bj
(yj) = sup
Yj
n
^
Bj
(yj)
o
;
0;otherwise:
Then the crisp output ofmean of maximum(MOM) defuzzication is
dened as
y
crisp
j
=

Yj
yj^
B

j
(yj) dyj

Yj
^
B

j
(yj) dyj
; (25)
where the fuzzy system must be dened so

Yj
^
B

j
(yj) dyj6= 0;8xi.
Notice that if^
B

j
(yj) = 1lies in a single interval
h
y
left
j
; y
right
j
i
Yj,
theny
crisp
j
=

y
left
j
+y
right
j

=2.

72. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Defuzzication on The Overall IFS: SOM and LOM
Denition 25 (Smallest of maximum)
Insmallest of maximum(SOM) defuzzication the outputy
crisp
j
is
computed as the minimal argument of the output universeYj, for which
the the overall implied fuzzy set
^
Bjreaches its maximum:
y
crisp
j
= min
"
arg sup
Yj
n
^
Bj
(yj)
o
#
: (26)
Denition 26 (Largest of maximum)
Inlargest of maximum(LOM) defuzzication the outputy
crisp
j
is
computed as the maximal argument of the output universeYj, for which
the the overall implied fuzzy set
^
Bjreaches its maximum:
y
crisp
j
= max
"
arg sup
Yj
n
^
Bj
(yj)
o
#
: (27)

73. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Defuzzication on The Overall IFS: Center of Area
Denition 27 (Center of area)
Incenter of area(COA) defuzzication the outputy
crisp
j
is computed
over the area of the MF of the overall implied fuzzy set
^
Bjas
y
crisp
j
=

Yj
yj^
Bj
(yj) dyj

Yj
^
Bj
(yj) dyj
; (28)
where the fuzzy system must be dened so

Yj
^
Bj
(yj) dyj6= 0;8xi.
Computationally expensive: overlapping implied fuzzy sets may
result in a overall implied fuzzy set with a sophisticated shape.
Computing the area of such shapes in real-time is not an easy task.

74. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Defuzzication: Example
For our symmetrical triangular MFs the area and center of area of the
implied fuzzy sets are easily calculated. If a symmetric triangle has a
height 1 and base widthw:
The area of a triangle with the top chopped o at heighthis
equal tow

h
h
2
2

The area of a triangle with heighthis equal to
1
2
wh
Herewis the support length of
^
B
i
j
andhis
premise(i)(x1; x2; : : : ; xn).10 200 10 200
min implication product implication
w w
h h

75. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Defuzzication: COG Example
For implication dened bymin:
u
crisp
=
bpl

U

b
pl(1)
(u)du+bps

U

cps(2)(u)du+bps

U

cps(3)(u)du+bz

U

bz(4)(u)du

U

b
pl(1)
(u)du+

U

cps(2)(u)du+

U

cps(3)(u)du+

U

bz(4)(u)du
=
=
(20)(1:1719)+(10)(4:6875)+(10)(1:1719)+(0)(2:1875)
1:1719+4:6875+1:1719+2:1875
=
82:032
9:2188
= 8:90
For implication dened by product:
u
crisp
=
bpl

U

b
pl(1)
(u)du+bps

U

cps(2)(u)du+bps

U

cps(3)(u)du+bz

U

bz(4)(u)du

U

b
pl(1)
(u)du+

U

cps(2)(u)du+

U

cps(3)(u)du+

U

bz(4)(u)du
=
=
(20)(0:625)+(10)(3:75)+(10)(0:625)+(0)(1:25)
0:625+3:75+0:625+1:25
=
56:25
6:25
= 9:0

76. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Defuzzication: Center-Average Example
For implication dened by product:
u
crisp
=
bplsup
u
n

b
pl(1)
(u)
o
+bpssup
uf
cps(2)(u)g+bpssup
uf
cps(3)(u)g+bzsup
uf
bz(4)(u)g
sup
u
n

b
pl(1)
(u)
o
+sup
uf
cps(2)(u)g+sup
uf
cps(3)(u)g+sup
uf
bz(4)(u)g
=
=
(20)(0:125)+(10)(0:75)+(10)(0:125)+(0)(0:25)
0:125+0:75+0:125+0:25
=
11:25
1:25
= 9:0
The supremum of a reduced triangular MF its its single peak which is
equal to
premise(i)(x1; x2; : : : ; xn).

77. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
FIS Input-Output Curve
Portrays the dependency of FIS output on its inputs. MATLAB command:gensurf-1.5
-1
-0.5
0
0.5
1
1.5
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
-15
-10
-5
0
5
10
15
error
errorDot
force

78. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Mamdani Fuzzy Control of Inverted Pendulum 0 2 4 6 8 10 12 14 16 18 20
-0.2
-0.1
0
0.1
0.2
Mamdani fyzzy control of inverted pendulum
Angle (rad)


0 2 4 6 8 10 12 14 16 18 20
-1.5
-1
-0.5
0
0.5
Position (m)


0 2 4 6 8 10 12 14 16 18 20
-20
-10
0
10
20
Time (s)
Force (N)


Position
PositionDot
Theta
ThetaDot
Control influence
Interference

79. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Outline
1. Concepts of Fuzzy Logic
1.1 Introduction
1.2 Fuzzy Sets
1.3 Fuzzy Set Operations
1.4 Membership Functions
1.5 Fuzzy Rules
2. Fuzzy Inference Systems
2.1 Introduction
2.2 Mamdani FIS
2.3 Mamdani Inference
2.4 Defuzzication
2.5 Mamdani FIS Editor
2.6 Takagi-Sugeno FIS
2.7 Sugeno FIS Editor
3. Applications
3.1 Fuzzy PID Controller
3.2 Fuzzy %Anything%
3.3 Adaptive FIS
3.4 Adaptive Control
3.5 Clustering and Classication
3.6 Discussion

80. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
FIS Design in MATLAB: Editor Main

81. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
FIS Design in MATLAB: MF Editor

82. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
FIS Design in MATLAB: Rule Viewer

83. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
FIS Design in MATLAB: Input-Output Curve

84. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Outline
1. Concepts of Fuzzy Logic
1.1 Introduction
1.2 Fuzzy Sets
1.3 Fuzzy Set Operations
1.4 Membership Functions
1.5 Fuzzy Rules
2. Fuzzy Inference Systems
2.1 Introduction
2.2 Mamdani FIS
2.3 Mamdani Inference
2.4 Defuzzication
2.5 Mamdani FIS Editor
2.6 Takagi-Sugeno FIS
2.7 Sugeno FIS Editor
3. Applications
3.1 Fuzzy PID Controller
3.2 Fuzzy %Anything%
3.3 Adaptive FIS
3.4 Adaptive Control
3.5 Clustering and Classication
3.6 Discussion

85. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Takagi-Sugeno FIS
Takagi-Sugeno or simply Sugeno-type FIS has a dierent way of
computing the consequence and defuzzication.
A general Sugeno rule has a form
IFx1isA
k
1ANDx2isA
l
2AND: : :ANDxnisA
p
nTHENzi=fi():
Herez=f()may be any function (even another mapping, like
neural network, or another FIS)
Usuallyzi=fi(x1; x2; : : : ; xn)is used. If this function is a rst
order polynomial, i.e.
zi=anx1+an1x2+ +a1xn+a0;
the inference system is called arst-orderSugeno FIS. Whenfis
a constant, the system is called azero-orderSugeno FIS.

86. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Sugeno Inference Principles
The premises
premise(i)(x1; x2; : : : ; xn)are computed as in the
Mamdani FIS, incorporating fuzzication and linguistic operators.
The defuzzication is usually performed using weighted average:
y
crisp
=
P
R
i=1
zi
premise(i)(x1; x2; : : : ; xn)
P
R
i=1

premise(i)(x1; x2; : : : ; xn)
; (29)
where the fuzzy system is dened so that
P
R
i=1

premise(i)(x1; x2; : : : ; xn)6= 0;8xi.
Thus the Sugeno FIS can be used as a general mapper for a wide
variety of applications.

87. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Sugeno FIS Example
We will not dene the whole rule-base of the inverted pendulum
controller for the Sugeno FIS. Lets specifyzi=fi(e;_e)for the
rules that are on in our example:
z1=5e+ 4 _e+ 3 for the
z2=4e+ 2 _e+ 2 for the
z3=2e+ 1 _e+ 1 for the
z4=0:5e+ 0:5 _e+ 0for the
For the valuese=
9
20
and_e=
9
80
the functions take on values
z1=5


9
20


+ 4

9
80


+ 3 = 11:482
z2=4


9
20


+ 2

9
80


+ 2 = 8:362
z3=2


9
20


+ 1

9
80


+ 1 = 4:181
z4=0:5


9
20


+ 0:5

9
80


+ 0 = 0:884

88. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Sugeno FIS Example
Then the FIS crisp output will be
y
crisp
=
P
4
i=1
zi
premise(i)(e;_e)
P
4
i=1

premise(i)(e;_e)
=
11:4820:125 + 8:3620:75 + 4:1810:125 + 0:8840:25
0:125 + 0:75 + 0:125 + 0:25
= 9:03
Notice, that no implication and aggregation is used. This simplies
the inference process a lot.

89. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Sugeno Input-Output Curve -0.2
-0.1
0
0.1
0.2
0.3
-1
-0.5
0
0.5
1
-20
-15
-10
-5
0
5
10
15
20
in1
in2
out

90. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Sugeno Fuzzy Control of Inverted Pendulum 0 2 4 6 8 10 12 14 16 18 20
-2
-1
0
1
2
Sugeno fyzzy control of inverted pendulum
Position (m)


0 2 4 6 8 10 12 14 16 18 20
-0.4
-0.2
0
0.2
0.4
Angle (rad)


0 2 4 6 8 10 12 14 16 18 20
-10
-5
0
5
10
Time (s)
Force (N)


Ref. position
Act. position
Theta
Control influence

91. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Outline
1. Concepts of Fuzzy Logic
1.1 Introduction
1.2 Fuzzy Sets
1.3 Fuzzy Set Operations
1.4 Membership Functions
1.5 Fuzzy Rules
2. Fuzzy Inference Systems
2.1 Introduction
2.2 Mamdani FIS
2.3 Mamdani Inference
2.4 Defuzzication
2.5 Mamdani FIS Editor
2.6 Takagi-Sugeno FIS
2.7 Sugeno FIS Editor
3. Applications
3.1 Fuzzy PID Controller
3.2 Fuzzy %Anything%
3.3 Adaptive FIS
3.4 Adaptive Control
3.5 Clustering and Classication
3.6 Discussion

92. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
FIS Design in MATLAB: Sugeno FIS Editor

93. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
FIS Design in MATLAB: Sugeno Rules

94. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
FIS Tuning
As was mentioned, the testing and tuning is the last step of FIS
development. If testing fails, the FIS has to be tuned or even redesigned.r(t)
Process
u(t) y(t)
Fuzzy logic
controllerd
dt
Σ
-
+
g
1
g
2
h
External FIS tuning is performed via input and output scaling gains. The
gain values may be either constant or functions of some sort, e.g. bell or
Gaussian functions.
Internal tuning is performed by reviewing the membership functions and
the rule-base. Trying out dierent inference and defuzzication
operations is also a good practice.
The MATLAB FIS editor is a good tool for debugging. There you can
observe the reaction of your rule ring strengths, input-output curves,
etc. to the changes you make.

95. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Outline
1. Concepts of Fuzzy Logic
1.1 Introduction
1.2 Fuzzy Sets
1.3 Fuzzy Set Operations
1.4 Membership Functions
1.5 Fuzzy Rules
2. Fuzzy Inference Systems
2.1 Introduction
2.2 Mamdani FIS
2.3 Mamdani Inference
2.4 Defuzzication
2.5 Mamdani FIS Editor
2.6 Takagi-Sugeno FIS
2.7 Sugeno FIS Editor
3. Applications
3.1 Fuzzy PID Controller
3.2 Fuzzy %Anything%
3.3 Adaptive FIS
3.4 Adaptive Control
3.5 Clustering and Classication
3.6 Discussion

96. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Fuzzy PID Controller
The term Fuzzy PID controller can be understood in two ways:
A fuzzy FIS, which has the inputse(t),
d
dt
e(t),

e(t)dt
A crisp PID controller, theKP,KIandKDcoecients of
which are tuned by a fuzzy expert system
A tunable PID controller allows to:
Increase the robustness of the typical PID controller
Increase its dynamic range
Account for dierent scenarios of system operation

97. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Fuzzy PID Controller Example 1

98. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Fuzzy PID Controller Example 2

99. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Fuzzy PID Control Simulation 0 5 10 15 20 25 30 35 40
0
1
2
3
4
5
Fuzzy PID Control of Tank System


0 5 10 15 20 25 30 35 40
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time (s)


Control influence: PID
Control influence: Fuzzy PID
Set value
Upper limit
Lower limit
Liquid level: PID
Liquid level: Fuzzy PID

100. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Outline
1. Concepts of Fuzzy Logic
1.1 Introduction
1.2 Fuzzy Sets
1.3 Fuzzy Set Operations
1.4 Membership Functions
1.5 Fuzzy Rules
2. Fuzzy Inference Systems
2.1 Introduction
2.2 Mamdani FIS
2.3 Mamdani Inference
2.4 Defuzzication
2.5 Mamdani FIS Editor
2.6 Takagi-Sugeno FIS
2.7 Sugeno FIS Editor
3. Applications
3.1 Fuzzy PID Controller
3.2 Fuzzy %Anything%
3.3 Adaptive FIS
3.4 Adaptive Control
3.5 Clustering and Classication
3.6 Discussion

101. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Making Anything Fuzzy
If you have some time variant system with parameters that
cannot be statically specied...
If you cannot describe parameter variation mathematically but
you intuitively know how they should be changed...
Introduce a fuzzy expert or control system to do it!
We have seen it in the fuzzy PID example
It is generally applicable to any linear or nonlinear dynamic
system model

102. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Fuzzy Predictor
Having a discrete time series1; 2; : : : ; k, the task of a prediction
algorithm is to determine the next values of the given time series
^
k+1;
^
k+2; : : : ;
^
k+l.
The time series possesses certain dynamical propertiesk+1=k+!k,
where!kis the system perturbation of unknown distribution.
The observed value may be aected by external interference
~
k=k+k, wherekis referred to as observation noise.
The Kalman (exponential average) predictor is given by the recurrence
^
k+1=
^
k+ (1)
~
k;
where
^
k+1is the predicted value of the time series,
^
kis the last known
predicted value,
~
kis the last observed value and2[0;1]is the weight
parameter.

103. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Fuzzy Predictor Continued
Basic logic tells us that:
If the system is steady,
~
kinuences prediction more and!0.
On the other hand, if the system is not steady or noisy,
~
kis less
reliable than
^
kand!1.
Specify the FIS input aserrorek=



^
k
~
k


, then develop rules, e.g.
IF error is small THENis large
IF error is medium THENis medium
IF error is large THENis small
And, well, you know the rest.
P.S. Think, how introducing thechange-in-errorinto the FIS will improve
the situation.

104. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
General Linear Dynamic Model
The linear discrete-time dynamic system model takes the form
xk=Ak1xk1+qk1
yk=Hk1xk+rk1
;
wherexkis the system state vector at time stepk,ykis the measurement
vector atk,Ak1is the transition matrix of the dynamic model,Hk1is the
measurement matrix,qk1N (0;Qk1)is the process noise with covariance
Qk1andrk1N (0;Rk1)is the measurement noise with covarianceRk1.
For the majority of applications it is assumed that the noise has xed
variance and a normal distribution
What to do, if noise is time variant and has varying distribution?
One solution is the to develop a fuzzy system, which will estimate noise
parameters and tune the controller, lter, etc. online

105. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Outline
1. Concepts of Fuzzy Logic
1.1 Introduction
1.2 Fuzzy Sets
1.3 Fuzzy Set Operations
1.4 Membership Functions
1.5 Fuzzy Rules
2. Fuzzy Inference Systems
2.1 Introduction
2.2 Mamdani FIS
2.3 Mamdani Inference
2.4 Defuzzication
2.5 Mamdani FIS Editor
2.6 Takagi-Sugeno FIS
2.7 Sugeno FIS Editor
3. Applications
3.1 Fuzzy PID Controller
3.2 Fuzzy %Anything%
3.3 Adaptive FIS
3.4 Adaptive Control
3.5 Clustering and Classication
3.6 Discussion

106. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Adaptive Neuro-Fuzzy Inference System
Adaptive Neuro-Fuzzy Inference System (ANFIS) is a representation of
the Sugeno FIS in a form of a feed-forward neural network.A
11
A
12
A
21
A
22
x
1
x
2
Π
Π
N
N
f
1
f
2
Σ
x
2x
1
x
2x
1
y
Layer 1Layer 2 Layer 3 Layer 4 Layer 5
w
1
w
2
w
2
w
1
f
1w
1
f
2w
2

107. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
ANFIS Architecture
The rst-order Sugeno systemr-th rule takes the form
IFx1isA
k
1ANDx2isA
l
2AND: : :ANDxnisA
p
n
THENfr=pn;rx1+pn1;rx2+ +p1;rxn+p0;r
Lets take a system with two inputsx1,x2, one outputy, and two rules:
Rule1 : IFx1isA
1
1ANDx2isA
1
2THENf1=p2;1x1+p1;1x2+p0;1
Rule2 : IFx1isA
1
2ANDx2isA
2
2THENf2=p2;2x1+p1;2x2+p0;2
Layer 1: Everyi

-th node is an adaptive node with a function
O1;i
=
A
k
i
(xi); i

=ik:i= 1;2;k= 1;2:
The parameters of the node's MF are calledpremise parameters.A
11
A
12
A
21
A
22
x
1
x
2
Π
Π
N
N
f
1
f
2
Σ
x
2x
1
x
2x
1
y
Layer 1Layer 2 Layer 3 Layer 4 Layer 5
w
1
w
2
w
2
w
1
f
1w
1
f
2w
2

108. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
ANFIS Architecture Continued
Layer 2: Everyi

-th node is a xed node, which calculates the ring
strengths for each rule:
O2;i
=wi
=
A
k
1
(x1)
A
k
2
(x2); i

=k= 1;2:
Besides product, other operations for the linguistic AND may be used.
Layer 3: Everyi

-th node is a xed node, which computes the ratio of
thei

-th ring strength to the sum of allRrules ring strengths:
O3;i
=wi
=
wi

P
R
r=1
wr
; i

= 1;2:
The outputs of this layer are callednormalized ring strengths.A
11
A
12
A
21
A
22
x
1
x
2
Π
Π
N
N
f
1
f
2
Σ
x
2x
1
x
2x
1
y
Layer 1Layer 2 Layer 3 Layer 4 Layer 5
w
1
w
2
w
2
w
1
f
1w
1
f
2w
2

109. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
ANFIS Architecture Continued
Layer 4: Everyi

-th node is an adaptive node with a function
O4;i
=wrfr=wr(p2;rx1+p1;rx2+p0;r); i

=r= 1;2:
The parametersfp2;r; p1;r; p0;rgare calledconsequent parameters.
Layer 5: The single node is a xed node, which computes the overall
output as a summation of all incoming values:
O5;1=y=
R
X
r=1
wrfr=
P
R
r=1
wrfr
P
R
r=1
wr
:A
11
A
12
A
21
A
22
x
1
x
2
Π
Π
N
N
f
1
f
2
Σ
x
2x
1
x
2x
1
y
Layer 1Layer 2 Layer 3 Layer 4 Layer 5
w
1
w
2
w
2
w
1
f
1w
1
f
2w
2

110. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
ANFIS Training
ANFIS is trained by a hybrid learning algorithm. Each iteration makes
two passes:
During the forward pass node outputs go forward until layer 4 and
the consequent parameters are identied by the least-squares method
In the backward pass the error signals (i.e. reference minus layer 4
output) propagate backward and the premise parameters are
updated by gradient descent
Forward pass Backward pass
Premise parameters Fixed Gradient descent
Consequent parametersLeast-squares estimator Fixed
Signals Node outputs Error signals

111. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
MATLAB ANFIS Editor:anfisedit

112. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Adaptive FIS Applications
The advantage of adaptive fuzzy systems compared to, e.g.
Articial Neural Networks (ANN) is that they are gray box as
opposed to ANN, which are black box systems.
The application range is no less than of ANN:
Nonlinear system identication
Adaptive control (process control, inverse kinematics, etc.)
Adaptive machine scheduling
Clustering, classication and pattern recognition
Adaptive expert systems, predictors
Adaptive noise cancellation
And others!

113. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Outline
1. Concepts of Fuzzy Logic
1.1 Introduction
1.2 Fuzzy Sets
1.3 Fuzzy Set Operations
1.4 Membership Functions
1.5 Fuzzy Rules
2. Fuzzy Inference Systems
2.1 Introduction
2.2 Mamdani FIS
2.3 Mamdani Inference
2.4 Defuzzication
2.5 Mamdani FIS Editor
2.6 Takagi-Sugeno FIS
2.7 Sugeno FIS Editor
3. Applications
3.1 Fuzzy PID Controller
3.2 Fuzzy %Anything%
3.3 Adaptive FIS
3.4 Adaptive Control
3.5 Clustering and Classication
3.6 Discussion

114. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Direct Adaptive Control r(t)
Process
u(t) y(t)
Reference model,
fuzzy expert system
Controller
Adaptation
mechanism
Controller
parameters

115. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Indirect Adaptive Control r(t)
Process
u(t) y(t)
Adaptive mapper:
ANFIS or other
System
identification
Controller
Controller
parameters
Controller
designer
Process
parameters
Fuzzy expert system

116. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Outline
1. Concepts of Fuzzy Logic
1.1 Introduction
1.2 Fuzzy Sets
1.3 Fuzzy Set Operations
1.4 Membership Functions
1.5 Fuzzy Rules
2. Fuzzy Inference Systems
2.1 Introduction
2.2 Mamdani FIS
2.3 Mamdani Inference
2.4 Defuzzication
2.5 Mamdani FIS Editor
2.6 Takagi-Sugeno FIS
2.7 Sugeno FIS Editor
3. Applications
3.1 Fuzzy PID Controller
3.2 Fuzzy %Anything%
3.3 Adaptive FIS
3.4 Adaptive Control
3.5 Clustering and Classication
3.6 Discussion

117. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Fuzzy Clustering and Classication
Fuzzy clustering and fuzzy classication dier from conventional crisp clustering
and classication approaches in that:
In crisp clustering each element of a dataset has a degree of belonging 1 to
its assigned cluster and 0 to all other clusters
In fuzzy clustering each element of a dataset has a degree of belonging
ranging from 0 to 1 to each of the clusters
In crisp classication a classied pattern belongs to one of the pre-specied
classes with certainty 1 and with certainty 0 to all other classes
In fuzzy classication a classied pattern belongs to each of the
pre-specied classes with certainty ranging from 0 to 1
The most common fuzzy clustering algorithm is Fuzzy C-means clustering.
Fuzzy classication is performed applying any of the adaptive FIS structures,
e.g. ANFIS, ARIC, GARIC, NNDFR, NEFCLASS, etc.

118. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Fuzzy C-Means: MATLAB GUIfindcluster

119. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Fuzzy Clustering Demo:fcmdemo

120. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Outline
1. Concepts of Fuzzy Logic
1.1 Introduction
1.2 Fuzzy Sets
1.3 Fuzzy Set Operations
1.4 Membership Functions
1.5 Fuzzy Rules
2. Fuzzy Inference Systems
2.1 Introduction
2.2 Mamdani FIS
2.3 Mamdani Inference
2.4 Defuzzication
2.5 Mamdani FIS Editor
2.6 Takagi-Sugeno FIS
2.7 Sugeno FIS Editor
3. Applications
3.1 Fuzzy PID Controller
3.2 Fuzzy %Anything%
3.3 Adaptive FIS
3.4 Adaptive Control
3.5 Clustering and Classication
3.6 Discussion

121. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Mamdani vs Sugeno FIS
Advantages of the Mamdani Method
It is intuitive
It has widespread acceptance
It is well suited for human input
Advantages of the Sugeno Method
It is computationally more ecient
It works well with linear techniques (e.g. PID control)
It works well with optimization and adaptive techniques
It has guaranteed continuity of the output surface
It is well suited for mathematical analysis

122. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Criticism of Mamdani Fuzzy Control
Fuzzy control methods are parasitic: they simply implement
trivial interpolations of control strategies obtained by other means
99% of fuzzy feedback control applications deal with
essentially 1st or 2nd-order, overdamped, SISO systems
Attempt to emulate or duplicate human control behavior?
Human is a very poor controller for complex, multi-variable,
marginally stable dynamic plants
Very hard to generate multidimensional if-then rule tables
No guarantees of closed-loop stability, stability-robustness and
of performance in presence of uncertainty
Cannot generate dierential equation controller rules
M. Athans, Crisp Control Is Always Better Than Fuzzy Feedback Control, EUFIT '99 debate with prof. L.A. Zadeh

123. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Criticism of Sugeno Fuzzy Control
Approach developed to overcome criticism regarding
closed-loop stability guarantees
Design full-state feedback controllers for each linear model
(using crisp control methods) and interpolate using
membership functions
Given that a state space model is necessary, why bother to
introduce fuzzy ideas when conventional crisp control
methods can deal with the design problem directly?
Current methodology does not address stability-robustness
and performance-robustness issues
Current methodology does not address output feedback
requiring dynamic compensator designs

124. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Look on the Bright Side
The answers to some of the critical claims can be found in
Passino, Chapter 8
Although there are many unsolved problems with fuzzy
control, everyone may try and decide for himself, whether the
methodology suits him or not
Fuzzy systems are useful in many other elds of intelligent
computer systems besides process control
It is good to have this tool in your pocket
If you cannot express your view in equations, but you can
verbally go fuzzy!

125. Concepts of Fuzzy LogicFuzzy Inference SystemsApplications
Useful Literature
K. M. Passino and S. Yurkovich,Fuzzy Control.
Addison Wesley Longman, Menlo Park, CA, 1998
J.-S. R. Jang, C.-T. Sun and E. Mizutani,
Neuro-fuzzy and Soft Computing: A Computational
Approach to Learning and Machine Intelligence.
Prentice Hall, 1997