Sergei_Astapov_Fuzzy_Control_lecture_slides.pdf

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

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
Classification and decision-making algorithms
Room: U02-305
E-mail: sergei.astapov@ttu.ee

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 Defuzzification
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 Classification
3.6 Discussion

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 Defuzzification
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 Classification
3.6 Discussion

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?

The Fuzzy Way of Thinking
t (°C)
μ(t)
10 20 30 40
0
-10
-20
-30
-40
1 hot
warm
warm hot
t (°C)
μ(t)
10 20 30 40
0
-10
-20
-30
-40
1
cool
chilly
cold
freezing
freezing cold chilly cool
Binary logic
Fuzzy logic

The Concept of Fuzzy Logic
Fuzzy logic variables have a range of truthfulness from 0 to 1
Fuzzy logic operates with linguistic variables, like
“temperature” instead of t(◦C)
Each variable has a specific number of linguistic values, like
“hot” or “cold”
Fuzzy inference is performed using linguistic rules, e.g.
IF temperature is cold THEN dress warm
The linguistic values and their truth degree are quantified
using membership functions (MF)

A Little Bit of History
1964 : Lotfi 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
1965–1975 : 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 first fuzzy logic controller
1977 : Dubois applied fuzzy sets in a comprehensive study of traffic conditions
1976–1987 : Industrial application of fuzzy logic in Japan and Europe
1987–Present : Widespread application

Conventional and Fuzzy Sets
Let X be a space of objects and x be a generic element of X. A
classical set A, A ⊆ X, is defined as a collection of elements
x ∈ X, such that each element x can either belong or not belong
to the set A.
The classical set thus can be characterized as A = {x | x ∈ X} .
By defining a characteristic function for each x, we can represent
the classical set A by a set of ordered pairs (x, 0) or (x, 1), which
indicate x /
∈ A or x ∈ A respectively.
In a fuzzy set the characteristic function is allowed to have values
of membership between 0 and 1.

Fuzzy Set Definition
Definition 1 (Fuzzy set)
If X is a collection of objects x, then a fuzzy set A in X is
defined as a set of ordered pairs:
A = {(x, µA(x)) | x ∈ X} , (1)
where µA(x) is called the membership function (MF) for the
fuzzy set A.
In fuzzy set theory classical sets are referred to as crisp sets and
the values as crisp values.
X is usually referred to as the universe of discourse. It represents
the range of values the fuzzy variables may take.
Universes of discourse may be either discrete or continuous.

Linguistic Variables and Values
Fuzzy sets usually carry names appealing in our daily linguistic usage
The universe is called a linguistic variable and its sets are called
linguistic values
The universe of discourse X is partitioned into several fuzzy sets,
with MFs covering X in a more or less uniform manner
Example 1
Consider the universe X of linguistic variable “temperature”. The
universe may be defined differently, 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
µfreezing(x), µcold(x), µchilly(x), µcool(x), µwarm(x), µhot(x).

Relevant Properties of Fuzzy Sets
Definition 2 (Support)
The support of a fuzzy set A is the set of all points x ∈ X, such that
µA(x) > 0:
support(A) = {x | µA(x) > 0} . (2)
Definition 3 (Core)
The core of a fuzzy set A is the set of all points x ∈ X, such that
µA(x) = 1:
core(A) = {x | µA(x) = 1} . (3)
Definition 4 (Crossover points)
A crossover point of a fuzzy set A is a point x ∈ X, at which
µA(x) = 0.5:
crossover(A) = {x | µA(x) = 0.5} . (4)

Relevant Properties of Fuzzy Sets Continued
Definition 5 (Normality)
A fuzzy set A is normal if its core is nonempty, i.e. we can always
find a point x ∈ X, such that µA(x) = 1.
Definition 6 (Fuzzy singleton)
A fuzzy set, the support of which is a single point in X with
µA(x) = 1 is called a fuzzy singleton.
Definition 7 (Symmetry)
A fuzzy set A is symmetric if its MF is symmetric around a
certain point x = c, namely, µA(c + x) = µA(c − x), ∀x ∈ X.

Relevant Properties of Fuzzy Sets Continued
Definition 8 (Open left, open right, closed sets)
A fuzzy set A is:
open left if limx→−∞ µA(x) = 1, limx→+∞ µA(x) = 0;
open right if limx→−∞ µA(x) = 0, limx→+∞ µA(x) = 1;
and closed if limx→−∞ µA(x) = limx→+∞ µA(x) = 0.
warm hot
x
μ(x)
10 20 30 40
0
-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

Containment
Definition 9 (Containment or subset)
Fuzzy set A is contained in fuzzy set B (or A is a subset of B),
iff µA(x) ≤ µB(x) for all x:
A ⊆ B ⇐⇒ µA(x) ≤ µB(x). (5)
x
μ(x)
1
A
B

Complement
Definition 10 (Complement or negation)
The complement of a fuzzy set A, denoted by A or ¬A, or
NOT A is defined as
µA(x) = 1 − µA(x). (6)
x
μ(x)
1
A
x
μ(x)
1
NOT A

Union
Definition 11 (Union or disjunction)
The union of two fuzzy sets A and B is a fuzzy set C, written as
C = A ∪ B or C = A OR B, the MF of which is related to those
of A and B by
µC(x) = max (µA(x), µB(x)) = µA(x) ∨ µB(x). (7)
x
μ(x)
1
A B
x
μ(x)
1
A OR B

Intersection
Definition 12 (Intersection or conjunction)
The intersection of two fuzzy sets A and B is a fuzzy set C,
written as C = A ∩ B or C = A AND B, the MF of which is
related to those of A and B by
µC(x) = min (µA(x), µB(x)) = µA(x) ∧ µB(x). (8)
x
μ(x)
1
A B
x
μ(x)
1
A AND B

Cartesian Product and Co-product
Definition 13 (Cartesian product and co-product)
Let A and B be fuzzy sets in X and Y , respectively. The
Cartesian product of A and B, denoted by A × B, is a fuzzy set
in the product space X × Y with the membership function
µA×B(x, y) = min (µA(x), µB(y)) . (9)
Similarly, the Cartesian co-product A + B is a fuzzy set with the
membership function
µA+B(x, y) = max (µA(x), µB(y)) . (10)

Preface
A fuzzy set is completely characterized by its MF
As the universe most often consists of real values, X ⊆ R, it
is convenient to define MFs as continuous functions
For a single linguistic variable the MFs are one-dimensional
Combining the universes of different linguistic variables, MFs
of higher dimensions may be derived
Here the most commonly applied MF types are presented

Straight-Line MF:Triangular MF
Definition 14 (Triangular MF)
A triangular MF is specified by three parameters {a, b, c} as follows:
triangle (x; a, b, c) =









0, x ≤ a.
x−a
b−a , a ≤ x ≤ b.
c−x
c−b , b ≤ x ≤ c.
0, c ≤ x.
(11)
It may also be described by min and max as
triangle (x; a, b, c) = max

min

x − a
b − a
,
c − x
c − b

, 0

. (12)
The parameters a and c locate the “feet” of the triangle and b — its
peak.

Straight-Line MF: Trapezoidal MF
Definition 15 (Trapezoidal MF)
A trapezoidal MF is specified by four parameters {a, b, c, d} as follows:
trapezoid (x; a, b, c, d) =















0, x ≤ a.
x−a
b−a , a ≤ x ≤ b.
1, b ≤ x ≤ c.
d−x
d−c , c ≤ x ≤ d.
0, d ≤ x.
(13)
An alternative expression using min and max is
trapezoid (x; a, b, c, d) = max

min

x − a
b − a
, 1,
d − x
d − c

, 0

. (14)
The parameters a and d locate the “feet” of the trapezoid and b and c —
its “shoulders”.

Smooth MF: Gaussian and Bell MF
Definition 16 (Gaussian MF)
A Gaussian MF is specified by two parameters {c, σ} as follows:
gaussian (x; c, σ) = e−1
2 (x−c
σ )
2
. (15)
The parameter c represents the MF center and σ determines the
MF width.
Definition 17 (Generalized bell MF)
A generalized bell MF is specified by three parameters {a, b, c}
as follows:
bell (x; a, b, c) =
1
1 + x−c
a
2b
, (16)
where b is usually positive (if b 0, then the MF becomes an
upside-down bell).

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])

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’

Straight-line and Smooth MFs: Analysis
What are the advantages and drawbacks of straight-line and
smooth MFs?
Straight-line MFs
Simple formulas: computational efficiency
Zero points strictly defined:
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 flexibility
Best for automated tuning (adaptive systems)
Less straight-forward: more problems during initial design

Open Membership Functions
Definition 18 (Sigmoidal MF)
A sigmoidal MF is specified by two parameters {a, c} as follows:
sig (x; a, c) =
1
1 + e−a(x−c)
, (17)
where a controls the slope of the crossover point c.
An open triangular MF is obtained by specifying ± inf as 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])

Asymmetric Membership Functions
There are numerous ways to get asymmetric smooth MFs. One way
is taking the difference |y1 − y2| and product y1y2 of 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

Asymmetric MF: Left-Right MF
Definition 19 (Left-right MF)
A left-right MF is specified by three parameters {α, β, c} as
LR (x; α, β, c) =
(
FL
c−x
α

, x ≤ c,
FR

x−c
β

, x ≥ c,
(18)
where FL(x) and FR(x) are monotonically decreasing functions
defined on [0, ∞) with FL(0) = FR(0) = 1 and
limx→∞ FL(x) = limx→∞ FR(x) = 0.

Asymmetric MF: Left-Right MF Example
Example 2
Let FL(x) =
p
max (0, 1 − x2), FR = e−|x|3
. Then applying (18)
we can generate different 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)

Asymmetric MF: Two-Sided Gaussian MF
Definition 20 (Two-sided Gaussian MF)
A two-sided Gaussian MF is defined by four parameters {c1, σ1, c2, σ2}
as
gaussian2 (x; c1, σ1, c2, σ2) =







exp
h
−1
2

x−c1
σ1
i
, x ≤ c1,
1, c1 x ≤ c2,
exp
h
−1
2

x−c2
σ2
i
, c2 ≤ x,
(19)
where c1, σ1 are the parameters of the left-most curve and c2, σ2 are the
parameters of the right-most curve.
The two-sided Gaussian is essentially a mixture of two Gaussian
functions defined by (15). It is computed in MATLAB using the
gauss2mf function.

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 defining
one’s own MFs
Two-dimensional MFs are not discussed here, for further study
please refer to literature

Fuzzy IF-THEN Rules
Definition 21 (Fuzzy if-then rule)
A fuzzy if-then rule, also known as a fuzzy rule, fuzzy implication, or
fuzzy conditional statement, assumes the form
IF x is A THEN y is B, (20)
where A and B are linguistic values defined by fuzzy sets on universes of
discourse X and Y , respectively. The expression x is A is called the
antecedent or premise, while y is B is called the consequence or
conclusion.
Expression (20), which is abbreviated as A → B, can be defined as a binary
fuzzy relation R on the product space X × Y : R = A → B. R can be viewed
as a fuzzy set of two-dimensional MF
µR(x, y) = f (µA(x), µB(y)) ,
where the function f is called the fuzzy implication function, that transforms
the membership degrees of x in A and y in B into those of (x, y) in A → B.

Multiple Input Multiple Output Rules
Let premise linguistic variables xi, i = 1, 2, . . . , n and consequence
linguistic variables yj, j = 1, 2, . . . , m take on values of their universes of
discourse Xi and Yj, respectively. Let xi be characterized by a set of
linguistic values
Ai =

Ak
i : k = 1, 2, . . . , Ni ,
and yj be characterized by a set of linguistic values
Bj =

Bl
j : l = 1, 2, . . . , Mi .
Then a MIMO rule with number of inputs n and number of outputs m
can be written as
IF x1 is Ap
1 AND x2 is Aq
2 AND . . . AND xn is Ar
n
THEN y1 is Bs
1 AND y2 is Bu
2 AND . . . AND ym is Bv
m.
(21)

Linguistic Operators
A large number of operators may be applied to linguistic
terms in fuzzy rules
Negation, e.g. “not warm”
Connectives: and, or, either, neither, etc.
Hedges: too, very, more or less, quite, extremely, etc.
For example “more or less warm but not too warm”
Here only not, and, or operators are discussed as they are
most common and sufficient in the majority of applications
In practice we will use only the and operator

Fuzzy Inference Systems
A fuzzy inference system (FIS) or, as it is also known in different
application areas, fuzzy expert system, fuzzy model, fuzzy
associative memory and fuzzy 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
Classification and clustering
Pattern recognition
Decision analysis and expert systems

Generic FIS Structure
Defuzzification
Fuzzification
Inference
mechanism
Rule-base
...
...
x1
x2
xn
y1
y2
yn
Crisp
inputs
Crisp
outputs
Fuzzified
inputs
Fuzzified
conclusions
Fuzzification: 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
Defuzzification: transformation of output fuzzy sets to crisp
values

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

Controlled Process: Inverted Pendulum
θ
F
x
r(t)
Inverted
pendulum
u(t) y(t)
Fuzzy logic
controller
d
dt
Σ
-
+
r(t) — reference θ angle
u(t) — force (N)
y(t) — θ angle (rad)
e(t) = r(t) − y(t)
e(t)

Mamdani Type FIS
Was proposed as the first 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

FIS Linguistic Variables and Values
For our inverted pendulum example we choose the following
inputs and outputs:
“error” describes e(t) = r(t) − y(t)
“change-in-error” describes d
dt e(t)
“force” describes u(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”

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 negsmall
F F F
(a) (b) (c)

Fuzzy Rule-Base
The number of rules for a MISO FIS is at most
Qn
i=1 Ni, where Ni
is the number of linguistic values for the i-th linguistic premise
variable. (All possible combinations of premise linguistic values.)
In our case the number of rules is equal to 5 · 5 = 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 -1 0 1 2
error
-2 2 2 2 1 0
-1 2 2 1 0 -1
0 2 1 0 -1 -2
1 1 0 -1 -2 -2
2 0 -1 -2 -2 -2

Membership Functions
e(t) (rad)
π/4
0
-π/4
-π/2
zero
negsmall
neglarge possmall poslarge
π/2
de(t)/dt (rad/s)
zero
negsmall
u(t) (N)
10 20
0
-10
-20
zero
negsmall
-30 30
π/8
0
-π/8
-π/4 π/4

Fuzzification
Singleton fuzzification: apply a fuzzy singleton µfuz
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 fuzzification” a Gaussian is used as
a fuzzification function, which accounts for inconsistency in the input signal.
e(t) (rad)
π/4
0
-π/4
-π/2
zero
negsmall
π/2
de(t)/dt (rad/s)
zero
negsmall
π/8
0
-π/8
-π/4 π/4
e(t) = -9π/20
de(t)/dt = 9π/80
Crisp input e(t) = −9π/20:
µ
neglarge
(e) = min µneglarge(e), µfuz
1 (e)

=
min(0.75, 1) = 0.75;
µ
negsmall
(e) = min µnegsmall(e), µfuz
1 (e)

=
min(0.25, 1) = 0.25; all other zero.
Crisp input ė(t) = 9π/80:
µ d
zero(ė) = min µzero(ė), µfuz
2 (ė)

=
min(0.125, 1) = 0.125;
µ
possmall
(ė) = min µpossmall(ė), µfuz
2 (ė)

=
min(0.875, 1) = 0.875; all other zero.

Mamdani Inference Mechanism Steps
Calculate the firing strength for each rule in the rule-base
Determine which rules are on using the firing strengths
Determine implied fuzzy sets — perform fuzzy implication
Determine overall implied fuzzy set — perform fuzzy
aggregation*
*Performed in case of applying specific types of defuzzification.
If defuzzification uses implied fuzzy sets, the step is not performed.

Firing Strength of a Premise
The firing 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 variables x1 and x2 the typical operators are the following:
Fuzzy complement (NOT):
Defined in (6) as µÂk
1
(x1) = 1 − µÂk
1
(x1)
Fuzzy union (OR):
Defined in (7) as maximum µÂk
1 ∪Âl
2
(x1, x2) = max

µÂk
1
(x1) , µÂl
2
(x2)

Alternative: algebraic sum
µÂk
1 ∪Âl
2
(x1, x2) = µÂk
1
(x1) + µÂl
2
(x2) − µÂk
1
(x1) µÂl
2
(x2)
Fuzzy intersection (AND):
Defined in (8) as minimum µÂk
1 ∩Âl
2
(x1, x2) = min

µÂk
1
(x1) , µÂl
2
(x2)

Alternative: algebraic product µÂk
1 ∩Âl
2
(x1, x2) = µÂk
1
(x1) µÂl
2
(x2)

Firing Strength: More Complex Premises
In premises with more complex logic, the firing strength is calculated by
partitioning the premise into simpler terms.
Example 3
The premise
IF x1 is Â2
1 AND x2 is Â1
2 AND x3 is NOT Â5
3 OR x4 is Â3
4
yields the firing strength
µpremise (x1, x2, x3, x4) =
max
h
min

µÂ2
1
(x1) , µÂ1
2
(x2) , 1 − µÂ5
3
(x3)

, µÂ3
4
(x4)
i
.
Also there exists an option to use a “rule certainty” weight. This way, for
the i-th rule, the firing strength is multiplied by the weight wi, which
specifies how certain we are in this specific 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.

Which Rules Are On
The rule is considered being “on” if its premise is non-zero:
µpremise (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 most 103
= 1000. With the universes partitioned by so many
rules, the number of “on” rules at any given time will be quite small. If
for example 10 rules are on, then mark those rules and perform later
steps with 10 sets of parameters, instead of using the whole rule-base and
performing 100 times more computations, mainly with zeros.

Implied Fuzzy Sets: Fuzzy Implication
The implied fuzzy set of an output yj for a rule i, which has a
consequent Bk
j , and a premise degree of membership equal to
µpremise(i) (x1, x2, . . . , xn), is characterized by
µB̂k
j
(yj) = min

µpremise(i) (x1, x2, . . . , xn) , µBk
j
(yj)

.
Alternatively the algebraic product can be defined as the
implication operation:
µB̂k
j
(yj) = µpremise(i) (x1, x2, . . . , xn) µBk
j
(yj) .
An implied fuzzy set is computed for every rule that is “on”.

Overall Implied Fuzzy Set: Fuzzy Aggregation
The overall implied fuzzy set B̂j of an output yj, which
incorporates the implied fuzzy sets
n
B̂k
j , B̂l
j, . . . , B̂p
j
o
is
characterized by
µB̂j
(yj) = max

µB̂k
j
(yj) , µB̂l
j
(yj) , . . . , µB̂p
j
(yj)

.
Alternatively the algebraic sum can be defined as the aggregation
operation:
µB̂j
(yj) = µB̂k
j
(yj) + µB̂l
j
(yj) + · · · + µB̂p
j
(yj) −
− µB̂k
j
(yj) µB̂l
j
(yj) . . . µB̂p
j
(yj) .

Mamdani Inference: Example
e(t) (rad)
π/4
0
-π/4
-π/2
zero
negsmall
π/2
de(t)/dt (rad/s)
zero
negsmall
π/8
0
-π/8
-π/4 π/4
e(t) = -9π/20
de(t)/dt = 9π/80
u(t) (N)
10 20
0
-10
-20
zero
negsmall
-30 30
u(t) (N)
10 20
0
-10
-20
zero
negsmall
-30 30
u(t) (N)
10 20
0
-10
-20
zero
negsmall
-30 30
Apply implication (min)
Apply
AND
(min)
u(t) (N)
10 20
0
-10 30
Apply aggregation (max)

Mamdani Inference: Example Computations
From the fuzzification stage we have established that we have four fuzzy values:
µ
neglarge
(e) = 0.75; µ
negsmall
(e) = 0.25; µ d
zero(ė) = 0.125;
µ
possmall
(ė) = 0.875. Thus the rules that are on are:
IF error is neglarge AND change-in-error is zero THEN force is poslarge (red)
IF error is neglarge AND change-in-error is possmall THEN force is possmall (orange)
IF error is negsmall AND change-in-error is zero THEN force is possmall (green)
IF error is negsmall AND change-in-error is possmall THEN force is zero (blue)
Compute the firing strengths of the four rules using min for the AND operator:
µpremise(1) (e, ė) = min(µ
neglarge
(e), µ d
zero(ė)) = min(0.75, 0.125) = 0.125;
neglarge
(e), µ
possmall
(ė)) = min(0.75, 0.875) = 0.75;
negsmall
(e), µ d
zero(ė)) = min(0.25, 0.125) = 0.125;
negsmall
(e), µ
possmall
(ė)) = min(0.25, 0.875) = 0.25.

Mamdani Inference: Example Computations Continued
Implied fuzzy sets are derived from the rule premises using min as:
µ
poslarge(1)
(u) = min µpremise(1) (e, ė) , µposlarge (u)

=
= min(0.125, 1) = 0.125;
µ
possmall(2)
(u) = min µpremise(2) (e, ė) , µpossmall (u)

=
= min(0.75, 1) = 0.75;
µ
possmall(3)
(u) = min µpremise(3) (e, ė) , µpossmall (u)

=
= min(0.125, 1) = 0.125;
µ d
zero(4) (u) = min µpremise(4) (e, ė) , µzero (u)

= min(0.25, 1) = 0.25.
The overall implied fuzzy set is obtained by fuzzy aggregation using max as:
µ
overall
(u) =
max

µ
poslarge(1)
(u) , µ
possmall(2)
(u) , µ
possmall(3)
(u) , µ d
zero(4) (u)

.

Mamdani Inference: Principle Justification
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 20
0
-10
-20
zero
negsmall
-30 30 u(t) (N)
10 20
0
-10 30
Aggregation (max)
Inference (product)

Defuzzification
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 defuzzification is applied to the implied sets.
A number of defuzzification strategies exist, and it is not hard to
invent more, suiting your specific application.
Each provides a means to choose a crisp output ycrisp
j based on
either the implied fuzzy sets or the overall implied fuzzy set.
Reviewed defuzzification 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)

Defuzzification on IFS: Center of Gravity
Definition 22 (Center of gravity)
In Center of gravity (COG) defuzzification the output ycrisp
j is
computed using the center of area and area of each implied fuzzy set:
ycrisp
j =
PR
i=1 bj
i
´
Yj
µB̂i
j
(yj) dyj
PR
i=1
´
Yj
µB̂i
j
(yj) dyj
, (22)
where R is the number or rules, bj
i is the center of area of the MF of Bp
j
associated with the implied fuzzy set B̂i
j for the i-th rule and
ˆ
Yj
µB̂i
j
(yj) dyj
denotes the area under µB̂i
j
(yj).

Defuzzification 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 off while
using triangle MFs and min for implication.
Notice though, that for this method to be reliable the fuzzy system
must be defined such that
R
X
i=1
ˆ
Yj
µB̂i
j
(yj) dyj 6= 0
for all xi. 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 use open MFs for
output fuzzy sets.

Defuzzification on IFS: Center-Average
Definition 23 (Center-average)
In Center-average defuzzification the output ycrisp
j is computed using
the centers of each of the output MFs and the maximum certainty of
each of the implied fuzzy sets:
ycrisp
j =
PR
i=1 bj
i supyj
n
µB̂i
j
(yj)
o
PR
i=1 supyj
n
µB̂i
j
(yj)
o , (23)
where supyj
denotes the supremum (i.e. the least upper bound) of the
implied fuzzy set µB̂i
j
(yj).

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
µB̂i
j
(yj)
o
= max

µB̂i
j
(yj)

.
Notice though, that the fuzzy system must be defined such that
R
X
i=1
sup
yj
n
µB̂i
j
(yj)
o
6= 0
for all xi. This is achieved as in the case of COG.

If using normal MFs for output fuzzy sets, then for many inference
strategies we have
sup
yj
n
µB̂i
j
(yj)
o
= µpremise(i) (x1, x2, . . . , xn) ,
which is the firing strength of rule i. The formula for
defuzzification is then given by
ycrisp
j =
PR
i=1 bj
i µpremise(i) (x1, x2, . . . , xn)
PR
i=1 µpremise(i) (x1, x2, . . . , xn)
, (24)
where
PR
i=1 µpremise(i) (x1, x2, . . . , xn) 6= 0, ∀xi must be ensured.
The shape of the output MFs does not matter, as bounds of
supremum subsets can be defined using singletons.

Defuzzification on The Overall IFS: Maximum Criterion
For the MOM, SOM and LOM defuzzification the crisp output is
chosen as a point on the output universe Yj, for which the overall
implied fuzzy set B̂j reaches its maximum:
ycrisp
j ∈
(
arg sup
Yj
n
µB̂j
(yj)
o
)
.
MOM, SOM and LOM differ in the strategy of choosing the crisp
value from this subset.
u(t) (N)
10 20
0
-10 30
supremum
MOM
LOM
SOM

Defuzzification on The Overall IFS: MOM
Definition 24 (Mean of maximum)
Define a fuzzy set B̂∗
j ⊆ Yj with a MF defined as
µB̂∗
j
(yj) =
(
1, µB̂j
(yj) = supYj
n
µB̂j
(yj)
o
,
0, otherwise.
Then the crisp output of mean of maximum (MOM) defuzzification is
defined as
ycrisp
j =
´
Yj
yjµB̂∗
j
(yj) dyj
´
Yj
µB̂∗
j
(yj) dyj
, (25)
where the fuzzy system must be defined so
´
Yj
µB̂∗
j
(yj) dyj 6= 0, ∀xi.
Notice that if µB̂∗
j
(yj) = 1 lies in a single interval
h
yleft
j , yright
j
i
⊆ Yj,
then ycrisp
j =

yleft
j + yright
j

/2.

Defuzzification on The Overall IFS: SOM and LOM
Definition 25 (Smallest of maximum)
In smallest of maximum (SOM) defuzzification the output ycrisp
j is
computed as the minimal argument of the output universe Yj, for which
the the overall implied fuzzy set B̂j reaches its maximum:
ycrisp
j = min

arg sup
Yj
n
µB̂j
(yj)
o
#
. (26)
Definition 26 (Largest of maximum)
In largest of maximum (LOM) defuzzification the output ycrisp
j is
computed as the maximal argument of the output universe Yj, for which
the the overall implied fuzzy set B̂j reaches its maximum:
ycrisp
j = max

arg sup
Yj
n
µB̂j
(yj)
o
#
. (27)

Defuzzification on The Overall IFS: Center of Area
Definition 27 (Center of area)
In center of area (COA) defuzzification the output ycrisp
j is computed
over the area of the MF of the overall implied fuzzy set B̂j as
ycrisp
j =
´
Yj
yjµB̂j
(yj) dyj
´
Yj
µB̂j
(yj) dyj
, (28)
where the fuzzy system must be defined so
´
Yj
µB̂j
(yj) dyj 6= 0, ∀xi.
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.

Defuzzification: 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 width w :
The area of a triangle with the top “chopped off” at height h is
equal to w

h − h2
2

The area of a triangle with height h is equal to 1
2 wh
Here w is the support length of B̂i
j and h is µpremise(i) (x1, x2, . . . , xn).
10 20
0 10 20
0
min implication product implication
w w
h h

Defuzzification: COG Example
For implication defined by min:
ucrisp =
bpl
´
U µb
pl(1)
(u)du+bps
´
U µc
ps(2)(u)du+bps
´
U µc
ps(3)(u)du+bz
´
U µb
z(4)(u)du
´
U µb
pl(1)
(u)du+
´
U µc
ps(2)(u)du+
´
U µc
ps(3)(u)du+
´
U µb
z(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 defined by product:
ucrisp =
bpl
´
U µb
pl(1)
(u)du+bps
´
U µc
ps(2)(u)du+bps
´
U µc
ps(3)(u)du+bz
´
U µb
z(4)(u)du
´
U µb
pl(1)
(u)du+
´
U µc
ps(2)(u)du+
´
U µc
ps(3)(u)du+
´
U µb
z(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

Defuzzification: Center-Average Example
For implication defined by product:
ucrisp =
bpl supu
n
µb
pl(1)
(u)
o
+bps supu{µc
ps(2)(u)}+bps supu{µc
ps(3)(u)}+bz supu{µb
z(4)(u)}
supu
n
µb
pl(1)
(u)
o
+supu{µc
ps(2)(u)}+supu{µc
ps(3)(u)}+supu{µb
z(4)(u)}
=
= (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 µpremise(i) (x1, x2, . . . , xn).

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

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

FIS Design in MATLAB: Editor Main

FIS Design in MATLAB: MF Editor

FIS Design in MATLAB: Rule Viewer

FIS Design in MATLAB: Input-Output Curve

Takagi-Sugeno FIS
Takagi-Sugeno or simply Sugeno-type FIS has a different way of
computing the consequence and defuzzification.
A general Sugeno rule has a form
IF x1 is Ak
1 AND x2 is Al
2 AND . . . AND xn is Ap
n THEN zi = fi(·).
Here z = f(·) may be any function (even another mapping, like
neural network, or another FIS)
Usually zi = fi (x1, x2, . . . , xn) is used. If this function is a first
order polynomial, i.e.
zi = anx1 + an−1x2 + · · · + a1xn + a0,
the inference system is called a first-order Sugeno FIS. When f is
a constant, the system is called a zero-order Sugeno FIS.

Sugeno Inference Principles
The premises µpremise(i) (x1, x2, . . . , xn) are computed as in the
Mamdani FIS, incorporating fuzzification and linguistic operators.
The defuzzification is usually performed using weighted average:
ycrisp
=
PR
i=1 ziµpremise(i) (x1, x2, . . . , xn)
PR
i=1 µpremise(i) (x1, x2, . . . , xn)
, (29)
where the fuzzy system is defined so that
PR
i=1 µpremise(i) (x1, x2, . . . , xn) 6= 0, ∀xi.
Thus the Sugeno FIS can be used as a general mapper for a wide
variety of applications.

Sugeno FIS Example
We will not define the whole rule-base of the inverted pendulum
controller for the Sugeno FIS. Lets specify zi = fi (e, ė) for the
rules that are on in our example:
z1 = −5e + 4ė + 3 for the red rule;
z2 = −4e + 2ė + 2 for the orange rule;
z3 = −2e + 1ė + 1 for the green rule;
z4 = −0.5e + 0.5ė + 0 for the blue rule.
For the values e = − 9
20π and ė = 9
80π the functions take on values
z1 = −5 − 9
20π

+ 4 9
80π

+ 3 = 11.482
z2 = −4 − 9
20π

+ 2 9
80π

+ 2 = 8.362
z3 = −2 − 9
20π

+ 1 9
80π

+ 1 = 4.181
z4 = −0.5 − 9
20π

+ 0.5 9
80π

+ 0 = 0.884

Sugeno FIS Example
Then the FIS crisp output will be
ycrisp
=
P4
i=1 ziµpremise(i) (e, ė)
P4
i=1 µpremise(i) (e, ė)
=
11.482 · 0.125 + 8.362 · 0.75 + 4.181 · 0.125 + 0.884 · 0.25
0.125 + 0.75 + 0.125 + 0.25
= 9.03
Notice, that no implication and aggregation is used. This simplifies
the inference process a lot.

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

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

FIS Design in MATLAB: Sugeno FIS Editor

FIS Design in MATLAB: Sugeno Rules

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
controller
d
dt
Σ
-
+
g1
g2
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 different inference and defuzzification
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 firing strengths, input-output curves,
etc. to the changes you make.

Fuzzy PID Controller
The term Fuzzy PID controller can be understood in two ways:
A fuzzy FIS, which has the inputs e(t), d
dte(t),
´
e(t)dt
A crisp PID controller, the KP , KI and KD coefficients 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 different scenarios of system operation

Fuzzy PID Controller Example 1

Fuzzy PID Controller Example 2

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

Making Anything Fuzzy
If you have some time variant system with parameters that
cannot be statically specified...
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

Fuzzy Predictor
Having a discrete time series θ1, θ2, . . . , θk, the task of a prediction
algorithm is to determine the next values of the given time series
θ̂k+1, θ̂k+2, . . . , θ̂k+l.
The time series possesses certain dynamical properties θk+1 = θk + ωk,
where ωk is the system perturbation of unknown distribution.
The observed value may be affected by external interference
θ̃k = θk + νk, where νk is referred to as observation noise.
The Kalman (exponential average) predictor is given by the recurrence
θ̂k+1 = αθ̂k + (1 − α)θ̃k,
where θ̂k+1 is the predicted value of the time series, θ̂k is the last known
predicted value, θ̃k is the last observed value and α ∈ [0, 1] is the weight
parameter.

Fuzzy Predictor Continued
Basic logic tells us that:
If the system is steady, θ̃k influences prediction more and α → 0.
On the other hand, if the system is not steady or noisy, θ̃k is less
reliable than θ̂k and α → 1.
Specify the FIS input as error ek = θ̂k − θ̃k , then develop rules, e.g.
IF error is small THEN α is large
IF error is medium THEN α is medium
IF error is large THEN α is small
And, well, you know the rest.
P.S. Think, how introducing the change-in-error into the FIS will improve
the situation.

General Linear Dynamic Model
The linear discrete-time dynamic system model takes the form
xk = Ak−1xk−1 + qk−1
yk = Hk−1xk + rk−1
,
where xk is the system state vector at time step k, yk is the measurement
vector at k, Ak−1 is the transition matrix of the dynamic model, Hk−1 is the
measurement matrix, qk−1 ∼ N (0, Qk−1) is the process noise with covariance
Qk−1 and rk−1 ∼ N (0, Rk−1) is the measurement noise with covariance Rk−1.
For the majority of applications it is assumed that the noise has fixed
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, filter, etc. online

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.
A11
A12
A21
A22
x1
x2
Π
Π
N
N
f1
f2
Σ
x2
x1
x2
x1
y
Layer 1 Layer 2 Layer 3 Layer 4 Layer 5
w1
w2
w2
w1
f1
w1
f2
w2

ANFIS Architecture
The first-order Sugeno system r-th rule takes the form
IF x1 is Ak
1 AND x2 is Al
2 AND . . . AND xn is Ap
n
THEN fr = pn,rx1 + pn−1,rx2 + · · · + p1,rxn + p0,r
Lets take a system with two inputs x1, x2, one output y, and two rules:
Rule1 : IF x1 is A1
1 AND x2 is A1
2 THEN f1 = p2,1x1 + p1,1x2 + p0,1
Rule2 : IF x1 is A1
2 AND x2 is A2
2 THEN f2 = p2,2x1 + p1,2x2 + p0,2
Layer 1: Every i∗
-th node is an adaptive node with a function
O1,i∗ = µAk
i
(xi) , i∗
= i × k : i = 1, 2; k = 1, 2.
The parameters of the node’s MF are called premise parameters.
A11
A12
A21
A22
x1
x2
Π
Π
N
N
f1
f2
Σ
x2
x1
x2
x1
y
w1
w2
w2
w1
f1
w1
f2
w2

ANFIS Architecture Continued
Layer 2: Every i∗
-th node is a fixed node, which calculates the firing
strengths for each rule:
O2,i∗ = wi∗ = µAk
1
(x1) µAk
2
(x2) , i∗
= k = 1, 2.
Besides product, other operations for the linguistic AND may be used.
Layer 3: Every i∗
-th node is a fixed node, which computes the ratio of
the i∗
-th firing strength to the sum of all R rules firing strengths:
O3,i∗ = wi∗ =
wi∗
PR
r=1 wr
, i∗
= 1, 2.
The outputs of this layer are called normalized firing strengths.
A11
A12
A21
A22
x1
x2
Π
Π
N
N
f1
f2
Σ
x2
x1
x2
x1
y
w1
w2
w2
w1
f1
w1
f2
w2

ANFIS Architecture Continued
Layer 4: Every i∗
-th node is an adaptive node with a function
O4,i∗ = wrfr = wr (p2,rx1 + p1,rx2 + p0,r) , i∗
= r = 1, 2.
The parameters {p2,r, p1,r, p0,r} are called consequent parameters.
Layer 5: The single node is a fixed node, which computes the overall
output as a summation of all incoming values:
O5,1 = y =
R
X
r=1
wrfr =
PR
r=1 wrfr
PR
r=1 wr
.
A11
A12
A21
A22
x1
x2
Π
Π
N
N
f1
f2
Σ
x2
x1
x2
x1
y
w1
w2
w2
w1
f1
w1
f2
w2

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 identified 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 parameters Least-squares estimator Fixed
Signals Node outputs Error signals

MATLAB ANFIS Editor: anfisedit

Adaptive FIS Applications
The advantage of adaptive fuzzy systems compared to, e.g.
Artificial 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 identification
Adaptive control (process control, inverse kinematics, etc.)
Adaptive machine scheduling
Clustering, classification and pattern recognition
Adaptive expert systems, predictors
Adaptive noise cancellation
And others!

Direct Adaptive Control
r(t)
Process
u(t) y(t)
Reference model,
fuzzy expert system
Controller
Adaptation
mechanism
Controller
parameters

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

Fuzzy Clustering and Classification
Fuzzy clustering and fuzzy classification differ from conventional crisp clustering
and classification 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 classification a classified pattern belongs to one of the pre-specified
classes with certainty 1 and with certainty 0 to all other classes
In fuzzy classification a classified pattern belongs to each of the
pre-specified classes with certainty ranging from 0 to 1
The most common fuzzy clustering algorithm is Fuzzy C-means clustering.
Fuzzy classification is performed applying any of the adaptive FIS structures,
e.g. ANFIS, ARIC, GARIC, NNDFR, NEFCLASS, etc.

Fuzzy C-Means: MATLAB GUI findcluster

Fuzzy Clustering Demo: fcmdemo

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 efficient
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

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 “differential equation” controller rules
M. Athans, Crisp Control Is Always Better Than Fuzzy Feedback Control, EUFIT ’99 debate with prof. L.A. Zadeh

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

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 fields 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!

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

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