fuzzy system

Fakultät ETIT Institut für Automatisierungstechnik, Professur für Prozessleittechnik
26.01.2010
Dr. Engin YEŞİL
Introduction to
Fuzzy Modeling & Control

Fuzzy Systems:
•
•
A static or dynamic system which makes use of fuzzy
sets or fuzzy logic and of the corresponding
mathematical framework is called a fuzzy system. system
There are a number of ways fuzzy sets can be
involved in a system, such as:
1. In the description of the system,
2. In the specification of the system’s parameters,
3. The input, output and state variables of a system may
be fuzzy sets.
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1.
•
•
In the description of the system:
A system can be defined, for instance, as a collection
of IF-THEN IF THEN rules with fuzzy predicates, or as a fuzzy
relation.
An example of a fuzzy rule describing the relationship
between a heating power and the temperature trend
in a room may be:
If the heating power is high
then the temperature will increase fast
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2.
•
In the specification of the system’s parameters:
The system can be defined by an algebraic or
differential equation, in which the parameters are
fuzzy numbers instead of real numbers.
“Almost
3”
“About
3 5
•Fuzzy numbers express the uncertainty in the
parameter values.
5”
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3.
•
•
The input, output and state variables of a system
may be fuzzy sets:
Fuzzy inputs can be readings from unreliable sensors
(“noisy” data), or quantities related to human
perception, such as comfort, beauty, etc.
Fuzzy systems can process such information, which is
not the case with conventional (crisp) systems.
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Crisp and fuzzy information in systems
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Fuzzy Systems
•
•
•
Fuzzy systems can be regarded as a generalization of
interval-valued systems, which are in turn a
generalization of crisp systems.
Note that a function f:XY can be regarded as a
subset of the Cartesian product X x Y , i.e., as a
relation.
The evaluation of the function for a given input
proceeds in three steps:
1. extend the given input into the product space X x Y
2. find the intersection of this extension with the
relation,
3.project this intersection onto Y.
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crisp function
interval function
fuzzy function
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•
•
•
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Fuzzy Systems
Most common are fuzzy systems defined by means of
if-then rules: rule-based fuzzy systems.
In the rest of this lecture we will focus on these
systems only.
Fuzzy systems can serve different purposes, such as
– modeling,
– data analysis,
– prediction or
– control.
Here, a fuzzy rule-based system is for simplicity called
a fuzzy model, model regardless of its eventual purpose.
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Input
Scaling
Factor
Components of a Fuzzy Systems
CRISP
Fuzzifier
Fuzzy Rule-Base
Fuzzy
Inference
Engine
FUZZY FUZZY
Defuzzifier
CRISP
Output
Scaling
Factor
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•
•
•
In fuzzy systems, the relationships between variables are
represented by means of fuzzy “IF-THEN” rules of the
following general form:
IF antecedent proposition
THEN consequent proposition
The antecedent proposition is always a fuzzy proposition of
the type “x is A” where x is a linguistic variable and A is a
linguistic term.
The proposition’s truth value (a real number between zero
and one) depends on the degree of match (similarity)
between x and A.
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•
Depending on the form of the consequent, four types of
rule-based fuzzy models are distinguished:
Mamdani (Linguistic) fuzzy model,
Singleton fuzzy model,
Takagi-Sugeno (T-S) fuzzy model,
(Tsukamoto fuzzy model).
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•
Mamdani (Linguistic) fuzzy model:
The linguistic fuzzy model (Zadeh, 1973; Mamdani, 1977)
has been introduced as a way to capture available (semi-)
qualitative knowledge in the form of if–then rules:
The input
(antecedent)
linguistic variable
The antecedent
linguistic terms
The consequent
linguistic terms
The output (consequent)
linguistic variable
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More examples?
•
•
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•
If
the pressure is HIGH then
If you are CLOSE to the other car then
If the room is VERY COLD then
to FULL
the volume is SMALL
brake is MAX
switch on the heater
If the error is ZERO and the change of the error is
ZERO then the change in control signal is ZERO
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•
•
•
Fuzzy logic operators, such as the conjunction,
disjunction and negation (complement), can be used to
combine the propositions.
A MIMO model can be written as a set of MISO models.
Therefore, for the ease of notation, we will write the
rules for MISO systems.
Most common is the conjunctive form of the antecedent,
which is given by:
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Example
Max-min composition
R1
R2
R3
: If X is small then Y is small
: If X is medium then Y is medium
: If X is large then Y is large
and centroid
X = input [10, 10]
Y = output [0, 10]
defuzzification were used.
Overall input-output curve
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Example
Max-min composition
X,
Y,
Z
[5, 5]
R1: If X is small and Y is small then Z is negative large
R2: If X is small and Y is large then Z is negative small
R3: If X is large and Y is small then Z is positive small
R4: If X is large and Y is large then Z is positive large
and centroid
defuzzification were used.
Overall input-output curve
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•
•
Singleton fuzzy model:
A special case of the linguistic fuzzy model is obtained
when the consequent fuzzy sets Bi are singleton fuzzy
sets.
These sets can be represented simply as real numbers,
yielding the following rules:
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•
•
Takagi-Sugeno fuzzy model:
The Takagi–Sugeno (T-S) fuzzy model (Takagi and
Sugeno, 1985), uses crisp functions in the consequents.
Hence, it can be seen as a combination of linguistic and
mathematical regression modeling in the sense that the
antecedents describe fuzzy regions in the input space in
which consequent functions are valid. The T-S rules have
the following form:
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•
•
The functions fi are typically of the same structure, only
the parameters in each rule are different.
Generally, fi is a vector-valued function, but for the ease of
notation we will consider a scalar fi in the sequel. A simple
and practically useful parameterization is the affine (linear
in parameters) form, yielding the rules:
What happens when a i is ZERO?
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60
50
40
30
20
10
0
1.2
1
0.8
0.6
0.4
0.2
f 2(x)
How do you define this function?
f 1(x)
0 1 2 3 4 5 6 7
Small Big
0
0 1 2 3 4 5 6 7
RULE 1:
IF x is Small
RULE 2:
IF x is Big
THEN y(x)=f1(s) THEN y(x)=f2(s) Folie 21

Example
R1: If X is small then Y = 0.1X + 6.4
R2: If X is medium then Y = 0.5X + 4
R3: If X is large then Y = X – 2
X = input
unsmooth
[10, 10]
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Example
R1: If X is small then Y = 0.1X + 6.4
R2: If X is medium then Y = 0.5X + 4
R3: If X is large then Y = X – 2
X = input [10, 10]
If we have smooth membership functions (fuzzy rules) the
overall input-output curve becomes a smoother one.
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Example
X, Y [5, 5]
R1: if X is small and Y is small then z = x +y +1
R2: if X is small and Y is large then z = y +3
R3: if X is large and Y is small then z = x +3
R4: if X is large and Y is large then z = x + y + 2
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What defines the size of the rule-base?
An example:
# of inputs 3: A, B, C
# of outputs 1: D
# of membership functions for each inputs:
• A 5
• B 3
• C 4
How many rules do you need?
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MIMO -
•
MISO
Note that the above model is a special case of
as the fuzzy set Ai in (1) is obtained as the Cartesian
product of fuzzy sets.
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•
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A set of rules in the conjunctive
antecedent form divides the input
domain into a lattice of fuzzy
hyperboxes, parallel with the axes.
Each of the hyperboxes is an Cartesian
product-space intersection of the
corresponding univariate fuzzy sets.
The number of rules in the conjunctive
form, needed to cover the entire
domain.
K
p
i1
p is the dimension of the input space, and Ni is the number of linguistic
terms of the ith antecedent variable.
N
i
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•
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By combining
– conjunctions,
– disjunctions,
– negations
various partitions of the antecedent
space can be obtained.
The boundaries are, however,
restricted to the rectangular grid
defined by the fuzzy sets of the
individual variables.
β
1
μ A ( x1
) μ A ( x 2 )
13 21
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•
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The antecedent form with multivariate
membership functions is the most general
one, as there is no restriction on the
shape of the fuzzy regions.
The boundaries between these regions can
be arbitrarily curved and opaque to the
axes.
Also the number of fuzzy sets needed to
cover the antecedent space may be much
smaller than in the previous cases.
Hence, for complex multivariable systems,
this partition may provide the most
effective representation.
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Decomposition
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Another way to reducing the complexity of
multivariable fuzzy systems is the decomposition into
subsystems with fewer inputs per rule base.
The subsystems can be inter-connected in a
hierarchical (multi-layer) structure.
This cascade connection will lead to the reduction of
the total number of rules.
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Fuzzification
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The first stage in the fuzzy system computations is to
transform the numeric inputs into fuzzy sets.
This operation is called fuzzification.
fuzzification
A Fuzzifier, Fuzzifier F, converts a numeric value, x* into a
fuzzy set. We require that the fuzzy set fulfills the
following axioms:
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Types of fuzzifiers:
1.
2.
Singleton Fuzzifiers:
Non-Singleton Fuzzifiers:
•
•
•
Triangular
Gaussian
…
• The singleton fuzzifier is predominant in fuzzy control.
• The main reason for this is that this fuzzifier simplifies the
computations in the fuzzy system considerably.
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Defuzzification
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•
•
The result of the inference process is an output
represented by a fuzzy set. As discussed previously,
the output of the fuzzy system should be a numeric
value.
The transformation of a fuzzy set into a numeric value
is called defuzzification.
defuzzification
A Defuzzifier
Defuzzifier, D, maps a fuzzy set into a numeric
value y*. We require the defuzzification operation to
fulfill the following axiom:
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•
1.
Types of defuzzifiers:
Several defuzzification operations have been suggested in
the literature.
The Center of Gravity
(COG) defuzzifier specifies the
numeric value to be the center of gravity of the fuzzy set.
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Types of defuzzifiers:
2.
The Center Average
(CA) defuzzifier specifies the
numeric value to be the weighted average of the centers,
ў (l) of the individual consequent fuzzy sets.
The weights, w (l) are equal to the height of the respective
fuzzy sets.
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Types of defuzzifiers:
3.
Maximum defuzzifiers determines y* as a point in the
interval IM where the fuzzy set has its maximum
value.
Since the defuzzifier should infer a unique output,
there are three different Maximum defuzzifiers
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Types of defuzzifiers:
There are several factors which can influence the choice of
defuzzifier. We may for instance require that the defuzzifier
produces a plausible result in computationally simple
way.
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Scaling Factors
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•
The input/output scaling factors may be used for
normalization of the input/output.
For the design of a fuzzy controller, the scaling factors
act like the controller gains. Therefore, they have an
important effect on the system performance and
stability.
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Fuzzy System (Overview)
A fuzzy system of class , consists of
a rule base
a fuzzifier
an inference engine
a defuzzifier
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Let’s design a Fuzzy System!!!
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Time
Distance
Close Average Far
A bit late Medium Little Little
Late High High Medium
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