seminar introduction to fuzzy logic i - eneaseminar introduction to fuzzy logic i itziar...

SeminarIntroduction to Fuzzy Logic I

Itziar Garcıa-Honrado

European Centre for Soft-ComputingMieres (Asturias)

Spain

05/04/2011

Itziar Garcıa-Honrado (ECSC) Fuzzy Logic 05/04/2011 1 / 50

Figure: Crisp Figure: Fuzzy


Control Systems I

Figure: Washing machineFigure: Rice cooker


Control Systems II

Figure: Unmanned helicopter Figure: Tensiometer


Control Systems III

Figure: Inverted pendulum


Control Systems IV

Example

A system with two continuous variables x , y ∈ [0, 1], described by 2 rules,

r1 : If x is big, then y is small

r2 : If x is very small, then y is very big

What is the output for the input x=0.4?


Control Systems V

Esquema


How to represent predicates? I

≤P is perceptive relation

L-degrees µPi: X → L (i = 1, 2, 3), where L is an ordered structure.


How to represent predicates? II

Types of predicates:

Rigid: consists in two classes. For instance, “To be even”

Semirigid: consists in a finite number of classes

Imprecise: consists in infinite classes. For instance, tall.


How to represent predicates? III

Rigid≡ Crisp

P =“To be 8”


How to represent predicates? IV

Semirigid≡ Finite number of classes


How to represent predicates? V

Each predicate P, could be represented by many fuzzysets depending on its use)

“Philosophical investigations”, Wittgenstein (1953): The meaning of a predicate

is its use in Language

Different models of P=small in X = [0, 10], with L = [0, 1] depending onthe use of P.


How to represent predicates? VI

Each use of the predicate is represented by a curve (by a fuzzy set).

µ : X → [0, 1]

A [0, 1]-degree µP is any function X → [0, 1], such that

If x ≤P y , then µP(x) ≤ µP(y),

It the inverse order introduce by the predicate ≤−1P is defined as

x ≤−1P y ⇔ y ≤P x .

So, x =P y iff x ≤P and x ≤−1P .

If µP(x) = r , it is said that x ∈r P.


How to represent predicates? VII

Design the representation of the predicate: Context, purpose, use,...

Example

Predicate tall in two different contexts, in a school or in a basketballteam.


How to represent predicates? VIII

Family resemblance in [0, 1]X , fr ⊂ F∗(X )×F∗(X )(read (µ, σ) ∈ fr: µ and σ show family resemblance)

(µ, σ) ∈ fr ⇔

1) Z (µ) ∩ Z (σ) 6= ∅,S(µ) ∩ S(σ) 6= ∅2) µ is non-decreasing in A ⊂ X ⇔

σ is non-decreasing in A3) µ is decreasing in A ⊂ X ⇔ σ is decreasing in A

S(µ) = {x ∈ X ;µ(x) = 1} and Z (µ) = {x ∈ X ;µ(x) = 0}.


How to represent predicates? IX

example 1

Z (µ) ∩ Z (σ) = [0, 2]S(µ)∩S(σ) = {10} ⇒ (µ, σ) ∈ fr

example 2

S(µ) ∩ S(σ) = ∅ ⇒ (µ, σ) /∈ fr


Opposite and negate I

Opposite or antonym

P aPeven oddtall shortsmall big


Opposite and negate II

Concerning the opposite aP of P, this opposition is translated by

≤aP=≤−1P

Hence, ≤a(aP)=≤−1aP = (≤−1

P )−1 =≤P , that forces a(aP) = P.

For example, with P =tall, it is aP = short and a(aP) = tall.

This property of aP shows a way for obtaining µaP once µP is known.Let it A : X → X be a symmetry on X , that is a function such that

If x ≤P y , then A(y) ≤P A(x)

A ◦ A = idX ,


Opposite and negate III

Once µP : X → L is known, take µaP(x) = µP(A(x)), for all x in X , thatis, µaP = µP ◦ A. Function µaP = µP ◦ A is a degree for aP, since:

x ≤aP y ⇔ y ≤P x ⇒ A(x) ≤P A(y) ⇒ µP(A(x)) ≤ µP(A(y)),

and verifies,

µa(aP) = µ(aP) ◦ A = (µP ◦ A) ◦ A = µP ◦ (A ◦ A) = µP ◦ idX = µP .


Opposite and negate IV

Example

Opposite of small

A?


Opposite and negate V

Example 2

Opposite of around 4A(x) = 10− x


Opposite and negate VI

Example 2

Opposite of around 4

A(x) =

{4− x , if x ≤ 414− x , if otherwise


Opposite and negate VII

Let it P be a predicate in X , and P ′=notP its negate.Concerning the negation P ′ of P, this negation is translated by

≤P′⊂≤−1P

Since,

If x is less P than y , then y is less not P than x ,

or, equivalently, ≤P⊂≤−1P′ .


Opposite and negate VIII

Let it N : L → L be a function such that

1. If a ≤ b, then N(b) ≤ N(a), for all a, b in L

2. N(α) = ω, and N(ω) = α, being α the infimum and ω the supremumof L.

with such a function N, it is

µP′ = N ◦ µP

an L-degree for P ′, since

x ≤P′ y ⇒ y ≤P x ⇒ µP(y) ≤ µP(x) ⇒

N(µP(x)) ≤ N(µP(y)) ⇔ µP′(x) ≤ µP′(y).


Opposite and negate IX

Provided the negation function does verify

3. N ◦ N =idL,

then

µ(P′)′(x) = N(µP′(x)) = N(N(µP(x))) = (N◦N)(µP(x)) = idL(µP(x)) = µP(x),

for all x in X , or µ(P′)′ = µP .Functions N verifying (1), (2), and (3) are called strong negations


Opposite and negate X

If L = [0, 1], there is a family of strong negations widely used in fuzzy settheory, the so-called Sugeno’s negations:

Nλ(a) =1− a

1 + λa, with λ > −1, for all a ∈ [0, 1].

For example, N(a) = 1− a,N1(a) = 1−a1+a ,N−0.5 = 1−a

1−0.5a ,N2(a) = 1−a1+2a ,

etc.Since obviously,

Nλ1 ≤ Nλ2 ⇔ λ2 ≤ λ1,

it results:

If λ ∈ (−1, 0], then N0 ≤ Nλ

If λ ∈ (0,+∞], then Nλ < N0.


Opposite and negate XI

Graphically,


Opposite and negate XII

With N0(a) = 1− a, and A(x) = 10− x in X = [0, 10], if

µbig (x) =

0, if x ∈ [0, 4]x−44 , if ∈ [4, 8]

1, if x ∈ [8, 10],

results

µsmall(x) = µbig (10− x) =

0, if x ∈ [6, 10]6−x4 , if ∈ [2, 6]

1, if x ∈ [0, 2],

µnot big (x) = 1− µbig (x) =

1, if x ∈ [0, 4]8−x4 , if ∈ [4, 8]

0, if x ∈ [8, 10],

whose graphics


Opposite and negate XIII

show that the pair (big, small) is coherent, since µsmall ≤ µnot big .

Notice that µaP ≤ µP′


Opposite and negate XIV

Remarks

Notice that µaP ≤ µP′

a(aP) = P, but not always (P ′)′ = P. (empty/full)

Notice the main difference of a symmetry and a negation A : X → Xand N : L → L


Union and intersection I

Let P, Q be two predicates in X .Consider the new predicates ‘P and Q’, and ‘P or Q’, used by means of:

‘x is P and Q’ ⇔ ‘x is P’ and ‘x is Q’

‘x is P or Q’ ⇔ Not (Not ‘x is P ’ and Not ‘x is Q’). (Duality)


Union and intersection II

Intersection

Take L-degrees µP , µQ .Given (L,≤), let ∗ : L× L → L be an operation, verifying the properties

a ≤ b, c ≤ d ⇒ a ∗ c ≤ b ∗ d

a ∗ c ≤ a, a ∗ c ≤ c ,

Then, µP(x) ∗ µQ(x) is an L-degree for P and Q in X , since:

x ≤P and Q y ⇒ x ≤P y and x ≤Q y ⇒ µP(x) ≤ µP(y) and µQ(x) ≤µQ(y) ⇒ µP(x) ∗ µQ(x) ≤ µP(y) ∗ µQ(y).


Union and intersection III

Hence,µP(x) ∗ µQ(x) = µP and Q(x),

is an L-degree for ‘P and Q’ in X , and the operation ∗ can be called anand-operation.Notice that it is,

µP and Q(x) ≤ µP(x), and µP and Q(x) ≤ µQ(x).


Union and intersection IV

If ∗ is an and-operation, and N : L → L is a strong negation, define

a⊕ b = N(N(a) ∗ N(b)), for all a, b ∈ L.

Since a ≤ b, c ≤ d ⇒ N(b) ≤ N(a),N(d) ≤ N(c) ⇒ N(b) ∗ N(d) ≤N(a) ∗ N(c) ⇒ N(N(a) ∗ N(c)) ≤ N(N(b) ∗ N(d)), it resultsa⊕ c ≤ b ⊕ d .

Analogously, from N(a) ∗ N(b) ≤ N(a), it followsa ≤ N(N(a) ∗ N(b)) = a⊕ b, and b ≤ a⊕ b, for all a, b.


Union and intersection V

Then, µP(x)⊕ µQ(x) is an L-degree for P or Q.

x ≤P or Q y ⇔ x ≤(P′ and Q′)′ y ⇔

y ≤P′ and Q′ x ⇒ µP′ and Q′(y) ≤ µP′ and Q′(x) ⇔

µP′(y) ∗ µQ′(y) ≤ µP′(x) ∗ µQ′(x) ⇒

N(µP(y)) ∗ N(µQ(y)) ≤ N(µP(x)) ∗ N(µQ(x)) ⇒

N(N(µP(x)) ∗ N(µQ(x)) ≤ N(N(µP(y)) ∗ N(µQ(y)) ⇔

µP(x)⊕ µQ(x) ≤ µP(y)⊕ µQ(y).

Hence, µP(x)⊕ µQ(x) can be taken as an L-degree for ‘P or Q’ in X, andthe operation ⊕ can be called an or-operation.


Union and intersection VI

Remark

The existence of operations ∗ and ⊕ in L, warrants the existence ofL-degrees for and, or, respectively.


Union and intersection VIIIn fact, other basic properties of ∗ and ⊕ are collected in the definition ofA Basic Flexible Algebra (BFA).A BFA is a seven-tuple L = (L,≤, 0, 1; ∗,⊕,′ ), where L is a non-emptyset, and

1 (L,≤) is a poset with minimum 0, and maximum 1.2 ∗ and ⊕ are mappings (binary operations) L× L → L, such that:

1 a∗1 = 1∗a = a, a∗0 = 0∗a = 0, for all a ∈ L2 a⊕ 1 = 1⊕ a = 1, a⊕ 0 = 0⊕ a = a, for all a ∈ L3 If a ≤ b, then a∗c ≤ b∗c , c∗a ≤ c∗b, for all a, b, c ∈ L4 If a ≤ b, then a⊕ c ≤ b ⊕ c , c ⊕ a ≤ c ⊕ b, for all a, b, c ∈ L

3 ′ : L → L verifies1 0′ = 1, 1′ = 02 If a ≤ b, then b′ ≤ a′

4 It exists L0, {0, 1} ⊂ L0 L, such that with the restriction of theorder and the three operations ∗,⊕, and ′ of L,L0 = (L0,≤, 0, 1; ∗,⊕,′ ) is a boolean algebra


Union and intersection VIII

∗ for fuzzy sets (T-norms)

A continuous t-norm is with mappings T : [0, 1]× [0, 1] → [0, 1] verifyingthe following properties,

Continuity in both variables

Associativity T (T (x , y), z) = T (x ,T (y , z)), ∀x , y , z ∈ [0, 1]

Commutativity T (x , y) = T (y , x) ∀x ∈ [0, 1]

Monotonicity T (x , y) ≤ T (z , t) if x ≤ z , y ≤ t

T (x , 1) = x , ∀x ∈ [0, 1]

T (x , 0) = 0, ∀x ∈ [0, 1]


Union and intersection IX

⊕ for fuzzy sets (T-conorms)

A continuous t-conorm is with mappings S : [0, 1]× [0, 1] → [0, 1] verifyingthe following properties,

Continuity in both variables

Associativity S(S(x , y), z) = S(x ,S(y , z)), ∀x , y , z ∈ [0, 1]

Commutativity S(x , y) = S(y , x) ∀x ∈ [0, 1]

Monotonicity S(x , y) ≤ S(z , t) if x ≤ z , y ≤ t

S(x , 0) = x , ∀x ∈ [0, 1]

S(x , 1) = 1, ∀x ∈ [0, 1]


Union and intersection X

Medium

Given P, if MP = Not P and Not aP, with N for not, A for the opposite,and ∗ for and, results

µMP(x) = µP′ and (aP)′(x) = µP′(x)∗µ(aP)′(x) = N(µP(x))∗N(µP(A(x))),

for all x in X .


Modifiers I

Linguistic modifiers or linguistic hedges, m, are adverbs acting on P just inthe concatenated form mP. For example, with m =very and P= tall, it ismP = very tall.

Among linguistic modifiers there are two specially interesting types:

Expansive modifiers, verifying idµP(x) ≤ µm,

Contractive modifiers, verifying µm ≤ idµP(x).


Modifiers II

With the expansive, it results

idµP(x)(µP(x)) = µP(x) ≤ µm(µP(x)) = µmP(x) : µP(x) ≤ µmP(x), for allx in X .

With the contractive, it resultsµmP(x) = µm(µP(x)) ≤idµP(x)(µP(x)) = µP(x) : µmP(x) ≤ µP(x),for allx in X .

In L = [0, 1] with the Zadeh’s old definitions,

µmore or less(a) =√

a, µvery (a) = a2.

Which one is contractive?


Linguistic Variable I

A linguistic variable LV is formed after considering

1 Its principal predicate, P

2 One of the opposites of P, aP

3 Some linguistic modifiers m1, . . . ,mn,

and by adding:

4 Its negate (not P), or the middle-predicate (MP), or P and Q, or notm1P, or P and m2aP, ...


Linguistic Variable II

Then LV, is called the linguistic variable generated by P, and reflects thelinguistic granulation perceived for the concept. For example,

LV=Age, is Age= { young, old, middle-aged, not old, not veryyoung,..}LV=Temperature, is Temp= { cold, hot, warm, not cold, not veryhot,..}LV=Size, is Size= {large, small, medium, very large,..}


Linguistic Variable III

Usually in fuzzy control are used only three labels, for instance:


Extension Principle I

Extension Principle

× : R × R → R, can be extended to ⊗ : [0, 1]R × [0, 1]R → [0, 1]R , bymeans of Extension Principle

(µ⊗ σ)(t) = Supt=x×y

min(µ(x), σ(y)).


Extension Principle II

Example extending the operation of +

Let’s calculate the sum for the following fuzzy sets:

µ = 0/0 + 1/1 + 0/3 + 0/4

σ = 0/0 + 0/1 + 0.5/2 + 1/3 + 0/4

µ⊕ σ = 0/0 + 0/1 + 0/2 + 0.5/3 + 1/4 + 0.5/5 + 0/6

Since,

(µ⊗ σ)(0) = Sup0=x×y

min(µ(x), σ(y)) = Sup(min(µ(0), σ(0))).

(µ⊗ σ)(1) = Sup1=x×y

min(µ(x), σ(y)) =

Sup(min(µ(1), σ(0)),min(µ(0), σ(1))) = Sup(0, 0) = 0.

(µ⊗ σ)(2) = Sup2=x×y

min(µ(x), σ(y)) =

Sup(min(µ(2), σ(0)),min(µ(0), σ(2)),min(µ(1), σ(1))) =Sup(0, 0, 0) = 0.


Extension Principle III

(µ⊗ σ)(3) = Sup3=x×y

min(µ(x), σ(y)) =

Sup(min(µ(3), σ(0)),min(µ(0), σ(3)),min(µ(1), σ(2)),min(µ(2), σ(1))) = Sup(0, 0, 0.5, 0) = 0.5.

(µ⊗ σ)(4) = Sup3=x×y

min(µ(x), σ(y)) =

Sup(min(µ(4), σ(0)),min(µ(0), σ(4)),min(µ(1), σ(3)),min(µ(3), σ(1)),min(µ(2), σ(2))) = Sup(0, 0, 1, 0.5, 0) = 1.

...


Extension Principle IV


Thanks for your atention!


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