Introduction to Intelligent Control Part 3keffe006/Workshop/ECE4951-Lecture3.pdf · Introduction to Intelligent Control Part 3 ECE 4951 - Spring 2010 Prof. Marian S. Stachowicz

Introduction to Intelligent ControlPart 3

ECE 4951 - Spring 2010

Part 3

Prof. Marian S. StachowiczLaboratory for Intelligent Systems

ECE Department, University of Minnesota Duluth

January 26 - 29, 2010

Part 1: Outline

• TYPES OF UNCERTAINTY

• Fuzzy Sets and Basic Operations on Fuzzy Sets

• Further Operations on Fuzzy Sets

2Intelligent Control

References for reading

1. M.S. Stachowicz, Lance Beall, Fuzzy Logic Packag e,Version-2 for Mathematica 5.1, Wolfram Research, Inc ., 2003

- Demonstration Notebook: 2.1, 2.2, 2.7.1, 2.7.2, 2.8.1, 2.8.2, 2.8.3, 2.8.5, 2.8.6, 2.8.8, 2.8.9- Manual: 1.1, 1.3.1, 1.3.2, 1.3.4, 1.4, 1.5


Randomness versus Fuzziness

• Randomness refers to an event that my or may not occur. Randomness: frequency of car accidents . Randomness: frequency of car accidents .

Fuzziness refers the boundary of a set that is not precise. Fuzziness: seriousness of a car accident.

Prof. George Klir

PROFESSOR GEJ. 4Intelligent Control

TYPES OF UNCERTAINTY

• STOCHASTIC UNCERTAINTYTHE PROBABILITY OF HITTING THE TARGET IS 0.8.

• LEXICAL UNCERTAINTYWE WILL PROBABLY HAVE A SUCCESFUL FINANCIAL YEAR.


FUZZY SETS THEORY versus PROBABILITY THEORY

• Patients suffering from hepatitis show in 60% of all cases high fever, in 45 % of all 60% of all cases high fever, in 45 % of all cases a yellowish colored skin, and in 30% of all cases nausea .


What are Fuzzy Sets?

Fuzzy setLotfi A. Zadeh[1965]

• A fuzzy subset A of a universe of discourse U is characterized by a membership function

µµµµA : U ⇒⇒⇒⇒ [0,1]µµµµA : U ⇒⇒⇒⇒ [0,1]which associates with each element u of U a number µµµµA(u) in the interval [0,1], which µµµµA(u) representing the grade of membership of u in A.


Fuzzy Sets

Fuzzy set A defined in the universal space U is a function defined in U which assumes is a function defined in U which assumes values in the range [ 0,1 ].

A : U ⇒⇒⇒⇒ [ 0, 1]


Characteristic Function

A : U ⇒⇒⇒⇒ {0, 1}

Membership Function

M : U ⇒⇒⇒⇒ [0, 1]


Universal Set

• U - is the universe of discourse, or universal set, which contains all the possible elements set, which contains all the possible elements of concern in each particular context of applications.


Membership function

The membership function M maps each element of U to a membership grade element of U to a membership grade ( or membership value) between 0 and 1.


Presentation of a fuzzy set

• A fuzzy set M, in the universal set can be presented by:presented by:- list form,- rule form,- membership function form.


List form of a fuzzy set

M = {{1,1},{2,1},{3,0.9},{4,0.7},{5,0.3},{6,0.1},{7,0},{8,0},{9,0},{10,0},{11,0},{12,0}},

where M is the membership function (MF) for fuzzy where M is the membership function (MF) for fuzzy set M.

Note:The list form can be used only for finite sets.


Fuzzy Logic Package form [M.S. Stachowicz + Lance Beall ,1995 & 2003]

M={{1,1},{2,1},{3,0.9},{4,0.7},{5,0.3},{6,0.1}} andandU={1,2,3,4,5,6,7,8,9,10,11,12}


Rule form of a fuzzy set

M = { u ∈∈∈∈ Uu meets some conditions},

where symbol denotes the phrase “such as”.


Membership form of a fuzzy set

Let A1 be a fuzzy set named ”numbers closed to zero”

A1(u) =exp(-u2)A1(u) =exp(-u )

A1(0)=1

A1(2)=exp(-4)

A1(-2)=exp(-4)


Numbers closed to zero

Representation of a fuzzy set

A fuzzy set A in U may be represented as a set

of ordered pairs of generic element u and its membership

value A(u),

A = {{u, A(u)} |||| u ∈∈∈∈ U}

value A(u),



When U is continuous, A is commonly written as:

∫∫∫∫ A(u) ////u∫∫∫∫u A(u) ////u

where integral sign does not denote integration; it denotes

the collection of all points u ∈∈∈∈ U with the associated MF

A(u).



When U is discrete, A is commonly written as:

∑ A(u) ////u,

where the summation sign does not represent arithmetic

addition.


Example 1

• Let U be the integer from 1 to 10, that U={ 1,2,…,9,10}. The fuzzy set “ Several ”may be defined as using:- the summation notation

Several ={0.5/3 + 0.8/4 + 1/5 + 1/6 +0.8/7 + 0.5/8}- FLP notation

Several ={{3.0.5},{4,0.8},{5,1},{6,1},{7,0.8},{8,0.5}}


Basic concepts and terminology

• The concepts of support, fuzzy singleton, crossover point, height, normal FS, crossover point, height, normal FS, αααα-cut, and convex fuzzy set are defined as follows:


Core, support, and crossover pointCore, support, and crossover point

MF

.5

1

u

.5

0Core

Crossover points

Support

α α α α - cut

αααα


The support of fuzzy set A

• The support of a fuzzy set A in the universal set U is a crisp set that contains all the elements of U that have nonzero membership values in A, that is,values in A, that is,

supp(A)= {u ∈∈∈∈ U | A(u) > 0}


Core

• The core of a fuzzy set A is the set of all points u in U such that A(u) = 1

Core(A) = {u | A(u) = 1}Core(A) = {u | A(u) = 1}


Normality

• A fuzzy set A is normal if its core in nonempty.

∃∃∃∃ u ∈∈∈∈ U A(u) = 1


Example

• Several ={0.5/3 + 0.8/4 + 1/5 + 1/6 +0.8/7 + 0.5/8} in U={ 1,2,…,9,10}.

• Supp( Several ) = {3,4,5,6,7,8}


Fuzzy singleton

• A fuzzy singleton is a fuzzy set A(u)=1 whose support is a single point in U.


Crossover point

• The crossover point of a fuzzy set is the point in U whose membership value in A equals 0.5.


Height

• The height of a fuzzy set is the largest membership value attained by any point.

• If the height of fuzzy set equals one, it is • If the height of fuzzy set equals one, it is called a normal fuzzy set.


Alpha Cuts

Alpha-cut

• Alpha-cut of fuzzy set/fuzzy relation is the crisp set that contains all the elements of universal space whose membership grades in set/relation are greater than or equal to the specified value are greater than or equal to the specified value of alpha.

Crisp set ααααA = { x|A(x) ≥≥≥≥ αααα }


Strong alpha-cut

• Strong alpha-cut of fuzzy set/fuzzy relation is the crisp set that contains all the elements of universal space whose membership grades in set/relation are greater than the specified in set/relation are greater than the specified value of alpha.

Crisp set αααα+A = { x|A(x) >>>> αααα }


Level set

• The set of all alpha-cuts of a fuzzy set/fuzzy relation is called a level set of set/relation .relation is called a level set of set/relation .

L(A) = {αααα|A(x) = αααα for some x ∈∈∈∈ X


Level set

LevelSet [fs1]

L(fs1) = {0.2, 0.4, 0.6, 0.8, 1} 36Intelligent Control

Alpha-cuts

(0.2), {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16},(0.4), {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, (0.6), {5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, (0.6), {5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, (0.8), {6, 7, 8, 9, 10, 11, 12, 13}, (1.0), {8, 9, 10, 11, 12}.

∀ A if α1< α2 then α1A ⊇ α2A

α1A ∩ α2A = α2A α1A ∪ α2A = α1A


Decomposition of fuzzy sets

For any A ⊂ ℜ

A = ∪ α αA(x) for α ∈ [ 0,1]A = ∪ α αA(x) for α ∈ [ 0,1]


The relationship between fuzzy set and crisp set

• Each fuzzy set can be uniquely presented by the family of all its αααα-cuts.the family of all its αααα-cuts.

• This representation allows extending various properties of crisp set and operations on crisp sets to their fuzzy counterparts.


Decomposition of fuzzy sets

L(fs1) = {0.2, 0.4, 0.6, 0.8, 1}


Property of alpha-cuts

• ∀ A if α1< α2 then α1A ⊇ α2A

• α1A ∩ α2A = α2A • α1A ∩ α2A = α2A • α1A ∪ α2A = α1A


Convex fuzzy set

• A fuzzy set A is convex if and only if it’s αααα -cuts ααααA is a convex set for any αααα in the interval αααα ∈∈∈∈ (0,1].interval αααα ∈∈∈∈ (0,1].

A[λλλλ x1 + (1- λλλλ ) x2] ≥≥≥≥ min[A(x 1),A(x 2)]for all x 1, x2 ∈∈∈∈ Rn and all λλλλ ∈∈∈∈ [0,1].


The Mathematics of Fuzzy SystemsPart 2

ECE 5831 - Fall 2009

Part 2


ECE Department, University of Minnesota, USA

October 1, 2009


1. W. Pedrycz and F. Gomide, Fuzzy Systems Engineering,

J. Wiley & Sons, Ltd, 2007Chapter 2 and 3

2. M.S. Stachowicz, Lance Beall, Fuzzy Logic Package ,2. M.S. Stachowicz, Lance Beall, Fuzzy Logic Package ,Version-2 for Mathematica 5.1, Wolfram Research, In c., 2003


3. G.J. Klir, Ute H.St. Clair, Bo Yuan, Fuzzy Set Th eory, Prentice Hall, 1997

Chapters 1, 2


Operations on fuzzy sets

• Inclusion• Equality• Standard Complement• Standard Complement• Standard Union• Standard Intersection


Inclusion

• Let X and Y be fuzzy sets defined in the same universal space U. We say that the fuzzy set X is included in the fuzzy set Y if and only if:fuzzy set Y if and only if:for every u in the set U we have X(u) ≤ Y(u)


Subset and proper subset

• X is a subset of Y, or is smaller than or equal to Y if and only if X(u) ≤ Y(u) for all u.

X ⊆⊆⊆⊆ YX ⊆⊆⊆⊆ Y

• X is a proper subset of Y, or is smaller than Y if and only if X(u) < Y(u) for all u.

X ⊂⊂⊂⊂ Y


Equality

• Let X and Y be fuzzy sets defined in the same universal space U. We say that sets X and Y are equal, which is denoted X = Y if and only if for all u in the set U , denoted X = Y if and only if for all u in the set U , X(u) = Y(u).


Standard complement

• Let X be fuzzy sets defined in the universal space U. We say that the fuzzy set Y is a complement of the fuzzy set X, if and only if, for all u in the set U,

Y(u) = 1 - X(u).Y(u) = 1 - X(u).


Standard union

• Let X and Y be fuzzy sets defined in the space U. We define the union of those sets as the smallest (in the sense of the inclusion) fuzzy set that contains both X and Y.

∀∀∀∀ u∈∈∈∈U, (X ∪∪∪∪ Y)(u) = Max(X(u), Y(u)).


Standard union

• ∀∀∀∀ u ∈∈∈∈ U, (X ∪∪∪∪ Y)(u) = Max(X(u), Y(u)).


Standard intersection

• Let X and Y be fuzzy sets defined in the space U. We define the intersection of those sets as the greate st (in the sense of the inclusion) fuzzy set that includ ed both in X and Y.

∀∀∀∀ u∈∈∈∈U, (X ∩∩∩∩ Y)(u) = Min(X(u), Y(u)).


Standard intersection

• ∀∀∀∀ u∈∈∈∈U, (X ∩∩∩∩ Y)(u) = Min(X(u), Y(u)).


Properties of crisp set operations

[A ∩∩∩∩ B = B ∩∩∩∩ A

Associativity (A U B) U C = A U (B U C)(A ∩∩∩∩ B) ∩∩∩∩ C = A ∩∩∩∩ (B ∩∩∩∩ C)

Distributivity A ∩∩∩∩ (B U C) = (A ∩∩∩∩ B) U (A ∩∩∩∩ C)A U (B ∩∩∩∩ C) = (A U B) ∩∩∩∩ (A U C)

Idempotents A U A = A` A ∩∩∩∩ A = A


Properties of fuzzy set operations

Involution (A’)’= A

Commutativity A U B = B U AA ∩∩∩∩ B = B ∩∩∩∩ A

Associativity (A U B) U C = A U (B U C)Associativity (A U B) U C = A U (B U C)(A ∩∩∩∩ B) ∩∩∩∩ C = A ∩∩∩∩ (B ∩∩∩∩ C)

Distributivity A ∩∩∩∩ (B U C) = (A ∩∩∩∩ B) U (A ∩∩∩∩ C)A U (B ∩∩∩∩ C) = (A U B) ∩∩∩∩ (A U C)

Idempotents A U A = AA ∩∩∩∩ A = A



Absorption A U (A ∩∩∩∩ B) = AA ∩∩∩∩ (A U B) = A

Absorption by X and ∅∅∅∅ A U X = XA ∩∩∩∩ ∅∅∅∅ = ∅∅∅∅

Identity A U ∅∅∅∅ = AIdentity A U ∅∅∅∅ = AA ∩∩∩∩ X = A

De Morgan’s laws (A ∩∩∩∩ B)’= A’ U B’(A U B)’ = A’ ∩∩∩∩ B’



Absorption A U (A ∩∩∩∩ B) = AA ∩∩∩∩ (A U B) = A

Absorption by X and ∅∅∅∅ A U X = XA ∩∩∩∩ ∅∅∅∅ = ∅∅∅∅

Identity A U ∅∅∅∅ = AIdentity A U ∅∅∅∅ = AA ∩∩∩∩ X = A

De Morgan’s laws (A ∩∩∩∩ B)’= A’ U B’(A U B)’ = A’ ∩∩∩∩ B’



Law of contradiction A ∩∩∩∩ A’= ∅∅∅∅

Law of excluded middle A U A’= X Law of excluded middle A U A’= X



Law of contradiction A ∩∩∩∩ A’ ≠≠≠≠ ∅∅∅∅

Law of excluded middle A U A’≠≠≠≠ X


Law of excluded middle A U A’≠≠≠≠ X

FuzzyPlot[MEDIUM U Complement[MEDIUM]]


Law of contradiction A ∩∩∩∩ A’ ≠≠≠≠ ∅∅∅∅

FuzzyPlot[MEDIUM ∩∩∩∩ Complement[MEDIUM]]


A general principle of duality

• For each valid equation in set theory that is based on the union and intersection operations, there corresponds a dual equation, operations, there corresponds a dual equation, also valid, that is obtained by replacing ∅∅∅∅, U, and ∩∩∩∩ with X, ∩∩∩∩, and U, respectively, and vice versa.


Standard operators 1

• When the range of grade of membership is restricted to the set {0,1}, these functions perform like the corresponding operators for Cantor's sets.


Standard operators 2

• If any error e is associated with the grade of membership A(u) and B(u), then the maximum error associated with the grade of membership of u in A', Union[A, B], and Intersection[A, B] remains e.Intersection[A, B] remains e.


Characteristic function

• Ch. de la Valle Poussin [1950], Integrales de

Lebesque, fonction d'ensemble, classes de

Baire, 2-e ed., Paris, Gauthier-Villars.


Membership function

• L. A. Zadeh [1965], Fuzzy sets ,Information and Control, volume 8, pp. 338-353.


• Goguen, J.A.[1967] L-fuzzy sets,

J. of Math Analysis and Applications,

18(1), pp.145-174.


• M. S. Stachowicz and M. E. Kochanska [1982],Graphic interpretation of fuzzy sets and fuzzy relations , Mathematics at the Service of fuzzy relations , Mathematics at the Service of Man. Edited by A. Ballester, D. Cardus, and E. Trillas, based on materials of Second World Conference, Universidad Politecnica Las Palmas, Spain.


www.wolfram.com/fuzzylogic

M.S. STACHOWICZ and L. BEALL [1995, 2003]


Individual decision making

ECE 5831 - Fall 2009


ECE Department, University of Minnesota, USA

October 10, 2009


1. W. Pedrycz and F. Gomide, Fuzzy Systems Engineering,

J. Wiley & Sons, Ltd, 2007Chapter 2 and 3

2. M.S. Stachowicz, Lance Beall, Fuzzy Logic Package ,2. M.S. Stachowicz, Lance Beall, Fuzzy Logic Package ,Version-2 for Mathematica 5.1, Wolfram Research, In c., 2003


3. G.J. Klir, Ute H.St. Clair, Bo Yuan, Fuzzy Set Th eory, Prentice Hall, 1997

Chapters 1, 2


Individual decision making

A decision is characterized by components:• Universal space U of possible actions;• a set of goals G i (i ∈∈∈∈ Nn) defined on U;• a set constraints C j (j ∈∈∈∈ Nn) defined on U.

Decision is determined by an aggregation operator.


AG-H - Faculty of Management, 22-26 May, 2006


Example: Job Selection

• Suppose that Sebastian from AG-H needs to decide which of the four possible jobs, say

• g(a1) = $40,000• g(a1) = $40,000• g(a2) = $45,000• g(a3) = $50,000• g(a4) = $60,000

to choose.


The constraints

• His goal is to choose a job that offers a high salary under the constraints that the job is interesting and within close driving distance.


In this case, the goal and constraints are all uncertain concepts and we need to use fuzzy sets to represent these concepts. sets to represent these concepts.


Goal: High salary-indirect form

• CF = FuzzyTrapezoid[37, 64, 75, 75,

UniversalSpace -> {0, 80, 5}]

• FuzzySet [{{40, 0.11}, {45, 0.3}, {50, 0.48}, {55, 0.67}, {60, 0.85},

{65, 1}, {70, 1}, {75, 1}}{65, 1}, {70, 1}, {75, 1}}


Fuzzy goal G: High salary-direct form

G = FuzzySet[{{1, .11}, {2, .3}, {3, .48}, {4, .85}}]


Constraint C1: Interesting job

C1 = FuzzySet[{{1, .4}, {2, .6}, {3, .2}, {4, .2}}]


Constraint C2: Close driving

C2 = FuzzySet[{{1, .1}, {2, .9}, {3, .7}, {4, 1}}]


Concept of desirable job

D = Intersection[G, C1, C2] =

FuzzySet[{{1, 0.1}, {2, 0.3}, {3, 0.2}, {4, 0.2}}]


For Sebastian from AG-H $ 45,000.

• The final result from the above analysis is a2, which is the most desirable job among the four available jobs under the given goal G and available jobs under the given goal G and constraints C1 and C2.

g(a2) = $ 45,000 with D( a2) = 0.3


Example: Optimal dividends

• The board of directors of a company needs to determine the optimal dividend to paid to the shareholders. For financial reasons, the the shareholders. For financial reasons, the dividend should be attractive (goal G) ; for reasons of wage negotiations, it should be modest (constraint C).The U is a set of possible dividends actions.


In this case, the goal and constraint are both uncertain concepts and we need to use fuzzy sets to represent these concepts. sets to represent these concepts.


Example: Optimal dividends

C : modest G : attractive

û : optimal dividends

↑85Intelligent Control

Optimal dividends

• OptimalDividends = Core[Normalize[Intersection

[modest, attractive]]][modest, attractive]]]


Comments

• Since this method ignores information concerning any of the other alternatives, it may not be desirable in all situations.

• An averaging operator may be used to reflect a • An averaging operator may be used to reflect a some degree of positive compensation exists among goals and constrains.

• When U is defined on R, it is preferable to determine û by appropriate defuzzification method.


Several experts

• There are five springboard divers: 1, 2, 3, 4, 5.

• There are ten referees: R1, R2 ,…, R10. • There are ten referees: R1, R2 ,…, R10.

• We need to determine a membership function A that will capture the linguistic term excellent div er.


Alice Bonnie Cathy Dina Eva

R1 1 1 1 1 1

R2 0 0 1 1 1

R3 0 1 0 1 0

R4 1 0 1 1 1

The Diving Survey

R4 1 0 1 1 1

R5 0 0 1 1 1

R6 0 1 1 1 1

R7 0 0 0 0 0

R8 1 1 1 1 1

R9 0 0 0 1 0

R10 0 0 0 1 0 89Intelligent Control

• For every diver we calculate the membership grade of belonging to the fuzzy sets A by taking the ratio of the total number of favorable answers to the total number of referees.answers to the total number of referees.

• A = {{1, 0.3}, {2, 0.4}, {3, 0.6}, {4, 0.9}, {5, 0. 6}}


Several experts

FS1 = FuzzySet[{{1, .3}, {2, .4}, {3, .6}, {4, .9}, {5, .6}},

UniversalSpace -> {1, 5, 1}];


Questions ?

Documents

Introduction to Intelligent Control Part 3keffe006/Workshop/ECE4951-Lecture3.pdf · Introduction to Intelligent Control Part 3 ECE 4951 - Spring 2010 Prof. Marian S. Stachowicz