Slide 1

Sharif university of Technology Supervisor: Dr.Bagheri Shuraki Presenters: Maryam Fereydoon, Sepideh Hajipour 1 Course: Fuzzy System and Sets Spring,2009 Title: Integration of Fuzzy Logic and Chaos Theory

Slide 2

2

Slide 3

Lotf Ali Asgar Zadeh 3

Slide 4

Brief Biography Born: February 4, 1921,age 88 Nationality: Iranian, AmericanIranianAmerican Fields: MathematicsMathematics Institutions: U.C. BerkeleyU.C. Berkeley Known for Founder of Fuzzy Mathematics, fuzzy set theory, and fuzzy logic.Fuzzy Mathematics fuzzy setfuzzy logic He was born in Baku, of an Iranian father and mother. At the age of 10 the Zadeh family moved to Iran. Zadeh grew up in Iran, studied at Alborz High School. In 1942, he was graduated from the University of Tehran in electrical engineering (Fanni), and moved to the United States in 1944. He received an MS degree in electrical engineering from MIT in 1946, and a PhD in electrical engineering from Columbia University in 1949.BakuIranAlborz High SchoolUniversity of Tehranelectrical engineeringUnited StatesMITColumbia University He was promoted to full professor in 1957. He published his seminal work on fuzzy sets in 1965. In 1973 he proposed his theory of fuzzy logic.fuzzy setsfuzzy logic Zadeh is married to Fay Zadeh and has two children, Stella Zadeh and Norman Zadeh. Zadeh has a long list of achievements 4

Slide 5

Edward Lorenz 5

Slide 6

Brief Biography Edward Norton Lorenz (May 23, 1917 - April 16, 2008) was an American mathematician and meteorologist, and a pioneer of chaos theory He discovered the strange attractor notion and coined the term butterfly effect. mathematicianmeteorologistchaos theorystrange attractorcoinedbutterfly effect Lorenz was born in West Hartford. He studied mathematics at both Dartmouth College and Harvard University.Lorenz was Professor of the Massachusetts Institute of Technology. Lorenz became skeptical of atmospheric phenomena. His work on the topic culminated in the publication his paper, the foundation of Chaos theory. His description of the Butterfly effect followed in 1969 Dartmouth CollegeHarvard University Massachusetts Institute of Technology atmospheric phenomenaChaos theoryButterfly effect Lorenz died at his home in Cambridge at the age of 90, having suffered from cancer. cancer Lorenz built a mathematical model of the way air moves around in the atmosphere. As Lorenz studied weather patterns he began to realize that they did not always change as predicted. Minute variations in the initial values of variables in his twelve variable computer weather model (c. 1960) would result in grossly divergent weather patterns. This sensitive dependence on initial conditions came to be known as the butterfly effect.mathematical model atmosphereweathercomputerweather butterfly effect 6

Slide 7

Chaos is a mathematical subject and therefore isn't for everybody. However, to understand the fundamental concepts, you don't need a background of anything more than introductory courses in algebra, trigonometry, geometry, and statistics. That's as much as you'll need for this book. (on the other hand, does require integral calculus, partial differential equations, computer programming, and similar topics.) In fact, change and time are the two fundamental subjects that together make up the foundation of chaos. chaos can be important because its presence means that long-term predictions are worthless and futile 7

Slide 8

A Simple Example X t+1 = 1.9-(X t ) ^2 Input value (xt) New value (xt+1) etc. 1.00.9 1.09 0.712 1.393 8

Slide 9

On the following pages we see three figures, two produced by the computer and one a photograph, Which one strikes you as being the most complex? If you and a friend compare answers and do not agree, this is no cause for concern. The term "complexity" has almost as many definitions as "chaos." A small portion of the Mandelbrot set, with enough surrounding points to render it visible.(z n+1 = z n 2 + c for n = 0, 1, 2, 3, with the starting value z 0 = 0 ) Complexity 9

Slide 10

A variant of the Japanese attractor Some wind streaks in a field of packed snow 10

Slide 11

Fractality A fractal triangle, formed by dividing a square into four smaller squares and discarding the upper right square so produced, then dividing each remaining square into four still smaller squares and discarding each upper right square so produced, and repeating the process indefinitely. A fractal formed, except that in each retained square the corner to be discarded has been chosen randomly. 11

Slide 12

A fractal tree, produced by first drawing a vertical segment, and then, after this segment or any other one has been drawn, treating it as a "parent" segment and drawing two "offspring" segments, each six-tenths as long as the parent, and each extending at right angles from the end of the parent. 12

Slide 13

Butterfly Effect & Turbulance It says that a butterfly flapping its wings in, say Brazil, might create different "initial conditions" and trigger a later tornado elsewhere (e.g. Texas), at least theoretically. not seem to be a clear distinction between turbulence and chaos 13

Slide 14

fuzzy systems technology has achieved its maturity with widespread applications in many industrial, commercial, and technical fields, ranging from control, automation, and artificial intelligence to image/signal processing, pattern recognition, and electronic commerce. Chaos, on the other hand, was considered one of the three monumental discoveries of the twentieth century together with the theory of relativity and quantum mechanics. A deep seated reason to study the interactions between fuzzy logic and chaos theory is that they are related at least within the context of human reasoning and information processing. 14

Slide 15

fuzzy definition of chaos, fuzzy modeling and control of chaotic systems using both Mamdani and Takagi Sugeno models, fuzzy model identification using genetic algorithms and neural network schemes, bifurcation phenomena and self-referencing in fuzzy systems, complex fuzzy systems and their collective behaviors, as well as some applications of combining fuzzy logic and chaotic dynamics, such as fuzzy chaos hybrid controllers for nonlinear dynamic systems, and fuzzy model based chaotic cryptosystems. 15

Slide 16

2 papers in these interdisciplines are discussing: 1- Chaotic Dynamics with Fuzzy Systems: explorations have been carried on mainly in three directions: the fuzzy control of chaotic systems, the definition of an adaptive fuzzy system by data from a chaotic time series, and the study of the theoretical relations between fuzzy logic and chaos. 16

Slide 17

The peculiarities of a chaotic system can be listed as follows: 1. Strong dependence of the behavior on initial conditions 2. The sensitivity to the changes of system parameters 3. Presence of strong harmonics in the signals 4. Fractional dimension of space state trajectories 5. Presence of a stretch direction, represented by a positive Lyapunov exponent 17

Slide 18

Famous artificial chaotic systems are Chuas circuit: x(k + 1) = ax(k)(1 x(k)) 18 one or more nonlinear elements one or more locally active resistors three or more energy-storage elements

Slide 19

Fuzzy Sets: 19

Slide 20

20

Slide 21

21

Slide 22

22 2D

Slide 23

Rule Base: 23

Slide 24

Conclusion s 1. Simple fuzzy systems are able to generate complex dynamics. 2. The precision in the approximation of the time series depends only on the number and the shape of the fuzzy sets for x. 3. The chaoticity of the system depends only on the shape of the fuzzy sets for d. 4. The analysis of a chaotic system via a linguistic description allows a better understanding of the system itself. 5. Accurate generators of chaos with desired characteristics can be built using the fuzzy model. 6. Multidimensional chaotic maps in some cases do not need a large number of rules in order to be represented. 24

Slide 25

2- Chaos Control Using Fuzzy Controllers (Mamdani Model): There are two good reasons for using the fuzzy control: first, mathematical model is not required for the process, and second, the nonlinear controller can be developed empirically, without complicated mathematics. The fuzzy logic controller can be used when there is no mathematical model available for the process, this gives the robustness behavior for the proposed fuzzy logic controller design. 25

Slide 26

Membership Functions: 26

Slide 27

27

Slide 28

28

Slide 29

System response with some possible fuzzy rule in the rule-based system: 29

Slide 30

Uncontrolle d simulations of Lorenz equation: (Chuas Circuit) 30

Slide 31

Controlled system to get on a periodic solution using the fuzzy controller: 31

Slide 32

Controlled system to get on a constant solution using the fuzzy controller: 32

Slide 33

The mathematical model of the lorenz system: mathematical model of the Chuas circuit: 33

Slide 34

3- Bifurcation Phenomena in Elementary Takagi Sugeno Fuzzy Systems: Bifurcation: A small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden qualitative or topological change in its behavior. Local bifurcations Changes in the dynamics of a small region of the phase space, typically, a neighborhood of an equilibrium point. Global bifurcations For instance, when the interaction of a limit cycle with a saddle point is produced. 34

Slide 35

The bifurcation diagram of the logistic map 35

Slide 36

Takagi-Sugeno System with Affine Consequents 36

Slide 37

37

Slide 38

38

Slide 39

Conclusion Even very elementary T-S systems can display local and global bifurcations 39

Slide 40

4- Self-Reference, Chaos, and Fuzzy Logic: The Liar paradox: This sentence is false T F T F T F T F T F 40

Slide 41

Fuzzy Definitions Lukasiewicz: 41

Slide 42

The Liar paradox with fuzzy membership functions A seed value of 0.3: This sentence is false 42

Slide 43

Attractor and Repeller Fixed Points in the Phenomena of Self-Reference Modest Liar: This sentence is fairly false 43

Slide 44

Emphatic Liar: This sentence is very false 44

Slide 45

True Teller Modest True-Teller Emphatic True-Teller This sentence is fairly true This sentence is very true This sentence is true 45

Slide 46

Fuzzy Chaos The Chaotic Liar This sentence is true if and only if it is false This sentence is as true as it is false 46

Slide 47

It is very false that this sentence is true iff it is false. 47

Slide 48

Fuzzy Self-Reference in Two Dimensions The Dualist: Fuzzy variations on the Dualist: Socrates: What Plato is about to say is true. Plato: Socrates speaks falsely. X: Y is true Y: X is false X: X is true if and only if Y is true. Y: Y is true if and only if X is very false. 48

Slide 49

Sequential calculation Escape-time diagram for a Fuzzy Dualist with simultaneous calculation 49

Slide 50

50

Slide 51

Thanks for Your Attention 51