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模糊邏輯 郭耀煌 課程大綱 • • • • • • • Fuzzy Sets Fuzzy Arithmetic Fuzzy Relations Fuzzy Logic Fuzzy Measure (Possibility Theory) Design Process and Design Tools Applications: expert systems, fuzzy controllers, pattern recognition, databases and information retrieval, decision making. 教材 • Textbook: Fuzzy Sets and Fuzzy Logic, Theory and Applications; George J. Klir & Bo Yuan, Prentice Hall, 1995. • Ref. – Fuzzy sets, Uncertainty, and Information, G. J. Klir and Tina A. Floger, Prentice Hall, 1988. – Fuzzy Set Theory and Its Applications, H. -J. Zimmermann, 1991. – Fuzzy Logic: Intelligence, Control, and Information, John Yen, Reza Langari, Prentice Hall, 1999. – Fuzzy Engineering, Bart Kosko, Prentice Hall, 1997. – 模糊理論及其應用, 2003 選課要求 •期中考、期末考(各20%) •平時作業(20%) •實作作業(20%) •期末專題(20%) •上課出席狀況、發言提問等(15%) •助教:蔡仁勝、李振維 Background 1. Handle complexity is a common issue in the information society: complexity originates from huge information and huge uncertainty. 2. 手排比自排複雜:手排需要更多的知識,而 且不確定性程度也增加(不知何時需換檔) 3. We must deal between the information available to us and the amount of uncertainty we allow. 4. Sometimes we can obtain a more robust conclusion by presenting an uncertain description instead of a precise description. (e.g., the description of weather) 4. Fuzziness is one feature of natural language so does not necessarily imply the loss of meaningful semantics. 5. Application roadmap of information technology: numerical analysis, large database, knowledge management. So, we must first know the characteristics of the world and its knowledge, then explore the possibility and limitation of knowledge. 6. Even supercomputer still lacks for the capability of summarization, which is the basis of intelligence and competence of human being. due to the binary logic basis of modern computer model. wait for chemical computer, bio-computer and molecular computer. – 辨識莫札特的音樂人類無法清楚列出標準 – 辨識人種:非超級電腦可行之工作 – 有些事情縱使不明確指出其法則,一樣可以去做, 即工作法則是晦暗不明的 7. Traditional AI paradigms: first order logic (John McCarthy, Nilsson Kowalski); ad-hoc techniques and heuristic procedures. (Marvin Minsky (MIT), Roger Schank). L. Zadeh: using fuzzy logic (approximate reasoning, nondiscrete) instead of first order logic as the basis of AI in common sense reasoning. 8. 現今的電腦並非計算能力不足,而是因為電腦軟硬 體皆非以fuzzy knowledge(非discrete的)及 common-sense reasoning為導向而設計 9. Law of Incompatibility: As complexity rises, precise statements lose meaning and meaningful statements lose precision. • Fuzzy logic denotes a retreat form unrealistic requirement of precision.(不是精確的東西就不是科學) – 古典機率理論被統計技巧取代 – 以數值分析解法對微分方程求解,在3~40年前無法被 相信的 • Paradigm shift: certainty in science uncertainty in science (molecular; probability theory (statistics; microscopic macroscopic) • Organized simplicity (Newtonian mechanics, analyzed by Calculus) organized complexity (involve nonlinear systems with large no. of components and rich interactions among the components, which are usually nondeterministic, but not as a result of randomness) disorganized complexity (randomness) • Bremetmann limit: No data processing system, whether artificial or living, can process more than 2 1047 bits per second per gram of its mass. (quantum theory) transcomputational problems • How to deal with systems and associated problems whose complexities are beyond our information processing limits? Fuzzy logic and It’s Applications Contents: 1. Introduction of Fuzzy Set theory 2. Basic of Fuzzy Logic 3. Fuzzy Inference 4. Applications of Fuzzy Logic Introduction 1965 Fuzzy Set (Prof. Lotfi A.Zadeh,UCB) 1966 Fuzzy logic (Dr. Peter N.Marinos, Bell Lab) Fuzzy Set Fuzzy Event 1972 Fuzzy Measure (Prof.Michio Sugeno) Crisp Element • 1944年, Zadeh進入MIT,此時computer age已經發韌, – Nobert Winer: cybernetics maintaining order in systems – Claude Shannon: information theory – Warren Mculloch/Walter Pitts: network networks – All these theories would make it possible to create a world in which information plays a major role • Fuzzy logic combines set theory, vagueness philosophy, multi-valued logic, Max-Black’s word usage charts. • Core thinking of fuzzy logic: What is a class? – Categories 遍佈我們的思考,即使動物也隨時在做分類 –語言即是classes的最高表示,大部分的字都refer to categories • 1970年David Marr認為handling classes是腦灰色皮質的永久 角色 • 數學家及理則學者以formal models來描繪classes, fuzzy sets 即是這種model. (19世紀Cantor發展set theory) • 字需要有context方能給予涵義(semantics),集合亦然, universe of discourse即充當set的context. • Bart Kosko: everything is fuzzy except numbers. • 人們在面對complex information時,會利用summarization的 策略 – Brain 一直在做summarizing sense data, which reduces massive details to chunks of perception. we see an almost closed circle as a complete one. – 語言亦是一種summarization • Arthur Geoffrion質疑如何客觀地定義membership function • Kahan: What we need is more logical thinking, not less – 沒有一個問題不能被ordinary logic執行得更好 Introduction Knowledge Representation example: age (Man Old) Membership Function traditional Age (Man Gt 60) 1 30 60 Ages Introduction Fuzzy Membership Function Age (Man Old) 1 0.5 30 60 Ages Fuzzy Logic A( x) : membership of the element x in the fuzzy subset A x : an element of the reference set E A, B, Fuzzy subset of E a A ( x), b B ( x), a, b[0,1] a b MIN (a, b) a b MAX (a, b) a 1 a a b ( a b) ( a b) Fuzzy Logic Commutativity a ()b b ()a Associativ e (a b) c a (b c) Distributi vity a (b c) (a b) (a c) ( a ) a DeMorgan' stheorems (a b) a b (a b) a b Fuzzy Inference 二值理論推論形式 (事實) 麻雀是鳥 (規則) 鳥會飛 (結論) 麻雀會飛 AI Language as LISP,Prolog “Pattern Matching” Fuzzy推論形式: (事實) 這番茄很紅 (規則) 蕃茄若是紅了就熟了 (結論) 這蕃茄很熟了 Fuzzy Inference (facts) X is A (rule) if X is A then Y is B 希望得到的結論是 Mamdani 法 (result) Y is B 1 A A 1 B B 0 0 Application 2 Air Conditioner System TEMP. SENSOR TEMP. ERROR TEMP. CHANGE FUZZY INFERENCE INVERTER FREQ. FUZZY RULES COMP VALVE MEMBERSHIP FUNCTIONS • 50 RULES (HEATING&A/C) • • MAX-PRODUCT INFERENCING DEFUZZIFICATION: CENTROID METHOD FAN SPEED Application 3 Control laws of a Washing Machine Laundry volume (V) Low Mid High Soft S = Weak T = Short S = Weak T = Short S = STD T = STD More or less soft S = Weak T = Short S = STD T = STD S = STD T = STD S = Weak T = Short S = STD T = STD S = Strong T = Long S = Weak T = Short S = STD T = STD S = Strong T = Long fabric quality More or less (Q) Hard Hard Application 3 Fuzzy Automatic Washing Machine laundry volume Stream strength optimum water level Laundry volume (V) fabric quality (Q) fabric quality FUZZY CONTROL Washing time High Mid Low Stream strength = Weak Washing time = Short Hard Mid Soft Stream strength = Strong Washing time = Long Stream strength = Strong Washing time = Short (Optimal Washing Cycle) Application 3 Fuzzy-Neuro Washing Machine(Panasonic) (OUTPUT) (INPUT) Quantity Turbidity (Optical sensor) Water Level Water Stream Strength FUZZY INFERENCE Change Rate Of Turbidity Washing Cycle Time Rinse Cycle Time Drain Cycle Time Tuning membership functions NEURAL NET Application 3 Fuzzy-Neuro Washing Machine(Hitachi/Sanyo) (OUTPUT) (INPUT) Quality(4) Quantity(3) Water Stream Strength FUZZY INFERENCE Washing Cycle Time Rinse Cycle Time Drain Cycle Time Quality(4) Quantity(3) Conductivity Sensor(5) (Room Temp (8) – Sanyo) NEURAL NET COMPENSATION Advantages of fuzzy system modeling 1. 2. 3. 4. 5. 6. The ability to model highly complex business problems. Improved cognitive modeling of expert systems Need not crisply dichotomize rules at artificial boundary; Reduce overall cognitive dissonance The ability to model systems involving multiple experts. Reduced model complexity: a. Fewer rules, b. Representing rules closer to natural language Improved handling of uncertainty and possibilities, Less externally complex problems can be isolated and fixed sooner improved MTTR and MTBF. 表1 關於理論應用方面的控制問題 以往方式的 問題所在 感覺型問題 非線形型問題 • 應 用 例 • • 控制目標用數 值表現很難。 控制結果的好 壞必須要用感 覺評估。 地下鐵乘客的 • 心情控制 汽車的 SUSPENSION • 起重機的操作 控制 • 分類型問題 由於控制對象的 狀況常常變化, 故無法完全控制 目標值,故 OVERSH-OOT 很大。 雖然預設的狀 況很複雜但卻 無法記述所有 PATTERN的 對應方法。 溫度控制 AIRCONDITIONER 位置控制HARD DISK 速度控制 AUTOIRIS/ AUTOFOCUS 機能 PATTERN認 識 • •