2010/07/12. Content. Fuzzy? Oxford Dictionary: blurred, indistinct, confused, imprecisely defined

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1 Content Introduction Graduate School of Science and Technology Basic Concepts Fuzzy Control Eamples H. Bevrani Fuzzy GC Spring Semester, 2 2 The class of tall men, or the class of beautiful women, do not constitute classes or sets in the usual mathematical sense of these terms. Yet, the fact remains that such imprecisely defined classes play an important role in human thinking, particularly in the domains of pattern recognition, communication of information, and abstraction. Lotfi. Zadeh, 965 Fuzzy? Oford Dictionary: blurred, indistinct, confused, imprecisely defined Fuzzy Systems? Knowledge-based or rule-based systems: If-Then rules 3 s compleity rises, precise statements lose meaning and meaningful statements lose precision. Lotfi. Zadeh, 965 The theory of fuzzy sets is a theory of graded concepts, a theory in which everything is a matter of degree. Lotfi Zadeh, 973 Unlike two-valued Boolean logic, fuzzy logic is based on degrees of membership. It deals with degrees of truth (a) Boolean Logic. (b) Multi-valued Logic. 6

2 Set of tall men: It covers all men, but their degrees of membership depend on their height. D egree of M em bership. C risp S ets.8.6 Tall M en D egree of M em bership Fuzzy Sets H eight, cm H eight, cm () = if < o if o o if () = < () < if < < 2 if 2 2 Tall men set: consists of three sets: short, average and Degree of tall men. Crisp Sets M embership Short verage Short Tall Tall Men Height, cm Degree of M embership Fuzzy Sets Short verage Tall Tall 965, Zadeh: Fuzzy Sets paper 968, Zadeh: Fuzzy algorithms 97, Zadeh & Bellman: Fuzzy decision making 97, Zadeh: Fuzzy ordering 973, Zadeh: Fuzzy control 975, Mamdani: st application (Steam engine) 2 2

3 975-8: Slowly progressed! 98, Sugeno: Control of a Fuji Water Purification Plant 983, Sugeno & Nishida: Self-parking Car 984, Togai & Watanabe: st VLSI based Fuzzy controller 987, Hitachi Co.: Sendai Subway System 988, Hirota & : Fuzzy Robot arm 989: Lab. for Int. Fuzzy Eng. Research-Japan 99: Fuzzy Logic System Institute (FLSI)-Japan 992: st IEEE Int. Conf. on Fuzzy Systems 993: st Issue of the IEEE Trans. On Fuzzy Sys. 996, Hiyama: Fuzzy Power System Stabilizer Fuzzy Mathematics Fuzzy Control Fuzzy Decision Making Uncertainty and Information Fuzzy Logic and I Easy to understand Fleible Don t need precise data Can model nonlinear functions Based on natural language 7 8 3

4 Eample : Fuzzy-based utomatic Car Speed Control Eample 2: Digital Image Stabilizer If all the points in the picture are moving in the Same direction, Then the hand is shaking and compensate it. If only some points in the picture are moving, Then the hand is not shaking and leaves it alone. 2 Open-loop Control Closed-loop Control Pure Takagi-Sugeno-Kang (TSK) With Fuzzifier and Defuzzifier U and V are real-valued variables. U and V are fuzzy sets. In Eng. Systems, inputs/outputs are real values. Problem: Using mathematical formula against using the human knowledge 4

5 Not B Complement Containment B B and y are real-valued variables. Intersection Union ( ) ( ) B B Not Complement Containment ( ) ( ) B B B B Intersection Union Containment: Consider U = {, 2, 3} and sets and B =.3/ +.5/2 + /3 B =.5/ +.55/2 + /3 then is a subset of B, or B Complement: Crisp Sets: Who does not belong to the set? Fuzzy Sets: How much do elements not belong to the set? Fuzzification: crisp to fuzzy Fuzzification is the process where the crisp quantities are converted to fuzzy. Union: Crisp Sets: Which element belongs to either set? Fuzzy Sets: How much of the element is in either set? The conversion of fuzzy values is represented by the membership functions. There are various methods to assign membership values or the membership functions to fuzzy variables. 5

6 Membership Function Fuzziness in a fuzzy set is characterized by its membership functions. Membership value is between and. Popular functions: Membership Value ssignments Intuition: is based on the human s own intelligence and understanding to develop the membership functions. Inference: The membership function is formed from the facts known and knowledge. Rank Ordering: e. g., best car between five cars among people. ngular: is defined on the universe of angles, hence is repeating shapes every 2Π cycles. Triangular Trapezoid Gaussian NN: is used to determine the membership values of any input data in the different regions. G: The method involved in computing membership functions using G. Defuzzification Defuzzification Methods It convert fuzzy sets into a crisp output The output of an entire fuzzy process can be union of two or more fuzzy membership functions. To eplain this, consider a fuzzy output, which is formed by two parts: () Lambda-cut sets, (2) Centroid method, (3) Height method, (4) Weighted average method, (5) Mean ma method, (6) Centre of sums, (7) Centre of largest area, Centroid method Is the most popular one. It finds a point representing the centre of gravity (COG) COG b a b COG a d d Degree of Membership (2). (3456).2 (789) Z Fuzzy Rules In 973, Lotfi Zadeh published his second most influential paper: Capturing human knowledge in fuzzy rules. fuzzy rule can be defined as a conditional statement in the form: IF is, THEN y is B where and y are linguistic variables; and and B are linguistic values determined by fuzzy sets on the universe of discourses X and Y, respectively. 6

7 In a fuzzy system, all rules fire to some etent. If the antecedent is true to some degree of membership, then the consequent is also true to that same degree. Eample: IF height is tall THEN weight is heavy Degree of Membership Tall men Degree of Membership Heavy men Height, cm Weight, kg 38 Design Steps Eample: ir conditioner Fuzzification (of the input variables) Rule evaluation (Inference) ggregation (of the rule outputs) Defuzzification. Specify the problem and define linguistic variables Temperature, Fan Speed Consider five temperature control switches: COLD, COOL, PLESNT, WRM, HOT. The corresponding speeds of the motor controlling the fan on the air-conditioner has the graduations: MINIML, SLOW, MEDIUM, FST, BLST. 2. Construct fuzzy rules Temperature fuzzy set The rules governing the air-conditioner are as follows: RULE : IF TEMP is COLD THENSPEED is MINIML RULE 2: IF TEMP is COOL THENSPEED is SLOW RULE 3: IF TEMP is PLESNT THEN SPEED is MEDIUM RULE 4: IF TEMP is WRM THENSPEED is FST RULE 5: IF TEMP is HOT THENSPEED is BLST < (T)< (T)= Temp COLD COOL PLESNT WRM HOT ( C). Y* N N N N 5 Y Y N N N N Y N N N 2.5 N Y* N N N 5 N Y N N N 7.5 N N Y* N N 2 N N N Y N 22.5 N N N Y* N 25 N N N Y N 27.5 N N N N Y 3 N N N N Y* (T)= Y : temp value belongs to the set (< ()<) Y* : temp value is the ideal member to the set ( ()=) N : temp value is not a member of the set ( ()=) 7

8 The analytically epressed membership for the temperature are: Temperature Fuzzy Sets COLD: for t COLD (t) = t / + Cool: for t 2.5 COOL (t) = t / 2.5 Pleasant: for 2.5 t 7.5 PLESENT (t) = t / etc all based on the linear equation: y = a + b Truth Va alue Temperature Degrees C Cold Cool Pleasent Warm Hot Speed fuzzy set Rev/sec (RPM) MINIML SLOW MEDIUM FST BLST Y* N N N N Y N N N N 2 Y Y N N N 3 N Y* N N N The analytically epressed membership for the speed are: 4 N Y N N N 5 N N Y* N N 6 N N N Y N MINIML: for v 3 COLD (t) = v / 3 + COLD 7 N N N Y* N 8 N N N Y Y 9 N N N N Y N N N N Y* SLOW: for v 3 SLOW (t) = v / 2.5 for 3 v 5 SLOW (t) = v / Y : temp value belongs to the set (< ()<) Y* : temp value is the ideal member to the set ( ()=) etc all based on the linear equation: y = a + b N : temp value is not a member of the set ( ()=) 3. Evaluate fuzzy rules Truth Value Speed Fuzzy Sets Speed MINIML SLOW MEDIUM FST BLST Eample: Use the system to compute the optimal fan speed, for temperature of 6 o C. Fuzzification Inference Composition Defuzzification 8

9 Fuzzification ffected fuzzy sets: COOL and PLESNT COOL (T) = T / PLSNT (T) = T /2.5-6 = 6 / = 6 /2.5-6 =.3 =.4 Temp=6 COLD COOL PLESNT WRM HOT.3.4 Inference RULE : IF temp is cold THEN speed is minimal RULE 2: IF temp is cool THEN speed is slow RULE 3: IF temp is pleasant THEN speed is medium RULE 4: IF temp is warm THEN speed is fast RULE 5: IF temp is hot THEN speed is blast Inference Composition RULE 2: IF temp is cool (.3) THEN speed is slow (.3) speed is slow (.3) + speed is medium (.4) RULE 3: IF temp is pleasant (.4) THEN speed is medium (.4) Fuzzy Modeling Defuzzification + fuzzy logic Static fuzzy system COG =.25(2.5) +.25(5) +.3( ) +.4( ) +.25(57.5) () +.4(5) +.25 = 45.54rpm Dynamic fuzzy system + fuzzy logic 9

10 Modeling Eample The input-output relationship of the model: smoothed fuzzy description (circles) and piecewise linear description (dashed lines). Fuzzy Control Fuzzy Control Design Steps w e C u S y Define control problem and system characteristics e(t) e(t) FLC Internal Structure u(t) u(t) Define variables and Fuzzy sets Z - e(t) Z - typical fuzzy logic controller is described by the relationship between change of control u(t), at a given time t, on the one hand u(t) = f(e(t), e(t)) and the change in the error e(t): e(t) = e(t) e (t ) Define Inference Rules Define Defuzzification Method Design Eample Inverted Pendulum Problem Fuzzy Control System Input variables: ngle of the Pendulum, Rate of change of the angle Output variables: Position of the cart or cart speed (produced by a PWM signal) X ref X Control problem: Keep pendulum upright by moving cart left or right. State variables: ngle of the Pendulum, Rate of change of the angle, Position of the cart or cart speed 6

11 Membership functions Input fuzzy sets for angle of the pole and rate of change of angle MN SN ZE SP MP Pendulum ngle d Output fuzzy set Inputs Pendulum ngular Velocity LN MN SN ZE SP MP LP Output Cart Speed - Speed of vehicle Fuzzy Rules The fuzzy rules are If is MN and d is MN then output is LN If is ZE and d is ZE then output is ZE If is SP and d is SP then output is MP etc. IF angle is zero and angular velocity is zero THEN speed shall be zero. The rules are best summaries by the Fuzzy ssociative Memory (FM) table. Defuzzification Fill in the blanks in the table. if angle is zero and angular velocity is zero then \ d MP SP ZE SN MN MP LP SP MP SP? ZE ZE SN MN MN? LN

12 Fuzzy Logic GC if angle is zero and angular velocity is zero then. Fuzzy logic controller if angle is zero and angular velocity is negative low then if angle is positive low and angular velocity is zero then Defuzzification must now be done on fused output. if angle is positive low and angular velocity is negative low then general scheme for fuzzy logic based GC 2. Fuzzy based PI (PID) controller t u( t) k P CE ( t) k I CE ( τ) dτ n u( kt) kp CE( kt) ki T CE( it) i CE ( kt ) CE k ) T u( kt) u( kt) u( k ) T kp ki CE( kt) T ( general scheme for adaptive fuzzy logic GC Fuzzy PI control scheme 3. Fuzzy Tuner Fuzzy logic for tuning of PI based GC system 2