Fuzzy Systems. Example of a Mamdani/Larsen fuzzy controller. Defuzzification. Fuzzy Systems (Cont.) Fuzzy Systems (Cont.) Fuzzification.

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1 Fuy Systems Example of a Mamdani/Larsen fuy controller Defuification How can fuy systems be used in a world where measurements and actions are expressed as crisp values? * Fuy Systems Toolbox, M. Beale and H Demuth Fuy Systems (Cont.) Fuy Systems (Cont.) Input Fuifier Fuy System Defuifier output * Fuify crisp inputs to get the fuy inputs * Defuify the fuy outputs to get crisp outputs 9 Degree F. It is too hot! Turn the fan on high Set the fan at 9% speed * Fuy Systems Toolbox, M. Beale and H Demuth Fuification Fuification Process of making a crisp quantity fuy Fuification example Fuy singleton If it is assumed that input data do not contain noise of vagueness, a fuy singleton can be used µ F (x) x Membership grade of crisp input x in the fuy set If the data are vague or perturbed by noise, they should be converted into a fuy number µ F (x) x base Crisp input x 1

2 Fuy Approximation Theorem Build a Fuy Controller 3 Steps 1. Pick the linguistic variables Example: Let temperature (X) be input and motor speed (Y) be output 2. Pick the fuy sets Define fuy subsets of the X and Y 3. Pick the fuy rules Associate output to the input Example: Build a Fuy Controller Example: Build a Fuy Controller Goal: Design a motor speed controller for air conditioner Step 1: assign input and output variables Let X be the temperature in Fahrenheit Let Y be the motor speed of the air conditioner Step 2: Pick fuy sets Define linguistic terms of the linguistic variables temperature (X) and motor speed (Y) and associate them with fuy sets For example, 5 linguistic terms / fuy sets on X Cold, Cool, Just Right, Warm, and Hot Say 5 linguistic terms / fuy sets on Y Stop, Slow, Medium, Fast, and Blast Example: Build a Fuy Controller Input Fuy sets Example: Build a Fuy Controller Output Fuy sets 2

3 Example: Build a Fuy Controller Example: Build a Fuy Controller Step 3: Assign a motor speed set to each temperature set If temperature is cold then motor speed is stop If temperature is cool then motor speed is slow If temperature is just right then motor speed is medium If temperature is warm then motor speed is fast If temperature is hot then motor speed is blast A Fuy Relation expressed by a rule Example: Build a Fuy Controller Example: Build a Fuy Controller A Fuy controller with 5 patches! " #$% & Example: temp. 65 degree F. Example: temp. 63 degree F. If temperature is just right then motor speed is medium *Fuy Thinking, Bart Kosko *Fuy Thinking, Bart Kosko 3

4 Example: t 63 degree F. Example: t 63 degree F. Summed (MAXed) of the partially fired then-part fuy sets OR OUTPUT *Fuy Thinkring, Bart Kosko *Fuy Thinkring, Bart Kosko Example: t 63 degree F. Defuification Defuify to find the output motor speed Question: how to convert a fuy set into a crisp value? Converts a fuy set into a crisp output. Defuification is a process to get a non-fuy value that best represents the possibility distribution of an inferred fuy control action. There is no systematic procedure for choosing a good defuification strategy. Selection of defuification procedure depends on the properties of the application. *Fuy Thinkring, Bart Kosko & '(" )*+,-!")**./ & Defuification " Defuification k j 1 j k j : control action whose membership functions reach the maximum. k: number of such control actions. 4

5 Defuification Defuification Max-membership principal, also known as height method Weighted average method Valid for symmetrical output membership functions Produces results very close to the COA method Less computationally intensive Formed by weighting each functions in the output by its respective maximum membership value Bisector of the Area Defuification First (or last) of maxima Defuification The BOA generates the action ( ) which partitions the area into two regions with the same area Determine the smallest value of the domain with maximied membership degree α µ ( ) d C β α min{ W } β max{ W } µ ( ) d C Example: Defuification Find an estimate crisp output from the following 3 membership functions Example: Defuification CENTROID 5

6 Example: Defuification Weighted Average Mean-Max Example: Defuification Z* (6+7)/2 6.5 Example: Defuification First and Last of maxima Defuification Which defuification method is the best? The answer is context or problem-dependent. 4 criteria against which to measure the methods: #1 Continuity. small change in the input should not produce the large change in the output Defuification #2 Disambiguity. Defuification method should always result in a unique value, I.e. no ambiguity. #3 Plausibility. Z* should lie approximatly in the middle of the support region and have high degree of membership. #4 Computational simplicity. Summary 1. Fuification of inputs. 2. Linguistic variables, linguistic terms and associated fuy sets 3. Rules 4. Defuification 6