fuzzy logic and fuzzy control - nju.edu.cn

INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015

Fuzzy Logic and Fuzzy Control

n  Fuzzy Logic �•  Zadeh defines fuzzy logic (FL) �•  two important concepts "�

n  Fuzzy Rules�•  fuzzy rules�•  Example of Fuzzification �•  Defuzzification�

n  Fuzzy Controllers�l  Why fuzzy control? �l  Example of fuzzy controller


Fuzzy Logic

•  Zadeh defines fuzzy logic (FL) as a logical system, which is an extension of multivalued logic �

•  FL is not based on a closed world model (we don’t assume� everything is known) �•  FL (& crisp logic) state objective descriptions/measures�•  FL does not require statistical independence of variables�•  In FL, the absence of a fact doesn’t imply anything �•  A FL membership value persists after observation �

INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2014

Fuzzy Logic


Fuzzy Logic


The more complex a system is, the more it involves intelligent behavior, the more likely it is that fuzzy logic will provide a good approach.

Fuzzy Logic

two important concepts within FL : �

•  linguistic variable �

•  fuzzy if-then rule


linguistic variable �

Fuzzy Logic


•  Concentration hedges (e.g. very) �

•  Dilation hedges

Fuzzy Logic

hedges


Fuzzy Rules �•  Have antecedent part and consequent part�


Fuzzy Rules

�•  Mamdani-type fuzzy rule: �If X1 is A1 and ... and Xn is An then Y is Bj ��•  TSK-type fuzzy rule: �If X1 is A1 and ... and Xn is An then Y = p0 + p1X1 + ... + pnXn��The default type of rule we use is the Mamdani type. �


Mamdani-type fuzzy rule


TSK FUZZY RULE


Fuzzifica)on

Fuzzification is the combining of the antecedent sets (the if- side of the rule) ��We use a gas flow regulator for a furnace as an example. �• Input parameters: indoor temp., outdoor temp., 5-min. delta temp.�• Output parameter: change in gas flow �


Fuzzy Set Defini)on Now we must define the fuzzy sets over each parameter. We decide to use triangular membership func)ons. First decide how many sets per parameter, then decide range for each: •  For the InTemp parameter, we define three fuzzy sets: cool, comfortable, and

too_warm. •  For OutTemp, we have five fuzzy sets defined: very_cold, chilly, warm,

very_warm, and hot. •  For DeltaInTemp, we define five fuzzy sets: large_negaAve, small_negaAve,

near_zero, small_posiAve, and large_posAve. •  For our output parameter FlowChange, we define five fuzzy sets:

decrease_greatly, decrease_small, no_change, increase_small, and increase_greatly.

Number of membership func)ons for a parameter depends on the situa)on.


Define the Rule Set Possible rules: ��Rule 1: If InTemp is comfortable and DeltaInTemp is near_zero, then FlowChange is no_change. ��Rule 2: If OutTemp is chilly and DeltaInTemp is small_negative, then FlowChange is increase_small. ��Rule 3: If InTemp is too_warm and DeltaInTemp is large_positive, then FlowChange is decrease_greatly. ��Rule 4: If InTemp is cool and DeltaInTemp is near_zero, then FlowChange is increase_small. �… �


Define the Membership Func)ons

(We define only those we need for our example rules.) For inside temperature (intemp):

⎭⎬⎫

⎩⎨⎧ ++=

700

601

501cool

⎭⎬⎫

⎩⎨⎧ ++=

800

701

600ecomfortabl

⎭⎬⎫

⎩⎨⎧ ++=

901

801

700_ warmtoo


Define the Membership Func)ons, Cont’d. For DeltaInTemp:

⎭⎬⎫

⎩⎨⎧ ++=

⎭⎬⎫

⎩⎨⎧

+++

−=

⎭⎬⎫

⎩⎨⎧ +

−+

−=

61

41

20_arg

20

01

20_

00

21

40_

positiveel

zeronear

negativesmall

For OutTemp:

⎭⎬⎫

⎩⎨⎧ ++=

700

501

300chilly

(These are all we need for our rules.)


Example of Fuzzification

Assume inside temperature is 67.5 F, change in temperature last five minutes is -1.6 F, and outdoor temperature is 52 F.

Now find fuzzy values needed for our four example rules:

For InTemp,

0.0)5.67( and ,75.0)5.67(,25.0)5.67( _ === warmtooecomfortablcool µµµ. For DeltaInTemp,

0.0)6.1( and ,2.0)6.1( ,8.0)6.1( _arg__ =−=−=− positiveelzeronearnegativesmall µµµ

For OutTemp,

9.0)52( =chillyµ.


Fire the Fuzzy Rules Using Zadeh’s AND process:

Rule 1: changeno _20.020.075.0 µ==∩

Rule 2: smallincrease _8.08.09.0 µ==∩

Rule 3: greatlydecrease _0.00.00.0 µ==∩

so Rule 3 doesn’t produce any output. Rule 3 is said to have fired but not to have been activated. Rule 4: smallincrease _2.02.025.0 µ==∩

Note that two rules result in the activation of increase_small.

for FlowChange.

for FlowChange.

for FlowChange.

for FlowChange.


Determine Output Fuzzy Set Ac)va)ons

In our example, rules 1, 2, and 4 are activated. ��Two output fuzzy sets are activated: �no_change (by rule 1) �increase_small (by rules 2 and 4) ��Note that a fuzzy set can be activated by several rules

(often at different levels) �We can use various ways to obtain the output when

multiple rules fire; Zadeh’s OR is one common way. This results in an output level of 0.8 for increase_small�


Defuzzifica)on

•  Combines set of if-then rules into a specific value of a control�(output) variable�•  Defuzzification is the process of getting a scalar out of the fuzzy

system. ��

Note: These rules are for changes in flow. �


Defuzzifica)on, Con)nued

Chop off consequent linguistic variable membership functions at values obtained Note that membership functions must have been established with scalar values assigned to consequent variable scale A common way to defuzzify is with the centroid (clipped center of gravity) method.

∑

∑=

ii

iii

x

xxOutput

)(

)(

µ

µ

Must specify how to deal with areas of overlap: count them once or twice.


Defuzzifica)on of Furnace Gas Flow

DG, increase greatly; DS, decrease slightly; NC, no change;

IS, increase slightly; IG, increase greatly


Other Defuzzifica)on Methods (Each represents outputs as arrows pushing down on x-‐axis.) Max-‐membership: Take centroid of membership func)on with highest value (Output = 1.0 in example) Mean-‐max-‐membership: Average value on the domain axis (Output = 0.5 in example) Center-‐of-‐maximum: Each membership value represented by one arrow. (Like forces pushing on a lever.) (Output = 0.8 in example)


Measures of Fuzziness

• Fuzziness measures for discrete fuzzy sets exist

• Fuzziness measures are metrics of fuzzy uncertainty

• These measures es)mate average ambiguity in fuzzy sets

• Goal: to describe the rela)onship of elements to sets and sets to one another


Fuzzy Control

• Largest applica)on of fuzzy logic

• First implemented by Mamdani in laboratory in 1975

• Thousands of commercial and industrial applica)ons today


Why Fuzzy Control?

* Efficiently incorporates human expert informa)on * Does not require a mathema)cal model of system * Produces nonlinear controllers * Easy and inexpensive to design * Easy to understand


The Fuzzy Controller

System being controlled

Fuzzify Defuzzify


The Fuzzy Controller

* Provides set of inputs to produce desirable outputs * Controller’s ac)ons defined by rule base * To build rule base:

1. Iden)fy and name input variables and define ranges 2. Iden)fy and name output variables and define ranges 3. Define fuzzy membership func)on for each input variable 4. Build rule base to govern controller 5. Define how control ac)ons will be combined to provide plant inputs


Example Fuzzy Controller

•  Build system to control speed of train

•  Specifically, design system to smoothly slow and stop train from any speed and distance from sta)on


Iden)fy and name input variables and define ranges

SPEED Linguistic Range

Low High

Fast 26.5 70 Medium Fast 6.5 46.5 Slow 2.5 10.5 Very Slow 1 4 Stopped 0 2

DISTANCE Linguistic Range

Low High

Far 1,500 ∞ Medium Far 100 3,000 Near 3 200 Very Near 1 5 At 0 2


Iden)fy and name output variables and define ranges

THROTTLE Linguistic Range

Low High

Full 60% 100% Medium 20% 80% Slight 3% 30% Very Slight 1% 5% No 0 2%

BRAKE Linguistic Range

Low High

Full 98% 100% Medium 95% 99% Slight 70% 97% Very Slight 20% 80% No 0 40%


Fuzzy Membership Func)ons for Speed


Fuzzy Membership Func)ons for Brake

%


Build Rule Base Matrix

•  Build matrix of input variable combina)ons •  Each matrix posi)on may have values for output variables •  If one output variable is specified, all must be •  Shaded area on next figure corresponds to rules:

As speed is ‘stopped’ and as distance is ‘at’ then do ‘full brake’ As speed is ‘stopped’ and as distance is ‘at’ then do ‘no throdle’

•  “AND” together the inputs to produce each output


Fuzzy Rule Base Matrix

Shaded entry: IF (speed) IS (stopped) AND IF (distance) IS (at)

THEN (full brake) AND (no throdle)


Determine How Control Will Occur Usually use centroid defuzzification Example: speed = 3 km/hr; distance = 1.8 m Determine which membership functions are activated and

to what values:

speed: µVY-SLOW(3)=0.667

µSLOW(3)=0.125

µAT(1.8)=0.1

µVY-NEAR(1.8)=0.4

distance:


3 km/hr Ac)vates Very Slow and Slow


1.8 Meters Ac)vates at and very near


Rules Ac)vated with speed = 3 and distance = 1.8


Use Centroid Defuzzifica)on to Obtain Output


Summary of Controller Opera)on

•  Sample inputs

•  Determine membership values ac)vated by inputs

•  Determine which rules fire

•  Combine relevant membership values using AND operator

•  Trace ac)va)on membership values back through output func)ons

•  Use centroid method to defuzzify output to a scalar

fuzzy logic and fuzzy control - nju.edu.cn

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