Fuzzy Systems 2

Lecture 5: Fuzzy Systems

Alan [email protected]

University College Cork

Mar 2010, Universitat de Girona: Intelligent Systems

Fuzzy Rules Fuzzy inference Sugeno fuzzy inference Mamdami or Sugeno? Summary

Overview

1 Fuzzy RulesIntroductionReasoning with fuzzy rules

2 Fuzzy inferenceMamdami methodInference Steps

3 Sugeno fuzzy inferenceSugeno

4 Mamdami or Sugeno?How to decide


Introduction

What is a Fuzzy Rule?

Zadeh (1973) suggested capturing human knowledge usingFuzzy Rules. A Fuzzy Rule can be defined as a conditionalstatement of the form:IF x is ATHEN y is B,

where x and y are linguistic variables; and A and B arelinguistic values, determined by fuzzy sets on the universe ofdiscourses X and Y , respectively. For example:IF objectWeight is heavyTHEN effortToLift is high,

uses a linguistic variable, objectWeight , can include the fuzzysets light , medium and heavy to describe weight. Separaterules are not necessary for cut-off points. In practice, Fuzzyexpert systems merge the rules and reduce the number of rulesby at least 90%!


Reasoning with fuzzy rules

Fuzzy sets of tall and heavy men

The above fuzzy sets provide the basis for a weight estimationmodel. The model is based on a relationship between a mansheight and his weight:IF height is tallTHEN weight is heavy .



Reasoning

Recall that a linguistic variable carries with it the concept offuzzy set qualifiers, called hedges, that modify the shape offuzzy sets. They include adverbs such as very, somewhat,quite, more or less and slightly. Fuzzy rules relate fuzzy sets:

ExampleIF height is very tallTHEN weight is very heavy .

In a fuzzy system, all rules fire to some degree, or in otherwords they fire partially. If the antecedent is true to somedegree of membership, then the consequent is also true to thatsame degree.



Montonic Selection

The value of the output or a truth membership grade of the ruleconsequent can be estimated directly from a correspondingtruth membership grade in the antecedent. This form of fuzzyinference uses a method called monotonic selection.

Figure: Monotonic Selection of values for mans weight.



Multiple Parts

A fuzzy rule can have multiple antecedents, for example:IF traffic is light AND weather is goodTHEN travelTime is short

The consequent of a fuzzy rule can also include multiple parts:IF traffic is lightTHEN travelTime is short ;

numberOfAccidents is low

Fuzzy rule in JESS: If you are tall then you are heavyFuzzyValue condition = new FuzzyValue(height,tall);

FuzzyValue conclusion = new FuzzyValue(weight, heavy);

FuzzyRule rule1 = new FuzzyRule();

rule1.addAntecedent(condition); rule1.addConclusion(conclusion);


Overview






Mamdami method

Mamdami

The most commonly used fuzzy inference technique is theso-called Mamdani method. Professor Ebrahim Mamdani ofLondon University built one of the first fuzzy systems to controla steam engine and boiler combination. He applied a set offuzzy rules supplied by experienced human operators in 1975.

Mamdami 4-step method1 fuzzification of the input variables,2 rule evaluation,3 aggregation of the rule outputs,4 defuzzification.


Inference Steps

Two-input, one-output example

Rule: 1IF funding is adequate OR staffing is smallTHEN risk is low

Rule: 2IF funding is marginal AND staffing is largeTHEN risk is normal

Rule: 3IF funding is inadequateTHEN risk is high

(Note that funding, staffing and risk are linguistic variables,inadequate, marginal and adequate are linguistic valuesdetermined by fuzzy sets on the funding universe of discourse.Also, small and large are linguistic values determined by fuzzysets on the staffing UoD etc.)


Inference Steps

Step 1: Fuzzification

The first step is to take the crisp inputs, (let funding and staffingbe x1 and y1), and determine the degree to which these inputsbelong to each of the appropriate fuzzy sets. The crisp input isa numerical input. For instance, let the expert determine afigure between 0-100 to represent funding and staffing, say35% and 60%.


Inference Steps

Step 2: Rule Evaluation

The second step is to take the fuzzified inputs,(x=A1) = 0.5, (x=A2) = 0.2, (y=B1) = 0.1 and (y=B2) = 0.7,and apply them to the antecedents of the fuzzy rules. If a givenfuzzy rule has multiple antecedents, the fuzzy operator (AND orOR) is used to obtain a single number that represents the resultof the antecedent evaluation. This number (the truth value) isthen applied to the consequent membership function. Toevaluate the disjunction of the rule antecedents, we use the ORfuzzy operation. Typically, using the fuzzy operation union:

AB(x) = max [A(x), B(x)]

Similarly, in order to evaluate the conjunction of the ruleantecedents, we apply the AND fuzzy operation intersection:

AB(x) = min[A(x), B(x)]


Inference Steps

Step 2: Mamdami Rule Evaluation


Inference Steps

Step 2: Mamdami Rule Evaluation contd.

The result of the antecedent evaluation can be now beapplied to the membership function of the consequent.Clipping is a common method of correlating the ruleconsequent with the truth value of the rule antecedent is tocut the consequent membership function at the level of theantecedent truth.

Since the top of the membership function is sliced, the clippedfuzzy set loses some information. However, clipping is still oftenpreferred because it involves less complex and fastermathematics, and generates an aggregated output surface thatis easier to defuzzify (in Step 4).


Inference Steps

Step 2: Mamdami Rule Evaluation contd.

Scaling is a better approach for preserving the shape ofthe fuzzy set. The original membership function of the ruleconsequent is adjusted by multiplying its membershipdegrees by the truth value of the rule antecedent.


Inference Steps

Step 3: Aggregation of Rule Outputs

Aggregation is the process of unification of the outputs of allrules. We take the membership functions of all ruleconsequents previously clipped or scaled and combine theminto a single fuzzy set. The input of the aggregation process isthe list of clipped or scaled consequent membership functions,and the output is one fuzzy set for each output variable.


Inference Steps

Defuzzification

The last step in the fuzzy inference process is defuzzification.Fuzziness helps us to evaluate the rules, but the final output ofa fuzzy system has to be a crisp number. The input for thedefuzzification process is the aggregate output fuzzy set andthe output is a single number.There are several defuzzification methods, but probably themost popular one is the centroid technique. It finds the pointwhere a vertical line would slice the aggregate set into twoequal masses. Mathematically this centre of gravity (COG)can be expressed as:

COG =

ba A(x)xdx ba A(x)dx

(1)


Inference Steps

Defuzzification contd.

Centroid defuzzification method finds a point representingthe centre of gravity of the fuzzy set, A, on the interval, ab.A reasonable estimate can be obtained by calculating itover a sample of points.


Overview






Sugeno

Reducing inference time

Mamdani-style inference, as we have just seen, requiresus to find the centroid of a two-dimensional shape byintegrating across a continuously varying function. Ingeneral, this process is not computationally efficient.Michio Sugeno suggested to use a single spike, a fuzzysingleton, as the membership function of the ruleconsequent. This is a fuzzy set with a membershipfunction that is unity at a single particular point on theuniverse of discourse and zero everywhere else.Sugeno-style fuzzy inference is very similar to theMamdani method. Sugeno changed only a ruleconsequent. He used a mathematical function of the inputvariable (instead of a fuzzy set).


Sugeno

Format of Sugeno-style fuzzy rule

IF x is AAND y is BTHEN z is f (x , y)

,where x , y and z are linguistic variables; A and B are fuzzysets on universe of discourses X and Y , respectively; andf (x , y) is a mathematical function.The most commonly used zero-order Sugeno fuzzy modelapplies fuzzy rules whose THEN part takes the following form:THEN z is k , where k is a constant.

In this case, the output of each fuzzy rule is constant. Allconsequent membership functions are represented bysingleton spikes.


Sugeno

Sugeno-style rule evaluation


Sugeno

Sugeno-style aggregation

We can find a weighted average (WA):

WA =(0.1 20) + (0.2 50) + (0.5 80)

0.1 + 0.2 + 0.5= 65


Sugeno

Sugeno-style defuzzification


Overview






How to decide

Strengths and Weaknesses: Mamdami & Sugeno

Mamdani method is widely accepted for capturing expertknowledge. It allows us to describe the expertise in moreintuitive, more human-like manner. However,Mamdani-type fuzzy inference entails a substantialcomputational burden.Sugeno method is computationally efficient and works wellwith optimisation and adaptive techniques, which makes itvery attractive in control problems, particularly for dynamicnonlinear systems.


Summary

SummaryWe saw how fuzzy rules relate fuzzy sets, possibly usingmonotonic selection.JESS example.Fuzzy inference: Mamdami or Sugeno?How to decide between them.

Main PartFuzzy RulesIntroductionReasoning with fuzzy rules

Fuzzy inferenceMamdami methodInference Steps

Sugeno fuzzy inferenceSugeno

Mamdami or Sugeno?How to decide

Summary

Fuzzy Systems 2

Documents

Transcript of Fuzzy Systems 2