Fuzzy Rule Base and Inferences

Dr. K. GanesanDirector – TIFAC-CORE in

Automotive Infotronics,VIT University, Vellore – 632 014

kganesan@vit.ac.in

Fuzzy Rule Base and Approximate Reasoning

• Fuzzy logic uses linguistic variables. • The values of a linguistic variable are words or sentences in a

natural or artificial language. • For example, height is a linguistic variable if it takes values

such as tall, medium, short and so on. • The linguistic variable provides approximate characterization

of a complex problem. • A linguistic variable is characterized by• name of the variable (x), 2) terms set of the variable t(x), 3)

syntactic rule for generating the values of x 4) semantic rule for associating each value of x with its meaning.

• Apart from the linguistic variables, there exist what are called as linguistic hedges (linguistic modifiers).

• For example, in the fuzzy set “very tall”, the word “very” is a linguistic hedge. A few popular linguistic hedges are : very, high, slightly, moderately, plus, minus, fairly, rather.

Reasoning• Reasoning has logic as its basis, whereas

propositions are text sentences expressed in any language and are generally expressed in a canonical form as z is P where z = symbol of the subject and P = predicate designing the characteristics of the subject.

• For example, “London is in United Kingdom” is a proposition in which “London” is the subject and “in United Kingdom” is the predicate, which specifies a property of “London”, its geographical location in United Kingdom.

Logic operations over propositions• Every proportion has its opposite, called negation. • For assuming opposite truth values, a proposition and its

negation are required. • Truth tables define logic functions of 2 propositions. Let

X and Y be two propositions, either of which can be true or false.

• The basic logic operations performed over the propositions are the following:

• Conjunction (^): X AND Y• Disjunction( V): X OR Y• Implication or conditional (=>) : IF X THEN Y• Bidirectional or equivalence () : X IF AND ONLY IF Y.

Few inference ruels• On the basis of these operations on propositions,

inference rules can be formulated. • Few inference rules are as follows:• [X ^ (X => Y)] => Y• [Y’ ^ (X => Y)] => X’• [(X => Y) ^ (Y => Z)] => (X => Z)• The above rules produce certain propositions that are

always true irrespective of the true values of propositions X and Y.

• Such propositions are called tautologies. • An extension of set-theoretic bivalence logic is the fuzzy

logic where the truth values are terms of the linguistic variable “truth”.

Linguistic truth value

• The truth values of propositions in fuzzy logic are allowed to range over the unit interval [0,1].

• A truth value in fuzzy logic “very true” may be interpreted as a fuzzy set in [0,1].

• The truth value of the proposition “Z is A”, or simply the truth value of A, denoted by tv(A) is defined by a point in [0,1] (called the numerical truth value) or a fuzzy set in [0,1] (called the linguistic truth value).

Truth value of a proposition• The truth value of a proposition can be obtained

from the logic operations of other propositions whose truth values are known.

• If tv(X) and tv(Y) are numerical truth values of propositions X and Y, respectively, then

• tv(X AND Y) = tv(X) ^ tv(Y) = min[tv(X), tv(Y)]• tv(X OR Y) = tv(X) V tv(Y) = max[tv(X), tv(Y)]• tv(NOT X) = 1 – tv(X)• tv(X => Y) = tv(X) => tv(B) = max{1- tv(X),

min[tv(X), tv(Y)]}

Fuzzy propositions• For extending the reasoning capability, fuzzy logic uses fuzzy

predicates, fuzzy predicate modifiers, fuzzy quantifiers and fuzzy qualifiers in the fuzzy propositions.

• The fuzzy propositions make the fuzzy logic differ from classical logic. The fuzzy propositions are as follows:

• Fuzzy predicates: In fuzzy logic the predicates can be fuzzy, for example, tall, short, quick.

• Hence we have propositions like “Peter is tall”. It is obvious that most of the predicates in natural language are fuzzy rather than crisp.

• Fuzzy-predicate modifiers: In fuzzy logic, there exists a wide range of predicate modifiers that act as hedges, for example, very, fairly, moderately, rather slightly.

• These predicate modifiers are necessary for generating the values of a linguistic variable.

• An example can be the proposition “Climate is moderately cool”.

• Fuzzy quantifiers: The fuzzy quantifiers such as most, several, many, frequently are used in fuzzy logic.

• Employing these, we can have proposition like “Many people are educated”.

• A fuzzy quantifier can be interpreted as a fuzzy number or a fuzzy proportion, which provides an imprecise characterization of the cardinality of one or more fuzzy or nonfuzzy sets.

• Fuzzy quantifiers can be used to represent the meaning of propositions containing probabilities; as a result, they can be used to manipulate probabilities within fuzzy logic.

• Fuzzy qualifiers: There are 4 modes of qualification in fuzzy logic, which are as follows:

• Fuzzy truth qualification: It is expressed as “x is τ”, in which τ is a fuzzy truth value.

• A fuzzy truth value claims the degree of truth of a fuzzy proposition.

• Consider the example, (Paul is Young) is NOT VERY True.• Here the qualified proposition is (Paul is Young) and the

qualifying fuzzy truth value is “NOT Very True”.

Fuzzy probability qualification:• It is denoted as “x is λ”, where λ is fuzzy probability. In

conventional logic, probability is either numerical or an interval. • In fuzzy logic, fuzzy probability is expressed by terms such as

likely, very likely, unlikely, around and so on. • Consider the example, (Paul is Young) is likely.• Here the qualifying fuzzy probability is “Likely”. • These probabilities may be interpreted as fuzzy numbers, which

may be manipulated using fuzzy arithmetic.• Fuzzy possibility qualification: It is expressed “x is π”, where π

is a fuzzy possibility and can be of the following forms: possible, quite possible, almost impossible.

• These values can be interpreted as labels of fuzzy subsets of the real line.

• Consider the example, (Paul is Young) is Almost Impossible.• Here the qualifying fuzzy possibility is “Almost Impossible”.

Fuzzy usuality qualification:

• It is expressed as “usually (X) = usually (X is F)”, in which the subject X is a variable taking values in a universe of discourse U and and the predicate F is a fuzzy subset of U and interpreted as a usual value of X denoted by U(X) = F.

• The propositions that are usually true or the events that have high probability of occurrence are related by the concept of usuality qualification.

Formation of Rules• The general way of representing human

knowledge is by forming natural language expressions given by

• IF antecedent THEN consequent.• The above expression is called as the IF-THEN

rule based form. • There are 3 general forms that exist for any

linguistic variable. They are:• Assignment statements• Conditional statements• Unconditional statements

• Assignment statements: They are of the form:• y = small • Orange color = orange• a = s• Paul is not tall and not very short• Climate = autumn• Outside temperature = normal• Conditional statements: The following are some

examples• IF y is very cool THEN stop.• IF A is high THEN B is low ELSE B is not low.• IF temperature is high THEN climate is hot.• Unconditional statements: They can be of the form• Goto sum• Stop• Divide by a.• Turn the pressure low.

• The assignment statements limit the value of a variable to a specific quantity.

• The canonical rule formation for a fuzzy rule-based system is given below:

• The canonical form of fuzzy rule based system• Rule 1: If condition C1, THEN restriction R1• Rule 2: If condition C2, THEN restriction R2• …• Rule n: If condition Cn, THEN restriction Rn.• Generally, both unconditional as well as conditional

statements place some restrictions on the consequent of the rule-based process.

• Fuzzy sets and relations generally model the restrictions. • The restriction statements, irrespective of conditional or

unconditional statements, are usually connected by linguistic connectives such as “and” , “or”, or “else”.

• The restrictions denoted as R1, R2, .., Rn apply to the consequent of the rules.

Decomposition of Rules (Compound Rules)

• A compound rule is a collection of many simple rules combined together.

• Any compound rule structure may be decomposed and reduced to a number of simple canonical rule forms.

• The rules are generally based on natural language representations.

• The following are the methods used for decomposition of compound linguistic rules into simple canonical rules.

• 1) Multiple conjunctive antecedents• IF x is A1, A2, …, An THEN y is Bm.• Assume a new fuzzy set Am defined as• Am = A1 ∩ A2 ∩ … ∩ An• And expressed by means of membership function• μAm(x) = min[μA1(x), μA2(x), ..., μAn(x).• In view of the fuzzy intersection operation, the compound rule

may be rewritten as • IF Am THEN Bm.

2) Multiple disjunctive antecedents

• IF x is A1 OR x is A2, … OR x is An THEN y is Bm.

• This is can be written as• IF x is An THEN y is Bm,• Where the fuzzy set Am is defined as• Am = A1 U A2 U A3 U … U An.• The membership function is given by• μAm(x) = max[μA1(x), μA2(x), ..., μAn(x)• which is based on the fuzzy union opearation.

3) Conditional statements (with ELSE and UNLESS)

• Statements of the kind• IF A1 THEN (B1 ELSE B2)• Can be decomposed into two simple canonical rule forms,

connected or “OR”:• IF A1 THEN B1• OR• IF NOT A1 THEN B2• IF A1 (THEN B1) UNLESS A2 • Can be decomposed as• IF A1 THEN B1• OR• IF A2 THEN NOT B1• IF A1 THEN(B1) ELSE IF A2 THEN (B2)• Can be decomposed into the form• IF A1 THEN B1• OR• IF NOT A1 AND IF A2 THEN B2.

4) Nested-IF-THEN rules

• The rule “IF A1 THEN [IF A2 THEN (B1)]” can be of the form

• IF A1 AND A2 THEN B1.• Thus based on the above methods,

compound rules can be decomposed into series of canonical simple rules.

Aggregation of Fuzzy Rules• The rule based system involves more than one rule. • Aggregation of rules is the process of obtaining the overall

consequents from the individual consequents provided by each rule.

• The following 2 methods exists that aid in determining the aggregation of fuzzy rules:

• Conjunctive system of rules: For a system of rules to be jointly satisfied, the rules are connected by “and” connectives.

• Here the aggregated output, y is determined by the fuzzy intersection of all individual rule consequents, yi where i=1 to n, as

• y = y1 and y2 and … and yn• or• y = y1 ∩ y2 ∩ y3 ∩ … ∩ yn.• This aggregate output can be defined by the membership function• μy(y) = min[μy1(y), μy2(y),… μyn(y).

Disjunctive system of rules:• In this case, the satisfaction of at least one rule is

required. • The rules are connected by “or” connectives. • Here, the fuzzy union of all individual rule contributions

determines the aggregated output, as• y = y1 or y2 or … or yn• or• y = y1 U y2 U … U yn• Again it can be defined by the membership function• μy(y) = min[μy1(y), μy2(y),… μyn(y).

Fuzzy Reasoning (Approxímate Reasoning)

• In fuzzy logic both the antecedents and consequents are allowed to be fuzzy propositions.

• There exist four modes of fuzzy approximate reasoning, which include:

• Categorical reasoning• Qualitative reasoning• Syllogistic reasoning• Dispositional reasoning• Categorical reasoning• In this type of reasoning, the antecedents contain no fuzzy

quantifiers and fuzzy probabilities. • The antecedents are assumed to be in canonical form. Let

us use the notations:• L, M, N ,… = fuzzy variables taking in the universe U, V, ;• A, B, C , … = fuzzy predicates

• 1) The projection rule of inference is defined by• [L, M is R] / L is [R ↓ L]• Where [R ↓ L] denotes the projection of fuzzy relation R

on L.• 2) The conjunction rule of inference is given by• L is A, L is B L is A ∩ B• (L,M) is A, L is B (L, M) is A ∩ ( B x V)• (L,M) is A, (M, N) is B (L,M,N) = (A x W) ∩ ( U x B)• 3) The disjunction rule of inference is given by• L is A OR L is B L is A x B• L is A, M is B (L, M) is A x B

• 4) The negative rule of inference is given by• NOT (L is A) A’.• 5) The compositional rule of inference is given by• L is A, (L,M) is R M is A.R• Where A.R denotes the max-min composition of

a fuzzy set A and a fuzzy relation R given by• μA.R(v) = max.min[μA(u), μR(u,v)].• 6) The extensión principle is defined as• L is A f(L) is f(A)• Where “f” is a mapping from u to v so that L is

mapped into f(L); and based on the extension principle, the membership function of f(A) is defined as

• μ f(A)(v) = Sup μA(u), where u ε U, v ε V.

Qualitative Reasoning• In qualitative reasoning the input-output relationship of a

system is expressed as a collection of fuzzy IF THEN rules where the antecedents and consequents involve fuzzy linguistic variables.

• Qualitative reasoning is widely used in control system analysis. • Let A and B be the fuzzy input variables and C be the fuzzy

output variable. • The relation among A, B and C may be expressed as• If A is x1 AND B is y1, THEN C is z1• If A is x2 AND B is y2, THEN C is z2• …• If A is xn AND B is yn, THEN C is zn• Where xi, yi and zi, i = 1 to n, are fuzzy subsets of their

respective universe of discourse.

Syllogistic Reasoning• In syllogistic reasoning, antecedents with fuzzy

quantifiers are related to inference rules. • A fuzzy syllogism can be expressed as follows:• x = k1 A’s are B’s• y = k2 C’s are D’s• z = k3 E’s are F’s• In the above A,B,C,D, E and F are fuzzy predicates; k1

and k2 are the given fuzzy quantifiers and k3 is the fuzzy quantifier which has to be decided.

• All the fuzzy predicates provide a collection of fuzzy syllogisms.

• These syllogisms create a set of inference rules, which combines evidence through conjunction and disjunction.

Given below are some important fuzzy syllogisms.

• Produce syllogism: C.A ^ B, F = C ^ D.• Chaining syllogism: C = B, F = D, E = A• Consequent conjunction syllogism: F = B ^ D, A

= C = E• Consequent disjunction syllogism: F = B v D, A =

C = E• Precondition conjunction syllogism: E = A ^ C, B

= D = F• Precondition dijunction syllogism : E = A v C, B =

D = F.

Dispositional Reasoning• In this kind of reasoning, the antecedents are dispositions that

may contain, implicitly or explicitly, the fuzzy quantifier “usually”. • Usuality plays a major role in dispositional reasoning and it links

together the dispositional syllogistic modes of reasoning. • The important inference rules of dispositional reasoning are the

following:• 1) Dispositional projection rule of inference:• Usually ((L,M) is R) usually ( L is [ R ↓ L]).• Where [ R ↓ L] is the projection of fuzzy relation R on L.• 2) Dispositional chaining hypersyllogism:• k1 A’s are B’s , k2 B’s are C’s, usually ( B C A)• usually ( (Q1(,)Q(2)) A’s are C’s).• The fuzzy quantifier “usually” is applied to the containment

relation B C A.

• 3) Dispositional consequence conjunction syllogism:

• usually (A’s are B’s), usually (A’s or C’s) • (2 usually (-)1 (A’s are (B and C)’s)• Is a specific case of dispositional reasoning.• 4) Dispositional entailment rule of inference:• usually (x is A), A C B usually(x is B)• x is A, usually (A C B) usually (x is B)• usually (x is A), usually (A C B) usually2 (x is B)• is the dispositional entailment rule of inference. • Here “usually2” is less specific than “usually”

Fuzzy Inference System (FIS)• Fuzzy rule-based systems, fuzzy models, and fuzzy

expert systems are also known as fuzzy inference systems.

• The key unit of a fuzzy logic system is FIS. • The primary work of this system is decision making. • FIS uses “IF … THEN” rules along with connectors “OR”

or “AND” for making necessary decision rules. • The input to FIS may be fuzzy or crisp, but the output

from FIS is always a fuzzy set. • When FIS is used as a controller, it is necessary to have

crisp output. • Hence, there should be a defuzzification unit for

converting fuzzy variables into crisp variables along FIS.

Construction and working Principle of FIS

• A FIS is constructed of 5 functional blocks as shown in Figure below. They are:

• 1) A rule base that contains numerous fuzzy IF-THEN rules.

• 2) A database that defines the membership functions of fuzzy sets used in fuzzy rules.

• 3) Decision making unit that performs operation on the rules.

• 4) Fuzzification interface that converts the crisp quantities into fuzzy quantities.

• 5) Defuzzification interface that converts the fuzzy quantities into crisp quantities.

The working methodology of FIS is as follows:

• Initially, in the fuzzification unit, the crisp input is converted into fuzzy input.

• Various fuzzification methods are employed for this.

• After this process, rule base is formed. • Database and rule base are collectively called

the knowledge base. • Finally, defuzzification process is carried out to

produce crisp output. • Mainly, the fuzzy rules are formed in the rule

base and suitable decisions are made in the decision making unit.

Methods of FIS• There are 2 important types of FIS. They are:

– Mamdani FIS– Sugeno FIS

• The difference between the 2 methods lies in the consequence of fuzzy rules.

• Fuzzy sets are used as rule consequents in Mamdani FIS and linear function of input variables are used as rule consequents in Sugeno’s method.

• Mamdani’s rule finds a greater acceptance in all universal approximators than Sugeno’s method.

Mamdani FIS• Ebsahim Mamdani proposed this system in 1975

to control a stream engine and boiler combination by synthesizing a set of fuzzy rules obtained from people working on the system.

• In this case, the output membership functions are expected to be fuzzy sets.

• After aggregation process, each output variable contains a fuzzy set, hence defuzzification is important at the output stage.

• The following steps have to be followed to compute the output from this FIS.

• Step 1: Determine a set of fuzzy rules.• Step 2: Make the inputs fuzzy using input membership

functions.• Step 3: Combine the fuzzified inputs according to the

fuzzy rules for establishing a rule strength.• Step 4: Determine the consequent of the rule by

combining the rule strength and the output membership function.

• Step 5: Combine all the consequents to get an output distribution.

• Step 6: Finally, a defuzzified output distribution is obtained.

• The fuzzy rules are formed using “IF…THEN” statements and “AND/OR” connectives.

• The consequence of the rule can be obtained in 2 steps:• By computing the rule strength completely using the

fuzzified inputs from the fuzzy combination;• By clipping the output membership function at the rule

strength.• The outputs of all the fuzzy rules are combined to obtain

one fuzzy output distribution. • From FIS, it is desired to get only one crisp output. • This crisp output may be obtained from defuzzification

process. • The common technique of defuzzification used are centre of

mass and mean of maximum.

• Consider a two input Mamdani FIS with 2 rules. • The model fuzzifies the two inputs by finding the

intersection of two crisp input values with the input membership function.

• The minimum operation is used to compute the fuzzy input “and” for combining the two fuzzified inputs to obtain a rule strength.

• The output membership function is clipped a the rule strength.

• Finally, the maximum operator is used to compute the fuzzy output “or” for combining the output of the two rules.

Takagi- Sugeno Fuzzy Model (TS Method)

• Sugeno fuzzy method was proposed by Takagi, Sugeno and Kang in 1985.

• The format of the fuzzy rule of a Sugeno fuzzy model is given by:

• IF x is A and y is B THEN z = f(x,y)• Where A,B are fuzzy sets in the antecedents and z=f(x,y) is a

crisp function in the consequent. • Generally, f(x,y) is a polynomial in the input variables x and y. • If f(x,y) is a first-order polynomial, we get first order Sugeno

fuzzy model. • If f is a constant, we get zero-order Sugeno fuzzy model. • A zero order Sugeno fuzzy model is functionally equivalent to

a radial basis function network under certain minor constraints.

• The main steps of the fuzzy inference process namely, • fuzzifying the inputs• applying the fuzzy operator• are exactly the same. • The main difference between Mamdani’s and Sugeno’s

methods is that Sugeno output membership functions are either linear or constant.

• The rule format of Sugeno form is given by:• “If 3=x and 5=y then output is z = ax+by+c”.• For a Sugeno model of zero order, the output level z is a

constant. • The operation of a Sugeno rule is shown in Figure below:

• Sugeno’s method can act as an interpolating supervisor for multiple linear controllers, which are to be applied, because of the linear dependence of each rule on the input variables of a system.

• A Sugeno model is suited for smooth interpolation of linear gains that would be applied across the input space and for modeling nonlinear systems by interpolating between multiple linear models.

• The Sugeno System uses adaptive techniques for constructing fuzzy models.

• The adaptive techniques are used to customize the membership fuctions.

Comparison between Mamdani and Sugeno Method

• The main difference between Mamdani and Sugeno methods lies in the output membership fuctions.

• The Sugeno output membership functions are either linear or constant.

• The difference also lies in the consequents of their fuzzy rules and as a result their aggregation and defuzzification procedures differ suitably.

• A large number of fuzzy rules must be employed in Sugeno method for approximating periodic or highly oscillatory functions.

• The configuration of Sugeno fuzzy systems can be reduced and it becomes smaller than that of Mamdani fuzzy systems if nontriangular or nontrapezoidal fuzzy input sets are used.

• Sugeno controllers have more adjustable parameters in the rule consequent and the number of parameters grows exponentially with the increase of the number of input variables.

• There exist several mathematical results for Sugeno fuzzy controllers than for Mamdani controllers.

• Formation of Mamdani FIS is more easier than Sugeno FIS.

The Main advantages of Mamdani method are:

• It has widespread acceptance• It is well-suitable for human input• It is intuitive• On the other hand, the advantages of Sugeno

method include:• It is computationally efficient.• It is compact and works well with linear

technique, optimization technique and adaptive technique.

• It is best suited for mathematical analysis.• It has a guaranteed continuity of the output

surface.