# What is Fuzzy Inference System? – Concepts & Foundation

Fuzzy Inference System (FIS) is key component of any fuzzy controller. FIS consists of various functional block.

The fundamental task of any FIS is to apply the if-then rules on fuzzy input and produce the corresponding fuzzy output. The whole process is based on the computer paradigm including fuzzy set theory, if-then rules and the fuzzy reasoning process.

Fuzzy inference (reasoning) is the actual process of mapping from a given input to an output using fuzzy logic.

FIS have been successfully applied in fields such as automatic control, data classification, decision analysis, expert systems and many more

And because of its multi-disciplinary nature, the fuzzy inference system is also known as

• Fuzzy-rule-based system
• Fuzzy expert system
• Fuzzy model
• Fuzzy associative memory
• Fuzzy logic controller
• Fuzzy system

Functional block diagram of fuzzy Inference system is depicted in following diagram:

As shown in Figure, a fuzzy controller operates by repeating a cycle of the following four steps :

1. Compare the input variables with the membership functions on the antecedent part to obtain the membership values of each linguistic label. (this step is often called fuzzification.)
2. Combine (usually multiplication or min) the membership values on the premise part to get firing strength (decree of fulfillment) of each rule.
3. Generate the qualified consequents (either fuzzy or crisp) or each rule depending on the firing strength.
4. Aggregate the qualified consequents to produce a crisp output. (This step is called defuzzification.)

## Components of FIS:

Knowledge Base = Data Base + Rule Base

• A database which defines the membership functions of the fuzzy sets used in the fuzzy rules
• A rule base containing a number of fuzzy IF–THEN rules

Fuzzifier:

• Converts the crisp input to a linguistic variable using the membership functions stored in the fuzzy knowledge base.

Inference Engine:

• Using If-Then type fuzzy rules converts the fuzzy input to the fuzzy output.

Defuzzifier:

• Converts the fuzzy output of the inference engine to crisp value using membership functions analogous to the ones used by the fuzzifier.
• Some commonly used defuzzifying methods:

## Fuzzy Inference Method:

The most important two types of fuzzy inference  methods:

Linguistic Fuzzy: Mamdani fuzzy inference (Mamdani and Assilian (1975))

• If I1 is A AND I2 is B then O is C
• Mamdani approach follows linguistic fuzzy modeling and characterized by its high interpretability and low accuracy.

Precise Fuzzy Modelling:

• Sugeno or Takagi–Sugeno–Kang or TS fuzzy inference (Sugeno (1985))
• If I1 is A AND I2 is B then O is f(I1, I2) = a1I1 + b1I2 + c1
• On the other hand, Takagi and Sugeno’s approach follows precise fuzzy modeling and obtains high accuracy but at the cost of low interpretability.

The main difference between the two methods lies in the consequent of fuzzy rules.

## Fuzzy Rule Based System:

IF premise (antecedent), THEN conclusion (consequent)

Canonical forms of 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

## Aggregation of Fuzzy Rules

Conjunctive System of Rules:

• In this rule system, we must satisfy all the  rule
• The rules are connected by ‘and’ connectives
• In this case, the aggregated output, y, is found by the fuzzy intersection of all the individual rule consequents, yi where i=1, 2, 3, …, r
• y = y1 and y2 and y3 …  yr
• y = y1 ∩ y2 ∩ y3 ∩ … ∩ yr
• μy(y) = min⁡[μy1(y), μy2(y), …, μyL(y)]

Disjunctive System of Rules:

• In this rule system, we should satisfy at least one rule
• The rules are connected by the ‘or’ connectives
• In this case, the aggregated output, y, is found by the fuzzy union of all the individual rule consequents, yi where i=1, 2, 3, …, r
• y = y1 or y2 or y3 …  yr
• y = y1 ∪ y2 ∪ y3 ∪ … ∪ yr
• μy(y) = max⁡[μy1(y), μy2(y), …, μyL(y)]

## Fuzzy Expert System:

Fuzzy expert systems are different than the conventional expert systems which relies on the symbolic logic. Where as, fuzzy expert system is inclined towards numeric computing.

Fuzzy expert system perform reasoning from data using membership functions and fuzzy rules

Functional diagram of fuzzy expert system is depicted here:

Components of fuzzy expert systems:

Knowledge Base (Long-Term Memory)

• Fuzzy Production Rules (If-Then)

Data Base (Short-Term Memory)

• Fact from user or Parameters

Inference Engine

• Data Driven (Forward Chaining, Modus Ponens)
• Goal Driven (Backward Chaining, Modus Tollens)

Meta-Knowledge Base

Explanatory Interface

Knowledge Acquisition Module