What is Fuzzy Inference System? – Concepts & Foundation

by codecrucks · Published 22/08/2021 · Updated 30/04/2023

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

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 has 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

The functional block diagram of the fuzzy Inference system is depicted in the following diagram:

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

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.)
Combine (usually multiplication or min) the membership values on the premise part to get the firing strength (decree of fulfilment) of each rule.
Generate the qualified consequents (either fuzzy or crisp) or each rule depending on the firing strength.
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:
- Center of Sums (CoS) Method
- Centroid of area (COA) Method

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’s approach follows linguistic fuzzy modelling and is 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 modelling 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

Watch on YouTube: Fuzzy Inference System

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 … y^r
y = y¹ ∩ y² ∩ y³ ∩ … ∩ y^r
μ_y(y) = min⁡[μ_y1(y), μ_y2(y), …, μ_yL(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 … y^r
y = y¹ ∪ y² ∪ y³ ∪ … ∪ y^r
μ_y(y) = max⁡[μ_y1(y), μ_y2(y), …, μ_yL(y)]

Fuzzy Expert System:

Fuzzy expert systems are different from conventional expert systems which rely on symbolic logic. Whereas, a fuzzy expert system is inclined towards numeric computing.

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

A functional diagram of a 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