# What is Fuzzy Inference System? – Concepts & Foundation

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:

## Fuzzy Inference Method:

The most important two types of fuzzy inference methods:

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

- If I
_{1}is A AND I_{2}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 I
_{1}is A AND I_{2}is B then O is f(I_{1}, I_{2}) = a_{1}I_{1}+ b_{1}I_{2}+ c_{1} - 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 C^{1}then restriction R^{1}**Rule 2**: If condition C^{2}then restriction R^{2}- .
- .
**Rule n**: If condition C^{n}then restriction R^{n}

## 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, y^{i}where i=1, 2, 3, …, r - y = y
^{1}and y^{2}and y^{3}… y^{r} - y = y
^{1}∩ y^{2}∩ y^{3}∩ … ∩ 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, y^{i}where i=1, 2, 3, …, r - y = y
^{1}or y^{2}or y^{3}… y^{r} - y = y
^{1}∪ y^{2}∪ y^{3}∪ … ∪ 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