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 :

- 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*firing strength*(*decree of fulfillment) 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 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 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 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 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 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