Download This iss the computational intelligence document notes. and more Study notes Computational Techniques in PDF only on Docsity!
Unit I
Fuzzy Logic
Content:
Introduction to Fuzzy Set- Introduction, definition, membership Function,
Fuzzy operator, Fuzzy Set Characteristics, Fuzziness and Probability.
Fuzzy Logic and Reasoning–Fuzzy Logic: Linguistics Variables and Hedges,
Fuzzy Rules.
Fuzzy Inferencing: neuro inferencing Fuzzification, Defuzzification Fuzzy logic
Controllers: Fuzzy logic Controllers, Fuzzy logic Controller Types.
Introduction to Fuzzy Set-
Introduction:
Fuzzy set theory was introduced in 1965 by Lotfi A. Zadeh. It is an extension of classical set theory. In classical sets, an element can only fully belong (1) or not belong (0) to a set. But in fuzzy set theory, an element can partially belong to a set with a membership value between 0 and 1.
Fuzzy set theory is used to handle imprecision, vagueness, and uncertainty. It uses linguistic terms such as “young,” “hot,” or “fast,” instead of exact numerical values. This makes it useful in artificial intelligence, control systems, and decision-making systems.
For example, in classical sets, a person is either young or not young. But in fuzzy sets, a person can be young with degree 0.7 or 0.5. This represents gradual change, which is closer to human thinking.
Fuzzy sets model human reasoning and approximate reasoning in artificial intelligence and control systems. Unlike crisp sets that use a binary characteristic function (0 or 1), fuzzy sets define, for example, "young" as a gradual, continuous transition.
Definition:
A fuzzy set is defined as a collection of elements where each element has a degree of membership between 0 and 1.
Mathematically, a fuzzy set A in universe of discourse X is defined as:
A = {(x, μA(x)) | x ∈ X}
Where:
● x is an element of universe X
● μA(x) is the membership function
● μA(x) ∈ [0,1]
Example: Let X = {10, 20, 30, 40, 50} represent temperature.
Fuzzy set “Hot” can be: Hot = {(10,0), (20,0.2), (30,0.5), (40,0.8), (50,1)}
This shows that 50°C is fully hot, while 30°C is moderately hot.
Membership Function:
A membership function is a mathematical function that defines the degree to which an element belongs to a fuzzy set. It assigns a membership value to each element in the universe of discourse. This value shows how strongly the element belongs to the fuzzy set. Unlike classical sets, where membership is only 0 or 1, fuzzy sets allow partial membership using values between 0 and 1.
Range :
The value of a membership function always lies between 0 and 1.
0 ≤ μ(x) ≤ 1
● μ(x) = 0 → element does not belong to the fuzzy set
● μ(x) = 1 → element fully belongs to the fuzzy set
● 0 < μ(x) < 1 → element partially belongs to the fuzzy set
Example: For fuzzy set “Hot temperature”
● 20°C → μ = 0.2 (slightly hot)
● 30°C → μ = 0.6 (moderately hot)
● 40°C → μ = 1 (fully hot)
Fuzzy operator:
1. Union (OR operation)
Meaning: Combines two fuzzy sets. It shows how much an element belongs to either set A or set B.
Formula:
μA∪B(x)=max(μA(x),μB(x))
Explanation: Take the maximum membership value from both sets.
Example: If μA(x) = 0. μB(x) = 0.
Then μA∪B(x) = max(0.4, 0.7) = 0.
Use: Used when condition is A OR B
2. Intersection (AND operation)
Meaning: Finds the common part of two fuzzy sets. It shows how much an element belongs to both A and B.
Formula:
μA∩B(x)=min(μA(x),μB(x))
Explanation: Take the minimum membership value.
Example: If μA(x) = 0. μB(x) = 0.
Then μA∩B(x) = min(0.4, 0.7) = 0.
Use: Used when condition is A AND B
3. Complement (NOT operation)
Meaning: Represents the opposite of a fuzzy set. Shows how much an element does not belong to the set.
Formula:
μA′(x)=1−μA(x)
Explanation: Subtract the membership value from 1.
Example: If μA(x) = 0.
Then μA'(x) = 1 − 0.6 = 0.
Use: Used when condition is NOT A
Fuzzy Set Characteristics:
1. Support Support is the set of all elements whose membership value is greater than 0.
Support of a fuzzy set is the collection of all elements whose membership value is greater than zero. It includes every element that belongs to the fuzzy set either partially or completely. If the membership value of an element is 0, it is not included in the support. In simple words, support represents all the elements that have at least some degree of belonging to the fuzzy set.
Mathematical form:
Support(A)={x∣μA(x)>0}
Meaning: These elements partially or fully belong to the fuzzy set.
Meaning: These elements partially belong to the fuzzy set.
Example: μA(x) = 0.3, 0.6, 0.8 → boundary elements
Simple meaning: Boundary = elements that partially belong to the set.
4. Normal Fuzzy Set A fuzzy set is normal if at least one element has membership value = 1.
A fuzzy set is called normal if at least one element in the set has a membership value equal to
- This means the maximum membership value of the set is 1. If no element reaches membership value 1, then the set is not normal. In simple terms, a normal fuzzy set must have at least one element that completely belongs to it.
Meaning: The fuzzy set has a complete membership point.
Example: If maximum μA(x) = 1 → Normal set If maximum μA(x) = 0.8 → Not normal
Simple meaning: Normal set = has at least one fully belonging element.
5. Convex Fuzzy Set A fuzzy set is convex if membership values do not decrease and increase again.
A fuzzy set is said to be convex if its membership function does not decrease and then increase again between two points. In other words, for any two elements in the set, the membership value of any element between them should be greater than or equal to the minimum of their membership values. Convex fuzzy sets usually have smooth shapes like triangular or trapezoidal curves. In simple words, a convex fuzzy set has no dips or gaps in its membership graph.
Meaning: The shape of the membership function is smooth like a triangle or curve.
Mathematical condition:
μA(λx1 +(1−λ)x2 )≥min(μA(x1),μA(x2))
Example: Triangular and trapezoidal fuzzy sets are convex.
Simple meaning: Convex set = no gaps or dips in membership graph.
Fuzziness and Probability:
Fuzziness (Fuzzy Sets): Fuzziness represents vagueness or unclear boundaries of a concept. It describes how much an element belongs to a set using a membership value between 0 and 1. It does not represent chance, but degree of truth. For example, consider the set “Hot temperature.” A temperature of 30°C may belong to the set with membership 0.6, and 40°C may belong with membership 0.9. This shows gradual belonging. Fuzziness is used when concepts are not clearly defined, such as tall, fast, or heavy. It models human thinking and linguistic terms.
Probability: Probability represents the chance or likelihood that an event will occur in the future. It deals with randomness and uncertainty of events. The probability value also ranges from 0 to 1, where 0 means the event will not happen and 1 means it will definitely happen. For example, if the probability of rain tomorrow is 0.7, it means there is a 70% chance of rain. Probability is used in situations involving uncertainty about outcomes, such as weather prediction, gambling, and risk analysis.
Fuzzy Logic and Reasoning–Fuzzy Logic:
Linguistics Variables and Hedges:
Linguistic Variables
● A linguistic variable is a variable described using words instead of exact numerical values.
● It helps represent human thinking and natural language in mathematical form.
● Each linguistic variable has:
○ A name (e.g., Temperature, Age, Speed)
○ A set of linguistic values (e.g., Cold, Warm, Hot)
○ A universe of discourse (range of values, e.g., 0°C to 50°C)
Fuzzy Rules
● Fuzzy rules are written in IF–THEN format.
● The IF part is called the antecedent (condition).
● The THEN part is called the consequent (result/action).
● Rules may contain multiple conditions connected using:
○ AND (min operation)
○ OR (max operation)
● Rules form the rule base of a fuzzy inference system.
● More rules increase system accuracy but also increase complexity.
● They can model expert knowledge in simple language.
● They are easy to understand and modify.
Reasoning Process in Fuzzy Logic
The reasoning process consists of several steps:
(1) Fuzzification
● Fuzzification is the first step in the fuzzy reasoning process.
● It converts crisp numerical input values into fuzzy values.
● It uses membership functions to calculate the degree of membership.
● One input can belong to multiple fuzzy sets at the same time with different membership values.
● It helps convert exact data into linguistic terms like low, medium, high.
● It prepares input data so fuzzy rules can be applied.
(2) Rule Evaluation (Inference)
● In this step, the system applies IF–THEN fuzzy rules.
● The system checks which rules are relevant to the input.
● Each rule calculates its own output fuzzy value.
● Multiple rules can be activated at the same time.
● Logical operators like AND (min) and OR (max) are used.
● This step simulates human decision-making.
(3) Aggregation
● Aggregation means combining outputs from all active rules.
● Each rule gives a fuzzy output, and aggregation merges them into one final fuzzy set.
● It uses operations like max function to combine outputs.
● It ensures all rule results are considered in final decision.
● It produces a single fuzzy output representation.
(4) Defuzzification
● Defuzzification is the final step.
● It converts the fuzzy output into a crisp numerical value.
● This crisp value can be used in real-world systems.
● It helps the system produce a clear and exact output.
● It is necessary because machines need exact values to operate.
Fuzzy Inferencing:
Fuzzy inferencing is the decision-making process in a fuzzy logic system where fuzzy inputs are processed using fuzzy rules to produce fuzzy outputs.
More detailed points:
● It is also called the Fuzzy Inference Engine.
○ Reasoning ability → Fuzzy logic
● It is used in:
○ Pattern recognition
○ Medical diagnosis
○ Stock market prediction
○ Robotics
○ Intelligent control systems
Example: ANFIS (Adaptive Neuro-Fuzzy Inference System)
Fuzzification:
Fuzzification is the process of converting crisp (exact numerical) input values into fuzzy values using membership functions. It helps the fuzzy logic system understand real-world inputs in the form of linguistic terms like low, medium, and high.
● Fuzzification is the first and essential step in a fuzzy logic system.
● It converts precise numerical data into fuzzy linguistic values.
● It uses membership functions to determine how much an input belongs to different fuzzy sets.
● The output of fuzzification is called the degree of membership, which ranges from 0 to 1.
● It helps represent uncertain, vague, or imprecise information in a mathematical form.
● It allows the system to work with human-like language, such as hot, cold, fast, slow.
● One crisp input can belong to more than one fuzzy set at the same time with different membership values.
● It acts as a bridge between real-world input and fuzzy inference engine.
● It improves the system’s ability to handle complex and non-linear problems.
● It is widely used in control systems, AI, robotics, washing machines, air conditioners, and decision-making systems.
Types of Membership Functions used in Fuzzification:
● Triangular Membership Function: Simple and widely used, shaped like a triangle.
● Trapezoidal Membership Function: Similar to triangular but has a flat top.
● Gaussian Membership Function: Smooth curve, used for precise systems.
Example:
Suppose the input temperature is 35°C, and fuzzy sets are Cold, Warm, and Hot.
Membership values may be:
● Cold = 0.
● Warm = 0.
● Hot = 0.
This means the temperature partially belongs to both Warm and Hot sets.
Defuzzification:
Defuzzification is the process of converting the final fuzzy output (membership values) into a single crisp numerical value. It is necessary because real-world machines and control systems require exact numbers to operate.
● It is the last step in a Fuzzy Inference System (FIS). ● It converts the aggregated fuzzy output set into a single numerical value. ● It transforms linguistic results like Fast, Medium, Slow into exact values like 75 RPM. ● It makes fuzzy logic systems practically usable in real-world applications. ● It ensures the system produces a clear, precise, and actionable output. ● It is essential for control systems and automation. ● The accuracy of the system depends heavily on the defuzzification method used. ● It reduces multiple membership values into one final decision value. ● It helps in achieving smooth and stable system performance.
Fuzzy logic Controllers:
A Fuzzy Logic Controller (FLC) is a control system that uses fuzzy logic to make decisions and control machines. It works like human thinking by using linguistic terms such as low, medium, and high instead of exact mathematical values.
A Fuzzy Logic Controller is an intelligent control system that converts input data into fuzzy values, applies fuzzy rules, and produces a crisp output to control a process or system.
A control system is an arrangement of physical components designed to alter another physical system so that this system exhibits certain desired characteristics. There exist two types of control systems: open-loop and closed-loop control systems. In open-loop control systems, the input control action is independent of the physical system output. On the other hand, in a closed-loop control system, the input control action depends on the physical system output. Closed-Hoop control systems are also known as feedback control systems. The first step toward controlling any physical variable is to measure it. A sensor measures the controlled signal, A plant is a physical system under control. In a closed-loop control system, forcing signals of the system inputs are determined by the output responses of the system. The basic control problem is given as follows:
Control System Design:
Designing a controller for a complex physical system involves the following steps:
- Decomposing the large-scale system into a collection of various subsystems.
- Varying the plant dynamics slowly and linearizing the nonlinear plane dynamics about a set of operating points.
- Organizing a set of state variables, control variables, or output features for the system under consideration.
- Designing simple P, PD, PID controllers for the subsystems. Optimal controllers can also be designed.
Step 1: Identify Input and Output Variables
First, decide what are the inputs and outputs of the system. Inputs are the values given to the controller, and outputs are the results produced.
Example: Input → Temperature Output → Fan Speed
Step 2: Divide Variables into Fuzzy Sets
Next, divide each input and output variable into different fuzzy groups using linguistic labels.
These labels can be:
● Low
● Medium
● High
Each group represents a range of values.
Example: Temperature → Cold, Warm, Hot
Step 3: Define Membership Functions
Now, create membership functions for each fuzzy set. These functions show how much a value belongs to a fuzzy set.
Membership value ranges from 0 to 1.
Example: Temperature 35°C → Warm = 0.6, Hot = 0.
Step 4: Create Rule Base
In this step, create IF–THEN rules to define system behavior.
These rules connect inputs to outputs.
Example: IF Temperature is Hot THEN Fan Speed is Fast
These rules help the controller make decisions.
1. Mamdani Fuzzy Logic Controller
A Mamdani Fuzzy Logic Controller is a type of fuzzy controller that uses fuzzy words (like low, medium, high) in both the condition and the result part of the rule. It works similar to how humans think and make decisions. For example, a rule can be: “If temperature is high, then fan speed is fast.” Here, both “high” and “fast” are fuzzy terms, not exact numbers. This makes the Mamdani controller easy to understand and explain because it uses natural language.
The Mamdani controller is the most commonly used fuzzy controller. It uses fuzzy sets in both input and output.
● Proposed by Ebrahim Mamdani in 1975.
● Uses fuzzy sets for output, not exact numerical values.
● Rules are written in simple linguistic form.
● It closely follows human thinking and reasoning.
● It requires defuzzification to convert fuzzy output into crisp value.
● It is easy to understand and design.
● It is widely used in industrial and real-world applications.
Example: IF Temperature is Hot THEN Fan Speed is Fast
Advantages:
● Easy to understand
● Good for human decision modeling
● Highly interpretable
Disadvantages:
● Requires more computation
● Slower compared to Sugeno controller
Applications:
● Air conditioners
● Washing machines
● Traffic control systems
2. Takagi-Sugeno (TS) Fuzzy Logic Controller
A Takagi–Sugeno (TS) Fuzzy Logic Controller is a type of fuzzy controller that uses mathematical functions (like linear equations or constant values) in the result part of the rule instead of fuzzy words. For example, a rule can be: “If temperature is high, then fan speed = 2 × temperature + 5.” Here, the output is calculated using a formula, not words like slow or fast.
Takagi-Sugeno controller uses mathematical functions instead of fuzzy sets for output.
● Proposed by Takagi and Sugeno.
● Output is a mathematical function (linear or constant).
● Provides crisp output directly or with simple calculation.
● Requires less computation time.
● More suitable for real-time and dynamic systems.
● More accurate than the Mamdani controller.
Example: IF Temperature is Hot THEN Fan Speed = 5 × Temperature + 10
Advantages:
● Faster computation
● More efficient
● Suitable for real-time systems
Disadvantages:
