Fuzzy Logic and Fuzzy Sets

Sub-topics: Motivation History Fuzzy logic representation Crisp set Vs Fuzzy set Membership functions How Fuzzy logic is applied? Applications Conclusion

Motivation The term “fuzzy logic” refers to a logic of approximation. Boolean logic assumes that every fact is either entirely true or false. Fuzzy logic allows for varying degrees of truth. Computers can apply this logic to represent vague and imprecise ideas.

History Lotfi Zadeh, at the University of California at Berkeley, first presented fuzzy logic in the mid-1960's. Zadeh developed fuzzy logic as a way of processing data. Instead of requiring a data element to be either a member or non-member of a set, he introduced the idea of partial set membership. In 1974 Mamdani and Assilian used fuzzy logic to regulate a steam engine. In 1985 researchers at Bell laboratories developed the first fuzzy logic chip.

WHAT IS FUZZY LOGIC? Definition of fuzzy Definition of fuzzy logic Fuzzy – “not clear, distinct, or precise; blurred” Definition of fuzzy logic A form of knowledge representation suitable for notions that cannot be defined precisely, but which depend upon their contexts.

TRADITIONAL REPRESENTATION OF LOGIC Slow Fast Speed = 0 Speed = 1 bool speed; get the speed if ( speed == 0) { // speed is slow } else { // speed is fast

FUZZY LOGIC REPRESENTATION Slowest For every problem must represent in terms of fuzzy sets. [ 0.0 – 0.25 ] Slow [ 0.25 – 0.50 ] Fast [ 0.50 – 0.75 ] Fastest [ 0.75 – 1.00 ]

FUZZY LOGIC REPRESENTATION CONT. Slowest Slow Fast Fastest float speed; get the speed if ((speed >= 0.0)&&(speed < 0.25)) { // speed is slowest } else if ((speed >= 0.25)&&(speed < 0.5)) { // speed is slow else if ((speed >= 0.5)&&(speed < 0.75)) // speed is fast else // speed >= 0.75 && speed < 1.0 // speed is fastest

Misconceptions and Controversies Fuzzy logic is same as “imprecise logic Fuzzy logic is a new way of expressing probability. Fuzzy logic will be difficult to scale to larger problems.

Sets and Fuzzy Sets Classical sets – either an element belongs to the set or it does not. Classical sets are also called crisp (sets). Fuzzy Set Theory: An object is either in a set, not in a set, or partially in a set. In fuzzy sets, membership is based on a degree between 0 and 1 0 = item not in set 1 = item is in set If degree is between 0 and 1, then this degree is the degree to which the item is thought to be in the set

Membership Functions To determine what is the membership value of an object in a set, refer to membership functions of the object’s attribute(s). For example, we may define our membership functions for the three sets Short, Medium and Tall Attribute is height 1.0 Membership Short Medium Tall 0.5 0.0 Height (meters) 1.4 1.5 1.6 1.8 1.9 2.0 1.55

Crisp Logic Operations B A and B 1 AND OR NOT A B A or B 1 A not A 1

Fuzzy Logic Operations NOT: If Fuzzy Statement A is m true, then the statement “Not A” is (1.0 – m) true. AND: If Fuzzy Statement A is m true, and Fuzzy Statement B is n true, then the Fuzzy Statement “A and B” is k true, where k = min(m,n). OR: If Fuzzy Statement A is m true, and Fuzzy Statement B is n true, then the Fuzzy Statement “A or B” is k true, where k = max(m,n).

Rules Crisp rule: Example: “If Self is Tall and Enemy is Short, then Attack.” The Condition of a Rule: The condition for this rule is: “If Self is Tall and Enemy is Short” Fuzzy rule: The condition of the rule once again is: “If Self is Tall and Enemy is Short” Suppose that Self is 0.3 Tall, and Enemy is 0.6 Short, then this condition is 0.3 True. So, should we attack?

Building Fuzzy Systems Crisp Input Fuzzification Inference Composition Defuzzification Fuzzification Input Membership Functions Fuzzy Input Rule Evaluation Rules Fuzzy Output Defuzzification Output Membership Functions Crisp Output

Where is Fuzzy Logic used? Fuzzy logic is used directly in very few applications. Most applications of fuzzy logic use it as the underlying logic system for decision support systems.

Fuzzy System Applications Cement Kiln - first expert system to use fuzzy logic Sendai Subway - most celebrated fuzzy logic system Bullet train between Tokyo and Osaka

Applications Cont. ABS Brakes Expert Systems Video Cameras Dishwashers Washing machines Bus Time Tables

TEMPERATURE CONTROLLER The problem Change the speed of a heater fan, based on the room temperature and humidity. A temperature control system has four settings Cold, Cool, Warm, and Hot Humidity can be defined by: Low, Medium, and High Using this we can define the fuzzy set.

BENEFITS OF USING FUZZY LOGIC

Why Use Fuzzy Logic?  An Alternative Design Methodology Which Is Simpler, And Faster Fuzzy Logic reduces the design development cycle Fuzzy Logic simplifies design complexity Fuzzy Logic improves control performance Fuzzy Logic simplifies implementation Fuzzy Logic reduces hardware costs

Limitations of Fuzzy Logic Stability Learning Fuzzy Logic control may not scale well to large or complex problems Verification and Validation requires extensive testing (as in any expert system).

CONCLUSION Fuzzy logic provides an alternative way to represent linguistic and subjective attributes of the real world in computing. It is able to be applied to control systems and other applications in order to improve the efficiency and simplicity of the design process.