Fuzzy Logic

Fuzzy Logic B. Vasanth, Electrical and Electronics Department, Rajalakshmi Engineering College Thandalam, Chennai, India vasanth1508@gmail. com I. INTRODUCTION Fuzzy logic was developed by Lotfi A. Zadeh in the 1960s in order to provide mathematical rules and functions which permitted natural language queries. Fuzzy logic provides a means of calculating intermediate values between absolute true and absolute false with resulting values ranging between 0. 0 and 1. 0. With fuzzy logic, it is possible to calculate the degree to which an item is a member. Fuzzy logic has rapidly become one of the most successful of today’s technologies for developing sophisticated control systems.

The reason for which is very simple. Fuzzy logic addresses such applications perfectly as it resembles human decision making with an ability to generate precise solutions from certain or approximate information. It fills an important gap in engineering design methods left vacant by purely mathematical approaches (e. . linear control design), and purely logic-based approaches (e. g. expert systems) in system design. While other approaches require accurate equations to model real-world behaviours, fuzzy design can accommodate the ambiguities of real-world human language and logic. It provides both an intuitive method for describing systems in human terms and automates the conversion of those system specifications into effective models. II. HOW DOES FUZZY LOGIC WORK? Fuzzy Logic requires some numerical parameters in order to operate such as what is considered significant error and significant rate-of-change-of-error, but exact values of these numbers are usually not critical unless very responsive performance is required in which case empirical tuning would determine them. For example, a simple temperature control system could use a single temperature feedback sensor whose data is subtracted from the command signal to compute “error” and then time-differentiated to yield the error slope or rate-of-change-of-error, hereafter called “error-dot”. Error might have units of degs F and a small error considered to be 2F while a large error is 5F. The “error-dot” might then have units of degs/min with a small error-dot being 5F/min and a large one being 15F/min. These values don’t have to be symmetrical and can be “tweaked” once the system is operating in order to optimize performance.

Generally, FL is so forgiving that the system will probably work the first time without any tweaking. III. FUZZY SETS A fuzzy set is a set whose elements have degrees of membership. That is, a member of a set can be full member (100% membership status) or a partial member (e. g. ess than 100% membership and greater than 0% membership). •A fuzzy subset F of a set S can be defined as a set of ordered pairs.

The first element of the ordered pair is from the set S, and the second element from the ordered pair is from the interval [0, 1]. •The value zero is used to represent non-membership; the value one is used to represent complete membership and the values in between are used to represent degrees of membership. IV. FUZZY SET OPERATIONS ?Union The membership function of the Union of two fuzzy sets A and B with membership functions and respectively is defined as the maximum of the two individual membership functions. This is called the maximum criterion. The Union operation in Fuzzy set theory is the equivalent of the OR operation in Boolean algebra. ?Intersection The membership function of the Intersection of two fuzzy sets A and B with membership functions and respectively is defined as the minimum of the two individual membership functions. This is called the minimum criterion. The Intersection operation in Fuzzy set theory is the equivalent of the AND operation in Boolean algebra ? Complement The membership function of the Complement of a Fuzzy set A with membership function is defined as the negation of the specified membership function. This is caleed the negation criterion.

The Complement operation in Fuzzy set theory is the equivalent of the NOT operation in Boolean algebra. The following rules which are common in classical set theory also apply to Fuzzy set theory. ?De Morgan’s Law ?Associativity ?Commutativity ?Distributivity V. WHY USE FUZZY LOGIC? Fuzzy Logic offers several unique features that make it a particularly good choice for many control problems. 1) It is inherently robust since it does not require precise, noise-free inputs and can be programmed to fail safely if a feedback sensor quits or is destroyed. The output control is a smooth control function despite a wide range of input variations. 2) Since the Fuzzy Logic controller processes user-defined rules governing the target control system, it can be modified and tweaked easily to improve or drastically alter system performance. New sensors can easily be incorporated into the system simply by generating appropriate governing rules. 3) Fuzzy Logic is not limited to a few feedback inputs and one or two control outputs, nor is it necessary to measure or compute rate-of-change parameters in order for it to be implemented. Any sensor data that provides some indication of a system’s actions and reactions is sufficient. This allows the sensors to be inexpensive and imprecise thus keeping the overall system cost and complexity low. 4) Because of the rule-based operation, any reasonable number of inputs can be processed (1-8 or more) and numerous outputs (1-4 or more) generated, although defining the rule base quickly becomes complex if too many inputs and outputs are chosen for a single implementation since rules defining their interrelations must also be defined. It would be better to break the control system into smaller chunks and use several smaller Fuzzy Logic controllers distributed on the system, each with more limited responsibilities. 5) Fuzzy Logic can control nonlinear systems that would be difficult or impossible to model mathematically. This opens doors for control systems that would normally be deemed unfeasible for automation. VI. HOW IS FUZZY LOGIC USED? 1) Define the control objectives and criteria: What am I trying to control? What do I have to do to control the system? What kind of response do I need? What are the possible (probable) system failure modes? ) Determine the input and output relationships and choose a minimum number of variables for input to the Fuzzy Logic engine (typically error and rate-of-change-of-error). 3) Using the rule-based structure of Fuzzy Logic, break the control problem down into a series of IF X AND Y THEN Z rules that define the desired system output response for given system input conditions.

The number and complexity of rules depends on the number of input parameters that are to be processed and the number fuzzy variables associated with each parameter. If possible, use at least one variable and its time derivative. Although it is possible to use a single, instantaneous error parameter without knowing its rate of change, this cripples the system’s ability to minimize overshoot for a step inputs. 4) Create Fuzzy Logic membership functions that define the meaning (values) of Input/Output terms used in the rules. 5) Create the necessary pre- and post-processing Fuzzy Logic routines if implementing in S/W, otherwise program the rules into the Fuzzy Logic H/W engine. 6) Test the system, evaluate the results, tune the rules and membership functions, and retest until satisfactory results are obtained. VII. DEGREES OF TRUTH Fuzzy logic and probabilistic logic are mathematically similar – both have truth values ranging between 0 and 1 – but conceptually distinct, due to different interpretations. Fuzzy logic corresponds to “degrees of truth”, while probabilistic logic corresponds to “probability, likelihood”; as these differ, fuzzy logic and probabilistic logic yield different models of the same real-world situations. Both degrees of truth and probabilities range between 0 and 1 and hence may seem similar at first. It is essential to realize that fuzzy logic uses truth degrees as a mathematical model of the vagueness phenomenon while probability is a mathematical model of randomness. 1)Truth values A basic application might characterize subranges of a continuous varirable. For instance, a temperature measurement for anti-lock brakes might have several separate membership functions defining particular temperature ranges needed to control the brakes properly.

Each function maps the same temperature value to a truth value in the 0 to 1 range. These truth values can then be used to determine how the brakes should be controlled. Fuzzy logic temperature In this image, the meaning of the expressions cold, warm, and hot is represented by functions mapping a temperature scale. A point on that scale has three “truth values”—one for each of the three functions. The vertical line in the image represents a particular temperature that the three arrows (truth values) gauge. Since the red arrow points to zero, this temperature may be interpreted as “not hot”. The orange arrow (pointing at 0. 2) may describe it as “slightly warm” and the blue arrow (pointing at 0. 8) “fairly cold”. 2)Linguistic variables While variables in mathematics usually take numerical values, in fuzzy logic applications, the non-numeric linguistic variables are often used to facilitate the expression of rules and facts. A linguistic variable such as age may have a value such as young or its antonym old. However, the great utility of linguistic variables is that they can be modified via linguistic hedges applied to primary terms.

The linguistic hedges can be associated with certain functions. VIII. LIMITATIONS OF FUZZY LOGIC It is difficult to make arguments for certain theories if it can’t be shown to perform alongside existing and accepted theories. Things like propositional logic are exact. If a statement in propositional logic could be illustrated with fuzzy logic… and fuzzy logic did it better, then maybe fuzzy would be more widely accepted. Fuzzy logic cannot be used for unsolvable problems.

This seems fairly reasonable, but its perception of being a guessing game may lead people to believe that it can be used for anything. An obvious drawback to fuzzy logic is that it’s not always accurate. The results are perceived as a guess, so it may not be as widely trusted as an answer from classical logic. Certainly, though, some chances need to be taken. How else can dressmakers succeed in business by assuming the average height for women is 5’6″? Fuzzy logic can be easily confused with probability theory, and the terms used interchangeably. While they are similar concepts, they do not say the same things.

Probability is the likelihood that something is true. Fuzzy logic is the degree to which something is true (or within a membership set). Classical logicians argue that fuzzy logic is unnecessary. Anything that fuzzy logic is used for can be easily explained using classic logic. For example, true and false are discrete. Fuzzy logic claims that there can be a gray area between true and false. But classic logic says that the definition of terms is inaccurate, as opposed to the actual truth of the statement.

Fuzzy logic has traditionally low respectability. That is probably its biggest problem. While fuzzy logic may be the superset of all logic, people don’t believe it. Classical logic is much easier to agree with because it delivers precision. Open-mindedness on the part of those who use logic is needed in order to change the acceptance of fuzzy logic. IX. EXAMPLES EXAMPLE 1 Fuzzy set theory defines fuzzy operators on fuzzy sets.

The problem in applying this is that the appropriate fuzzy operator may not be known. For this reason, fuzzy logic usually uses IF-THEN rules, or constructs that are equivalent, such as fuzzy associative matrices. Rules are usually expressed in the form: IF variable IS property THEN action For example, a simple temperature regulator that uses a fan might look like this: IF temperature IS very cold THEN stop fan IF temperature IS cold THEN turn down fan IF temperature IS normal THEN maintain level IF temperature IS hot THEN speed up fan There is no “ELSE” – all of the rules are evaluated, because the temperature might be “cold” and “normal” at the same time to different degrees. The AND, OR, and NOT operators of Boolean logic exist in fuzzy logic, usually defined as the minimum, maximum, and complement; when they are defined this way, they are called the Zadeh operators. So for the fuzzy variables x and y: NOT x = (1 – truth(x)) x AND y = minimum(truth(x), truth(y)) x OR y = maximum(truth(x), truth(y)) There are also other operators, more linguistic in nature, called hedges that can be applied. These are generally adverbs such as “very”, or “somewhat”, which modify the meaning of a set using a mathematical formula. EXAMPLE 2 Consider the colour wheel. Remember that there are 3 primary colours: Red, Yellow, and Blue.

These colours, separately, represent crisp sets. For example, true red is a non-member of true blue and of true yellow; true blue is a non-member of true yellow and of true red; yellow is a non-member of true red and of true blue.

There is a crisp boundary between these primary colours. But, it is possible to mix these colours with varying amounts of the true colours resulting in different shades of non-true colours. For example, mixing true red with true blue in equal portions of each will result in violet with a membership degree of 0. 5 in true red and 0. 5 in true blue. Different amounts of true red and true blue will result in varied membership values for the violet. The different violets represent the fuzzy boundaries between true red and true blue! EXAMPLE 3 Here is an example describing a set of young people using fuzzy sets. In general, young people range from the age of 0 to 20. But, if we use this strict interval to define young people, then a person on his 20th birthday is still young (still a member of the set). But on the day after his 20th birthday, this person is now old (not a member of the young set). How can one remedy this? By RELAXING the boundary between the strict separation of young and old.

This separation can easily be relaxed by considering the boundary between young and old as “fuzzy”. The figure below graphically illustrates a fuzzy set of young and old people. Notice in the figure that people whose ages are >= zero and 20 and < 30 are partial members of the young set. For example, a person who is 25 would be young to the degree of 0. 5. Finally people whose ages are >= 30 are non-members of the young set.

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