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+ICT619 Intelligent SystemsTopic 3: Fuzzy SystemsPART A Introduction Applications Fuzzy sets and fuzzy logic Probability and fuzzy logic
Design of a fuzzy controllerPART B Building fuzzy systems Advantages and limitations of fuzzy systems Case Studies
Most rules in real life are designed to deal with attributes and outcomes that are meant to be clearly distinguishable. Most categories of objects and properties are determined by using thresholds or hard boundaries separating distinct ranges of values, with each such range associated with any one category or property.
But in many situations it is difficult to apply such clear cut boundaries. Application of such boundaries, even if possible, quite often goes against common sense. For example, if temperatures of 26 degrees or above are regarded as warm, and those below 26 degrees as cool, this temperature classification rule makes more sense for a temperature of 17 degrees than it does for a temperature of 25 degrees. This rule dictates something for certain temperatures that goes against common sense.
It is obvious that rules of categorisation with hard boundaries are unsatisfactory as they fail to reflect the inherent imprecision of real life objects and attributes. This has led to the conclusion that imprecision can be handled more effectively if rules themselves are defined using imprecise or fuzzy boundaries. For example, a temperature of 25 degrees is warm to a much greater extent than it is cool. Expressions in everyday language often use imprecise terms such as strong, cold, very and low without actually defining exact boundaries within which they apply simply because such definitions would require us to stipulate hard boundaries that would go against common sense.
Rules with hard boundaries are based on traditional set theory and two-valued (binary) logic where any proposition is either true or false, any element either belongs to a set or it does not. In 1965 Lotfi Zadeh of the University of California proposed an alternative fuzzy logic and associated fuzzy set theory, which can be used for approximate reasoning. In it, elements can belong to a set to a degree, giving rise to degrees of memberships, which can be any value between 0 (meaning definitely not a member) to 1 (definitely a member). Instead of being either true or false, a proposition in fuzzy logic can be both true and false to certain degrees!
Fuzzy logic and fuzzy rule based systems, which are based on fuzzy logic, provide a methodology for reasoning that can handle imprecision in rules and assertions expressed by human experts. Fuzzy logic enables reasoning using approximations but this does not imply vague or fuzzy outputs for problem solutions. The imprecisely stated inputs are transformed into precise numerical inputs that eventually produce precise numerical outputs.
Regardless of the controversy generated about the mathematical foundation of fuzzy systems, intelligent control and decision support systems based on fuzzy logic have proved their superiority over conventional hard logic based systems.
Fuzzy system applications
Fuzzy logic and systems based on it are an actively investigated topic in many fields including engineering and computer science, business and management, psychology, philosophy, and mathematics.
Fuzzy system applications may be grouped into two categories:
1. Fuzzy control systems
2. Fuzzy decision support systems
A control system continuously receives sensor inputs such as speed, distance, temperature, pressure etc, and utilises these to vary some output variables such as amount of electric current flowing into an electric motor or rate of fuel supply to an engine. The objective is to achieve a smoother, more effective and efficient operation, for example, a self-regulated air conditioner that maintains desirable temperature and humidity with less fluctuation and greater energy efficiency.
Fuzzy decision support systems, unlike their control system counterparts are not as time critical, and are used to make recommendations to aid decision making, usually in a business environment.
Probably the most renowned fuzzy system in use today is the subway in the Japanese city of Sendai. Since 1987, a fuzzy control system has been in operation to keep the trains running safely, smoothly and more efficiently than possible under manual control.
Fuzzy control systems have appeared in numerous products such as domestic appliances and motor cars manufactured by the Japanese. Matsushitas fuzzy vacuum cleaner, washing machine and camcorders are examples of such appliances.
Mitsubishi designed a fuzzy control system for lifts, improving their efficiency at handling crowds all wanting to be in the lift at the same time. Nissan uses fuzzy automatic transmission and fuzzy antilock brakes in its cars. Sonys fuzzy TV set automatically adjusts sharpness, contrast, brightness and colour.
Fuzzy intelligent systems in business
Although it is relatively early days, fuzzy logic, in the form of fuzzy expert systems, is proving to be a powerful tool in business knowledge based systems and decision support. Fuzzy ES have been successfully applied to large real world applications in
Managed health care
Financial services such as insurance risk assessment and company stability analysis
Product marketing and sales analysis
Extraction of information from databases (data mining)
Resource and project management
In general, intelligent systems based on fuzzy logic are less complex, smaller, easier to build, implement, maintain and extend than similar systems built using conventional expert systems.
Fuzzy sets and fuzzy logicFuzzy sets
In classical logic, the boundary between what is in a set and what is outside is sharp. For example, all people earning $75,000 or higher may belong to (be a member of) the set high-income earner. Anyone earning less than $75,000 is not in the set high-income earner, even if that person earns $74,900! Because of the sharpness of the set boundary, such sets are known as crisp sets.
The nature of the membership graph for a crisp set is illustrated using the above example in the diagram below.
As the domain (in this case income) value increases, the degree of membership in the set high-income earner remains false (zero), but suddenly jumps to 1 (true) as income reaches $75000.
For a fuzzy set, the membership value lies within the range zero (no membership) to 1 (complete membership). For example, the membership graph of the fuzzy set high-income earner may have the shape shown below. According to this membership function, someone earning $30,000 will have a membership value of 0.1, and an earning of $74,900 will give a membership value of 0.998. All incomes at or below $25,000 have the membership value zero and all those at or above $75,000 have membership value 1.
The diagram below shows the membership functions of two example fuzzy sets warm and hot. These two membership functions are bell-shaped, Depending on the application, fuzzy set membership functions can have different shapes including S-shape, triangle, trapezoid and so on.
By allowing continuous valued degrees of membership, fuzzy sets enable the handling of imprecise concepts such as high, weak, warm and medium, which are commonly encountered in real life problems.
Fuzzy logic and probability
There has been a lot of discussion about the nature of fuzzy logic ever since it appeared in the 70s. Many are quite dismissive about fuzzy logic and regard it as just a form of probability. They are also sceptical about the soundness of its basis and its reliability and the term fuzzy has not helped in this regard. But as outlined below, despite their apparent similarity, there is an important difference between the two paradigms.
Both fuzzy logic and probability deal with the issue of uncertainty. Both use a continuous 0 to 1 scale for measuring uncertainty, but with a fundamental difference.
Probability deals with likelihood the chance of something happening or something having a certain property. Fuzzy logic deals not with likelihood of something having a certain property, but the degree to which it has that property. For example, if we toss a coin, probability attempts to predict the likelihood of it landing with the head or tail face up. Once the coin has landed, there is nothing ambiguous about the outcome of the tossing event probability is no longer applicable, nor necessary.
If on the other hand, we had a badly defaced coin, there would be uncertainty as to whether a face is actually the head or the tail. It is in this situation that fuzzy logic plays its role by stating the degree to which the face in question might be the head or tail. In other words, fuzzy logic concerns not the likelihood of the event (head or tail), but the degree to which the event has happened.
Fuzzy set theory and fuzzy logic provide a mathematical tool for handling uncertainty as described above. Despite the controversy it has generated, there is certainly no fuzziness about its usefulness as a powerful tool for solving problems.
The fuzzy model of a problem consists of a series of unconditional and conditional fuzzy propositions. These are similar in form to those in conventional expert systems. An unconditional fuzzy proposition has the form
x is Y
where x is a linguistic vari