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th industrial and consumer applications. In 1988 Japan established the Laboratory for International Fuzzy Engineering LIFE, a cooperative arrangement between 48 panies to pursue fuzzy researchMatsushita vacuum cleaners use micro controllers running fuzzy algorithms to interrogate dust sensors and adjust suction power accordingly. Hitachi washing machines use fuzzy controllers to loadweight, fabricmix, and dirt sensors and automatically set the wash cycle for the best use of power, water, and detergentCanon developed an autofocusing camera that uses a chargecoupled device CCD to measure the clarity of the image in six regions of its field of view and use the information provided to determine if the image is in focus. It also tracks the rate of change of lens movement during focusing, and controls its speed to prevent camera39。s fuzzy control system uses 12 inputs: 6 to obtain the current clarity data provided by the CCD and 6 to measure the rate of change of lens movement. The output is the position of the lens. The fuzzy control system uses 13 rules and requires kilobytes of memoryAs another example of a practical system, an industrial air conditioner designed by Mitsubishi uses 25 heating rules and 25 cooling rules. A temperature sensor provides input, with control outputs fed to an inverter, a pressor valve, and a fan motor. Compared to the previous design, the fuzzy controller heats and cools five times faster, reduces power consumption by 24%, increases temperature stability by a factor of two, and uses fewer sensors. The enthusiasm of the Japanese for fuzzy logic is reflected in the wide range of other applications they have investigated or implemented: character and handwriting recognition。 optical fuzzy systems。 robots, voicecontrolled robot helicopters Work on fuzzy systems is also proceeding in the US and Europe. The US Environmental Protection Agency has investigated fuzzy control for energyefficient motors, and NASA has studied fuzzy control for automated space docking: simulations show that a fuzzy control system can greatly reduce fuel consumption. Firms such as Boeing, General Motors, AllenBradley, Chrysler, Eaton, and Whirlpool have worked on fuzzy logic for use in lowpower refrigerators, improved automotive transmissions, and energyefficient electric motors. In 1995 Maytag introduced an intelligent dishwasher based on a fuzzy controller and a onestop sensing module that bines a thermistor, for temperature measurement。 a conductivity sensor, to measure detergent level from the ions present in the wash。 a turbidity sensor that measures scattered and transmitted light to measure the soiling of the wash。 and a magostrictive sensor to read spin rate. The system determines the optimum wash cycle for any load to obtain the best results with the least amount of energy, detergent, and water Research and development is also continuing on fuzzy applications in software, as opposed to firmware, design, including fuzzy expert systems and integration of fuzzy logic with neuralwork and socalled adaptive geic software systems, with the ultimate goal of building selflearning fuzzy control systems. Fuzzy sets The input variables in a fuzzy control system are in general mapped into by sets of membership functions similar to this, known as fuzzy sets. The process of converting a crisp input value to a fuzzy value is called fuzzification. A control system may also have various types of switch, or ONOFF, inputs along with its analog inputs, and such switch inputs of course will always have a truth value equal to either 1 or 0, but the scheme c