Fuzzy logic has been used in numerous applications such as facial pattern recognition, air conditioners, washing machines, vacuum cleaners, anti-skid braking systems, transmission systems, control of subway systems and automatic (unmanned) helicopters, knowledge-based systems for multi-objective optimization of power systems, weather forecasting systems, models for new product pricing or project risk assessment, medical diagnosis and treatment plans,stock trading and much more.

Fuzzy logic has been successfully used in numerous fields such as control systems engineering, image processing, power engineering, industrial automation, robotics, consumer electronics, and optimization.

One of the most famous applications of fuzzy logic is that of the Sendai Subway system in Sendai, Japan.Fuzzy logic based PLCs have been developed by companies like Moeller.

Regarding the field of fuzzy control I would say that two major properties of fuzzy set theory are most important:

1. The approximation property, which means (roughly speaking) that a smooth function f(x) can be approximated by a number of fuzzy sets to an arbitrary degree of approximation (--> universal approximator).

2. The interpolation property ( follows from the approximation property) which leads to the nice feature that inherently nonlinear problems like approximation of nonlinear functions, modeling of nonlinear technical and non-technical problems (fuzzy modeling) etc. can be solved using a finite number of rules blending each other in a smooth way.

In control tasks both properties lead to the fact that expert knowledge can be formulated by rules reflecting both the system's behavior and the regarding controller where the expert knowledge need not be mathematically formulated but rather in a more linguistic manner (--> Mamdani controllers).

On the other hand nonlinear control problems can be locally linearized with respect to the state space of the system and the design of local controllers (--> Takagi Sugeno Kang controllers). In this case system and controller are mathematically formulated by state equations and regarding control laws. Problems of stability, performance, adaptation etc. are still discussed in the control community.

Newton and derivatives are good for laws of physics  and artificial neural networks. But human brain can not calculate on these principles. It seems that natural neurons work on the principle of contraction rule. When for want get some entropy it is possible mathematically with Newton tangent method or secants and contractions. For medicine also is better work with fuzzy because of uncertainty ( many situations in human body are " normal " under some intervals ) . Words are better then numbers in some cases.

Words have also ability of bifurcations as it is in some cases at equations ( linear or nonlinear) . Then people must use quasi-regularization for next treatment.

Simply we can say, where the uncertainty occur, the fuzzy can play a major role.