Fuzzy Control
#Fuzzy Control Basics
###Crisp Sets
- Crisps Sets are those sets from which every value either has a 0 (nothing) contribution or a 1 (full) contribution.
- For example : Consider a Universe of numbers U. Select the set {1,2,3,4,5} out of it. Now, if this set is a crisp set, we can say that 1,2,3,4,5 have a 1 (full) contribution to this set, while other numbers in the universe don't have any contribution to the set.
- Such contribution are mathematically termed as membership. Each set has it's own function of such contribution by it's constituent elements, also known as membership functions.
- So, in the example mentioned above, membership function has 1 at 1,2,3,4,5 and zero at all other values in the universe.
- The only difference is here we can have any value of membership - from 0 to 1 - of any value in the universe. Anything.
- Owing to this reason Fuzzy sets can be classified into two sub-categories : Discrete and Continuous
- Discrete Fuzzy set has discrete values in the set, each having it's own discrete membership value. Example of the membership function of a discrete fuzzy set :
- Continuous Fuzzy Set has continuous and the membership function is also continuously defined. There are various shapes of such membership functions which are explained later. Example of the Trapezoidal membership function of a continuous fuzzy set :
where, mu(x) is the membership value at any x on the x-axis.