Fuzzy control - centroid defuzzification using max-min inferencing

Defuzzification is the process of producing a quantifiable result in crisp logic, given

fuzzy set In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined a ...

s and corresponding membership degrees. It is the process that maps a fuzzy set to a crisp set. It is typically needed in

fuzzy control A fuzzy control system is a control system based on fuzzy logic—a mathematical system that analyzes analog input values in terms of logical variables that take on continuous values between 0 and 1, in contrast to classical or digital logic, ...

systems. These systems will have a number of rules that transform a number of variables into a fuzzy result, that is, the result is described in terms of membership in

fuzzy sets In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined ...

. For example, rules designed to decide how much pressure to apply might result in "Decrease Pressure (15%), Maintain Pressure (34%), Increase Pressure (72%)". Defuzzification is interpreting the membership degrees of the fuzzy sets into a specific decision or real value. The simplest but least useful defuzzification method is to choose the set with the highest membership, in this case, "Increase Pressure" since it has a 72% membership, and ignore the others, and convert this 72% to some number. The problem with this approach is that it loses information. The rules that called for decreasing or maintaining pressure might as well have not been there in this case. A common and useful defuzzification technique is ''center of gravity''. First, the results of the rules must be added together in some way. The most typical fuzzy set membership function has the graph of a

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

. Now, if this triangle were to be cut in a straight horizontal line somewhere between the top and the bottom, and the top portion were to be removed, the remaining portion forms a

trapezoid A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a Convex polygon, convex quadri ...

. The first step of defuzzification typically "chops off" parts of the graphs to form trapezoids (or other shapes if the initial shapes were not triangles). For example, if the output has "Decrease Pressure (15%)", then this triangle will be cut 15% the way up from the bottom. In the most common technique, all of these trapezoids are then superimposed one upon another, forming a single

geometric shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie on ...

. Then, the

centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...

of this shape, called the ''fuzzy centroid'', is calculated. The ''x'' coordinate of the centroid is the defuzzified value.

Methods

There are many different methods of defuzzification available, including the following: * AI (adaptive integration) * BADD (basic defuzzification distributions) * BOA (bisector of area) * CDD (constraint decision defuzzification) * COA (center of area) * COG (center of gravity) * ECOA (extended center of area) * EQM (extended quality method) * FCD (fuzzy clustering defuzzification) * FM (fuzzy mean) * FOM (first of maximum) * GLSD (generalized level set defuzzification) * ICOG (indexed center of gravity) * IV (influence value){{ cite journal , author1=Madau, D. P. , author2=Feldkamp, L. A. , title = Influence value defuzzification method , journal = Fuzzy Systems , year = 1996 , volume = 3 , pages = 1819–1824 , doi = 10.1109/FUZZY.1996.552647 * LOM (last of maximum) * MeOM (mean of maxima) * MOM (middle of maximum) * QM (quality method) * RCOM (random choice of maximum) * SLIDE (semi-linear defuzzification) * WFM (weighted fuzzy mean) The maxima methods are good candidates for fuzzy reasoning systems. The distribution methods and the area methods exhibit the property of continuity that makes them suitable for fuzzy controllers.

Methods

Notes

See also