A First Course in FUZZY and NEURAL CONTROL

4.2. MAIN APPROACHES TO FUZZY CONTROL 139

also be assessed by using a concept known asVapnik-Chervonenki dimensionof
a class of functions, just as in the case ofneuralnetworks[23].Thisisdiscussed
in Section 5.3.
We will discuss some well-known methods for constructing fuzzy control
systems. We lookfirst at the Mamdani and Larsen methods.

4.2.1 MamdaniandLarsenmethods ...............

Thefirst application of fuzzy logic was developed by E. H. Mamdani in 1974.
He designed an experimental fuzzy control system for a steam engine and boiler
combination by synthesizing a set of linguistic control rules obtained from expe-
rienced human operators. The popular method known today as the Mamdani
method is very similar to his original design.
For the Mamdani and Larsen methods, a typical rule base is of the form

Rj:IfxisAjthenuisBj,j=1, 2 ,...,r

wherex=(x 1 ,...,xn)∈Xandu∈U,andAj(x)=mini=1...n{Aji(xi)}for

Aj = A 1 j◊A 2 j◊∑∑∑◊Anj:X=X 1 ◊X 2 ◊∑∑∑◊Xn→[0,1]

Bj : U→[0,1]

How can we interpret the rule ìRj:IfAj(x)thenBj(u)î? One common
view is that each ruleRjrepresents a fuzzy relation onX◊U.Inotherwords,
when the input isx, then the degree to which a valueu∈Uis consistent (or
compatible) with the meaning ofRjis

Cj(x,u)=T(Aj(x),Bj(u))

whereTis some t-norm. Thus,Cjis a fuzzy subset ofX◊U. For a given input
x,Cjinduces a fuzzy subset ofU,namely

Cxj(x):u−→Cj(x,u),u∈U,j=1, 2 ,...,N

Thus, the output of each ruleRjis a fuzzy subset ofU. The Mamdani method
uses minimum for the t-norm

Cjx(u)=Cj(x,u)=Aj(x)∧Bj(u) (4.1)

and the Larsen method uses the product for the t-norm

Cxj(u)=Cj(x,u)=Aj(x)Bj(u) (4.2)

Having translated each ruleRjintoCjx, the next task is to fuse all the rules
together. We face the following problem: GivenNfuzzy subsetsC 1 x,...,CNxof
U,weneedtocombinethemtoproduceanoveralloutput. Fromasemantic
viewpoint, of course, it is a question of how the rules are connected. From a
mathematical viewpoint, we want to form a single fuzzy subset ofUfrom the