A First Course in FUZZY and NEURAL CONTROL

102 CHAPTER 3. FUZZY LOGIC FOR CONTROL

mean

√

xy,harmonicmeanxy/(x+y), and median. All averaging operations
can be extended to operations on fuzzy sets via

(A⊕B)(x)=A(x)⊕B(x)

The quasi-arithmetic mean is a direct generalization of the arithmetic mean.

Definition 3.5Aquasi-arithmetic meanis a strictly monotonic, continu-
ous averaging operator satisfying

4.(x⊕y)⊕(z⊕w)=(x⊕z)⊕(y⊕w)(uisbisymmetric).

Any quasi-arithmetic mean⊕can be written in the form

x⊕y=f−^1

μ

f(x)+f(y)

2

∂

wherefis a continuous, strictly increasing functionf:[0,1]→[0,1]satisfying
f(0) = 0andf(1) = 1. Technically, every quasi-arithmetic mean isisomorphic
to the arithmetic mean. Examples of quasi-arithmetic means include the power
and logarithmic means:

x⊕y =

μ

xa+ya

2

∂ (^1) a
,a> 0
x⊕y =loga(ax+ay),a> 1
Thequasi-arithmeticmeanfornpointsx 1 ,x 2 ,...,xnis given by
f−^1
μPn
i=1f(xi)
n

∂

Averaging operators suggested by Werners [80] combine the minimum and
maximum operators with the arithmetic mean.

Definition 3.6Theìfuzzy andîoperator⊕àof Werners is given by

x⊕ày=γ(x∧y)+(1−γ)

x+y

2

and theìfuzzy orîoperator⊕ˇof Werners is given by

x⊕ˇy=γ(x∨y)+(1−γ)

x+y

2

for someγ∈[0,1].

Asγranges from 0 to 1 , the ìfuzzy andî ranges from the arithmetic mean
to the minimum, and the ìfuzzy orî ranges from the arithmetic mean to the
maximum.
An operator, suggested by Zimmermannand Zysno [88], that is more general
in the sense that the compensation between intersection and union is expressed
by a parameterγ, is called ìcompensatory and.î This is not an averaging oper-
ator in the sense of our definition, since it is not idempotent. However, it plays
a similar role in decision making.