126 CHAPTER 3. FUZZY LOGIC FOR CONTROL
Proof.The theorem follows immediately from the equalities below.
f(Aα)={f(u):A(x)≥α}
= {y∈Y:A(x)≥α,f(u)=y}
(∨Af−^1 )α = {y∈Y:∨Af−^1 (y)≥α}
= {y∈Y:∨{A(x):f(x)=y}≥α}Of course, for someα,itmaynotbetruethat∨A(f−^1 (y)) =αfor any
y.Thefunction∨Af−^1 is sometimes writtenf(A), and in this notation, the
theorem relatesf(Aα)andf(A)α.
The situationX=X 1 ◊X 2 ◊∑∑∑◊Xnis of special interest. In that case,
letA(i)be a fuzzy subset ofXi.ThenA(1)◊∑∑∑◊A(n)is a fuzzy subset of
X, and trivially
°
A(1)◊∑∑∑◊A(n)¢
α=A(1)
α ◊...◊A
(n)
α. The fuzzy subset
∨(A(1)◊∑∑∑◊A(n))f−^1 is sometimes writtenf(A(1),...,A(n)). In this notation,
thethirdpartofthetheoremmaybestatedas
ïf(A
(1)
α ,...,A(n)
α )=f(A(1),...,A(n))αfor allα>^0 if and only if for each
y∈Y,
∨(A(1)◊∑∑∑◊A(n))(f−^1 (y))A(1)◊∑∑∑◊A(n))(x)
for somex∈f−^1 (y).WhenX=Y◊Y,fis a binary operation onY.IfAandBare fuzzy
subsets ofY and the binary operation is denoted◦and written in the usual
way, then the theorem specifies exactly whenAα◦Bα=(A◦B)α,namelywhen
certain supremums are realized.
3.11Universalapproximation.......................
As we will see in the next two chapters, the design of fuzzy systems or of neural
networks is aimed at approximating some idealistic input-output maps. As
such, a general question is this. Iff:Rn→Ris an arbitrary function, can
we construct a fuzzy system or a neural network to approximatef?Ofcourse,
this question should be formulated in more specific terms. A positive answer to
this question will put fuzzy and neural control on afirm theoretical basis, since
it provides a guideline for design of successful controllers. Note however that if
this is only an existence theorem, it does not actually tell us how tofind the
desired approximator forf.
A simple form in the theory of approximation of functions that is suitable
for our purpose here is as follows. It is well-known that any continuous function
f:[a, b]→R,defined on a closed and bounded (compact) interval[a, b],canbe
approximated uniformly by polynomials ñ that is, for any degree of accuracy
ε> 0 , there is a polynomialp(x)on[a, b]such thatsupx∈[a,b]|p(x)−f(x)|<ε.
This is the classical Weierstrass theorem. The extension of this theorem, known