128 CHAPTER 3. FUZZY LOGIC FOR CONTROL
Theorem 3.8 (Stone-Weierstrass theorem)Let(X,d)be a compact met-
ric space. LetH⊆C(X)such that
1.HisasubalgebraofC(X):Ifa∈Randf,g∈H,thenaf∈H and
f+g∈H.2.Hvanishes at no point ofX:Foranyx∈X,thereisanh∈Hsuch that
h(x) 6 =0.3.HseparatespointsofX:Ifx,y∈H,x 6 =y, then there is anh∈Hsuch
thath(x) 6 =h(y).ThenH isdenseinC(X)ñthatis,foranyf∈C(x)and anyε> 0 ,
there is anh∈Hsuch thatkf−hk<ε,wherek∑kis the sup-norm onC(X).
This theorem says that one can approximate elements ofC(X)by elements
of the subclassHarbitrarily closely. The application of this theorem to the
setting of fuzzy and neural control is this. If our idealistic input-output map
is a continuous function defined on a compact set of some metric space, then
it is possible to approximate it to any degree of accuracy by fuzzy systems or
neural networks, provided, of course, that we design our fuzzy systems or neural
networks to satisfy the conditions of the Stone-Weierstrass theorem.
Even if we design fuzzy systems or neural networks to satisfy the conditions of
the Stone-Weierstrass theorem, we still do not know which fuzzy system or which
specific neural network architecture to choose as the desired approximator, since,
as we already said, this theorem is only an existence theorem and is not a
constructive one. Its usefulness is to guide us to set up approximation models
that contain a ìgoodî approximator. In real-world applications, how tofind
that good approximator is the main task. As we will see, in the context of fuzzy
and neural control, this task is calledtuning. Since it is possible to design fuzzy
systems and neural networks to satisfy the conditions of the Stone-Weierstrass
theorem, but fuzzy systems and neural networks can approximate arbitrary
continuous functions definedoncompactsets.Sothese two approaches possess
the so-calleduniversal approximation property.
3.12Exercisesandprojects ........................
- For ordinary (crisp) subsetsAandBof a setX, the intersection, union,
and complement were defined in Equations 3.3 and 3.4. Prove that the
intersection, union, and complement for ordinary setsA, B:X→{ 0 , 1 }
satisfy all the following equations.(Each of these equations has been used
as the basis for a generalization of intersection, union, and complement to
fuzzy sets.)
(a) intersection:
i.(A∩B)(x)=A(x)∧B(x)