246 CHAPTER 7. FUZZY-NEURAL AND NEURAL-FUZZY CONTROL
be implemented by some kind of clustering. Clustering using neural networks
belongs to the domain of unsupervised learning that relies on input data and
no corresponding outputs, as opposed to the supervised learning that we have
considered so far. Since the problems ofmembership function determination
and of extraction of fuzzy rules from numerical data by neural networks are
essentially based on unsupervised learning methods, which are not treated in
this text, we elaborate only slightly onthese problems in order for the reader
to be aware of useful methods and some references.
As in conventional clustering, the goal is to group data points that are similar
to each other in some way ñ that is, forming clusters. Given a set of crisp input-
output tuples, or training data(xi,yi),i=1,...,n, fuzzy clustering techniques
utilizing neural networks are applied to the input data to determine a collection
of fuzzy clusters. Each fuzzy cluster represents one fuzzy ìIf... then... î rule,
where the fuzzy membership functions in the rule are obtained by projecting
the cluster to input and output spaces.
We refer readers who are interested in more detail on utilization of neural
networksinproducingmembershipfunctionstoaworklikeAdeliandHung[2].
See also any of [1, 24, 36, 39, 68].
7.5 Exercises and projects
- A multi-input/multi-output system is characterized by the following set
of nonlinear difference equations:
∑
y 1 (k+1)
y 2 (k+1)
∏
=
∑
0. 8 − 0. 3
− 0. 20. 5
∏∑
y 1 (k)
y 2 (k)∏
+
∑
− 0. 1 − 0. 2
0. 0 − 0. 1
∏∑
y 12 (k−1)
y 22 (k−1)∏
+
∑
0. 70. 1
0. 10. 5
∏∑
u 1 (k)
u 2 (k)∏
The system is stable for inputsu 1 =u 2 ∈[− 0. 1 , 0 .1].(a) Generate a set of data fory 1 andy 2 using uniformly distributed
random inputsu 1 andu 2.
(b) Using the ANFIS editor, generate a trained set of fuzzy inference
rules.
(c) For the inputsu 1 (k)=Asin(2∗πk/5)andu 2 (k)=Bcos(2∗πk/5),
test the performance of the trained ANFIS forA, B∈[− 10 ,10].
(d) Is ANFIS a suitable approach to identify the nonlinear system? Ex-
plain why or why not.
(e) Discuss the effects of sample size on ANFIS learning.