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where A11, A12, A21, A22, B1 and B2 are given fuzzy numbers.

4.6. The signal is described as follows

where Ω and Φ are Gaussian fuzzy numbers with the membership functions

Determine X(t). Plot its membership function for selected time instants (t). Study the same problem assuming that Ω and Φ are intervals defined as follows

Compare the results of computing using fuzzy numbers and intervals.

4.7. Consider a fuzzy model governed by the expression

where A and B are triangular fuzzy numbers with the membership functions A =(0.5, 1, 2) and B = (0.9, 1, 1.1).

(i)  Assume that X is a singleton. Plot Y as a function of x.
(ii)  Treat the above expression as an equation for A, B and Y given. Solve it with respect to X.

4.8. Consider an addition of two images (fuzzy relations A and B),

see Fig. 4.34. Propose pertinent computational formulas. Illustrate your findings for Gaussian membership functions.


Figure 4.34  Addition of two images

4.9. Consider an iterative process of subtracting a triangular fuzzy number A = (1, 0.8, 1.2) from a numeric value B = {10},

Analyze the results of successive iterations and discuss an effect of accumulation of fuzziness. Repeat the same analysis for an iterative multiplication of A and B.

4.10. The classic quadratic map is described as

Assuming that A is a triangular fuzzy number, A = (0.5, 0.45, 0.55), analyze the results of iteration of A through this nonlinear mapping.

4.11. Consider the sup-product composition used in the extension principle and perform the addition of two triangular fuzzy numbers. What is the result of this operation? Comment whether you still could retain a triangular fuzzy number as the outcome of this operation. This generalized version of addition of fuzzy numbers has been intensively studied by Dubois and Prade (1981).

4.12. Show that if A1 and A2 are two fuzzy numbers with triangular membership functions with 1/2 overlap and

where a1 and a2 are real numbers, then the above expression treated as a function of x, y = y(x) is a piecewise linear function.

4.13. Generalize a triangular membership function to a case of many variables. Assuming the standard way of computing an activation (matching) level of this linguistic label, express your finding as a certain distance function.

4.14. Linguistic quantifiers are fuzzy sets defined in [0, 1]

They act on fuzzy sets by transforming their membership functions in the following way

Discuss the role of the quantifiers: more or less (τ(v) = v0.5), very (τ(v) = v2) and unknown (τ(v) = 1.0) in transforming fuzzy sets.

4.15. A small robot is equipped with infrared sensors that are used to sense eventual obstacles. Based on the signals of these sensors, the robot can navigate throughout the environment by using two motors that allow it to turn right, left or move forward. Propose some navigation rules. Could you think of some defaults in this navigation protocol?

4.9. References

1.  Astrom, K., B. Wittenmark, Computer Controlled Systems: Theory and Design, Prentice-Hall, Englewood Cliffs, NJ, 1984.
2.  J. P. Aubin, H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, 1990.
3.  J. G. Dijkman, H. van Haeringen, S. I. De Lange, Fuzzy numbers, J. Math Anal. and Appl. 92, 1983, 301 - 341.
4.  D. Dubois, H. Prade, Fuzzy Sets and Systems, Academic Press, New York, 1980, Chapter 2.
5.  D. Dubois, H. Prade, Operations on fuzzy numbers, Int. J. Systems Science, 9, 1979, 613 -626.
6.  D. Dubois, H. Prade, Fuzzy real algebra - some results, Fuzzy Sets and Systems, 2, 1979, 327 - 348.
7.  D. Dubois, H. Prade, Additions of interactive fuzzy numbers, IEEE Trans. on Automatic Control, 26, 1981, 926 - 936.
8.  M. L. Ginsberg (ed), Readings in Nonmonotonic Reasoning, Morgan Kaufman, Los Altos, CA, 1987.
9.  A. Kaufmann, M. M. Gupta, Fuzzy Mathematical Models in Engineering and Management Science, North Holland, Amsterdam, 1988.
10.  H. K.Khalil, Nonlinear Systems, 2nd Edition, Prentice Hall, Upper Saddle River, NJ, 1996.
11.  Kosko, B., Neural Netwoks and Fuzzy Systems: A Dynamical Systems Approach to Machine Intelligence, Prentice-Hall, NJ, 1992.
12.  R. E. Moore, Interval Analysis, Prentice Hall, Englewood Cliffs, NJ, 1966.
13.  M. Mizumoto, K. Tanaka, The four operations of arithmetic on fuzzy numbers. Syst. -Comput. -Controls, 7, 1976, 73 - 81.
14.  W. Pedrycz, Fuzzy Control and Fuzzy Systems, 2nd Edition, Research Studies Press, Taunton, UK, 1993.
15.  W. Pedrycz, Fuzzy Sets Engineering, CRC Press, Boca Raton, FL, 1995.
16.  R. R. Yager, Using approximate reasoning to represent default knowledge, Artificial Intelligence, 31, 1987, 99 - 112.
17.  R. R.Yager, D. Filev, Essentials of Fuzzy Modeling and Control, Wiley Interscience, New York, 1994.
18.  L. A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. on Systems, Man, and Cybernetics, 2, 1973, 28 - 44.
19.  H. J. Zimmermann, Fuzzy Set Theory and Its Applications, 2nd Edition, Kluwer Academic Publishers, Dordrecht, 1993.


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