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8.11. Problems
8.1. In the rule-based approximation problem discuss an issue of a number of approximation knots. It may be expected that with an increasing number of knots the quality of approximation may increase. If this is so, are there any other reasons for which we would like to keep the number of knots low.
8.2. Consider a differentiable function f defined in [a, b] ∴ R and a single approximation knot m1. Determine its optimal location so that the approximation error

becomes minimized and where

How does this choice depend upon the form of the membership function in the antecedent of the rule?
8.3. The classic problem of control engineering concerns an optimal control of linear systems

with a quadratic performance index. Under these two essential assumptions an optimal control assumes a form of a linear feedback

where K is a gain matrix. The pertinent design method is readily available in the literature. Discuss how you can exploit the same idea of optimal control in the case of a nonlinear system governed by the formula

where f is a differentiable function of its arguments.
8.4. Show an example of the direct binary encoding of triangular fuzzy sets with 1/2 overlap between adjacent linguistic terms. Show how the binary crossover and mutation work in this case. Do these operations affect the number of obtained linguistic terms?
8.12. References
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- 2. K. Balakrishnian, V. Honavar, Evolutionary design of neural architectures. A preliminary taxonomy and guide to literature, TR 95-01, Dept. of Computer Science, Iowa University, Ames.
- 3. L.B. Booker, D.E. Goldberg, J. H. Holland, Classifier systems and genetic algorithms, Artificial Intelligence, 40, 1989, 235 - 282.
- 4. B. P. Buckles, F.E. Petry (eds.), Genetic Algorithms, IEEE Computer Society Press, Los Alamitos, CA, 1992.
- 5. L. Davis, Handbook of Genetic Algorithms, van Nostrand Reinhold, New York, 1991.
- 6. K. DeJong, An Analysis of the Behavior of a Class of Genetic Adaptive Systems, Ph.D. thesis, Department of Computer and Communications Sciences, University of Michigan, Ann Arbor, MI, 1975.
- 7. F. Grau, Neural network synthesis using cellular encoding and the genetic algorithm, Ph. D. thesis, Ecole Normale Superiere de Lyon, 1994.
- 8. J. J. Grefenstette, Optimization of control parameters for genetic algorithms, IEEE Transactions on Systems, Man and Cybernetics, SMC-16, 122-128.
- 9. N.H. Hassoun, J. Song, Hybrid genetic/gradient search for multilayer perception training, Optical Memory and Neural Networks, 2, 1993, 1-5.
- 10. F. Herrera, M. Lozano, J. L. Verdegay, The use of fuzzy connectives to design real-coded genetic algorithms, Mathware and Soft Computing, 1, 1995, 239 - 251.
- 11. E. Herrera, M. Lozano, Adaptation of genetic algorithm parameters based on fuzzy logic controllers. In: F. Herrera, J. L. Verdegay (eds.) Genetic Algorithms and Soft Computing, Physica-Verlag, Heidelberg, 1996, pp. 95-124.
- 12. J.H. Holland, J.S. Reitman, Cognitive systems based on adaptive algorithms, In: D.A. Waterman, F. Hayes-Roth (eds) Pattern Directed Inference Systems, Academic Press, New York, 1978. pp. 313-329.
- 13. V. Honvar, L. Uhr, Generative learning structures for generalized connectionist networks, Information Sciences, 70, 1993, 75-108.
- 14. C. L. Karr, Design of an adaptive fuzzy logic controller using a genetic algorithm, Proc. 4th Int. Conf. on Genetic Algorithms, San Diego, July 13-16, 1991, pp. 450-457.
- 15. C. L. Karr, D. A. Stanley, Fuzzy logic and genetic algorithms in time-varying control problems, Proc. North American Fuzzy Information Processing Society, Columbia, MO, May 14-17, 1991, pp. 285-290.
- 16. C. L. Karr, E. J. Gentry, Fuzzy control of pH using genetic algorithms, IEEE Trans. on Fuzzy Systems, 1, 1993, 46-53.
- 17. Z. Michalewicz, Genetic Algorithms+ Data Structures=Evolution Programs, Springer-Verlag, Berlin, 1992.
- 18. G.F. Miller, P.M. Todd, S.V. Hegde, Designing neural networks using genetic algorithms. Proc. of the 3rd Int. Conference on Genetic Algorithms, Morgan Kaufmann, San Mateo, 1989, pp. 379-384.
- 19. H. Nomura, I. Hayashi, N. Wakami, A self-tuning method of fuzzy reasoning by genetic algorithm, Proc. Int. Fuzzy Systems and Intelligent Control Conference, Lousville, KY, 1992, pp. 236-245.
- 20. J. D. Schaffer, R. A. Caruana, L. J. Eshelman, R. Das, A study of control parameters affecting online performance of genetic algorithms for function optimization, Proc. of the 3rd International Conference on Genetic Algorithms and their Applications, Arlington, 1989.
- 21. S.F. Smith, A learning system based on genetic adaptive algorithms, Ph.D. thesis, Univ. of Pittsburgh, 1980.
- 22. H. Valenzuela-Rendon, The fuzzy classifier system: a classifier system for continuously varying variables, Proc. 4th Int. Conf. on Genetic Algorithms, 1991, pp. 346-353.
- 23. X. Yao, A review of evolutionary artificial neural networks, Int. J. of Intelligent Systems, 8, 993, pp. 539-567.
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