Abstract
This work deals with the problems of the Continuous Extremal Fuzzy Dynamic System (CEFDS) optimization and briefly discusses the results developed by Sirbiladze (Int J Gen Syst 34(2):107–138, 2005a; 34(2):139–167, 2005b; 34(2):169–198, 2005c; 35(4):435–459, 2006a; 35(5):529–554, 2006b; 36(1): 19–58, 2007; New Math Nat Comput 4(1):41–60, 2008a; Mat Zametki, 83(3):439–460, 2008b). The basic properties of extended extremal fuzzy measures and Sugeno’s type integrals are considered and several variants of their representation are given. Values of extended extremal conditional fuzzy measures are defined as a levels of expert knowledge reflections of CEFDS states in the fuzzy time intervals. The notions of extremal fuzzy time moments and intervals are introduced and their monotone algebraic structures that form the most important part of the fuzzy instrument of modeling extremal fuzzy dynamic systems are discussed. A new approach in modeling of CEFDS is developed. Applying the results of Sirbiladze (Int J Gen Syst 34(2) 107–138, 2005a; 34(2):139–167, 2005b), fuzzy processes with possibilistic uncertainty, the source of which are expert knowledge reflections on the states on CEFDS in extremal fuzzy time intervals, are constructed (Sirbiladze in Int J Gen Syst 34(2):169–198, 2005c). The dynamics of CEFDS’s is described. Questions of the ergodicity of CEFDS are considered. A fuzzy-integral representation of a continuous extremal fuzzy process is given. Based on the fuzzy-integral model, a method and an algorithm are developed for identifying the transition operator of CEFDS. The CEFDS transition operator is restored by means of expert data with possibilistic uncertainty, the source of which is expert knowledge reflections on the states of CEFDS in the extremal fuzzy time intervals. The regularization condition for obtaining quasi-optimal estimator of the transition operator is represented by the theorems. The corresponding calculating algorithm is provided. The results obtained are illustrated by an example in the case of a finite set of CEFDS states.
Similar content being viewed by others
References
Bellman R. E. (1947) Dynamic programming. Princeton University Press, Princeton
Bellman R.E., Esogbue A.O. (1984) Fuzzy dynamic programming and its extensions. Fuzzy Sets Secision Analysis, Studies Management Science, 20. North-Holland, Amsterdam, pp 147–167
Bellman R. E., Zadeh L. A. (1970) Decision-making in a fuzzy environment. Management Science Series B 17: 141–164
Buckley J.J., Feuring J. (2000) Fuzzy differential equations. Fuzzy Sets and Systems 110: 43–54
Castillo O., Melin P. (2001) Soft computing for control of non-linear dynamic systems. Studies in Fuzziness and Soft Computing, (Vol. 73). Physica-Verlag, Wulzburg
Diamond P., Kloeden P. (1994) Metric spaces of Fuzzy sets: Theory and applications. World Scientific, Singapore
Ding Z., Ma M., Kandel A. (1998) On the observativity of fuzzy dynamical control systems (I). Fuzzy Sets and Systems 95: 53–65
Dubois D., Prade H. (1988) Possibility theory. Plenum Press, New York
Dumitrescu D., Haloiu C., Dumitrescu A. (2000) Generators of fuzzy dynamical systems. Fuzzy Sets and Systems 113: 447–451
Feng Y. (2000) Fuzzy stochastic differential systems. Fuzzy Sets and Systems 115: 351–363
Freidman M., Henne M., Kandel A. (1997) Most typical values for fuzzy sets. Fuzzy Sets and Systems 87: 27–37
Grabisch M. (1995) Fuzzy integral in multicriteria decision making. Fuzzy sets and Systems 69: 279–298
Grabisch, M., Murofushi, T., Sugeno, M. (eds) (2000) Fuzzy measures and integrals. Theory and applications. Studies in Fuzziness and Soft Computing, 40. Physica-Verlag, Heidelberg
Higashi M., Klir G. J. (1984) Identification of fuzzy relation systems. IEEE Transactions on Systems, Man, and Cybernetics 14(2): 349–355
Kaleva O. (1987) Fuzzy differential equations. Fuzzy Sets and Systems 24: 301–317
Kandel A. (1978) Fuzzy statistics and forecast evaluation. IEEE Transactions on Systems, Man, and Cybernetics, SMG-8 5: 396–401
Kandel A., Byatt W. J. (1980) Fussy processes. Fuzzy Sets and Systems 4: 117–152
Klir G. J. (1985) Architecture of systems problem solving. Plenum Press, New York and London
Klir G. J. (1992) Fuzzy measure theory. Plenum Press, New York
Klir G. J., Folger T. A. (1988) Fuzzy sets, uncertainty and information. Prentice Hale, England
Klir G. J., Wang Z., Harmanec D. (1997) Construction of fuzzy measures in expert systems. Fuzzy Sets and Systems 92: 251–264
Kloeden P. E. (1982) Fuzzy dynamical systems. Fuzzy Sets and Systems 7(3): 275–296
Kurano M., Yasuda M., Nakagami J., Yoshida Y. (1999) A fuzzy relational equation in dynamic fuzzy systems. Fuzzy Sets and Systems 101: 439–443
Liu B. (2002) Toward fuzzy optimization without mathematical ambiguity. Fuzzy Optimization Decision Making 1(1): 43–63
Melin, P., & Castillo, O. (2002). Modelling, simulation and control of non-linear dynamical systems. An intelligent approach using soft computing and fractal theory. With 1 IBM-PC floppy disk (3.5 inch; HD). Numerical Insights, 2. London: Taylor & Francis, Ltd.
Ming M., Friedman M., Kandel A. (1997) General fuzzy least squares. Fuzzy Sets and Systems 88(1): 107–118
Pack J. Y., Han H. K. (2000) Fuzzy differential equations. Fuzzy Sets and Systems 110: 63–77
Pap, E. (1995). Null-additive Set Functions. Mathematics and its Applications, 337. Dordrecht: Kluwer Academic Publishers Group; Bratislava: Ister Science.
Piegat, A. (2001). Fuzzy modeling and control. Studies in Fuzziness and Soft Computing 69. Heidelberg: Physica-Verlag.
Ruan D., Kacprzyk J., Fedrizzi M. (2001) Soft computing for risk evaluation and management. Applications in technology, environment and finance. Physica-Verlag, Heidelberg
Sirbiladze G. (2005a) Modeling of extremal fuzzy dynamic systems. I. Extended extremal fuzzy measures. International Journal of General Systems 34(2): 107–138
Sirbiladze G. (2005b) Modeling of extremal fuzzy dynamic systems. II. Extended extremal fuzzy measures on composition products of measurable spaces. International Journal of General Systems 34(2): 139–167
Sirbiladze G. (2005c) Modeling of extremal fuzzy dynamic systems. III. Modeling of extremal and controllable extremal fuzzy processes. International Journal of General Systems 34(2): 169–198
Sirbiladze G. (2006a) Modeling of extremal fuzzy dynamic systems. IV. Identification of fuzzy-integral models of extremal fuzzy processes. International Journal of General Systems 35(4): 435–459
Sirbiladze G. (2006b) Modelling of extremal fuzzy dynamic systems. V. Optimization of continuous controllable extremal fuzzy processes and the choice of decisions. International Journal of General Systems 35(5): 529–554
Sirbiladze G. (2007) Modeling of extremal fuzzy dynamic systems. VI. Problems of states estimation (filtration) of extremal fuzzy process. International Journal of General Systems 36(1): 19–58
Sirbiladze G. (2008a) On fuzzy optimal controls in the weakly structurable continuous dynamic systems (WSCDS). New Mathematics and Natural Computation 4(1): 41–60
Sirbiladze, G. (2008b). Transformation theorems for extended lower and upper Sugeno integrals (Russian). Matematicheskie Zametki, 83(3), 439–460. translation in Mathematical Notes, 83(3–4), 399–419.
Sirbiladze, G. (2008c) Bellman‘s optimality principle in the weakly structurable dynamic systems. In: 9th WSEAS interneational conference on Fuzzy systems [FS‘08]. Sofia, Bulgaria, May 2–4. Advanced Topics on Fuzzy Systems, pp. 33–41.
Sirbiladze, G. (2010) Fuzzy dynamic programming problem for extremal fuzzy dynamic systems. In: W. A. Lodwick, & J. Kacprzyk (Eds.), Fuzzy optimization: Recent developments and applications, Studies in Fuzziness and Soft Computing, (to be published).
Sirbiladze G., Gachechiladze T. (2005) Restored fuzzy measures in expert decision-making. Information Science 169(1/2): 71–95
Sirbiladze G., Ghvaberidze B., Latsabidze T., Matsaberidze B. (2009) Using a minimal fuzzy covering in decision-making problems. Information Science 179: 2022–2027
Sirbiladze G., Sikharulidze A. (2003a) Weighted fuzzy averages in fuzzy environment. I. Insufficient expert data and fuzzy averages. International Journal of Uncertainty. Fuzziness Knowledge-Based Systems 11(2): 139–157
Sirbiladze G., Sikharulidze A. (2003b) Weighted fuzzy averages in fuzzy environment. II. Generalized weighted fuzzy expected values in fuzzy environment. International Journal of Uncertainty. Fuzziness Knowledge-Based Systems 11(2): 159–172
Sirbiladze, G., Sikharulidze, A., Ghvaberidze, B., & Matsaberidze, B. (to be published). Fuzzy-probabilistic aggregations in the discrete covering problem. International Journal of General Systems.
Srivastava P., Khare M., Srivastava Y. K. (2000) Fuzzy dynamical systems-inverse and direct spectra. Fuzzy Sets and Systems 113: 439–445
Sugeno, M. (1974). Theory of fuzzy integrals and its applications. Ph.D. Thesis of Tokyo Insitute of Technology.
Tanaka H., Ishibuchi H., Yoshikawa S. (1995) Exponential possibility regression analysis. Fuzzy Sets and Systems 69(3): 305–318
Teodorescu H.-N., Kandel A., Schneider M. (1999) Fuzzy modeling and dynamics (preface). Fuzzy Sets and Systems 106: 1–2
Terano T., Asai K., Sugeno M. (1992) Fuzzy systems theory and its applications. (Translated from the Japanese.). Academic Press, Inc, Boston, MA
Vachkov G., Fukuda T. (2000) Simplified fuzzy model based identification of dynamical systems. International Journal of Fuzzy Systems (Taiwan) 2(4): 229–235
Yoshida Y. (1994) Markov chains with a transition possibility measure and fuzzy dynamic programming. Fuzzy Sets and Systems 66: 39–57
Yoshida, Y. (eds) (2001) Dynamical aspects in Fuzzy decision making. Studies in Fuzziness and Soft Computing, (Vol. 73). Physica-Verlag, Wurzburg
Yoshida Y., Yasuda M., Nakagami J., Kyrano M. (1998) A limit theorem in dynamic fuzzy systems with a monotone property. Fuzzy Sets and Systems 94: 109–119
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sirbiladze, G. Fuzzy identification problem for continuous extremal fuzzy dynamic system. Fuzzy Optim Decis Making 9, 233–274 (2010). https://doi.org/10.1007/s10700-010-9081-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10700-010-9081-2