Transparent Fuzzy Systems - Modeling and Control by Riid A.

By Riid A.

This thesis completely investigates the problems concerning transparency. Fuzzy platforms are usually divided into periods. it's proven right here that for those sessions various definitions of transparency practice. for traditional fuzzy structures that use fuzzy propositions in EF-THEX principles, specific transparency constraints were derived. in response to those constraints, exploitation/modification schemes of current id algorithms are instructed, in addition, a brand new set of rules for education common fuzzy structures has been proposed, with a substantial power to lessen the distance among accuracy and transparency in fuzzy modeling. For 1st order Takagi-Sugeno structures which are interpreted when it comes to neighborhood linear types, such stipulations can't be derived because of method structure and its bad interpolation homes of 1st order TS platforms. it's. notwithstanding, attainable to unravel the transparency protection challenge within the context of modeling with one other proposed process that advantages from rale activation measure exponents.

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By Riid A.

This thesis completely investigates the problems concerning transparency. Fuzzy platforms are usually divided into periods. it's proven right here that for those sessions various definitions of transparency practice. for traditional fuzzy structures that use fuzzy propositions in EF-THEX principles, specific transparency constraints were derived. in response to those constraints, exploitation/modification schemes of current id algorithms are instructed, in addition, a brand new set of rules for education common fuzzy structures has been proposed, with a substantial power to lessen the distance among accuracy and transparency in fuzzy modeling. For 1st order Takagi-Sugeno structures which are interpreted when it comes to neighborhood linear types, such stipulations can't be derived because of method structure and its bad interpolation homes of 1st order TS platforms. it's. notwithstanding, attainable to unravel the transparency protection challenge within the context of modeling with one other proposed process that advantages from rale activation measure exponents.

Show description

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* 1 Henri Cartan Les travaux de Koszul, I (Lie algebra cohomology)
* 2 Claude Chabauty Le théorème de Minkowski-Hlawka (Minkowski-Hlawka theorem)
* three Claude Chevalley L'hypothèse de Riemann pour les corps de fonctions algébriques de caractéristique p, I, d'après Weil (local zeta-function)
* four Roger Godement Groupe complexe unimodulaire, I : Les représentations unitaires irréductibles du groupe complexe unimodulaire, d'après Gelfand et Neumark (representation concept of the complicated certain linear group)
* five Léo Kaloujnine Sur l. a. constitution de p-groupes de Sylow des groupes symétriques finis et de quelques généralisations infinies de ces groupes (Sylow theorems, symmetric teams, countless crew theory)
* 6. Pierre Samuel l. a. théorie des correspondances birationnelles selon Zariski (birational geometry)
* 7 Jean Braconnier Sur les suites de composition d'un groupe et los angeles travel des groupes d'automorphismes d'un groupe fini, d'après H. Wielandt (finite groups)
* eight Henri Cartan, Les travaux de Koszul, II (see 1)
* nine Claude Chevalley, L'hypothèse de Riemann pour les groupes de fonctions algébriques de caractéristique p, II,, d'après Weil (see 3)
* 10 Luc Gauthier, Théorie des correspondances birationnelles selon Zariski (see 6)
* eleven Laurent Schwartz, Sur un mémoire de Petrowsky : "Über das Cauchysche challenge für ein method linearer partieller Differentialgleichungen im gebiete nichtanalytischen Funktionen" (partial differential equations)
* 12 Henri Cartan, Les travaux de Koszul, III (see 1)
* thirteen Roger Godement, Groupe complexe unimodulaire, II : los angeles transformation de Fourier dans le groupe complexe unimodulaire à deux variables, d'après Gelfand et Neumark (see 4)
* 14 Marc Krasner, Les travaux récents de R. Brauer en théorie des groupes (finite groups)
* 15 Laurent Schwartz, Sur un deuxième mémoire de Petrowsky : "Über das Cauchysche challenge für procedure von partiellen Differentialgleichungen" (see 11)
* sixteen André Weil Théorèmes fondamentaux de los angeles théorie des fonctions thêta, d'après des mémoires de Poincaré et Frobenius (theta functions)
* 17 André Blanchard, Groupes algébriques et équations différentielles linéaires, d'après E. Kolchin (differential Galois theory)
* 18 Jean Dieudonné, Géométrie des espaces algébriques homogènes, d'après W. L. Chow (algebraic geometry)
* 19 Roger Godement, Sommes maintains d'espaces de Hilbert, I (functional research, direct integrals)
* 20 Charles Pisot, Démonstration élémentaire du théorème des nombres premiers, d'après Selberg et Erdös (prime quantity theorem)
* 21 Georges Reeb, Propriétés des trajectoires de certains systèmes dynamiques (dynamical systems)
* 22 Pierre Samuel, Anneaux locaux ; advent à l. a. géométrie algébrique (local rings)
* 23 Marie-Hélène Schwartz, Compte-rendu de travaux de M. Heins sur diverses majorations de l. a. croissance des fonctions analytiques et sous-harmoniques (complex research, subharmonic functions)
* 24 Charles Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable (connections on fiber bundles)
* 25 Roger Godement, Sommes keeps d'espaces de Hilbert, II (see 19)
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* 27 Jean-Pierre Serre, Extensions de groupes localement compacts, d'après Iwasawa et Gleason (locally compact groups)
* 28 René Thom, Les géodésiques dans les variétés à courbure négative, d'après Hopf (geodesics)
* 29 Armand Borel, Groupes localement compacts, d'après Iwasawa et Gleason (see 27)
* 30 Jacques Dixmier, Facteurs : class, size, hint (von Neumann algebras)
* 31 Jean-Louis Koszul, Algèbres de Jordan (Jordan algebras)
* 32 Laurent Schwartz, Sur un mémoire de ok. Kodaira : "Harmonic fields in riemannian manifolds (generalized strength theory)", II (see 26)
* 33 Armand Borel, Sous-groupes compacts maximaux des groupes de Lie, d'après Cartan, Iwasawa et Mostow (maximal compact subgroups)
* 34 Henri Cartan, Espaces fibrés analytiques complexes (analytic geometry, fiber bundles)
* 35 Charles Ehresmann, Sur les variétés presque complexes (almost-complex manifolds)
* 36 Samuel Eilenberg, Exposition des théories de Morse et Lusternick-Schnirelmann (Morse idea, Lyusternik-Schnirelmann category)
* 37 Luc Gauthier, Quelques variétés usuelles en géométrie algébrique (algebraic geometry)
* 38 Jean-Louis Koszul, Cohomologie des espaces fibrés différentiables et connexions (Chern-Weil theory)
* 39 Jean Delsarte, Nombre de strategies des équations polynomiales sur un corps fini, d'après A. Weil (Weil conjectures)
* forty Jacques Dixmier, Anneaux d'opérateurs et représentations des groupes (operator algebras, illustration theory)
* forty-one Roger Godement, Théorie des caractères dans les groupes unimodulaires (unimodular groups)
* forty two Pierre Samuel, Théorie du corps de periods neighborhood selon G. P. Hochschild (local type box theory)
* forty three Laurent Schwartz, Les théorèmes de Whitney sur les fonctions différentiables (singularity theory)
* forty four Jean-Pierre Serre, Groupes d'homotopie (homotopy groups)
* forty five Armand Borel, Cohomologie des espaces homogènes (cohomology of homogeneous areas of Lie groups)
* forty six Samuel Eilenberg, Foncteurs de modules et leurs satellites, d'après Cartan et Eilenberg (homological algebra)
* forty seven Marc Krasner, Généralisations non-abéliennes de l. a. théorie locale des corps de sessions (local fields)
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Additional resources for Transparent Fuzzy Systems - Modeling and Control

Example text

9). If the intersection point of two interpolating local models falls into the interpolation area (di < q < ci+1), V-type interpolation is the case. Otherwise, Sinterpolation occurs. 2. 2. Comparison of V- and S-type interpolation. 2 neither of the interpolation types has clear advantage over another. In (Babuska et. al. 1996), however, preference seems to be given to V-type and weighted-mean defuzzification algorithm is replaced by another functional - smoothing maximum. This replacement can be considered a 42 Transparent fuzzy systems: modeling and control deviation from the "classic" TS inference algorithm, and is not accepted here because of computational complexities of smoothing maximum.

36) i =1 Modified is also the 6th step - defuzzification. With TS systems the implication and aggregation operators are product and sum respectively, using which center-of-gravity defuzzification reduces to an algorithm that is known as fuzzy c-means defuzzification (FcM). FcM, in fact, combines the aggregation and defuzzification into one operation and is thus more than a defuzzification method (Jager 1995). R results in homogeneous TS system. 39). 39). Sometimes FcM defuzzification is applied to standard fuzzy systems so that before performing the weighted sum, each output fuzzy set is represented by its numerical representation br, which is normally chosen to be the center of gravity of the given output set.

Note that the conclusions are valid for 0th order TS systems, too. The overlap of input MFs is also one of the most important factors influencing interpolation in fuzzy systems. It is reported (Shaw 1998) that a suggested minimum of 25% and a maximum of 75% have been established experimentally. Frequently, 50% overlap is a reasonable compromise. The effect of overlap to the interpolation can be most conveniently observed in twodimensional space that we do by constructing five otherwise equivalent SISO fuzzy systems, made up of 6 rules with 0%, 25%, 50%, 75% and 100% overlap degree, respectively.

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