Fuzzy Transform and Its Applications to Problems of Engineering Practice

Mgr. Martin ŠTĚPNIČKA

Doctoral thesis

Abstract:

Teorie fuzzy mnoˇzin iniciov´ana L.A. Zadehem poskytla matematik°um vhodn´y n´astroj k modelov´an´? fenom´enu v´agnosti a inˇzn´yr°um vnesla nov´e svˇetlo do teorie ˇr´?zen´? (regulace). Pozdˇeji, v roce 1985, T. Takagi a M. Sugeno navrhli fuzzy model, kter´y se stal obl´?ben´y d´?ky sv´ym aproximaˇcn´?m moˇznostem. Koneˇcnˇe v 90-t´ych letech se nˇekolik prac´? zab´yvalo aproximaˇcn´?mi moˇznostmi dalˇs´?ch obecnˇe pouˇz´?van´ych fuzzy model°u. Na z´akladˇe tohoto historick´eho v´yvoje byly poloˇzeny z´aklady nov´eho oboru zvan´eho fuzzy aproximace, kter´y se zab´yv´a aproximaˇcn´?mi vlastnostmi fuzzy model°u a jejich rozvojem. Fuzzy transformace (F-transformace) jakoˇzto konkr´etn´? z´astupce metod tohoto oboru je hlavn´?m objektem studia t´eto pr´ace. Fuzzy transformace byla navrˇzena jako pilotn´? fuzzy aproximaˇcn´? technika s c´?lem b´yt aplikov´ana v dosud netradiˇcn´?ch oblastech jako naˇr´?klad numercik´e ˇreˇsen´? diferenci ´aln´?ch rovnic. Na druhou stranu, takov´e techniky nestoj´? mimo ostatn´? techniky implementovan´e ve fuzzy syst´emech. Naopak, pokud jsou korektnˇe postaveny tj. respektuj´? pravidla a z´akonitosti fuzzy logik a fuzzy syst´em°u, poskytuj´? n´am mocn´e n´astroje pro pr´aci s probl´emy typick´ymi pro implementaci fuzzy syst´em°u. C´?lem t´eto pr´ace je v´yzkum fuzzy transformace z aproximaˇcn´?ho hlediska a jej´? pouˇzit´? v numerick´ych metod´ach. Obecnˇe pak mluv´?me o numerick´ych metod´ach na z´akladˇe fuzzy aproximaˇcn´?ch model°u. D´ale je technika studov´ana v kontextu ostatn´?ch fuzzy model°u a nakonec implementov´ana v automatick´emˇr´?zen´? tj. oblasti velmi typick´e pro fuzzy pˇr´?stupy. Struktura pr´ace je n´asleduj´?c´?. Kapitola 1 je ´uvodem do studia a ´uvodem do oblast´? uvaˇzovan´ych v pr´aci. Kapitola 2 pˇripom´?n´a z´akladn´? definice a fakta o metodˇe F-transformace a uv´ad´? nov´e v´ysledky pouˇzit´e v dalˇs´?ch kapitol´ach. Kapitola 3 se zab´yv´a rozˇs´?ˇren´?m F-transformace na funkce dvou a v´?ce promˇenn´ych a uv´ad´? v´ysledky analogick´e jednodimenzion´aln´?mu pˇr´?padu. V kapitole 4 a je studov´ana moˇznost aplikace metody na numerick´y v´ypoˇcet parci´aln´?ch diferenci´aln´?ch. Kapitola 5 je vˇenov´ana tzv. aditivn´?m norm´aln´?m form´am. Pˇredstavuje tˇr´?du aditivn´?ch norm´aln´?ch forem motivovanou konkr´etn´?m pˇr´?padem publikovan´ym v 8 [59]. Reprezentace F-transformace jako specieln´?ho pˇr´?padu aditivn´? norm´aln´? formy, jej´? rozˇs´?ˇren´? pro dalˇs´? operace a studium pˇrirozen´ych ot´azek o vztahu F-transformace a dalˇs´?ch fuzzy aproximaˇcn´?ch technik je hlavn´?m pˇr´?nosem kapitoly. Kapitola 6 pˇredstavuje aditivn´? interpretace b´az´? fuzzy pravidel a zamˇeˇruje se na jejich fuzzy interpolaˇcn´? vlastnosti. Moˇzn´y zp°usob identifikace b´aze s takovou interpretac ´? vyuˇz´?vaj´?c´? F-transformaci je zde uveden tak´e. Kapitola 7 je aplikaˇcn´? ˇc´ast´? pr´ace, kter´a pracuje s heuristick´ym n´avrhem pro fuzzy regulaci vych´azej´?c´?m z aditivn ´?ch norm´aln´?ch forem a F-transformace. Tato kapitola pˇr´?mo pouˇz´?v´a v´ysledky pˇredchoz´?ch kapitol a demonstruje vlastnosti navrhovan´e metody. Dobr´e chov´an´? procesu ˇr´?zen´eho navrhovanou metodou jsou demonstrov´any na re´aln´e aplikaci - na ˇr´?zen´? autonomn´?ho robota. Koneˇcnˇe, kapitola 8 uv´ad´? pˇr´?stup k F-transformaci z pohledu neuronov´ych s´?t´?. Poslednˇi kapitola shrnuje v´ysledky pr´ace.

Abstract:

The fuzzy set theory initiated by L.A. Zadeh provided mathematicians with an appropriate tool for modelling the vagueness phenomenon and shed new light into the control theory for engineers. Later, in 1985, T. Takagi and M. Sugeno invented a particular fuzzy model which became very popular due to its approximation ability. Finally, in the 1990?s, several studies aimed at approximation properties of the other widely used fuzzy models. Based on the historical development briefly recapitulated above, a new field called fuzzy approximation focusing on approximation properties of fuzzy models and development of new methods using fragments of the fuzzy set theory has been established. Fuzzy transform (F-transform), a particular method belonging to this field, is the main object of the study in this thesis. Fuzzy transform has been proposed as a pilot fuzzy approximation technique with the aim of being applied in up to now unusual application fields such as numerical solution of differential equations, for example. On the other hand, such techniques, including the fuzzy transform, are not excluded from the other techniques involved in fuzzy systems. Vice-versa, if they are correctly built (i.e. if they respect rules of law of fuzzy logics and the state of arts in fuzzy system), they provide us with powerful tools for dealing with typical problems for implementations of fuzzy system. The goal of this thesis is to investigate the fuzzy transform from the approximation point of view and to incorporate it in further numerical methods. In general, we talk about numerical methods on the basis of fuzzy approximation models. Moreover, the technique is studied in the context of other fuzzy models and finally implemented in automatic control, a typical field for fuzzy approaches. The structure of the work is as follows. Chapter 1 provides an introduction to the study and a brief state of the art of the studied fields. Chapter 2 recalls basic definitions and facts about the F-transform method and introduces new results useful for further chapters. Chapter 3 focuses on an extension of the F-transform for function with two or more variables and presents analogous results to those from the one dimension. In Chapter 4, a possible application to numerical computation 6 of partial differential equations is studied. Chapter 5 is devoted to the so-called additive normal forms. It introduces a class of additive normal forms stemming from the one defined in [59]. A representation of the F-transform as a special case of the additive normal form, its extension to a normal form with other possible operations and answering natural questions about the relations between F-transform and other fuzzy approximation techniques is the main aim of the chapter. Chapter 6 introduces additive interpretations of fuzzy rule bases and aims at their fuzzy interpolation properties. It presents a possible way how to identify a rule base with the additive interpretation using the F-transform. Chapter 7 is the application part of the Thesis which deals with a heuristic proposal for fuzzy control stemming from additive normal forms and the F-transforms. This chapter explicitly uses results from the previous chapters to demonstrate properties of the proposed method. Good behaviour of a system controlled by the proposed method is demonstrated on a real application: control of an autonomous robot. Finally, Chapter 8 provides a neural network point of view to the F-transform and presents experiments justifying this approach. The last chapter summarizes the results from the thesis and briefly discusses them. Keywords: Fuzzy transform, Approximation, Numerical methods, Partial differential equations, Normal forms, Fuzzy rule based systems. 7

Keywords

Kl´?ˇcov´a slova, Fuzzy transformace, Aproximace, Numerick´e metody, Parcil´aln´?

Language used: Czech

Date on which the thesis was submitted / produced: 20. 6. 2007

Accessible from:: 20. 6. 2007

Thesis defence

• Date of defence: 26. 2. 2008
• Supervisor: Prof. Irina Perfiljeva, CSc.

University of Ostrava

Faculty of Science

Doctoral programme / field:
Applied Mathematics / Fuzzy Modelling