Analog Realization of a Fractional Recursive Variable-Type and Order Operator for a Particular Switching Strategy

Dominik Sierociuk,Wiktor Malesza,Michal Macias,Michał Sławomir Wiraszka

doi:10.3390/electronics9050855

Dominik Sierociuk, Wiktor Malesza + Show 2 more

Open Access

https://doi.org/10.3390/electronics9050855

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Journal: Electronics	Publication Date: May 21, 2020
Citations: 2	License type: CC BY 4.0

Affiliation: Warsaw University of Technology

Abstract

In this paper, we propose a method of practical realization and an actual, physical hardware implementation of a fractional variable-type and order difference operator that switches between two (i.e., B - and D -type) variable-order definitions. After the theoretical model of such a switch, we report the experimental validation on an analog model to prove its adequacy. The tests prove with great certainty that the proposed model and the realization behave correctly. They also let the authors assume that the proposed method is the only one suitable for this case, based on the counterexamples presented.

Highlights

Fractional calculus is a generalization of traditional differential calculus for cases in which orders of the differentiation and integration are real or even complex numbers
The heat transfer is a typical diffusion problem and its modeling based on fractional calculus for a solid beam was presented in [5]
It is worth noting that an origin of the investigated analog model was analytically proved

Summary

Introduction

Fractional calculus is a generalization of traditional differential calculus for cases in which orders of the differentiation and integration are real or even complex numbers. The energy storage process in these devices is based on diffusion of ions; that is why fractional-order models are more accurate than the integer-order ones [9]. The iterative and recursive variable-order definitions are connected together by duality property As it was presented in [24], only for the compositions of particular type definitions and opposite signs of orders, it is possible to obtain the expected result—an original function. This occurs when two corresponding iterative and recursive (dual) definitions are taken into consideration. Appendix A contains short discussion about extension of this method for other order changing strategies

Fractional Constant-Type Variable-Order Differences

Fractional Variable-Type and Order Difference

Analog Model of D-Type Variable-Type and Order Operator

Experimental Results

Conclusions