Fractional Calculus as a Simple Tool for Modeling and Analysis of Long Memory Process in Industry

Petráš, Ivo; Terpák, Ján

doi:10.3390/math7060511

Cited by 32 publications

(17 citation statements)

References 13 publications

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“…In recent years, fractional-order differential operators are introduced into nonlinear dynamical system, and the study of chaos in fractional-order nonlinear dynamical systems becomes a hot topic. At present, there are many definitions of the fractional derivatives, including Grünwald-Letnikov (G-L) definition [38,39], Riemann-Liouville (R-L) definition [40,41] and Caputo definition [42,43].…”

Section: The Definitions Of the Fractional Derivativesmentioning

confidence: 99%

Complexity Analysis of Three-Dimensional Fractional-Order Chaotic System Based on Entropy Theory

Zhang

Yang

2021

IEEE Access

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Section: The Definitions Of the Fractional Derivativesmentioning

confidence: 99%

Complexity Analysis of Three-Dimensional Fractional-Order Chaotic System Based on Entropy Theory

Zhang

Yang

2021

IEEE Access

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“…Consider the definition of Grünwald-Letnikov fractional derivative as defined above, some important helpful properties must be referred. 26 Property 1. For β ¼ 0, it has…”

Section: Preliminarymentioning

confidence: 99%

“…In the subject of fractional calculus, the Grünwald‐Letnikov definition is considered one of the most frequently used definitions. Considering its good usability for discrete control algorithms and its wide application in engineering, 25 the Grünwald‐Letnikov fractional derivative are employed as our main tools in this study.Definition The Grünwald‐Letnikov fractional derivative with

αth - order

on the half axis

ℝ^{+}

of the function

normalφ ((), t)

is elaborated as follows 26

{D_{t_{0}}^{italicGL}}_{t}^{β} φ ((), t) goodbreak= \lim_{h \to 0} h^{- β} \sum_{j = 0}^{k} {(- 1)}^{j} ((), \binom{β}{j}) φ ((), italickh goodbreak- italicjh), for ((), β \in R),

where

Γ (() .)

represents Gamma function and

{D_{t_{0}}^{italicGL}}_{t}^{β}

is the Grünwald‐Letnikov derivative operator, which is abbreviated as

D^{β}

when

t_{0} = 0

.…”

Section: Preliminarymentioning

confidence: 99%

Design of neural network fractional‐order backstepping controller for MPPT of PV systems using fractional‐order boost converter

Youcef

Khettab

Bensafia

2021

Int Trans Electr Energ Syst

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The main objective of this article is to apply the fractional calculus for establishing a novel design of photovoltaic (PV) system. In order to enhance the efficiency and robustness of the maximum power point tracking (MPPT) approach, a fractional-order (FO) DC-DC boost converter is proposed for a PV system. Due to the nonlinearity of the PV module, an artificial neural network (ANN) loop has been used to consistently generate an optimal reference voltage. Using FO control, an incommensurate FO backstepping controller (FO-BSC) has been ultimately integrated for tracking the maximum power point in the presence of tremendously atmospheric conditions and load changes. In this

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“…Fractional calculus is the study of noninteger order derivatives and integrals where the order can be rational, real, or even complex [1]. Over the last few decades, researchers showed a huge interest in the study of FOS due to their flexibility and their ability to model systems with memory dependency [3]. Also, another advantage of fractional-order modeling is that it regenerates closer The associate editor coordinating the review of this manuscript and approving it for publication was Jun Shen .…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulations and FPGA Implementations of Fractional-Order Systems Based on Product Integration Rules

et al. 2020

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Product integration (PI) rules are well known numerical techniques that are used to solve differential equations of integer and, recently, fractional orders. Due to the high memory dependency of the PI rules used in solving fractional-order systems (FOS), their hardware implementation is very difficult and resources-demanding. In this paper, modified versions of the PI rules are introduced to facilitate their digital implementations. The studied rules are PI rectangular, PI trapezoidal, and predict-evaluate-correctevaluate (PECE) rules. The three modified versions of the PI rules are validated using a benchmark system of differential equations for different sizes of the memory window to show the effect of the window size on the solution accuracy. Additionally, the three modified versions of the PI rules are used to simulate a novel fractional-order chaotic system (FOCS) where its bifurcation diagrams are discussed with the window length parameter. The chaotic system is then implemented on a Field Programmable Gate Array (FPGA). There are only a few trials in literature to implement the fractional PECE algorithm on FPGA, nevertheless, the proposed FPGA realization is compared with the most recent of these trials. The FPGA implementations of the three PI rules are made using Xilinx ISE 14.7 on Artix 7 kit. The achieved throughput are 1280 Mbits/sec for PI rectangular, 128.8 Mbits/sec for PI trapezoidal, and 129.12 Mbits/sec for fractional-order PECE.

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Fractional Calculus as a Simple Tool for Modeling and Analysis of Long Memory Process in Industry

Cited by 32 publications

References 13 publications

Complexity Analysis of Three-Dimensional Fractional-Order Chaotic System Based on Entropy Theory

Complexity Analysis of Three-Dimensional Fractional-Order Chaotic System Based on Entropy Theory

Design of neural network fractional‐order backstepping controller for MPPT of PV systems using fractional‐order boost converter

Numerical Simulations and FPGA Implementations of Fractional-Order Systems Based on Product Integration Rules

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