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1

A New Class of Fractional Cumulative Residual Entropy - Some Theoretical Results

Benmahmoud Slimane

Journal of Telecommunications and Information Technology

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2023

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nr 1

25--29

EN

In this paper, by differentiating the entropy’s generating function (i.e., h(t) = R SX̄F tX (x)dx) using a Caputo fractional-order derivative, we derive a generalized non-logarithmic fractional cumulative residual entropy (FCRE). When the order of differentiation α → 1, the ordinary Rao CRE is recovered, which corresponds to the results from first-order ordinary differentiation. Some properties and examples of the proposed FCRE are also presented.

2

Finite length triple estimation algorithm and its application to gyroscope mems noise identification

Macias Michał, Sierociuk Dominik

Acta Mechanica et Automatica

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2023

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Vol. 17, no. 2

219--229

EN

The noises associated with MEMS measurements can significantly impact their accuracy. The noises characterised by random walk and bias instability errors strictly depend on temperature effects that are difficult to specify during direct measurements. Therefore, the paper aims to estimate the fractional noise dynamics of the stationary MEMS gyroscope based on finite length triple estimation algorithm (FLTEA). The paper deals with the state, order and parameter estimation of fractional order noises originating from the MEMS gyroscope, being part of the popular Inertial Measurement Unit denoted as SparkFun MPU9250. The noise measurements from 𝑥,𝑦 and 𝑧 gyroscope axes are identified using a modified triple estimation algorithm (TEA) with finite approximation length. The TEA allows a simultaneous estimation of the state, order and parameter of fractional order systems. Moreover, as it is well-known that the number of samples in fractional difference approximations plays a key role, we try to show the influence of applying the TEA with various approximation length constraints on final estimation results. The validation of finite length TEA in the noise estimation process coming from MEMS gyroscope has been conducted for implementation length reduction achieving 50% of samples needed to estimate the noise with no implementation losses. Additionally, the capabilities of modified TEA in the analysis of fractional constant and variable order systems are confirmed in several numerical examples.

3

Analysis of dynamic characteristics of selected pneumatic systems with rractional calculus. simulation and laboratory research

Luft M., Krzysztoszek K., Pietruszczak D., Nowocień A.

TransNav : International Journal on Marine Navigation and Safety of Sea Transportation

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2021

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Vol. 15 No. 4

835--843

EN

The section of the paper on simulation studies presents the application of fractional calculus to describe the dynamics of pneumatic systems. In the construction of mathematical models of the analysed dynamic systems, the Riemann-Liouville definition of differ-integral of non-integer order was used. For the analysed model, transfer function of integer and non-integer order was determined. Functions describing characteristics in time and frequency domains were determined, whereas the characteristics of the analysed systems were obtained by means of computer simulation. MATLAB were used for the simulation research. The section of the paper on laboratory research presents the results of the laboratory tests of the injection system of the internal combustion engine with special attention to the verification of simulated tests of selected pneumatic systems described with the use of fractional calculus.

4

A new numerical scheme for solving the two dimensional fractional diffusion equation

Altan Koç Dilara, Gülsu Mustafa

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 2

5--16

EN

In this study, the locally one dimensional (LOD) method is used to solve the two dimensional time fractional diffusion equation. The fractional derivative is the Caputo fractional derivative of order α. The rate of convergence of the finite difference method is presented. It is seen that this method is in agreement with the obtained numerical solutions with acceptable central processing unit time (CPU time). Error estimates, numerical and exact results are tabulated. The graphics of errors are given.

5

Time-optimal control of linear fractional systems with variable coefficients

Matychyn Ivan, Onyshchenko Viktoriia

International Journal of Applied Mathematics and Computer Science

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2021

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Vol. 31, no. 3

375--386

EN

Linear systems described by fractional differential equations (FDEs) with variable coefficients involving Riemann–Liouville and Caputo derivatives are examined in the paper. For these systems, a solution of the initial-value problem is derived in terms of the generalized Peano–Baker series and a time-optimal control problem is formulated. The optimal control problem is treated from the convex-analytical viewpoint. Necessary and sufficient conditions for time-optimal control similar to that of Pontryagin’s maximum principle are obtained. Theoretical results are supported by examples.

6

The IoT gateway with active queue management

Domański Adam, Domańska Joanna, Czachórski Tadeusz, Klamka Jerzy, Szyguła Jakub, Marek Dariusz

International Journal of Applied Mathematics and Computer Science

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2021

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Vol. 31, no. 1

165--178

EN

As the traffic volume from various Internet of things (IoT) networks increases significantly, the need for adapting the quality of service (QoS) mechanisms to the new Internet conditions becomes essential. We propose a QoS mechanism for the IoT gateway based on packet classification and active queue management (AQM). End devices label packets with a special packet field (type of service (ToS) for IPv4 or traffic class (TC) for IPv6) and thus classify them as priority for real-time IoT traffic and non-priority for standard IP traffic. Our AQM mechanism drops only non-priority packets and thus ensures that real-time traffic packets for critical IoT systems are not removed if the priority traffic does not exceed the maximum queue capacity. This AQM mechanism is based on the PIα controller with non-integer integration order. We use fluid flow approximation and discrete event simulation to determine the influence of the AQM policy on the packet loss probability, queue length and its variability. The impact of the long-range dependent (LRD) traffic is also considered. The obtained results show the properties of the proposed mechanism and the merits of the PIα controller.

7

Stability and robustness analysis of discrete-time fractional-piecewise-constant-order PID controller

Oziablo Piotr, Mozyrska Dorota, Wyrwas Malgorzata

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2021

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Vol. 69, nr 5

art. no. e137937

EN

In the paper we propose a fractional-piecewise-constant-order PID controller and discuss the stability and robustness of a closed loop system. In stability analysis we use the transform method and include the Nyquist-like criteria. Simulations for designed controllers are performed for the second-order plant with a delay.

8

Mathematical models of a pneumatic cascade and pneumatic membrane actuator described with a fractional calculus

Luft M., Łukasik Z., Pietruszczak D.

TransNav : International Journal on Marine Navigation and Safety of Sea Transportation

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2020

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Vol. 14. no. 3

643--648

EN

The paper presents the analysis of dynamic properties of pneumatic systems such as: cascade connection of membrane pressure transmitters and a pneumatic membrane actuator by means of differential equations of integer and non-integer order. The analyzed systems were described from the time perspective by means of step response, and in terms of frequency with the help of the Bode plot, i.e. logarithmic magnitude and phase responses. Each response was determined using differential equations of non-integer order. To determine the responses, the interactive Simulink package was an irreplaceable programming tool built on the basis of the MATLAB program, which enables the analysis and synthesis of continuous dynamic systems.

9

The fourth-order ordinary differential equation with the fractional initial/boundary conditions

Siedlecki Jarosław

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 1

79--87

EN

The initial/boundary value problem for the fourth-order homogeneous ordinary differential equation with constant coefficients is considered. In this paper, the particular solutions an ordinary differential equation with respect to the set of boundary conditions are studied. At least one of the boundary conditions is described by a fractional derivative. Finally, a few illustrative examples of particular solutions to the considered problem are shown.

10

Application of fractional calculus in iterative sliding mode synchronization control

Zhang Xin, Wen-Ru Lu, Zhang Liang, Xu Wen-Bo

Archives of Electrical Engineering

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2020

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Vol. 69, nr 3

499--519

EN

In order to control joints of manipulators with high precision, a position tracking control strategy combining fractional calculus with iterative learning control and sliding mode control is proposed for the control of a single joint of manipulators. Considering the coupling between joints of manipulators, a fractional-order iterative sliding mode crosscoupling control strategy is proposed and the theoretical proof of its progressive stability is given. The paper takes a two-joint manipulator as the research object to verify the control strategy of a single-joint manipulator. The results show that the control strategy proposed in this paper makes the two-joint mechanical arm chatter less and the tracking more accurate. The synchronous control of the manipulator is verified by a three-joint manipulator. The results show that the angular displacement adjustment times of the threejoint manipulator are 0.11 s, 0.31 s and 0.24 s, respectively. 3.25 s > 5 s, 3.15 s of a PD cross-coupling control strategy; 2.85 s, 2.32 s, 4.22 s of a PD iterative cross-coupling control strategy; 0.14 s, 0.33 s, 0.28 s of a fractional-order sliding mode cross-coupling control strategy. The root mean square error of the position error of the designed control strategy is 6.47 × 10−6 rad, 3.69 × 10−4 rad, 6.91 × 10−3 rad, respectively. The root mean square error of the synchronization error is 3.96×10−4 rad, 1.36×10−3 rad, 7.81×10−3 rad, superior to the other three control strategies. The results illustrate the effectiveness of the proposed control method.

11

On the existence of optimal consensus control for the fractional Cucker-Smale model

Almeida R., Kamocki R., Malinowska A. B., Odzijewicz T.

Archives of Control Sciences

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2020

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Vol. 30, No. 4

625--651

EN

This paper addresses the nonlinear Cucker-Smale optimal control problem under the interplay of memory effect. The aforementioned effect is included by employing the Caputo fractional derivative in the equation representing the velocity of agents. Sufficient conditions for the existence of solutions to the considered problem are proved and the analysis of some particular problems is illustrated by two numerical examples.

12

Description of selected pneumatic elements and systems using fractional calculus

Pietruszczak Daniel, Nowocień Artur

Journal of Automation, Electronics and Electrical Engineering

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2019

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Vol. 1, No. 1

43--49

EN

The paper presents the application of fractional calculus to describe the dynamics of selected pneumatic elements and systems. In the construction of mathematical models of the analysed dynamic systems, the Riemann-Liouville definition of differintegral of non- integer order was used. For the analysed model, transfer function of integer and non-integer order was determined. Functions describing characteristics in frequency domains were determined, whereas the characteristics of the elements and systems were obtained by means of computer simulation. MATLAB programme were used for the simulation research.

PL

W artykule przedstawiono zastosowanie rachunku różniczkowego niecałkowitych rzędów (ang. fractional calculus) do opisu dynamiki zjawisk układów pneumatycznych wybranych elementów i układów. W budowie modeli matematycznych, analizowanych układów dynamicznych, wykorzystano definicję Riemanna–Liouville’a pochodno–całki niecałkowitego rzędu. Dla analizowanego modelu, wyznaczono transmitancję operatorową całkowitego i niecałkowitego rzędu. Wyznaczono zależności opisujące charakterystyki częstotliwościowe, na drodze symulacji komputerowej uzyskano charakterystyki analizowanych układów. Do badań symulacyjnych wykorzystano oprogramowanie MATLAB.

13

Investigation of free vibration and buckling of Timoshenko nano-beam based on a general form of eringen theory using conformable fractional derivative and Galerkin method

Mohammadi Farogh Souf, Rahimi Zaher, Sumelka Wojciech, Yang Xiao-Jun

Engineering Transactions

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2019

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Vol. 67, iss. 3

347--367

EN

The purpose of this paper is to study the free vibration and buckling of a Timoshenko nano-beam using the general form of the Eringen theory generalized based on the fractional derivatives. In this paper, using the conformable fractional derivative (CFD) definition the generalized form of the Eringen nonlocal theory (ENT) is used to consider the effects of integer and noninteger stress gradients in the constitutive relation and also to consider small-scale effect in the vibration of a Timoshenko nano-beam. The governing equation is solved by the Galerkin method. Free vibration and buckling of a Timoshenko simply supported (S) nano-beam is investigated, and the influence of the fractional and nonlocal parameters is shown on the frequency ratio and buckling ratio. In this sense, the obtained formulation allows for an easier mapping of experimental results on nano-beams. The new theory (fractional parameter) makes the modeling more flexible. The model can conclude all of the integer and non-integer operators and is not limited to the special operators such as ENT. In other words, it allows to use more sophisticated/flexible mathematics to model physical phenomena.

14

Optimal input signal design for fractional-order system identification

Jakowluk W.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2019

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Vol. 67, nr 1

37--44

EN

The optimal design of excitation signal is a procedure of generating an informative input signal to extract the model parameters with maximum pertinence during the identification process. The fractional calculus provides many new possibilities for system modeling based on the definition of a derivative of noninteger-order. A novel optimal input design methodology for fractional-order systems identification is presented in the paper. The Oustaloup recursive approximation (ORA) method is used to obtain the fractional-order differentiation in an integer order state-space representation. Then, the presented methodology is utilized to solve optimal input design problem for fractional-order system identification. The fundamental objective of this approach is to design an input signal that yields maximum information on the value of the fractional-order model parameters to be estimated. The method described in this paper was verified using a numerical example, and the computational results were discussed.

15

Fractional order model of measured quantity errors

Cioć R., Chrzan M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2019

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Vol. 67, nr 6

1023--1030

EN

The paper presents an interpretation of fractional calculus for positive and negative orders of functions based on sampled measured quantities and their errors connected with digital signal processing. The derivatives as a function limit and the Grunwald-Letnikov differintegral are shown in chapter 1 due to the similarity of the presented definition. Notation of fractional calculus based on the gradient vector of measured quantities and its geometrical and physical interpretation of positive and negative orders are shown in chapter 2 and 3.

16

Problems in solving fractional differential equations in a microcontroller implementation of an FOPID controller

Matusiak Mariusz, Ostalczyk Piotr

Archives of Electrical Engineering

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2019

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Vol. 68, nr 3

565--577

EN

The article focuses on the fractional-order backward difference, sum, linear time-invariant equation analysis, and difficulties of the fractional calculus microcontroller implementation with regard to designing a fractional-order proportional integral derivative (FOPID) controller. In opposite to the classic proportional integral derivative (PID), the FOPID controller is defined by five independent parameters. Hence, it is more customizable and, potentially, more precise on condition that the values of fractional integration and differentiation orders are properly selected. However, a number of operations and the time required to calculate the output signal continuously increase. This can be a significant problem considering the limitations of a microcontroller, including memory size and a constant sampling time of the set-up analog-to-digital (ADC) converters. In the article, three solutions are considered, and results obtained in the experiments are presented.

17

Axial vibration of bars using fractional viscoelastic material models

Łabędzki P., Pawlikowski R., Radowicz A.

Vibrations in Physical Systems

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2018

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Vol. 29

art. no. 2018009

EN

This paper presents dynamic analysis of a bar with one end fixed and other free, loaded with force at its free end. The viscoelastic material of the bar is described by fractional models (Scot-Blair, Voigt, Maxwell and Zener models). Rayleigh-Ritz and Laplace transform methods were applied to obtain closed-form solution of the considered problem.

18

Two-stage scheme for identification of parallel Scott-Blair fractional model of viscoelastic biological materials

Stankiewicz A.

ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

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2018

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Vol. 7, No 1

87--96

EN

Fractional calculus considers derivatives and integrals of an arbitrary order. This article focuses on fractional parallel Scott-Blair model of viscoelastic biological materials, which is a generalization of classic Kelvin-Voight model to non-integer order derivatives suggested in the previous paper. The parallel Scott-Blair model admit the closed form of analytical solution in terms of two power functions multiplied by Debye type weight function. To build a parallel Scott-Blair model when only discrete-time measurements of the relaxation modulus are accessible for identification is a basic concern. Based on asymptotic models a two-stage approach is proposed for fitting the measurement data, which means that in the first stage the data are fitted by solving two dependent, but simple, linear least-squares problems in two separate time intervals. Next, at the second stage of the identification procedure the exact parallel Scott-Blair model optimal in the least-squares sense is computed. The log-transformed relaxation modulus data is used in the first stage of identification scheme, while the original relaxation modulus data is applied for the second stage identification. A complete identification procedure is presented. The usability of the method to find the parallel Scott-Blair fractional model of real biological material is demonstrated. The parameters of the parallel Scott-Blair model of a sample of sugar beet root, which very closely approximate the experimental relaxation modulus data, are given.

19

Parallel Scott-Blair fractional model of viscoelastic biological materials

Stankiewicz A.

ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

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2018

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Vol. 7, No 1

107--115

EN

Fractional calculus is a mathematical approach dealing with derivatives and integrals of arbitrary and also complex orders. Therefore, it adds a new means to understand and describe the nature and behavior of complex dynamical systems. Here we use the fractional calculus for modeling mechanical viscoelastic properties of materials. In the present work, after reviewing some of the main viscoelastic fractional models, a new parallel model is employed, connecting in parallel two Scott-Blair models with additional multiplicative weight functions. The model is presented in terms of two power functions weighted by Debye-type functions extend representation, understanding and description of complex systems viscoelastic properties. Monotonicity of the model relaxation modulus is studied and some upper bounds for the minimal time value, above which the model relaxation modulus is monotonically decreasing are given and compared both analytically and numerically. The comparison with the results of relaxation tests executed on some real phenomena has shown that the parallel Scott-Blair model involving fractional derivatives has been in a good agreement.

20

On the monotonicity of the relaxation spectrum of fractional Maxwell model of viscoelastic materials

Stankiewicz A.

ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

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2018

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Vol. 7, No 1

117--125

EN

This article focuses on the relaxation spectrum of fractional Maxwell model, which is a generalization of classic viscoelastic Maxwell model to non-integer order derivatives. The analytical formula for the spectrum of relaxation frequencies is derived. Theoretical analysis of the relaxation spectrum monotonicity is conducted by using simple analytical methods and illustrated by means of numerical examples. The necessary and sufficient conditions for the existence and uniqueness of the maximum of relaxation spectrum are stated and proved. The analytical formulas for minimum and maximum of the relaxation spectrum are derived. Also, a few useful properties concerning the relaxation spectrum monotonicity and concavity are given in the mathematical form of simple inequalities expressed directly in terms of the fractional Maxwell model parameters, which can be used to simplify the calculations and analysis.