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Number of results: 9
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Abstract

In this paper the controllability properties of the convex linear combination of fractional, linear, discrete-time systems are characterized and investigated. The notions of linear convex combination and controllability in the context of fractional-order systems are recalled. Then, the controllability property of such a linear combination of discrete-time, linear fractional systems is proven. Further, the reduction of an infinite problem of transition matrix derivation is reduced to a finite one, which greatly simplifies the numerical burden of the controllability issue. Examples of controllable and uncontrollable, single-input, linear systems are presented. The possibility of extension of the considerations to multi-input systems is shown.
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Authors and Affiliations

Tadeusz Kaczorek
1
ORCID: ORCID
Jerzy Klamka
2
ORCID: ORCID
Andrzej Dzieliński
3
ORCID: ORCID

  1. Bialystok University of Technology, Faculty of Electrical Engineering, ul. Wiejska 45D, Bialystok, Poland
  2. Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, ul. Bałtycka 5, Gliwice, Poland
  3. Warsaw University of Technology, Faculty of Electrical Engineering, ul. Koszykowa 75, Warsaw, Poland
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Abstract

The paper investigates the controllability of fractional descriptor linear systems with constant delays in control. The Caputo fractional derivative is considered. Using the Drazin inverse and the Laplace transform, a formula for solving of the matrix state equation is obtained. New criteria of relative controllability for Caputo’s fractional descriptor systems are formulated and proved. Both constrained and unconstrained controls are considered. To emphasize the importance of the theoretical studies, an application to electrical circuits is presented as a practical example.
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Authors and Affiliations

Beata Sikora
ORCID: ORCID

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Abstract

In this paper cluster consensus is investigated for general fractional-order multi agent systems with nonlinear dynamics via adaptive sliding mode controller. First, cluster consensus for fractional-order nonlinear multi agent systems with general formis investigated. Then, cluster consensus for the fractional-order nonlinear multi agent systems with first-order and general form dynamics is investigated by using adaptive sliding mode controller. Sufficient conditions for achieving cluster consensus for general fractional-order nonlinear multi agent systems are proved based on algebraic graph theory, Lyapunov stability theorem andMittag-Leffler function. Finally, simulation examples are presented for first-order and general form multi agent systems, i.e. a single-link flexible joint manipulator which demonstrates the efficiency of the proposed adaptive controller.

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Authors and Affiliations

Zahra Yaghoubi
Heidar Ali Talebi
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Abstract

It is shown how a stability test, alternative to the classical Routh test, can profitably be applied to check the presence of polynomial roots inside half-planes or even sectors of the complex plane. This result is obtained by exploiting the peculiar symmetries of the root locus in which the basic recursion of the test can be embedded. As is expected, the suggested approach proves useful for testing the stability of fractional-order systems. A pair of examples show how the method operates. It is believed that the suggested geometric approach can also be of some didactic value in introducing basic control-system tools to engineering students.
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Bibliography

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[2] A.T. Azar, A.G. Radwan, and S.Vaidyanathan, Eds.: Mathematical Techniques of Fractional Order Systems, Elsevier, Amsterdam, The Netherlands, 2018.
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Authors and Affiliations

Daniele Casagrande
1
Wiesław Krajewski
2
Umberto Viaro
1

  1. Polytechnic Department of Engineering and Architecture, University of Udine, via delle Scienze 206, 33100 Udine, Italy
  2. Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01–447 Warsaw, Poland
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Abstract

This paper presents equivalent impedance and operator admittance systems for fractional order elements. Presented models of fractional order elements of the type: s α L α and 1/s α C α, (0 α 1) were obtained using the Laplace transform based on the expansion of the factor sign to an infinite fraction with varying degrees of accuracy – the continued fraction expansion method (CFE). Then circuit synthesis methods were applied. As a result, equivalent circuit diagrams of fractional order elements were obtained. The obtained equivalent schemes consist both of classical RLC elements, as well as active elements built based on operational amplifiers. Numerical experiments were conducted for the constructed models, presenting responses to selected input signals.
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Authors and Affiliations

Sebastian Różowicz
1
ORCID: ORCID
Maciej Włodarczyk
1
ORCID: ORCID
Andrzej Zawadzki
1
ORCID: ORCID

  1. Kielce Universityof Technology, Department of Industrial Electrical Engineering and Automatic Control, TysiacleciaPanstwa Polskiego 7, 25-314 Kielce, Poland
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Abstract

This paper focuses on the global practical Mittag-Leffler feedback stabilization problem for a class of uncertain fractional-order systems. This class of systems is a larger class of nonlinearities than the Lipschitz ones. Based on the quasi-one-sided Lipschitz condition, firstly, we provide sufficient conditions for the practical observer design. Then, we exhibit that practical Mittag-Leffler stability of the closed loop system with a linear, state feedback is attained. Finally, a separation principle is established and we prove that the closed loop system is practical Mittag-Leffler stable.
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Authors and Affiliations

Imed Basdouri
1
ORCID: ORCID
Souad Kasmi
2
Jean Lerbet
3

  1. Gafsa University, Faculty of Sciences of Gafsa, Department of Mathematics, Zarroug Gafsa 2112 Tunisia
  2. Sfax University, Faculty of Sciences of Sfax, Department of Mathematics, BP 1171 Sfax 3000 Tunisia
  3. Laboratoire de Mathématiques et de Modélisation d’Evry, Univ d’Evry, Université Paris Saclay, France
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Abstract

In the paper a new, state space, fully discrete, fractional model of a heat transfer process in one dimensional body is addressed. The proposed model derives directly from fractional heat transfer equation. It employes the discrete Grünwald-Letnikov operator to express the fractional order differences along both coordinates: time and space. The practical stability and numerical complexity of the model are analysed. Theoretical results are verified using experimental data.
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Bibliography

  1.  S. Das, Functional Fractional Calculus for System Identification and Controls, Springer, Berlin, 2010.
  2.  R. Caponetto, G. Dongola, L. Fortuna, and I. Petras, “Fractional order systems: Modeling and Control Applications”, in World Scientific Series on Nonlinear Science, ed. L.O. Chua, pp. 1–178, University of California, Berkeley, 2010.
  3.  A. Dzieliński, D. Sierociuk, and G. Sarwas, “Some applications of fractional order calculus”, Bull. Pol. Ac.: Tech. 58(4), 583– 592 (2010).
  4.  C.G. Gal and M. Warma, “Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions”, Evol. Equ. Control Theory 5(1), 61–103 (2016).
  5.  E. Popescu, “On the fractional Cauchy problem associated with a feller semigroup”, Math. Rep. 12(2), 81–188 (2010).
  6.  D. Sierociuk et al., “Diffusion process modeling by using fractional-order models”, Appl. Math. Comput. 257(1), 2–11 (2015).
  7.  J.F. Gómez, L. Torres, and R.F. Escobar (eds.), “Fractional derivatives with Mittag-Leffler kernel trends and applications in science and engineering”, in Studies in Systems, Decision and Control, vol. 194, ed. J. Kacprzyk, pp. 1–339. Springer, Switzerland, 2019.
  8.  M. Dlugosz and P. Skruch, “The application of fractional-order models for thermal process modelling inside buildings”, J. Build Phys. 1(1), 1–13 (2015).
  9.  A. Obrączka, Control of heat processes with the use of noninteger models. PhD thesis, AGH University, Krakow, Poland, 2014.
  10.  A. Rauh, L. Senkel, H. Aschemann, V.V. Saurin, and G.V. Kostin, “An integrodifferential approach to modeling, control, state estimation and optimization for heat transfer systems”, Int. J. Appl. Math. Comput. Sci. 26(1), 15–30 (2016).
  11.  T. Kaczorek, “Singular fractional linear systems and electri cal circuits”, Int. J. Appl. Math. Comput. Sci. 21(2), 379–384 (2011).
  12.  T. Kaczorek and K. Rogowski, Fractional Linear Systems and Electrical Circuits, Bialystok University of Technology, Bialystok, 2014.
  13.  I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  14.  B. Bandyopadhyay and S. Kamal, “Solution, stability and realization of fractional order differential equation”, in Stabilization and Control of Fractional Order Systems: A Sliding Mode Approach, Lecture Notes in Electrical Engineering 317, pp. 55–90, Springer, Switzerland, 2015.
  15.  D. Mozyrska, E. Girejko, M. Wyrwas, “Comparison of hdifference fractional operators”, in Advances in the Theory and Applications of Non- integer Order Systems, eds. W. Mitkowski et al., pp. 1–178. Springer, Switzerland, 2013.
  16.  P. Ostalczyk, “Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains”, Int. J. Appl. Math. Comput. Sci. 22(3), 533–538 (2012).
  17.  E.F. Anley and Z. Zheng, “Finite difference approximation method for a space fractional convection–diffusion equation with variable coefficients”, Symmetry 12(485), 1–19 (2020).
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Authors and Affiliations

Krzysztof Oprzędkiewicz
1
ORCID: ORCID

  1. AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059 Kraków, Poland
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Abstract

This paper introduces a fractional-order PD approach (F-oPD) designed to control a large class of dynamical systems known as fractional-order chaotic systems (F-oCSs). The design process involves formulating an optimization problem to determine the parameters of the developed controller while satisfying the desired performance criteria. The stability of the control loop is initially assessed using the Lyapunov’s direct method and the latest stability assumptions for fractional-order systems. Additionally, an optimization algorithm inspired by the flight skills and foraging behavior of hummingbirds, known as the Artificial Hummingbird Algorithm (AHA), is employed as a tool for optimization. To evaluate the effectiveness of the proposed design approach, the fractional-order energy resources demand-supply (Fo-ERDS) hyperchaotic system is utilized as an illustrative example.
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Authors and Affiliations

Ammar Soukkou
1
Yassine Soukkou
2
Sofiane Haddad
1
Mohamed Benghanem
3
Abdelhamid Rabhi
4

  1. Renewable Energy Laboratory, Faculty of Science and Technology, Department of Electronics, University of MSBY Jijel, BP. 98, Ouled Aissa, Jijel, Algeria
  2. Research Center in Industrial Technologies CRTI, P. O. Box. 64, Cheraga 16014, Algiers, Algeria
  3. Physics Department, Faculty of Science, Islamic University of Madinah, Madinah, KSA
  4. Modeling, Information and Systems Laboratory, University of Picardie Jules Verne, Amiens, France.

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