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Abstract

In the paper positive fractional continuous-time linear systems are considered. Positive fractional systems without delays and positive fractional systems with a single delay in control are studied. New criteria for approximate and exact controllability of systems without delays as well as a relative controllability criterion of systems with delay are established and proved. Numerical examples are presented for different controllability criteria. A practical application is proposed.
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Bibliography

[1] A. Abdelhakim and J. Tenreiro Machado: A critical analysis of the conformable derivative, Nonlinear Dynamics, 95 (2019), 3063–3073, DOI: 10.1007/s11071-018-04741-5.
[2] K. Balachandran, Y. Zhou and J. Kokila: Relative controllability of fractional dynamical systems with delays in control, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 3508–3520, DOI: 10.1016/j.cnsns.2011.12.018.
[3] K. Balachandran, J. Kokila, and J.J. Trujillo: Relative controllability of fractional dynamical systems with multiple delays in control, Computers and Mathematics with Apllications, 64 (2012), 3037–3045, DOI: 10.1016/j.camwa.2012.01.071.
[4] P. Duch: Optimization of numerical algorithms using differential equations of integer and incomplete orders, Doctoral dissertation, Lodz University of Technology, 2014 (in Polish).
[5] C. Guiver, D. Hodgson and S. Townley: Positive state controllability of positive linear systems. Systems and Control Letters, 65 (2014), 23–29, DOI: 10.1016/j.sysconle.2013.12.002.
[6] R.E. Gutierrez, J.M. Rosario and J.T. Machado: Fractional order calculus: Basic concepts and engineering applications, Mathematical Problems in Engineering, 2010 Article ID 375858, DOI: 10.1155/2010/375858.
[7] T. Kaczorek: Positive 1D and 2D Systems, Communications and Control Engineering, Springer, London 2002.
[8] T. Kaczorek: Fractional positive continuous-time linear systems and their reachability, International Journal of Applied Mathematics and Computer Science, 18 (2008), 223–228, DOI: 10.2478/v10006-008-0020-0.
[9] T. Kaczorek: Positive linear systems with different fractional orders, Bulletin of the Polish Academy of Sciences: Technical Sciences, 58 (2010), 453–458, DOI: 10.2478/v10175-010-0043-1.
[10] T. Kaczorek: Selected Problems of Fractional Systems Theory, Lecture Notes in Control and Information Science, 411, 2011.
[11] T. Kaczorek: Constructability and observability of standard and positive electrical circuits, Electrical Review, 89 (2013), 132–136.
[12] T. Kaczorek: An extension of Klamka’s method of minimum energy control to fractional positive discrete-time linear systems with bounded inputs, Bulletin of the Polish Academy of Sciences: Technical Sciences, 62 (2014), 227–231, DOI: 10.2478/bpasts-2014-0022.
[13] T. Kaczorek: Minimum energy control of fractional positive continuoustime linear systems with bounded inputs, International Journal of Applied Mathematics and Computer Science, 24 (2014), 335–340, DOI: 10.2478/amcs-2014-0025.
[14] T. Kaczorek and K. Rogowski: Fractional Linear Systems and Electrical Circuits, Springer, Studies in Systems, Decision and Control, 13 2015.
[15] T. Kaczorek: A class of positive and stable time-varying electrical circuits, Electrical Review, 91 (2015), 121–124. DOI: 10.15199/48.2015.05.29.
[16] T. Kaczorek: Computation of transition matrices of positive linear electrical circuits, BUSES – Technology, Operation, Transport Systems, 24 (2019), 179–184, DOI: 10.24136/atest.2019.147.
[17] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, 2006.
[18] J. Klamka: Controllability of Dynamical Systems, Kluwer Academic Publishers, 1991.
[19] T.J.Machado,V. Kiryakova and F. Mainardi: Recent history of fractional calculus, Communications in Nonlinear Science and Numerical Simulation, 6 (2011), 1140–1153, DOI: 10.1016/j.cnsns.2010.05.027.
[20] K.S. Miller and B. Ross: An Introduction to the Fractional Calculus and Fractional Differential Calculus, Villey, 1993.
[21] A. Monje, Y. Chen, B.M. Viagre, D. Xue and V. Feliu: Fractional-order Systems and Controls. Fundamentals and Applications, Springer-Verlag, 2010.
[22] K. Nishimoto: Fractional Calculus: Integrations and Differentiations of Arbitrary Order, University of New Haven Press, 1989.
[23] K.B. Oldham and J. Spanier: The Fractional Calculus, Academic Press, 1974.
[24] I. Podlubny: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, In: Mathematics in Science and Engineering, Academic Press, 1999.
[25] S.G. Samko, A.A. Kilbas and O.I. Marichev: Fractional Integrals and Derivatives: Theory and Applications, Gordan and Breach Science Publishers, 1993.
[26] J. Sabatier, O.P. Agrawal and J.A. Tenreiro Machado: Advances in Fractional Calculus, In: Theoretical Developments and Applications in Physics and Engineering, Springer-Verlag, 2007.
[27] B. Sikora: Controllability of time-delay fractional systems with and without constraints, IET Control Theory & Applications, 10 (2016), 1–8, DOI: 10.1049/iet-cta.2015.0935.
[28] B. Sikora: Controllability criteria for time-delay fractional systems with a retarded state, International Journal of Applied Mathematics and Computer Science, 26 (2016), 521–531, DOI: 10.1515/amcs-2016-0036.
[29] B. Sikora and J. Klamka: Constrained controllability of fractional linear systems with delays in control, Systems and Control Letters, 106 (2017), 9–15, DOI: 10.1016/j.sysconle.2017.04.013.
[30] B. Sikora and J. Klamka: Cone-type constrained relative controllability of semilinear fractional systems with delays, Kybernetika, 53 (2017), 370–381, DOI: 10.14736/kyb-2017-2-0370.
[31] B. Sikora: On application of Rothe’s fixed point theorem to study the controllability of fractional semilinear systems with delays, Kybernetika, 55 (2019), 675–689, DOI: 10.14736/kyb-2019-4-0675.
[32] T. Schanbacher: Aspects of positivity in control theory, SIAM J. Control and Optimization, 27 (1989), 457–475.
[33] B. Trzasko: Reachability and controllability of positive fractional discretetime systems with delay, Journal of Automation Mobile Robotics and Intelligent Systems, 2 (2008), 43–47.
[34] J. Wei: The controllability of fractional control systems with control delay, Computers and Mathematics with Applications, 64 (2012), 3153–3159, DOI: 10.1016/j.camwa.2012.02.065.
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Authors and Affiliations

Beata Sikora
1
ORCID: ORCID
Nikola Matlok
1

  1. Department of Applied Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland
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Abstract

In this paper,we start by the research of the existence of Lyapunov homogeneous function for a class of homogeneous fractional Systems, then we shall prove that local and global behaviors are the same. The uniform Mittag-Leffler stability of homogeneous fractional time-varying systems is studied. A numerical example is given to illustrate the efficiency of the obtained results.
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Bibliography

[1] V. Andrieu, L. Praly, and A. Astolfi: Homogeneous approximation, recursive observer design, and output feedback. SIAM Journal on Control and Optimization, 47(4), (2008), 1814–1850, DOI: 10.1137/060675861.
[2] A. Bacciotti and L. Rosier: Liapunov Functions and Stability in Control Theory. Lecture Notes in Control and Inform. Sci, 267 (2001), DOI: 10.1007/b139028.
[3] K. Diethelm: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Series on Complexity, Nonlinearity and Chaos, Springer, Heidelberg, 2010.
[4] M.A. Duarte-Mermoud, N. Aguila-Camacho, J.A. Gallegos, and R. Castro-Linares: Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun. Nonlinear Sci. Numer. Simul., 22(1-3) (2015), 650–659, DOI: 10.1016/j.cnsns. 2014.10.008.
[5] H. Hermes: Homogeneous coordinates and continuous asymptotically stabilizing feedback controls. In: Differential equations: stability and control, Proc. Int. Conf., Colorado Springs/CO (USA) 1989, Lect. Notes Pure Appl. Math. 127, 249-260 (1990).
[6] H. Hermes: Nilpotent and high-order approximations of vector field systems. SIAM Rev, 33, (1991), 238–264, DOI: 10.1137/1033050.
[7] Y. Li, Y. Chen, and I. Podlubny: Stability of fractional-order nonlinear dynamic system: Lyapunov direct method and generalized Mittag- Leffler stability. Comput. Math. Appl, 59(5) (2010), 1810–1821, DOI: 10.1016/j.camwa.2009.08.019.
[8] Y. Li, Y. Chen, and I. Podlubny: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica, 45 (2009), 1965–1969, DOI: 10.1140/epjst/e2011-01379-1.
[9] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo: Theory and applications of fractional differential equations. North-Holland Mathematics Studies, 204 , Elsevier Science B.V., Amsterdam 2006, DOI: 10.1016/s0304- 0208(06)80001-0.
[10] T. Menard, E. Moulay, and W. Perruquetti: Homogeneous approximations and local observer design. ESAIM: Control, Optimization and Calculus of Variations, 19 (2013), 906–929, DOI: 10.1051/cocv/2012038.
[11] K.B. Oldham and J. Spanier: The Fractional Calculus. Academic Press, New-York, 1974.
[12] I. Podlubny: Fractional Differential Equations. Mathematics in Sciences and Engineering. Academic Press, San Diego, 1999.
[13] H. Rios, D. Efmov, L. Fridman, J. Moreno, and W. Perruquetti: Homogeneity based uniform stability analysis for time-varying systems. IEEE Transactions on automatic control, 61(3), (2016), 725–734, DOI: 10.1109/TAC.2015.2446371.
[14] R. Rosier: Homogeneous Lyapunov function for homogeneous continuous vector field. System & Control Letters, 19 (1992), 467–473, DOI: 10.1016/0167-6911(92)90078-7.
[15] H.T. Tuan and H. Trinh: Stability of fractional-order nonlinear systems by Lyapunov direct method. IET Control Theory Appl, 12 (2018), DOI: 10.1049/ict-cta.2018.5233.
[16] F. Zhang, C. Li, and Y.Q. Chen: Asymptotical stability of nonlinear fractional differential system with Caputo derivative. Int. J. Differ. Equ., (2011), 1–12, DOI: 10.1155/2011/635165.
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Authors and Affiliations

Tarek Fajraoui
1
Boulbaba Ghanmi
1
ORCID: ORCID
Fehmi Mabrouk
1
Faouzi Omri
1

  1. University of Gafsa, Tunisia, Faculty of Sciences of Gafsa, Department of Mathematics, University campus Sidi Ahmed Zarroug 2112 Gafsa, Tunisia
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Abstract

In this paper we present and discuss a new class of singular fractional systems in a multidimensional state space described by the Roesser continuous-time models. The necessary and sufficient conditions for the asymptotic stability and admissibility by the use of linear matrix inequalities are established. All the obtained results are simulated by some numerical examples to show the applicability and accuracy of our approach.
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Authors and Affiliations

Kamel Benyettou
1
Djillali Bouagada
1
ORCID: ORCID

  1. Department of Mathematics and Computer Science, ACSY Team-Laboratory of Pure and Applied Mathematics, Abdelhamid Ibn Badis University Mostaganem, P.O.Box 227/118 University of Mostaganem, 27000 Mostaganem, Algeria
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Abstract

In this paper we study the dynamical behavior of linear discrete-time fractional systems. The first main result is that the norm of the difference of two different solutions of a time-varying discrete-time Caputo equation tends to zero not faster than polynomially. The second main result is a complete description of the decay to zero of the trajectories of one-dimensional time-invariant stable Caputo and Riemann-Liouville equations. Moreover, we present Volterra convolution equations, that are equivalent to Caputo and Riemann-Liouvile equations and we also show an explicit formula for the solution of systems of time-invariant Caputo equations.

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Authors and Affiliations

P.T. Anh
A. Babiarz
A. Czornik
M. Niezabitowski
S. Siegmund
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Abstract

In the paper finite-dimensional semilinear dynamical control systems described by fractional-order state equations with the Hilfer fractional derivative are discussed. The formula for a solution of the considered systems is presented and derived using the Laplace transform. Bounded nonlinear function �� depending on a state and controls is used. New sufficient conditions for controllability without constraints are formulated and proved using Rothe’s fixed point theorem and the generalized Darbo fixed point theorem. Moreover, the stability property is used to formulate constrained controllability criteria. An illustrative example is presented to give the reader an idea of the theoretical results obtained. A transient process in an electrical circuit described by a system of Hilfer type fractional differential equations is proposed as a possible application of the study.
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Authors and Affiliations

Beata Sikora
1
ORCID: ORCID

  1. Department of Applied Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland

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