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Minimum energy control of fractional positive continuous-time linear systems using Caputo-Fabrizio definition

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Języki publikacji
EN
Abstrakty
EN
The Caputo-Fabrizio definition of the fractional derivative is applied to minimum energy control of fractional positive continuous- time linear systems with bounded inputs. Conditions for the reachability of standard and positive fractional linear continuous-time systems are established. The minimum energy control problem for the fractional positive linear systems with bounded inputs is formulated and solved.
Rocznik
Strony
45--51
Opis fizyczny
Bibliogr. 33 poz., rys., wykr.
Twórcy
autor
  • Faculty of Electrical Engineering, Bialystok University of Technology, 45D Wiejska St., 15-351 Białystok, Poland
Bibliografia
  • [1] L. Farina and S. Rinaldi, Positive Linear Systems; Theory and Applications, J. Wiley, New York, 2000.
  • [2] T. Kaczorek, Positive 1D and 2D systems, Springer Verlag, London, 2001.
  • [3] K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
  • [4] P. Ostalczyk, Epitome of the Fractional Calculus: Theory and its Applications in Automatics, Wydawnictwo Politechniki Łódzkiej, Łódź, 2008 [in Polish].
  • [5] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • [6] T. Kaczorek, “Fractional positive continuous-time systems and their reachability”, Int. J. Appl. Math. Comput. Sci. 18 (2), 223-228 (2008).
  • [7] T. Kaczorek, “Positive linear systems consisting of n subsystems with different fractional orders”, IEEE Trans. Circuits and Systems 58 (6), 1203-1210 (2011).
  • [8] T. Kaczorek, “Positivity and reachability of fractional electrical circuits”, Acta Mechanica et Automatica 5 (2), 42-51 (2011).
  • [9] T. Kaczorek, Selected Problems of Fractional Systems Theory, Springer-Verlag, Berlin, 2012.
  • [10] M. Busłowicz, “Stability of linear continuous time fractional order systems with delays of the retarded type”, Bull. Pol. Ac.: Tech. 56 (4), 319-324 (2008).
  • [11] T. Kaczorek, “Practical stability of positive fractional discrete- time linear systems”, Bull. Pol. Ac.: Tech. 56 (4), 313-317 (2008).
  • [12] Ł. Sajewski, “Descriptor fractional discrete-time linear systems with two different fractional orders and its solution”, Bull. Pol. Ac.: Tech. 64 (1), 15-20 (2016).
  • [13] T. Kaczorek, “Analysis of positivity and stability of fractional discrete-time nonlinear systems”, Bull. Pol. Ac.: Tech. 64 (3), 491-494 (2016).
  • [14] T. Kaczorek, “Drazin inverse matrix method for fractional descriptor discrete-time linear systems”, Bull. Pol. Ac.: Tech. 64 (2), 395-399 (2016).
  • [15] A. Dzieliński, D. Sierociuk, and G. Sarwas, “Ultracapacitor parameters identification based on fractional order model”, Proc. ECC’09, Budapest, (2009).
  • [16] A.G. Radwan, A.M. Soliman, A.S. Elwakil, and A. Sedeek, “On the stability of linear systems with fractional-order elements”, Chaos, Solitons and Fractals 40 (5), 2317-2328 (2009).
  • [17] E.J. Solteiro Pires, J.A. Tenreiro Machado, and P.B. Moura Oliveira, “Fractional dynamics in genetic algorithms”, Workshop on Fractional Differentiation and its Application 2, 414-419 (2006).
  • [18] J. Klamka, “Relative controllability and minimum energy control of linear systems with distributed delays in control”, IEEE Trans. Autom. Contr. 21 (4), 594-595 (1976).
  • [19] J. Klamka, “Minimum energy control of 2D systems in Hilbert spaces”, System Sciences 9 (1-2), 33-42 (1983).
  • [20] J. Klamka, Controllability of Dynamical Systems, Kluwer Academic Press, Dordrecht, 1991.
  • [21] T. Kaczorek and J. Klamka, “Minimum energy control of 2D linear systems with variable coefficients”, Int. J. of Control 44 (3), 645-650 (1986).
  • [22] J. Klamka, “Controllability and minimum energy control problem of fractional discrete-time systems”, in: New Trends in Nanotechnology and Fractional Calculus, pp. 503-509, eds. D. Baleanu, Z.B. Guvenc, and J.A. Tenreiro Machado, Springer- Verlag, New York, (2010).
  • [23] T. Kaczorek, “Minimum energy control of fractional positive continuous-time linear systems”, Proc. of 18th International Conference of Methods and Models in Automation and Robotics, Międzyzdroje (2013).
  • [24] T. Kaczorek, “Minimum energy control of fractional positive electrical circuits”, Proc. of 22th European Signal Processing Conference, (2014).
  • [25] T. Kaczorek, “Minimum energy control of positive continuous- time linear systems with bounded inputs”, Archives of Electrical Engineering 63 (1), 19-27 (2014).
  • [26] T. Kaczorek, “An extension of Klamka’s method of minimum energy control to fractional positive discrete-time linear systems with bounded inputs”, Bull. Pol. Ac.: Tech. 62 (2), 227-231 (2014).
  • [27] T. Kaczorek, “Minimum energy control of descriptor positive discrete-time linear systems”, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering 33 (3), 976-988 (2014).
  • [28] Ł. Sajewski, “Reachability, observability and minimum energy control of fractional positive continuous-time linear systems with two different fractional orders”, Multidim. Syst. Sign. Process 27 (1), 27-41 (2016).
  • [29] A. Benzaouia, A. Hmamed, F. Mesquine, M. Benhayoun, and F. Tadeo, “Stabilization of continuous-time fractional positive systems by using Lyapunov function”, IEEE Trans. Autom. Control 59 (8), 2203-2208 (2014).
  • [30] F. Mesquine, A. Hmamed, M. Benhayoun, A. Benzaouia, and F. Tadeo, “Robust stabilization of constrained uncertain continuous- time fractional positive systems”, Journal of the Franklin Institute 352 (1), 259-270 (2015).
  • [31] A. Hmamed, F. Mesquine, M. Benhayoun, A., Benzaouia, and F. Tadeo, “Continuous-time fractional positive systems with bounded states”, IEEE 52nd Annual Conference on Decision and Control (CDC), Florence, 2127-2132 (2013).
  • [32] M. Caputo and M. Fabrizio, “A new definition of fractional derivative without singular kernel”, Progr. Fract. Differ. Appl. 1 (2), 1-13 (2015).
  • [33] J. Losada and J. Nieto, “Properties of a new fractional derivative without singular kernel”, Progr. Fract. Differ. Appl. 1 (2), 87-92 (2015).
Uwagi
PL
Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).
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Bibliografia
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