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An extension of Klamka’s method of minimum energy control to fractional positive discrete-time linear systems with bounded inputs

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Abstrakty
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The Klamka’s method of minimum energy control problem is extended to fractional positive discrete-time linear systems with bounded inputs. Sufficient conditions for the existence of solution to the problem are established. A procedure for solving of the problem is proposed and illustrated by numerical example.
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autor
  • Faculty of Electrical Engineering, Bialystok University of Technology, 45D Wiejska St., 15-351 Bialystok, 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, WPŁ, 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, “Positivity and reachability of fractional electrical circuits”, Acta Mechanica et Automatica 5 (2), 42–51 (2011).
  • [8] T. Kaczorek, “Positive linear systems consisting of n subsystems with different fractional orders”, IEEE Trans. Circuits and Systems 58 (6), 1203–1210 (2011).
  • [9] T. Kaczorek, Selected Problems of Fractional Systems Theory, Springer-Verlag, Berlin, 2012.
  • [10] A. Dzieliński and D. Sierociuk, “Stability of discrete fractional order state-space systems”, J. Vibrations and Control 14 (9/10), 1543–1556 (2008).
  • [11] 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).
  • [12] T. Kaczorek, “Asymptotic stability of positive fractional 2D linear systems”, Bull. Pol. Ac.: Tech. 57 (3), 289–292 (2009).
  • [13] T. Kaczorek, “Practical stability of positive fractional discretetime linear systems”, Bull. Pol. Ac.: Tech. 56 (4), 313–317 (2008).
  • [14] T. Kaczorek, “Practical stability and asymptotic stability of positive fractional 2D linear system”, Asian J. Control 12 (2), 200–207 (2010).
  • [15] A. Ruszewski, “Stability regions of closed loop system with time delay internal plant of fractional order and fractional order PI controller”, Bull. Pol. Acad. Sci. Tech. 56 (4), 329–332 (2008).
  • [16] M. Busłowicz, “Stability of state-space models of linear continuous-time fractional order systems”, Acta Mechanica et Automatica 5 (2), 15–22 (2011).
  • [17] M. Busłowicz, “Stability analysis of continuous-time linear systems consisting of n subsystem with different fractional orders”, Bull. Pol. Ac.: Tech. 60 (2), 279–284 (2012).
  • [18] M. Busłowicz, “Simple analytic conditions for stability of fractional discrete-time linear system with diagonal state matrix”, Bull. Pol. Ac.: Tech. 60 (4), 809–814 (2012).
  • [19] M. Busłowicz and T. Kaczorek, “Simple conditions for practical stability of linear positive fractional discrete-time linear systems”, Int. J. Appl. Math. Comput. Sci. 19 (2), 263–269 (2009).
  • [20] A. Dzieliński, D. Sierociuk, and G. Sarwas, “Ultracapacitor parameters identification based on fractional order model”, Proc. ECC’09 1, CD-ROM (2009).
  • [21] T. Kaczorek, “Controllability and observability of linear electrical circuits”, Electrical Review 87 (9a), 248–254 (2011).
  • [22] T. Kaczorek, “Reachability and controllability to zero tests for standard and positive fractional discrete-time systems”, Journal Europ´een des Syst`emes Automatis´es, JESA 42 (6–8), 769–787 (2008).
  • [23] A.G. Radwan, A.M. Soliman, A.S. Elwakil, and A. Sedeek, “On the stability of linear systems with fractional-order elements”, Chaos, Solitones and Fractals 40 (5), 2317–2328 (2009).
  • [24] E.J. Solteiro Pires, J.A. Tenreiro Machado, and P.B. Moura Oliveira, “Functional dynamics in genetic algorithms”, Workshop on Fractional Differenation and Its Application 2, 414–419 (2006).
  • [25] A. Dzieliński, D. Sierociuk, and G. Sarwas, “Some applications of fractional order calculus”, Bull. Pol. Ac.: Tech. 58 (4), 583–592 (2010).
  • [26] J. Klamka, Controllability of Dynamical Systems, Kluwer Academic Press, Dordrecht, 1991.
  • [27] J. Klamka, “Minimum energy control of 2D systems in Hilbert spaces”, System Sciences 9 (1–2), 33–42 (1983).
  • [28] J. Klamka, “Relative controllability and minimum energy control of linear systems with distributed delays in control”, IEEE rans. Autom. Contr. 21 (4), 594–595 (1976).
  • [29] T. Kaczorek and J. Klamka, “Minimum energy control of 2D linear systems with variable coefficients”, Int. J. Control 44 (3), 645–650 (1986).
  • [30] J. Klamka, “Controllability and minimum energy control problem of fractional discrete-time systems”, in New Trends in Nanotechology and Fractional Calculus, eds.: D. Baleanu, Z.B. Guvenc, and J.A. Tenreiro Machado, pp. 503–509, Springer-Verlag, New York, 2010.
  • [31] T. Kaczorek, “Minimum energy control of fractional positive continuous-time linear systems”, Proc. Conf. MMAR 1, CDROM (2013).
  • [32] T. Kaczorek, “Minimum energy control of descriptor positive discrete-time linear systems”, Compel 23 (2), 205–211 (2013).
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Bibliografia
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