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Stability analysis of positive linear systems by decomposition of the state matrices into symmetrical and antisymmetrical parts

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The stability of positive linear continuous-time and discrete-time systems is analyzed by the use of the decomposition of the state matrices into symmetrical and antisymmetrical parts. It is shown that: 1) The state Metzler matrix of positive continuous-time linear system is Hurwitz if and only if its symmetrical part is Hurwitz; 2) The state matrix of positive linear discrete-time system is Schur if and only if its symmetrical part is Hurwitz. These results are extended to inverse matrices of the state matrices of the positive linear systems.
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  • Bialystok University of Technology, Faculty of Electrical Engineering, Wiejska 45D, 15-351 Bialystok
Bibliografia
  • [1] A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, 1994.
  • [2] M. Busłowicz and T. Kaczorek, “Simple conditions for practical stability of positive fractional discrete-time linear systems”, Int. J. Appl. Math. Comput. Sci., 19(2), 263‒169 (2009).
  • [3] L. Farina and S. Rinaldi, Positive Linear Systems; Theory and Applications, J. Wiley, New York 2000.
  • [4] T. Kaczorek, “Analysis and comparison of the stability of discrete-time and continuous-time linear systems”, Archives of Control Sciences, 26(4), 441‒432 (2016).
  • [5] T. Kaczorek, “Analysis of positivity and stability of discrete-time and continuous-time nonlinear systems”, Computational Prob-lems of Electrical Engineering, 5(1), 11‒16 (2015).
  • [6] T. Kaczorek, “Fractional positive continuous-time linear systems and their reachability”, Int. J. Appl. Math. Comput. Sci., 18(2), 223‒228 (2008).
  • [7] T. Kaczorek, Positive 1D and 2D Systems, Springer-Verlag, London 2002.
  • [8] T. Kaczorek, “Positive fractional continuous-time linear systems with singular pencils”, Bull. Pol. Ac.: Tech., 60(1), 9‒12 (2012).
  • [9] T. Kaczorek, “Positive singular discrete-time linear systems”, Bull. Pol. Ac.: Tech., 45(4), 619‒631 (1997).
  • [10] T. Kaczorek, “Positivity and stability of discrete-time nonlinear systems”, IEEE 2nd International Conference on Cybernetics, 156‒159 (2015).
  • [11] T. Kaczorek, “Stability of fractional positive nonlinear systems”, Archives of Control Sciences, 25(4), 491‒496 (2015).
  • [12] T. Kaczorek, Selected Problems of Fractional Systems Theory, Springer, Berlin 2012.
  • [13] T. Kaczorek, “Stability of fractional positive continuous-time linear systems with state matrices in integer and rational powers”, Bull. Pol. Ac.: Tech., 65(3), 305‒311 (2017).
  • [14] T. Kaczorek, “Stability of interval positive fractional discrete-time linear systems:, Int.J.Math.Comput.Sci., 28(3), 451‒456 (2018).
  • [15] T. Kaczorek, and K. Rogowski, Fractional Linear Systems and Electrical Circuits. Studies in Systems, Decision and Control, vol. 13, Springer 2015.
  • [16] W. Mitkowski, “Remarks on stability of positive linear systems”, Control and Cybernetics, 29(1), 295‒304 (2000).
  • [17] W. Mitkowski, “Dynamical properties of Metzler systems”, Bull. Pol. Ac.: Tech. 56(1), 309‒314 (2008).
  • [18] M.D. Ortigueira, Fractional Calculus for Scientists and Engineers, Springer 2011.
  • [19] P. Ostalczyk, Discrete fractional calculus, World Science Publ. Co., New Jersey 2016.
  • [20] I. Podlubny, Fractional Differential Equations. Academic Press: San Diego 1999.
  • [21] Ł. Sajewski, “Descriptor fractional discrete-time linear system with two different fractional orders and its solution”, Bull. Pol. Ac.: Tech. 64(1), 15‒20 (2016).
  • [22] M.A. Rami and D. Napp, “Characterization and Stability of Autonomous Positive Descriptor Systems”, IEEE Transactions on Automatic Control, 57(10), 2668‒2673 (2012).
  • [23] E. Virnik, “Stability analysis of positive descriptor systems”, Linear Algebra Appl. 429, 2640‒2659 (2008).
Uwagi
PL
Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).
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Bibliografia
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