Abstract
Perfect nonlinear fractional descriptor observers for fractional descriptor continuous-time nonlinear systems are proposed. Necessary and sufficient conditions for the existence of the observers are established. The design procedure of the nonlinear fractional observers is given. It is based on the elementary row (column) operations and reducing the singular matrix of the system to upper (lower) triangular form. The effectiveness of the procedure is demonstrated on a numerical example.
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W. Cuihong, New delay-dependent stability criteria for descriptor systems with interval time delay. Asian Journal of Control. 14, No 1, (2012), 197–206; DOI: 10.1002/asjc.287.
L. Dai, Singular Control Systems, Lecture Notes in Control and Information Sciences. Springer-Verlag, Berlin (1989), DOI: 10.1007/BFb0002475.
M. Dodig, M. Stosic, Singular systems, state feedbacks problem. Linear Algebra and its Applications. 431, No 8, (2009), 1267–1292; DOI: 10.1016/j.laa.2009.04.024.
G. Duan, Analysis and Design of Descriptor Linear Systems. Springer-Verlag, New York (2010), DOI: 10.1007/978-1-4419-6397-0.
M.M. Fahmy, J. O’Reilly, Matrix pencil of closed-loop descriptor systems: infinite-eigenvalue assignment. Int. J. Control. 49, No 4, (1989), 1421–1431; DOI: 10.1080/00207178908559713.
F.R. Gantmacher, The Theory of Matrices. Chelsea Publishing Co, New York (1960).
T. Kaczorek, Checking of the positivity of descriptor linear systems with singular pencils. Archives of Control Sciences. 22, No 1, (2012), 77–86; DOI: 10.2478/v10170-011-0013-3.
T. Kaczorek, Descriptor fractional linear systems with regular pencils. Asian Journal of Control. 15, No 4, (2013), 1051–1064; DOI: 10.1002/asjc.579.
T. Kaczorek, Fractional descriptor observers for fractional descriptor continuous-time linear system. Archives of Control Sciences. 24, No 1, (2014), 39–52; DOI: 10.2478/acsc-2014-0003.
T. Kaczorek, Fractional positive continuous-time linear systems and their reachability. Int. J. Appl. Math. Comput. Sci. 18, No 2, (2008), 233–228; DOI: 10.2478/v10006-008-0020-0.
T. Kaczorek, Full-order perfect observers for continuous-time linear systems. Bull. Pol. Acad. Sci.: Tech. 49, No 4, (2001), 549–558.
T. Kaczorek, Infinite eigenvalue assignment by an output feedback for singular systems. Int. J. Appl. Math. Comput. Sci. 14, No 1, (2004), 19–23; DOI: 10.3182/20050703-6-CZ-1902.00564.
T. Kaczorek, Linear Control Systems: Analysis of Multivariable Systems. J. Wiley & Sons, New York (1992).
T. Kaczorek, Perfect observers of fractional descriptor continuous-time linear system In: Advances in Modelling and Control of Non-integer Order Systems. 320 (2015), 3–12, Springer; DOI: 10.1007/978-3-319-09900-2_1.
T. Kaczorek, Positive fractional continuous-time linear systems with singular pencils. Bull. Pol. Ac.: Tech. 60, No 1, (2012), 9–12; DOI: 10.2478/v10175-012-0002-0.
T. Kaczorek, Positive linear systems consisting of nsubsystems with different fractional orders. IEEE Trans. on Circuits and Systems. 58, No 6, (2011), 1203–1210; DOI: 10.1109/TCSI.2010.2096111.
T. Kaczorek, Reduced-order fractional descriptor observers for fractional descriptor continuous-time linear system. Bull. Pol. Acad. Sci.: Tech. 62, No 4, (2014), 889–895; DOI: 10.2478/bpasts-2014-0098.
T. Kaczorek, Selected Problems of Fractional Systems Theory. Springer-Verlag, Berlin (2011), DOI: 10.1007/978-3-642-20502-6.
R. Kociszewski, Observer synthesis for linear discrete-time systems with different fractional orders. Measurements Automation Robotics (PAR). 17, No 2, (2013), 376–381 (in Polish, CD-ROM).
V. Kucera, P. Zagalak, Fundamental theorem of state feedback for singular systems. Automatica. 24, No 5, (1988), 653–658; DOI: 10.1016/0005-1098(88)90112-4.
F.L. Lewis, Descriptor systems: Expanded descriptor equation and Markov parameters. IEEE Trans. Autom. Contr. 28, No 5, (1983), 623–627; DOI: 10.1109/TAC.1983.1103285.
D.G. Luenberger, Dynamic equations in descriptor form. IEEE Trans. Autom. Contr. 22, No 3, (1977), 312–321; DOI: 10.1109/TAC.1977.1101502.
D.G. Luenberger, Time-invariant descriptor systems. Automatica. 14, No 5, (1978), 473–480; DOI: 10.1016/0005-1098(78)90006-7.
I. N’Doye, M. Darouach, H. Voos, M. Zasadzinski, Design of unknown input fractional-order observers for fractional-order systems. Int. J. Appl. Math. Comput. Sci. 23, No 3, (2013), 491–500; DOI: 10.2478/amcs-2013-0037.
I. Podlubny, Fractional Differential Equations. Academic Press, New York (1999).
P. Van Dooren, The computation of Kronecker’s canonical form of a singular pencil. Linear Algebra and its Applications. 24 (1979), 103–140; DOI: 10.1016/0024-3795(79)90035-1.
E. Virnik, Stability analysis of positive descriptor systems. Linear Algebra and its Applications. 429, No 10, (2008), 2640–2659; DOI: 10.1016/j.laa.2008.03.002.
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Kaczorek, T. Perfect nonlinear observers of fractional descriptor continuous-time nonlinear systems. FCAA 19, 775–784 (2016). https://doi.org/10.1515/fca-2016-0041
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DOI: https://doi.org/10.1515/fca-2016-0041