×

Application of the Drazin inverse to the analysis of descriptor fractional discrete-time linear systems with regular pencils. (English) Zbl 1293.93496

Summary: The Drazin inverse of matrices is applied to find the solutions of the state equations of descriptor fractional discrete-time systems with regular pencils. An equality defining the set of admissible initial conditions for given inputs is derived. The proposed method is illustrated by a numerical example.

MSC:

93C55 Discrete-time control/observation systems
15A22 Matrix pencils
93C05 Linear systems in control theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bru, R. , Coll, C., Romero-Vivo S. and Sanchez, E. (2003). Some problems about structural properties of positive descriptor systems, in L. Benvenuti, A. Santis and L. Farina , Positive Systems, Lecture Notes in Control and Information Sciences, Vol. 294, Springer, Berlin, pp. 233-240. · Zbl 1060.93065
[2] Bru, R., Coll, C. and Sanchez, E. (2000). About positively discrete-time singular systems, in N.E. Mastorakis System and Control: Theory and Applications, Electrical and Computer Engineering Series, World Scientific and Engineering Society, Athens, pp. 44-48.
[3] Bru, R., Coll, C. and Sanchez, E. (2002). Structural properties of positive linear time-invariant difference-algebraic equations, Linear Algebra and Applications 349(1-3): 1-10. · Zbl 1006.93006 · doi:10.1016/S0024-3795(02)00277-X
[4] Campbell, S.L.,Meyer, C.D. and Rose, N.J. (1976). Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients, SIAM Journalon Applied Mathematics 31(3): 411-425. · Zbl 0341.34001 · doi:10.1137/0131035
[5] Commalut, C. and Marchand, N. (2006). Positive Systems, Lecture Notes in Control and Information Sciences, Vol. 341, Springer-Verlag, Berlin.
[6] Dai, L. (1989). Singular Control Systems, Lectures Notes in Control and Information Sciences, Vol. 118, Springer-Verlag, Berlin. · Zbl 0669.93034
[7] Dodig, M. and Stosic, M. (2009). Singular systems state feedbacks problems, Linear Algebra and Its Applications431(8): 1267-1292.
[8] Fahmy, M.H, and O’Reill, J. (1989). Matrix pencil of closed-loop descriptor systems: Infinite-eigenvalues assignment, International Journal of Control 49(4): 1421-1431. · Zbl 0681.93036 · doi:10.1080/00207178908961323
[9] Farina, L. and Rinaldi, S. (2000). Positive Linear Systems, J. Willey, New York, NY. · Zbl 0988.93002
[10] Gantmacher, F.R. (1960). The Theory of Matrices, Chelsea Publishing Co., New York, NY. · Zbl 0088.25103
[11] Kaczorek, T. (1992). Linear Control Systems, Vol. 1, Research Studies Press, J. Wiley, New York, NY. · Zbl 0784.93002
[12] Kaczorek, T. (2002). Positive 1D and 2D Systems, Springer-Verlag, London. · Zbl 1005.68175
[13] Kaczorek, T. (2004). Infinite eigenvalue assignment by an output/feedback for singular systems, International Journalof Applied Mathematics and Computer Science 14(1): 19-23. · Zbl 1171.93331
[14] Kaczorek, T. (2007a). Polynomial and Rational Matrices. Applicationsin Dynamical Systems Theory, Springer-Verlag, London. · Zbl 1114.15019
[15] Kaczorek, T. (2007b). Realization problem for singular positive continuous-time systems with delays, Control and Cybernetics36(1): 47-57. · Zbl 1293.93378
[16] Kaczorek, T. (2010). Positive linear systems with different fractional orders, Bulletin of the Polish Academy of Sciences:Technical Sciences 58(3): 453-458. · Zbl 1220.78074 · doi:10.2478/v10175-010-0043-1
[17] Kaczorek, T. (2011a). Checking of the positivity of descriptor linear systems by the use of the shuffle algorithm, Archivesof Control Sciences 21(3): 287-298. · Zbl 1264.93096 · doi:10.2478/v10170-010-0044-1
[18] Kaczorek, T. (2011b). Selected Problems of Fractional SystemsTheory, Springer-Verlag, Berlin.
[19] Kaczorek T. (2011c). Singular fractional discrete-time linear systems, Control and Cybernetics 40(3): 1-8. · Zbl 1318.93058
[20] Kaczorek T. (2011d). Reduction and decomposition of singular fractional discrete-time linear systems, Acta Mechanica etAutomatica 5(4): 1-5. · Zbl 1318.93058
[21] Kucera, V. and Zagalak, P. (1988). Fundamental theorem of state feedback for singular systems, Automatica 24(5): 653-658. · Zbl 0661.93033 · doi:10.1016/0005-1098(88)90112-4
[22] Van Dooren, P. (1979). The computation of Kronecker’s canonical form of a singular pencil, Linear Algebra andIts Applications 27: 103-140. · Zbl 0416.65026 · doi:10.1016/0024-3795(79)90035-1
[23] Virnik, E. (2008). Stability analysis of positive descriptor systems, Linear Algebra and Its Applications 429(10): 2640-2659. · Zbl 1147.93033 · doi:10.1016/j.laa.2008.03.002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.