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On a Non-homogeneous Singular Linear Discrete Time System with a Singular Matrix Pencil

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Abstract

In this article, we study the initial value problem of a non-homogeneous singular linear discrete time system whose coefficients are either non-square constant matrices or square with an identically zero matrix pencil. By taking into consideration that the relevant pencil is singular, we provide necessary and sufficient conditions for existence and uniqueness of solutions. More analytically we study the conditions under which the system has unique, infinite and no solutions. Furthermore, we provide a formula for the case of the unique solution. Finally we provide some numerical examples based on a singular discrete time real dynamical system to justify our theory.

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References

  1. H. Akcay, Frequency domain subspace identification of discrete-time singular power spectra. Signal Process. 92(9), 2075–2081 (2012)

    Article  Google Scholar 

  2. S.L. Campbell, Singular systems of differential equations, vol. 1 (Pitman, San Francisco, 1980). Vol. 2 (1982)

    MATH  Google Scholar 

  3. S.L. Campbell, C.D. Meyer, N.J. Rose, Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients. SIAM J. Appl. Math. 31(3), 411–425 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  4. S.L. Campbell, Comments on 2-D descriptor systems. Automatica 27(1), 189–192 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Cao, S. Zhong, Y. Hu, A descriptor system approach to robust stability of uncertain degenerate systems with discrete and distribute delays. J. Control Theory Appl. 5(4), 357–364 (2007) (English)

    Article  MathSciNet  MATH  Google Scholar 

  6. P. Cui, C. Zhang, H. Zhang, H. Zhao, Indefinite linear quadratic optimal control problem for singular discrete-time system with multiple input delays. Automatica 45(10), 2458–2461 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. L. Dai, in Singular Control Systems, ed. by M. Thoma, A. Wyner. Lecture Notes in Control and Information Sciences, (1988)

    Google Scholar 

  8. I.K. Dassios, On stability and state feedback stabilization of singular linear matrix difference equations. Adv. Differ. Equ. 2012, 75 (2012)

    Article  MathSciNet  Google Scholar 

  9. I. Dassios, On non-homogeneous generalized linear discrete time systems. Circuits Syst. Signal Process. 31(5), 1699–1712 (2012)

    Article  MathSciNet  Google Scholar 

  10. I.K. Dassios, On a boundary value problem of a class of generalized linear discrete-time systems. Adv. Differ. Equ. 2011, 51 (2011)

    Article  MathSciNet  Google Scholar 

  11. I.K. Dassios, Perturbation and robust stability of autonomous singular linear matrix difference equations. Appl. Math. Comput. 218, 6912–6920 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Debbouche, Fractional nonlocal impulsive quasilinear multi-delay integro-differential systems. Adv. Differ. Equ. 2011, 5 (2011)

    Article  MathSciNet  Google Scholar 

  13. A. Debbouche, D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems. Comput. Math. Appl. 62(3), 1442–1450 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Debbouche, D. Baleanu, R.P. Agarwal, Nonlocal nonlinear integrodifferential equations of fractional orders. Adv. Differ. Equ. 2012, 78 (2012)

    Article  Google Scholar 

  15. R.F. Gantmacher, The Theory of Matrices I, II (Chelsea, New York, 1959)

    Google Scholar 

  16. E. Grispos, S. Giotopoulos, G. Kalogeropoulos, On generalised linear discrete-time regular delay systems. J. Inst. Math. Comput. Sci., Math. Ser. 13(2), 179–187 (2000)

    MathSciNet  MATH  Google Scholar 

  17. E. Grispos, G. Kalogeropoulos, M. Mitrouli, On generalised linear discrete-time singular delay systems, in Proceedings of the 5th Hellenic-European Conference on Computer Mathematics and Its Applications. HERCMA 2001, September 20–22, 2001, ed. by E.A. Lipitakis (LEA, Athens, 2002), pp. 484–486. 2 volumes

    Google Scholar 

  18. E. Grispos, G. Kalogeropoulos, I. Stratis, On generalised linear discrete-time singular delay systems. J. Math. Anal. Appl. 245(2), 430–446 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. E. Grispos, Singular generalised autonomous linear differential systems. Bull. Greek Math. Soc. 34, 25–43 (1992)

    MathSciNet  MATH  Google Scholar 

  20. J.-e. Feng, P. Cui, Z. Hou, Singular linear quadratic optimal control for singular stochastic discrete-time systems. Optim. Control Appl. Methods (2012)

  21. T. Kaczorek, General response formula for two-dimensional linear systems with variable coefficients. IEEE Trans. Autom. Control Ac-31, 278–283 (1986)

    Article  MathSciNet  Google Scholar 

  22. T. Kaczorek, Singular Roesser model and reduction to its canonical form. Bull. Pol. Acad. Sci., Tech. Sci. 35, 645–652 (1987)

    MATH  Google Scholar 

  23. T. Kaczorek, Equivalence of singular 2-D linear models. Bull. Polish Academy Sci., Electr. Electrotech. 37 (1989)

  24. G.I. Kalogeropoulos, Matrix pencils and linear systems. Ph.D. Thesis, City University, London (1985)

  25. G. Kalogeropoulos, I.G. Stratis, On generalized linear regular delay systems. J. Math. Anal. Appl. 237(2), 505–514 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. N. Karcanias, G. Kalogeropoulos, Geometric theory and feedback invariants of generalized linear systems: a matrix pencil approach. Circuits Syst. Signal Process. 8(3), 375–397 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Klamka, Controllability of Dynamical Systems (Kluwer Academic, Dordrecht, 1991)

    MATH  Google Scholar 

  28. J. Klamka, Controllability of dynamical systems. Mat. Stosow. 50(9), 57–75 (2008)

    MathSciNet  Google Scholar 

  29. J. Klamka, Controllability of nonlinear discrete systems. Int. J. Appl. Math. Comput. Sci. 12(2), 173–180 (2002)

    MathSciNet  MATH  Google Scholar 

  30. J. Klamka, Controllability and minimum energy control problem of fractional discrete-time systems, in Monograph New Trends in Nanotechnology and Fractional Calculus, ed. by D. Baleanu, Z.B. Guvenc, J.A. Tenreiro Machado (Springer, New York, 2010), pp. 503–509

    Chapter  Google Scholar 

  31. F.L. Lewis, A survey of linear singular systems. Circuits Syst. Signal Process. 5, 3–36 (1986)

    Article  MATH  Google Scholar 

  32. F.L. Lewis, Recent work in singular systems, in Proc. Int. Symp. Singular systems, Atlanta, GA (1987), pp. 20–24

    Google Scholar 

  33. F.L. Lewis, A review of 2-D implicit systems. Automatica 28(2), 345–354 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  34. H. Mejhed, N. El Houda Mejhed, A. Hmamed, Componentwise stability of the singular discrete time system using the methodology of the Drazin inverse. J. Trans. Syst. Control 4(12), 571–580 (2009)

    Google Scholar 

  35. M. Mitrouli, G. Kalogeropoulos, A compound matrix algorithm for the computation of the Smith form of a polynomial matrix. Numer. Algorithms 7(2–4), 145–159 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  36. K. Ogata, Discrete Time Control Systems (Prentice Hall, New York, 1987)

    Google Scholar 

  37. W.J. Rugh, Linear System Theory (Prentice Hall International, London, 1996)

    MATH  Google Scholar 

  38. J.T. Sandefur, Discrete Dynamical Systems (Academic Press, San Diego, 1990)

    MATH  Google Scholar 

  39. G.W. Steward, J.G. Sun, Matrix Perturbation Theory (Oxford University Press, London, 1990)

    Google Scholar 

  40. D.N. Vizireanu, A fast, simple and accurate time-varying frequency estimation method for single-phase electric power systems. Measurement 45(5), 1331–1333 (2012)

    Article  Google Scholar 

  41. D.N. Vizireanu, S.V. Halunga, Simple, fast and accurate eight points amplitude estimation method of sinusoidal signals for DSP based instrumentation. J. Instrum. 7(4) (2012)

  42. W. Zhou, H. Lu, C. Duan, M. Li, Delay-dependent robust control for singular discrete-time Markovian jump systems with time-varying delay. Int. J. Robust Nonlinear Control 20(10), 1112–1128 (2010)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We are very grateful to the anonymous referees.

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  1. Department of Mathematics, University of Athens, Athens, Greece

    Ioannis Dassios & Grigoris Kalogeropoulos

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  1. Ioannis Dassios
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  2. Grigoris Kalogeropoulos
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Correspondence to Ioannis Dassios.

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Dassios, I., Kalogeropoulos, G. On a Non-homogeneous Singular Linear Discrete Time System with a Singular Matrix Pencil. Circuits Syst Signal Process 32, 1615–1635 (2013). https://doi.org/10.1007/s00034-012-9541-8

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  • DOI: https://doi.org/10.1007/s00034-012-9541-8

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