Circuits, Systems and Signal Processing

, Volume 29, Issue 5, pp 971–997 | Cite as

Convergence of Unilateral Laplace Transforms on Time Scales

  • John M. Davis
  • Ian A. Gravagne
  • Robert J. MarksII
Article

Abstract

A time scale is any closed subset of the real line. Continuous time and discrete time are special cases. The unilateral Laplace transform of a signal on a time scale subsumes the continuous-time unilateral Laplace transform, and the discrete-time unilateral z-transform as special cases. The regions of convergence (ROCs) time scale Laplace transforms are determined by the time scale’s graininess. For signals with finite area, the ROC for the Laplace transform resides outside of a Hilger circle determined by the time scales’s smallest graininess. For transcendental functions associated with the solution of linear time-invariant differential equations, the ROCs are determined by function parameters (e.g., sinusoid frequency) and the largest and smallest graininess values in the time scale. Since graininess always lies between zero and infinity, there are ROCs applicable to a specified signal on any time scale. All ROCs reduce to the familiar half-plane ROCs encountered in the continuous-time unilateral Laplace transform and circle ROCs for the unilateral z-transform. If a time scale unilateral Laplace transform converges at some point in the transform plane, a region of additional points can be identified as also belonging to the larger ROC.

Keywords

Time scales Laplace transform z-transforms Region of convergence Hilger circle 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications (Birkhäuser, Boston, 2001) MATHGoogle Scholar
  2. 2.
    M. Bohner, A. Peterson (eds.), Advances in Dynamic Equations on Time Scales (Birkhäuser, Boston, 2002) Google Scholar
  3. 3.
    J.M. Davis, J. Henderson, K.R. Prasad, W.K.C. Yin, Solvability of a nonlinear second order conjugate eigenvalue problem on a time scale. Abstr. Appl. Anal. 5, 91–99 (2000) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    J.M. Davis, J. Henderson, K.R. Prasad, Upper and lower bounds for the solution of the general matrix Riccati differential equation on a time scale. J. Comput. Appl. Math. 141, 133–145 (2002) CrossRefMathSciNetGoogle Scholar
  5. 5.
    J.M. Davis, I.A. Gravagne, B.J. Jackson, R.J. Marks II, A.A. Ramos, The Laplace transform on time scales revisited. J. Math. Anal. Appl. 332, 1291–1307 (2007) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    I.A. Gravagne, J.M. Davis, J. Dacunha, R.J. Marks II, Bandwidth sharing for controller area networks using adaptive sampling, in Proc. Int. Conf. Robotics and Automation (ICRA), New Orleans, LA, April 2004, pp. 5250–5255 Google Scholar
  7. 7.
    G. Guseinov, Integration on time scales. J. Math. Anal. Appl. 285(1), 107–127 (2003) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    S. Hilger, Ein Masskettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. Ph.D. thesis, Universität Würzburg, Germany (1988) Google Scholar
  9. 9.
    S. Hilger, Special functions: Laplace and Fourier transform on measure chains. Dyn. Syst. Appl. 8, 471–488 (1999) MATHMathSciNetGoogle Scholar
  10. 10.
    R.J. Marks II, I. Gravagne, J.M. Davis, J.J. DaCunha, Nonregressivity in switched linear circuits and mechanical systems. Math. Comput. Model. 43, 1383–1392 (2006) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    R.J. Marks II, I.A. Gravagne, J.M. Davis, A generalized Fourier transform and convolution on time scales. J. Math. Anal. Appl. 340(2), 901–919 (2008) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    R.J. Marks II, Handbook of Fourier Analysis and Its Applications (Oxford University Press, London, 2009) MATHGoogle Scholar
  13. 13.
    J. Seiffertt, S. Sanyal, D.C. Wunsch, Hamilton–Jacobi–Bellman equations and approximate dynamic programming on time scales. IEEE Trans. Syst. Man Cybern. Part B 38(4), 918–923 (2008) CrossRefGoogle Scholar
  14. 14.
    E. Talvila, The distributional Denjoy integral. Real Anal. Exch. 33(1), 51–82 (2007/2008) MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • John M. Davis
    • 1
  • Ian A. Gravagne
    • 2
  • Robert J. MarksII
    • 2
  1. 1.Department of MathematicsBaylor UniversityWacoUSA
  2. 2.Department of Electrical and Computer EngineeringBaylor UniversityWacoUSA

Personalised recommendations