Multidimensional Systems and Signal Processing

, Volume 5, Issue 4, pp 455–462 | Cite as

Robust stability of multidimensional difference equations with shift-variant coefficients

  • S. A. Yost
  • P. H. Bauer


This paper addresses the asymptotic stability of multidimensional systems represented by first hyperquadrant causal linear difference equations whose coefficients are shift-varying. The results extend previous 1-D results, and include the derivation of a fixed region of stability in the parameter space, as well as a sequence of shift-variant parameter regions. In the case of a fixed parameter region, the largest stable hyperdiamond centered at the origin will be obtained. For the shift-variant case, it will be shown that the instantaneous stable parameter region always includes this hyperdiamond.

Key Words

m-D stability robustness shift-variant systems structured uncertainties asymptotic stability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E.I. Jury, “Stability of Multidimensional Systems and Related Problems,” inMultidimensional Systems: Techniques and Applications (S.G. Tzafestas, ed.), New York: Marcel Dekker, 1986, Chap. 3.Google Scholar
  2. 2.
    C.V. Hollot and A.C. Bartlett, “Some Discrete-Time Counterparts to Kharitonov's Stability Criterion for Uncertain Systems,”IEEE Trans. Automat. Control, vol. 31, 1986, pp. 355–356.Google Scholar
  3. 3.
    F. Kraus, B.D.O. Anderson, E.I. Jury, and M. Mansour, “On the Robustness of Low-Order Schur Polynomials,”IEEE Trans. Circuits Systems, 1988, vol. 35, pp. 570–577.Google Scholar
  4. 4.
    J.E. Ackerman and B.R. Barmish, “Robust Schur Stability of a Polytope of Polynomials,”IEEE Trans. Automat. Control, vol. 33, 1988, pp. 984–986.Google Scholar
  5. 5.
    F.J. Kraus and M. Mansour, “On Robust Stability of Discrete Systems,” inProc. 29th IEEE Conf. Decision Contr., Honolulu, HI, 1990, pp. 421–422.Google Scholar
  6. 6.
    Y.K. Foo and Y.C. Soh, “Schur Stability of Interval Polynomials,”IEEE Trans. Automat. Control, vol. 38, 1993, pp. 943–946.Google Scholar
  7. 7.
    V.L. Kharitonov, “Asymptotic Stability of an Equilibrium Position of a Family of Systems of Linear Differential Equations,”Differential'nye Uravnenia, vol. 14, 1978, pp. 2086–2088.Google Scholar
  8. 8.
    P.H. Bauer, M. Mansour, and J. Durán, “Stability of Polynomials with Time-Variant Coefficients,”IEEE Trans. Circuits Systems, vol. 40, 1993, pp. 423–426.Google Scholar
  9. 9.
    P.H. Bauer, K. Premaratne, and J. Durán, “A Necessary and Sufficient Condition for Robust Asymptotic Stability of Time-Variant Discrete Systems,”IEEE Trans. Automat. Control, vol. 38, 1993, pp. 1427–1430.Google Scholar
  10. 10.
    E. Yaz and X. Niu, “Stability Robustness of Linear Discrete-Time Systems in the Presence of Uncertainties,”IJC, vol. 50, 1989, pp. 173–182.Google Scholar
  11. 11.
    S.R. Kolla, R.A. Yedavalli, and J.B. Farison, “Robust Stability Bounds of Time-Varying Perturbations for State Space Models of Linear Discrete-Time Systems,”IJC, vol. 50, 1989, pp. 151–159.Google Scholar
  12. 12.
    W.-S. Lu, “2-D Stability Test via 1-D Stability Robustness Analysis,”IJC, vol. 48, 1988, pp. 1735–1741.Google Scholar
  13. 13.
    P.H. Bauer and K. Premaratne, “Robust Stability of Time-Variant Interval Matrices,” inProc. 29th IEEE Conf. Decision Contr., Honolulu, HI, 1990, pp. 434–435.Google Scholar
  14. 14.
    P.H. Bauer and E.I. Jury, “BIBO-Stability of Multidimensional (mD) Shift-Variant Discrete Systems,”IEEE Trans. Automat. Control, vol. 36, 1991, pp. 1057–1061.Google Scholar
  15. 15.
    P.H. Bauer and E.I. Jury, “A Stability Analysis of Two-Dimensional Nonlinear Digital State-Space Filters,”IEEE Trans. Acoust., Speech, Signal Process, vol. 38, 1990, pp. 1578–1586.Google Scholar
  16. 16.
    P.H. Bauer, “Robustness and Stability Properties of First-Order Multidimensional (m-D) Discrete Processes,”Multidimens. Syst. Signal Proc., 1990, vol. 1, pp. 75–86.Google Scholar
  17. 17.
    L.M. Silverman and B.D.O. Anderson, “Controllability, Observability and Stability of Linear Systems,”SIAM J. Control, vol. 6, 1968, pp. 121–130.Google Scholar
  18. 18.
    C.R. Johnson, “Adaptive IIR Filtering: Current Results and Open Issues,”IEEE Trans. Inform. Theory, vol. IT-30, 1984, pp. 237–250.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • S. A. Yost
    • 1
  • P. H. Bauer
    • 1
  1. 1.Department of Electrical EngineeringUniversity of Notre DameNotre Dame

Personalised recommendations