On the Stability of Linear Discrete Systems and Related Problems

  • M. Mansour
  • E. I. Jury


In this report some developments in the area of stability of discrete systems originally motivated by a paper by Kalman and Bertram are overviewed. It is shown that an analog to the Schwarz-form was developed for discrete systems. This form was applied in determining the margin of stability, Identification, signal processing using lattice filters and model reduction of one-dimensional and multi-dimensional systems


Discrete System Model Reduction Discrete Time System Linear Discrete System Lattice Filter 
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  1. [1]
    Wall, H.S.: Polynomials whose zeros have negative real parts. Amer Math Monthly, 52 (1945), pp 308–322Google Scholar
  2. [2]
    Schwarz, H.: Ein Verfahren zur Stabilitätsfrage bei Matrizen-Eigenwertproblemen, Zeit f angew Math u Physik, 1956, pp 473–500Google Scholar
  3. [3]
    Kalman, R., Bertram, J.: Control System analysis and design via the second method of Lyapunov. J of Basic Engineering, June 1960, p 371Google Scholar
  4. [4]
    Parks, P.: A new proof of the Hurwitz-stability criterion by the second method of Lyapunov with applications to optimum transfer functions. Fourth Joint Automatic Control Conference, June 1963Google Scholar
  5. [5]
    Gantmacher, F.R.: Applications of the theory of matrices. Interscience, New York, 1959MATHGoogle Scholar
  6. [6]
    Mansour, M.: Stability criteria of linear systems and the second method of Lyapunov. Scientia Electrica, Vol XI, 1965, pp 65–104Google Scholar
  7. [7]
    Mansour, M.: Die Stabilität linearer Abtastsysteme und die zweite Methode von Lyapunov. Regelungstechnik, Heft 12, 1965, pp 592–596Google Scholar
  8. [8]
    Chen, C.F., Chu, H.: A matrix for evaluating Schwarz form. IEEE Trans on Aut Control, 1966, pp 303–305Google Scholar
  9. [9]
    Puri, N., Weygandt, C.: Second method of Lyapunov and Routh canonical form. J. of the Franklin Institute, Nov 1963, p 365Google Scholar
  10. [10]
    Anderson, B.D.O., Jury, E.I., Mansour, M.: Schwarz-Matrix Properties for Continous and Discrete Time Systems. Int J Control, Vol 23, 1976, pp 1–16MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Schur, I.: Ueber Potenzreihen, die im Innern des Einheitskreises beschränkt sind. J. Reine u. Angewandte Math. 147 (1917), pp 205–232Google Scholar
  12. [12]
    Cohn, A.: Ueber die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreis, Math Zeit 14 (1922), pp 110–148MATHCrossRefGoogle Scholar
  13. [13]
    Jury, E.I.: Theory and Applications of the z-Transform Method. Krieger, 1982Google Scholar
  14. [14]
    Mansour, M., Jury, E.I., Chaparro, L.F.: Estimation of the margin of stability for linear continuous and discrete systems. Int J Control, Vol 30, 1979, pp 49–69MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Mansour, M.: A note on the stability of Linear Discrete Systems and Lyapunov Method. IEEE Trans on Aut Control, Vol AC-27, No 3, June 1982, pp 707–708CrossRefGoogle Scholar
  16. [16]
    Dourdoumas, N.: Ein Beitrag zur Identifikation und Approximation von Systemen mit Hilfe linearer diskreter mathematischer Modelle. Archiv für Elektrotechnik 62 (1980), pp 1–4Google Scholar
  17. [17]
    Takizawa, M., Kishi, H., Hamada, N.: Synthesis of Lattice digital filters by the state variable method. Electron Commun Japan, 65-A, pp 27–36Google Scholar
  18. [18]
    Badreddin, E., Mansour, M.: Model reduction of discrete time systems using the Schwartz canonical form. Electron. Lett., Vol 16, No 20, Sep 25, 1980, pp 782–783CrossRefGoogle Scholar
  19. [19]
    Badreddin, E., Mansour, M.: A multivariable normal form for model reduction of discrete-time sytems. Syst. Control. Lett. 4 (1983), pp 271–285MathSciNetCrossRefGoogle Scholar
  20. [20]
    Badreddin, E., Mansour, M.: A second multivariable normal form for model reduction of discrete-time systems. Syst Control Lett 4 (1984), pp 109–117MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    Anderson, B.D.O., Jury, E.I., Mansour, M.: On Model Reduction of Discrete Time Systems. Automatica, Vol 22, No 6, 1986, pp 717–721MATHCrossRefGoogle Scholar
  22. [22]
    Antoulas, A.C., Mansour, M.: On Stability and the Cascade Structure. Technical Report 1990Google Scholar
  23. [23]
    Jury, E.I., Premaratne, K.: Model Reduction of Two-Dimensional Discrete Systems. IEEE Trans on Circuit and Systems, Vol CAS-33, No 5, May 1986, pp 558–562CrossRefGoogle Scholar
  24. [24]
    Premaratne, K., Jury, E.I., Mansour, M.: Multivariable Canonical Forms for Model Reduction of 2D Discrete Time Systems. Vol CAS-37, No 4, IEEE Trans on Circuits and Systems, April 1990, pp 488–501Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • M. Mansour
    • 1
  • E. I. Jury
    • 2
  1. 1.Institute of Automatic Control, Swiss Federal Institute of TechnologyETH-ZürichZürichSwitzerland
  2. 2.Department of Electrical and Computer EngineeringUniversity of MiamiCoral GablesUSA

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