Multidimensional Systems and Signal Processing

, Volume 4, Issue 4, pp 355–391 | Cite as

Stability tests and performance bounds for a class of 2D linear systems

  • E. Rogers
  • D. H. Owens


Repetitive, or multipass, processes are a class of 2D systems characterized by a recursive action with interaction between successive outputs or pass profiles. This interpass interaction is the source of the unique control problem for these processes in that it can cause the output sequence to exhibit oscillations which increase in amplitude from pass to pass. Previous work has developed an abstract stability theory and applied it to subclasses, such as discrete nonunit memory linear processes which are considered here, to produce basic stability tests. This article begins by reviewing the known stability tests and concludes that, at best, they only produce highly qualitative indicators of relative stability and performance. Hence, unlike (say) Bode and Nyquist tests for standard linear systems, they are of limited appeal as a basis for computer-aided control systems design. To remove this difficulty, step response data is used to develop new simulation-based tests which yield, at no extra cost, unique computable performance measures. Further, the undoubted advantages of having such measures available is clearly shown by developing a (virtually) complete solution to controller design for one subclass.

Key Words

repetitive dynamics stability performance bounds 


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  1. 1.
    D.H. Owens, “Stability of Linear Multipass Processes,”Proceedings of the IEE, vol. 124, pp. 1079–1982.Google Scholar
  2. 2.
    E. Rogers and D.H. Owens,Stability Analysis for Linear Repetitive Processes, Lecture Notes in Control and Information Sciences, vol. 175, Springer-Verlag, 1992.Google Scholar
  3. 3.
    E. Rogers and K.J. Smyth, “Modelling of Industrial Repetitive Processes,” Research Report No. DC 25, University of Strathclyde, 1990.Google Scholar
  4. 4.
    E. Rogers and D.H. Owens, “2D Transfer Functions and Stability Tests for Discrete Linear Repetitive Processes,” inRealization and Modelling in Systems Theory, M. A. Kaashoek, et al., eds., Boston: Birkhauser, 1992, pp. 351–356.Google Scholar
  5. 5.
    F.M. Boland and D.H. Owens, “Linear Multipass Processes—A Two Dimensional Interpretation,”Proceedings of the IEE, vol. 127, 1980, pp. 189–193.Google Scholar
  6. 6.
    E. Rogers and D.H. Owens, “Stability Analysis for Discrete Linear Multipass Processes with Non-Unit Memory,”IMA Journal of Mathematical Control and Information, vol. 6, 1989, pp. 399–409.Google Scholar
  7. 7.
    E.I. Jury,Inners and the Stability of Dynamic Systems, New York: Wiley, 1974.Google Scholar
  8. 8.
    D.D. Siljak, “Algebraic Criterion for Positive Realness Relative to the Unit Circle,”Journal of the Franklin Institute, vol. 295, 1973, pp. 469–476.Google Scholar
  9. 9.
    F.R. Gantmacher,The Theory of Matrices, vols. I, II, New York: Chelsea, 1959.Google Scholar
  10. 10.
    E. Rogers and D.H. Owens. “Lyapunov Based Stability Tests for Repetitive Processes,” in Preparation.Google Scholar
  11. 11.
    W.S. Lu and E.B. Lee, “Stability Analysis for Two-Dimensional Filters via a Lyapunov Approach,”IEEE Transactions on Circuits and Systems, vol. CAS 32, 1985, pp. 61–68.Google Scholar
  12. 12.
    P. Agathoklis, E.I. Jury, and M. Mansour, “An Algebraic Test for Internal Stability of 2D Discrete Systems,” inRealization and Modelling in Systems Theory, M.A. Kaashoek et al. eds., Boston: Birkhauser, 1990, pp. 303–310.Google Scholar
  13. 13.
    B.D.O. Anderson, P. Agathoklis, E.I. Jury, and M. Mansour, “Stability and the Matrix Lyapunov Equation for Discrete 2-Dimensional Systems,”IEEE Transactions on Circuits and Systems, vol. CAS 33, 1986, pp. 261–266.Google Scholar
  14. 14.
    P. Agathoklis, “Lower Bounds for the Stability Margin of Discrete Two-Dimensional Systems Based on the Two-Dimensional Lyapunov Equation,”IEEE Transactions on Circuits and Systems, vol. CAS 35, 1988, pp. 745–749.Google Scholar
  15. 15.
    K.J. Smyth and E. Rogers, “Requirements for Controller Design for Linear Repetitive Processes,” Research Report No. D.C. 37, University of Strathclyde, 1990.Google Scholar
  16. 16.
    K.J. Smyth, “Computer Aided Analysis for Linear Repetitive Processes,” Ph.D. thesis, University of Strathclyde, 1991.Google Scholar
  17. 17.
    D.H. Owens and A. Chotai, “Robust Controller Design for Linear Dynamic Systems Using an Approximate Model,”Proceedings of the IEE, vol. 130, pp. 45–56.Google Scholar
  18. 18.
    D.H. Owens,Multivariable and Optimal Systems, London: Academic Press, 1981.Google Scholar
  19. 19.
    E. Rogers and D.H. Owens, “Multivariable Controller Design for Industrial Repetitive Processes,”Proc. Control 91, IEE, Edinburgh, 1991, pp. 1267–1270.Google Scholar
  20. 20.
    E. Rogers and P. Rocha, “Non-unit Memory Linear Repetitive Processes—A 2D Systems Approach to Analysis and Control,” in preparation.Google Scholar
  21. 21.
    D.H. Owens and E. Rogers, “391-1 Norm Minimization and the Stabilization of Systems with Repetitive Dynamics, Trans. Inst. M.C., vol. 14, 3, pp. 126–129, 1992.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • E. Rogers
    • 1
  • D. H. Owens
    • 2
  1. 1.Advanced Systems Research Group, Department of Aeronautics and AstronauticsUniversity of SouthamptonSouthamptonU.K.
  2. 2.Centre for Systems and Control Engineering, School of EngineeringUniversity of ExeterExeterU.K.

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