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

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Abstract

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.

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References

  1. D.H. Owens, “Stability of Linear Multipass Processes,”Proceedings of the IEE, vol. 124, pp. 1079–1982.

  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.

  3. E. Rogers and K.J. Smyth, “Modelling of Industrial Repetitive Processes,” Research Report No. DC 25, University of Strathclyde, 1990.

  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. 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. 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. E.I. Jury,Inners and the Stability of Dynamic Systems, New York: Wiley, 1974.

    Google Scholar 

  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. F.R. Gantmacher,The Theory of Matrices, vols. I, II, New York: Chelsea, 1959.

    Google Scholar 

  10. E. Rogers and D.H. Owens. “Lyapunov Based Stability Tests for Repetitive Processes,” in Preparation.

  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. 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. 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. 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. K.J. Smyth and E. Rogers, “Requirements for Controller Design for Linear Repetitive Processes,” Research Report No. D.C. 37, University of Strathclyde, 1990.

  16. K.J. Smyth, “Computer Aided Analysis for Linear Repetitive Processes,” Ph.D. thesis, University of Strathclyde, 1991.

  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.

  18. D.H. Owens,Multivariable and Optimal Systems, London: Academic Press, 1981.

    Google Scholar 

  19. E. Rogers and D.H. Owens, “Multivariable Controller Design for Industrial Repetitive Processes,”Proc. Control 91, IEE, Edinburgh, 1991, pp. 1267–1270.

  20. E. Rogers and P. Rocha, “Non-unit Memory Linear Repetitive Processes—A 2D Systems Approach to Analysis and Control,” in preparation.

  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 

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Authors and Affiliations

  1. Advanced Systems Research Group, Department of Aeronautics and Astronautics, University of Southampton, SO9 5NH, Southampton, U.K.

    E. Rogers

  2. Centre for Systems and Control Engineering, School of Engineering, University of Exeter, EX4 4QF, Exeter, U.K.

    D. H. Owens

Authors
  1. E. Rogers
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  2. D. H. Owens
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Rogers, E., Owens, D.H. Stability tests and performance bounds for a class of 2D linear systems. Multidim Syst Sign Process 4, 355–391 (1993). https://doi.org/10.1007/BF00989651

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