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Decoupling of linear systems by dynamic output feedback

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Abstract

Necessary and sufficient conditions for decoupling of linear systems by dynamic output feedback are derived, under the requirement that the decoupled system be internally stable. The conditions are stated in terms of quantities which are directly related to the transfer matrix of the given system. The main issue is resolved through the introduction of a new concept—the strict adjoint. The strict adjoint is a “minimal” polynomial matrix that diagonalizes a given matrix.

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References

  1. G. Basile, and G. Marro, A state-space approach to noninteracting controls,Richerche di Automatica, 1, 68–77 (1970).

    Google Scholar 

  2. M. M. Bayoumi, and T. L. Duffield, Output feedback decoupling and pole placement in linear time invariant systems,Trans. IEEE, AC-22, 143–142 (1977).

    Google Scholar 

  3. R. Bellman,Introduction to Matrix Analysis, McGraw-Hill, (1970).

  4. R. M. Brasch, and J. B. Pearson, Pole placement using dynamic compensators,Trans. IEEE, AC-15, 34–43 (1970).

    Google Scholar 

  5. C. A. Desoer, and W. S. Chan, The feedback interconnection of lumped linear time invariant systems,J. Franklin Inst., 300, 335–351 (1975).

    Google Scholar 

  6. P. Falb, and W. Wolovich, Decoupling in the design of multivariable control systems,IEEE Transactions on Automatic Control, AC-12, 651–659 (1967).

    Google Scholar 

  7. E. G. Gilbert, The decoupling of multivariable systems by state feedback,SIAM Journal on Control, 7, 50–63 (1969).

    Google Scholar 

  8. I. C. Gokhberg, and M. G. Krein, Systems of integral equations, on a half line with kernels depending on the difference of arguments,American Mathematical Society: Translations, 14, 2:217–287 (1960).

    Google Scholar 

  9. J. Hammer, Stability and stable precompensation: An algebraic approach,Math. Systems Theory 16, 265–296 (1983).

    Google Scholar 

  10. J. Hammer, Linear dynamic output feedback: Invariants and stability,IEEE Trans. on Automatic Control, April 1983(b).

  11. M. L. J. Hautus and M. Heymann, Linear feedback—an algebraic approach,SIAM Journal on Control and Optimization, 16, 83–105 (1978).

    Google Scholar 

  12. M. L. J. Hautus and M. Heymann, Linear feedback decoupling—transfer function analysis, Preprint, Department of Electrical Engineering, Technion, Israel Institute of Technology, Haifa, Israel (1980).

    Google Scholar 

  13. J. H. Howze, Necessary and sufficient conditions for decoupling using output feedback,IEEE Transactions on Automatic Control, AC-18, 44–46 (1973).

    Google Scholar 

  14. J. H. Howze, and J. B. Pearson, Decoupling and arbitrary pole-placement in linear systems using output feedback,IEEE Transactions on Automatic Control, AC-15, 660–663 (1970).

    Google Scholar 

  15. C. C. Macduffee,The Theory of Matrices, Chelsea, New York (1934).

    Google Scholar 

  16. B. S. Morgan, The synthesis of linear multivariable systems by state variable feedback,IEEE Transactions on Automatic Control, AC-9, 405–411 (1964).

    Google Scholar 

  17. A. S. Morse, System invariants under feedback and cascade control,Lecture notes in Mathematical Systems, Vol. 131 (G. Marchesini and S. Mitter, editors) Springer Verlag, Berlin (1975).

    Google Scholar 

  18. A. S. Morse, and W. M. Wonham, Decoupling and pole assignment by dynamic compensation,SIAM Journal on Control, 8, 317–337 (1970).

    Google Scholar 

  19. L. Pernebo, An algebraic theory for the design of controllers for linear multivariable systems-Part I: Structure matrices and feed-forward design,Trans. IEEE, AC-26, 171–183 (1981).

    Google Scholar 

  20. Z. V. Rekasius, Decoupling of multivariable systems by means of state variable feedback, Proceedings of the Third Annual Allerton Conference, pp. 439–448 (1965).

  21. L. M. Silverman, Decoupling with state feedback and precompensation,IEEE Transactions on Automatic Control, AC-15, 487–489 (1970).

    Google Scholar 

  22. L. M. Silverman, and H. J. Payne, Input-output structure of linear systems with application to the decoupling problem,SIAM Journal on Control and Optimization, 9, 199–233 (1971).

    Google Scholar 

  23. S. H. Wang, and E. J. Davison, Design of decoupled systems: a frequency domain approach,International Journal of Control, 21, 529–536 (1975).

    Google Scholar 

  24. W. A. Wolovich, Output feedback decoupling,IEEE Transactions on Automatic Control, AC-20, 148–149 (1975).

    Google Scholar 

  25. W. A. Wolovich, Skew prime polynomial matrices,IEEE Transactions on Automatic Control, AC-23, 880–887 (1978).

    Google Scholar 

  26. W. A. Wolovich, and P. Ferreira, Output regulation and tracking in linear multivariable systems,IEEE Transactions on Automatic Control, AC-24, 460–465 (1979).

    Google Scholar 

  27. W. M. Wonham,Linear Multivariable Control, Springer, Berlin (1974).

    Google Scholar 

  28. W. M. Wonham, and A. S. Morse, Decoupling and pole-assignment in linear multivariable systems: a geometric approach,SIAM Journal on Control, 8, 1–18, (1970).

    Google Scholar 

  29. B. F. Wyman,Linear systems over commutative rings, Lecture notes, Stanford University, Stanford, California, U.S.A. (1972).

    Google Scholar 

  30. D. C. Youla, On the factorization of rational matrices,IRE Trans. on Information Theory, pp. 172–189 (1961).

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Author information

Authors and Affiliations

  1. Center for Mathematical System Theory, University of Florida, Gainesville, Florida, USA

    Jacob Hammer

  2. Department of Electrical Engineering, University of Florida, Gainesville, Florida, USA

    Pramod P. Khargonekar

Authors
  1. Jacob Hammer
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  2. Pramod P. Khargonekar
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Additional information

This research was supported in part by US Army Research Grant DAAG29-80-C0050 and US Air Force Grant AFOSR76-3034D through the Center for Mathematical System Theory, University of Florida, Gainesville, Florida 32611, U.S.A.

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Hammer, J., Khargonekar, P.P. Decoupling of linear systems by dynamic output feedback. Math. Systems Theory 17, 135–157 (1984). https://doi.org/10.1007/BF01744437

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  • DOI: https://doi.org/10.1007/BF01744437

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