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The Role of Poles and Zeros in Multivariable Feedback Theory

  • A. G. J. MacFarlane

Abstract

The increasing interest in large-scale systems with complex control structures, together with the widespread use of state-space models as the basic form of system description, naturally leads one to wonder what relevance the basic ideas of classical control theory (poles, zeros, transfer functions, Nyquist diagrams, root loci) have to such problems. Classical single-variable feedback theory revolves round the properties of poles and zeros of scalar-valued functions of a complex variable. Rosenbrock’s pioneering work (Rosenbrock, 1970,1974) showed that algebraic definitions could be given for multivariable poles and zeros, and that multivariable frequency-response design methods could be developed. Work by MacFarlane (1975), Kouvaritakis (1975a, 1975b), Karcanias (1975) and Shaked (1975)has shown that generalisations exist of the Nyquist (1932)-Bode(1945) frequency response approaches and of the root locus method (Evans 1954). Almost all of this work however is either algebraic ,using concepts such as the Smith-McMillan (McMillan,1952)t (Rosenbrock,1970) form of a transfer function matrix, or geometric, using concepts such as the null-space of the output map of a state-space description.

Keywords

Riemann Surface Algebraic Function Root Locus Feedback Connection Rectilinear Motion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Bliss, G.S., 1966 (reprint of 1933 original): Algebraic functions ( New York: Dover )Google Scholar
  2. Bode, H.W., 1945, Network Analysis and Feedback Amplifier Design ( New York: Van Nostrand )Google Scholar
  3. Evans, W.R.,1954, Control System Dynamics (New York: McGraw-Hill)Google Scholar
  4. Hsu, C.H. and Chen, C.T.,1968, A proof of the stability of multivariable feedback systems, Proc. IEEE, 56, 2061–2062CrossRefGoogle Scholar
  5. Karcanias, N., 1975, Geometric theory of zeros and its use in feedback analysis, Ph.D. Thesis, University of ManchesterGoogle Scholar
  6. Kontakos,T.,19 73 Algebraic and geometric aspects of multivariable feedback control systems, Ph.D. Thesis, University of ManchesterGoogle Scholar
  7. Kouvaritakis, B. and Macfarlane, A.G.J., 1975a, Geometric method for computing zeros of square matrix transfer functions, to be published in Int.Jnl.Control.Google Scholar
  8. Kouvaritakis, B. and Macfarlane, A.G.J., 1975b, Geometric method of computing and synthesizing zeros for general non-square matrix transfer functions, to be published in Int. Jnl.ControlGoogle Scholar
  9. Kouvaritakis, B. and Shaked, U., 1975, Asymptotic behaviour of root loci of linear multivariable systems, to be published in Int. Jnl.ControlGoogle Scholar
  10. Macfarlane, A.G.J., 1970, The return-difference and return-ratio matrices and their use in the analysis and design of multivariable feedback control systems, Proc.IEE, 117, 2037–2059Google Scholar
  11. Macfarlane, A.G. J., 1975, Relationships between recent developments in linear control theory and classical design techniques. Measurement and Control, 8, 179–187, 219–223, 278–284, 319–324 and 371–375Google Scholar
  12. McMillan, B., 1952, Introduction to formal realizability theory - II, Bell Syst. Tech. J., 31, 541–600Google Scholar
  13. Nyquist, H., 1932, Regeneration theory, Bell Syst.Tech. J., 11, 126–147Google Scholar
  14. Rosenbrock, H.H., 1970, State space and multivariable theory ( London: Nelson )Google Scholar
  15. Rosenbrock, H.H., 1973, The zeros of a system, Int.J. Control, 18 297–299CrossRefGoogle Scholar
  16. Rosenbrock, H.H., 1974, Correction to ‘The zeros of a system’, Int.Jnl.Control, 20, 525–527CrossRefGoogle Scholar
  17. Rosenbrock, H.H., 1974, Computer-aided control system design ( London: Academic Press).Google Scholar
  18. Springer, G., 1957, Introduction to Riemann Surfaces (Reading, Mass: Addison-Wesley).Google Scholar
  19. Wolovich, W.A., 1973, On determining the zeros of a state-space system, IEEE Trans, on Aut. Control,AC-18, 542–544Google Scholar
  20. Zadeh, L.A., and Desoer, C.A., 1963, Linear system theory ( New York: McGraw-Hill )Google Scholar

Copyright information

© Plenum Press, New York 1976

Authors and Affiliations

  • A. G. J. MacFarlane
    • 1
  1. 1.Engineering DepartmentUniversity of CambridgeCambridgeEngland

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