The Role of Poles and Zeros in Multivariable Feedback Theory

  • A. G. J. MacFarlane


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.


Riemann Surface Algebraic Function Root Locus Feedback Connection Rectilinear Motion 
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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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