Properties and calculation of transmission zeros of linear multivariable systems

  • Authors:

  • E.J Davison;S.H Wang

  • Affiliations:

  • Department of Electrical Engineering, University of Toronto, Canada;College of Engineering and Applied Science, University of Colorado, Colorado Springs, Colorado 80907, U.S.A.

  • Venue:
  • Automatica (Journal of IFAC)
  • Year:
  • 1974

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Abstract

A new definition of transmission zeros for a linear, multivariable, time-invariant system is made which is shown to be equivalent to previous definitions. Based on this new definition of transmission zeros, new properties of transmission zeros of a system are then obtained; in particular, it is shown that a system with an unequal number of inputs and outputs almost always has no transmission zeros and that a system with an equal number of inputs and outputs almost always has either n-1 or n transmission zeros, where n is the order of the system; transmission zeros of cascade systems are then studied, and it is shown how the transmission zeros of a system relate to the poles of a closed loop system subject to high gain output feedback. An application of transmission zeros to the servomechanism problem is also included. A fast, efficient, numerically stable algorithm is then obtained which enables the transmission zeros of high order multivariable systems to be readily obtained. Some numerical examples for a 9th order system are given to illustrate the algorithm.