On the relevance of abstract algebra to control theory

  • Authors:

  • M. A. Arbib;H. P. Zeiger

  • Affiliations:

  • Department of Electrical Engineering, Stanford University, Stanford, California USA;Department of Aerospace Engineering, University of Colorado, Boulder, Colorado USA

  • Venue:
  • Automatica (Journal of IFAC)
  • Year:
  • 1969

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Abstract

We show that many concepts of linear system theory are better understood in an abstract algebraic framework which also applies to the non-linear machines studied in automata theory. In particular, reachability and observeability properties are shown to hold without any restrictions on linearity, and we present a general identification procedure which reduces in the linear case to a generalization of Ho's algorithm. We place convolution theory and Kalman's module-theoretic approach in this general context. Finally, we introduce a new notion of duality for systems, and relate it to reachability and observability.