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High gain feedback and the stabilizability of multivariable systems

  • Session 1 Algebraic And Geometric System Theory
  • Conference paper
  • First Online:
Book cover Analysis and Optimization of Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 44))

Abstract

The problems of determining the minimal order of a stabilizing compensator for a fixed linear, multivariable system and for the generic p × m system of fixed degree are considered. An elementary geometric argument gives sufficient conditions for the generic stabilizability by a compensator of order ≤ q. A more delicate geometric argument, involving pole-placement in the high gain limit, is then used to derive necessary conditions, obtained jointly with B.D.O. Anderson, for the lower bound q ≥ 1. Taken together, these results determine the minimal order in certain low dimensional cases. The general upper bound, however, is not always tight and in many cases can be improved upon by more powerful techniques. For example, based on a geometric model for finite and infinite gains, sufficient conditions for q=0 are derived in this paper in terms of a topological invariant (of the "gain space") introduced by Ljusternick and Šnirel'mann in the calculus of variations. Using the Schubert calculus, an estimate of the Ljusternick-Šnirel'mann category is obtained, yielding a stabilizability criterion which, to my knowledge, contains the previous results in the literature on stabilizability by constant gain output feedback, as special cases.

Research partially supported by the NASA under Grant No. NSG-2265, the National Science Foundation under Grant No. ECS-81-21428, the Air Force Office of Scientific Research under Grant No. AFOSR-81-0054, and the Office of Naval Research under JSEP Contract No. N00014-75-C-0648.

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Authors and Affiliations

  1. Division of Applied Sciences and Department of Mathematics, Harvard University, 02138, Cambridge, MA

    Christopher I. Byrnes

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  1. Christopher I. Byrnes
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A. Bensoussan J. L. Lions

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© 1982 Springer-Verlag

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Byrnes, C.I. (1982). High gain feedback and the stabilizability of multivariable systems. In: Bensoussan, A., Lions, J.L. (eds) Analysis and Optimization of Systems. Lecture Notes in Control and Information Sciences, vol 44. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0044378

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  • DOI: https://doi.org/10.1007/BFb0044378

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12089-6

  • Online ISBN: 978-3-540-39526-3

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