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Guardian maps and the generalized stability of parametrized families of matrices and polynomials

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Abstract

The generalized stability of families of real matrices and polynomials is considered. (Generalized stability is meant in the usual sense of confinement of matrix eigenvalues or polynomial zeros to a prescribed domain in the complex plane, and includes Hurwitz and Schur stability as special cases.) Guardian maps and semiguardian maps are introduced as a unifying tool for the study of this problem. These are scalar maps which vanish when their matrix or polynomial argument loses stability. Such maps are exhibited for a wide variety of cases of interest corresponding to generalized stability with respect to domains of the complex plane. In the case of one- and two-parameter families of matrices or polynomials, concise necessary and sufficient conditions for generalized stability are derived. For the general multiparameter case, the problem is transformed into one of checking that a given map is nonzero for the allowed parameter values.

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Author information

Author notes

  1. Lahcen Saydy

    Present address: Department of Electrical Engineering, University of Maryland, 20742, College Park, Maryland, USA

  2. André L. Tits

    Present address: INRIA (project META2), Rocquencourt, France

Authors and Affiliations

  1. Département de Physique, Faculté des Sciences, Université Cadi Ayyad, Marrakech, Morocco

    Lahcen Saydy & Eyad H. Abed

  2. the Systems Research Center, University of Maryland, 20742, College Park, Maryland, USA

    André L. Tits

  3. the Lund Institute of Technology, Lund, Sweden

    André L. Tits & Eyad H. Abed

Authors
  1. Lahcen Saydy
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  2. André L. Tits
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  3. Eyad H. Abed
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Additional information

This research was supported in part by the National Science Foundation’s Engineering Research Centers Program, NSFD CDR 8803012, and was also supported by the NSF under Grants ECS-86-57561, DMC-84-51515, and by the Air Force Office of Scientific Research under Grant AFOSR-87-0073.

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Saydy, L., Tits, A.L. & Abed, E.H. Guardian maps and the generalized stability of parametrized families of matrices and polynomials. Math. Control Signal Systems 3, 345–371 (1990). https://doi.org/10.1007/BF02551375

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  • DOI: https://doi.org/10.1007/BF02551375

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