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Authors
Rodriguez-Angeles, E.; Torres-Munoz, J. A.; Mendez-Barrios, C. F.

Digital Object Identifier (DOI)
10.3182/20071017-3-BR-2923.00036

Page Numbers:
214-219

Index Terms
stability criteria,bivariate polynomials,bilinear systems

Abstract
An algorithmic criterion is proposed for checking the stability of a bivariate polynomial in the sense of the so-called Stable class. For this, a novel graphical test for the Schur Stable bivariate class is introduced. The approach relies on the fact that both the Schur Stable and the Stable bivariate classes are nicely related by a bilinear transformation.

References

[1] Basu, S. and A. Fettweis (1987). New results on
    stable multidimensional polynomials part II:
    Discrete case. IEEE Transactions on Circuits
    and Systems 34, 1264-1274.
[2] Bhattacharyya, S.P., H. Chapellat and L.H. Keel
    (1995). Robust Control: The Parametric Approach
    . Prentice Hall.
[3] Bistritz, Y. (1999). Stability testing of two-dimensional
     discrete linear systems polynomials
     by two-dimensional tabular form.
    IEEE Transactions on Circuits and Systems-I
     46, 666-676.
[4] Bistritz, Y. (2002). On testing stability of 2-d
    discrete systems by a finite collection of 1-d
    stability tests. IEEE Transactions on Circuits
    and Systems-I, 49, 1634-1638.
[5] Bose, N.K. (1982). Applied Multidimensional Systems
     Theory. Van Nostrand Reinhold Company.
    
[6] Fettweis, A. and S. Basu (1987). New results
    on stable multidimensional polynomials part
    i: Continuous case. IEEE Transactions on
    Circuits and Systems 34, 1221-1232.
[7] Huang, T.S. (1972). Stability of two-dimensional
    recursive digital filters. IEEE Transactions on
    Audio and Electroacoustics 20, 158-163.
[8] Jury, E.I. (1965). A modified stability table for
    linear discrete systems. Proceedings of the
    IEEE 53, 184-185.
[9] Jury, E.I. (1988). Modified stability table for 2-d
    digital filters. IEEE Transactions on Circuits
    and Systems-I 35, 116-119.
[10] Kharitonov, V.L. and J.A. Torres-Muñoz (1999).
     Robust stability of multivariate polynomials
     part 1: Small coefficient perturbations. Multidimensional
      Systems and Signal Processing
     10, 7-20.
[11] Parks, P.C. and V. Hahn (1992). Stability Theory.
     Prentice Hall.
[12] Torres-Muñoz, J.A., E. Rodríguez-Angeles and
     V.L. Kharitonov (2006). On schur stable multivariate
      polynomials. IEEE Transactions on
     Circuits and Systems-I 53, 1166-1173.
[13] Zeheb, E. and E. Walach (1981). Zero sets of
     multiparameter functions and stability of
     multidimensional systems. IEEE Transactions
     on Acoustic, Speech and Signal Processing
     29, 197-206.