Robustness of pole location in perturbed systems

Abstract In this paper we present some results on robustness of location of roots of polynomials in given regions of the complex plane for unknown but bounded perturbations on the polynomial coefficients. A geometric approach in coefficient space is exploited to derive maximal deviations (in a given class of admissible perturbations) of characteristic polynomial coefficients of an uncertain linear system from their nominal values preserving system poles in a given region of the complex plane. It is also shown that the solution of this problem can be used to give computationally feasible necessary and sufficient conditions such that all the roots of the members of a family of polynomials lie in a given open region of the complex plane. This last result can be considered an extension of the result of the well-known theorem of Kharitonov. It is also outlined how the proposed technique can be used to deal with families of polynomials with linearly correlated coefficient perturbations.

[1]  Antonio Vicino,et al.  Maximal polytopic stability domains in parameter space for uncertain systems , 1989 .

[2]  Shankar P. Bhattacharyya,et al.  Robust Stabilization Against Structured Perturbations , 1987 .

[3]  B. Anderson,et al.  On robust Hurwitz polynomials , 1987 .

[4]  James S. Meditch,et al.  A canonical parameter space for linear systems design , 1978 .

[5]  Ezra Zeheb,et al.  On robust Hurwitz and Schur polynomials , 1986 .

[6]  J. Ackermann.  Parameter space design of robust control systems , 1980 .

[7]  Jürgen Garloff,et al.  Convex combinations of stable polynomials , 1985 .

[8]  B. Barmish.  Invariance of the strict Hurwitz property for polynomials with perturbed coefficients , 1983, The 22nd IEEE Conference on Decision and Control.

[9]  J. Dieudonné,et al.  La théorie analytique des polynomes d'une variable (à coefficients quelconques) , 1938 .

[10]  C. B. Soh,et al.  On the stability properties of polynomials with perturbed coefficients , 1985 .

[11]  S. Bhattacharyya,et al.  Robust stability with structured real parameter perturbations , 1987 .

[12]  J. Garloff,et al.  Stability of polynomials under coefficient perturbation , 1985 .

[13]  Antonio Vicino,et al.  Some results on robust stability of discrete-time systems , 1988 .

[14]  Christopher V. Hollot,et al.  Stability of families of polynomials: geometric considerations in coefficient space , 1987 .

[15]  W. Karl,et al.  Comments on ‘ A necessary and sufficient condition for the stability of interval matrices ’† , 1984 .

[16]  C. B. Soh,et al.  Addendum to "On the stability properties of polynomials with perturbed coefficients" , 1987 .

[17]  A. Vicino,et al.  Robustness analysis for uncertain dynamical systems with structured perturbations , 1988, Proceedings of the 27th IEEE Conference on Decision and Control.

[18]  Christopher V. Hollot,et al.  Root Locations of an Entire Polytope of Polynomials: It Suffices to Check the Edges , 1987 .

[19]  J. Guiver,et al.  Strictly Hurwitz property invariance of quartics under coefficient perturbation , 1983 .