Eigenvector Sensitivity: A Kharitonov Result

DOI: 10.4236/ajcm.2013.32024   PDF   HTML   XML   5,095 Downloads   7,889 Views  


This study is motivated by a need to effectively determine the difference between a system fault and normal system operation under parametric uncertainty using eigenstructure analysis. This involves computational robustness of eigenvectors in linear state space systems dependent upon uncertain parameters. The work involves the development of practical algorithms which provide for computable robustness measures on the achievable set of eigenvectors associated with certain state space matrix constructions. To make connections to a class of systems for which eigenvalue and characteristic root robustness are well understood, the work begins by focusing on companion form matrices associated with a polynomial whose coefficients lie in specified intervals. The work uses an extension of the well known theories of Kharitonov that provides computational efficient tests for containment of the roots of the polynomial (and eigenvalues of the companion matrices) in desirable regions, such as the left half of the complex plane.

Share and Cite:

V. Winstead, "Eigenvector Sensitivity: A Kharitonov Result," American Journal of Computational Mathematics, Vol. 3 No. 2, 2013, pp. 158-168. doi: 10.4236/ajcm.2013.32024.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] R. Smith, “Eigenvalue Perturbation Models for Robust Control,” IEEE Transactions on Automatic Control, Vol. 40, No. 6, 1995, pp. 1063-1066. doi:10.1109/9.388684
[2] T. Alt and F. Jabbari, “Robustness Bounds for Linear Systems under Uncertainty: Eigenvalues Inside a Wedge,” Journal of Guidance, Control, and Dynamics, Vol. 16, No. 4, 1993, pp. 695-701. doi:10.2514/3.21069
[3] S. Wang, “Robust Schur Stability and Eigenvectors of Uncertain Matrices,” Proceedings of the American Control Conference, Vol. 5, 1997, pp. 3449-3454.
[4] P. Michelberger, P. Varlaki, A. Keresztes and J. Bokor, “Design of Active Suspension System for Road Vehicles: An Eigenstructure Assignment Approach,” Proceedings of the 23rd Fédération Internationale des Sociétés d’Ingénieurs des Techniques de l’Automobile (FISITA) World Automotive Congress, Torino, 1990, pp. 213-218.
[5] D. Hinrichson and A. J. Pritchard, “Mathematical Systems Theory I: Modelling, State Space Analysis, Stability and Robustness,” Springer, Berlin, 2005.
[6] S.-G. Wang and S. Lin, “Eigenvectors and Robust Stability of Uncertain Matrices,” International Journal of Control and Intelligent Systems, Vol. 30, No. 3, 2002, pp. 126-133.
[7] R. Patton and J. Chen, “On Eigenstructure Assignment for Robust Fault Diagnosis,” International Journal of Robust and Nonlinear Control, Vol. 10, No. 14, 2000, pp. 1193-1208. doi:10.1002/1099-1239(20001215)10:14<1193::AID-RNC523>3.0.CO;2-R
[8] T. Kato, “Perturbation Theory for Linear Operators,” Springer-Verlag, New York, 1966. doi:10.1007/978-3-642-53393-8
[9] J. H. Wilkinson, “The Algebraic Eigenvalue Problem,” Oxford University Press, Oxford, 1965.
[10] B. Barmish, “New Tools for Robustness of Linear Systems,” Macmillan Publishing Company, New York, 1994.
[11] C. Goffman, “Calculus of Several Variables,” Harper and Row Publishers, New York, 1965.
[12] D. Delchamps, “State Space and Input-Output Linear Systems,” Springer-Verlag, New York, 1988. doi:10.1007/978-1-4612-3816-4
[13] S. Eisenstat and I. Ipsen, “Three Absolute Perturbation Bounds for Matrix Eigenvalues Imply Relative Bounds,” SIAM Journal on Matrix Analysis and Applications, Vol. 20, No. 1, 1998, pp. 149-158. doi:10.1137/S0895479897323282
[14] F. Bazán, “Matrix Polynomials with Partially Prescribed Eigenstructure: Eigenvalue Sensitivity and Condition Estimation,” Computational and Applied Mathematics, Vol. 24, No. 3, 2005, pp. 365-392. doi:10.1590/S0101-82052005000300003
[15] A. Edelman and H. Murakami, “Polynomial Roots From Companion Matrix Eigenvalues,” Mathematics of Computation, Vol. 64, No. 210, 1995, pp. 763-776. doi:10.1090/S0025-5718-1995-1262279-2

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.