Abstract
The problem of checking robust stability of interval matrices has been proved to be NP-hard. However a closely related problem can be effectively solved in the framework of the stochastic approach [1]. Moreover the deterministic interval robust stability radius happens to be very conservative for large dimensions from the probabilistic point of view.
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References
B. R. Barmish, B. T. Polyak, A new approach to open robustness problems based on probabilistic prediction formulae, Proceedings of the 13th IFAC World Congress,San-Francisco,CA,July 1996, Vol. H, p. 1–6.
B. R. Barmish, C. V. Hollot, Counter-example to a recent result on the stability of interval matrices by S. Bialas, Intern. Journ. Contr., 39, p. 1103–1104, 1984.
J. D. Cobb, C. L. DeMarco, The minimal dimension of stable faces to guarantee stability of a matrix polytope, IEEE Trans. Autom. Contr., 34, p. 1990–1992, 1989.
A. S. Nemirovskii, Several NP-hard problems arising in robust stability analysis, Math. Contr.,Sign., Syst., 6, 1, p. 99–105, 1994.
L. H. Keel, S. P. Bhattacharyya, Robust stability of interval matrices: a computational approach, Intern. Journ. Contr., 62, 6, p. 1491–1506, 1995.
A. Geman, A limit theorem for the norm of random matrices, Ann. Prob., 8, 2, p. 252–261, 1980.
L. Qiu, B. Bernhardsson, A. Rantzer, E. J. Davison, P. M. Young, J. C. Doyle, A formula for computation of the real stability radius, Automatica, 31, 6, p. 878–890, 1995.
Y. Q. Yin, Z. D. Bai, P. R. Krishnaiah, On the limit of the largest eigenvalue of the large dimensional sample covariance matrix, Prob. Theory and Related Fields, 78, 4, p. 509–521, 1988.
V. L. Girko, The circular law, Prob. Theory and Appl., 29, 4, p. 669–679, 1984.
V. L. Girko, The Circular Low: ten years later, Random Oper. and Stoch. Equ., 2, 3, p. 235–276, 1994.
B. R. Barmish, C. M. Lagoa, The uniform distribution: a rigorous justification for its use in robustness analysis, Proceedings of the 35th CDC, Kobe, Japan, December 1996.
B. Shafai, M. Kothandaraman, J. Chen, Real and complex stability radii for nonnegative and Metzlerian systems, Proceedings of the 32nd CDC, San-Antonio, TX, p. 3482–3484, 1993.
S. P. Bhattacharyya, H. Chapellat, L. H. Keel, Robust Control: the Parametric Approach, Prentice Hall, 1995.
N. K. Son, D. Hinrichsen, Robust stability of positive continuous time systems, Numer. Fùnct. Anal. and Optimiz., 17, 5 /6, p. 649–659, 1996.
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Polyak, B.T. (1998). Robust Stability of Interval Matrices: a Stochastic Approach. In: Marti, K., Kall, P. (eds) Stochastic Programming Methods and Technical Applications. Lecture Notes in Economics and Mathematical Systems, vol 458. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45767-8_12
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DOI: https://doi.org/10.1007/978-3-642-45767-8_12
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