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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 396759, 11 pages
Stability of Matrix Polytopes with a Dominant Vertex and Implications for System Dynamics
Department of Automatic Control and Applied Informatics, Technical University “Gheorghe Asachi” of Iasi, Boulevard Mangeron 27, 700050 Iasi, Romania
Received 1 November 2012; Accepted 21 March 2013
Academic Editor: Gani Stamov
Copyright © 2013 Octavian Pastravanu and Mihaela-Hanako Matcovschi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- S. Białas, “A necessary and sufficient condition for the stability of convex combinations of stable polynomials or matrices,” Bulletin of the Polish Academy of Sciences, vol. 33, no. 9-10, pp. 473–480, 1985.
- B. R. Barmish, M. Fu, and S. Saleh, “Stability of a polytope of matrices: counterexamples,” IEEE Transactions on Automatic Control, vol. 33, no. 6, pp. 569–572, 1988.
- M. Fu and B. R. Barmish, “Maximal unidirectional perturbation bounds for stability of polynomials and matrices,” Systems & Control Letters, vol. 11, no. 3, pp. 173–179, 1988.
- C. B. Soh, “Schur stability of convex combination of matrices,” Linear Algebra and its Applications, vol. 128, pp. 159–168, 1990.
- Q. G. Wang, “Necessary and sufficient conditions for stability of a matrix polytope with normal vertex matrices,” Automatica, vol. 27, no. 5, pp. 887–888, 1991.
- N. Cohen and I. Lewkowicz, “A necessary and sufficient criterion for the stability of a convex set of matrices,” IEEE Transactions on Automatic Control, vol. 38, no. 4, pp. 611–615, 1993.
- V. V. Monov, “On the spectrum of convex sets of matrices,” IEEE Transactions on Automatic Control, vol. 44, no. 5, pp. 1009–1012, 1999.
- Z. Zahreddine, “Matrix measure and application to stability of matrices and interval dynamical systems,” International Journal of Mathematics and Mathematical Sciences, vol. 2, pp. 75–85, 2003.
- P.-A. Bliman, “A convex approach to robust stability for linear systems with uncertain scalar parameters,” SIAM Journal on Control and Optimization, vol. 42, no. 6, pp. 2016–2042, 2004.
- G. Chesi, “Establishing stability and instability of matrix hypercubes,” Systems & Control Letters, vol. 54, no. 4, pp. 381–388, 2005.
- L. Grman, D. Rosinová, V. Veselý, and A. Kozáková, “Robust stability conditions for polytopic systems,” International Journal of Systems Science, vol. 36, no. 15, pp. 961–973, 2005.
- B. T. Polyak and E. N. Gryazina, “Stability regions in the parameter space: -decomposition revisited,” Automatica, vol. 42, no. 1, pp. 13–26, 2006.
- L. Gurvits and A. Olshevsky, “On the NP-hardness of checking matrix polytope stability and continuous-time switching stability,” IEEE Transactions on Automatic Control, vol. 54, no. 2, pp. 337–341, 2009.
- L. Kolev and S. Petrakieva, “Assessing the stability of linear time-invariant continuous interval dynamic systems,” IEEE Transactions on Automatic Control, vol. 50, no. 3, pp. 393–397, 2005.
- L. X. Liao, “Necessary and sufficient conditions for stability of a class of interval matrices,” International Journal of Control, vol. 45, no. 1, pp. 211–214, 1987.
- S. H. Lin, Y. T. Juang, I. K. Fong, C. F. Hsu, and T. S. Kuo, “Dynamic interval systems analysis and design,” International Journal of Control, vol. 48, no. 5, pp. 1807–1818, 1988.
- J. Chen, “Sufficient conditions on stability of interval matrices: connections and new results,” IEEE Transactions on Automatic Control, vol. 37, no. 4, pp. 541–544, 1992.
- P. H. Bauer and K. Premaratne, “Time-invariant versus time-variant stability of interval matrix systems,” in Fundamentals of Discrete Time Systems: A Tribute to Professor E.I. Jury, M. Jamshidi, M. Mansour, and B. D. O. Anderson, Eds., pp. 181–188, TSI Press, Albuquerque, NM, USA, 1993.
- M. E. Sezer and D. D. Šiljak, “On stability of interval matrices,” IEEE Transactions on Automatic Control, vol. 39, no. 2, pp. 368–371, 1994.
- K. Wang, A. N. Michel, and D. R. Liu, “Necessary and sufficient conditions for the Hurwitz and Schur stability of interval matrices,” IEEE Transactions on Automatic Control, vol. 39, no. 6, pp. 1251–1255, 1994.
- D. Liu and A. Molchanov, “Criteria for robust absolute stability of time-varying nonlinear continuous-time systems,” Automatica, vol. 38, no. 4, pp. 627–637, 2002.
- A. P. Molchanov and D. Liu, “Robust absolute stability of time-varying nonlinear discrete-time systems,” IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications, vol. 49, no. 8, pp. 1129–1137, 2002.
- W.-J. Mao and J. Chu, “Quadratic stability and stabilization of dynamic interval systems,” IEEE Transactions on Automatic Control, vol. 48, no. 6, pp. 1007–1012, 2003.
- T. Alamo, R. Tempo, D. R. Ramírez, and E. F. Camacho, “A new vertex result for robustness problems with interval matrix uncertainty,” Systems & Control Letters, vol. 57, no. 6, pp. 474–481, 2008.
- G. P. Barker, A. Berman, and R. J. Plemmons, “Positive diagonal solutions to the Lyapunov equations,” Linear and Multilinear Algebra, vol. 5, no. 4, pp. 249–256, 1978.
- A. Berman and D. Hershkowitz, “Matrix diagonal stability and its implications,” Society for Industrial and Applied Mathematics, vol. 4, no. 3, pp. 377–382, 1983.
- E. Kaszkurewicz and A. Bhaya, Matrix Diagonal Stability in Systems and Computation, Birkhäuser, Boston, Mass, USA, 2000.
- O. Pastravanu and M. Voicu, “Generalized matrix diagonal stability and linear dynamical systems,” Linear Algebra and its Applications, vol. 419, no. 2-3, pp. 299–310, 2006.
- O. Pastravanu and M. H. Matcovschi, “Diagonal stability of interval matrices and applications,” Linear Algebra and its Applications, vol. 433, no. 8–10, pp. 1646–1658, 2010.
- M.-H. Matcovschi, O. Pastravanu, and M. Voicu, “On some properties of diagonally stable polytopic systems,” in Proceedings of the 15th International Conference on System Theory, Control and Computing (ICSTCC '11), pp. 353–358, 2011.
- M. Arcak and E. D. Sontag, “Diagonal stability of a class of cyclic systems and its connection with the secant criterion,” Automatica, vol. 42, no. 9, pp. 1531–1537, 2006.
- O. Mason and R. Shorten, “On the simultaneous diagonal stability of a pair of positive linear systems,” Linear Algebra and its Applications, vol. 413, no. 1, pp. 13–23, 2006.
- O. Pastravanu and M. H. Matcovschi, “Matrix measures in the qualitative analysis of parametric uncertain systems,” Mathematical Problems in Engineering, vol. 2009, Article ID 841303, 17 pages, 2009.
- H. K. Wimmer, “Diagonal stability of matrices with cyclic structure and the secant condition,” Systems & Control Letters, vol. 58, no. 5, pp. 309–313, 2009.
- M. Arcak, “Diagonal stability on cactus graphs and application to network stability analysis,” IEEE Transactions on Automatic Control, vol. 56, no. 12, pp. 2766–2777, 2011.
- W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath, Boston, Mass, USA, 1965.
- F. Blanchini and S. Miani, Set-Theoretic Methods in Control, Birkhäuser, Boston, Mass, USA, 2008.
- R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991.