Robust Stability of D-symmetrizable Hyperbolic Systems

• Autorzy: Ziółko M. , Białas S.
• Języki publikacji: EN

Abstrakty

EN

The systems under consideration are governed by a set of first-order linear partial differential hyperbolic equations together with boundary conditions. The Lyapunov method is used to verify the stability of the initial-boundary value problem. Necessary and sufficient conditions for stability are obtained under the assumption that the matrix coefficients in the differential equations and in the boundary conditions are D-symmetrizable. The considered systems have an interesting property: Hurwitz type stability and Schur type stability occur in one system simultaneously. The stability of the conditions type system is a stability of wave propagation. The stability of the discrete type system is a stability of the boundary feedback and the boundary reflections. Necessary and sufficient conditions for the robust stability of an initial-boundary value problem are obtained for the case where the matrix coefficients belong to a convex hull of stable and D-symmetrizable matrices.

Słowa kluczowe

EN

robust stability; hyperbolic equations; Hurwitz stability; Schur stability; symmetrizable matrices

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2001
• Tom: vol. 49, nr 1
• Strony: 167--176
• Opis fizyczny: bibliogr. 13 poz.

Twórcy

autor

Ziółko M.

autor

Białas S.

• Department of Electronics, University of Mining and Metallurgy, Al. Mickiewicza 30, 30-059 Kraków, Poland (Wydział Elektroniki Akademii Górniczo-Hutniczej)

Bibliografia

• [1] S. Białas, A necessary and sufficient condition for the stability of convex combinations of stable polynomials or matrices, Bull, Pol. Ac: Tech, 33 (1985) 473-480.
• [2] S. Białas, M. Ziółko, Necessary and sufficient condition for robust D-symmetrizability of matrices, Bull. Pol. Ac.: Tech., 49, 1, (2001) 177-182.
• [3] N. Cohen, T. Lewkowicz, A necessary and sufficient criterion for the stability of a convex set of matrices, IEEE Trans. Automat. Contr., AC-38 (1993) 611-615.
• [4] R. Courant, D. Hilbert, Methods of mathematical physics, John Wiley & Sons, New York 1962.
• [5] L. H. Keel, Bhattacharyya, Robust stability of interval matrices, Int. J. Cont., 40, 6, (1995) 1491-1506.
• [6] H. K. Khalil, On the existence of positive diagonal P such that PA + ATP < 0, IEEE Trans. Automat. Contr., AC-27 (1982) 181-184.
• [7] V. Lakshmikantham, S. Leela, Nonlinear differential equations in abstract spaces, Pergamon Press, Oxford 1981.
• [8] J. Rohn, Stability of interval matrices: the real eigenvalue case, IEEE Trans. Automat. Contr., AC-37 (1992) 1604-1605.
• [9] J. Rohn, An algorithm for checking stability of symmetric interval matrices, IEEE Trans. Automat. Contr., AC-41 (1996) 133-136.
• [10] R. X. Qian, C. L. DeMarco, An approach to robust stability of matrix polytopes through copositive homogeneous polynomials, IEEE Trans. Automat. Contr., AC-37 (1992) 848-852.
• [11] D. L. Russell, Quadratic performance criteria in boundary control of linear symmetric hyperbolic systems, SIAM J. Contr., 11 (1973) 475-509.
• [12] K. Wang, A. N. Michel, D. Liu, Necessary and sufficient conditions for the Hurwitz and Schur stability of interval matrices, IEEE Trans. Automat Contr., AC-39 (1994) 1251-1255.
• [13] M. Ziółko, Application of Lyapunow functional to studying stability of linear hyperbolic systems, IEEE Trans. Automat. Contr., AC-35 (1990) 1173-1176.