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A novel LMI-based optimization algorithm for the guaranteed estimation of the domain of attraction using rational Lyapunov functions. (English) Zbl 1269.34062
Summary: In this article we deal with the classical problem of estimating the domain of attraction (DOA) of autonomous dynamical systems. We propose a new LMI estimation method based on recent results from the mathematical theory of moments. In contrast to previous works we exploit the advantages of rational Lyapunov functions to enhance the estimates. Several examples illustrate the estimation method.
Reviewer: Reviewer (Berlin)

MSC:
34D20 Stability of solutions to ordinary differential equations
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
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[1] Lyapunov, A., Problème général de la stabilité du mouvement, Ann. fac. sci. Toulouse, 9, 203-474, (1907), (Translation of a paper published in Comm. Soc. math. Kharkow 1893, in Russian). · JFM 38.0738.07
[2] Hahn, W., Stability of motion, die grundlehren der mathematischen wissenschaften, band 138, (1967), Springer Berlin
[3] Khalil, H.K., Nonlinear systems, (1992), MacMillan New York · Zbl 0626.34052
[4] G. Chesi, A. Tesi, A. Vicino, Computing optimal quadratic Lyapunov functions for polynomial nonlinear systems via LMIs, Proceedings of the 15th IFAC Trienial World Congress, Barcelona, Spain, 2002.
[5] Chiang, H.D.; Thorp, J.S., Stability regions of nonlinear dynamical systems: a constructive methodology, IEEE trans. automat. control, 34, 12, 1229-1241, (1989) · Zbl 0689.93046
[6] K. Forsman, Constructive commutative algebra in nonlinear control theory, Linköping studies in science and technology, Dissertations, No. 261, 1991.
[7] Genesio, R.; Tartaglia, M.; Vicino, A., On the estimation of aymptotic stability regions: state of the art and new proposals, IEEE trans. automat. control, 30, 8, 747-755, (1985) · Zbl 0568.93054
[8] O. Hachicho, B. Tibken, Estimating domains of attraction of a class of nonlinear dynamical systems with LMI methods based on the theory of moments, Proceedings of the 41st Conference on Decision and Control CDC 2002, Las Vegas, Nevada, 2002, pp. 3150-3155.
[9] Hewit, J.R.; Storey, C., Comparison of numerical methods in stability analysis, Int. J. control, 10, 6, 687-701, (1969) · Zbl 0184.18603
[10] Levin, A., An analytical method for estimating the domain of attraction for polynomial differential equations, IEEE trans. automat. control, 36, 12, 2471-2475, (1994) · Zbl 0825.93680
[11] Michel, A.N.; Sarabudla, N.R.; Miller, R.K., Stability analysis of complex dynamical systems—some computational methods, Circuits systems signal process., 1, 2, 171-202, (1982) · Zbl 0493.93040
[12] Tesi, A.; Villoresi, F.; Genesio, R., On the stability domain estimation via a quadratic Lyapunov function: convexity and optimality properties for polynomial systems, IEEE trans. automat. control, 41, 1650-1657, (1996) · Zbl 0870.34057
[13] Tibken, B.; Hofer, E.P.; Demir, C., Guaranteed regions of attraction for dynamical polynomial systems, (), 119-128
[14] B. Tibken, Estimation of the domain of attraction for polynomial systems via LMI’s, Proceedings of the IEEE Conference on Decision and Control, CDC’00, Sydney, Australia, December 12-15, 2000, pp. 3860-3864.
[15] B. Tibken, O. Hachicho, Estimation of the domain of attraction for polynomial systems using multidimensional grids, Proceedings of the IEEE Conference on Decision and Control, CDC’00, 2000, pp. 3870-3874.
[16] Vannelli, A.; Vidysagar, M., Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems, Automatica, 21, 1, 69-80, (1985) · Zbl 0559.34052
[17] Zubov, V.I., Questions of the theory of Lyapunov’s second method, construction of the general solution in the region of asymptotical stability, Prikl. mat. meh., 19, 179-210, (1955)
[18] O. Hachicho, Stability analysis of polynomial dynamical systems with semidefinite optimization, Doctoral Dissertation, University of Wuppertal, Der Andere Verlag, Osnabrück, Germany, 2003.
[19] Aneke, S.J., Mathematical modelling of drug resistant malaria parasites and vector populations, Math. meth. appl. sci., 25, 4, 335-346, (2002) · Zbl 0994.92025
[20] J.M. Epstein, Nonlinear dynamics, mathematical biology, and social science, Santa Fe Institute Studies in the Sciences of Complexity, Lecture Notes, vol. IV, Advanced Book Program, Addison-Wesley Publishing Company, Reading, MA, 1997.
[21] FitzHugh, R., Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1, 445-466, (1961)
[22] Hénon, M.; Heiles, C., The applicability of the third integral of motion: some numerical experiments, Astron. J., 69, 1, 73-79, (1964)
[23] Hindmarsh, J.L.; Rose, R.M., A model of the nerve impulse using two first-order differential equations, Nature, 296, 162-164, (1982)
[24] Keener, J.; Sneyd, J., Mathematical physiology, (1998), Springer New York, Berlin, Heidelberg · Zbl 0913.92009
[25] Koch, Ch., Biophysics of computation, (1999), Oxford University Press Oxford
[26] R.A. Meyers (Ed.) in chief, Encyclopedia of Physical Science and Technology, Academic Press, San Diego, 2002.
[27] O. Föllinger, Nichtlineare Regelungen I, Oldenbourg Verlag, 1993.
[28] U. Sieber, Ljapunow-Synthese nichtlinearer Systeme durch Gütemass-angleichung, VDI-Verlag, 1990.
[29] Bronstein, I.; Semendjajew, K.A.; Musiol, G.; Mühlig, H., Taschenbuch der Mathematik, (2001), Verlag Harri Deutsch Thun und Frankfurt am Main · Zbl 1121.00301
[30] F. Alizadeh, J.P. Haeberly, M.V. Nayakkankuppam, M.L. Overton, SDPPack user’s guide. \(\langle\)http://www.cs.nyu.edu/faculty/overton/sdppack/sdppack.html⟩, 1997.
[31] Boyd, S.; El Ghaoui, S.L.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in systems and control theory, SIAM, Philadelphia, 1994, invariance under coefficient perturbation, IEEE trans. acoust. speech signal process, 28, 6, 660-665, (1980)
[32] Y. Nesterov, A. Nemirovskii, Interior point polynomial algorithms in convex programming, Studies in Applied Mathematics, vol. 13, SIAM, Philadelphia, 1994. · Zbl 0824.90112
[33] P.A. Parrilo, Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization, Ph.D. Thesis at California Institute of Technology Pasadena, California, 2000.
[34] P.A. Parrilo, On a decomposition of multivariable forms via LMI methods, Proceedings of the American Control Conference Chicago, 2000, pp. 322-326.
[35] J.F. Sturm, Using SeDuMi 1.0x, a Matlab Toolbox for optimization over symmetric cones. \(\langle\)http://fewcal.kub.nl/sturm/software/sedumi.html\(\rangle\), 1999. · Zbl 0973.90526
[36] Vandenberghe, L.; Boyd, S., Semidefinite programming, SIAM rev., 38, 1, 49-95, (1996) · Zbl 0845.65023
[37] Davison, E.J.; Kurak, E.M., A computational method for determining quadratic Lyapunov functions for non-linear systems, Automatica, 7, 627-636, (1971) · Zbl 0225.34027
[38] Lasserre, J.B., Global optimization with polynomials and the problem of moments, SIAM J. optim., 11, 3, 796-817, (2001) · Zbl 1010.90061
[39] La Salle, J.; Lefschetz, S., Stability by Liapunov’s direct method, (1961), Academic Press New York, London · Zbl 0098.06102
[40] D. Henrion, J.B. Lasserre, Gloptipoly: global optimization over polynomials with Matlab and SeDuMi. \(\langle\)http://www.laas.fr/henrion/⟩, 2002. · Zbl 1070.65549
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