zbMATH — the first resource for mathematics

Estimating the domain of attraction via union of continuous families of Lyapunov estimates. (English) Zbl 1109.37012
Summary: This paper proposes a new approach to estimate the domain of attraction of equilibrium points of polynomial systems. The idea consists of estimating the domain of attraction via the union of a continuous family of Lyapunov estimates rather than via one Lyapunov estimate only as done in existing methods. This family is obtained through a convex LMI optimization by deriving a stability condition which takes simultaneously into account all Lyapunov functions considered. Moreover, inner approximations to the union of this family via a set with simple shape are derived, too.

37B25 Stability of topological dynamical systems
90C25 Convex programming
Full Text: DOI
[1] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory, (1994), SIAM Philadelphia · Zbl 0816.93004
[2] Chesi, G., Computing output feedback controllers to enlarge the domain of attraction in polynomial systems, IEEE trans. automat. control, 49, 10, 1846-1850, (2004) · Zbl 1365.93204
[3] Chesi, G., Estimating the domain of attraction for uncertain polynomial systems, Automatica, 40, 11, 1981-1986, (2004) · Zbl 1067.93055
[4] Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A., Solving quadratic distance problems: an LMI-based approach, IEEE trans. automat. control, 48, 2, 200-212, (2003) · Zbl 1364.90240
[5] Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A., LMI-based computation of optimal quadratic Lyapunov functions for odd polynomial systems, Internat. J. nonlinear robust control, 15, 1, 35-49, (2005) · Zbl 1056.93059
[6] M. Choi, T. Lam, B. Reznick, Sums of squares of real polynomials, in: Proceedings of Symposia in Pure Mathematics, 1995, pp. 103-126. · Zbl 0821.11028
[7] Genesio, R.; Tartaglia, M.; Vicino, A., On the estimation of asymptotic stability regions: state of the art and new proposals, IEEE trans. automat. control, 30, 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, in: Proceedings of 41st IEEE Conference on Decision and Control, Las Vegas, Nevada, 2002, pp. 3150-3155.
[9] Z. Jarvis-Wloszek, R. Feeley, W. Tan, K. Sun, A. Packard, Some controls applications of sum of squares programming, in: Proceedings of 42nd IEEE Conference on Decision and Control, Maui, Hawaii, 2003, pp. 4676-4681.
[10] Khalil, H.K., Nonlinear systems, (2001), Prentice-Hall Englewood Cliffs, NJ · Zbl 0626.34052
[11] J. Löfberg, YALMIP: a toolbox for modeling and optimization in MATLAB, in: Proceedings of the CACSD Conference, Taipei, Taiwan, 2004, available from \(\langle\)http://control.ee.ethz.ch/\(\sim\)joloef/yalmip.php⟩
[12] Nesterov, Y.; Nemirovsky, A., Interior point polynomial methods in convex programming: theory and applications, (1993), SIAM Philadelphia
[13] P.A. Parrilo, Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization, Ph.D. Thesis, California Institute of Technology, 2000.
[14] Sturm, J.F., Using sedumi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optim. methods software, 11-12, 625-653, (1999) · Zbl 0973.90526
[15] B. Tibken, Estimation of the domain of attraction for polynomial systems via LMI’s, in: Proceedings of 39th IEEE Conference on Decision and Control, Sydney, Australia, 2000, pp. 3860-3864.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.