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LMI-based computation of optimal quadratic Lyapunov functions for odd polynomial systems. (English) Zbl 1056.93059
Summary: The problem of estimating the domain of attraction (DA) of equilibria is considered for odd polynomial systems. Specifically, the computation of the optimal quadratic Lyapunov function (OQLF), i.e. the quadratic Lyapunov function (QLF) which maximizes the volume of the largest estimate of the DA (LEDA), is addressed. In order to tackle this double non-convex optimization problem, a relaxation approach based on homogeneous polynomial forms is proposed. The first contribution of the paper shows that a lower bound of the LEDA for a fixed QLF can be obtained via linear matrix inequalities (LMIs) based procedures. Also, condition for checking tightness of the lower bound are provided. The second contribution is a strategy for selecting a good starting point for the OQLF search, which is based on the volume maximization of the region where the time derivative of the QLFs is negative and is given in terms of LMIs. Several application examples are presented to illustrate the numerical behaviour of the proposed approach.

93D30 Lyapunov and storage functions
93B40 Computational methods in systems theory (MSC2010)
Full Text: DOI
[1] Nonlinear Systems. McMillan: New York; 1992.
[2] Genesio, IEEE Transactions on Automatic Control 30 pp 747– (1985)
[3] Davison, Automatica 7 pp 627– (1971)
[4] Michel, Circuit Systems and Signal Processing 1 pp 561– (1982) · Zbl 0493.93040 · doi:10.1007/BF01600051
[5] Optimal ellipsoidal stability domain estimates for odd polynomial systems. Proceedings of the 36th IEEE CDC, San Diego, December 1997; 3528-3529.
[6] New convexification techniques and their applications to the analysis of dynamic systems and vision problems. Ph.D. Thesis, Università di Bologna, 2001.
[7] Chesi, IEEE Transactions on Automatic Control 48 pp 200– (2003)
[8] Lie algebra and Lie groups in control theory. In Geometric Methods in Systems Theory, (eds). Reidel: Dordrecht, 1973; 43-82. · doi:10.1007/978-94-010-2675-8_2
[9] Applied Multidimensional Systems Theory. Van Rostrand Reinhold: New York, 1982. · Zbl 0574.93031
[10] Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. Ph.D. Thesis, California Institute of Technology, 2000.
[11] Linear Matrix Inequalities in System and Control Theory. SIAM: Philadelphia, 1994. · Zbl 0816.93004 · doi:10.1137/1.9781611970777
[12] Inequalities (2nd edn). Cambridge University Press: Cambridge, 1988. · Zbl 0634.26008
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