zbMATH — the first resource for mathematics

Local stability analysis using simulations and sum-of-squares programming. (English) Zbl 1155.93417
Summary: The problem of computing bounds on the region-of-attraction for systems with polynomial vector fields is considered. Invariant subsets of the region-of-attraction are characterized as sublevel sets of Lyapunov functions. Finite-dimensional polynomial parametrizations for Lyapunov functions are used. A methodology utilizing information from simulations to generate Lyapunov function candidates satisfying necessary conditions for bilinear constraints is proposed. The suitability of Lyapunov function candidates is assessed solving linear sum-of-squares optimization problems. Qualified candidates are used to compute invariant subsets of the region-of-attraction and to initialize various bilinear search strategies for further optimization. We illustrate the method on small examples from the literature and several control oriented systems.

93D30 Lyapunov and storage functions
90C22 Semidefinite programming
Full Text: DOI
[1] Boyd, S.; Vandenberghe, L., Convex optimization, (2004), Cambridge Univ. Press · Zbl 1058.90049
[2] Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A., \sclmi-based computation of optimal quadratic \sclyapunov functions for odd polynomial systems, International journal of robust nonlinear control, 15, 35-49, (2005) · Zbl 1056.93059
[3] Chiang, H.-D.; Thorp, J.S., Stability regions of nonlinear dynamical systems: A constructive methodology, IEEE transactions on automatic control, 34, 12, 1229-1241, (1989) · Zbl 0689.93046
[4] Davison, E.J.; Kurak, E.M., A computational method for determining quadratic \sclyap. functions for nonlinear systems, Automatica, 7, 627-636, (1971) · Zbl 0225.34027
[5] Genesio, R.; Tartaglia, M.; Vicino, A., On the estimation of asymptotic stability regions: state of the art and new proposals, IEEE transactions on automatic control, 30, 8, 747-755, (1985) · Zbl 0568.93054
[6] Hachicho, O., & Tibken, B. (2002). Estimating domains of attraction of a class of nonlinear dynamical systems with LMI methods based on the theory of moments. In Proc. CDC (pp. 3150-3155)
[7] Hauser, J., & Lai, M. C. (1992). Estimating quadratic stability domains by nonsmooth optimization. In Proc. ACC (pp. 571-576)
[8] Koc˘vara, M., & Stingl, M. (2005). PENBMI user’s guide
[9] Papachristodoulou, A. (2005). Scalable analysis of nonlinear systems using convex optimization. Ph.D. dissertation. Caltech
[10] Parrilo, P., Semidefinite programming relaxations for semialgebraic problems, Mathematical programming series B, 96, 2, 293-320, (2003) · Zbl 1043.14018
[11] Prokhorov, D. V., & Feldkamp, L. A. (1999). Application of \scSVM to \scLyapunov function approximation. In Proc. int. joint conf. on neural networks
[12] Serpen, G. (2005). Search for a Lyapunov function through empirical approximation by artificial neural nets: Theoretical framework. In Proc. int. joint conf. on artificial neural networks. (pp. 735-740)
[13] Sturm, J., Using \scse\scdu\scmi 1.02, a \scmatlab toolbox for optimization over symmetric cones, Optimization methods and software, 11, 625-653, (1999)
[14] Tan, W. (2006). Nonlinear control analysis and synthesis using sum-of-squares programming. Ph.D. dissertation. UC, Berkeley
[15] Tan, W., & Packard, A. (2006). Stability region analysis using sum of squares programming. In Proc. ACC (pp. 2297-2302)
[16] Tempo, R.; Calafiore, G.; Dabbene, F., Randomized algorithms for analysis and control of uncertain systems, (2005), Springer · Zbl 1079.93002
[17] Tibken, B. (2000). Estimation of the domain of attraction for polynomial systems via LMIs. In Proc. CDC (pp. 3860-3864)
[18] Tibken, B., & Fan, Y. (2006). Computing the domain of attraction for polynomial systems via BMI optimization methods. In Proc. ACC (pp. 117-122)
[19] Toker, O., & Ozbay, H. (1995). On the \scNP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback. In Proc. ACC (pp. 2525-2526)
[20] Vannelli, A.; Vidyasagar, M., Maximal \sclyapunov functions and domains of attraction for autonomous nonlinear systems, Automatica, 21, 1, 69-80, (1985) · Zbl 0559.34052
[21] Vidyasagar, M., Nonlinear systems analysis, (1993), Prentice Hall · Zbl 0900.93132
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.