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A semi-algebraic approach for asymptotic stability analysis. (English) Zbl 1217.93153
Summary: This paper deals with the problem of computing Lyapunov functions for asymptotic stability analysis of autonomous polynomial systems of differential equations. We propose a new semi-algebraic approach by making advantage of the local property of the Lyapunov function as well as its derivative. This is done by first constructing a semi-algebraic system and then solving this semi-algebraic system in an adaptive way. Experimental results show that our semi-algebraic approach is more efficient in practice, especially for low-order systems.

93D20 Asymptotic stability in control theory
93B25 Algebraic methods
Full Text: DOI
[1] Nguyen, T.V.; Mori, T.; Mori, Y., Existence conditions of a common quadratic Lyapunov function for a set of second-order systems, Transactions of the society of instrument and control engineers, 42, 3, 241-246, (2006)
[2] Nguyen, T.V.; Mori, T.; Mori, Y., Relations between common Lyapunov functions of quadratic and infinity-norm forms for a set of discrete-time LTI systems, IEICE transactions on fundamentals of electronics, communications and computer sciences, E89-A, 6, 1794-1798, (2006)
[3] Tarski, A., A decision method for elementary algebra and geometry, (1951), Univ. of California Press Berkeley · Zbl 0044.25102
[4] Collins, G.E., Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition, (), 134-183
[5] Yang, L.; Xia, B., Automated deduction in real geometry, (), 248-298 · Zbl 1082.65054
[6] She, Z.; Xia, B.; Xiao, R., A semi-algebraic approach for the computation of Lyapunov functions, (), 7-12
[7] Hahn, W., Stability of motion, (1967), Springer · Zbl 0189.38503
[8] Henzinger, T.A.; Kopke, P.W.; Puri, A.; Varaiya, P., What’s decidable about hybrid automata, Journal of computer and system sciences, 57, 94-124, (1998) · Zbl 0920.68091
[9] S. Pettersson, B. Lennartson, An LMI approach for stability analysis of nonlinear systems, in: Proc. of the 4th European Control Conference, 1997
[10] Branicky, M.S., Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE transactions on automatic control, 43, 4, 475-482, (1998) · Zbl 0904.93036
[11] Decarlo, R.A.; Branicky, M.S.; Pettersson, S.; Lennartson, B., Perspectives and results on the stability and stabilizability of hybrid systems, Proceedings of the IEEE, 88, 7, (2000)
[12] Burchardt, H.; Oehlerking, J.; Theel, O., The role of state-space partitioning in automated verification of affine hybrid system stability, (), 187-192
[13] Collins, G.E.; Hong, H., Partial cylindrical algebraic decomposition for quantifier elimination, Journal of symbolic computation, 12, 299-328, (1991) · Zbl 0754.68063
[14] Rudin, W., Principles of mathematical analysis, (1976), McGraw-Hill · Zbl 0148.02903
[15] Wang, D.; Xia, B., Computer algebra, (2004), Tsinghua Univ. Press Beijing
[16] Parrilo, P.A., Semidefinite programming relaxations for semialgebraic problems, Mathematical programming series. B, 96, 2, 293-320, (2003) · Zbl 1043.14018
[17] Gallo, G.; Mishra, B., Efficient algorithms and bounds for Wu-Ritt characteristic sets, (), 119-142 · Zbl 0747.13019
[18] A. Papachristodoulou, S. Prajna, On the construction of Lyapunov functions using the sum of squares decomposition, in: Proc. of the IEEE Conf. on Decision and Control, 2002 · Zbl 1138.93391
[19] G. Ghesi, A. Tesi, A. Vicino, On the optimal quadratic Lyapunov functions for polynomial systems, in: 15 th Int. Symp. on Mathematical Theory of Networks and Systems, 2002
[20] K. Forsman, Optimization, stability and cylindrical decomposition, Technical report, Automatic Control Group in Linköping, 1993
[21] S. Prajna, A. Papachristodoulou, P.A. Parrilo, Introducing SOSTOOLS: A general purpose sum of squares programming solver, in: Proc. of the IEEE Conf. on Decision and Control, CDC, 2002
[22] K. Forsman, Construction of Lyapunov functions using Gröbner bases, in: Proc. of the 30th Conf. on Decision and Control, 1991, pp. 798-799
[23] Wang, D.; Xia, B., Stability analysis of biological systems with real solution classification, (), 354-361 · Zbl 1360.92048
[24] Shields, D.N.; Storey, C., The behaviour of optimal Lyapunov functions, International journal of control, 21, 4, 561-573, (1975) · Zbl 0318.93024
[25] Genesio, R.; Tartaglia, M.; Vicino, A., On the estimation of asympototic stability regions: state of the art and new proposals, IEEE transactions on automatic control, 30, 8, 747-755, (1985) · Zbl 0568.93054
[26] T.-C. Wang, S. Lall, M. West, Polynomial level-set methods for nonlinear dynamical systems analysis, in: Proceedings of the Allerton Conference on Communication, Control and Computing, 2005
[27] Lakshmikantham, V.; Leela, S.; Martynyuk, A., Practical stability of nonlinear systems, (1990), World Scientific · Zbl 0753.34037
[28] S. Ratschan, Z. She, Providing a basin of attraction to a target region by computation of Lyapunov-like functions, in: Proc. of the 4th IEEE Int. Conf. on Computational Cybernetics, 2006, pp. 245-249 · Zbl 1215.65188
[29] Ratschan, S.; She, Z., Safety verification of hybrid systems by constraint propagation based abstraction refinement, ACM transactions on embedded computing systems, 6, 4, 1-23, (2007), Article No. 8
[30] Podelski, A.; Wagner, S., Model checking of hybrid systems: from reachability towards stability, () · Zbl 1178.93077
[31] She, Z.; Zheng, Z., Tightened reachability constraints for the verification of linear hybrid systems, Nonlinear analysis: hybrid systems, 2, 4, 1222-1231, (2008) · Zbl 1163.93006
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