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Partial Stabilization of a Class of Nonlinear Systems Via Center Manifold Theory
Authors: Ping He
Abstract:This paper addresses the problem of the partial state feedback stabilization of a class of nonlinear systems. In order to stabilization this class systems, the especial place of this paper is to reverse designing the state feedback control law from the method of judging system stability with the center manifold theory. First of all, the center manifold theory is applied to discuss the stabilization sufficient condition and design the stabilizing state control laws for a class of nonlinear. Secondly, the problem of partial stabilization for a class of plane nonlinear system is discuss using the lyapunov second method and the center manifold theory. Thirdly, we investigate specially the problem of the stabilization for a class of homogenous plane nonlinear systems, a class of nonlinear with dual-zero eigenvalues and a class of nonlinear with zero-center using the method of lyapunov function with homogenous derivative, specifically. At the end of this paper, some examples and simulation results are given show that the approach of this paper to this class of nonlinear system is effective and convenient.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1328686Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1513
 A. Bacciotti, "Local stabilization of nonlinear systems," in Ser. Adv. Math. Appl. Sci. , vol. 8, River Edge, NJ: World Scientific, 1992.
 J. H. Fu, "Lyapunov functions and stability criteria for nonlinear systems with multiple critical eigenvalues," Mathematics of Control, Signals, and Systems, no. 7, pp. 255-278, 1994.
 E. D. Sontag, "Feedback stabilization of nonlinear systems," in Robust Control of Linear Systems and Nonlinear Control, vol. 4, M. A. Kaashoek, J. H. van Schuppen, and A. C. M. Ran, Ed. Boston, MA: Birkhauser, 1990, Progress in Systems and Control Theory Series, pp. 61-81.
 H. Hermes, "Asymptotically stabilizing feedback controls," Journal of Differential Equations, Vol. XX, no. 92, pp. 76-89, Apr. 1991.
 H. Hermes, "Resonance and feedback stabilization," in Proc. IFAC Nonlinear Control Systems Design Symp. (NOLCOS 95), Tahoe City, NV, 1997, pp. 47-52.
 W. P. Dayawansa, "Recent advances in the stabilization of nonlinear systems for low-dimensional systems," in Proc. IFAC Nonlinear Control Systems Design Symp. (NOLCOS 92), M. Fliess, Ed., Bordeaux, France, 1992, pp. 1-8.
 A. Bacciotti, P. Boieri, and L. Mazzi, "Linear stabilization of nonlinear cascade systems," Mathematics of Control, Signals, and Systems, Vol. 6, pp. 146-165, Jun. 1993.
 D. Aeyels, "Stabilization of a class of non-linear systems by a smooth feedback control," Systems & Control Letters, Vol. 5, pp. 289-294, Apr. 1985.
 J. Carr, Applications of Center Manifold Theory, New York: Springer- Verlag, 1981.
 H. K. Khalil, Nonlinear Systems, Second etition, Prentice Hall, New Jersey, 1996, ch. 4.
 A. Isidori, Nonlinear Control Systems, Third etition, London: Springer- Verlag, 1995, Appendix B. 1 .
 S. Behtash and D. Dastry, "Stabilization of non-linear systems with uncontrollable linearization," IEEE Transactions on Automatic Control, Vol. 33, pp. 585-590, Jun. 1988.
 R. Sepulchre and D. Aeyels, "Homogeneous Lyapunov functions and necessary conditions for stabilization," Mathematics of Control, Signals, and Systems, vol. 9, pp. 34-58, 1996.
 D. Cheng, "Stabilization via polynomial Lyapunov function," in W. Kang, M. Xiao, C. Borges Ed. New Trends in Nonlinear Dynamics and Control, Springer, Berlin, 2003, pp. 161-173.
 D. Cheng and C. Martin, "Stabilization of Nonlinear Systems via Designed Center Manifold," IEEE Transactions on Automatic Control, Vol. 46, pp. 1372-1383, Sep. 2001.
 D. Cheng, "Stabilization of a class of nonlinear non-minimum phase systems," Asian Journal of Control, Vol. 2, pp. 132-139, Jun. 2000.
 D. Cheng and Y. Guo, "Stabilization of nonlinear systems via the center manifold approach," Systems & Control Letters, Vol. 57, pp. 511-518, Jun. 2008.
 D. Liaw and C. Lee, "Stabilization of Nonlinear Critical Systems by Center Manifold Approach," in Conf. Rec. 2005 IEEE International Conference On Systems & Signals, pp. 610-613.
 D. CHENG, Q. HU and H. QIN, "Feedback diagonal canonical form and its application to stabilization of nonlinear systems," Science in China Series F: Information Sciences, Vol. 48, pp. 201-210, Apr. 2005.
 C. Juan and A. Manuel, "Passivity based stabilization of non-minimum phase nonlinear systems," Kybernetika, Vol. 45, pp. 417-426, Jun. 2009.
 F. Amato, C. Cosentino, and A. Merola, "Sufficient conditions for finitetime stability and stabilization of nonlinear quadratic systems," IEEE Transactions on Automatic Control, Vol. 55, pp. 430-434, Feb. 2010.
 J. Tsinias, "Remarks on asymptotic controllability and sampled-data feedback stabilization for autonomous systems", IEEE Transactions on Automatic Control, Vol. 55, pp. 721-726, March 2010.
 K. Kalsi, J. M. Lian, and S. H. Zak, "Decentralized Dynamic Output Feedback Control of Nonlinear Interconnected Systems," IEEE Transactions on Automatic Control, Vol. 55, pp. 1964-1970, Aug. 2010.
 B. B. Sharma, and I. N. Kar, "Contraction theory-based recursive design of stabilising controller for a class of non-linear systems," IET Control Theory & Applications, Vol. 4, pp. 1005-1018, June 2010.
 M. S. Koo, H. L. Choi, and J. T. Lim, "Global regulation of a class of uncertain nonlinear systems by switching adaptive controller," IEEE Transactions on Automatic Control, Vol. 55, pp. 2822-2827, Dec. 2010.
 W. Li, Y. Jing, and S. Zhang, "Decentralised stabilisation of a class of large-scale high-order stochastic non-linear systems," IET Control Theory & Applications, Vol. 4, pp. 2441-2453, Nov. 2010.
 Z. Zhang, S. Xu, and Y. Chu, "Adaptive stabilisation for a class of nonlinear state time-varying delay systems with unknown time-delay bound," IET Control Theory & Applications, Vol. 4, pp. 1905-1913, Oct. 2010.
 M. S. Mahmoud, and Y. Xia, "New stabilisation schemes for discrete delay systems with uncertain non-linear perturbations," IET Control Theory & Applications, Vol. 4, pp. 2937-2946, Dec. 2010.
 X. Cai, Z. Han, and J. Huang, "Stabilisation for a class of non-linear systems with uncertain parameter based on centre manifold," IET Control Theory & Applications, Vol. 4, pp. 1558-1568, Sep. 2010.
 Z. T. Ding, "Global output feedback stabilisation of a class of nonlinear systems with unstable zero dynamics," in 2010 American Control Conference, pp. 4241-4246.
 H. S. Wu, "Adaptive robust stabilization for a class of uncertain nonlinear systems with external disturbances," in 36th Annual Conference of the IEEE Industrial Electronics Society, Glendale, AZ, USA, 2010, pp. 53-58.
 C. T. Chen, Linear System Theory and Design, Holt, Rinehart and Winston, New York, 1984.
 B. Hamzi, W. Kang, and A. J. Krener, "Control of center manifolds," in the 42nd IEEE Conference on Decision & Control, Maui, Hawaii, USA, Dec. 2003, pp. 2065-2070.
 D. C Liaw, Feedback stabilization via center manifold reduction with application to tethered satellites, Ph. D. Dissertation. University of Maryland, 1990.
 C. A. Schwartz and A. Yan, "Systematic construction of Lyapunov function for nonlinear systems in critical cases", in Proceedings of the 34th IEEE Conference on Decision and Control, New Orleans, LA, 1995, pp. 3779-3784.
 M. B. William, An Introduction to Differentiable Manifolds and Riemannian Geometry, Singapore: Elsevier, 1986.
 E. Panteley, and A. Loria, "Growth rate conditions for uniform asymptotic stability of cascaded time-varying systems," Automatic, Vol. 42, pp. 453-460, Sep. 2001.
 R. Sepulchre, M. Arcak, and A. R. Teel "Trading the stability of finite zeros for global stabilization of nonlinear cascade systems," IEEE Transactions on Automatic Control, Vol. 47, pp. 521-525, Mar. 2002.
 S. H. Li, and Y. P. Tian "Finite-time stability of cascaded time-varying systems," International Journal of Control, Vol. 80, pp. 646-657, Apr. 2007.
 S. H. Ding, S. H. Li, and Q. Li "Globally uniform stability of a class of continuous cascaded systems," Acta Automatic Sinica, Vol. 34, pp. 1268-1274, Oct. 2008.