Partial Stabilization of a Class of Nonlinear Systems Via Center Manifold Theory
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.
[1] A. Bacciotti, "Local stabilization of nonlinear systems," in Ser. Adv. Math.
Appl. Sci. , vol. 8, River Edge, NJ: World Scientific, 1992.
[2] 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.
[3] 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.
[4] H. Hermes, "Asymptotically stabilizing feedback controls," Journal of
Differential Equations, Vol. XX, no. 92, pp. 76-89, Apr. 1991.
[5] H. Hermes, "Resonance and feedback stabilization," in Proc. IFAC
Nonlinear Control Systems Design Symp. (NOLCOS 95), Tahoe City, NV,
1997, pp. 47-52.
[6] 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.
[7] 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.
[8] 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.
[9] J. Carr, Applications of Center Manifold Theory, New York: Springer-
Verlag, 1981.
[10] H. K. Khalil, Nonlinear Systems, Second etition, Prentice Hall, New
Jersey, 1996, ch. 4.
[11] A. Isidori, Nonlinear Control Systems, Third etition, London: Springer-
Verlag, 1995, Appendix B. 1 .
[12] 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.
[13] 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.
[14] 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.
[15] 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.
[16] D. Cheng, "Stabilization of a class of nonlinear non-minimum phase
systems," Asian Journal of Control, Vol. 2, pp. 132-139, Jun. 2000.
[17] D. Cheng and Y. Guo, "Stabilization of nonlinear systems via the center
manifold approach," Systems & Control Letters, Vol. 57, pp. 511-518,
Jun. 2008.
[18] 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.
[19] 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.
[20] C. Juan and A. Manuel, "Passivity based stabilization of non-minimum
phase nonlinear systems," Kybernetika, Vol. 45, pp. 417-426, Jun. 2009.
[21] 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.
[22] 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.
[23] 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.
[24] 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.
[25] 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.
[26] 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.
[27] 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.
[28] 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.
[29] 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.
[30] 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.
[31] 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.
[32] C. T. Chen, Linear System Theory and Design, Holt, Rinehart and
Winston, New York, 1984.
[33] 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.
[34] D. C Liaw, Feedback stabilization via center manifold reduction with
application to tethered satellites, Ph. D. Dissertation. University of
Maryland, 1990.
[35] 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.
[36] M. B. William, An Introduction to Differentiable Manifolds and Riemannian
Geometry, Singapore: Elsevier, 1986.
[37] 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.
[38] 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.
[39] 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.
[40] 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.
[1] A. Bacciotti, "Local stabilization of nonlinear systems," in Ser. Adv. Math.
Appl. Sci. , vol. 8, River Edge, NJ: World Scientific, 1992.
[2] 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.
[3] 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.
[4] H. Hermes, "Asymptotically stabilizing feedback controls," Journal of
Differential Equations, Vol. XX, no. 92, pp. 76-89, Apr. 1991.
[5] H. Hermes, "Resonance and feedback stabilization," in Proc. IFAC
Nonlinear Control Systems Design Symp. (NOLCOS 95), Tahoe City, NV,
1997, pp. 47-52.
[6] 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.
[7] 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.
[8] 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.
[9] J. Carr, Applications of Center Manifold Theory, New York: Springer-
Verlag, 1981.
[10] H. K. Khalil, Nonlinear Systems, Second etition, Prentice Hall, New
Jersey, 1996, ch. 4.
[11] A. Isidori, Nonlinear Control Systems, Third etition, London: Springer-
Verlag, 1995, Appendix B. 1 .
[12] 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.
[13] 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.
[14] 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.
[15] 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.
[16] D. Cheng, "Stabilization of a class of nonlinear non-minimum phase
systems," Asian Journal of Control, Vol. 2, pp. 132-139, Jun. 2000.
[17] D. Cheng and Y. Guo, "Stabilization of nonlinear systems via the center
manifold approach," Systems & Control Letters, Vol. 57, pp. 511-518,
Jun. 2008.
[18] 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.
[19] 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.
[20] C. Juan and A. Manuel, "Passivity based stabilization of non-minimum
phase nonlinear systems," Kybernetika, Vol. 45, pp. 417-426, Jun. 2009.
[21] 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.
[22] 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.
[23] 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.
[24] 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.
[25] 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.
[26] 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.
[27] 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.
[28] 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.
[29] 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.
[30] 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.
[31] 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.
[32] C. T. Chen, Linear System Theory and Design, Holt, Rinehart and
Winston, New York, 1984.
[33] 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.
[34] D. C Liaw, Feedback stabilization via center manifold reduction with
application to tethered satellites, Ph. D. Dissertation. University of
Maryland, 1990.
[35] 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.
[36] M. B. William, An Introduction to Differentiable Manifolds and Riemannian
Geometry, Singapore: Elsevier, 1986.
[37] 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.
[38] 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.
[39] 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.
[40] 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.
@article{"International Journal of Information, Control and Computer Sciences:49715", author = "Ping He", title = "Partial Stabilization of a Class of Nonlinear Systems Via Center Manifold Theory", 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.", keywords = "Partial stabilization, Nonlinear critical systems, Centermanifold theory, Lyapunov function, System reduction.", volume = "5", number = "3", pages = "219-13", }
