Partial stabilization of a class of continuous nonlinear control systems with separated variables

Abstract

In this paper, the partial stabilization problem for a class of nonlinear continuous control systems with separated variables is investigated. Several stabilizing controllers are constructed based on the partial stability theory of Lyapunov and the property of M-matrix, and some of these stabilizing controllers are only related to partial state variables. The controllers constructed here are shown to guarantee partial asymptotic stability of the closed-loop systems and these sufficient conditions may give some instructions to actual engineering application. A example is also given to illustrate the design method.

Keywords

Nonlinear control system / separated variables / partial stability / stabilizing controller / M-matrix

References

[1] Alexander Z.. Partial asymptotic stabilization of nonlinear distributed parameter systems. Automatica, 2005, 41(1): 1-10.

[2] Alexander, L.Zuyev, “Application of control Lyapunov functions technique for partial stabilization”, Proceeding of the 2001 IEEE International Conference on Control Applications, 2001, Mexico City, pp509–513.

[3] Bondi P., Fergola P., Gambardella L.. Partial stability of large-scale systems. IEEE Transactions on Automatic Control, 1979, AC-24(1): 94-97.

[4] Elaydi S. N.. An Iintroduction to Difference Equations, 1999, Second Edition New York: Springer-Verlag, Inc

[5] Ge Z.M., Chen Y.S.. Synchronization of unidirectional coupled chaotic systems via partial stability. Chaos, Solitons and Fractals, 2004, 21(1): 101-111.

[6] Jian J.G., Liao X.X.. Partial exponential stability of nonlinear time-varying large-scale systems. Nonlinear Analysis, 2004, 59(5): 789-800.

[7] Kolesnichenko, O., A.S Shiriaev, “Extension of Pozharitsky theorem for partial stabilization of a system with several first integrals”, Proceedings of the 41 ^st IEEE Conference on Dceision and Control, Las Vegus, Nerada USA, pp3512–3517, 2002.

[8] Liao X.X.. Mathematical Theory and Application of Stability, 2001, Wuhan: Huazhong Normal University Press

[9] Liao X.X.. Stability, boundedness, dissipation of partial variables for nonlinear systems with separating variables. Science in China, 1992, 35(9): 1025-1039.

[10] Liao X.X., Li J.. Robust interval stability, persistence and partial stability on Lotka-Volterra systems with time-delay. Applied Mathematics and Computation, 1996, 75: 103-115.

[11] Liao X.X., Chen G.R.. Chaos synchronization of general Lurie systems via time-delay feedback control. International Journal of Bifurcation and Chaos, 2003, 13(1): 207-213.

[12] Liu B.Y., Gui W.H.. Stabilization controller for a class of nonlinear discrete control systems with separated variables. Mathematical Theory and Application, 2003, 23(3): 53-55.

[13] Vorotnikov V.I.. Partial Stability and Control, 1998, Boston: Birkhauser

[14] Vorotnikov V.I.. Problems of stability with respect to part of the variables. Appl. Maths Mechs, 1999, 63(5): 695-703.

[15] Wang Z.S.. Performance analysis and control of uncertain systems. PhD Thesis, 2003, Wuhan: Huazhong University of Science & Technology

[16] Wassim M.H., Tomohisa H., Vijaysekhar C.. Robust adaptive control for nonlinear uncertain systems. Automatica, 2003, 39: 551-556.

[17] Xu S.Y., Yang C.. Stabilization of discrete time singular systems: a matrix inequalities approach. Automatica, 1999, 35(9): 1613-1617.