Abstract
Stability is a fundamental property for dynamic systems. In most engineering projects unstable systems are useless. Therefore in system analysis and control design the stability and stabilization become the first priority to be consider. This chapter considers the stability of dynamic systems and the stabilization and stabilizer design of nonlinear control systems. In Section 7.1 the concepts about stability of dynamic systems are presented. Section 7.2 considers the stability of nonlinear systems via its linear approximation. The Lapunov direct method is discussed in Section 7.3. Section 7.4 presents the LaSalle’s invariance principle. The converse theory of Lyapunov stability is introduced in Section 7.5. Section 7.6 is about the invariant set. The input-output stability of control systems is discussed in Section 7.7. In section 7.8 the semi-tensor product is used to find the region of attraction. Many results in this chapter are classical, hence the proofs are omitted.
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References
Boothby W. An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edn. Orlando: Academic Press, 1986.
Carr J. Applications of Centre Manifold Theory. New York: Springer, 1981.
Cheng D. On Lyapunov mapping and its applications. Communications on Information and Systems, 2001, 1(5): 195–212.
Chiang H, Hirsch M, Wu F. Stability regions of nonlinear autonomous dynamical systems. IEEE Trans. Aut. Contr., 1988, 33(1): 16–27.
Chiang H, Wu F. Foundations of the potential energy boundary surface method for power system transient stability analysis. IEEE Trans. Circ. Sys., 1988, 35(6): 712–728.
Hahn W. Stability of Motion. Berlin: Springer, 1967.
Horn R, Johnson C. Matrix Analysis. New York: Cambbridge Univ. Press, 1985.
Khalil H. Nonlinear Systems, 3rd edn. New Jersey: Prentice Hall, 2002.
Mu X, Cheng D. On stability and stabilization of time-varying nonlinear control systems. Asian J. Contr., 2005, 7(3): 244–255.
Saha S, Fouad A, Kliemamm W, et al. Stability boundary approximation of a power system using the real normal form of vector fields. IEEE Trans. Power Sys., 1997, 12(2): 797–802.
Sastry S. Nonlinear Systems. New York: Springer, 1999.
Varaiya P, Wu F, Chen R. Direct methods for transient stability analysis of power systems: Recent results. Proceedings of the IEEE, 1985, 73(12): 1703–1715.
Vidyasagar M. Decomposition techniques for large-scale systems with nonadditive interactions: Stability and stabilizability. IEEE Trans. Aut. Contr., 1980, 25(4): 773–779.
Vidyasagar M. Nonlinear Systems Analysis. New Jersey: Prentice Hall, 1993.
Zaborszky J, Huang J, Zheng B, et al. On the phase protraits of a class of large nonlinear dynamic systems such as the power systems. IEEE Trans. Aut. Contr., 1988, 33(1): 4–15.
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Cheng, D., Hu, X., Shen, T. (2010). Stability and Stabilization. In: Analysis and Design of Nonlinear Control Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11550-9_7
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DOI: https://doi.org/10.1007/978-3-642-11550-9_7
Publisher Name: Springer, Berlin, Heidelberg
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