Abstract
We consider the robust linear stabilization problem for a family of nonlinear controllable systems that contains functional-parametric uncertainties and depends on the control nonlinearly. We obtain sufficient conditions for robust stabilization and synthesize state linear controllers that perform robust stabilization. We also obtain necessary conditions for robust stabilization that are close to sufficient. The synthesis is based on the method of Lyapunov functions.
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References
Krasovskii, N.N., Stabilization Problems for Controllable Motions, in Malkin, I.G., Teoriya ustoichivosti dvizheniya (Theory of Motion Stability), Moscow: Nauka, 1966, pp. 475–514.
Korobov, V.I., Metod funktsii upravlyaemosti (The Method of Controllability Function), Moscow-Izhevsk: NITs “Regulyarnaya i Khaoticheskaya Dinamika,” 2007.
Barabanov, N.E., On Quadratic Stabilizability of Linear Dynamical Systems, Sib. Mat. Zh., 1996, vol. 37, no. 1, pp. 3–19.
Voronov, K.V., Koroleva, O.I., and Nikiforov, V.O., Robust Control of Nonlinear Objects with Functional Uncertainties, Autom. Remote Control, 2001, vol. 62, no. 2, pp. 269–277.
Miroshnik, I.V., Nikiforov, V.O., and Fradkov, A.V., Nelineinoe i adaptivnoe upravlenie slozhnymi dinamicheskimi sistemami (Nonlinear and Adaptive Control for Complex Dynamical Systems), St. Petersburg: Nauka, 2000.
Polyak, B.T. and Shcherbakov, P.S., Robastnaya ustoichivost’ i upravlenie (Robust Stability and Control), Moscow: Nauka, 2002.
Battilotti, S., Robust Stabilization of Nonlinear Systems with Pointwise Norm—Bounded Uncertainties: A Control Lyapunov Function Approach, IEEE Trans. Automat. Control, 1999, vol. 44, no. 1, pp. 3–17.
Bobtsov, A.A. and Nikolaev, N.A., Fradkov Theorem-Based Design of the Control of Nonlinear Systems with Functional and Parametric Uncertainties, Autom. Remote Control, 2005, vol. 66, no. 1, pp. 108–118.
Boyd, S.L., El Ghaoui, L., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in Systems and Control Theory, Philadelphia: SIAM, 1994.
Amato, F., Robust Control of Linear Systems Subject to Uncertain Time-Varying Parameters, Berlin: Springer-Verlag, 2006.
Jabri, D., Guelton, K., Manamanni, N., Jaadari, A., and Chinh, C., Robust Stabilization of Nonlinear Systems Based on a Switched Fuzzy Control Law, CEAI, 2012, vol. 14, no. 2, pp. 40–49.
Khalil, N.K., Nonlinear Systems, New York: Prentice Hall, 2002.
Nguang, S. and Fu, M., Global Quadratic Stabilization of a Class of Nonlinear Systems, Int. J. Robust Nonlinear Control, 1998, vol. 8, pp. 483–497.
Zak, S.H., Systems and Control, Oxford: Oxford Univ. Press, 2002.
Zuber, I.E. and Gelig, A.Kh., Synthesis of Robust Stabilizing Control for Nonlinear Systems, ENOC-2008, St. Petersburg, 2008, vol. 4.
Yu, M., Wang, L., and Chu, T., Robust Stabilization of Nonlinear Sampled-Data Systems, Am. Control Conf., 2005, pp. 3421–3426.
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Original Russian Text © V.I. Korobov, A.V. Lutsenko, 2014, published in Avtomatika i Telemekhanika, 2014, No. 8, pp. 99–112.
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Korobov, V.I., Lutsenko, A.V. Robust stabilization of one class of nonlinear systems. Autom Remote Control 75, 1433–1444 (2014). https://doi.org/10.1134/S00051179140800074
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DOI: https://doi.org/10.1134/S00051179140800074