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Finite-time stabilization of uncertain non-autonomous chaotic gyroscopes with nonlinear inputs. (English) Zbl 1233.93081

Summary: Gyroscopes are one of the most interesting and everlasting nonlinear nonautonomous dynamical systems that exhibit very complex dynamical behavior such as chaos. In this paper, the problem of robust stabilization of the nonlinear non-autonomous gyroscopes in a given finite time is studied. It is assumed that the gyroscope system is perturbed by model uncertainties, external disturbances, and unknown parameters. Besides, the effects of input nonlinearities are taken into account. Appropriate adaptive laws are proposed to tackle the unknown parameters. Based on the adaptive laws and the finite-time control theory, discontinuous finite-time control laws are proposed to ensure the finite-time stability of the system. The finite-time stability and convergence of the closed-loop system are analytically proved. Some numerical simulations are presented to show the efficiency of the proposed finite-time control scheme and to validate the theoretical results.

MSC:

93D21 Adaptive or robust stabilization
93C10 Nonlinear systems in control theory
93C40 Adaptive control/observation systems
34C28 Complex behavior and chaotic systems of ordinary differential equations
34H10 Chaos control for problems involving ordinary differential equations
37N35 Dynamical systems in control
70Q05 Control of mechanical systems
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[1] Ott, E., Grebogi, C., and Yorke, J. A. Using chaos to direct trajectories to targets. Phys. Rev. Lett., 65, 3215–3218 (1990)
[2] Hu, M. F. and Xu, Z. Y. Spatio-temporal chaotic synchronization for modes coupled two Ginzburg-Landau equations. Appl. Math. Mech. -Engl. Ed., 27, 1149–1156 (2006) DOI 10.1007/s10486-006-0816-y · Zbl 1163.37397
[3] Ahn, C. K. Generalized passivity-based chaos synchronization. Appl. Math. Mech. -Engl. Ed., 31, 1009–1018 (2010) DOI 10.1007/s10483-008-1005-y · Zbl 1210.34068
[4] Liu, Y. and Lü, L. Synchronization of N different coupled chaotic systems with ring and chain connections. Appl. Math. Mech. -Engl. Ed., 29, 1299–1308 (2008) DOI 10.1007/s10483-008-1005-y · Zbl 1163.93025
[5] Pourmahmood, M., Khanmohammadi, S., and Alizadeh, G. Synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller. Commun. Nonlinear Sci. Numer. Simulat., 16, 2853–2868 (2011) · Zbl 1221.93131
[6] Aghababa, M. P., Khanmohammadi, S., and Alizadeh, G. Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique. Appl. Math. Model., 30, 3080–3091 (2001) · Zbl 1219.93023
[7] Tang, L. J., Li, D., and Wang, H. X. Lag synchronization for fuzzy chaotic system based on fuzzy observer. Appl. Math. Mech. -Engl. Ed., 30, 803–810 (2009) DOI 10.1007/s10483-009-0615-y · Zbl 1180.34086
[8] Bobtsov, A., Nikolaev, N., and Slita, O. Control of chaotic oscillations of a satellite. Appl. Math. Mech. -Engl. Ed., 28, 893–900 (2007) DOI 10.1007/s10483-007-0706-z · Zbl 1231.93044
[9] Chen, H. K. Chaos and chaos synchronization of a symmetric gyro with linearplus-cubic damping. J. Sound Vib., 255, 719–740 (2002) · Zbl 1237.70094
[10] Van Dooren, R. Comments on chaos and chaos synchronization of a symmetric gyro with linear-plus-cubic damping. J. Sound Vib., 268, 632–634 (2008) · Zbl 1236.70034
[11] Ge, Z. M. and Chen, H. K. Bifurcations and chaos in a rate gyro with harmonic excitation. J. Sound Vib., 194, 107–117 (1996)
[12] Tong, X. and Mrad, N. Chaotic motion of a symmetric gyro subjected to a harmonic base excitation. J. Appl. Mech. Trans. Amer. Soc. Mech. Eng., 68, 681–684 (2001) · Zbl 1110.74711
[13] Leipnik, R. B. and Newton, T. A. Double strange attractors in rigid body motion with linear feedback control. Phys. Lett. A, 86, 63–67 (1981)
[14] Ge, Z. M., Chen, H. K., and Chen, H. H. The regular and chaotic motion of a symmetric heavy gyroscope with harmonic excitation. J. Sound Vib., 198, 131–147 (1996) · Zbl 1235.70016
[15] Lei, Y., Xu, W., and Zheng, H. Synchronization of two chaotic nonlinear gyros using active control. Phys. Lett. A, 343, 153–158 (2005) · Zbl 1194.34090
[16] Hung, M., Yan, J., and Liao, T. Generalized projective synchronization of chaotic nonlinear gyros coupled with dead-zone input. Chaos Solitons & Fractals, 35, 181–187 (2008)
[17] Yau, H. Nonlinear rule-based controller for chaos synchronization of two gyros with linear-pluscubic damping. Chaos Solitons & Fractals, 34, 1357–1365 (2007) · Zbl 1142.93375
[18] Yau, H. Chaos synchronization of two uncertain chaotic nonlinear gyros using fuzzy sliding mode control. Mech. Syst. Signal Pr., 22, 408–418 (2008)
[19] Yan, J. J., Hung, M. L., and Liao, T. L. Adaptive sliding mode control for synchronization of chaotic gyros with fully unknown parameters. J. Sound Vib., 298, 298–306 (2006) · Zbl 1243.93097
[20] Yan, J. J., Hung, M. L., Lin, J. S., and Liao, T. L. Controlling chaos of a chaotic nonlinear gyro using variable structure control. Mech. Syst. Signal Pr., 21, 2515–2522 (2007)
[21] Bwidehat, S. P. and Bernstein, D. S. Finite-time stability of continuous autonomous systems. SIAM J. Control Optim., 38, 751–766 (2008)
[22] Curran, P. F. and Chua, L. O. Absolute stability theory and the synchronization problem. Int. J. Bifurcat. Chaos, 7, 1357–1382 (1997) · Zbl 0910.34054
[23] Wang, H., Han, Z., Xie, Q., and Zhang. W. Finite-time chaos control via nonsingular terminal sliding mode control. Commun. Nonlinear Sci. Numer. Simulat., 14, 2728–2733 (2009) · Zbl 1221.37225
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