Abstract
An impulsive control scheme of the Lur’e system and several theorems on stability of impulsive control systems was presented, these theorems were then used to find the conditions under which the Lur’e system can be stabilized by using impulsive control with varying impulsive intervals. The parameters of Lur’e system and impulsive control law are given, a theory of impulsive synchronization of two Lur’e system is also presented. A numerical example is used to verify the theoretical result.
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References
Lakshmikantham V, Bainov D D, Simeonov P S.Theory of Impulsive Differential Equations[M]. Singapore: World Scientific, 1989.
Samoilenko A M, Perestyuk N A.Impulsive Differential Equations[M]. Singapore: World Scientific, 1995.
Yang T. Impulsive control[J].IEEE Trans Automat Contr, 1999,44(5):1081–1083.
Schweizer J, Kennedy M P. Predictive Poincare control: A control theory for chaotic systems[J].Phys Rev E, 1995,52(5):4865–4867.
Hunt E R, Johnson G. Keeping chaos at bay[J].IEEE Spectrum, 1993,30(11):32–36.
Stojanovski T, Kocarev L, Parlitz U. Driving and synchronizing by chaotic impulses[J].Phys Rev E, 1996,43(9):782–785.
Yang T, Yang L B, Yang C M. Impulsive synchronization of Lorenz systems[J].Phys Lett A, 1997,226(6):349–354.
Amirtkar R E, Gupte N. Synchronization of chaotic orbits: The effect of a finite time step[J].Phys. Rev E (Statistical Physics, Plasmas, Fluids and Related Interdisciplinary, Topics), 1993,47(6): 3889–3895.
Yang T, Chua L O. Impulsive control and synchronization of nonlinear dynamical systems and application to secure communication[J].Int J Bifur Chaos, 1997,7(3):645–664.
Yang T, Chua L O. Impulsive control and synchronization of chaotic systems and secure communication[R]. Electronics Research Laboratory, College of Engineering, University of California, Berkeley, CA 94720, Memorandum No. UCN/ERL M97/12, 29 January 1997.
Yang T, Yang L B, Yang C M. Impulsive control of Lorenz system[J].Phys D, 1997,110: 18–24.
Yang T, Yang C M, Yang L B. Control of Rossler system to periodic motions using impulsive control methods[J].Phys Lett A, 1997,232:356–361.
Yang T, Chua L O. Impulsive stabilization for control and synchronization of chaotic systems: Theory and application to secure communication[J].IEEE Trans, Circuits Syst I, 1997,44(10): 976–988.
Xie W X, Wen C Y, Li Z G. Impulsive control for the stabilization and synchronization of Lorenz systems[J].Phys Lett A, 2000,275:67–72.
Li Z G, Wen C Y, Soh Y C. Analysis and design of impulsive control systems[J].IEEE Trans Automat Contr, 2001,46(6):894–903.
Vidyasagar M.Nonlinear Systems Analysis[M]. London: Prentice-Hall, 1993.
Grujic L J T, Petkovski D J. On robustness of Lur’e systems with multiple non-linearities[J].Automatica, 1987,23:327–334.
Tesk A, Vicino A. Robust absolute stability of Lur’e control systems in parameter space[J].Automatica, 1991,27(1):147–151.
Dahleh M, Tesk A, Vicino A. On the robust Popov criterion for interval Lur’e system[J].IEEE Tansactions on Automatic Control, 1993,38:1400–1405.
Mori T, Nishimura T, Kuroe Y, et al. Comments on ‘On the robust Popov criterion for interval Lur’ee system’[J].IEEE Tansactions on Automatic Control, 1995,40:136–137.
Wada T, Ikeda M, Ohta Y, et al. Parametric absolute stability of Lur’e systems[J].IEEE Tansactions on Automatic Control, 1998,43(11):1649–1653.
Konishi K, Kokame H. Robust stability of Lur’e systems with time-varying uncertainties: a linear matrix inequality approach[J].Int J Systems Science, 1999,30(1):3–9.
Ugrinovskii V A, Petersen I R. Guaranteed cost control of uncertain systems via Lur’e-Postnikov Lyapunov functions[J].Automatica, 2000,36:279–285.
Wada T, Ikeda M, Ohta Y, et al. Parametric absolute stability of multivariable Lur’e systems[J].Automatica, 2000,36:1365–1372.
SUN Ji-tao, DENG Fei-qi, LIU Yong-qing. Robust absolute stability of interval Lur’e type systems with time delay[J].J Syst Engineering and Electronics, 2001,23(3):47–50. (in Chinese)
SUN Ji-tao, DENG Fei-qi, LIU Yong-qing. On the robust stability of uncertain Lur’e control system[J].J Syst Engineering and Electronics, 2001,23(8):58–60. (in Chinese)
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Communicated by DAI Shi-qiang
Foundation items: National 973 Program of China (2002CB312200); the National Natural Science Foundation of China (79970030, 60104004); the Municipal Natural Science Foundation of Shanghai, China (03ZR14095)
Biography: SUN Ji-tao (1963∼), Professor, Doctor
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Ji-tao, S., Qi-di, W. Impulsive control for the stabilization and synchronization of Lur’e systems. Appl Math Mech 25, 322–328 (2004). https://doi.org/10.1007/BF02437335
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DOI: https://doi.org/10.1007/BF02437335