Chaos Suppression of a Class of Fractional-Order Chaotic Systems with Order Lying in (1, 2)
It is shown that fractional-order (FO) nonlinear systems can also show higher nonlinearity and complex dynamics. FO chaotic systems have wider applications in secure communication, signal processing, financial field due to FO chaos has larger key space and more complex random sequences than integer-order chaos. Thanks to the lack of the effective analytical methods and controller design methods of integer-order chaotic systems can not be applied directly to FO chaos systems, to control chaos of FO chaotic systems is a very interesting and difficult problem, especially for FO chaotic system with order α:1<α<2. Based on the stability theory of FO systems and the linear state feedback control, an LMI criterion for controlling a class of fractional-order chaotic systems with fractional-order α:1<α<2 is addressed in this paper. The proposed method can be easily verified and resolved by using the Matlab LMI toolbox. Moreover, the proposed controller is linear, easy to implement and overcome some defects in the recent literature, which have improved the existing results. The method employed in this letter can effectively avoid control cost and inaccuracy in the literatures, and can be be applied to FO hyperchaos systems and synchronization controller design of FO chaotic system. Theoretical analysis and numerical simulations are presented to demonstrate the validity and feasibility of the proposed methods.
Chaos Suppression of a Class of Fractional-Order Chaotic Systems with Order Lying in (1, 2), Mathematics Letters.
Vol. 4, No. 3,
2018, pp. 51-58.
Andrievskii, B. R. and A. L. Fradkov, “Control of Chaos: Methods and Applications. I. Methods,” Automation and Remote Control, 2003, vol. 64, pp. 673-713.
Andrievskii, B. R. and Fradkov, A. L., “Control of chaos: Methods and applications II. Applications,” Automation and Remote Control, 2004, pp. 505-533.
Chen, G. R., Controlling chaos and bifurcations in engineering systems, CRC Press, New York, 1999.
Wang, G., Yu, X. and Chen, S., The control, synchronization and application of chaos, The National Defence Industry Press, Beijing, 2001.
DePaula, A. S. and Savi, M. A., “A multiparameter chaos control method based on OGY approach,” Chaos Solitons Fractals, 2009, pp. 1376-1390.
Matouk, A. E., “Chaos, feedback control and synchronization of a fractional-order modified Autonomous VanderPol-Duffing circuit,” Communications in Nonlinear Science and Numerical Simulation, 2011, pp. 975-986.
Zhong, Q. S., Bao, J. F., Yu, Y. B., et al, “Impulsive control for fractional-order chaotic systems,” Chinese Physics Letters, 2008, pp. 2812-2815.
Njah, A. N. and Sunday, O. D., “Generalization on the chaos control of 4-D chaotic systems using recursive backstepping nonlinear controller,” Chaos Solitons Fractals, 2009, pp. 2371-2376.
Odibat, Z. M., “Adaptive feedback control and synchronization of non-identical chaotic fractional order systems,” Nonlinear Dynamics, 2010, pp. 479-487.
Starrett, J., “Control of chaos by occasional bang-bang,” Physical Review E, 2003, pp. 036203.
Dadras, S. and Momeni, H. R., “Control of a fractional-order economical system via sliding mode,” Physica A, 2010, pp. 2434-2442.
Ahmad, W. M. and Harb, A. M., “On nonlinear control design for autonomous chaotic systems of integer and fractional orders,” Chaos Solitons Fractals, 2003, pp. 693-701.
Nbendjo, B., Salissou, Y. and Woafo, P., “Active control with delay of catastrophic motion and horseshoes chaos in a single well Duffing oscillator,” Chaos Solitons Fractals, 2005, pp. 809-816.
Ge, Z. M. and Ou, C. Y., “Chaos in a fractional order modified Duffing system,” Chaos solitons Fractals, 2007, pp. 262-291.
Hartley, T. T., Lorenzo, C. F. and Qammer, H. K., “Chaos in a fractional order Chua’s system,” IEEE Transactions on Circuits and Systems I-Fundamental Theory and Applications, 1995, pp. 485-490.
Li, C. G. and Chen, G. R., “Chaos and hyperchaos in the fractional-order Rössler equations,” Physica A, 2004, pp. 55-61.
Li, C. P. and Peng, G. J., “Chaos in Chen’s system with a fractional order,” Chaos Solitons Fractals, 2004, pp. 443-450.
Grigorenko, I. and Grigorenko, E., “Chaotic dynamics of the fractional lorenz system,” Physical Review Letters, 2003, 034101.
Deng, W. H. and Li, C. P., “Chaos synchronization of the fractional Lü system,” Physica A, 2005, pp. 61-72.
Wang, X. Y. and Wang, M. J., “Dynamic analysis of the fractional-order Liu system and its synchronization,” Chaos, 2007, pp. 033106.
Chen, W. C., “Nonlinear dynamics and chaos in a fractional-order financial system,” Chaos, Solitons Fractals, 2008, pp. 1305-1314.
Petr¨¢, I., “Chaos in the fractional-order Volta¡¯s system: modeling and simulation,” Nonlinear dynamics, 2009, pp. 157-170.
Wu, X. J. and Lu, Y., “Generalized projective synchronization of the fractional-order Chen hyperchaotic system,” Nonlinear Dynamics, 2009, pp. 25-35.
Wang, T. S. and Wang, X. Y., “Generalized synchronization of fractional order hyperchaotic Lorenz system,” Modern Physics Letters B, 2009, pp. 2167-2178.
Yu, Y. G. and Li, H. X., “The synchronization of fractional-order Rössler hyperchaotic systems,” Physica A, 2008, pp. 1393-403.
Wajdi, M. A and Reyad, E. K, “Fractional-order dynamical models of love Chaos,” Solitons and Fractals, 2007, pp. 1367-1375.
Li, C. G. and Chen, G. R., “Chaos in the fractional order Chen system and its control,” Chaos Solitons Fractals, 2004, pp: 549-554.
Wang, X. Y. and He, Y. Y., Wang, M. J, “Chaos control of a fractional order modified coupled dynamos system,” Nonlinear Analysis, 2009, pp: 6126-6134.
Zhang, K., Wang, H. and Fang, H., “Feedback control and hybrid projective synchronization of a fractional-order Newton-Leipnik system,” Communications in Nonlinear Science and Numerical Simulation, 2011, pp: 317-328.
Matouk, A. E., “Stability conditions, hyperchaos and control in a novel fractional order hyperchaotic system,” Physics Letters A, 2009, pp. 2166-2173.
Lu, J. G. and Chen, Y. Q., “Robust stability and stabilization of fractional-order interval systems with the fractional order α: the 0<α<1 case,” IEEE Transactions on Automatic Control, 2010, pp. 152-158.
Matgnon, D., “Proceedings multi conference on computational engineering in systems and application,” IMICS, IEEE-SMC, Lille, France, 1996, pp. 963-968.
Chilali, M., Gahinet, P. and Apkarian, P., “Robust pole placement in LMI regions,” IEEE Transactions on Automatic Control, 1999, pp. 2257-2270.
Khargonekar, P. P., Petersen, I. R. and Zhou, K., “Robust stabilization of uncertain linear system: quadratic stabilizability and h_∞ control theory,” IEEE Transactions on Automatic Control, 1990, pp. 356-361.
Boyd, S., Ghaoui, L., Feron, E., Balakrishnan V, Linear matrix inequalities in system and control theory, Philadelphia, SIAM, 1994.
El-Khazali, R., Ahmad, W. and Al-Assaf, Y., “Sliding mode control of generalized fractional chaotic systems,” International Journal Bifurcation and chaos, 2006, pp. 3113-3125.