Recently, control and synchronization of fractional chaotic systems have increasingly attracted much attention in the fractional control community. In this paper we introduce a novel class of fractional chaotic systems in the pseudo state space and propose an adaptive sliding mode control scheme to stabilize the chaotic systems in the presence of uncertainties and external disturbances whose bounds are unknown. To verify the effectiveness of the proposed adaptive sliding mode control technique, numerical simulations of control design of fractional Lorenz's system and Chen's system are presented. 1. Introduction Fractional calculus is an old and yet novel topic whose infancy dates back to the end of the 17th century, the time when Newton and Leibniz established the foundations of classical calculus. For three centuries, fractional calculus developed mainly as a pure theoretical mathematical field without applications. However, in the last two decades it has attracted the interest of researchers in several areas including mathematics, physics, chemistry, material, engineering, finance, and even social science. The stability of fractional differential equations (FDEs) and fractional control have both gained rapid development very recently [1–3]. One of the most important areas of application is the chaos theory. In recent years, fractional chaotic systems have intensively attracted a great deal of attention due to the ease of their electronics implementations and the rapid development of the stability of FDEs. More and more fractional dynamics described in the pseudo state space exhibiting chaos have been found, such as the fractional Chua circuit [4], the fractional Van der Pol oscillator [5–7], the fractional Lorenz system [8, 9], the fractional Chen system [10–12], the fractional Lü system [13], the fractional Liu system [14], the fractional R？ssler system [15, 16], the fractional Arneodo system [17], the fractional Newton-Leipnik system [18–20], the fractional Lotka-Volterra system [21, 22], the fractional finance system [20, 23], and the fractional Rucklidge system [24]. Most of the above papers have used numerical methods to present chaotic behaviors. In particular, control and synchronization of fractional chaotic systems have increasingly attracted much attention in the fractional control community. Moreover, several control and synchronization methods have been proposed based on the stability of fractional differential equations in the pseudo state space [22]. The linear state feedback control algorithm based upon the stability criterion of linear
References
[1]
J. T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140–1153, 2011.
[2]
C. P. Li and F. R. Zhang, “A survey on the stability of fractional differential equations,” European Physical Journal, vol. 193, no. 1, pp. 27–47, 2011.
[3]
Y. Q. Chen, I. Petrá？, and D. Xue, “Fractional order control—tutorial,” in Proceedings of the American Control Conference (ACC '09), pp. 1397–1411, St. Louis, Mo, USA, June 2009.
[4]
I. Petrá？, “A note on the fractional-order chua’s system,” Chaos, Solitons and Fractals, vol. 38, no. 1, pp. 140–147, 2008.
[5]
J.-H. Chen and W.-C. Chen, “Chaotic dynamics of the fractionally damped van der Pol equation,” Chaos, Solitons and Fractals, vol. 35, no. 1, pp. 188–198, 2008.
[6]
R. S. Barbosa, J. A. T. MacHado, B. M. Vinagre, and A. J. Caldern, “Analysis of the Van der Pol oscillator containing derivatives of fractional order,” Journal of Vibration and Control, vol. 13, no. 9-10, pp. 1291–1301, 2007.
[7]
M. S. Tavazoei, M. Haeri, M. Attari, S. Bolouki, and M. Siami, “More details on analysis of fractional-order Van der Pol oscillator,” Journal of Vibration and Control, vol. 15, no. 6, pp. 803–819, 2009.
[8]
I. Grigorenko and E. Grigorenko, “Chaotic dynamics of the fractional Lorenz system,” Physical Review Letters, vol. 91, no. 3, 4 pages, 2003.
[9]
Y. Yu, H.-X. Li, S. Wang, and J. Yu, “Dynamic analysis of a fractional-order Lorenz chaotic system,” Chaos, Solitons and Fractals, vol. 42, no. 2, pp. 1181–1189, 2009.
[10]
C. Li and G. Chen, “Chaos in the fractional order Chen system and its control,” Chaos, Solitons and Fractals, vol. 22, no. 3, pp. 549–554, 2004.
[11]
J. G. Lu and G. Chen, “A note on the fractional-order Chen system,” Chaos, Solitons and Fractals, vol. 27, no. 3, pp. 685–688, 2006.
[12]
T. Ueta and G. Chen, “Bifurcation analysis of Chen's equation,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 10, no. 8, pp. 1917–1931, 2000.
[13]
J. G. Lu, “Chaotic dynamics of the fractional-order Lü system and its synchronization,” Physics Letters A, vol. 354, no. 4, pp. 305–311, 2006.
[14]
X.-Y. Wang and M.-J. Wang, “Dynamic analysis of the fractional-order Liu system and its synchronization,” Chaos, vol. 17, no. 3, 6 pages, 2007.
[15]
C. Li and G. Chen, “Chaos and hyperchaos in the fractional-order R？ssler equations,” Physica A, vol. 341, no. 1-4, pp. 55–61, 2004.
[16]
W. Zhang, S. Zhou, H. Li, and H. Zhu, “Chaos in a fractional-order R？ssler system,” Chaos, Solitons and Fractals, vol. 42, no. 3, pp. 1684–1691, 2009.
[17]
J. G. Lu, “Chaotic dynamics and synchronization of fractional-order Arneodo's systems,” Chaos, Solitons and Fractals, vol. 26, no. 4, pp. 1125–1133, 2005.
[18]
Y. Kang, K.-T. Lin, J.-H. Chen, L.-J. Sheu, and H.-K. Chen, “Parametric analysis of a fractional-order Newton-Leipnik system,” Journal of Physics, vol. 96, no. 1, Article ID 012140, 2008.
[19]
L.-J. Sheu, H.-K. Chen, J.-H. Chen et al., “Chaos in the Newton-Leipnik system with fractional order,” Chaos, Solitons and Fractals, vol. 36, no. 1, pp. 98–103, 2008.
[20]
M. S. Tavazoei and M. Haeri, “Chaotic attractors in incommensurate fractional order systems,” Physica D, vol. 237, no. 20, pp. 2628–2637, 2008.
[21]
E. Ahmed, A. M. A. El-Sayed, and H. A. A. El-Saka, “Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 542–553, 2007.
[22]
I. Petrá？, Fractional-Order Nonlinear Systems, Springer, Heidelberg, Germany, 2011.
[23]
W.-C. Chen, “Nonlinear dynamics and chaos in a fractional-order financial system,” Chaos, Solitons and Fractals, vol. 36, no. 5, pp. 1305–1314, 2008.
[24]
Y. Zhang and T. Zhou, “Three schemes to synchronize chaotic Fractional-order Rucklidge systems,” International Journal of Modern Physics B, vol. 21, no. 12, pp. 2033–2044, 2007.
[25]
C. P. Li, W. H. Deng, and D. Xu, “Chaos synchronization of the Chua system with a fractional order,” Physica A, vol. 360, no. 2, pp. 171–185, 2006.
[26]
J. Yan and C. Li, “The synchronization of three fractional differential systems,” Chaos, Solitons & Fractals, vol. 32, no. 2, pp. 751–757, 2007.
[27]
Z. M. Odibat, N. Corson, M. A. Aziz-Alaoui, and C. Bertelle, “Synchronization of chaotic fractional-order systems via linear control,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 20, no. 1, pp. 81–97, 2010.
[28]
S. Kuntanapreeda, “Robust synchronization of fractional-order unified chaotic systems via linear control,” Computers & Mathematics with Applications, vol. 63, no. 1, pp. 183–190, 2012.
[29]
G. Peng, “Synchronization of fractional order chaotic systems,” Physics Letters A, vol. 363, no. 5-6, pp. 426–432, 2007.
[30]
L.-G. Yuan and Q.-G. Yang, “Parameter identification and synchronization of fractional-order chaotic systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 305–316, 2012.
[31]
A. S. Hegazi, E. Ahmed, and A. E. Matouk, “On chaos control and synchronization of the commensurate fractional order Liu system,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 5, pp. 1193–1202, 2013.
[32]
H. Zhu, S. Zhou, and Z. He, “Chaos synchronization of the fractional-order Chen's system,” Chaos, Solitons and Fractals, vol. 41, no. 5, pp. 2733–2740, 2009.
[33]
A. E. Matouk, “Chaos, feedback control and synchronization of a fractional-order modified autonomous Van der Pol-Duffing circuit,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 2, pp. 975–986, 2011.
[34]
J. Yan and C. Li, “On chaos synchronization of fractional differential equations,” Chaos, Solitons and Fractals, vol. 32, no. 2, pp. 725–735, 2007.
[35]
S. Bhalekar, V. Daftardar-Gejji, and S. Das, “Synchronization of different fractional order chaotic systems using active control,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 11, pp. 3536–3546, 2010.
[36]
M. M. Asheghan, M. T. H. Beheshti, and M. S. Tavazoei, “Robust synchronization of perturbed Chen’s fractional-order chaotic systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 6, pp. 1044–1051, 2011.
[37]
L. Pan, W. Zhou, L. Zhou, and K. Sun, “Chaos synchronization between two different fractional-order hyperchaotic systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 6, pp. 2628–2640, 2011.
[38]
Z. Odibat, “A note on phase synchronization in coupled chaotic fractional order systems,” Nonlinear Analysis: Real World Applications, vol. 13, no. 2, pp. 779–789, 2012.
[39]
M. Mahmoudian, R. Ghaderi, A. Ranjbar, and J. Sadati, “Synchronization of fractional-order chaotic system via adaptive PID controller,” in New Trends in Nanotechnologyand Fractional Calculus Applications, pp. 445–452, Springer, Heidelberg, Germany, 2010.
[40]
L. Jie, L. Xinjie, and Z. Junchan, “Prediction-control based feedback control of a fractional order unified chaotic system,” in Proceedings of the Chinese Control and Decision Conference (CCDC '11), pp. 2093–2097, Mianyang, China, May 2011.
[41]
A. G. Radwan, K. Moaddy, K. N. Salama, S. Momani, and I. Hashimb, “Control and switching synchronization of fractionalorder chaotic systems using active control technique,” Journal of Advanced Research, 2013.
[42]
J. J. E. Slotine and W. Li, Applied Nonlinear Control, vol. 1, Prenticehall, New Jersey, NJ, USA, 1991.
[43]
C. Yin, S.-m. Zhong, and W.-f. Chen, “Design of sliding mode controller for a class of fractional-order chaotic systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 356–366, 2012.
[44]
M. P. Aghababa, “Robust stabilization and synchronization of a class of fractional-order chaotic systems via a novel fractional sliding mode controller,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2670–2681, 2012.
[45]
S. Dadras and H. R. Momeni, “Control of a fractional-order economical system via sliding mode,” Physica A, vol. 389, no. 12, pp. 2434–2442, 2010.
[46]
D.-Y. Chen, Y.-X. Liu, X.-Y. Ma, and R.-F. Zhang, “Control of a class of fractional-order chaotic systems via sliding mode,” Nonlinear Dynamics, vol. 67, no. 1, pp. 893–901, 2012.
[47]
M. R. Faieghi, H. Delavari, and D. Baleanu, “Control of an uncertainfractional-order Liu system via fuzzy fractional-order sliding modecontrol,” Journal of Vibration and Control, vol. 18, pp. 1366–1374, 2012.
[48]
M. R. Faieghi, H. Delavari, and D. Baleanu, “A note on stability of sliding mode dynamics in suppression of fractional-order chaotic systems,” Computers & Mathematics with Applications, vol. 66, no. 5, pp. 832–837, 2013.
[49]
C. Yin, S. Dadras, and S.-M. Zhong, “Design an adaptive sliding mode controller for drive-response synchronization of two different uncertain fractional-order chaotic systems with fully unknown parameters,” Journal of the Franklin Institute, vol. 349, no. 10, pp. 3078–3101, 2012.
[50]
Y. Jian, S. Bao, J. Wenqiang, and P. Tetie, “Sliding mode control of the fractional order unified chaotic system,” Abstract and Applied Analysis. In press.
[51]
J. Sabatier, C. Farges, and J. C. Trigeassou, “Fractional systems state space description: some wrong ideas and proposed solutions,” Journal of Vibration and Control, 2013.
[52]
J. Trigeassou and N. Maamri, “The initial conditions of Riemman-Liouville and Caputo derivatives: an integrator interpretation,” in Proceedings of the Food and Drug Administration (FDA'10), Badajoz, Spain, 2010.
[53]
J. C. Trigeassou, N. Maamri, J. Sabatier, and A. Oustaloup, “State variables and transients of fractional order differential systems,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3117–3140, 2012.
[54]
J. C. Trigeassou, N. Maamri, J. Sabatier, and A. Oustaloup, “Transients of fractional-order integrator andderivatives Signal,” Image and Video Processing, vol. 6, no. 3, pp. 359–372, 2012.
[55]
J. C. Trigeassou and N. Maamri, “Initial conditions and initialization of linear fractional differential equations,” Signal Processing, vol. 91, no. 3, pp. 427–436, 2011.
[56]
C. F. Lorenzo and T. T. Hartley, “Initialization of fractional-order operators and fractional differential equations,” Journal of Computational and Nonlinear Dynamics, vol. 3, no. 2, Article ID 021101, 2008.
[57]
T. T. Hartley and C. F. Lorenzo, “The error incurred using the Caputo-derivative Laplace transform,” in Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (DETC '07), vol. 87648, San Diego, Calif, USA, September 2009.
[58]
C. F. Lorenzo and T. T. Hartley, “Time-varying initialization and Laplace transformof the Caputo derivative: with order between zero and one,” in Proceedings of the ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (IDETC/CIE '11), pp. 28–31, Washington, DC, USA, August 2011.
[59]
J. C. Trigeassou, N. Maamri, J. Sabatier, and A. Oustaloup, “A lyapunov approach to the stability of fractional differential equations,” Signal Processing, vol. 91, no. 3, pp. 437–445, 2011.
[60]
Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, 2007.
[61]
I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, vol. 198, Academic Press, San Diego, Calif, USA, 1999.
[62]
R. Caponetto, Fractional Order Systems, vol. 72, World Scientific Publishing Company, 2010.
[63]
C. A. Monje, Y. Chen, B. M. Vinagre, D. Xue, and V. Feliu, Fractional-Order Systems and Controls: Fundamentals and Applications, Springer, London, UK, 2010.
[64]
M. Roopaei, B. Ranjbar Sahraei, and T.-C. Lin, “Adaptive sliding mode control in a novel class of chaotic systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp. 4158–4170, 2010.
[65]
L. Fridman, J. Moreno, and R. Iriarte, Sliding Modes After the First Decade of the 21st Century: State of the Art, vol. 412, Springer, 2011.
