CHAOS CONTROL AND SYNCHRONIZATION USING SYNERGETIC CONTROLLER WITH FRACTIONAL AND LINEAR EXTENDED MANIFOLD
In this manuscript, for the first time, a fractional-order manifold in a synergetic approach using a fractional order controller is introduced. Furtheremore, in the synergetic theory a macro variable is expended into a linear combination of state variables. An aim is to increase the convergence rate as well as time response of the whole closed loop system. Quality of the proposed controller is investigated to control and synchronize a nonlinear chaotic Coullet system in comparison with an integer order manifold synergetic controller. The stability of the proposed controller is proven using the Lyapunov method. In this regard stabilizing control effort is yielded. Simulation result confirm convergence of states towards zero. This is achieved through a control effort with fewer oscillations and lower amplitude of signls which confirm feasibility of the control effort in practice.
KEYWORDS: synergetic control theory; fractional order system; synchronization; nonlinear chaotic Coullet system; chaos control
 Djennoune S, Bettayeb M. (2013) Optimal synergetic control for fractional-order systems. Automatica, 49(7):2243-2249.
 Jiang Z. (2009) Design of a nonlinear power system stabilizer using synergetic control theory. Electric Power Sys. Res., 79(6):855-862.
 Jiang Z, Dougal RA. (2004) Synergetic control of power converters for pulse current charging of advanced batteries from a fuel cell power source. IEEE Trans. Power Electronic, 19(4):1140-1150.
 Santi E, et al. (2003) Synergetic control for DC-DC boost converter: implementation options.IEEETrans.Ind.App., 39(6):1803-1813. http://dx.doi.org/10.1109/TIA.2003.818967
 Ni J, et al. (2014) Variable speed synergetic control for chaotic oscillation in power system. Nonlinear Dynamics, 78(1):681-690.
 Delavari H, Lanusse P, Sabatier J. (2013) Fractional order controller design for a flexible link manipulator robot. Asian J. Control, 15(3):783-795.
 El-Khazali R. (2013) Fractional-order controller design. Computers & Mathematics with Applications, 66(5):639-646.
 Faieghi MR, Delavari H, Baleanu D. (2012) Control of an uncertain fractional-order Liu system via fuzzy fractional-order sliding mode control. J. Vib. Control, 18(9), 1366-1374.
 Kilbas AAA, Srivastava HM, Trujillo JJ. (2006) Theory and applications of fractional differential equations, vol. 204, Elsevier Science Limited.
 Magin RL. (2006) Fractional calculus in bioengineering. Begell House, Rodding.
 Mainardi F. (1997) Fractional calculus. Springer.
 Monje, CA, et al. (2010) Fractional-order systems and controls: fundamentals and applications. Springer.
 Odibat Z, Momani S. (2006) Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. Num. Simulation, 7(1):27-34.
 Podlubny I. (1998) 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.
 Sabatier J, Agrawal OP, Machado JT. (2007) Advances in fractional calculus. Springer.
 Luo Y, et al. (2010) Tuning fractional order proportional integral controllers for fractional order systems. J. Process Control, 20(7):823-831.
 Meng L, Wang DF, Han P. (2012) Identification of fractional order system using particle swarm optimization. IEEE International Conference on Machine Learning and Cybernetics (ICMLC).
 Saxena R, Mathai A, Haubold H. (2004) Unified fractional kinetic equation and a fractional diffusion equation. Astrophysics and Space Science, 290(3-4):299-310.
 Bagley RL, Torvik P. (1983) A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheology, 27(3):201-210.
 Di Paola M, Zingales M. (2012) Exact mechanical models of fractional hereditary materials. J. Rheology, 56(5):983-1004.
 Hilfer R, et al. (2000) Applications of fractional calculus in physics. vol. 128. World Scientific.
 Poinot T, Trigeassou J. (2004) Identification of fractional systems using an output-error technique. Nonlinear Dynamics, 38,133-154.
 Cois O, Oustaloup A, Battaglia E. (2000) Non-integer model from modal decomposition for time domain system identification. 12th IFAC Symposium on System Identification, Santa Barbara, CA, USA.
 Sabatier J, et al. (2002) CRONE control: Principles and extension to time-variant plants with asymptotically constant coefficients. Nonlinear Dynamics, 29(1-4):363-385.
 Podlubny I. (1999) Fractional-order systems and PI/sup/spl lambda//D/sup/spl mu//-controllers. IEEE Trans. Autom. Control, 44(1):208-214.
 Nechadi E, et al. (2012) Type-2 fuzzy based adaptive synergetic power system control. Electric Power Sys. Res., 88:9-15.
 Santi E, et al. (2004) Synergetic control for power electronics applications: a comparison with the sliding mode approach. J. Circ. Sys. Comp., 13(4):737-760.
 Caputo M. (1969) Elasticita´ e dissipazione. Zanichelli. Bologna.
 Caputo M. (1989) The rheology of an anelastic medium studied by means of the observation of the splitting of its eigenfrequencies. J. Acoustical Soc. America, 86(5):1984-1987.
 El-Sayed AM. (1995) Fractional order evolution equations. J. Fract. Calc, 7:89-100.
 El-Sayed AM, Ibrahim A. (1995) Multivalued fractional differential equations. App. Math. Comp., 68(1):15-25.
 Vinagre B, et al. (2002) Using fractional order adjustment rules and fractional order reference models in model-reference adaptive control. Nonlinear Dynamics, 29(1-4):269-279.
 Xue D, Zhao C, Chen Y. (2006) Fractional order PID control of a DC-motor with elastic shaft: a case study. Proceedings of American Control Conference.
 Jesus IS, Machado JT. (2008) Fractional control of heat diffusion systems. Nonlinear Dynamics, 54(3):263-282.
 Arneodo A, Coullet P, Tresser C. (1981) Possible new strange attractors with spiral structure. Comm. Math. Phys., 79(4):573-579.
 Liu Y, Chen L. (2013) Chaos in attitude dynamics of spacecraft. Springer.
 Petržela J, Kolka Z, Hanus S. (2011) Simple chaotic oscillator: from mathematical model to practical experiment. Models and Applications of Chaos Theory in Modern Sciences, p. 317.
The IIUM Engineering journal follows the open access policy.