Observer-Based Approach for Fractional-Order Chaotic Synchronization and Secure Communication

This paper presents a method based on the state observer design for constructing a chaotically synchronized systems. Fractional-order direct Lyapunov theorem is used to derive the closed-loop asymptotic stability. The gains of the observer and observer-based controller are obtained in terms of linear matrix inequalities (LMIs) formulation. The proposed approach is then applied to secure communications. The method combines chaotic masking and chaotic modulation, where the information signal is injected into the transmitter and simultaneously transmitted to the receiver. Chaotic synchronization and chaotic communication are achieved simultaneously via a state observer design technique. The fractional-order chaotic Lorenz and Lü systems are given to demonstrate the applicability of the proposed approach.

[1]  M. Boutayeb,et al.  Generalized state-space observers for chaotic synchronization and secure communication , 2002 .

[2]  S. Bhalekar,et al.  Synchronization of different fractional order chaotic systems using active control , 2010 .

[3]  N. Laskin.  Fractional market dynamics , 2000 .

[4]  S. Manabe.  A Suggestion of Fractional-Order Controller for Flexible Spacecraft Attitude Control , 2002 .

[5]  A. Méhauté,et al.  Introduction to transfer and motion in fractal media: The geometry of kinetics , 1983 .

[6]  Jürgen Kurths,et al.  Synchronization - A Universal Concept in Nonlinear Sciences , 2001, Cambridge Nonlinear Science Series.

[7]  Ivo Petras,et al.  A note on the fractional-order Chua’s system , 2008 .

[8]  Darong Lai,et al.  Generalized synchronization of the fractional-order chaos in weighted complex dynamical networks with nonidentical nodes , 2012 .

[9]  Gérard Scorletti,et al.  Control of rational systems using linear-fractional representations and linear matrix inequalities , 1996, Autom..

[10]  Weihua Deng,et al.  Short memory principle and a predictor-corrector approach for fractional differential equations , 2007 .

[11]  R. Bagley,et al.  Fractional order state equations for the control of viscoelasticallydamped structures , 1991 .

[12]  Zidong Wang,et al.  Pinning control of fractional-order weighted complex networks. , 2009, Chaos.

[13]  Mohammad Saleh Tavazoei,et al.  Limitations of frequency domain approximation for detecting chaos in fractional order systems , 2008 .

[14]  Ivo Petras,et al.  A note on the fractional-order Volta’s system , 2010 .

[15]  Guoqing Chen,et al.  An RLC interconnect model based on fourier analysis , 2005, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[16]  P. Khargonekar,et al.  Robust stabilization of uncertain linear systems: quadratic stabilizability and H/sup infinity / control theory , 1990 .

[17]  Saïd Mammar,et al.  Observer for Lipschitz nonlinear systems: Mean Value Theorem and sector nonlinearity transformation , 2012, 2012 IEEE International Symposium on Intelligent Control.

[18]  Chunguang Li,et al.  Chaos in the fractional order Chen system and its control , 2004 .

[19]  A. S. MorseCenter.  Certainty Equivalence Implies Detectability , 1998 .

[20]  R. Bagley,et al.  The fractional order state equations for the control of viscoelastically damped structures , 1989 .

[21]  B. Onaral,et al.  Linear approximation of transfer function with a pole of fractional power , 1984 .

[22]  Jürgen Kurths,et al.  Synchronization: Phase locking and frequency entrainment , 2001 .

[23]  J. Tsinias.  Observer design for nonlinear control systems , 1989 .

[24]  Teh-Lu Liao,et al.  An observer-based approach for chaotic synchronization with applications to secure communications , 1999 .

[25]  Victor George Jenson,et al.  Mathematical Methods in Chemical Engineering , 1978 .

[26]  R. Rajamani.  Observers for Lipschitz nonlinear systems , 1998, IEEE Trans. Autom. Control..

[27]  N. Engheta.  On fractional calculus and fractional multipoles in electromagnetism , 1996 .

[28]  Eduardo Sontag,et al.  Output-to-state stability and detectability of nonlinear systems , 1997 .

[29]  Igor Podlubny,et al.  Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation , 2001, math/0110241.

[30]  Jürgen Kurths,et al.  Complex Dynamics in Physiological Systems: From Heart to Brain , 2009 .

[31]  Mohammad Saleh Tavazoei,et al.  A necessary condition for double scroll attractor existence in fractional-order systems , 2007 .

[32]  Mohamed Darouach,et al.  Observer-based control for fractional-order continuous-time systems , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[33]  J García-Ojalvo,et al.  Spatiotemporal communication with synchronized optical chaos. , 2000, Physical review letters.

[34]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[35]  M. Darouach,et al.  Full-order observers for linear systems with unknown inputs , 1994, IEEE Trans. Autom. Control..

[36]  Hamid Reza Momeni,et al.  Observer Based Control of a Class of Nonlinear Fractional Order Systems Using LMI , 2012 .

[37]  Naser Pariz,et al.  A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter , 2009 .

[38]  Wang Zicai.  Observer Design for a Class of Nonlinear Systems , 1998 .

[39]  J. Sprott.  Chaos and time-series analysis , 2001 .

[40]  J. Gauthier,et al.  A simple observer for nonlinear systems applications to bioreactors , 1992 .

[41]  I. Podlubny.  Fractional differential equations , 1998 .

[42]  Yangquan Chen,et al.  Computers and Mathematics with Applications Stability of Fractional-order Nonlinear Dynamic Systems: Lyapunov Direct Method and Generalized Mittag–leffler Stability , 2022 .

[43]  Mona E. Zaghloul,et al.  Improved masking algorithm for chaotic communications systems , 1996 .

[44]  B. d'Andrea-Novel,et al.  Observer-based controllers for fractional differential systems , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[45]  K. Cole.  ELECTRIC CONDUCTANCE OF BIOLOGICAL SYSTEMS , 1933 .

[46]  Gildas Besançon.  2 Observer Design for Nonlinear Systems , 2006 .

[47]  I. Petráš.  Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation , 2011 .

[48]  W. Deng,et al.  Chaos synchronization of the fractional Lü system , 2005 .