An Extended Kalman Filtering Approach to Nonlinear Time-Delay Systems: Application to Chaotic Secure Communications

An observer design problem for a class of nonlinear time-delay systems subject to white Gaussian noise is studied. The problem addressed is the design of a full-state observer based on the extended Kalman filter (EKF), in which two terms are added to the Riccati differential equation. One of those makes the system robust to noise influences, and the other one is utilized for dealing with the delayed state. A Lyapunov-Krasovskii functional is used for giving sufficient conditions for local stability of the reconstruction error dynamics. A hybrid chaotic cryptosystem procedure based on the n-shift cipher is combined with synchronization based on the EKF for developing a secure communication system. A semiconductor laser with optical feedback is employed as a chaotic signal generator in which a full-state estimation via the EKF is achieved.

[1]  R. Lang,et al.  External optical feedback effects on semiconductor injection laser properties , 1980 .

[2]  P. Krishnaprasad,et al.  Dynamic observers as asymptotic limits of recursive filters , 1982, CDC 1982.

[3]  V. Kolmanovskii,et al.  Stability of Functional Differential Equations , 1986 .

[4]  Jesper Mørk,et al.  Bistability and low-frequency fluctuations in semiconductor lasers with optical feedback: a theoretical analysis , 1988 .

[5]  T. B. Fowler,et al.  Application of stochastic control techniques to chaotic nonlinear systems , 1989 .

[6]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

[7]  Alan V. Oppenheim,et al.  Synchronization of Lorenz-based chaotic circuits with applications to communications , 1993 .

[8]  Morgül,et al.  Observer based synchronization of chaotic systems. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[9]  Gavrielides,et al.  Lang and Kobayashi phase equation. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[10]  Henk Nijmeijer,et al.  An observer looks at synchronization , 1997 .

[11]  Leon O. Chua,et al.  Cryptography based on chaotic systems , 1997 .

[12]  Konrad Reif,et al.  An EKF-Based Nonlinear Observer with a Prescribed Degree of Stability , 1998, Autom..

[13]  J. S. Thorp,et al.  PDMA-1: chaotic communication via the extended Kalman filter , 1998 .

[14]  Ali A. Minai,et al.  Communicating with noise: How chaos and noise combine to generate secure encryption keys. , 1998, Chaos.

[15]  K. Alan Shore,et al.  Controlling dynamics in external-cavity laser diodes with electronic impulsive delayed feedback , 1998 .

[16]  Konrad Reif,et al.  The extended Kalman filter as an exponential observer for nonlinear systems , 1999, IEEE Trans. Signal Process..

[17]  K A Shore,et al.  Demonstration of optical synchronization of chaotic external-cavity laser diodes. , 1999, Optics letters.

[18]  R. Unbehauen,et al.  Stochastic stability of the continuous-time extended Kalman filter , 2000 .

[19]  Henry Leung,et al.  An aperiodic phenomenon of the extended Kalman filter in filtering noisy chaotic signals , 2000, IEEE Trans. Signal Process..

[20]  K. A. Shore,et al.  Controlling chaos in a semiconductor laser by external optical injection , 2000 .

[21]  Kevin M. Short,et al.  Reconstructing the keystream from a chaotic encryption scheme , 2001 .

[22]  W. Schwarz,et al.  Chaos and cryptography , 2001 .

[23]  Booncharoen Sirinaovakul,et al.  Introduction to the Special Issue , 2002, Comput. Intell..

[24]  Keith J. Burnham,et al.  On designing observers for time-delay systems with non-linear disturbances , 2002 .

[25]  Antonio Barreiro,et al.  Sonar-based robot navigation using nonlinear robust observers , 2003, Autom..

[26]  M. Boutayeb,et al.  Observers-Based Synchronization and Input Recovery for a Class of Nonlinear Chaotic Models , 2006, IEEE Transactions on Circuits and Systems I: Regular Papers.

[27]  Laurent Larger,et al.  Chaos-based communications at high bit rates using commercial fibre-optic links , 2005, Nature.

[28]  Henk Nijmeijer,et al.  Prediction of chaotic behavior , 2005, IEEE Transactions on Circuits and Systems I: Regular Papers.

[29]  K. Alan Shore,et al.  Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers , 2005 .

[30]  Carlos Pantaleón,et al.  Comments on "An aperiodic phenomenon of the extended Kalman filter in filtering noisy chaotic signals" , 2005, IEEE Trans. Signal Process..

[31]  F. Allgower,et al.  An EKF-based observer for nonlinear time-delay systems , 2006, 2006 American Control Conference.

[32]  E Schöll,et al.  All-optical noninvasive control of unstable steady states in a semiconductor laser. , 2006, Physical review letters.

[33]  Hieu Minh Trinh,et al.  State and Input Simultaneous Estimation for a Class of Time-Delay Systems With Uncertainties , 2007, IEEE Transactions on Circuits and Systems II: Express Briefs.

[34]  Zidong Wang,et al.  A note on chaotic synchronization of time-delay secure communication systems , 2008 .

[35]  Stephen P. Banks,et al.  Stabilisation of chaotic dynamics in semiconductor lasers with optical feedback using optimal control , 2008 .

[36]  Kia Fallahi,et al.  An application of Chen system for secure chaotic communication based on extended Kalman filter and multi-shift cipher algorithm , 2008 .

[37]  Jamal Daafouz,et al.  A connection between chaotic and conventional cryptography , 2008, IEEE Transactions on Circuits and Systems I: Regular Papers.

[38]  Jinde Cao,et al.  Adaptive Stabilization and Synchronization for Chaotic Lur'e Systems With Time-Varying Delay , 2008, IEEE Transactions on Circuits and Systems I: Regular Papers.

[39]  Xin-Chu Fu,et al.  Projective Synchronization of Driving–Response Systems and Its Application to Secure Communication , 2009, IEEE Transactions on Circuits and Systems I: Regular Papers.

[40]  A. Zemouche,et al.  Nonlinear-Observer-Based ${\cal H}_{\infty}$ Synchronization and Unknown Input Recovery , 2009, IEEE Transactions on Circuits and Systems I: Regular Papers.