Chaotic observer-based synchronization under information constraints.

Limitations of observer-based synchronization systems under information constraints (limited information capacity of the coupling channel) are evaluated. We give theoretical analysis for multidimensional drive-response systems represented in the Lurie form (linear part plus nonlinearity depending only on measurable outputs). It is shown that the upper bound of the limit synchronization error (LSE) is proportional to the upper bound of the transmission error. As a consequence, the upper and lower bounds of LSE are proportional to the maximum rate of the coupling signal and inversely proportional to the information transmission rate (channel capacity). Optimality of the binary coding for coders with one-step memory is established. The results are applied to synchronization of two chaotic Chua systems coupled via a channel with limited capacity.

[1]  Robin J. Evans,et al.  Stabilizability of Stochastic Linear Systems with Finite Feedback Data Rates , 2004, SIAM J. Control. Optim..

[2]  Giorgio Parisi,et al.  Physica A: Statistical Mechanics and its Applications: Editorial note , 2005 .

[3]  Seth Lloyd,et al.  Information-theoretic approach to the study of control systems , 2001, physics/0104007.

[4]  Alexander L. Fradkov.  Cybernetical Physics: From Control of Chaos to Quantum Control , 2007 .

[5]  Alexander L. Fradkov,et al.  On self-synchronization and controlled synchronization , 1997 .

[6]  S. Boccaletti,et al.  Synchronization of chaotic systems , 2001 .

[7]  M. Hasler,et al.  Persistent clusters in lattices of coupled nonidentical chaotic systems. , 2003, Chaos.

[8]  T. Carroll,et al.  Master Stability Functions for Synchronized Coupled Systems , 1998 .

[9]  Touchette,et al.  Information-theoretic limits of control , 1999, Physical review letters.

[10]  Guanrong Chen,et al.  Secure synchronization of a class of chaotic systems from a nonlinear observer approach , 2005, IEEE Transactions on Automatic Control.

[11]  V. Astakhov,et al.  Information theoretic approach to quantify complete and phase synchronization of chaos. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

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

[14]  R. Brockett,et al.  Systems with finite communication bandwidth constraints. I. State estimation problems , 1997, IEEE Trans. Autom. Control..

[15]  H. Fujisaka,et al.  Stability Theory of Synchronized Motion in Coupled-Oscillator Systems , 1983 .

[16]  Aleksandr L. Fradkov,et al.  Application of cybernetic methods in physics , 2005 .

[17]  M. Rabinovich,et al.  Stochastic synchronization of oscillation in dissipative systems , 1986 .

[18]  Sanjoy K. Mitter,et al.  Endcoding complexity versus minimum distance , 2005, IEEE Transactions on Information Theory.

[19]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[20]  M Palus,et al.  Synchronization as adjustment of information rates: detection from bivariate time series. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  C. K. Yuen,et al.  Introduction to Digital Filtering , 1978, IEEE Transactions on Systems, Man, and Cybernetics.

[22]  C. Grebogi,et al.  Using geometric control and chaotic synchronization to estimate an unknown model parameter. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  Ljupco Kocarev,et al.  Applications of symbolic dynamics in chaos synchronization , 1997 .

[24]  Axthonv G. Oettinger,et al.  IEEE Transactions on Information Theory , 1998 .

[25]  Valentin Afraimovich,et al.  Synchronization in lattices of coupled oscillators with Neumann/periodic boundary conditions , 1998 .

[26]  L. Ljung,et al.  Control theory : multivariable and nonlinear methods , 2000 .

[27]  Wen-Wei Lin,et al.  Asymptotic Synchronization in Lattices of Coupled Nonidentical Lorenz equations , 2000, Int. J. Bifurc. Chaos.

[28]  Daniel Liberzon,et al.  Hybrid feedback stabilization of systems with quantized signals , 2003, Autom..

[29]  Nathan van de Wouw,et al.  Convergent dynamics, a tribute to Boris Pavlovich Demidovich , 2004, Syst. Control. Lett..

[30]  Alexander L. Fradkov,et al.  Introduction to Control of Oscillations and Chaos , 1998 .

[31]  Shui-Nee Chow,et al.  Synchronization in lattices of coupled oscillators , 1997 .

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

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

[34]  Ned J Corron,et al.  Information flow in chaos synchronization: fundamental tradeoffs in precision, delay, and anticipation. , 2003, Physical review letters.

[35]  Sekhar Tatikonda,et al.  Control under communication constraints , 2004, IEEE Transactions on Automatic Control.

[36]  Daniel Liberzon,et al.  Quantized feedback stabilization of linear systems , 2000, IEEE Trans. Autom. Control..

[37]  Robin J. Evans,et al.  Topological feedback entropy and Nonlinear stabilization , 2004, IEEE Transactions on Automatic Control.

[38]  M. Hasler,et al.  Connection Graph Stability Method for Synchronized Coupled Chaotic Systems , 2004 .

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

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

[41]  Henk Nijmeijer,et al.  A dynamical control view on synchronization , 2001 .