Practically stable observer-based synchronization of discrete-time chaotic systems over the limited-band communication channel Alexander L. Fradkov, Boris Andrievsky and Alexey Andrievsky Abstract— Solution to the problem of observer-based syn- chronization over the limited bandwidth communication link for a class of discrete-time chaotic systems is presented. The result is demonstrated on the synchronization of chaotic Henon systems via a channel with limited capacity. It is shown that the proposed synchronization system is practically stable in the sense that the limit synchronization error can be made depending only on the number of digits in the computer. Key words: Chaotic behavior, Communication constraints, Synchronization, Practical stability I. I NTRODUCTION Chaotic synchronization has attracted the attention of researchers since the 1980s [1]–[4] and is still an area of active research [5]–[9]. Recently information-theoretic con- cepts were applied to analyze and quantify synchronization [10]–[14]. In [11], [12] mutual information measures were introduced for evaluating the degree of chaotic synchro- nization. In [10], [13] the methods of symbolic dynamics were used to relate synchronization precision to capacity of the information channel and to the entropy of the drive system. Baptista and Kurths [14] introduced the concept of a chaotic channel as a medium formed by a network of chaotic systems that enables information from a source to pass from one system (transmitter) to another system (receiver). They characterized a chaotic channel by the mutual information (difference between the sum of the positive Lyapunov expo- nents corresponding to the synchronization manifold and the sum of positive exponents corresponding to the transverse manifold). However, in existing papers limit possibilities for the precision of controlled synchronization have not been analyzed. Recently the limitations of control under constraints im- posed by a finite capacity information channel have been in- vestigated in detail in the control theoretic literature, see [15] and references therein. It was shown that stabilization under information constraints is possible if and only if the capacity of the information channel exceeds the entropy production of the system at the equilibrium [16]–[18]. In [19], [20] a general statement was proposed, claiming that the difference between the entropies of the open loop and the closed loop systems cannot exceed the information introduced by the A. L. Fradkov and B. Andrievsky are with the Institute for Problems of Mechanical Engineering, Russian Academy of Sciences, 61, V.O. Bolshoy Ave., 199178, Saint Petersburg, Russia. {bandri,alf}@control.ipme.ru A. Andrievsky is with the Control Systems Department, Baltic State Technical University, Saint Petersburg, Russia alexeyandrievsky@mail.ru Corresponding author: Prof. Boris Andrievsky. controller, including the transmission rate of the information channel. However, results of the mentioned works on control sys- tem analysis and design under information constraints do not apply to synchronization systems since in a synchronization problem trajectories in the phase space converge to a set (a manifold) rather than to a point, i.e. in the general case the problem cannot be reduced to simple stabilization. The problem is still more complicated for nonlinear systems, for incomplete state measurements and in the presence of uncertainty. Specifically, almost nothing is known about limit possibilities of estimation and control under information constraints for the partial stabilization, or set stabilization problem. Such a problem arises if one needs to stabilize a limit cycle or a chaotic attractor, which is important for the control of oscillatory modes in engineering systems [5], [21], [22]. However, analytical performance estimates of chaotic control systems are known only for a few cases, even without information constraints, see, e.g. [23], [24]; their develop- ment requires a sophisticated mathematical apparatus. Observer-based synchronization systems are used in the case of incomplete measurements, when all phase vari- ables are not available for measurement and coupling. Such systems are well studied without information constraints [25]–[27]. Observer-based synchronization of continuous- time chaotic systems under information constraints is studied in [28], [29], where limit possibilities are established. Similar results for adaptive synchronization were recently obtained in [30]. The papers [28]–[30] deal with synchronization of continuous-time chaotic systems over the digital communi- cation link with finite capacity. The overall system can be naturally viewed as a hybrid one, i.e., system described by a coupling between continuous and discrete dynamics [31]. In the mentioned works the sampling rate was considered as a design parameter and the one-step-memory coder was used. Based on these conditions, it was established in [28]– [30] that the binary coding procedure minimizes the bit-per- second data rate over the channel, and also the ratio between the optimal sample time and the upper bound of the limit synchronization error was found. The present paper is devoted to the synchronization prob- lem of the discrete-time chaotic systems. In this case, the sampling time is not considered as a system parameter, and the bit-per-step rate is used as a measure of the channel capacity. Besides, the overall system for the discrete case is not a hybrid one. Therefore, the errors of modeling continuous-time systems by difference equations do not occur. This makes possible to use the full-order coders,