In this chapter we consider the problem of state space reconstruction given noisy signals from known chaotic systems. This problem is equivalent to chaos synchronization and is solved with the help of tools from optimal control theory, namely the Kalman filter. The presented approach is introduced for chaotic Lur'e systems to replace the widely used error feedback synchronization. This is in fact an extension of the previously used methods and allows to take into account knowledge about observation noise on the strange attractor to be filtered. We show that the filtering performance is superior to that of error feedback synchronization, in particular if the noise level is non-negligible. Since most of the chaotic electronic circuits are indeed of Lur'e type, the proposed method is suitable for filtering and reconstructing the state of such circuits as used for chaos applications in signal processing or telecommunications.

24.1 Introduction

The discovery that chaotic systems can be synchronized ^[12] can be considered a key event for the research and application of chaotic systems. It has inspired a number of chaos-based master-slave systems, in particular for communication applications ^[4]. Early attempts are typically based on some form of error feedback synchronization. However, it was found that the extreme sensitivity to noise on the transmission channel (of the employed synchronization schemes) practically prohibits a competitive use of this technique in practical information transmission systems.