Adaptive observer-Based Synchronization for Communication

In recent years a growing interest was observed in the problem of synchronizing chaotic systems [Afraimovich et al., 1987; Pecora & Carroll, 1990; Cuomo et al., 1993; Blekhman et al., 1995; Nijmeijer & Mareels, 1997; Pogromsky & Nijmeijer, 1999]. It was motivated not only by scientific interest in the problem, but also by practical applications in different fields [Blekhman, 1988; Lindsey, 1972], particularly in telecommunications [Kocarev et al., 1992; Cuomo et al., 1993; Dedieu et al., 1993]. However most design methods were suggested and justified under conditions that all the system parameters are known and states are available for measurement. Also, some methods apply only to low-dimensional systems. Of practical interest is the problem of synchronizing two or more systems when not only initial state but also some parameters are not known to the designer of the receiver. This more complicated problem, which may correspond to the case where parameter modulation is used for message transmission, is referred to as adaptive synchronization [Fradkov, 1994, 1995; Wu et al., 1996; Markov & Fradkov, 1997]. Control theory opens new horizons in the synchronization problem and allows to give general framework for its study [Blekhman et al., 1997]. This paper is devoted to design of an adaptive observer oriented to the synchronization for the purpose of communications. A simple design of an adaptive observer for estimating the unknown parameters of the transmitter is proposed based on the design of the Lyapunov function for error system. It provides necessary and sufficient conditions for the existence of a quadratic Lyapunov function for