# Adaptive, Observer-Based Synchronization of Different Chaotic Systems
## 1 Introduction  
As chaotic behavior is so sensitive to initial conditions, it is hard to believe that two chaotic systems may achieve synchronous motion. Fortunately, as it was demonstrated in [1], proper control can solve this problem. Despite 30 years of research, synchronization of chaos remains a hot research theoretical topic and stimulates numerous applications. It is a well-established fact that effective synchronization of chaotic systems is wildly used in secure communication [2; 3; 4]. Chaotic motion is observed in chemical processes, and synchronization of chaotic oscillations with periodic motion is crucial for certain chemical technologies [5]. The understanding of chaotic dynamics of biological and ecological systems [6; 7] helps to apply proper methods to synchronize chaotic motion and to improve the system welfare, despite the destructive activity of humankind. Chaos synchronization in laser systems [8; 9] is investigated because of potential applications in secure communication, new technologies such as chaotic Lidar, or random number generators. The mentioned areas of research on chaos synchronization and control are just a few of many reported. Especially, synchronization of two quite different chaotic systems is a new, interesting fundamental problem and may lead to important applications [10; 11]. Numerous biological systems (such as circulatory and respiratory systems [12]) behave synchronously, although they are quite different, and the achievement of this synchronous behavior determines health or disease. Hence, synchronization of different chaotic systems may be considered as a treatment restoring health and welfare. Synchronization of different chaotic systems will open new opportunities in secure communication and any other applications mentioned above. Any real system's parameters are inevitably perturbed by external factors and cannot be exactly known. Therefore, synchronization of two different chaotic systems inthe presence of unknown parameters is more essential and useful in real-life applications, and this problem will be reflected in this contribution. We can distinguish two main approaches to the synchronization of chaotic systems. The first approach is to propose a proper control for a slave system to follow the desired trajectory, generated by a master system. Numerous control techniques are reported: sliding mode control [13; 14], adaptive backstepping [15; 16], dynamic surface control (DCS) [17; 18], etc. Commonly, measurement of the master system output is not enough to synthesize the controller. Higher derivatives of the master output are usually necessary to synthesize the controller, but these signals are rarely available. Using the second approach, the slave system is made to be an observer for the master system [19; 20; 21]. Therefore, the structure of the slave system is defined by the master system, and the possibilities to synchronize two different chaotic systems are limited. In this contribution, we connect both approaches by proposing the novel structure shown in Figure 1. The observer provides estimated state variables to contribute to the control law while tracking errors are fed back from the controller to improve the observer. Several nonlinear adaptive observers are reported in the literature: Luenberger-type observer [22; 23; 24], Kalman filter [25], sliding mode observers [26; 27], high gain [28; 29], etc. Some of them require special conditions for a nonlinear system, for example, Lipschitz  
Figure 1: General scheme of the proposed approach. type [30; 31] or quadratic-type [32] constraints. After careful consideration, we selected a classical solution reported in [33] based on K filters presented in [34]. Such observers are smooth and free from application difficulties common for sliding mode and high-gain observers, do not require any global constraints for system nonlinearities (as it is assumed in [35], for example), and a linear approximation of system nonlinearities (as in [36]) is not necessary.