# Adaptive, Observer-Based Synchronization of Different Chaotic Systems
## 2 Adaptive Observer for an Output Nonlinear Parametric System  
We consider a general nonlinear system (nonlinear in states and inputs), but we assume that it is transformable (by a certain nonlinear state transformation, which may depend on unknown parameters) into a nonlinear system described by state and output equations as follows:  
\[\dot{\mathbf{z}}=\mathbf{A}\mathbf{z}+\mathbf{\varphi}(y,\mathbf{r})+\mathbf{F}(y,\mathbf{r})^{T}\mathbf{\theta} \tag{1}\]  
\[y=c^{T}\mathbf{z}, \tag{2}\]  
where \(\mathbf{z}\) denotes a vector of \(n\) state variables; \(y\) is a scalar, measurable output; \(\mathbf{r}\) is an external, measurable input. The matrices \(\mathbf{A}\), \(c^{T}\) of appropriate dimensions (\(n\times n\), \(1\times n\)) are known, as well as nonlinear functions \(\mathbf{\varphi}(y,\mathbf{r})\), \(\mathbf{F}(y,\mathbf{r})\), while \(\mathbf{\theta}\) represents a vector of \(p\) unknown, constant, bounded parameters. The design of an adaptive observer for the system (1), (2) is discussed in this section. Let us call the assumed system model (1), (2) an output-nonlinear parametric system (ONP). The model (1), (2) is restrictive as the nonlinearities and terms with unknown parameters in the state equation depend on the measured signals only (that is why we call it output-nonlinear). We have to stress that this restriction is not imposed because of the adaptive control. Even if the parameters are known, the class of systems globally stabilizable by output feedback is not much wider than the class provided by (1), (2) [33; 40]. Despite this, many important chaotic systems may be transformed to the form (1), (2), so we claim that the proposed approach is general enough. The system form (1), (2) corresponds with the selected observer design technique, and this was widely discussedand compared with other opportunities in Section 1. The features of ONP systems and the systems transformable into the ONP form are an interesting research topic, although outside the scope of this paper. Let us mention that the system obtained by a linear state transformation from any ONP system remains in the ONP form. The system (1), (2) may be chaotic or not. We assume that there exists an output-feedback-gain matrix \(\mathbf{k}\), such that \(\mathbf{A}_{o}:=\mathbf{A}-\mathbf{k}\mathbf{c}^{T}\) is stable, and therefore, for any positive definite matrix \(\mathbf{Q}\), the Lyapunov equation  
\[\mathbf{A}_{o}^{T}\mathbf{R}+\mathbf{R}\mathbf{A}_{o}=-\mathbf{Q} \tag{3}\]  
possesses a positive definite solution \(\mathbf{R}\). Of course, working with a single output system (\(\mathbf{k}\) is a column vector with \(n\) parameters) is a special case of multidimensional output. Multioutput case means that more than \(n\) feedback parameters may be used to satisfy (3). Therefore, the single output is the most restrictive case. Transmission of a single signal is also an advantage in secure communication, which is the main application field of chaos synchronization. Motivated by the idea of K filters presented in [33; 34], we introduce an observer, so that we are able to reconstruct the state variables \(\mathbf{z}\) despite unknown parameters \(\mathbf{\theta}\).