# Adaptive, Observer-Based Synchronization of Different Chaotic Systems
## 2 Adaptive Observer for an Output Nonlinear Parametric System
Let us define the observer state variables \(\hat{\mathbf{z}}\) and the observer state equation as  
\[\dot{\hat{\mathbf{z}}}=\mathbf{A}_{o}\hat{\mathbf{z}}+\mathbf{\varphi}(\mathbf{y},\mathbf{r})+\mathbf{k} \mathbf{y}+\mathbf{P}^{T}\dot{\hat{\mathbf{\theta}}}+\mathbf{F}(\mathbf{y},\mathbf{r})^{T}\hat{\mathbf{ \theta}}+\mathbf{s} \tag{4}\]  
where  
* The \(p\times n\) matrix variable \(\mathbf{P}\) is an output of a linear filter to be defined;
* The unknown parameters \(\mathbf{\theta}\) are substituted by adaptive parameters \(\hat{\mathbf{\theta}}\), tuned according to adaptive law \[\dot{\hat{\mathbf{\theta}}}=\mathbf{\Gamma}\mathbf{P}\mathbf{e}_{1}-\mathbf{\Gamma}\tau_{1}\] (5) where \(e_{1}:=y-\mathbf{c}^{T}\hat{\mathbf{z}}\); \(\mathbf{\Gamma}\) is a symmetric, positive definite matrix of design parameters;
* The component \(\mathbf{s}\) and the tuning function \(\tau_{1}\) are used to modify the observer dynamics according to the slave system tracking errors, and meanwhile may be assumed equal to zero. The adaptive law (5) will be justified by the analysis of the Lyapunov function including the complete closed-loop system. The form of (5) is standard: It contains a component with an observer error \(e_{1}\) and a tuning component \(\tau_{1}\), which is used to avoid overparameterization. In this way, the observer is prepared for cooperation with the adaptive tracking controller of a slave system, although at this moment, the specific choice of (5) is not necessary. We denote the state estimation error as  
\[\mathbf{e}=\mathbf{z}-\hat{\mathbf{z}}, \tag{6}\]  
the parameter adaptation error as  
\[\widetilde{\mathbf{\theta}}=\mathbf{\theta}-\hat{\mathbf{\theta}}, \tag{7}\]  
and the \"composite\" error as  
\[\mathbf{\varepsilon}=\mathbf{e}-\mathbf{P}^{T}\widetilde{\mathbf{\theta}}. \tag{8}\]  
Differentiation of (8) performed after plugging in (1), (4), and \(\dot{\widetilde{\mathbf{\theta}}}=-\dot{\hat{\mathbf{\theta}}}\) provides that the time derivative of \(\mathbf{\varepsilon}\) is given by  
\[\begin{split}\dot{\mathbf{\varepsilon}}&=\dot{\hat{\mathbf{z} }}-\dot{\hat{\mathbf{z}}}^{T}\widetilde{\mathbf{\theta}}+\mathbf{P}^{T}\dot{\hat{\mathbf{ \theta}}}=\mathbf{A}\mathbf{z}+\mathbf{\varphi}(\mathbf{y},\mathbf{r})+\mathbf{F}(\mathbf{y},  
Therefore, if the \(p\times n\) matrix variable \(\mathbf{P}\) is modified according to the differential equation,  
\[\dot{\mathbf{P}}^{T}=\mathbf{A}_{\mathbf{o}}\mathbf{P}^{T}+\mathbf{F}(y,\mathbf{r})^{T}, \tag{10}\]  
we obtain from (9)  
\[\dot{\mathbf{\varepsilon}}=\mathbf{A}_{\mathbf{o}}\mathbf{\varepsilon}-\mathbf{s}, \tag{11}\]  
and this result does not depend on any specific adaptive law (5). Hence, in the nonadaptive case, when \(\widetilde{\mathbf{\theta}}=\mathbf{0},\;\mathbf{s}=0\), the error \(\mathbf{\varepsilon}=\mathbf{e}\) converges to zero for any initial value of \(\mathbf{e}\). For the adaptive case, we may use the Lyapunov function  
\[y=c^{T}\mathbf{z}, \tag{2}\]0  
which is positive definite as a function of \(\mathbf{\varepsilon},\;\widetilde{\mathbf{\theta}}\). For \(\mathbf{\tau}_{1}=\mathbf{0}\) and \(\mathbf{s}=\mathbf{0}\), as \(\widetilde{\mathbf{\theta}}^{T}\mathbf{P}\mathbf{c}=\mathbf{c}^{T}\mathbf{P}^{T}\widetilde{\mathbf{ \theta}}=e_{1}-\mathbf{c}^{T}\mathbf{\varepsilon}\), we obtain  
\[