# Adaptive, Observer-Based Synchronization of Different Chaotic Systems
## 5 Adaptive Control
### Stage 1  
Let us denote the tracking error (also synchronization error) by  
\[v_{1}=x_{1}-w_{1} \tag{24}\]  
and observe that  
\[\dot{v}_{1}=\dot{x}_{1}-\dot{w}_{1}=x_{2}-w_{2} \tag{25}\]  
may be represented as  
\[\dot{v}_{1}=\mathbf{\alpha}^{T}\mathbf{z}-w_{2}=\mathbf{\alpha}^{T}\Big{(}\mathbf{\dot{z}}+\mathbf{ \varepsilon}+\mathbf{P}^{T}\mathbf{\widetilde{\theta}}\Big{)}-w_{2} \tag{26}\]  
(because it follows from (6) and (8) that \(\mathbf{z}=\mathbf{\dot{z}}+\mathbf{\varepsilon}+\mathbf{P}^{T}\mathbf{\widetilde{\theta}}\)). Let \(w_{2d}\) denote the desired trajectory of \(w_{2}\), which acts as a virtual control in (26). The tracking error for \(w_{2}\) is defined by  
\[v_{2}=w_{2d}-w_{2}. \tag{27}\]  
After selecting  
\[w_{2d}=\mathbf{\alpha}^{T}\mathbf{\dot{z}}+K_{1}v_{1}, \tag{28}\]  
with a positive design parameter \(K_{1}\), the error dynamics Equation (26) becomes  
\[\dot{v}_{1}=-K_{1}v_{1}+\mathbf{\alpha}^{T}\Big{(}\mathbf{\varepsilon}+\mathbf{P}^{T}\mathbf{ \widetilde{\theta}}\Big{)}+v_{2} \tag{29}\]  
The tuning function \(\mathbf{\tau}_{1}\) in (5) and the component \(\mathbf{s}\) in (4) are selected from the analysis of Lyapunov function  
\[V_{1}=V+\frac{1}{2}v_{1}^{2}=\mathbf{\varepsilon}^{T}\mathbf{R}\mathbf{\varepsilon}+\frac {1}{2}\mathbf{\widetilde{\theta}}^{T}\mathbf{\Gamma}^{-1}\mathbf{\widetilde{\theta}}+ \frac{1}{2}v_{1}^{2} \tag{30}\]  
Taking \(\mathbf{\tau}_{1}\) and \(\mathbf{s}\) into account leads us to have, instead of (13), the following equation:  
\[\dot{V}=-\mathbf{\varepsilon}^{T}\bigg{(}\mathbf{Q}-\frac{1}{2}\mathbf{\varepsilon}\mathbf{ \varepsilon}^{T}\bigg{)}\mathbf{\varepsilon}-\frac{1}{2}\epsilon_{1}^{2}-\frac{1 }{2}\Big{(}\epsilon_{1}-\mathbf{\varepsilon}^{T}\mathbf{\varepsilon}\Big{)}^{2}+\mathbf{ \widetilde{\theta}}^{T}\mathbf{\tau}_{1}-2\mathbf{\varepsilon}^{T}\mathbf{R}\mathbf{s} \tag{31}\] \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Adaptive Observer** & **Adaptive Controller** \\ \hline \(\mathbf{\dot{z}}\) & Estimated state variables & \(v_{1}\) & Synchronization error, first-loop tracking error \\ \(\mathbf{e}\) & Estimation error & \(w_{2d}\) & First-loop stabilizing function (desired trajectory for \(w_{2}\)) \\ \(e_{1}\) & First-state variable estimation error & \(v_{2}\) & Tracking error for \(w_{2}\) \\ \(\mathbf{\varepsilon}\) & “Composite” error & \(w_{3d}\) & Second-loop stabilizing function (desired trajectory for \(w_{3}\)) \\ \(\mathbf{\dot{\theta}}\),\(\mathbf{\widetilde{\theta}}\) & Adaptive parameters and adaptation error & \(v_{3}\) & Tracking error for \(w_{3}\) \\ \(\mathbf{P}\) & Auxiliary matrix variable & \(\omega\) & Filter state variable \\ \(\mathbf{k}\) & Design matrix responsible for observer dynamics & \(v_{3f}\) & Filter tracking error \\ \(\mathbf{Q}\), \(\mathbf{R}\) & Auxiliary positive definite matrices used to & \(x_{i}-w_{i},\ i=1,2,3\) & Synchronization errors \\ \(\mathbf{\Gamma}\), \(\sigma\) & Design parameters responsible for adaptation & \(u\) & Control input \\ \(\mathbf{\tau}_{1}\), \(\mathbf{s}\), \(\mathbf{\tau}_{2}\), \(\mathbf{s}_{1}\) & Corrective signals from adaptive controller & \(\mathbf{\Theta}\), \(\mathbf{\widetilde{\theta}}\) & Adaptive parameters and adaptation error \\  & & & Design parameters shaping trajectories of \\  & & & \(v_{1},v_{2},v_{3f}\) \\  & & & Filter parameter \\  & & \(\mathbf{\Gamma}_{1}\), \(\sigma_{1}\) & Design parameters responsible for adaptation \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Important signals and parameters.