# Adaptive, Observer-Based Synchronization of Different Chaotic Systems
## 5 Adaptive Control
### Stage 3  
Considering (39), the time derivative \(\dot{w}_{3d}\) is rather complicated, although it is not impossible to write it down in an explicit form. Obtaining approximation of this derivative from a linear filter  
\[\dot{\omega}=\Omega(w_{3d}-\omega) \tag{46}\]  
(where \(\omega\) is the filter state variable, and \(\Omega\) denotes the filter parameter) allows simplification of the controller. The filter output \(\omega\) is the response of a stable linear system (46) to the input \(w_{3d}\); therefore, \(w_{3d}-\omega\) is bounded, for example, \(\left(w_{3d}-\omega\right)^{2}\leq\rho\). The filter tracking error is denoted by  
\[v_{3f}=\omega-w_{3} \tag{47}\]  
and this, together with (36), allows us to use \(v_{3}=w_{3d}+v_{3f}-\omega\) and to express (45) as  
\[ \begin{split}\dot{V}_{2}&=-\epsilon^{T}\Big{(} \mathbf{Q}-\tfrac{1}{2}\mathbf{c}\mathbf{c}^{T}\Big{)}\mathbf{\varepsilon}-\tfrac{1}{2} \epsilon_{1}^{2}-\tfrac{1}{2}\big{(}e_{1}-\mathbf{c}^{T}\mathbf{\varepsilon}\big{)}^ {2}-K_{1}v_{1}^{2}-K_{2}v_{2}^{2}\\ &\qquad\qquad+v_{3f}v_{2}+(w_{3d}-\omega)v_{2}+\sigma\widetilde{ \theta}^{T}\hat{\mathbf{\theta}},\end{split}  \tag{48}\]  
and (39) as  
\[\dot{v}_{2}=-K_{2}v_{2}-v_{1}+\frac{1}{2}\mathbf{\pi}^{T}\mathbf{R}^{-1}\mathbf{s_{1}}- \frac{1}{2}\mathbf{\pi}^{T}\mathbf{R}^{-1}\mathbf{\alpha}v_{2}+K_{1}\mathbf{\pi}^{T}\Big{(} \mathbf{\varepsilon}+\mathbf{P}^{T}\widetilde{\mathbf{\theta}}\Big{)}+v_{3f}+w_{3d}-\omega. \tag{49}\]  
As \(\dot{v}_{3f}=\dot{\omega}-\dot{w}_{3}\), Equation (46) and the last equation in (22) allow us to write down  
\[\dot{v}_{3f}=\Omega(w_{3d}-\omega)-\mathbf{\Theta}^{T}\mathbf{H}(\mathbf{w})-h(\mathbf{w})-u \tag{50}\]  
The control \(u\) is used to cancel inconvenient nonlinearities in (50), to stabilize the trajectory \(v_{3f}\), and to original components, canceling inconvenient terms in the Lyapunov function derivative. Unknown slave system parameters \(\mathbf{\Theta}\) are substituted by adaptive parameters \(\hat{\mathbf{\Theta}}\), and the adaptation error is defined as \(\widetilde{\mathbf{\Theta}}=\mathbf{\Theta}-\hat{\mathbf{\Theta}}\). All those reasons explain the following selection:  
\[u=\Omega(w_{3d}-\omega)-\hat{\mathbf{\Theta}}^{T}\mathbf{H}(\mathbf{w})-h(\mathbf{w})+K_{3}v_{ 3f}+v_{2}. \tag{51}\]Under control (51), Equation (50) is simplified as  
\[\dot{v}_{3f}=-\widetilde{\mathbf{\Theta}}^{T}\mathbf{H}(\mathbf{w})-K_{3}v_{3f}-v_{2}. \tag{52}\]  
The Lyapunov function for the whole system is chosen as  
\[V_{3}=V_{2}+\frac{1}{2}v_{3f}^{2}+\frac{1}{2}\widetilde{\mathbf{\Theta}}^{T}\mathbf{ \Gamma}_{1}^{-1}\widetilde{\mathbf{\Theta}}=\frac{1}{2}v_{1}^{2}+\frac{1}{2}v_{2}^ {2}+\frac{1}{2}v_{3f}^{2}+\frac{1}{2}\widetilde{\mathbf{\theta}}^{T}\mathbf{\Gamma}^{- 1}\widetilde{\mathbf{\theta}}+\epsilon^{T}\mathbf{R}\epsilon+\frac{1}{2}\widetilde\mathbf{{\Theta}}^{T}\mathbf{\Gamma}_{1}^{-1}\widetilde{\mathbf{\Theta}}. \tag{53}\]  
As \(\dot{V}_{3}=\dot{V}_{2}+v_{3f}\dot{v}_{3f}-\widetilde{\mathbf{\Theta}}^{T}\mathbf{ \Gamma}_{1}^{-1}\dot{\mathbf{\Theta}}\), after plugging in (48) and (52), we obtain  
\[ \begin{array}{l}\dot{V}_{3}=-\epsilon^{T}\Big{(}\mathbf{Q}-\frac{1}{2}\epsilon c ^{T}\Big{)}\epsilon-\frac{1}{2}e_{1}^{2}-\frac{1}{2}\big{(}e_{1}-\epsilon^{T} \epsilon\big{)}^{2}-K_{1}v_{1}^{2}-K_{2}v_{2}^{2}+v_{3f}v_{2}\\ \quad+(w_{3d}-\omega)v_{2}+\sigma\widetilde{\mathbf{\theta}}^{T}\mathbf{\hat{\theta}} +v_{3f}\Big{(}-\widetilde{\mathbf{\Theta}}^{T}\mathbf{H}(\mathbf{w})-K_{3}v_{3f}-v_{2} \Big{)}-\widetilde{\mathbf{\Theta}}^{T}\mathbf{\Gamma}_{1}^{-1}\dot{\mathbf{\Theta}}.\end{array}  \tag{54}\]  
The robust adaptive law \(\dot{\hat{\mathbf{\Theta}}}\) is used to cancel the components containing \(\widetilde{\mathbf{\Theta}}\); hence,  
\[\dot{\hat{\mathbf{\Theta}}}=\mathbf{\Gamma}_{1}\Big{(}-\mathbf{H}(\mathbf{w})v_{3f}-\sigma_{1 }\mathbf{\hat{\Theta}}\Big{)} \tag{55}\]  
and \(\sigma_{1}>0\) is a small design parameter.