# Adaptive, Observer-Based Synchronization of Different Chaotic Systems
## 5 Adaptive Control
### Stage 2  
As \(\dot{v}_{2}=\dot{w}_{2d}-\dot{w}_{2}\), using (28), (4), (29), (34), and (22), together with representing virtual control \(w_{3}\) as  
\[w_{3}=w_{3d}-v_{3}, \tag{36}\]  
where \(w_{3d}\) is the desired trajectory for \(w_{3}\) and \(v_{3}\) denotes the tracking error, provides  
\[\dot{v}_{2}=\mathbf{\alpha}^{T}\dot{\mathbf{z}}+K_{1}\dot{v}_{1}-\dot{w}_ {2}=\mathbf{\alpha}^{T}A_{\mathbf{o}}\dot{\mathbf{z}}+\mathbf{\alpha}^{T}\bigg{\{}\mathbf{\varphi} (y,r)+\mathbf{k}y+\mathbf{F}(y,r)^{T}\hat{\mathbf{\theta}}\bigg{\}} \tag{37}\] \[+\mathbf{\alpha}^{T}\mathbf{P}^{T}\underbrace{\left\{\mathbf{\Gamma}\mathbf{P} \mathbf{e}e_{1}-\mathbf{\Gamma}\underbrace{-\mathbf{P}\mathbf{\alpha}v_{1}+\mathbf{\tau}_{2}}_{ \mathbf{\tau}_{1}}\right\}}_{\mathbf{\tau}_{1}}+\mathbf{\alpha}^{T}\left\{\underbrace{ \frac{1}{2}\mathbf{R}^{-1}(\mathbf{\alpha}v_{1}+\mathbf{s}_{1})}_{\mathbf{s}}\right\}-w_{3d}+v _{3}\] \[+K_{1}\underbrace{\left\{-K_{1}v_{1}+\mathbf{\alpha}^{T}\Big{(} \varepsilon+\mathbf{P}^{T}\widetilde{\mathbf{\theta}}\Big{)}+v_{2}\right\}}_{\dot{v}_ {1}}=G+\tfrac{1}{2}\mathbf{\alpha}^{T}\mathbf{R}^{-1}\mathbf{s_{1}}+K_{1}\mathbf{\alpha}^{T} \Big{(}\varepsilon+\mathbf{P}^{T}\widetilde{\mathbf{\theta}}\Big{)}-w_{3d}+v_{3},\]  
where  
\[G=\mathbf{\alpha}^{T}\Big{\{}\mathbf{A}_{\mathbf{o}}\dot{\mathbf{z}}+\mathbf{\varphi}(y,r)+\mathbf{k}y +\mathbf{P}^{T}\{\mathbf{\Gamma}\mathbf{P}\mathbf{e}_{1}-\mathbf{\Gamma}\{-\mathbf{P}\mathbf{\alpha}v_{1}+ \mathbf{\tau}_{2}\}\}+\mathbf{F}(y,r)^{T}\hat{\mathbf{\theta}}+\tfrac{1}{2}\mathbf{R}^{-1}\mathbf{ \alpha}v_{1}\Big{\}}+K_{1}v_{2}. \tag{38}\]  
This motivates us to form the desired stabilizing function \(w_{3d}\) as  
\[w_{3d}=K_{2}v_{2}+G+v_{1}+\tfrac{1}{2}K_{1}\mathbf{\alpha}^{T}\mathbf{R}^{-1}\mathbf{ \alpha}v_{2}\ \, \tag{39}\]  
with the positive design parameter \(K_{2}\). The first component in (39) stabilizes the dynamics of \(v_{2}\), the second compensates for the nonlinearities, and the two last equations are used to cancel some unnecessary terms in the Lyapunov function derivative. Finally, (37) is simplified to  
\[\dot{v}_{2}=-K_{2}v_{2}-v_{1}+\tfrac{1}{2}\mathbf{\alpha}^{T}\mathbf{R}^{-1}\mathbf{s_{1}} \ \ -\frac{1}{2}\mathbf{\alpha}^{T}\mathbf{R}^{-1}\mathbf{\alpha}v_{2}+K_{1}\mathbf{\alpha}^{T} \Big{(}\varepsilon+\mathbf{P}^{T}\widetilde{\mathbf{\theta}}\Big{)}+v_{3}. \tag{40}\]  
The Lyapunov function mentioned above is selected as  
\[V_{2}=V_{1}+\frac{1}{2}v_{2}^{2}=\frac{1}{2}v_{1}^{2}+\frac{1}{2}v_{2}^{2}+ \frac{1}{2}\widetilde{\mathbf{\theta}}^{T}\mathbf{\Gamma}^{-1}\widetilde{\mathbf{\theta}}+ \varepsilon^{T}\mathbf{R}\varepsilon, \tag{41}\]  
where \(V_{1}\) is as defined in (30). Plugging in \(\dot{V}_{1}\) from (35) and \(\dot{v}_{2}\) from (40) into \(\dot{V}_{2}=\dot{V}_{1}+v_{2}\dot{v}_{2}\) results in \[ \begin{split}\dot{V}_{2}&=-\epsilon^{T}\Big{(}\mathbf{Q}- \tfrac{1}{2}\epsilon\mathbf{c}^{T}\Big{)}\epsilon-\tfrac{1}{2}\epsilon_{1}^{2}- \tfrac{1}{2}\big{(}e_{1}-\mathbf{c}^{T}\mathbf{\varepsilon}\big{)}^{2}-K_{1}v_{1}^{2}+ \widetilde{\theta}^{T}\mathbf{\tau}_{2}+v_{1}v_{2}-\epsilon^{T}\mathbf{s_{1}}-K_{2}v_{ 2}^{2}\\ &\qquad-v_{1}v_{2}+\tfrac{1}{2}v_{2}\mathbf{\pi}^{T}\mathbf{R}^{-1}\mathbf{s_ {1}}-\tfrac{1}{2}K_{1}\mathbf{\pi}^{T}\mathbf{R}^{-1}\mathbf{\alpha}v_{2}^{2}+K_{1}v_{2} \mathbf{\pi}^{T}\Big{(}\mathbf{\varepsilon}+\mathbf{P}^{T}\widetilde{\mathbf{\theta}}\Big{)}+v_ {2}v_{3}.\end{split}  \tag{42}\]  
Selecting  
\[\mathbf{s_{1}}=K_{1}\mathbf{\alpha}v_{2} \tag{43}\]  
cancels \(K_{1}v_{2}\mathbf{\pi}^{T}\mathbf{\varepsilon}\) by \(\epsilon^{T}\mathbf{s_{1}}\), and \(0.5v_{2}\mathbf{\pi}^{T}\mathbf{R}^{-1}\mathbf{s_{1}}\) is canceled by the term \(0.5K_{1}\mathbf{\pi}^{T}\mathbf{R}^{-1}\mathbf{\alpha}v_{2}^{2}\), which was intentionally initiated in (39). The tunning function \(\mathbf{\tau}_{2}\) is used to cancel components containing \(\widetilde{\mathbf{\theta}}\) in (42); therefore,  
\[\mathbf{\tau}_{2}=-K_{1}v_{2}\mathbf{P}\mathbf{\pi}+\sigma\hat{\mathbf{\theta}}, \tag{44}\]  
where \(\sigma>0\) is a small design parameter.