# Synchronization and Application of a Novel Hyperchaotic System Based on Adaptive Observers
## 4 Adaptive Observer Based Synchronization of the Proposed Hyperchaotic System
This design is illustrated in (11):  
\[\hat{x}=A\hat{x}+Bf(\hat{x})+Bg(\hat{x})\hat{\theta}+\frac{1}{2}\overline{k}B( y_{2}-C_{2}\hat{x})+L(y_{1}-C_{1}\hat{x})+\alpha(t) \tag{13}\]  
Here, \(\hat{\theta}\) and \(\overline{k}\) symbolize the answers to two adaptation laws that are still to be found, and \(\alpha(t)\) is a further control rule that will be defined later [37]. The error signals are \(e=x-\hat{x}\), \(\widetilde{\theta}=\theta-\hat{\theta}\) and \(\widetilde{k}=k-\overline{k}\). Here, \(k\) is a constant to be determined. Utilizing (10) and (13), the error dynamics are expressed as follows:  
\[\dot{e}=(A-LC_{1})e+B(f(x)\ -f(\hat{x}))+B(g(x)\theta-g(\hat{x})\hat{\theta})- \frac{1}{2}\overline{k}B(y_{2}-C_{2}\hat{x})+B\eta-\alpha \tag{14}\]  
The synchronization problem thus transforms into ensuring the stability of the error dynamics. The response system can achieve global synchronization with the drive system if it attains global stability at the origin. We assume the validity of the assumptions for both the driver system and the response system. The response system can synchronize with the driver system under certain conditions. Firstly, the adaptation laws for \(\hat{\theta}\) and \(\overline{k}\) are selected as given in (15) and (16):  
\[\dot{\hat{\theta}}=l_{\theta}g(\hat{x})^{T}(y_{2}-C_{2}\hat{x}) \tag{15}\]  
\[\dot{\overline{k}}=l_{k}\|y_{2}-C_{2}\hat{x}\|^{2} \tag{16}\]  
Here, the positive constants \(l_{\theta}\) and \(l_{k}\) are chosen appropriately. The additional control law \(\alpha(t)\) is chosen as (17):  
\[\alpha(t)=r(t)B(y_{2}-C_{2}\hat{x}) \tag{17}\]The value of \(r(t)\) used in (17) is shown in (18):  
\[r(t)\geq\frac{\varepsilon_{\eta}}{2(y_{2}-C_{2}\hat{x})^{T}} \tag{18}\]  
Building upon Assumption 3 and the Kalman and Yakubovich Lemma, two positive definite matrices, denoted as \(P=P^{T}\) and \(Q=Q^{T}\)[38], are introduced. These matrices are considered valid for the algebraic equation. \[\left(A-LC_{1}\right)^{T}P+P(A-LC_{1})=-Q,\ \ \ \ \ B^{T}P=C_{2} \tag{19}\]  
To prove the stability of the error dynamics given in (14), let us consider the Lyapunov function candidate given in (20):  
\[V=e^{T}Pe+\frac{1}{2}l_{k}^{-1}\overset{\sim}{k}+\frac{1}{l_{\theta}}\overset{ \sim}{\theta} \tag{20}\]  
The derivative of (20) with respect to time becomes (21):  
\[\dot{V}= e^{T}\Big{[}\big{(}A-LC_{1}\big{)}^{T}P+P(A-LC_{1})\Big{]}e+l_{k}^{-1 }\overset{\sim}{k}\overset{\sim}{k}+\frac{2}{l_{\theta}}\overset{\sim}{ \theta}+2e^{T}PB(f(x)-f(\hat{x})) \tag{21}\] \[+2e^{T}PB\eta-2e^{T}P\alpha+2e^{T}PB\big{(}g(x)\theta-g(\hat{x}) \big{)}-\overline{k}e^{T}PB(y_{2}-C_{2}\hat{x})\]  
If we then rewrite Equation (21) taking into account the law of adaptation given in (15), we obtain (22):  
\[\dot{e}=(A-LC_{1})e+B(f(x)\ -f(\hat{x}))+B(g(x)\theta-g(\hat{x})\hat{\theta})- \frac{1}{2}\overline{k}B(y_{2}-C_{2}\hat{x})+B\eta-\alpha \tag{14}\]0 \[\dot{e}=(A-LC_{1})e+B(f(x)\ -f(\hat{x}))+B(g(x)\theta-g(\hat{x})\hat{\theta})- \frac{1}{2}\overline{k}B(y_{2}-C_{2}\hat{x})+B\eta-\alpha \tag{14}\]1  
Given the first assumption, we obtain the inequalities shown in (23):  
\[\dot{e}=(A-LC_{1})e+B(f(x)\ -f(\hat{x}))+B(g(x)\theta-g(\hat{x})\hat{\theta})- \frac{1}{2}\overline{k}B(y_{2}-C_{2}\hat{x})+B\eta-\alpha \tag{14}\]2 \[\dot{e}=(A-LC_{1})e+B(f(x)\ -f(\hat{x}))+B(g(x)\theta-g(\hat{x})\hat{\theta})- \frac{1}{2}\overline{k}B(y_{2}-C_{2}\hat{x})+B\eta-\alpha \tag{14}\]3 \[\dot{e}=(A-LC_{1})e+B(f(x)\ -f(\hat{x}))+B(g(x)\theta-g(\hat{x})\hat{\theta})- \frac{1}{2}\overline{k}B(y_{2}-C_{2}\hat{x})+B\eta-\alpha \tag{14}\]4  
Here, \(\varepsilon_{f}\), \(\varepsilon_{g}\) and \(\varepsilon_{\eta}\) are three appropriate positive constants.