## 4 Stability Analysis  
**Definition 1**. : _The equilibrium point \(x=0\)is said to be semi-globally uniformly exponentially stable if for each \(r>0\) and for all \((t_{0},x(t_{0}))\in\Re_{+}\times B_{r}\), a \(k>0\) and \(\gamma>0\) exist such that \(\|x(t_{0})\|\leq ke^{-\gamma(t-t_{0})}\forall t\geq t_{0}\geq 0\)._  
According to Equation (14), we can select the observer gains as \(\big{[}\alpha_{1}\quad\alpha_{2}\quad\alpha_{3}\big{]}=\big{[}\omega_{0}\beta_ {1},\omega_{0}{}^{2}\beta_{2},\omega_{0}{}^{3}\beta_{3}\big{]}\), where \(\omega_{0}\) is a constant greater than zero, \(\beta_{1}i=1,2,3\). To ensure  
Figure 2: The control structure diagram. the characteristic polynomial \(\mathrm{S}^{3}+\beta_{1}\mathrm{S}^{2}+\beta_{2}\mathrm{S}+\beta_{3}\) be a Hurwitz polynomial, let \(\mathrm{S}^{3}+\beta_{1}\mathrm{S}^{2}+\beta_{2}\mathrm{S}+\beta_{3}=\left( \mathrm{S}+1\right)^{3}\), where \(\beta_{\mathrm{i}}=\frac{\left(\mathrm{n}+1\right)! }{\mathrm{i}\left(\mathrm{n }+1-\mathrm{i}\right)!},\mathrm{i}=1,2,3\). Then, the characteristic polynomial of the linear observer can be expressed as:  
\[\lambda(\mathrm{S})=\mathrm{S}^{3}+\omega_{\mathrm{o}}\beta_{1}\mathrm{S}^{2} +{\omega_{\mathrm{o}}}^{2}\beta_{2}\mathrm{S}+{\omega_{\mathrm{o}}}^{3}\beta_{ 3}=\left(\mathrm{S}+\omega_{\mathrm{o}}\right)^{3}, \tag{21}\]  
Define \(\mathrm{z}_{\mathrm{ei}}=\mathrm{x}_{\mathrm{i}}-\mathrm{z}_{\mathrm{i}}, \mathrm{i}=1,2,3\) as the estimation errors, and according to (11), the observer can be expressed as:  
\[\left\{ \begin{array}{l}\dot{z}_{\mathrm{e}1}=\mathrm{z}_{\mathrm{e}2}-\omega _{\mathrm{o}}\beta_{1}\mathrm{z}_{\mathrm{e}1}\\ \dot{z}_{\mathrm{e}2}=\mathrm{z}_{\mathrm{e}3}-\omega_{\mathrm{o}}^{2}\beta_{ 2}\mathrm{z}_{\mathrm{e}1}\\ \dot{z}_{\mathrm{e}3}=\mathrm{z}_{f}-\dot{z}_{f}-\omega_{\mathrm{o}}^{3}\beta_ {3}\mathrm{z}_{\mathrm{e}1}\end{array} \right., \tag{22}\]  
Defining \(\delta_{\mathrm{i}}=\frac{\mathrm{z}_{\mathrm{ei}}}{{\omega_{\mathrm{o}}}^{ \mathrm{i}-1}},\mathrm{i}=1,2,3\), we obtain:  
\[\dot{\delta}= \omega_{\mathrm{o}}\mathrm{M}\delta+\mathrm{N}\frac{\mathrm{z}_{f }-\dot{z}_{f}}{{\omega_{\mathrm{o}}}^{2}}, \tag{23}\]  
where \(\mathrm{M}= \begin{bmatrix}-\beta_{1}&1&0\\ -\beta_{2}&0&1\\ -\beta_{3}&0&0\end{bmatrix} \), \(\mathrm{N}= \begin{bmatrix}0&0&1\end{bmatrix} ^{\mathrm{T}}\), \(\mathrm{M}\) is the Hurwitz matrix of \(\beta\). **Theorem 1**. : _Assuming that \(\mathrm{z}_{f}\)is globally Lipschitz with respect to \(\mathrm{z}_{m}\), then there exists a constant \(\omega_{\mathrm{o}}\)\(>0\) which makes \(\lim\limits_{\mathrm{t}\rightarrow\infty}\mathrm{z}_{\mathrm{ei}}(\mathrm{t})=0, \mathrm{i}=1,2,3\)._  
**Prove of Theorem 1. ** M is a Hurwitz matrix, there exists a unique positive definite matrix \(\mathrm{P}\), which makes \(\mathrm{M}^{\mathrm{T}}\mathrm{P}+\mathrm{PM}=-\mathrm{I}\). Selecting the Lyapunov equation \(\mathrm{V}(\delta)=\delta^{\mathrm{T}}\mathrm{P}\delta\), the derivation gives \(\dot{\mathrm{V}}(\delta)=\frac{\partial\mathrm{V}(\delta)}{\partial\delta}\dot{\delta}\), where \(\frac{\partial\mathrm{V}(\delta)}{\partial\delta}=\frac{\partial\delta^{ \mathrm{T}}\mathrm{P}\delta}{\partial\delta}=2\delta^{\mathrm{T}}\mathrm{P}\).