## 3 Observer Design
### Observer Design for the Interconnected System
#### 3.1.2 State Estimator Design for the New Process Subsystem  
The following system can be viewed as a transformed form of the process subsystem described in Equation (1):  
\[\Sigma_{\mathsf{P}^{\prime}}:\left\{ \begin{array}{l}\dot{\xi}_{1}=\bar{\mathsf{ f}}_{1}(\xi_{1},\xi_{2},\mathsf{u})\\ y_{1}=\xi_{11}\end{array} \right. \tag{9}\]  
where \(\bar{\mathsf{f}}_{1}(\xi_{1},\xi_{2},\mathsf{u}):=\left( \begin{array}{c} \mathsf{f}(\xi_{11})+\mathsf{g}(\xi_{11})\xi_{12}\\ \varepsilon\left(\mathsf{u},\dot{\mathsf{u}},\xi_{2}\right)\end{array} \right)\). The new process subsystem in Equation (9) can be expressed as follows:  
\[\Sigma_{\mathsf{P}^{\prime}}:\left\{ \begin{array}{l}\dot{\xi}_{1}=\mathrm{G} (\xi_{11})\xi_{1}+\mathrm{F}(\xi_{11})+\bar{\tau}(\mathsf{u},\dot{\mathsf{u}}, \xi_{2})\\ y_{1}=\mathrm{C}\xi_{1}\end{array} \right. \tag{10}\]  
where \(\mathrm{G}(\xi_{11})=\left( \begin{array}{cc}0&\mathsf{g}(\xi_{11})\\ 0&0\end{array} \right)\), \(\mathrm{F}(\xi_{11})=\left( \begin{array}{c}\mathsf{f}(\xi_{11})\\ 0\end{array} \right)\), \(\mathrm{C}\left( \begin{array}{cc}\mathsf{I}_{\mathsf{n}}&0\end{array} \right)\), \(\bar{\tau}(\mathsf{u},\dot{\mathsf{u}},\xi_{2})=\left[ \begin{array}{cc}0& \varepsilon\left(\mathsf{u},\dot{\mathsf{u}},\xi_{2}\right)\end{array}  \right]^{\mathrm{T}}\),and \(\mathsf{I}_{\mathsf{n}}\) is the \(\mathsf{n}\times\mathsf{n}\) identity matrix. As demonstrated in [30], supposing that the following assumptions related to the boundedness of the states, signals, and functions are satisfied, an extended, high-gain observer for the system in Equation (10) can be formed. **Assumption 2**. : _It states that there exist finite real numbers \(\varrho\), \(\tau\) with \(0<\varrho\leq\tau\), and that \(\varrho^{2}I_{n}\leq F^{\mathrm{T}}(\xi_{11})F(\xi_{11})\leq\tau^{2}I_{n}\)._  
**Assumption 3**. : _Is that \(F(\xi_{11})\) is a global Lipchitz, with respect to \(\xi_{11}\), locally and uniformly with respect to \(\mathsf{u}\)._  
**Assumption 4**. : _It states that \(g(\xi_{11})\) is a global Lipchitz with respect to \(\xi_{11}\)._  
Then, an extended, high-gain observer for the system in Equation (10) can be given as  
\[\Sigma_{\mathsf{P}\mathsf{O}^{\prime}}:\left\{ \begin{array}{l}\dot{\xi}_{1 1}=\mathrm{G}\big{(}\dot{\xi}_{11}\big{)}\dot{\xi}_{1}+\mathrm{F}\big{(}\dot{ \xi}_{11}\big{)}+\bar{\tau}\big{(}\mathsf{u},\dot{\mathsf{u}},\dot{\xi}_{2} \big{)}-\Lambda^{-1}\big{(}\dot{\xi}_{1}\big{)}S_{\theta}^{-1}\mathrm{C}^{ \mathrm{T}}\big{(}\mathrm{C}\dot{\xi}_{1}-\mathsf{y}_{1}\big{)}\\ y_{1}=\mathrm{C}\dot{\xi}_{1}\end{array} \right. \tag{11}\]  
where \(\mathrm{H}\big{(}\dot{\xi}_{1}\big{)}:=\Lambda^{-1}\big{(}\dot{\xi}_{1}\big{)} S_{\theta}^{-1}\mathrm{C}^{\mathrm{T}}\) is the gain function and \(\Lambda\big{(}\dot{\xi}_{1}\big{)}:=\left[ \begin{array}{cc}\mathrm{I}&0\\ 0&\mathsf{g}\big{(}\dot{\xi}_{11}\big{)}\end{array} \right]\), \(\mathrm{S}_{\theta}\) is the unique symmetric positive definite matrix, satisfying the following algebraic Lyapunov equation:  
\[\mathrm{S}_{\theta}+\mathrm{A}^{\mathrm{T}}\mathrm{S}_{\theta}+\mathrm{S}_{ \theta}\mathrm{A}-\mathrm{C}^{\mathrm{T}}\mathrm{C}=0 \tag{12}\]  
where \(\mathrm{A}=\left[ \begin{array}{cc}0&\mathrm{I}\\ 0&0\end{array} \right]\), \(\theta>0\) is a parameter defined by Equation (12), and the solution is  
\[\mathrm{S}_{\theta}=\left[ \begin{array}{cc}\frac{1}{\theta}\mathrm{I}&- \frac{1}{\theta^{2}}\mathrm{I}\\ -\frac{1}{\theta^{2}}\mathrm{I}&\frac{2}{\theta^{3}}\mathrm{I}\end{array} \right] \tag{13}\]  
Then, the gain of the estimator can be given by  
\[\mathrm{H}\big{(}\dot{\xi}_{1}\big{)}=\Lambda^{-1}\big{(}\dot{\xi}_{1}\big{)} \mathrm{S}_{\theta}^{-1}\mathrm{C}^{\mathrm{T}}=\left[ \begin{array}{cc}2\theta \mathrm{I}\\ \theta^{2}\mathsf{g}^{-1}\big{(}\dot{\xi}_{11}\big{)}\end{array}