## 3 Observer Design
### Observer Design for the Interconnected System  
#### 3.1.1 Interconnected System Extension  
For the interconnected system described by Equations (1) and (2), in order to facilitate analysis, the unknown interconnection \(\mathsf{u}_{\mathsf{a}}\) is extended as a new state \(\mathsf{x}_{\mathsf{u}}\):  
\[\left\{ \begin{array}{l}\mathsf{x}_{\mathsf{u}}:\stackrel{{ \vartriangle\mathsf{a}}}{{=}}\mathsf{u}_{\mathsf{a}}\\ \dot{\mathsf{x}}_{\mathsf{u}}:=\dot{\mathsf{u}}_{\mathsf{a}}\end{array} \right. \tag{3}\]  
To get a function for \(\dot{\mathsf{x}}_{\mathsf{u}}\), inspired by the work proposed in [29], let us derivate the output \(\mathsf{u}_{\mathsf{a}}\) in Equation (2) to get the following function:  
\[\dot{\mathsf{x}}_{\mathsf{u}}:=\dot{\mathsf{u}}_{\mathsf{a}}=\varepsilon \big{(}\mathsf{u},\dot{\mathsf{u}},\mathsf{x}_{\mathsf{a}}\big{)}=\frac{ \partial\mathsf{h}_{\mathsf{a}}}{\partial\mathsf{u}}(\mathsf{u},\mathsf{x}_{ \mathsf{a}})\dot{\mathsf{u}}+\frac{\partial\mathsf{h}_{\mathsf{a}}}{\partial \mathsf{x}_{\mathsf{a}}}(\mathsf{u},\mathsf{x}_{\mathsf{a}})\mathsf{f}_{ \mathsf{a}}(\mathsf{u},\mathsf{x}_{\mathsf{a}}) \tag{4}\]  
where the function \(\varepsilon\big{(}\mathsf{u},\dot{\mathsf{u}},\mathsf{x}_{\mathsf{a}}\big{)}\) is with respect to the time derivative of the output \(\mathsf{u}_{\mathsf{a}}\) in Equation (2). We can define Assumption 1 as follows: For any function \(\mathsf{u}\in\mathcal{U}(\mathsf{t},\hat{\mathsf{x}}_{\mathsf{a}})\in \mathcal{AC}\big{(}\Re^{+},\Re\big{)}\), there exists a real constant satisfied by \(\gamma_{1}\):  
\[\parallel\varepsilon\big{(}\mathsf{u},\dot{\mathsf{u}},\hat{\mathsf{x}}_{ \mathsf{a}}\big{)}-\varepsilon\big{(}\mathsf{u},\dot{\mathsf{u}},\mathsf{x}_{ \mathsf{a}}\big{)}\parallel\leq\gamma_{1}\parallel\mathsf{x}_{\mathsf{a}}- \hat{\mathsf{x}}_{\mathsf{a}}\parallel \tag{5}\]  
**Assumption 1**. : _It refers to the global Lipchitz-type condition of function \(\varepsilon\), although this condition seems restrictive, becomes much lower since \(u\) and \(u_{\mathsf{a}}\) are bounded, which is usually the case in physical situations._  
Moreover, this boundedness can be found by introducing saturation in the argument of \(\mathsf{h}_{\mathsf{a}}\). According to [29], if \(\mathsf{u}\) and \(\mathsf{h}_{\mathsf{a}}\) belong to a compact set \(\dot{\mathsf{u}},\mathfrak{V}\), then the global Lipchitz-type condition of function \(\varepsilon\) can be replaced by local smoothness by using saturations. Thus, a new interconnected system is constituted of \(\Sigma^{\prime}:=\Sigma_{\mathsf{P}}+\Sigma_{\mathsf{a}}+\mathsf{x}_{\mathsf{ u}}\):  
\[\Sigma^{\prime}:\left\{ \begin{array}{l}\dot{\mathsf{x}}=\mathsf{f}(\mathsf{ x})+\mathsf{g}(\mathsf{x})\mathsf{x}_{\mathsf{u}}\\ \dot{\mathsf{x}}_{\mathsf{u}}=\varepsilon\big{(}\mathsf{u},\dot{\mathsf{u}}, \mathsf{x}_{\mathsf{a}}\big{)}\\ \dot{\mathsf{x}}_{\mathsf{a}}=\mathsf{f}_{\mathsf{a}}(\mathsf{x}_{\mathsf{a}}, \mathsf{u})\\ \mathsf{y}=\mathsf{x}\\ \mathsf{x}(\mathsf{t}_{0})=\mathsf{x}_{0};\,\mathsf{x}_{\mathsf{a}}(\mathsf{t }_{0})=\mathsf{x}_{\mathsf{a}0};\,\mathsf{x}_{\mathsf{u}}=\mathsf{h}_{ \mathsf{a}}(\mathsf{x}_{\mathsf{a}},\mathsf{u})\end{array} \right. \tag{6}\]  
where the input of the system is \(\mathsf{u}\), the output is \(\mathsf{y}\), and \(\mathsf{x}_{\mathsf{u}}\) is an unmeasured state. Let \(\xi_{1}=[\ \ \xi_{11}\ \ \ \xi_{12}\ ]=[\ \ \mathsf{x}\ \ \ \mathsf{x}_{ \mathsf{u}}\ \ ]\), \(\xi_{2}=\mathsf{x}_{\mathsf{a}}\). Thus, the above system becomes  
\[\Sigma^{\prime}:\left\{ \begin{array}{l}\dot{\xi}_{1}=\bar{\mathsf{f}}_{1}( \xi_{1},\xi_{2},\mathsf{u})\\ \dot{\xi}_{2}=\bar{\mathsf{f}}_{2}(\xi_{2},\mathsf{u})\\ \left(\begin{array}{l}\mathsf{y}_{1}\\ \mathsf{y}_{2}\end{array}\right)=\left(\begin{array}{l}\mathsf{h}_{1}(\xi_{ 11})\\ \mathsf{h}_{2}(\xi_{2})\end{array}\right)\end{array} \right. \tag{7}\]  
where \(\bar{\mathsf{f}}_{1}(\xi_{1},\xi_{2},\mathsf{u}):=\left( \begin{array}{l} \mathsf{f}(\xi_{11})+\mathsf{g}(\xi_{11})\xi_{12}\\ \varepsilon\big{(}\mathsf{u},\dot{\mathsf{u}},\xi_{2}\big{)}\end{array}