## 3 Observer Design
### Observer Design for the Interconnected System
#### 3.1.3 State Estimator Design for the New Actuator Subsystem  
Equation (18) can be viewed as a transformed form of the actuator subsystem described in Equation (2):  
\[\Sigma_{a^{\prime}}:\left\{ \begin{array}{l}\dot{\xi}_{2}=\overline{\mathrm{ f}}_{2}(\xi_{2},\mathrm{u})\\ y_{2}=\mathrm{h}_{2}(\xi_{2},\mathrm{u})\end{array} \right. \tag{18}\]  
where \(\overline{\mathrm{f}}_{2}(\xi_{2},\mathrm{u}):=\mathrm{f}_{\mathrm{a}}(\xi_{ 2},\mathrm{u})\). A convergence observer can be designed for the system in Equation (18) as follows:  
\[\Sigma_{a\mathcal{O}}:\left\{ \begin{array}{l}\dot{\xi}_{2}=\overline{\mathrm{ f}}_{2}\big{(}\dot{\xi}_{2},\mathrm{u}\big{)}+\kappa_{2}\big{(}\sigma_{2}, \dot{\xi}_{2}\big{)}\big{(}\mathrm{h}_{2}\big{(}\dot{\xi}_{2}\big{)}-y_{2} \big{)}\\ \dot{\sigma}_{2}=\Theta_{2}\big{(}\dot{\xi}_{2},\mathrm{u},\sigma_{2}\big{)} \end{array} \right. \tag{19}\]  
where \(\kappa_{2}\) and \(\Theta_{2}\) are smooth gain functions, with respect to their arguments, the state variable \(\big{(}g_{2},\dot{\xi}_{2}\big{)}\) belongs to \((\Re^{\mathrm{n}}\times\Theta_{2})\), and \(\Theta_{2}\) is a subset of \(\Re^{\mathrm{n}}\), which is positively invariant by the second equation of (19). The state estimation error is defined as  
\[\mathrm{e}_{\mathrm{a}}:=\dot{\xi}_{2}(\mathrm{t})-\xi_{2}(\mathrm{t}) \tag{20}\]  
Then, by subtracting the corresponding Equations (18) and (19), we get the following error dynamics:  
\[\dot{\mathrm{e}}_{\mathrm{a}}(\mathrm{t},\mathrm{e}_{\mathrm{a}})=\overline{ \mathrm{f}}_{2}(\xi_{2},\mathrm{u})-\overline{\mathrm{f}}_{2}\big{(}\dot{\xi}_ {2},\mathrm{u}\big{)}-\kappa_{2}\big{(}g_{2},\dot{\xi}_{2}\big{)}\big{(} \mathrm{h}_{2}\big{(}\dot{\xi}_{2}\big{)}-y_{2}\big{)} \tag{21}\]  
where \(\mathrm{K}_{2}\big{(}\mathrm{u},\dot{\xi}_{2},y_{2}\big{)}:=\kappa_{2}\big{(}g _{2},\dot{\xi}_{2}\big{)}\big{(}\mathrm{h}_{2}\big{(}\dot{\xi}_{2}\big{)}-y_{2} \big{)}\). In order to formulate a solution to the convergence of the above observer, we need to follow Assumption 6, with respect to the error Lyapunov function introduced in [26]. This error Lyapunov function shows the equivalence of the existence of an error Lyapunov function and the existence of a converging observer. **Assumption 6**. : _Is that for any \(u\in\mathfrak{U}(t,e_{a})\in\mathcal{AC}(\Re^{+},\Re)\), there exists a continuously differentiable function \(V_{a}\) and positive constants \(a_{2},\ \beta_{2},\ \gamma_{21},\ \gamma_{22}\) to satisfy_  
\[\left\{ \begin{array}{l}\left(\mathrm{a}\right)\gamma_{21}\|\;\mathrm{e}_{a} \;\|^{2}\leq V_{a}(\mathrm{t},\mathrm{e}_{a})\leq\gamma_{22}\|\;\mathrm{e}_{a} \;\|^{2}\\ \left(\mathrm{b}\right)\frac{\partial V_{a}}{\partial\mathrm{t}}(\mathrm{t}, \mathrm{e}_{a})+\frac{\partial V_{a}}{\partial\mathrm{e}_{a}}(\mathrm{t}, \mathrm{e}_{a})\dot{\mathrm{e}}_{a}\leq-\alpha_{2}\|\;\mathrm{e}_{a}\;\|^{2} \\ \left(\mathrm{c}\right)\;\|\;\frac{\partial V_{a}}{\partial\mathrm{e}_{a}}( \mathrm{t},\mathrm{e}_{a})\;\|\leq\beta_{2}\;\|\;\mathrm{e}_{a}\;\|\end{array} \right.. \tag{22}\]  
The observer defined by Equation (19) is a converging observer if Assumption 6 is satisfied. However, the observer in Equation (19) can only be realized when the output \(\mathrm{y}_{2}\) is measurable, which is not the case. Due to this, \(\mathrm{y}_{2}\) in our design represents the output of the actuator subsystem. It is assumed to be unmeasured and, therefore, \(\mathrm{y}_{2}\) must be replaced by an estimate \(\widetilde{\mathrm{y}}_{2}\) using the available measurements. Fortunately, that estimation of \(\widetilde{\mathrm{y}}_{2}\) is available in the process subsystem observer in Equation (11). By substituting \(\mathrm{u}_{a}\) for \(\widetilde{\mathrm{y}}_{2}\), we can now implement the observer in Equation (19) for an actuator subsystem as  
\[\Sigma_{a}\mathrm{O}^{\prime\prime}:\left\{