## 3 Observer Design
### Observer Design for the Interconnected System
#### 3.1.3 State Estimator Design for the New Actuator Subsystem
\begin{array}{l}\dot{\xi}_{2}= \widetilde{\mathrm{f}}_{2}(\hat{\xi}_{2},\mathrm{u})+\kappa_{2}\big{(}\sigma_ {2},\hat{\xi}_{2}\big{)}\big{(}\mathrm{h}_{2}\big{(}\hat{\xi}_{2}\big{)}- \widetilde{\mathrm{y}}_{2}\big{)}\\ \dot{\sigma}_{a}=\Theta_{2}\big{(}\hat{\xi}_{2},\mathrm{u},\sigma_{2}\big{)} \end{array} \right. \tag{23}\]  
where \(\mathrm{K}_{2}\big{(}\mathrm{u},\hat{\xi}_{2},\widetilde{\mathrm{y}}_{2} \big{)}:=\kappa_{2}\big{(}g_{2},\hat{\xi}_{2}\big{)}\big{(}\mathrm{h}_{2} \big{(}\hat{\xi}_{2}\big{)}-\widetilde{\mathrm{y}}_{2}\big{)}\). The estimation error is produced again by subtracting the corresponding equation in Equations (18) and (23), and the new error dynamics are achieved as follows:  
\[ \begin{array}{rl}\dot{\widetilde{\mathrm{e}}}_{a}(\mathrm{t},\widetilde{ \mathrm{e}}_{a})&=\widetilde{\mathrm{f}}_{2}(\hat{\xi}_{2},\mathrm{u})- \widetilde{\mathrm{f}}_{2}\big{(}\hat{\xi}_{2},\mathrm{u})-\mathrm{K}_{2} \big{(}\mathrm{u},\hat{\xi}_{2},\widetilde{\mathrm{y}}_{2}\big{)}\\ &=\widetilde{\mathrm{f}}_{2}(\hat{\xi}_{2},\mathrm{u})-\widetilde{\mathrm{f}}_ {2}\big{(}\hat{\xi}_{2},\mathrm{u})-\mathrm{K}_{2}\big{(}\mathrm{u},\hat{\xi}_ {2},\widetilde{\mathrm{y}}_{2}\big{)}+\mathrm{K}_{2}\big{(}\mathrm{u},\hat{\xi }_{2},\mathrm{y}_{2}\big{)}-\mathrm{K}_{2}\big{(}\mathrm{u},\hat{\xi}_{2}, \widetilde{\mathrm{y}}_{2}\big{)}\\ &=\dot{\mathrm{e}}_{a}(\mathrm{t},\mathrm{e}_{a})+\mathrm{K}_{2}\big{(} \mathrm{u},\hat{\xi}_{2},\mathrm{y}_{2}\big{)}-\mathrm{K}_{2}\big{(}\mathrm{u },\hat{\xi}_{2},\widetilde{\mathrm{y}}_{2}\big{)}\end{array}  \tag{24}\]  
In order to ensure the stability of the error dynamics in Equation (24), an assumption is required with respect to the sensitivity of \(\mathrm{K}_{2}\big{(}\mathrm{u},\hat{\xi}_{2},\mathrm{y}_{2}\big{)}\), with changes of \(\mathrm{y}_{2}\). **Assumption 7**. : _It provides a sufficient condition for achieving this purpose, stating that for any \(u\in\mathfrak{U}\), \(\big{(}t,\hat{\xi}_{2},\widetilde{\mathrm{y}}_{2}\big{)}\in\mathcal{AC}(\Re^ {+},\Re)\), there exists a real constant \(\gamma_{2}\) to satisfy_  
\[\|\;\mathrm{K}_{2}\big{(}\mathrm{u},\hat{\xi}_{2},\widetilde{\mathrm{y}}_{2} \big{)}-\mathrm{K}_{2}\big{(}\mathrm{u},\hat{\xi}_{2},\mathrm{y}_{2}\big{)}\;\| \leq\gamma_{2}\;\|\;\mathrm{y}_{2}-\widetilde{\mathrm{y}}_{2}\;\|\;. \tag{25}\]  
Similar to Assumption 1, Assumption 7 implies a global Lipchitz-type condition on function \(\mathrm{K}_{2}\) such that, in a physical problem, \(\mathrm{u},\mathrm{y}_{2}\) are bounded. Therefore, it can also be replaced by a local smoothness condition. In addition to asking that the state estimation error \(\mathrm{e}_{a}\) converge to \(0\) in the absence of disturbances, we want it to still converge to \(0\) if a disturbance is present, but converge to \(0\) and remain bounded if the disturbance is bounded. Therefore, Assumption 7 implies that the definition of \(\dot{\widetilde{\mathrm{e}}}_{a}(\mathrm{t},\widetilde{\mathrm{e}}_{a})\) in (24) is not affected. In particular, since output \(\widetilde{\mathrm{y}}_{2}\), used in the observer in Equation (23), is in fact a virtual measurement which is estimated by the output of the process subsystem, an estimation error becomes unavoidable. This estimation error can be viewed as a bounded disturbance to the real output of the actuator subsystem \(\mathrm{y}_{2}\). Therefore, the basic problem addressed in this work is the design of nonlinear observers that possesses robustness to the disturbance affecting the real output. **Theorem 2**. : _It says that if Assumptions 6 and 7 are satisfied, then the observer described in Equation (23) is a converging observer for the actuator subsystem described in Equation (18)._  
The proof is given in Appendix B. Let us assume that  
\[\varsigma=\min\{\alpha_{2},(\theta-\eta_{1})\}\]  
In this case,  
\[V^{*}\leq 2\varsigma V \tag{34}\]  
It should also be noted that  
\[ \begin{array}{ll}V^{*}_{p}+V^{*}_{a}&\geq 2\sqrt{V^{*}_{p}}\sqrt{V^{*}_{a}}\\ &=2\sqrt{\alpha_{2}(\theta-\eta_{1})}\sqrt{V_{p}}\sqrt{V_{a}}\end{array}