## 3 Main Results
### Observer-Based Output Feedback Control with Event-Triggered Sampling  
Considering the plant (1), our objective is to design an observer-based output-feedback control law using event-based sampling in this subsection. Our analysis approach aims to identify a broad range of output feedback control rates that satisfy the conditions and achieve practical convergence of the system under existing event-triggered estimation,while incorporating control design and event-driven separation. An output feedback control signal is provided by:  
\[\hat{u}_{i}(t)=\gamma_{i}(\varsigma_{i}) \tag{15}\]  
The function \(\gamma_{i}\) presented here is a universally applicable output feedback expression that relies on the system state. Our original design intention was to leverage these states. As long as the control rate of Proposition 1 is satisfied, it can be seamlessly integrated with our event-driven approach to achieve control convergence. Further elucidation regarding the properties of \(\gamma_{i}\) will be provided in subsequent context. Based on the active disturbance rejection control approach, the controller we use, in fact, is:  
\[u_{i}(t)=\hat{u}_{i}(t)-\phi_{i}(t) \tag{16}\]  
The following dynamics is given:  
\[\left[ \begin{array}{c}\dot{r}_{i}(t)\\ \dot{r}_{i}(t)\\ \dot{\xi}_{i}(t)\\ \dot{\psi}_{i}(t)\\ \dot{\phi}_{i}(t)\\ \end{array} \right]=\left[ \begin{array}{cccc}0&1&0&0&0\\ 0&0&0&0\\ \ell_{i}\beta_{i1}&0&-\ell_{i}\beta_{i1}&1&0\\ \ell_{i}^{2}\beta_{i2}&0&-\ell_{i}^{2}\beta_{i2}&0&1\\ \ell_{i}^{3}\beta_{i3}&0&-\ell_{i}^{3}\beta_{i3}&0&0\\ \end{array} \right]\left[ \begin{array}{c}r_{i}(t)\\ \dot{r}_{i}(t)\\ \dot{\varsigma}_{i}(t)\\ \psi_{i}(t)\\ \end{array} \right]+\left[ \begin{array}{cccc}0&0&0\\ 1&0&1\\ 0&\ell_{i}\beta_{i1}&0\\ 0&\ell_{i}^{2}\beta_{i2}&1\\ 0&\ell_{i}^{3}\beta_{i2}&0\\ \end{array} \right]\left[ \begin{array}{c}h_{i}(t)-\phi_{i}(t)\\ r_{i}(t_{k})-r_{i}(t)\\ \dot{a}_{i}(t)\\ \end{array} \right] \tag{17}\]  
Here, we make the following transformation:  
\[\left[ \begin{array}{c}\frac{\dot{r}_{i}(t)}{\ell_{i}}\\ \frac{\dot{r}_{i}(t)}{\ell_{i}^{2}}\\ \frac{\dot{\xi}_{i}(t)}{\ell_{i}^{2}}\\ \frac{\dot{\phi}_{i}(t)}{\ell_{i}^{2}}\\ \frac{\dot{\phi}_{i}(t)}{\ell_{i}^{2}}\\ \frac{\dot{\phi}_{i}(t)}{\ell_{i}^{2}}\\ \frac{\dot{\phi}_{i}(t)}{\ell_{i}^{2}}\\ \frac{\dot{\phi}_{i}(t)}{\ell_{i}^{2}}\\ \end{array} \right]=\left[ \begin{array}{cccc}0&\ell_{i}&0&0&0\\ 0&0&0&0\\ \ell_{i}\beta_{i1}&0&-\ell_{i}\beta_{i1}&\ell_{i}&0\\ \ell_{i}\beta_{i2}&0&-\ell_{i}\beta_{i2}&0&\ell_{i}\\ \ell_{i}\beta_{i3}&0&0\\ \end{array} \right]\left[ \begin{array}{c}h_{i}(t)-\phi_{i}(t)\\ r_{i}(t_{k})-r_{i}(t)\\ \dot{a}_{i}(t)\\ \end{array} \right] \tag{18}\]  
For the sake of convenience, we define a new state \(\Theta_{i}=\left[ \begin{array}{cccc}\frac{r_{i}}{\ell_{i}}&\frac{\dot{r}_{i }}{\ell_{i}^{2}}&\frac{\dot{\varsigma}_{i}}{\ell_{i}^{2}}&\frac{\dot{\varsigma }_{i}}{\ell_{i}^{2}}&\frac{\dot{\varsigma}_{i}}{\ell_{i}^{2}}&\frac{\dot{ \varsigma}_{i}}{\ell_{i}^{2}}&\frac{\dot{\varsigma}_{i}}{\ell_{i}^{2}}\\ \end{array} \right]^{T}\) and a new function \(F(\Theta_{i})\):  
\[F(\Theta_{i})=\left[ \begin{array}{cccc}0&1&0&0&0\\ 0&0&0&0\\ \beta_{i1}&0&-\beta_{i1}&1&0\\ \beta_{i2}&0&-\beta_{i2}&0&1\\ \beta_{i3}&0&-\beta_{i3}&0&0\\ \end{array} \right]\Theta_{i}(t)+\left[ \begin{array}{c}0\\ \frac{1}{\ell_{i}^{3}}\\ 0\\ \frac{1}{\ell_{i}^{3}}\\ 0\\ \end{array} \right]\hat{a}_{i}(t) \tag{19}\]  
The introduced controller (15) satisfies the following proposition. Proposition 1: _There exists a Lyapunov function \(W(\Theta_{i})\) which satisfies the following conditions:_  
\[a_{1}\|\Theta_{i}\|^{2}\leq W(\Theta_{i})\leq a_{2}\|\Theta_{i}\|^{2} \tag{20}\]  
\[\frac{\partial W}{\partial\Theta_{i}}F(\Theta_{i})\leq-a_{3}\|\Theta_{i}\|^{2} \tag{21}\]  
\[\left\|\frac{\partial W}{\partial\Theta_{i}}\right\|\leq a_{4}\|\Theta_{i}\| \tag{22}\]  
_where \(a_{1 form of the output feedback algorithm employed here remains undefined, various optimization control methods can ultimately be applied.