## 6 Robustness of the Developed Event-Triggered Controller  
In this section, we study the robustness of the developed ETC mechanism against external disturbances, measurement noises, and computational errors. Consider the following system:  
\[\dot{x}(t) =Ax(t)+Bu(t)+Ew(t) \tag{29}\] \[y(t) =Cx(t)+\
u(t),\]where \(x\in\mathbb{R}^{n_{x}}\) is the state, \(u\in\mathbb{R}^{n_{u}}\) is the control input, \(y\in\mathbb{R}^{n_{y}}\) is the measured output, \(w\in\mathbb{R}^{n_{w}}\) is the external disturbance, and \(\
u\in\mathbb{R}^{n_{v}}\) is the measurement noise, which is assumed to be differentiable. Here, we assume that only some parts of the state can be measured, not the full state, as considered in Section 3, which is more challenging. Matrices \(A,B,E\) are given in (4) and the matrix \(C\) is given by  
\[C=\begin{bmatrix}1&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&1&0&0&0&0&0&0,\end{bmatrix} \tag{30}\]  
which implies that only the system frequency \(x_{1}=f(t)\) and wind generation frequency \(x_{5}=f_{w}(t)\) can be measured. The plant (29) is stabilized by the following observer-based controller  
\[\dot{\zeta}(t) = A\zeta(t)+Bu(t)+F(y(t_{k})-C\zeta(t)) \tag{31}\] \[u(t) = -K\zeta(t)+v(t),\]  
where \(\zeta\in\mathbb{R}^{n_{\zeta}}\) is the estimated state, \(v\in\mathbb{R}^{n_{v}}\) is the computational error, such as glitch or controller disturbances, and \(F\) is the gain matrix of the observer. Note that the plant output \(y(t)\) is only sent to the controller at triggering instants \(t_{k}\), as indicated by the term \(y(t_{k})\) in (31). Now, we define the sampling error of the output measurement \(y(t)\) as follows:  
\[e(t)=y(t_{k})-y(t)\qquad\forall t\in[t_{k},t_{k+1}). \tag{32}\]  
Then, systems (29) and (31) become  
\[\dot{x}(t) = Ax(t)-BK\zeta(t)+Bv(t)+Ew(t) \tag{33}\] \[\dot{\zeta}(t) = A\zeta(t)-BK\zeta(t)+Bv(t)+F(y(t)+e(t))-FC\zeta(t)\] \[= (A-BK-FC)\zeta+FCx(t)+Fv(t)+Fe(t)+Bv(t).\]  
Consequently, the dynamics of \(e(t)\) become  
\[\dot{e}(t) = -C\dot{x}+\dot{v}(t) \tag{34}\] \[= -CAx(t)+CBK\zeta(t)-CBv(t)-CEw(t)+\dot{v}(t).\]  
Let \(\chi(t):=(x(t),\zeta(t))\), \(\omega(t):=(w(t),v(t),v(t),\dot{v}(t))\). Then, closed-loop systems (33) and (34) can be written as  
\[\dot{\chi}(t) = \begin{bmatrix}A&-BK\\ FC&A-BK-FC\end{bmatrix}\chi(t)+\begin{bmatrix}0\\ F\end{bmatrix}e(t)+\begin{bmatrix}E&B&0&0\\ 0&B&F&0\end{bmatrix}\omega(t) \tag{35}\] \[= :\mathcal{A}_{1}\chi(t)+\mathcal{B}_{1}e(t)+\mathcal{E}_{1} \omega(t),\]  
and  
\[\dot{e}(t) = \begin{bmatrix}-CA&CBK\end{bmatrix}\chi(t)+\begin{bmatrix}-CE&0&- CB&1\end{bmatrix}\omega(t) \tag{36}\] \[= :\mathcal{A}_{2}\chi(t)+\mathcal{E}_{1}\omega(t).\]  
Then, if we define \(\xi=(\chi,e,\tau)\), we obtain the hybrid model (21) with \(\mathcal{B}_{2}=0\) and \(\mathcal{A}_{1},\mathcal{B}_{1},\mathcal{E}_{1},\mathcal{A}_{2},\mathcal{E}_{2}\), and the flow and jump sets are modified as follows:  
\[\mathcal{C} = \left\{\xi(t):|e(t)|\leq\sigma|y(t)|\;\;\text{or}\;\;\tau\in[0,T]\right\} \tag{37}\] \[\mathcal{D} = \left\{\xi(t):|e(t)|\geq\sigma|y(t)|\;\;\text{and}\;\;\tau\geq T \right\},\]where the triggering threshold is now a function of the measured output \(y\) rather than the full state, as in (22). Then, the obtained result in Theorem 1 applies. Indeed, in this case, the obtained ETC parameters and the guaranteed \(\mathcal{L}_{2}\)-gain will be different, according to the existing disturbances.