## 4 State-Feedback Control
### Event-Triggering Rule  
After designing the gain matrix \(K\), we consider the effect of sampling and we synthesize an event-triggering mechanism to generate the transmission times. The sampling error is defined as follows:  
\[e(t)=x(t_{k})-x(t)\qquad\forall t\in[t_{k},t_{k+1}). \tag{8}\]  
It should be noted that \(e(t)\) is reset to zero at each \(t_{k}\) by updating the last sent value \(x(t_{k})\) to the current value \(x(t)\). Hence, we have  
\[\dot{e}(t) = -\dot{x}(t) \tag{9}\] \[e(t_{k}^{+}) = 0.\]  
From (8) and (19), we obtain  
\[\dot{x}(t) = (A-BK)x(t)-BKe(t)+Ew(t)\] \[=: \mathcal{A}_{1}x(t)+\mathcal{B}_{1}e(t)+\mathcal{E}_{1}w(t).\]  
Consequently, the dynamics of the sampling error \(e(t)\) are  
\[\dot{e}(t) = -(A-BK)x(t)+BKe(t)-Ew(t)\] \[=: \mathcal{A}_{2}x(t)+\mathcal{B}_{2}e(t)+\mathcal{E}_{2}w(t).\]  
An intuitive design of the ETC will be \(t_{k+1}=\inf\{t\geq t_{k}:\ |e(t)|\geq\sigma|x(t)|\}\), where \(\sigma>0\) is a design parameter to be specified later. However, in this case, an infinite number of transmissions can possibly occur within a finite time, leading to Zeno sampling, particularly since the plant is affected by external disturbances, which is not implementable in practice due to hardware constraints. By employing the time-regularization technique, one efficient way to solve this problem is to impose a strictly positive minimum duration \(T\) on the inter-transmission intervals; see, e.g., [26; 28; 45]. In this way, the ETC condition is modified to  
\[t_{k+1}=\inf\{t\geq t_{k}+T:\ |e(t)|\geq\sigma|x(t)|\}, \tag{12}\]  
where \(T>0\) is a design parameter. Hence, according to (12), an updated transmission \(t_{k+1}\) is only permitted when a certain amount of time \(T\) has passed since the last triggering instant \(t_{k}\), and such that \(|e(t)|\geq\sigma|x(t)|\) is satisfied. The enforced dwell time \(T\) is designed as the maximal allowable transmission interval (MATI) in the time-triggered control [23]. To that end, based on [23], the following conditions are required to derive the MATI bound. **Assumption 1**. : _Consider the system (10), (11). There exist \(\varepsilon,\gamma,\eta>0\), and a real matrix \(P=P^{T}>0\), such that the following condition holds:_  
\[ \begin{bmatrix}\mathcal{A}_{1}^{T}\mathbf{P}+\mathbf{P}\mathcal{A}_{1}+\mathcal{A}_{2 }^{T}\mathcal{A}_{2}+(1+\varepsilon)\mathbb{I}_{n_{x}}&\star&\star\\ B_{1}^{T}\mathbf{P}&-\gamma^{2}\mathbb{I}_{n_{x}}&\star\\ \mathcal{E}_{1}^{T}\mathbf{P}+\mathcal{E}_{2}^{T}\mathcal{A}_{2}&0&\mathcal{E}_{2 }^{T}\mathcal{E}_{2}-\eta\mathbb{I}_{n_{x}}\end{bmatrix} <0. \tag{13}\]Condition (13) imposes an \(\mathcal{L}_{2}\)-gain stability from \((e,w)\) to \(x\) with an \(\mathcal{L}_{2}\)-gain equal to \(\sqrt{\eta}\). In other words, by defining \(W(e):=|e|\) and \(V(x):=x^{T}Px\), it holds that  
\[\langle\
abla W(e),\mathcal{A}_{2}x(t)+\mathcal{B}_{2}e(t)+ \mathcal{E}_{2}w(t)\rangle \leq H(x,w)+L|e|, \tag{14}\]  
where \(H(x,w):=|\mathcal{A}_{2}x+\mathcal{E}_{2}w|\) and \(L:=|\mathcal{B}_{2}|\). Consequently, by post- and pre-multiplying (13) by \((x,e,w)\) and its transpose, the feasibility of (13) implies that  
\[\langle\
abla V(x),\mathcal{A}_{1}x(t)+\mathcal{B}_{1}e(t)+ \mathcal{E}_{1}w(t)\rangle\leq-(1+\varepsilon)|x|^{2}-H^{2}(x,w)+\gamma^{2}W^ {2}(e)+\eta|w|^{2}. \tag{15}\]  
Then, according to [23], the stability of the closed-loop system \((x,e)\) is maintained under time-triggered control if the time \(T\) is taken, such that \(T\in\mathcal{T}(\gamma,L)\), where \(\mathcal{T}(\gamma,L)\), and is given by  
\[\mathcal{T}(\gamma,L):=\left\{ \begin{array}{ll}\frac{1}{T^{\prime}}\arctan (r)&\gamma>L\\ \frac{1}{T}&\gamma=L\\ \frac{1}{L^{\prime}}\arctan(r)&\gamma<L\end{array} \right. \tag{16}\]  
with \(r:=\sqrt{|(\frac{\gamma}{L})^{2}-1|}\) and \(\gamma,L\) from Assumption 1.