## 2 System Description and Preliminaries
### Network Control Based on Dynamic Event-Triggered Scheme  
Benefit from the rapid development and wide application of network communication technology, the combination of control system and network transmission has attracting more and more attentions of scholars [28; 29]. The traditional time-triggered/periodic-triggered control scheme has the shortcoming of wasting network resources (CPU for computing) which information are less important, the static event-triggered control in Figure 3 is introduced to solve this problem, which greatly alleviates the pressure of network bandwidth and reduces the waste of communication resources while implementing effective control. In this article, a dynamic event-triggered scheme in Figure 4 is proposed to further save the network bandwidth. An intelligent sensor is composed of a sensor, sampler, event generator and memory. The sensor is responsible for collecting the sensitive information of the controlled object, and the sampler collects the continuous signals transmitted by the sensor at a fixed period. The event generator screens the received sampled data according to the preset event trigger conditions, and the sampled signals that meet the trigger conditions will be sent to the controller, otherwise they will not be sent. Sampling sequence \(\mathbb{H}_{1}=\{0,h,2h,\cdots,kh\}\), \(k\in\{0,1,2,3,\cdots\}\), the time series of successful signal release is \(\mathbb{H}_{2}=\{t_{1},t_{2},t_{3},\cdots,t_{k}\},k\in\{0,1,2,3,\cdots\}\). Clearly \(\mathbb{H}_{2}\in\mathbb{H}_{1}\), the real control signal \(u(t)\) for the controller is provided by  
\[u(t)\ =\ K_{i}x(t_{k}),\quad t\in[t_{k},t_{k+1}), \tag{11}\]  
For \(t\in\left[t_{k}+\tau_{t_{k}},t_{k+1}+\tau_{t_{k+1}}\right]\), motivated by [30], we can easily prove that there must be a positive integer \(d_{M}(\geq 1)\) such that  
\[t_{k}+d_{M}h+\tau<r_{k+1}+\tau_{t_{k+1}}\leq t_{k}+(d_{M}+1)h+\tau\]  
\(\left[t_{k}+\tau_{t_{k}},t_{k+1}+\tau_{t_{k+1}}\right]\) is divided into the following \(d_{M}+1\) subintervals, where \(t_{k}^{i}=t_{k}+ih\), \(i=0,1,2,\cdots,d_{M}\), we define  
\[\left\{ \begin{array}{rl}\Delta_{0}&=\left[t_{k}+\tau_{t_{k}},t_{k}^{1}+ \bar{\tau}\right]\\ \Delta_{i}&=\left[t_{k}^{i}+\bar{\tau},t_{k}^{i}+h+\bar{\tau}\right],\quad i=1,2,\cdots,d_{M}-1\\ \Delta_{d_{M}}&=\left[t_{k}^{d_{M}}+\bar{\tau},t_{k+1}h+\tau_{t_{k+1}}\right] \end{array} \right. \tag{12}\]  
Figure 4: Dynamic event-triggered control model. Figure 3: Static event-triggered control model. easy to know \(\left[t_{k}+\tau_{t_{k}},t_{k+1}+\tau_{t_{k+1}}\right]=\cup_{i=0}^{d_{M}}\Delta_{i}\). Define a function \(\tau(t)\):  
\[\tau(t)=t-t_{k^{\prime}}^{i},t\in\Delta_{i},i=0,1,2,\cdots,d_{M} \tag{13}\]  
According to the above equation:  
\[\left\{ \begin{array}{ll}\tau_{t_{k}}\leq\tau(t)\leq h+\bar{\tau}&t\in\Delta _{0}\\ \tau_{t_{k}}\leq\bar{\tau}\leq\tau(t)\leq h+\bar{\tau}&t\in\Delta_{i}\\ \tau_{t_{k}}\leq\tau\leq\tau(t)\leq h+\bar{\tau}&t\in\Delta_{d_{M}}\end{array} \right. \tag{14}\]  
when \(t\in\left[t_{k}+\tau_{t_{k}},t_{k+1}+\tau_{t_{k+1}}\right],\tau(t)\in\left[ \tau_{1},\tau_{2}\right]\), where \(\tau_{2}=h+\bar{\tau}\). Furthermore, let us define an error vector as following:  
\[e(t)=\left\{ \begin{array}{cl}0,&t\in\Delta_{0},\\ x(t_{k}^{i})-x(t_{k}),&t\in\Delta_{i},i=1,2,\cdots,d_{M}-1,\\ x(t_{k}^{\Delta_{M}})-x(t_{k}),&t\in\Delta_{M}.\end{array} \right. \tag{15}\]  
Then, dynamic event-triggered conditions will be written as follows. \[\varrho\kappa(t)+\sigma x^{T}(t-\tau(t))WX(t-\tau(t))-e^{T}(t)We(t)\leq 0 \tag{16}\]  
where \(\sigma\in(0,1),\varrho\in\mathbb{R}_{0}^{+}\) are parameters to be designed; \(W>0\) is an undetermined matrix with appropriate dimensions. Furthermore, the internal variable \(\kappa(t)\) that fulfills the following differential equation is  
\[\left\{ \begin{array}{ll}\dot{\kappa}(t)&=&-\varsigma(\kappa(t))-e^{T}(t)We( t)+\sigma x^{T}(t-\tau(t))Wx(t-\tau(t))\\ \kappa(0)&=&\kappa_{0}\end{array} \right.