## 2 Problem Formulation  
Consider a single-input single-output discrete-time system:  
\[ \begin{split} y_{k}&=a_{1}u_{k}+a_{2}u_{k-1}+\cdots+a_{n }u_{k-n+1}+d_{k}\\ &=\phi_{k}^{T}\theta+d_{k},\end{split}  \tag{1}\]  
where \(d_{k}\) is the system noise; \(u_{k}\) is the system input; \(\phi_{k}=[u_{k},\ldots,u_{k-n+1}]^{T}\) is the regression vector consisting of the system input; \(\theta=[a_{1},\ldots,a_{n}]^{T}\in\mathbb{R}^{n}\) is the unknown parameter; \(y_{k}\) is the system output, measured by a binary sensor with a threshold of \(C\), which can be represented by an indicator function as  
\[s_{k}=I_{\{y_{k}\leq C\}}=\left\{ \begin{array}{ll}1,&y_{k}\leq C;\\ 0,&\text{else}.\end{array} \right. \tag{2}\]  
As illustrated in Figure 1, the transmission process of \(s_{k}\) requires passing through an event trigger. It is indicated by \(\gamma_{k}^{e}=1\) that the trigger was triggered and sent for the current moment of data; otherwise, it is indicated by \(\gamma_{k}^{e}=0\). Moments are divided into \(n\) groups based on their congruence modulo \(n\), and let \(\omega_{0,i}=0\), \(i=1,2,\ldots,n\) represent the initial values of cumulative sums for \(n\) groups. At moments of congruent modulo \(n\), the event trigger computes and updates the cumulative sums for \(n\) groups and decides whether to trigger according to their relationship to the adjustment factor \(m\). CSTCS is as follows:  
\[\gamma_{k}^{e} = I_{\{s_{k}=1\}}\sum_{i=1}^{n}I_{\{\text{rem}(k,n)=\text{rem}(i,n) \}}I_{\{\text{rem}(\omega_{k},i,m)=0\}}, \tag{3}\] \[\omega_{k,i} = \omega_{k-1,i}+s_{k}I_{\{\text{rem}(k,n)=\text{rem}(i,n)\}},\;\;i =1,2,\ldots,n, \tag{4}\]  
where \(m\) is a given positive integer and \(\text{rem}(k,n)\) denotes the remainder of k divided by n.  
**Remark 1**. : _For a positive integer \(n\), integers \(z_{1}\) and \(z_{2}\) are congruent modulo \(n\) when their difference is a multiple of \(n\)._  
**Remark 2**. : _In (4), the cumulative sum of \(s_{k}\) in \(n\) groups is essentially divided and computed through an indicator function \(I_{\{\text{rem}(k,n)=\text{rem}(i,n)\}}\). For CSTCS (3) to be triggered, the current value of \(s_{k}\) must equal \(1\) and the cumulative sum \(\omega_{k,i}\) of the group corresponding to \(k\) must be divisible by the adjustment factor \(m\); otherwise, it is not triggered._  
Data \(s_{k}\) may be lost during transmission to the remote estimator owing to uncertainty in the communication network. Denote the packet loss process with a Bernoulli random variable:  
\[\gamma_{k}^{d}=\left\{ \begin{array}{ll}1,&\text{no packet loss};\\ 0,&\text{packet loss}.\end{array} \right. \tag{5}\]  
Therefore, at the \(k\) moment, let \(\gamma_{k}=\gamma_{k}^{d}\gamma_{k}^{e}\) be used to indicate whether or not data are received at the estimation center. From CSTCS (3) and (4), it is known that \(s_{k}=1\) when \(\gamma_{k}^{e}=1\); then, there is \(\gamma_{k}^{d}\gamma_{k}^{e}s_{k}=\gamma_{k}\). And when \(\gamma_{k}=0\), indicating that the remote center  
Figure 1: System structure. has not received any data, it is indiscernible whether this is due to a packet loss occurring during transmission or the event-triggered mechanism not being triggered. The sequence of system noise, denoted as \(\{d_{k}\}\), consists of random variables that are independent and identically distributed. They follow a probability distribution function, represented by \(F(\cdot)\). Furthermore, the inverse function of the distribution, \(F^{-1}(\cdot)\), not only exists but is also twice continuously differentiable. (1) \(\gamma_{k}^{d}\) follows a Bernoulli distribution with a probability of \(p\), where \(\mathrm{P}(\gamma_{k}^{d}=0)=p\), \(\mathrm{P}(\gamma_{k}^{d}=1)=1-p\). (2) The packet loss sequence \(\{\gamma_{k}^{d}\}\) is statistically independent of the system noise sequence \(\{d_{k}\}\). These assumptions are common in system identification and are utilized by numerous references to facilitate the analysis of systems [20,30].