## 2 Problem Formulation
### The Relationship between Convergence Rate and \(\alpha\)  
In a scenario where the probability of packet loss is unknown, the estimation of the probability during the identification process not only wastes some communication resources but also affects the convergence property of the estimation algorithm. Use the mean square convergence rate \(N\mathbb{E}(\widehat{\theta}_{N}-\theta)(\widehat{\theta}_{N}-\theta)^{T}\) as an index to measure the performance of the algorithm. Due to the introduction of the packet loss process, the randomness of the event-triggered mechanism is enhanced, making it difficult to obtain an explicit expression for the convergence rate of the algorithm. However, the relationship between the convergence rate and \(\alpha\) can be fitted through numerical fitting, which, in turn, can solve the trade-off issue between the network resources and the convergence property of the algorithm. (1) The method of selecting the subset \(Z_{N}\)  
When the probability of packet loss is unknown, knowing that \(\alpha\in(0,1)\), the subset is obtained differently depending on whether \(\alpha\) is a rational or irrational number. If \(\alpha\) is a rational number, \(\alpha\) can be expressed as a fraction in the form of \(a/b\), and the selection of the subset is as follows:  
\[O_{N,i}^{0}=\{x:x\in O_{N,i},\text{rem}(\frac{x-i}{n},b)<a\} \tag{38}\]  
where \(O_{N,i}=\{i,n+i,\dots,(L_{N}-1)n+i\}\), \(i=1,2,\dots,n\). \(Z_{N}\) is given by (22). If \(\alpha\) is an irrational number, it is selected using Algorithm 1 according to [PERSON]'s equidistribution theorem [33]. ```
0: Given an irrational number \(\alpha\), the data length \(N\), the input period \(n\). 0:\(Z_{N}\) is a subset of \(\{1,2,\dots,N\}\) and its density is close to \(\alpha\). 1:\(L_{N}\leftarrow\lfloor\frac{N}{n}\rfloor\)
2:\(Z_{N}\leftarrow\emptyset\)
3:for\(i=1\) to \(n\)do
4:\(O_{N,i}^{0}\leftarrow\textsc{SelectSubset}(i,n,L_{N},\alpha)\)
5:\(Z_{N}\gets Z_{N}\cup O_{N,i}^{0}\)
6:endfor
7:return\(Z_{N}\)
8:functionSelectSubset(\(i,n,L_{N},\alpha\))
9:\(S\leftarrow\) empty set
10:for\(l=1\) to \(L_{N}\)do
11:\(cur\gets l\alpha-\lfloor l\alpha\rfloor\)
12:if\(cur<\alpha\)then
13:\(S.\textsc{add}((l-1)*n+i)\)
14:endif
15:endfor
16:return\(S\)
17:endfunction
```  
**Algorithm 1** Generation of subset \(Z_{N}\) when \(\alpha\) is an irrational number  
(2) Data generation  
According to Theorem 2, a sufficiently large \(N\) is given to guarantee the convergence of the estimation algorithm (25)-(27). Based on the above subset selection method to obtain \(Z_{N}\) for different values of \(\alpha\), generate the data for sufficient number of trajectories. Use the estimation algorithm (25)-(27) to calculate the parameter estimation values of each trajectory and obtain the average of these parameter estimation values \(\bar{\theta}\). Then, \(\bar{\theta}\) is used to replace the true value \(\theta\) in the calculation of the mean square convergence rate. Also, calculate the average value of the trace of \(N(\widehat{\theta}_{N}-\bar{\theta})(\widehat{\theta}_{N}-\bar{\theta})^{T}\) for the multiple trajectory data to replace the mean-square convergence rate \(N\mathbb{E}(\widehat{\theta}_{N}-\bar{\theta})^{T}\) as an index for evaluating the algorithm performance. The specific data generation process is described as follows. Given the data length \(N\) and the adjustment factor \(m\), obtain parameter estimation values according to the identification algorithm, and repeat to acquire data for \(M\) trajectories, where \(M\) is a large positive integer. Denote each parameter estimation value as \(\widehat{\theta}_{N,j},j=1,2,\dots,M\); then, the average parameter estimate is  
\[\bar{\theta}=\frac{1}{M}\sum_{j=1}^{M}\widehat{\theta}_{N,j},\ \ j=1,2,\dots,M. \tag{39}\]  
Use the above equation to replace the true parameter value in the calculation of the substitution index for the mean square convergence rate, i.e., the average value of the approximate covariance matrix trace for multiple trajectory data.