Congruential Summation-Triggered Identification of FIR Systems under Binary Observations and Uncertain Communications

Abstract

With the advancement of network technology, there has been an increase in the volume of data being transmitted across networks. Due to the bandwidth limitation of communication channels, data often need to be quantized or event-triggered mechanisms are introduced to conserve communication resources. On the other hand, network uncertainty can lead to data loss and destroy data integrity. This paper investigates the identification of finite impulse response (FIR) systems under the framework of stochastic noise and the combined effects of the event-triggered mechanism and uncertain communications. The study provides a reference for the application of remote system identification under transmission-constrained and packet loss scenarios. First, a congruential summation-triggered communication scheme (CSTCS) is introduced to lower the communication rate. Then, parameter estimation algorithms are designed for scenarios with known and unknown packet loss probabilities, respectively, and their strong convergence is proved. Furthermore, an approximate expression for the convergence rate is obtained by data fitting under the condition of uncertain packet loss probability, treating the trade-off between convergence performance and communication resource usage as a constrained optimization problem. Finally, the rationality and correctness of the algorithm are verified by numerical simulations.

1. Introduction

With the development of communication technology and control theory, networked systems have been widely used in a variety of fields, such as distributed power systems, remote surgery, factory automation, and unmanned aerial vehicles [1,2,3,4,5]. Unlike traditional point-to-point systems, their feedback control exists within a network loop. We refer to such systems as networked control systems (NCSs). NCSs can be used not only for long-distance control systems, but also to greatly reduce unnecessary cabling as signals are transmitted through the network.

Due to the limited precision and cost of sensors, as well as network bandwidth constraints, the information collected by sensors needs to be quantized before transmission [6]. The quantization process reduces the amount of information in the data, increasing the difficulty of identification, making research on the identification of quantized systems highly significant. Guo et al. [7] studied the recursive identification of finite impulse response (FIR) systems, designed the algorithm using stochastic approximation under binary quantization and unreliable communication, and evaluated its convergence performance. Guo et al. [8] discussed the identification algorithm for FIR systems where both input and output are quantized, proposing a maximum likelihood-based recursive algorithm for data loss scenarios. Lei et al. [9] explored the adaptive tracking problem in strict-feedback switched nonlinear systems with quantized inputs, providing new solutions to enhance system robustness and tracking precision. Guo et al. [10] discussed the security issues in identifying FIR systems with binary observations and provided an optimal defense strategy. Ding et al. [11] considered the identification of nonlinear systems with Hammerstein structures and employed a Gaussian process to model the system impulse response. Guo et al. [12] investigated defense against data-tampering attacks within the framework of system identification with quantized observations and designed the corresponding identification algorithm.

For short sampling periods, the amount of data to be transmitted significantly increases, leading to a need for greater bandwidth and more efficient transmission channels. Thus, saving communications resources is essential, and event-triggered mechanisms are an effective solution. Shi et al. [13] presented a discrete event-driven mechanism, studied the event-driven $H_{\infty }$ control of networked singular systems with state and input quantization, and further analyzed the performance. In Seifullaev et al. [14], an event-triggered control for nonlinear systems with intermittent packet transmission was investigated, and it was demonstrated through a numerical example that the method substantially reduced the number of transmissions compared to periodic sampling. Guo et al. [15] introduced an event-triggered communication scheme for binary-valued observations, to save communication resources; they proposed an algorithm for estimating unknown parameters based on weighted least-squares optimization and discussed the trade-off between average communication rate and estimation performance. Cui et al. [16] researched the remote state estimation for linear discrete-time invariant systems with event-driven mechanisms introduced by communication, employing Gaussian-conserving, event-based sensor scheduling to optimally balance communication costs and estimation accuracy. Huang and Liu [17] introduced new adaptive tracking methods for systems within two variants of event-driven architectures.

NCSs bring numerous conveniences by incorporating communication networks, but a network is not a reliable communication medium. Data packet loss, delays, and sequence disorders inevitably occur during network transmission [18]. It is a struggle to directly apply traditional control theories to the analysis of NCSs, due to these issues. In NCSs, data loss occurs when buffer overflows, router congestion, or connection interruptions happen at network nodes. Packet loss is unpredictable, leading to a lack of data integrity and adverse effects on the system. The parameter identification of network control systems relies on the observation of output data, which, being transmitted over a network, inevitably introduces elements of communication uncertainty.

In NCSs, packet loss is typically addressed through three main approaches. The first approach employs stochastic systems and statistical principles, where the packet loss is mathematically described either by using the average packet loss probability or by assuming that the packet loss adheres to a specific probability distribution. The second approach involves a design method based on time-lag systems, which is applied under conditions where the total extent of the packet loss between two sampling moments is known. The third approach transforms the periods during which packet loss occurs into interruptions in the corresponding transmission channel connections. This is integrated with the theory of motion switching systems, thereby treating NCSs experiencing packet loss as if they were dynamic systems with switching functionalities [19,20,21]. Significant advancements have been made in research focused on control and estimation under conditions of uncertain communication. Lu et al. [22] primarily investigated the fundamental issues of linear quadratic Gaussian control in NCSs with unreliable communication channels, as well as the stability of such systems under conditions of input delays, packet loss, and observation delays. Caballero-Águila et al. [23] explored the distributed fusion estimation of multi-sensor outputs under uncertain network conditions, introducing a sequence of Bernoulli random variables to describe observations for updating estimates at each sampling time, and they used predictions to compensate for information loss due to packet loss. Mohammadzadeh et al. [24] emphasized the significance of packet loss information in NCSs for enhancing control performance and identifying communication failures or attacks, proposing two methods: one involving recursive filters based on NCSs input–output models for packet loss estimation and Kalman filtering for state estimation, and the other using state space models of NCS for simultaneous estimation of controlled object states and packet loss scenarios. Yang et al. [25] investigated hierarchical fusion estimation in multi-sensor networks, employing an asynchronous sampling scheme to improve estimation performance under data packet loss and transmission delays. Sun et al. [26] created a new model for describing random transmission delays and packet loss in communication channels, introducing an optimal linear filter for estimation based on this model.

Quantization identification, event-triggered mechanisms, and system control and estimation under communication uncertainty have made significant progress. In most existing studies, uncertain channels and event-triggered mechanisms are typically applied in the control domain, yet are rarely discussed in system identification. This paper investigates FIR system identification under the combined impact of these three factors. The advantages of FIR systems in complex systems and intelligent algorithm applications are evident. With appropriate design and application, FIR systems can process and analyze complex data structures and system behaviors [27,28,29]. This makes them ideal for many signal-processing and system-identification applications. Packet loss greatly impacts the received data sequence, and the quantization process coupled with event-triggered mechanisms further reduces the amount of information in the data, creating a highly nonlinear interaction that poses great challenges for system parameter identification.

To overcome these challenges, the congruential summation-triggered communication scheme (CSTCS) and quantization are introduced, with the aim of reducing communication frequency while retaining as much information as possible available to the system. The data packet loss process is established using a set of Bernoulli random variables. Under the premise of strong correlation in the received data, an identification algorithm is designed using estimation compensation and statistical methods, and its convergence is proved in conjunction with the law of large numbers for martingales. In cases where the mean-square convergence rate is difficult to express theoretically, the relationship between the convergence rate and the proportion of data used for packet loss estimation is represented using numerical fitting methods. Furthermore, for FIR systems under the proposed communication scheme and data packet loss, the communication rate is given and the trade-off between convergence property and network resources is discussed.

The main contributions of this paper are summarized below:

• Differently from most of the existing studies, this paper simultaneously considers the combined effects of quantization, data packet loss, and event-triggered mechanisms.
• When the probability of packet loss is unknown, CSTCS is redesigned to obtain a new identification algorithm.
• Numerical fitting is used to obtain an approximate expression for the mean-square convergence rate that is difficult to obtain directly, which in turn allows the trade-off between convergence property and network resources to be resolved.

The remainder of this paper is organized as follows. Section 2 describes the system model, introduces CSTCS, and describes the communication uncertainty mechanism. Section 3 designs the identification algorithm in situations where the packet loss probability is known and proves its convergence. Section 4 redesigns the identification algorithm for the scenario where the probability of packet loss is unknown and analyzes its convergence performance. Section 5 discusses the communication rate in the context of unknown packet loss probability and the trade-off between this and convergence performance. Section 6 validates the theoretical results through numerical simulations. Finally, Section 7 concludes the paper and suggests directions for future work.

2. Problem Formulation

Consider a single-input single-output discrete-time system:

$$\begin{array}{cc}\hfill y_k& =a_1{u}_k+a_2{u}_{k-1}+\cdots +a_n{u}_{k-n+1}+d_k\hfill \\ \hfill & ={\varphi }_k^T\theta +d_k,\hfill \end{array}$$

(1)

where $d_k$ is the system noise; $u_k$ is the system input; ${\varphi }_k={[u_k,\dots ,u_{k-n+1}]}^T$ is the regression vector consisting of the system input; $\theta ={[a_1,\dots ,a_n]}^T\in {\mathbb{R}}^n$ 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

$$s_k=I_{\{y_k\le C\}}=\left\{\begin{array}{cc}1,\hfill & y_k\le C;\hfill \\ 0,\hfill & \mathrm{else}.\hfill \end{array}\right.$$

(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 ${\omega }_{0,i}=0$, $i=1,2,\dots ,n$ 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:

$$\begin{array}{ccc}\hfill {\gamma }_k^e& =& I_{\{s_k=1\}}\sum _{i=1}^n{I}_{\left\{\mathrm{rem}\right(k,n)=\mathrm{rem}(i,n\left)\right\}}{I}_{\{\mathrm{rem}({\omega }_{k,i},m)=0\}},\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill {\omega }_{k,i}& =& {\omega }_{k-1,i}+s_k{I}_{\left\{\mathrm{rem}\right(k,n)=\mathrm{rem}(i,n\left)\right\}},i=1,2,\dots ,n,\hfill \end{array}$$

(4)

where m is a given positive integer and $\mathrm{rem}(k,n)$ 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 $I_{\left\{\mathrm{rem}\right(k,n)=\mathrm{rem}(i,n\left)\right\}}$. 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:

$$\begin{array}{c}\hfill {\gamma }_k^d=\left\{\begin{array}{cc}1,\hfill & \mathrm{no} \mathrm{packet} \mathrm{loss};\hfill \\ 0,\hfill & \mathrm{packet} \mathrm{loss}.\hfill \end{array}\right.\end{array}$$

(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 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.

Assumption 1.

The sequence of system noise, denoted as $\left\{d_k\right\}$, consists of random variables that are independent and identically distributed. They follow a probability distribution function, represented by $F(·)$. Furthermore, the inverse function of the distribution, $F^{-1}(·)$, not only exists but is also twice continuously differentiable.

Assumption 2.

(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 $\left\{{\gamma }_k^d\right\}$ is statistically independent of the system noise sequence $\left\{d_k\right\}$.

Remark 3.

These assumptions are common in system identification and are utilized by numerous references to facilitate the analysis of systems [20,30]. These assumptions are not fundamental and they can be modified or extended to other noise models. For the non-stochastic noise, with reference to the methods in Wang et al. [30], it is possible to generalize the methods of this paper to the case of deterministic noise by adjusting the model structure.

3. Parameter Estimation Algorithm and Its Convergence: The Scenario with Known Probability of Packet Loss

Assume that the system input $\left\{u_k\right\}$ is n-periodic, i.e., $u_{k+n}=u_k,\forall k\ge 1$. Define $\begin{array}{c}{\pi }_1={\varphi }_1^T,\dots ,{\pi }_n={\varphi }_n^T.\end{array}$ Then, the circulant matrix formed by $\left\{u_k\right\}$ is

$$\Phi ={[{\pi }_1,\dots ,{\pi }_n]}^T.$$

(6)

Based on the input period n, the data length N is divided as follows:

$$L_N=⌊\frac{N}{n}⌋,$$

(7)

where $\lfloor ·\rfloor$ denotes the floor function, representing the greatest integer less than or equal to the number “·”. For the data length N, when the probability of packet loss p is known, an algorithm is designed to estimate the unknown parameter $\theta$:

$$\begin{array}{ccc}\hfill {\widehat{\theta }}_N& =& {\Phi }^{-1}{\left[C-F^{-1}\left({\xi }_{N,1}\right),\dots ,C-F^{-1}\left({\xi }_{N,n}\right)\right]}^T,\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill {\xi }_{N,i}& =& \frac{m}{L_N}\sum _{l=1}^{L_N}\frac{{\gamma }_{(l-1)n+i}}{1-p},i=1,2,\dots ,n,\hfill \end{array}$$

(9)

where $F^{-1}(·)$ is given by Assumption 1.

Remark 4.

The effectiveness of the algorithm is predicated on the assumption that Φ is invertible. By integrating the designability of inputs with the characteristics of the circulant matrix, a condition is provided to ensure the invertibility of Φ. Specifically, it is required that $u_1+u_{2}x+u_3{x}^2+\cdots +u_n{x}^{n-1}=0$ does not have any unit roots.

For ease of presentation, for $i=1,2,\dots ,n$, the following notations are introduced:

$$\begin{array}{ccc}\hfill F_i& =& F(C-{\pi }_i\theta ),\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill f_i& =& f(C-{\pi }_i\theta ),\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill \mu (l,i)& =& (l-1)n+i,\hfill \end{array}$$

(12)

where $f(·)$ is the probability density function of the noise.

Theorem 1.

Under uncertain communication (5), consider the system (1) that introduces CSTCS (3) and (4) and binary quantization (2). Provided that Assumptions 1 and 2 are valid, the estimate ${\widehat{\theta }}_N$ derived from (8) and (9) strongly converges to the true value, i.e.,

$$\begin{array}{c}{\widehat{\theta }}_N\to \theta ,asN\to \infty \mathrm{w}.\mathrm{p}.1.\end{array}$$

Proof. 

Under Assumptions 1 and 2, it can be seen that

$$\begin{array}{ccc}\hfill \mathrm{P}(s_k=1)& =& \mathrm{P}(y_k\le C)\hfill \\ & =& \mathrm{P}({\varphi }_k^T\theta +d_k\le C)\hfill \\ & =& \mathrm{P}(d_k\le C-{\varphi }_k^T\theta )\hfill \\ & =& F(C-{\varphi }_k^T\theta ),\hfill \\ \hfill \mathrm{P}({\gamma }_k^d=1)& =& 1-p.\hfill \end{array}$$

Then, we have

$$\begin{array}{ccc}\hfill \mathbb{E}\left(s_k\right)& =& \mathrm{P}(s_k=1)=F(C-{\varphi }_k^T\theta ),\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill \mathbb{E}\left({\gamma }_k^d\right)& =& 1-p.\hfill \end{array}$$

(14)

By the Kolmogorov law of large numbers [31], it can be ascertained that

$$\begin{array}{ccc}\hfill \frac{1}{L_N}\sum _{l=1}^{L_N}{s}_{\mu (l,i)}& \to & F_i,N\to \infty ,i=1,2,\dots ,n,\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill \frac{1}{L_N}\sum _{l=1}^{L_N}{\gamma }_{\mu (l,i)}^d& \to & 1-p,N\to \infty ,i=1,2,\dots ,n.\hfill \end{array}$$

(16)

From (3) and (4), we have $\frac{1}{L_N}{\sum }_{l=1}^{L_N}{s}_{\mu (l,i)}=\frac{m}{L_N}{\sum }_{l=1}^{L_N}{\gamma }_{\mu (l,i)}^e+\frac{G_{N,i}}{L_N}$, where

$$G_{N,i}=\sum _{l=1}^{L_N}{s}_{\mu (l,i)}-m\sum _{l=1}^{L_N}{\gamma }_{\mu (l,i)}^e,i=1,2,\dots ,n.$$

(17)

Noting that $G_{N,i}\in \{0,1,2,\dots ,m-1\}$, it follows that

$$\frac{1}{L_N}\sum _{l=1}^{L_N}{s}_{\mu (l,i)}-\frac{m}{L_N}\sum _{l=1}^{L_N}{\gamma }_{\mu (l,i)}^e\to 0,N\to \infty .$$

(18)

Denote ${\mathcal{F}}_k$ as the $\sigma$-algebra formed by $d_1,d_2,\dots ,d_k$. Then, from (14) we have

$$\mathbb{E}[{\gamma }_{\mu (l,i)}^d-(1-p)|{\mathcal{F}}_{\mu (l,i)}]=\mathbb{E}[{\gamma }_{\mu (l,i)}^d-(1-p)]=0.$$

(19)

Define ${\tilde{\gamma }}_{\mu (l,i)}={\gamma }_{\mu (l,i)}-(1-p){\gamma }_{\mu (l,i)}^e$. According to the definition of ${\gamma }_{\mu (l,i)}^e$, it can be seen that ${\gamma }_{\mu (l,i)}^e$ is ${\mathcal{F}}_{\mu (l,i)}$-measurable and finite. Consequently, by Theorem 3 in ([32], p. 217) and the tower property of conditional expectation, it follows that

$$\begin{array}{ccc}& & \mathbb{E}\left[{\tilde{\gamma }}_{\mu (l,i)}\right|{\mathcal{F}}_{\mu (l,i)-1}]\hfill \\ & =& \mathbb{E}\left[\mathbb{E}\left[{\tilde{\gamma }}_{\mu (l,i)}\right|{\mathcal{F}}_{\mu (l,i)}]|{\mathcal{F}}_{\mu (l,i)-1}\right]\hfill \\ & =& \mathbb{E}\left[\mathbb{E}[{\gamma }_{\mu (l,i)}-(1-p){\gamma }_{\mu (l,i)}^e|{\mathcal{F}}_{\mu (l,i)}]|{\mathcal{F}}_{\mu (l,i)-1}\right]\hfill \\ & =& \mathbb{E}\left[\mathbb{E}\left[({\gamma }_{\mu (l,i)}^d-(1-p)){\gamma }_{\mu (l,i)}^e\right|{\mathcal{F}}_{\mu (l,i)}]|{\mathcal{F}}_{\mu (l,i)-1}\right]\hfill \\ & =& \mathbb{E}\left[{\gamma }_{\mu (l,i)}^e\mathbb{E}\left[({\gamma }_{\mu (l,i)}^d-(1-p))\right|{\mathcal{F}}_{\mu (l,i)}]|{\mathcal{F}}_{\mu (l,i)-1}\right]\hfill \\ & =& \mathbb{E}\left[0|{\mathcal{F}}_{\mu (l,i)-1}\right]\hfill \\ & =& 0,i=1,2,\dots ,n.\hfill \end{array}$$

Therefore, for $i=1,2,\dots ,n$, $\{{\tilde{\gamma }}_{\mu (l,i)},{\mathcal{F}}_{\mu (l,i)},l\ge 1\}$ is a martingale difference sequence.

Given ${\gamma }_{\mu (l,i)}\in \{0,1\}$ and knowing that ${\tilde{\gamma }}_{\mu (l,i)}\le p$, it can be derived that $\mathbb{E}{\left({\sum }_{l=1}^{L_N}{\tilde{\gamma }}_{\mu (l,i)}\right)}^2$$<\infty$, and, furthermore,

$$\begin{array}{c}\hfill \sum _{l=1}^{\infty }\frac{\mathbb{E}\left({\tilde{\gamma }}_{\mu (l,i)}^2\right)}{l^2}\le \sum _{l=1}^{\infty }\frac{\mathbb{E}\left(p^2\right)}{l^2}=\frac{p^2{\pi }^2}{6}<\infty .\end{array}$$

By the law of large numbers for the martingale difference sequence [32], it can be seen that

$$\begin{gathered} \begin{array}{ccc}\hfill \frac{1}{L_N}\sum _{l=1}^{L_N}{\tilde{\gamma }}_{\mu (l,i)}& =& \frac{1}{L_N}\sum _{l=1}^{L_N}{\gamma }_{\mu (l,i)}-\frac{1}{L_N}\sum _{l=1}^{L_N}(1-p){\gamma }_{\mu (l,i)}^e\hfill \end{array} \\ \begin{array}{ccc}& \to & 0,N\to \infty \mathrm{w}.\mathrm{p}.1.\hfill \end{array} \end{gathered}$$

(20)

From the above and (18), we have

$$\begin{array}{c}\hfill \frac{1}{L_N}\sum _{l=1}^{L_N}{s}_{\mu (l,i)}-\frac{m}{L_N}\sum _{l=1}^N\frac{{\gamma }_{\mu (l,i)}}{1-p}\to 0,N\to \infty ,i=1,2,\dots ,n.\end{array}$$

Together with (15), it follows that

$$\begin{array}{c}\hfill \frac{m}{L_N}\sum _{l=1}^{L_N}\frac{{\gamma }_{\mu (l,i)}}{1-p}\to F_i,N\to \infty ,i=1,2,\dots ,n.\end{array}$$

(21)

Combining (6) and (8), the theorem is proved.    □

4. Parameter Estimation Algorithm and Its Convergence Performance: The Scenario with Unknown Probability of Packet Loss

4.1. Parameter Estimation Algorithm and Its Convergence

Select the subset $O_{N,i}^0$ from $O_{N,i}=\{i,n+i,\dots ,(L_N-1)n+i\}$ for $i=1,2,\dots ,n$, ensuring that the number of elements in the subsets corresponding to different i is the same. Combine these subsets into a new set:

$$\begin{array}{c}\hfill Z_N=\bigcup _{i=1}^n{O}_{N,i}^0.\end{array}$$

(22)

Let the number of elements in $Z_N$ be $N_0$, and $N_0/N\to \alpha ,N\to \infty ,\alpha \in (0,1)$. In the scenario where p is unknown, improve CSTCS as follows:

$$\begin{array}{ccc}\hfill {\gamma }_k^e& =& \left\{\begin{array}{cc}1,\hfill & k\in Z_N;\hfill \\ I_{\{s_k=1\}}\sum _{i=1}^n{I}_{\left\{\mathrm{rem}\right(k,n)=\mathrm{rem}(i,n\left)\right\}}{I}_{\{\mathrm{rem}({\omega }_{k,i},m)=0\}},\hfill & k\in {\overline{Z}}_N,\hfill \end{array}\right.\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill {\omega }_{k,i}& =& {\omega }_{k-1,i}+s_k{I}_{\{\mathrm{rem}(k,n)=\mathrm{rem}(i,n),k\in {\overline{Z}}_N\}},i=1,2,\dots ,n,\hfill \end{array}$$

(24)

where ${\omega }_{0,i}=0,i=1,2,\dots ,n$, and ${\overline{Z}}_N$ represents the complement of $Z_N$ within $\{1,2,\dots ,N\}$.

The unknown parameter $\theta$ can be estimated by the following algorithm:

$$\begin{array}{ccc}\hfill {\widehat{\theta }}_N& =& {\Phi }^{-1}{\left[C-F^{-1}\left({\zeta }_{N,1}\right),\dots ,C-F^{-1}\left({\zeta }_{N,n}\right)\right]}^T,\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill {\zeta }_{N,i}& =& \frac{m}{L_N-N_0/n}\sum _{k=1}^N\frac{{\gamma }_k{I}_{\{k\in {\overline{O}}_{N,i}^0\}}}{1-{\widehat{p}}_N},i=1,2,\dots ,n,\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill {\widehat{p}}_N& =& 1-\frac{1}{N_0}\sum _{k=1}^N{\gamma }_k{I}_{\{k\in Z_N\}},\hfill \end{array}$$

(27)

where ${\overline{O}}_{N,i}^0$ denotes the complement of $O_{N,i}^0$ within $O_{N,i}$ for $i=1,2,\dots ,n$.

Lemma 1

([32]). If the sequences of random variables $\left\{X_k\right\}$, $\left\{Y_k\right\}$, $\left\{Z_k\right\}$ satisfy $X_k\stackrel{\mathrm{d}}{\to }X$, $Y_k\stackrel{\mathrm{P}}{\to }{\lambda }_1$, and  $Z_k\stackrel{\mathrm{P}}{\to }{\lambda }_2$, then we have

$$X_k{Y}_k+Z_k\stackrel{\mathrm{d}}{\to }{\lambda }_{1}X+{\lambda }_2,k\to \infty ,$$

where ${\lambda }_1,{\lambda }_2$ are finite constants.

Theorem 2.

Under the uncertain communication (5), consider the system (1) that introduces the improved CSTCS (23) and (24) and binary quantization (2). If Assumptions 1 and 2 are true, the estimate ${\widehat{\theta }}_N$ derived from (25)–(27) strongly converges to the true value, i.e.,

$$\begin{array}{c}{\widehat{\theta }}_N\to \theta ,asN\to \infty ,\mathrm{w}.\mathrm{p}.1.\end{array}$$

Proof. 

From (13) and (14), we have

$$\begin{array}{ccc}\hfill \mathbb{E}\left(s_k\right)& =& \mathrm{P}(s_k=1)=F(C-{\varphi }_k^T\theta ),\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill \mathbb{E}\left({\gamma }_k^d\right)& =& 1-p.\hfill \end{array}$$

(29)

And then, by the Kolmogorov law of large numbers [31], one can obtain

$$\begin{array}{ccc}\hfill \frac{1}{L_N-N_0/n}\sum _{k=1}^N{s}_k{I}_{\{k\in {\overline{O}}_{N,i}^0\}}& \to & F_i,N\to \infty ,i=1,2,\dots ,n,\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill \frac{1}{L_N-N_0/n}\sum _{k=1}^N{\gamma }_k^d{I}_{\{k\in {\overline{O}}_{N,i}^0\}}& \to & 1-p,N\to \infty ,i=1,2,\dots ,n,\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill \frac{1}{N_0}\sum _{k=1}^N{\gamma }_k^d{I}_{\{k\in Z_N\}}& \to & 1-p,N\to \infty .\hfill \end{array}$$

(32)

When $k\in Z_N$, ${\gamma }_k^e=1$, and, thus, from (32) it follows that

$$\begin{array}{c}\hfill {\widehat{p}}_N\to p,N\to \infty .\end{array}$$

(33)

From (23) and (24), we can know $\frac{1}{L_N-N_0/n}{\sum }_{k=1}^N{s}_k{I}_{\{k\in {\overline{O}}_{N,i}^0\}}=\frac{m}{L_N-N_0/n}{\sum }_{k=1}^N{\gamma }_k^e{I}_{\{k\in {\overline{O}}_{N,i}^0\}}$$+\frac{{\overline{G}}_{N,i}}{L_N-N_0/n}$, where

$${\overline{G}}_{N,i}=\sum _{k=1}^N{s}_k{I}_{\{k\in {\overline{O}}_{N,i}^0\}}-m\sum _{k=1}^N{\gamma }_k^e{I}_{\{k\in {\overline{O}}_{N,i}^0\}},i=1,2,\dots ,n,$$

(34)

Noting that ${\overline{G}}_{N,i}\in \{0,1,2,\dots ,m-1\}$, we have

$$\begin{array}{c}\hfill \frac{1}{L_N-N_0/n}\sum _{k=1}^N{s}_k{I}_{\{k\in {\overline{O}}_{N,i}^0\}}-\frac{m}{L_N-N_0/n}\sum _{k=1}^N{\gamma }_k^e{I}_{\{k\in {\overline{O}}_{N,i}^0\}}\to 0,N\to \infty .\end{array}$$

(35)

According to (29), when $k\in {\overline{Z}}_N$,

$$\begin{array}{c}\hfill \mathbb{E}[{\gamma }_k^d-(1-p)|{\mathcal{F}}_k]=\mathbb{E}[{\gamma }_k^d-(1-p)]=0.\end{array}$$

(36)

Denote ${\tilde{\gamma }}_k={\gamma }_k-(1-p){\gamma }_k^e$. Then, by Theorem 3 in ([32], p. 217), it can be ascertained that

$$\begin{array}{ccc}\hfill \mathbb{E}\left[{\tilde{\gamma }}_k\right|{\mathcal{F}}_{k-1}]& =& \mathbb{E}\left[\mathbb{E}\left[{\tilde{\gamma }}_k\right|{\mathcal{F}}_k]|{\mathcal{F}}_{k-1}\right]\hfill \\ & =& \mathbb{E}\left[\mathbb{E}\left[({\gamma }_k^d-(1-p)){\gamma }_k^e\right|{\mathcal{F}}_k]|{\mathcal{F}}_{k-1}\right]\hfill \\ & =& \mathbb{E}\left[{\gamma }_k^e\mathbb{E}\left[({\gamma }_k^d-(1-p))\right|{\mathcal{F}}_k]|{\mathcal{F}}_{k-1}\right]\hfill \\ & =& \mathbb{E}\left[0|{\mathcal{F}}_{k-1}\right]\hfill \\ & =& 0,k\in {\overline{Z}}_N.\hfill \end{array}$$

Therefore, $\{{\tilde{\gamma }}_k,{\mathcal{F}}_k,k\in {\overline{Z}}_N\}$ is a martingale difference sequence. Given ${\gamma }_k\in \{0,1\}$, it is known that ${\tilde{\gamma }}_k\le p$. Thus, we have $\mathbb{E}{\left({\sum }_{k=1}^N{\tilde{\gamma }}_k{I}_{\{k\in {\overline{Z}}_N\}}\right)}^2<\infty$ and

$$\begin{array}{c}\hfill \sum _{k=1}^{\infty }\frac{\mathbb{E}\left({\tilde{\gamma }}_k^2\right)}{k^2}{I}_{\{k\in {\overline{Z}}_N\}}\le \sum _{k=1}^{\infty }\frac{\mathbb{E}\left(p^2\right)}{k^2}=\frac{p^2{\pi }^2}{6}<\infty .\end{array}$$

According to the law of large numbers for the martingale difference sequence, it can be seen that

$$\begin{array}{ccc}& & \frac{1}{L_N-N_0/n}\sum _{k=1}^N{\tilde{\gamma }}_k{I}_{\{k\in {\overline{O}}_{N,i}^0\}}\hfill \\ & =& \frac{1}{L_N-N_0/n}\sum _{k=1}^N{\gamma }_k{I}_{\{k\in {\overline{O}}_{N,i}^0\}}-\frac{1}{L_N-N_0/n}\sum _{k=1}^N(1-p){\gamma }_k^e{I}_{\{k\in {\overline{O}}_{N,i}^0\}}\hfill \\ & \to & 0,N\to \infty ,i=1,2,\dots ,n\mathrm{w}.\mathrm{p}.1.\hfill \end{array}$$

(37)

Combined with (35), we have

$$\begin{array}{ccc}& & \frac{1}{L_N-N_0/n}\sum _{k=1}^N{s}_k{I}_{\{k\in {\overline{O}}_{N,i}^0\}}-\frac{m}{L_N-N_0/n}\sum _{k=1}^N\frac{{\gamma }_k{I}_{\{k\in {\overline{O}}_{N,i}^0\}}}{1-p}\hfill \\ & \to & 0,N\to \infty ,i=1,2,\dots ,n.\hfill \end{array}$$

By Lemma 1, combining (30) and (33), we can obtain

$$\begin{array}{c}\hfill \frac{m}{L_N-N_0/n}\sum _{k=1}^N\frac{{\gamma }_k{I}_{\{k\in {\overline{O}}_{N,i}^0\}}}{1-{\widehat{p}}_N}\to F_i,N\to \infty ,i=1,2,\dots ,n.\end{array}$$

The proof is completed.    □

4.2. 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:

$$O_{N,i}^0=\{x:x\in O_{N,i},\mathrm{rem}(\frac{x-i}{n},b)<a\}$$

(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 Weyl’s equidistribution theorem [33].

Algorithm 1 Generation of subset $Z_N$ when $\alpha$ is an irrational number
Input:
	Given an irrational number $\alpha$, the data length N, the input period n.
Output:
	$Z_N$ 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:	$Z_N\leftarrow \varnothing$
3:	for $i=1$ to n do
4:	$O_{N,i}^0\leftarrow$SelectSubset($i,n,L_N,\alpha$)
5:	$Z_N\leftarrow Z_N\cup O_{N,i}^0$
6:	end for
7:	return $Z_N$
8:	function SelectSubset($i,n,L_N,\alpha$)
9:	$S\leftarrow \mathrm{empty}\mathrm{set}$
10:	for $l=1$ to $L_N$ do
11:	$cur\leftarrow l\alpha -\lfloor l\alpha \rfloor$
12:	if $cur<\alpha$ then
13:	$S.\mathrm{ADD}\left(\right(l-1)*n+i)$
14:	end if
15:	end for
16:	return S
17:	end function

(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 $\overline{\theta }$. Then, $\overline{\theta }$ 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 $N({\widehat{\theta }}_N-\overline{\theta }){({\widehat{\theta }}_N-\overline{\theta })}^T$ for the multiple trajectory data to replace the mean-square convergence rate $N\mathbb{E}{({\widehat{\theta }}_N-\overline{\theta })}^T$ 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

$$\overline{\theta }=\frac{1}{M}\sum _{j=1}^M{\widehat{\theta }}_{N,j},j=1,2,\dots ,M.$$

(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. For convenience, denote $\Lambda (\alpha ;{\widehat{\theta }}_N)=N({\widehat{\theta }}_N-\overline{\theta }){({\widehat{\theta }}_N-\overline{\theta })}^T$ and define the average value of the matrix trace for multiple trajectories as ${\Omega }_{N,b}=\mathrm{tr}\left(\overline{\Lambda }({\alpha }_b;{\widehat{\theta }}_{N,{\alpha }_b})\right),b=1,2,\dots ,B$, where B is the number of different values of $\alpha$.

Through the above method, a set of convergence rate data $\{({\alpha }_1,{\Omega }_{N,1}),({\alpha }_2,{\Omega }_{N,2}),\dots ,$ $({\alpha }_B,{\Omega }_{N,B})\}$ can be obtained under the same m and unknown-but-consistent packet loss conditions for different values of $\alpha$. The data are used for subsequent model fitting and solving optimization problems. The data generation process is illustrated in Figure 2:

(3)

Model order determination

Use polynomial fitting to model the relationship between the index and $\alpha$. Since the specific order of the relationship expression is unknown, the Akaike information criterion (AIC) is used to determine the order of polynomial fitting. Suppose the fitting expression is

$${\Omega }_N\left(\alpha \right)=c_h{\alpha }^h+c_{h-1}{\alpha }^{h-1}+\cdots +c_1\alpha +c_0,$$

(40)

where h is the order of the polynomial. Denote ${\nu }_h=[c_0,c_1,\dots ,c_h]$ as the polynomial coefficients,

$$\begin{array}{c}\hfill X_h=\left[\begin{array}{cccc}1& {\alpha }_1& \cdots & {\alpha }_1^h\\ ⋮& ⋮& \ddots & ⋮\\ 1& {\alpha }_B& \cdots & {\alpha }_B^h\end{array}\right],Y=\left[\begin{array}{c}{\Omega }_{N,1}\\ ⋮\\ {\Omega }_{N,B}\end{array}\right].\end{array}$$

Then, the least squares solution for the polynomial coefficients can be obtained as ${\widehat{\nu }}_h={\left(X_h^T{X}_h\right)}^{-1}{X}_h^{T}Y$.

Denote the sum of the squared residuals:

$$e_h=\sum _{b=1}^B{\left({\Omega }_{N,b}-({\widehat{c}}_h{\alpha }_b^h+{\widehat{c}}_{h-1}{\alpha }_b^{h-1}+\cdots +{\widehat{c}}_1{\alpha }_b+{\widehat{c}}_0)\right)}^2.$$

In the calculation of AIC, the likelihood function is inversely proportional to the sum of the squared residuals. Therefore, replacing the likelihood function with the sum of the squared residuals, the calculation formula becomes

$$Bln\left(\frac{e_h}{B}\right)+2(h+1).$$

For each order, use the least squares method to find the best polynomial coefficients, and calculate the AIC values for different orders. The final chosen fitting order is

$$h^{*}=arg\underset{h\ge 1}{min}\{Bln\left(\frac{e_h}{B}\right)+2(h+1)\}.$$

(41)

The corresponding polynomial coefficients are

$${\widehat{\nu }}_{h^{*}}={\left(X_{h^{*}}^T{X}_{h^{*}}\right)}^{-1}{X}_{h^{*}}^{T}Y.$$

(42)

Using the above method, the relationship between the trace of the approximate covariance matrix and $\alpha$ is obtained as

$${\Omega }_N\left(\alpha \right)={\widehat{\nu }}_{h^{*}}{[1,\alpha ,\dots ,{\alpha }^{h^{*}}]}^T.$$

Denote the minimum value of ${\Omega }_N\left(\alpha \right)$ in the interval $(0,1)$ as D. From the identification algorithm (25)–(27), it is known that the parameter estimate fails to converge to the true value when $\alpha$ is 0 or 1. So, when ${\Omega }_N\left(\alpha \right)$ in the $(0,1)$ interval of the minimum value does not exist—that is, when the minimum value is obtained approaching the endpoints—there is no solution. Then, the solution that makes the convergence rate optimal is

$$\begin{array}{c}\hfill {\alpha }_N^{*}=\left\{\begin{array}{cc}\mathrm{No}\mathrm{solution},\hfill & \mathrm{if}D\mathrm{does}\mathrm{not}\mathrm{exist};\hfill \\ D,\hfill & \mathrm{if}D\mathrm{exists}.\hfill \end{array}\right.\end{array}$$

5. Trade-Off between Communication Rate and Convergence Performance

5.1. Communication Rate with Unknown Packet Loss Probability

Theorem 3.

Under the condition that the probability of packet loss is unknown, the communication rate of the improved CSTCS (23) and (24) is

$$\frac{1}{N}\sum _{k=1}^N{\gamma }_k^e\to \overline{\gamma }=\alpha +\frac{1-\alpha }{mn}\sum _{i=1}^n{F}_i,N\to \infty .$$

(43)

Proof. 

From (35), it is known that

$$\begin{array}{c}\hfill \frac{1}{N-N_0}\sum _{k=1}^N{\gamma }_k^e{I}_{\{k\in {\overline{O}}_{N,i}^0\}}-\frac{1}{m(N-N_0)}\sum _{k=1}^N{s}_k{I}_{\{k\in {\overline{O}}_{N,i}^0\}}\to 0,N\to \infty .\end{array}$$

(44)

Combining with (30), we can obtain

$$\begin{array}{c}\hfill \frac{1}{N-N_0}\sum _{k=1}^N{\gamma }_k^e{I}_{\{k\in {\overline{O}}_{N,i}^0\}}\to \frac{F_i}{mn},N\to \infty .\end{array}$$

(45)

Then, it follows that

$$\begin{array}{ccc}\hfill \frac{1}{N}\sum _{k=1}^N{\gamma }_k^e& =& \frac{1}{N}\sum _{k=1}^N\left({\gamma }_k^e{I}_{\{k\in Z_N\}}+{\gamma }_k^e{I}_{\{k\in {\overline{Z}}_N\}}\right)\hfill \\ & =& \frac{1}{N}\left(N_0+\sum _{i=1}^n\sum _{k=1}^N{\gamma }_k^e{I}_{\{k\in {\overline{O}}_{N,i}^0\}}\right)+\frac{1}{N}({\gamma }_{n{L}_N+1}^e+\dots +{\gamma }_N^e)\hfill \\ & =& \frac{N_0}{N}+\sum _{i=1}^n\frac{N-N_0}{N}\left(\frac{1}{N-N_0}\sum _{k=1}^N{\gamma }_k^e{I}_{\{k\in {\overline{O}}_{N,i}^0\}}\right)+\frac{1}{N}({\gamma }_{n{L}_N+1}^e+\dots +{\gamma }_N^e)\hfill \\ & \to & \alpha +\frac{1-\alpha }{mn}\sum _{i=1}^n{F}_i,N\to \infty .\hfill \end{array}$$

The proof is completed.    □

5.2. Trade-Off between Convergence Performance and Communication Rate

From Theorem 3 and Section 4.2, it is clear that in scenarios where p is unknown both the communication rate and convergence rate are related to $\alpha$. To represent their relationship with $\alpha$, the communication rate is denoted as $\overline{\gamma }=\overline{\gamma }\left(\alpha \right)$. $\Omega \left(\alpha \right)$ serves as an indicator to describe the performance of the algorithm (25)–(27), which represents the trace of the covariance matrix $N\mathbb{E}({\widehat{\theta }}_N-\theta ){({\widehat{\theta }}_N-\theta )}^T$, as N tends to infinity.

What is the optimal selection of $\alpha$ that minimizes $\overline{\gamma }\left(\alpha \right)$ while meeting the performance requirements of the algorithm? This can be regarded as a constrained optimization problem, as follows:

$$\begin{array}{ccc}& \underset{\alpha }{min}& \overline{\gamma }\left(\alpha \right),\hfill \end{array}$$

(46)

$$\begin{array}{ccc}& s.t.& \Omega \left(\alpha \right)\le \epsilon ,\alpha \in (0,1),\hfill \end{array}$$

(47)

where $\epsilon >0$ is a constant to bound the algorithm performance. Define the solution set of $\Omega \left(\alpha \right)\le \epsilon$ in $(0,1)$ as $A_{\epsilon }\ne \varnothing$ and denote ${\alpha }^{*}$ as the solution to (46) and (47).

Remark 5.

The index $\Omega \left(\alpha \right)$ that portrays the convergence rate can be obtained approximately by using the method in Section 4.2 or by employing other nonlinear modeling methods, such as exponential and logarithmic curve fitting.

Theorem 4.

Under the conditions of Theorem 1, the solution to the optimization problem (46) and (47) is provided by

$${\alpha }^{*}=min{A}_{\epsilon },$$

(48)

where $min{A}_{\epsilon }$ denotes the minimum value in the set $A_{\epsilon }$. At this point, the communication rate of the improved CSTCS (23) and (24) is

$$\overline{\gamma }\left({\alpha }^{*}\right)={\alpha }^{*}+\frac{1-{\alpha }^{*}}{mn}\sum _{i=1}^n{F}_i.$$

(49)

Proof. 

From Theorem 3, it is known that

$$\frac{\mathrm{d}\left(\overline{\gamma }\left(\alpha \right)\right)}{\mathrm{d}\alpha }=1-\frac{1}{mn}\sum _{i=1}^n{F}_i.$$

Combining $m\ge 1$ and $F_i<1,i=1,2,\dots ,n$, we have $\frac{\mathrm{d}\left(\overline{\gamma }\left(\alpha \right)\right)}{\mathrm{d}\alpha }>0$. Furthermore, considering the solution set $A_{\epsilon }\ne \varnothing$ of $\Omega \left(\alpha \right)\le \epsilon$, the solution to (46) and (47) can be obtained as

$${\alpha }^{*}=min{A}_{\epsilon }.$$

(50)

Substituting ${\alpha }^{*}$ into (43) gives the communication rate.    □

6. Numerical Simulation

Consider the following system:

$$\left\{\begin{array}{c}{y}_k={\varphi }_k^T\theta +d_k;\hfill \\ \hfill s_k=I_{\{y_k\le C\}},k=1,\dots ,N,\end{array}\right.$$

where the noise $d_k$ obeys a Gaussian distribution with zero mean and a variance of 225, satisfying the conditions in Assumption 1; $\theta ={[5,20]}^T$, $C=70$, $N=10000$; the minimum positive period of the input is 2, and one complete period is $[2,3]$.

The data are transmitted under data packet loss and CSTCS. Set the adjustment factor m to 5 and the packet loss probability p during data transmission to 0.3.

A portion of the data are used for estimating p when the estimation center is unaware of the packet loss probability. Figure 3 shows the event-triggered mechanism triggering and the data packet loss situation in the interval [950, 1000]. At the moments when data are utilized for estimating p, the value of ${\gamma }_k^e$ is equal to 1. The subset $Z_N$ is selected as described in Section 4.2, and the data proportion used for estimating the packet loss probability is set to $\alpha =0.5$.

The parameter $\theta$ is estimated under the unknown and known packet loss probability, respectively. Figure 4 and Figure 5 show the convergence of the parameter under the above values of $m,p,\alpha$, with both ${\widehat{\theta }}_N$ converging gradually to the true value ${[5,20]}^T$. This verifies the conclusions of Theorems 1 and 2. Figure 6 shows the relationship between convergence performance and p in the context of known packet loss probabilities, particularly when the packet loss probabilities are set at $0.1,0.3,0.5$. The performance measurement standard is based on the average value of $\left|\right|{\widehat{\theta }}_N-\theta \left|\right|$ generated from 200 trajectories. The higher the probability of packet loss, the worse the convergence performance.

Three different types of noise are selected that satisfy the conditions of Assumption 1 and are different from Gaussian noise: uniform distribution noise over the interval $[-20,20]$, Laplace noise with zero mean and a scale parameter of 100, and Cauchy noise with a location parameter of 0 and a scale parameter of 40. Figure 7 displays the convergence of the identification algorithm under these three different types of noise when the packet loss probability is known, while Figure 8 shows the convergence of the algorithm when the packet loss probability is unknown. The fact that the algorithm still converges under these three types of noise indicates that it is not limited to Gaussian noise. Subsequent simulations and analyses are still based on Gaussian noise.

Based on Theorem 3, the calculated communication rates for $\alpha =0.1,0.3,0.5$ are 0.2207, 0.3939, and 0.5671, respectively. From Figure 9, it is clear that as N increases, the average communication rate progressively approaches the theoretical value. Furthermore, the larger $\alpha$ results in the higher communication rate, which is consistent with the result of Theorem 3.

Adjusting m to 10, the relationship between the convergence rate and $\alpha$ is obtained according to the subset selection, data generation, and model fitting methods described in Section 4.2. Figure 10 shows that the order of the model is determined to be 5 according to the AIC criterion. Figure 11 describes, under the current simulation conditions, how the index ${\Omega }_N$ approximating the convergence rate varies with the proportion $\alpha$ of data used for estimating the probability of packet loss. In this case, the optimal value of ${\alpha }_N^{*}$ is 0.16. Take $\alpha ={\alpha }_N^{*}$ and any $\alpha$ that is different from ${\alpha }_N^{*}$, such as $\alpha =0.1,0.3,0.5$, respectively. Figure 12 further verifies the optimality of ${\alpha }_N^{*}$ by using the average of $\left|\right|{\widehat{\theta }}_N-\theta \left|\right|$ calculated from 200 trajectories as a representation of the convergence performance. The convergence performance is best when $\alpha ={\alpha }_N^{*}$.

7. Conclusions

This paper explored the congruential summation-triggered identification of FIR systems with quantized observations under uncertain communications. CSTCS was introduced, according to system order and cumulative effect, and the packet loss process model was established using a set of Bernoulli random variables. Parameter estimation algorithms were designed for scenarios with known and unknown packet loss probabilities, respectively, and their strong convergence was demonstrated. The relationship between the convergence rate and the proportion of data used for packet loss estimation was expressed using numerical fitting, representing the trade-off between the communication rate and the convergence property as a constrained optimization problem. These provided a solution approach for system identification and optimization analysis under network congestion conditions, and, due to the strong scalability of FIR systems, the work in this paper could be extended to more complex systems. Also, the consideration of multi-threshold quantization and other aspects of communication uncertainty, such as data errors or transmission delays, is a worthwhile direction for research.