## 1 Introduction
However, due to the limited communication capacity of the network and the lack of channel protection measures, various network-induced phenomena are inevitably encountered in the process of data transmission, such as time delays, cyber-attacks and packet loss. Network-induced phenomena can raise the instability factor of the system and reduce the performance of the estimator [29]. Therefore, estimating these phenomena timely and reducing their influence on estimation performance as far as possible has important practical meaning. For a variety of network-induced phenomena that occur in engineering practice, such as various measurement and state value anomalies (state saturation [30], constrained measurement [31], measurement outliers [32]) and various cyber-attacks (malicious attacks that can bypass anomaly detectors [33], deception attack [34], joint false data injection attack [35], replay attack [36], DoS attack [37; 38] and mixed-attacks [39]), the corresponding estimation schemes have been designed to reduce the influence of the above phenomena on the estimation performance. Scholars have conducted research on the problem of set-membership estimation with network-induced phenomena from different perspectives. For example, to overcome the communication constraints, refs. [40; 41] have designed the corresponding SMF in the case of introducing redundant channels. In view of the incomplete data caused by various factors in the transmission process, refs. [42; 43] have used censored measurement to describe this phenomenon and have designed the corresponding SMF. For the actual engineering scenarios, a state estimation approach that utilizes the Kalman filter and recursive systematic convolutional code has been proposed in [44] to effectively estimate the state of the smart grid and detect attacks, which guarantees the normal operation of the smart grid. In [45], a particle filter based on the Paillier encryption-decryption scheme has been proposed for the multi-machine power systems for confronting the potential risk of cyber-attacks, which makes the attacker unable to attack the encrypted data to ensure the precision of state estimation during the operation of the multi-machine power systems. In fact, several studies have considered the various sensor-related problems and network-induced phenomena mentioned above, and have given hypotheses for corresponding approaches. For MMs, ref. [46] has given the hypothesis that MMs must satisfy the binary exchange sequence of the conditional probability distribution. Reference [27] has given the hypothesis that MMs need to satisfy Bernoulli distribution. Reference [28] has given the hypothesis that the occurrence of different MMs in each sensor node can be described by a diagonal matrix, which is more realistic. Regarding cyber-attacks, many scholars have hypothesized that cyber-attacks obey the hidden Markov model [47]. However, since the Bernoulli distribution can directly indicate whether the attack occurs through the value of the random variable (0 or 1) [48], which is more suitable to describe the deception attack. Therefore, this paper also hypothesizes that the cyber-attacks follow the Bernoulli distribution and uses a diagonal matrix to describe the continuous occurrence of the MMs on different sensor nodes. From the aforementioned and related literature, we can see that current studies only consider the set-membership estimation of MNNs encountering either various sensor related problems or various network-induced phenomena. As far as we know, we have not yet seen the research results on how to conduct set-membership estimation and reduce the influence on state estimation performance when the MNNs encounter multiple sensor-related problems and multiple network-induced phenomena simultaneously.