## 1 Introduction  
With the rapid development of the internet and cloud computing technology, the demand for the integration of the physical world and the digital world is increasing [1]. The efficient, green, and intelligent characteristics of CPSs are highly compatible with the needs of the times, putting them in a booming and prosperous period under the information age, and are rapidly being applied in areas such as industrial production, national defense and military, smart grids, smart healthcare, and other fields [2, 3, 4, 5]. In industrial production, CPS technology is deeply intertwined with Industry 4.0, based on embedded systems and information technology, and it can monitor the status of physical equipment in factories in real time to deeply perceive the production process, and its development is highly consistent with that of smart factories [6, 7]. In recent years, with the rapid promotion of automatic driving, the highly integrated information technology of the car can combine and interact with computer and network technology, and vehicle operation in the physical world using functions such as perception technology, real-time data processing, and control algorithms [8, 9]. However, the integration of wireless communication technology in CPSs gives the possibility of remote control of the system, and therefore wireless communication has the advantages of convenient control, but also makes the communication between systemshighly open and vulnerable to malicious attackers [10; 11; 12]. Such attacks can be categorized into data tampering attacks and denial of service (DoS) attacks according to their effects, among which data tampering attacks can be realized in the form of data modification attacks, false data injection (FDI) attacks, and data replay attacks [13]. In CPSs, data tampering attacks can hijack and tamper with the normal data in the communication between the sender and the receiver, so that the data used by the receiver are not the true data of the system, which in turn causes the control link to give the wrong control commands, leading to the malfunctioning of the whole system [14; 15; 16]. The study of data tampering attacks is conducive to the defender to make timely and appropriate decisions on the attack strategy, issue correct control instructions, and minimize the damage of data tampering attacks on the CPS, so there have been related studies. [PERSON]. [17] proposed a linear spoofing attack strategy for the false data injection attack and gave the corresponding feasibility constraints to ensure that the attacker could successfully inject false data without being detected. [PERSON]. [18] investigated the problem of designing FDI attacks for a class of CPSs with a state estimator and an attack detector, where the desired sequence of non-perfect attacks could be designed by analyzing the maximum eigenvalues of the auxiliary matrices and the corresponding eigenvectors in the absence of a priori knowledge of the estimator. And unlike previous zero-mean Gaussian distribution attack strategies, ref. [19] proposed an optimal attack strategy based on arbitrary mean Gaussian distribution. With an understanding of the attacker's attack pattern, the study of attack detection from the defender's perspective focused on how to identify and respond to ongoing or potential attacks. In terms of FDI attack detection, ref. [20] designed a nonlinear local joint estimator, and a learning-based fusion criterion was proposed for multi-sensor FDI attacks to simultaneously estimate the system state and the attack signal. For covert FDI attacks in train ground communication systems, ref. [21] used an authentication mechanism to design an automatically generated multiplicative coding scheme to detect intrusions, dynamically updated the coding sequences online to encrypt and decrypt the raw train measurements, and then built a defense model based on the navigational projection algorithm to reconstruct the train location information corrupted by the attacker. For detecting FDI attacks in smart grids, ref.