Fast Detection of FDI Attacks and State Estimation in Unmanned Surface Vessels Based on Dynamic Encryption

Abstract

Wireless sensor networks (WSNs) are used for data acquisition and transmission in unmanned surface vessels (USVs). However, the openness of wireless networks makes USVs highly susceptible to false data injection (FDI) attacks during data transmission, which affects the sensors’ ability to receive real data and leads to decision-making errors in the control center. In this paper, a novel dynamic data encryption method is proposed whereby data are encrypted prior to transmission and the key is dynamically updated using historical system data, with a view to increasing the difficulty for attackers to crack the ciphertext. At the same time, a dynamic relationship is established among ciphertext, key, and auxiliary encrypted ciphertext, and an attack detection scheme based on dynamic encryption is designed to realize instant detection and localization of FDI attacks. Further, an $H_{\infty }$ fusion filter is designed to filter external interference noise, and the real information is estimated or restored by the weighted fusion algorithm. Ultimately, the validity of the proposed scheme is confirmed through simulation experiments.

1. Introduction

Cyber–physical systems (CPSs) are becoming increasingly important in modern science and technology, especially in areas such as ocean monitoring and defense [1,2,3]. CPSs closely integrate computational, cyber, and physical processes, enabling systems to sense and respond to environmental changes in real time [4,5,6]. Wireless sensor networks (WSNs), as integrated systems, usually integrate multiple sensors and information sources, such as Global Navigation Satellite Systems (GNSSs), Doppler Velocimeters (DVLs), Inertial Navigation Systems (INSs), and so on, at their core [7,8]. Through the effective combination of CPSs and wireless sensor networks, CPSs show great potential in data acquisition, processing, and transmission. Unmanned surface vessels (USVs), as a key application of this system [9,10,11], rely on WSNs for real-time data collection and transmission for efficient monitoring and decision making in the marine environment [12,13].

The reliability of data is directly related to the USV’s ability to perform its mission [14,15,16], especially in the complex maritime environment and under potential cyberattacks, where ensuring the authenticity and integrity of the data is of particular importance. In USV applications, WSNs enable these vessels to efficiently perform data acquisition and real-time monitoring in a vast marine environment. By integrating data from different sensors, USVs are able to quickly adapt to complex marine environments, enhancing their operational efficiency and safety. During data measurement and transmission, the data are susceptible to a variety of external disturbances, including marine environmental disturbances (e.g., the effects of wind, waves, currents, etc., on USVs) and signal disturbances, which can lead to variations in the data, further exacerbating the uncertainty in the system’s decision making and thus affecting the safety and efficiency of USVs. To make data transmission more accurate, the use of WSNs and $H_{\infty }$ fusion filtering techniques is beneficial. Recently, the rapid development of $H_{\infty }$ filtering technology has provided a new solution for data processing in complex systems [17,18,19]. With its superior anti-interference ability, this technology can effectively filter external interference noise and improve the reliability and accuracy of data. By fusing data from different sensors, $H_{\infty }$ fusion filtering can significantly improve the accuracy of the data, thus enhancing the autonomous decision-making level of USVs. This is essential to the autonomous navigation and operation of USVs in dynamic and uncertain marine environments.

Although USVs show great advantages in data acquisition and transmission, their high reliance on wireless communication also exposes them to potential cybersecurity threats, especially in terms of false data injection (FDI) attacks [20,21,22]. FDI attacks are cyberattacks that mislead the system’s decision making by tampering with or falsifying data, and the attacker can influence the behavior of the USVs through these false data, leading to wrong navigation and decision making. FDI attacks are highly stealthy, and malicious attackers are potentially able to inject false information into USVs through wireless networks without being detected [23,24,25]. Such attacks can not only mislead USVs’ mission execution, e.g., wrong course selection or improper action strategies, but also lead to more serious consequences, such as personnel casualties. In addition, FDI attacks may also cause a chain reaction to other USV-dependent data systems, affecting the overall efficiency of maritime surveillance and defense. Therefore, the development of effective protective measures and data processing strategies has become urgent.

To date, multiple researchers have considered FDI attacks. For example, Chen et al. [26] developed a framework capable of simultaneously estimating system states and FDI attack signals under FDI and denial-of-service attacks and implementing compensatory control. At the same time, a novel co-design methodology was developed. The method is capable of estimating and compensating unknown FDI attacks without relying on any a priori constraints on the frequency, duration, or derivatives of the FDI attack signals and effectively suppressing their negative impacts. The compensation method is effective against jamming attacks; however, the FDI attack is detected only after the injection into the system, i.e., after the system state information has been tampered with.

To address the above FDI detection issues, Xia et al. [27] proposed a distributed detection method which consists of two steps: a kernel entropy-based discrimination scheme identifies secure nodes and attacked nodes by calculating the kernel entropy of the measured data between nodes, while a state-aware scheme enables each node to monitor its own state and filter reliable neighboring nodes. The method theoretically derives the conditions that guarantee the stability of the algorithm’s mean value and reveals how its performance changes under an FDI attack.

Zhang et al. [28] introduced an adaptive backstepping technique into nonlinear CPSs and integrated a dynamic surface control technique which successfully overcomes the computational complexity (combinatorial explosion) problem existing in the traditional backstepping method. The method significantly reduces the computational burden by bypassing the direct complex computation of higher-order systems through the stepwise design of virtual control inputs. In addition, they developed a novel defense strategy centered on a nonlinear interference observer. The strategy is capable of accurately estimating the external composite disturbances and substantially improving the system robustness. The nonlinear disturbance observer dynamically estimates the magnitude of the disturbance and its trend by monitoring the system state and output in real time, enabling the controller to compensate accordingly in real time, thus effectively enhancing the system’s stability when it is subjected to attacks and disturbances. Although the above methods have a certain degree of fault tolerance and FDI attack resistance, they do not solve the privacy protection problem in the data transmission process.

Li et al. [29] designed a low-computational-complexity encryption scheme for data based on random matrices to achieve instant detection and localization of FDI attacks during local estimation transmission. Using the results of the encryption detection scheme, a secure fusion estimation algorithm was proposed. The algorithm is able to discard the attacked local estimation information and use only the unattacked local estimation information for fusion estimation. Although the method proposed in the article has significant advantages in terms of privacy protection of data, the static key it uses lacks forward security, and the entire cryptographic detection scheme will fail once it is intercepted on the transmission link or restored through brute force decryption.

Hence, improving the ability of USVs to protect data privacy against FDI attacks and ensuring the accuracy of data estimation through dynamic key management and data encryption techniques have become very meaningful and urgent research focuses. The primary contributions of this paper can be outlined as follows:

(1)

A novel dynamic data encryption scheme for privacy protection during data transmission is proposed. The scheme encrypts the data before data transmission and dynamically updates the key by utilizing historical system data, which increases the difficulty for attackers to crack the ciphertext compared with the literature [29].

(2)

Establishment of a dynamic relationship between the ciphertext, the key, and the auxiliary encrypted ciphertext. Once the ciphertext or auxiliary encrypted ciphertext is maliciously tampered with, the dynamic relationship is destroyed. Based on this feature, we design an attack detection scheme based on dynamic data encryption. The scheme can detect FDI attacks in real time and immediately discard the damaged data.

(3)

$H_{\infty }$ fusion filters are proposed to suppress external noise and interference during data measurement and transmission. The weighted fusion algorithm fuses the data that have not received the attack to more accurately estimate and restore the real signal, ensuring the information integrity and reliability of USVs during data transmission.

The organization of this paper is as follows: Section 2 develops the system model for the problem under consideration; Section 3 details the proposed dynamic data encryption scheme and the FDI attack detection method; Section 4 introduces the design of the $H_{\infty }$ fusion filter and explores its applications; Section 5 employs simulation experiments to validate the performance of the proposed scheme.

Notations: $R^{n\times n}$ denotes the n-dimensional Euclidean space, and I and 0 represent the identity matrix and the zero matrix of appropriate dimensions, respectively. The transpose of a matrix is indicated by the superscript T. For any matrix $X\in R^{n\times n}$, the condition $X>0( X<0)$ defines X as a real symmetric positive definite matrix (or a real symmetric negative definite matrix). ⊕ represents the XOR operator.

2. System Description

Since USVs are susceptible to interference by unpredictable environmental factors such as wind and wave currents during navigation, they not only directly change their motion attitude but also introduce uncertainty noise in their motion control system and sensor measurement data. As shown in Figure 1, when USVs transmit data to the sensors via a wireless network, the communication link may be subjected to an FDI attack, which tampers with the transmitted data, resulting in the distortion of the sensor reception, which in turn leads to wrong decision making by the control center.

Based on [14], the mathematical model of the motion of USVs can be described as $\dot{\eta }=J\left(\phi \right)v$, $M\dot{v}=-Dv+\tau +d\left(t\right)$. In which $\eta ={[u,v,r]}^T$ represents the position vector of the vessel and consisting of north-east position $(\chi ,\mu )$ and yaw angle $\sigma \in [0, 2\pi ]$, and $v={\left[u,\nu ,r\right]}^T$ denotes the USV velocity vector in the vessel coordinate system, which consists of longitudinal u, transverse $\nu$, and yaw angular velocities r. The rotation matrix $J\left(\phi \right)$ is expressed as

$$J\left(\phi \right)=\left[\begin{array}{ccc}cos\phi & -sin\phi & 0\\ sin\phi & cos\phi & 0\\ 0& 0& 1\end{array}\right]$$

(1)

Based on [30], a class of discrete-time multi-sensor network systems are modeled as follows:

$$\left\{\begin{array}{c}{x}_{k+1}=A{x}_k+B{\omega }_k\hfill \\ y_{i,k}=H_{i,k}{x}_k+v_{i,k}\hfill \\ z_k=L{x}_k\hfill \end{array}\right.$$

(2)

where $x_k={[P_{x,k},P_{y,k},{\sigma }_k]}^T$ denotes the position state information, $x_k={\eta }_k$, $(P_{x,k},P_{y,k})$ denotes the position coordinates, ${\sigma }_k$ denotes the yaw. $x_k$ and $y_{i,k}$ are the system state and measured output of the ith sensor at moment k, respectively. $z_k$ is the state to be estimated. ${\omega }_k\in R^{n_{\omega }}$ is an unknown external disturbance, and $v_{i,k}$ is measurement noise belonging to $l_2\left[0, \infty \right)$. Matrices A, B, $H_{i,k}$, and L have the appropriate dimensions.

At time instant k, each sensor receives a packet from the USV containing the ciphertext and its auxiliary encrypted ciphertext. The detector applies an attack detection scheme to the packet and discards any packet that has been maliciously altered by an attacker; subsequently, it performs a decryption operation on the ciphertext in any legitimate packet. However, marine environmental disturbances (e.g., the effects of wind, waves, and currents on USVs) are inevitable during data transmission. These disturbances introduce system state noise, which in turn leads to biased data. Therefore, $H_{\infty }$ fusion filtering is used to filter the interference noise and estimate or restore the real information by the weighted fusion algorithm.

As shown in Figure 1, during the transmission of data packets, a malicious attacker may launch an FDI attack on unknown data in the communication links to tamper with these data., whereas in a multi-sensor system, due to the limited resources of the attackers, it is unlikely that they can attack all communication links at the same time [31]. Therefore, it is reasonable to assume that there is at least one unattacked communication link in a multi-sensor system. This assumption ensures that the sensors receive at least one true system state. If all of them were attacked, then the sensors would not be able to obtain reliable data.

3. Encryption Attack Detection Scheme

This section designs an attack detection method incorporating dynamic data encryption, aiming to achieve real-time identification of FDI attacks. First, a dynamic data encryption mechanism with low computational overhead is constructed to ensure the security of transmitted data. Subsequently, based on this encryption mechanism, a corresponding attack detection scheme is designed. The specific process is shown in Figure 2.

3.1. Dynamic Data Encryption Scheme

As shown in Figure 1, local data $x_{i,k}$ is encrypted using a key $c_{i,k}$ before transmission. Once it reaches the decryptor, it will be restored from it using the decryption scheme. The specific encryption scheme is designed as follows:

$$\left\{\begin{array}{c}{c}_{i,0}=s_{i,0}\hfill \\ x_{i,k}^c=x_{i,k}-\phi \left(c_{i,k}\right)\hfill \\ E_{i,k}^c=x_{i,k}\oplus c_{i,k}-\phi \left(c_{i,k}\right)\hfill \\ c_{i,k+1}=x_{i,k}+s_{i,k}\hfill \end{array}\right.$$

(3)

where $c_{i,k}\in {\mathrm{R}}^{n\times \left(n+1\right)}$ is the key in the encryption process, $s_{i,k}\in {\mathrm{R}}^{n\times \left(n+1\right)}$ is a matrix of random numbers with suitable dimensions, $c_{i,k}$ is updated each time by summing it with $s_{i,k}$ based on the historical system data to achieve dynamic encryption, and $c_{i,0}=s_{i,0}$ is the given initial key. $x_{i,k}^c$ is the ciphertext corresponding to the encryption of $x_{i,k}$, and $E_{i,k}^c$ is the auxiliary encrypted ciphertext used for detecting FDI attacks. The function $\phi \left(c_{i,k}\right):{\mathrm{R}}^{n\times \left(n+1\right)}\to {\mathrm{R}}^{n\times \left(n+1\right)}$ is an arbitrary encryption function about $c_{i,k}$. For security reasons, the function $\phi \left(c_{i,k}\right)$ will be physically implanted during the initial deployment phase. In addition, the encryption function $\phi \left(c_{i,k}\right)$ can be selected according to the application scenarios as well as the requirements of computational complexity. After the encryption is completed, the encryptor sends the packet $\{x_{i,k}^c,E_{i,k}^c\}$ to the decryptor through the communication network.

To recover the original data $x_{i,k}$ from $x_{i,k}^c$, the decryption scheme is designed as follows:

$$\left\{\begin{array}{c}{d}_{i,0}=s_{i,0}\hfill \\ x_{i,k}^d=x_{i,k}^c+\phi \left(d_{i,k}\right)\hfill \\ D_{i,k}^d=E_{i,k}^c+\phi \left(d_{i,k}\right)\hfill \\ d_{i,k+1}=x_{i,k}^d+s_{i,k}\hfill \end{array}\right.$$

(4)

where $d_{i,k}\in {\mathrm{R}}^{n\times \left(n+1\right)}$ is the key in the decryption process, $x_{i,k}^d$ is the decrypted value of $x_{i,k}$, and $D_{i,k}^d$ is an auxiliary value to be used in an attack detection scheme.

Notations: Under encryption method (3) and decryption method (4), a random matrix $s_{i,k}$ is introduced, which has to be consistent during encryption and decryption. In order to ensure that $d_{i,k}$ is equal to $c_{i,k}$, the ciphertext $x_{i,k}^c$ is recovered as $x_{i,k}$. In fact, since the use of the same seed when using a random number generator produces unique random sequences, it is possible to use in (3) and (4) the same random number generator seed to ensure the consistency of $s_{i,k}$, and this is a widely used solution in cryptography [32]. For security reasons, physical implantation of seeds will be used during the initial deployment phase.

In the attack-free case, the following propositions about dynamic data encryption schemes (3) and (4) can be obtained.

Proposition 1.

In the absence of FDI attacks, the following conclusions can be drawn using dynamic data encryption schemes (3) and (4):

$$\left\{\begin{array}{c}{x}_{i,k}^d=x_{i,k}\hfill \\ x_{i,k}^d\oplus D_{i,k}^d=d_{i,k}=c_{i,k}\hfill \end{array}\right.$$

(5)

Proof. 

According to dynamic data encryption schemes (3) and (4), $d_{i,k}=c_{i,k}$ is established; when the decryptor receives the data packet $\{x_{i,k}^c,E_{i,k}^c\}$, the decryption process is shown in (6):

$$\left\{\begin{array}{c}{x}_{i,k}^d=x_{i,k}^c+\phi \left(d_{i,k}\right)=x_{i,k}-\phi \left(c_{i,k}\right)+\phi \left(d_{i,k}\right)=x_{i,k}\hfill \\ D_{i,k}^d=E_{i,k}^c+\phi \left(d_{i,k}\right)=x_{i,k}\oplus c_{i,k}-\phi \left(c_{i,k}\right)+\phi \left(d_{i,k}\right)=x_{i,k}\oplus c_{i,k}\hfill \end{array}\right.$$

(6)

The decrypted data is then examined using the detection scheme to obtain the following results:

$$x_{i,k}^d\oplus D_{i,k}^d=x_{i,k}\oplus x_{i,k}\oplus c_{i,k}=c_{i,k}=d_{i,k}$$

(7)

According to the mathematical description of Equations (6) and (7), the correct data can be obtained in the ideal case of no FDI attack, and the attack detection scheme does not detect that the packet suffers from an FDI attack.    □

3.2. Scheme for Detection of Encryption-Based Attacks

Note that from Proposition 1, it is clear that in dynamic data encryption schemes (3) and (4), $x_{i,k}^d\oplus D_{i,k}^d=d_{i,k}$; this feature enables us to determine whether the system data have been tampered with by a malevolent attacker. Based on dynamic data encryption schemes (3) and (4), an attack detection scheme as shown in Algorithm 1 is proposed, which can detect whether the data are under FDI attack in a very short time.

Theorem 1.

In the absence of FDI attacks, the original data $x_{i,k}$ can be recovered using Algorithm 1.

Proof. 

In the absence of a cyberattack, $x_{i,k}^d=x_{i,k}$ is known from Proposition 1, which implies that the original data $x_{i,k}$ can be recovered using Algorithm 1. Also, $x_{i,k}^d\oplus D_{i,k}^d=d_{i,k}=c_{i,k}$ implies that Case 1 in Algorithm 1 is always satisfied.   □

Remark 1.

In encryption schemes (3) and (4), only XOR operations and basic arithmetic operations are employed, featuring low computational complexity. It enhances data security and keeps the system performance overhead within a negligible range.

Remark 2.

From Theorem 1, the system exists with a specific dynamic relation, i.e., $x_{i,k}^d\oplus D_{i,k}^d=d_{i,k}$, when there is no FDI attack. This dynamic relation will be changed when subjected to an FDI attack. This property can be used as a sufficient criterion for attack detection and provides a theoretical basis for the cryptographic detection algorithm proposed in this paper.

Algorithm 1: Attack detection algorithm based on dynamic data encryption

Step 1 (encryption process): At moment k, the encryptor executes the encryption process, generates the ciphertext $x_{i,k}^c$ and the auxiliary encryption ciphertext $E_{i,k}^c$, and encapsulates them into a data packet $\{x_{i,k}^c,E_{i,k}^c\}$.

Step 2 (data transmission): The encryptor sends the packet to the decryptor through the communication network.

Step 3 (decryption process): When the decryptor receives the received packet, it performs the decryption process.

Step 4 (attack Detection): After the decryption process, the following detection scheme is executed.

Case 1: If $x_{i,k}^d\oplus D_{i,k}^d=d_{i,k}$, this received packet is not attacked by FDI. $k\to k+1$, return to step 1.

Case 2: If $x_{i,k}^d\oplus D_{i,k}^d\ne d_{i,k}$, then this received packet is subject to FDI attack. Discard the data packets received on this communication link this time. $k\to k+1$, return to step 1.

4. ${\mathit{H}}_{\infty }$ Fusion Estimation Under FDI Attack

In data transmission, the effects of complex and variable marine environmental disturbances (e.g., wind-induced offsets and wave-induced impacts) on USVs are unavoidable. These powerful environmental factors can cause unpredictable attitude changes and position drifts in USVs, resulting in increased system state noise and biased data. For this reason, the $H_{\infty }$ fusion filtering technique is used to filter out the noise, and the real state information is estimated by a weighted fusion algorithm.

4.1. The Effectiveness of the Encryption-Based Attack Detection Scheme in Detecting FDI Attacks

In Algorithm 1, $x_{i,k}^c$ and $E_{i,k}^c$ are packaged and sent to the decryptor via the communication network. It is assumed that the attacker can recognize $x_{i,k}^c$ and $E_{i,k}^c$ to implement tampering. The attack strategy consists of the following three scenarios:

• Scenario 1: The attacker modifies only $x_{i,k}^c$ as follows:

  $$x_{i,k}^{\prime c}=x_{i,k}^c+a_{i,k}$$

  (8)
• Scenario 2: The attacker modifies only $E_{i,k}^c$ as follows:

  $$E_{i,k}^{\prime c}=E_{i,k}^c+b_{i,k}$$

  (9)
• Scenario 3: The attacker modifies both $x_{i,k}^c$ and $E_{i,k}^c$ as follows:

  $$\left\{\begin{array}{c}{x}_{i,k}^{\prime c}=x_{i,k}^c+a_{i,k}\hfill \\ E_{i,k}^{\prime c}=E_{i,k}^c+b_{i,k}\hfill \end{array}\right.$$

  (10)

where $a_{i,k}$ and $b_{i,k}$ are arbitrary non-zero attack signals of the appropriate dimensions.

Theorem 2.

For Algorithm 1, with the FDI attacks in Equations (8)–(10) in Scenarios 1–3 at time k, the encryption-based attack detection algorithm will trigger an alert immediately from time k onwards.

Proof. 

If the FDI attack in Equation (8) in Scenario 1 starts at time k, the decryption process is as follows:

$$\left\{\begin{array}{c}{x}_{i,k}^{\prime d}=x_{i,k}^{\prime c}+\phi \left(d_{i,k}\right)\hfill \\ =x_{i,k}+a_{i,k}\hfill \\ D_{i,k}^d=E_{i,k}^c+\phi \left(d_{i,k}\right)\hfill \\ =x_{i,k}\oplus c_{i,k}\hfill \end{array}\right.$$

(11)

□

Then, the improved format can be obtained as

$$x_{i,k}^{\prime d}\oplus D_{i,k}^d=\left[x_{i,k}+a_{i,k}\right]\oplus \left[x_{i,k}+c_{i,k}\right]=a_{i,k}\oplus c_{i,k}\ne d_{i,k}$$

(12)

If the FDI attack in (9) in Scenario 2 begins at time k, the decryption process is defined as follows:

$$\left\{\begin{array}{c}{x}_{i,k}^d=x_{i,k}^c+\phi \left(d_{i,k}\right)=x_{i,k}\hfill \\ D_{i,k}^{\prime d}=E_{i,k}^{\prime c}+\phi \left(d_{i,k}\right)\hfill \\ =x_{i,k}\oplus c_{i,k}+b_{i,k}\end{array}\right.$$

(13)

Thus, it can be derived that

$$x_{i,k}^{\prime d}\oplus D_{i,k}^{\prime d}=x_{i,k}\oplus x_{i,k}\oplus c_{i,k}+b_{i,k}=c_{i,k}+b_{i,k}\ne d_{i,k}$$

(14)

From the mathematical description of Equations (11)–(14), it can be seen that Scenario 1 and Scenario 2 always satisfy the Case 2 decision condition in Algorithm 1, which ensures that the encryption-based attack detection algorithm achieves a zero-delay alarm at attack initiation moment k. For the FDI attack corresponding to Scenario 3 (Equation (10)), the consistency conclusion can be obtained by the same methodological derivation, and its proof process is omitted here.

4.2. $H_{\infty }$ Fusion Estimation Based on Fast Attack Detection

Although attack detection algorithms can effectively filter out data tampered with by malicious attackers, the external noise interference inherent in the measurement and transmission process can still lead to biased system state information obtained by the sensors, which in turn affects the accuracy of control decisions. Therefore, the aim of this study is to design an $H_{\infty }$ filter that mitigates the effect of noise on the estimated system state. The $H_{\infty }$ filter is able to effectively cope with the noise and ensure that the system can still provide accurate state estimation and control in the event of noise interference. Then, a secure fusion estimation algorithm is proposed to perform optimal weighted fusion of unattacked data, thereby significantly improving the accuracy of state estimation while ensuring security and enabling more precise estimation and reconstruction of the true signal.

Consider the following local filter structure on sensor i:

$$\left\{\begin{array}{c}{\widehat{x}}_{i,k+1}=A_{pi}{\widehat{x}}_{i,k}+B_{pi}{y}_{i,k}\hfill \\ {\widehat{z}}_{i,k}=C_{pi}{\widehat{x}}_{i,k}\hfill \end{array}\right.$$

(15)

where ${\widehat{x}}_k$ and ${\widehat{z}}_k$ are the estimated values of $x_k$ and $z_k$, respectively. $A_{pi}$, $B_{pi}$, and $C_{pi}$ are filter parameter matrices to be probe-based.

By introducing the local error ${\tilde{z}}_{i,k}=z_k-{\widehat{z}}_{i,k}$ and combining Equations (2) and (15), the local filtering error system for sensor i is established as follows:

$$\left\{\begin{array}{c}{\varsigma }_{i,k}={\overline{A}}_i{\varsigma }_{i,k-1}+{\overline{B}}_i{\omega }_k-1+{\overline{D}}_i{v}_{i,k-1}\\ {\tilde{z}}_{i,k}={\overline{C}}_i{\varsigma }_{i,k}\end{array}\right.$$

(16)

In order to facilitate the derivation of the formula for the filter performance, the symbols in Equation (16) are denoted as follows: ${\varsigma }_{i,k}\stackrel{\Delta }{=}\left[\begin{array}{c}{x}_k\\ {\widehat{x}}_{i,k}\end{array}\right];{\overline{A}}_i=\left[\begin{array}{cc}A& 0\\ B_{pi}{H}_{i,k}& A_{pi}\end{array}\right];{\overline{B}}_i=\left[\begin{array}{c}B\\ 0\end{array}\right];{\overline{C}}_i=\left[\begin{array}{cc}L& -C_{pi}\end{array}\right];{\overline{D}}_i=\left[\begin{array}{c}0\\ B_{pi}\end{array}\right].$

${\varsigma }_{i,k}$ represents the state vector of local filtering error, ${\omega }_k\in R^{n_{\omega }}$ is the unknown external interference, $v_{i,k}$ is the measurement noise belonging to $l_2\left[0, \infty \right)$, and ${\overline{C}}_i$ is the gain matrix of the localized filter.

In the case of non-zero noise interference, the stability of the system can be determined by the $H_{\infty }$ filter performance criterion. Specifically, when the upper bound of the filter output energy is smaller than a given performance criterion, the filter system is proved to be asymptotically stable. Based on the actual noise characteristics and filter structure analysis, the $H_{\infty }$ performance criterion adopted in this chapter is expressed as follows:

$$\sum _{k=0}^{\infty }{\parallel {\tilde{z}}_{i,k}\parallel }^2<l^2\left(\sum _{k=0}^{\infty }\parallel {\omega }_k{\parallel }^2+\sum _{k=0}^{\infty }{\parallel v_{i,k}\parallel }^2+x_0^T{\theta }_1{x}_0+{(x_0-{\widehat{x}}_0)}^T{\theta }_2(x_0-{\widehat{x}}_0)\right)$$

(17)

where ${\theta }_1$ and ${\theta }_2$ are the two given positive definite real symmetric auxiliary matrices, and these two parameter matrices achieve boundedness control of the system paradigm through a weight adjustment mechanism.

Theorem 3.

For a scalar $\rho >0$ and real positive definite matrices ${\theta }_1$ and ${\theta }_2$, system (16) is asymptotically stable if there exists a real symmetric matrix Q satisfying Equation (18).

Proof. 

According to the auxiliary matrices P and $\theta$ given below, an inequality satisfying the filter performance criterion is obtained:

$$\left\{\begin{array}{c}\left[\begin{array}{ccc}{\overline{A}}_i^{T}Q{\overline{A}}_i+{\overline{C}}_i^T{\overline{C}}_i-Q& {\overline{A}}_i^{T}Q{\overline{B}}_i& {\overline{A}}_i^{T}Q{\overline{D}}_i\\ {\overline{B}}_i^{T}Q{\overline{A}}_i& {\overline{B}}_i^{T}Q{\overline{B}}_i-{\rho }^{2}I& {\overline{B}}_i^{T}Q{\overline{D}}_i\\ {\overline{D}}_i^{T}Q{\overline{A}}_i& {\overline{D}}_i^{T}Q{\overline{B}}_i& {\overline{D}}_i^{T}Q{\overline{D}}_i-{\rho }^{2}I\end{array}\right]\hfill \\ Q<{\rho }^2\theta \hfill \end{array}\right.<0$$

(18)

where

$$\mathrm{P}=\left[\begin{array}{ccc}{\overline{C}}_i^T{\overline{C}}_i-Q& 0& 0\\ 0& -{\rho }^{2}I& 0\\ 0& 0& -{\rho }^{2}I\end{array}\right];\theta =\left[\begin{array}{cc}{\theta }_1+{\theta }_2& -{\theta }_1\\ -{\theta }_1& -{\theta }_1\end{array}\right].$$

(19)

□

Proof. 

We set up the Lyapunov function:

$$V_k={\varsigma }_{i,k}^{T}Q{\varsigma }_{i,k}$$

(20)

□

There are two scenarios that need to be discussed here.

(1)

When ${\omega }_k=0$ and $v_{i,k}=0$,

$$\begin{array}{c}\Delta V_k\hfill \\ =V_{k+1}-V_k\hfill \\ ={\varsigma }_{i,k}^T({\overline{A}}_i^{T}Q{\overline{A}}_i-Q){\varsigma }_{i,k}\hfill \end{array}$$

(21)

Considering ${\overline{C}}_i^T{\overline{C}}_i\ge 0$ and combining it with Equation (18) show the inequality ${\overline{A}}_i^{T}Q{\overline{A}}_i-Q<0$.

Therefore, $\Delta V_k<0$ guarantees the asymptotic stability of the filter under noiseless conditions.

(2)

When ${\omega }_k>0$ and $v_{i,k}>0$,

$$\begin{array}{c}\Delta V_k\hfill \\ =V_{k+1}-V_k\hfill \\ ={\varsigma }_{i,k+1}^{T}Q{\varsigma }_{i,k+1}-{\varsigma }_{i,k}^{T}Q{\varsigma }_{i,k}\hfill \\ ={({\overline{A}}_i{\varsigma }_{i,k}+{\overline{B}}_i{\omega }_k+{\overline{D}}_i{v}_k)}^{T}Q({\overline{A}}_i{\varsigma }_{i,k}+{\overline{B}}_i{\omega }_k+{\overline{D}}_i{v}_k)-{\varsigma }_{i,k}^{T}Q{\varsigma }_{i,k}\hfill \\ ={\varsigma }_{i,k}^T{\overline{A}}_i^{T}Q{\overline{A}}_i{\varsigma }_{i,k}+{\varsigma }_{i,k}^T{\overline{A}}_i^{T}Q{\overline{B}}_i{\omega }_k+{\varsigma }_{i,k}^T{\overline{A}}_i^{T}Q{\overline{D}}_i{v}_k\hfill \\ +{\omega }_k^T{\overline{B}}_i^{T}Q{\overline{A}}_i{\varsigma }_{i,k}+{\omega }_k^T{\overline{B}}_i^{T}Q{\overline{B}}_i{\omega }_k+{\omega }_k^T{\overline{B}}_i^{T}Q{\overline{D}}_i{v}_k\hfill \\ +v_k^T{\overline{D}}_i^{T}Q{\overline{A}}_i{\varsigma }_{i,k}+v_k^T{\overline{D}}_i^{T}Q{\overline{B}}_i{\omega }_k+v_k^T{\overline{D}}_i^{T}Q{\overline{D}}_i{v}_k-{\varsigma }_{i,k}^{T}Q{\varsigma }_{i,k}\hfill \\ ={\vartheta }_{i,k}^T\mathrm{\Xi }\left(Q\right){\vartheta }_{i,k},\hfill \end{array}$$

(22)

Here, ${\vartheta }_{i,k}=\left[\begin{array}{c}{\varsigma }_k\\ {\omega }_k\\ v_{i,k}\end{array}\right];\mathrm{\Xi }\left(Q\right)=\left[\begin{array}{ccc}{\overline{A}}_i^{T}Q{\overline{A}}_i-Q& {\overline{A}}_i^{T}Q{\overline{B}}_i& {\overline{A}}_i^{T}Q{\overline{D}}_i\\ {\overline{B}}_i^{T}Q{\overline{A}}_i& {\overline{B}}_i^{T}Q{\overline{B}}_i& {\overline{B}}_i^{T}Q{\overline{D}}_i\\ {\overline{D}}_i^{T}Q{\overline{A}}_i& {\overline{D}}_i^{T}Q{\overline{B}}_i& {\overline{D}}_i^{T}Q{\overline{D}}_i\end{array}\right].$

Continuing the derivation yields

$$\begin{array}{cc}\hfill \Delta V_k& =V_{k+1}-V_k\hfill \\ \hfill & ={\vartheta }_{i,k}^T{\rm Y}\left(k\right){\vartheta }_{i,k}+{\rho }^2{\omega }_k^T{\omega }_k+{\rho }^2{v}_{i,k}^T{v}_{i,k}+{\tilde{z}}_{i,k}^T{\tilde{z}}_{i,k}\hfill \end{array}$$

(23)

where

$${\rm Y}\left(k\right)=\left[\begin{array}{ccc}{\overline{A}}_i^{T}Q{\overline{A}}_i+{\overline{C}}_i^T{\overline{C}}_i-Q& {\overline{A}}_i^{T}Q{\overline{B}}_i& {\overline{A}}_i^{T}Q{\overline{D}}_i\\ {\overline{B}}_i^{T}Q{\overline{A}}_i& {\overline{B}}_i^{T}Q{\overline{B}}_i-{\rho }^{2}I& {\overline{B}}_i^{T}Q{\overline{D}}_i\\ {\overline{D}}_i^{T}Q{\overline{A}}_i& {\overline{D}}_i^{T}Q{\overline{B}}_i& {\overline{D}}_i^{T}Q{\overline{D}}_i-{\rho }^{2}I\end{array}\right]$$

Combined with Equation (18), the inequality can be written as

$$\Delta V_k-{\rho }^2{\omega }_k^T{\omega }_k-{\rho }^2{v}_{i,k}^T{v}_{i,k}+{\tilde{z}}_{i,k}^T{\tilde{z}}_{i,k}<0$$

(24)

Then,

$${\vartheta }_{i,k}^T{\rm Y}\left(k\right){\vartheta }_{i,k}<0$$

(25)

which satisfies

$$\begin{array}{cc}\hfill & \sum _{k=0}^{\infty }\left(\Delta V_k-{\rho }^2{\omega }_k^T{\omega }_k+{\rho }^{2}v{i,k}^{T}vi,k+{\tilde{z}}_{i,k}^T{\tilde{z}}_{i,k}\right)\hfill \\ \hfill & =\sum _{k=0}^{\infty }\left(\left|\right|{\tilde{z}}_{i,k}{\left|\right|}^2-{\rho }^2\left(\right||{\omega }_k{\left|\right|}^2+{\left|\right|vi,k\left|\right|}^2)\right)-v_1<0\hfill \end{array}$$

(26)

Further,

$$\sum _{k=0}^{\infty }\left|\right|{\tilde{z}}_{i,k}{\left|\right|}^2<{\rho }^2\sum _{k=0}^{\infty }\left(\right||{\omega }_k{\left|\right|}^2+\left|\right|v_{i,k}{\left|\right|}^2)+v_1$$

(27)

Since $Q<{\rho }^2\theta$,

$$\begin{gathered} \sum _{k=0}^{\infty }\parallel {\tilde{z}}_{i,k}{\parallel }^2<{\rho }^2\sum _{k=0}^{\infty }(\parallel {\omega }_k{\parallel }^2+\parallel v_{i,k}{\parallel }^2)+{\rho }^2\left[\begin{array}{cc}{x}_k& {\widehat{x}}_{i,k}\end{array}\right]\theta \left[\begin{array}{c}{x}_k\\ {\widehat{x}}_{i,k}\end{array}\right] \\ ={\rho }^2\sum _{k=0}^{\infty }(\parallel {\omega }_k{\parallel }^2+\parallel v_{i,k}{\parallel }^2)+{\rho }^2\left[\begin{array}{cc}{x}_k& {\widehat{x}}_{i,k}\end{array}\right]\left[\begin{array}{cc}{\theta }_1+{\theta }_2& -{\theta }_1\\ -{\theta }_1& -{\theta }_1\end{array}\right]\left[\begin{array}{c}{x}_k\\ {\widehat{x}}_{i,k}\end{array}\right] \\ ={\rho }^2\left(\sum _{k=0}^{\infty }(\parallel {\omega }_k{\parallel }^2+\parallel v_{i,k}{\parallel }^2)+x_1^T{\theta }_2{x}_1+{(x_1-{\widehat{x}}_1)}^T{\theta }_2(x_1-{\widehat{x}}_1)\right) \end{gathered}$$

(28)

Thus,

$$\sum _{k=0}^{\infty }{\parallel {\tilde{z}}_{i,k}\parallel }^2<{\rho }^2\left(\sum _{k=0}^{\infty }(\parallel {\omega }_k{\parallel }^2+\parallel v_{i,k}{\parallel }^2)+x_1^T{\theta }_2{x}_1+{(x_1-{\widehat{x}}_1)}^T{\theta }_2(x_1-{\widehat{x}}_1)\right)$$

(29)

It can be concluded that $V_k$ satisfies the $H_{\infty }$ fusion filtering performance criteria of this paper.

We define ${\Lambda }_k$ and ${\overline{\Lambda }}_k$ as the set of attacked local estimation indexes and the set of unattacked local estimation indexes, ${\Lambda }_k\cup {\overline{\Lambda }}_k=\left\{1,2,\dots ,n\right\}$, respectively, at time k. For the multi-sensor configuration of system (2), the proposed distributed $H_{\infty }$ fusion filter design is presented below:

$${\widehat{z}}_k=\sum _{i=1}^n{\iota }_i{\widehat{z}}_{i,k}$$

(30)

$${\iota }_i={\displaystyle \frac{1/{\kappa }_i^2}{{\displaystyle \sum _{j=1}^n}{\kappa }_j^2}}$$

(31)

where ${\widehat{z}}_k$ denotes the fused global estimate, ${\iota }_i>0$ denotes the fused weights and satisfies the normalization condition ${\displaystyle \sum _{i\in {\overline{\Lambda }}_k}}{\iota }_i=1$, n denotes the total number of sensors, and ${\kappa }_i$ represents the filter error.

5. Simulations

To verify the effectiveness of the dynamic data encryption fusion estimation strategy for USV-oriented FDI attacks proposed, a simulation experiment platform is constructed in this study. Based on the dynamics model of a USV and the established system model, it is assumed that the sensor network contains six nodes. In order to enhance security, the first three sensors receive the first state component of the system, and the last three sensors receive the second state component of the system. During the measurement process, the communication data are affected by measurement noise:

$$\begin{array}{cc}\hfill v_{1,k}& =0.022sin\left(k\right),v_{2,k}=0.044sin\left(k\right).\hfill \end{array}$$

(32)

The external interference noise is

$${\omega }_k={\left[\begin{array}{cc}0.2cos\left(2k\right)& 0.1sin\left(k\right)\end{array}\right]}^T.$$

(33)

It is used to simulate the effect of wind, waves, and other factors on data acquisition and transmission in the marine environment. The simulation duration is set to [0, 100] seconds, and the initial state value of the system is

$$x_0={\left[\begin{array}{cc}1& 2\end{array}\right]}^T.$$

(34)

The system parameter matrix is defined as

$$A=\left[\begin{array}{cc}0.77& 0\\ 0& 0.77\end{array}\right],B=\left[\begin{array}{cc}1.1& 0\\ 0& 1.1\end{array}\right].$$

(35)

It is assumed that there exists at least one sensor that is not subject to the FDI attack in the first three sensor groups and the last three sensor groups at any time, and the location of the FDI attack occurs randomly. In addition, we set the value of the attack signal to a random value in the interval [40, 80]. In the dynamic data encryption scheme, the initial key matrix $c_{i,0}$ is randomly generated, and the key $c_{i,k}$ is updated according to the historical system data during the subsequent encryption process, with the update step set to 1. The encryption function $\phi \left(c_{i,k}\right)=-c_{i,k}$. The following $H_{\infty }$ fusion filter parameters are given according to the Monte Carlo method:

$$\begin{array}{cccc}\hfill A_{pi}& =\left[\begin{array}{cc}0.4& 0\\ 0& 0.4\end{array}\right],\hfill & \hfill B_{pi}& =\left[\begin{array}{cc}0.4& 0\\ 0& 0.4\end{array}\right],\hfill \\ \hfill C_{pi}& =\left[\begin{array}{cc}1.005& 0\\ 0& 1.005\end{array}\right],\hfill & \hfill F_{pi}& =\left[\begin{array}{cc}0.6& 0\\ 0& 0.6\end{array}\right].\hfill \end{array}$$

(36)

To address the issue of privacy protection during data transmission, we use dynamic encryption methods for protection. Figure 3 demonstrates the effect of the decryptor on the recovery of key state quantities of the system. Figure 3a depicts the horizontal position of the system ($x_{1,k}$) dynamic: the black dashed line represents the true horizontal position, and the red line represents the decrypted position ($x_{1,k}^d$) obtained by the decryptor. Figure 3b depicts the velocity of the system ($x_{2,k}$) dynamic: the black dashed line represents the true velocity, and the blue solid line represents the decryption velocity ($x_{2,k}^d$) obtained by the decryptor. The comparison shows that the positions and velocities output by the decryptor can effectively track and approximate their corresponding true values, indicating that the decryptor has a excellent performance in recovering the position and velocity states of the system.

Figure 4 and Figure 6 together show the state observation effect of the multi-sensor system under FDI attack. Figure 4a marks the true horizontal position of the system ($x_{1,k}$) by the black dashed line, while presenting the observed values of sensor 1/2/3 pairs of $x_{1,k}$ by the red, blue, and orange solid lines, respectively. And the red block in Figure 4b marks the FDI attack occurring at the corresponding moment. Figure 6a similarly presents the true speed ($x_{2,k}$; black dashed line) with the three sensors’ observations of $x_{2,k}$ (red/blue/orange solid lines). Figure 6b synchronizes the distribution of FDI attacks in the $x_{2,k}$ channel. The comparison between the two sets of images clearly shows that during the normal operation period without being attacked, there is very little error between the data obtained by the sensor and the real data, while there is a significant deviation during the 40 to 80 FDI attack period (red block), which verifies the destructive impact of the attack on the data. Figure 5 and Figure 7 show the FDI attack detection results of sensors 1−3 and 4−6, respectively. The green dots in the figures indicate the alerts issued when an attack is detected, and the results show that the proposed scheme in this paper has a 99% FDI detection rate and a 1% false alarm rate.

Figure 4. Comparison of USV location observation and FDI attack distribution.

Figure 4. Comparison of USV location observation and FDI attack distribution.

Figure 5. FDI detection results of sensors 1−3.

Figure 5. FDI detection results of sensors 1−3.

Figure 6. Comparison of USV velocity observation and FDI attack distribution.

Figure 6. Comparison of USV velocity observation and FDI attack distribution.

Figure 7. FDI detection results of sensors 4−6.

Figure 7. FDI detection results of sensors 4−6.

Figure 8 and Figure 9 show the true values and estimated values of the horizontal position and velocity of the USV, respectively. As shown in Figure 8, the tracking error of this method (red solid line) for the true position (black dashed line) is significantly lower than the CI fusion result (blue dotted line) in reference [29]. Especially at moments [50, 80], the protection advantage of the dynamic key mechanism is particularly significant: the position offset of this method is strictly controlled within $\pm 0.15$ m, whereas the reference [29] method results in a maximum offset of $\pm 0.82$ m due to static key vulnerability. The speed estimation results in Figure 9 further indicate that this method (solid red line) effectively suppresses the influence of disturbances through $H_{\infty }$ filtering, and the speed estimation error is strictly controlled within the range of $\pm 0.12$ m/s; in contrast, the method in reference [29] has a maximum error of $\pm 0.76$ m/s due to static key vulnerability.

Table 1. Quantitative table of computational efficiency.

Module	Average Processing Latency (ms)	Proportion of Resources
Dynamic encryption	0.45	7.2%
Attack detecting	0.18	2.9%
$H_{\infty }$ fusion filter	1.23	19.7%

Attention: Based on AMD Ryzen 7 5800H platform testing.

summarizes the processing latency and resource allocation of different modules in the experimental scenario.

Table 2. Standard deviation (SD) and mean square error (MSE) of the position with respect to the comparison literature [29].

	Standard Deviation	Mean Square Error
Method of comparison	0.1559	0.0243
This method	0.0603	0.0036

summarizes the performance comparison results of different methods for USV position estimation in experimental scenarios with FDI attacks. Among them, the standard deviation of position estimation reflects the degree of dispersion of position estimation values around their mean. The smaller the SD, the better the stability and consistency of the estimation results. The mean square error of position estimation measures the average square of the difference between the estimated position and the true position. The smaller the MSEs, the higher the accuracy of the estimation and the more accurate the tracking of the true value.

Table 3. Comparison with the single execution time of the comparative literature [29].

	Single Execution Time (s)
Method of comparison	12.8
This method	10.3

summarizes the single run times for the different methods. The comparison shows that the mean square error value obtained from this paper’s method is smaller, the single run time is shorter, and the robustness is better, indicating that the method has a stronger ability to resist FDI attacks.

6. Conclusions

The innovative security solution for the problem of FDI attacks faced by USVs during data transmission in CPSs is investigated. Due to the open nature of WSNs, attackers can tamper with sensor data, leading to poor decision making by the control center. To solve this problem, a dynamic data encryption method for encrypting the data before transmission and dynamically updating the key using historical data to enhance the difficulty of cracking is developed. At the same time, we construct a dynamic correlation mechanism among the ciphertext, the key, and the auxiliary encrypted ciphertext. The goal is to create a detection system that can quickly identify and map FDI attacks. In addition, external noise interference is filtered by the $H_{\infty }$ fusion filter, and the weighted fusion algorithm is used to restore the actual information. The results show that the position estimation error is strictly controlled within a range ±0.15 m and the velocity estimation error is controlled within a range ±0.12 m/s. Compared with the scheme in the literature [29] (maximum position offset of ±0.82 m and velocity offset of ±0.76 m/s), the accuracy improvement is significant. In summary, this scheme provides an effective solution for the secure transmission and reliable state estimation of USVs in open-WSN environments through the collaborative design of dynamic key encryption, real-time attack detection and localization, and $H_{\infty }$ filtering and weighted fusion.