Enhancing Unmanned Marine Vehicle Security: A Periodic Watermark-Based Detection of Replay Attacks

Abstract

This paper explores a periodic watermark-based replay attack detection method for Unmanned Marine Vehicles modeled in the framework of the Takagi–Sugeno fuzzy system. The precise detection of replay attacks is crucial for ensuring the security of Unmanned Marine Vehicles; however, traditional timestamp-based or encoded measurement-dependent detection approaches often sacrifice system performance to achieve higher detection rates. To reduce the potential performance degradation, a periodic watermark-based detection scheme is developed, in which a compensation signal together with a periodic Gaussian watermark signal is integrated into the actuator. By compensation calculations conducted with all compensatory signals in each period, the position corresponding to a minimum value of the detection function can be derived. Then, the time that the attacks occurred can be ensured with the aid of the comparison between this position with the watermark signal in the same period. An application on a UMV is shown to demonstrate the effectiveness of the presented scheme in detecting replay attacks while minimizing control costs.

1. Introduction

Unmanned Marine Vehicles (UMVs) represent a category of highly automated, intelligent, cost-effective, and efficient maritime vessels that operate without human intervention, which have been found to have widespread applications in the area of the observation of marine hydrological data, detection of marine environments, and exploration of marine resources etc. [1]. Their developments have significantly advanced marine technologies [2,3,4], and also showcased enormous potential in scientific research, military applications [5,6], and civilian uses. In contrast to traditional vessels, UMVs can operate under harsh weather conditions and undertake monotonous and hazardous operations in complicated marine environments [7]. Such versatility has fostered the development and research within the domains of navigation control, fleet coordination, trajectory tracking control etc. [8,9].

The rapid development of UMVs has been inseparable from the significant growth of modern network communications [10]. UMVs are transitioning from reliance on traditional communication channels to integrated networks to enhance system efficiency. However, this transition also introduces significant cybersecurity challenges, such as network attacks [11,12]. Network attacks are not only increasingly common but also growing in complexity, leading to substantial risks to the UMVs systems [13,14]. Hence, the security challenges faced by UMVs are crucial, and the detection of attacks has significant research significance. In an open network environment, cybersecurity problems targeting UMV systems primarily manifest as denial of service (DoS) attacks and replay attacks [15]. Extensive investigations by Zhang et al. [16] and Fei et al. [17] have concentrated on countering DoS attacks on UMV systems. Unfortunately, unlike DoS attacks, replay attackers do not require prior knowledge of the control systems. These attackers typically retransmit system measurement data from periods of stability, rendering the attacks inherently covert and difficult to detect through standard methods [18]. The covert nature of replay attacks makes them challenging to detect with traditional methods, which contributes to their frequent occurrence in contemporary UMV systems.

The rising occurrence of replay attacks has drawn the interest of scholars, especially in the aspect of attack detection. Existing methods primarily include timestamp techniques and the addition of physical watermarks to control signals. Timestamp-based replay attack detection was first proposed in [19], where Kerberos leverages a reputable third-party authentication service to authenticate user identities. Subsequently, Greene et al. [20] proposed an enhanced remote keyless entry system employing timestamping and exclusive OR (EOR) encoding to robustly defend against replay attacks, significantly improving security over systems. Farha et al. [21] introduced a timestamp-based method to enhance ZigBee network security against replay attacks, ensuring effectiveness without impacting network efficiency. Furthermore, a Promela-based model checking approach for verifying timestamp-utilizing security protocols against replay attacks, effectively identifying vulnerabilities in the protocol, was proposed in [22]. Nonetheless, it had been suggested by Yang et al. [23] and Zhu et al. [24] that transmitted signals are vulnerable to interception by replay attackers, who can alter the timestamps, leading to noticeable decline in the efficiency of the detection strategy mentioned in [19,20,21,22]. To address this issue, Du et al. [25] added the continuous addition of Gaussian random noise to the control signals, causing noticeable variations in data residuals before and after an attack, therefore facilitating the detection of such attacks. Then, an approach for designing optimal physical watermark signals to detect replay attacks in linear time-invariant systems with unknown parameters was studied in [26]. These strategies adjusted the system’s optimal control signals, trading off control costs for improved detection of attacks. Therefore, an optimized watermarking method that balances improved detection of replay attacks against control costs in networked control systems was discussed in [27]. Ma et al. [28] presented a watermarking method for efficiently detecting transient covert attacks in industrial cyber-physical systems, reducing system performance cost while detecting detection accuracy. A recursive watermark algorithm significantly safeguarding against replay attacks with remarkable efficiency and saving the control cost by embedding rapid, detectable watermarks in hard real-time channels was introduced in [29]. In [30], a watermark-based encoding and decoding method were also proposed to detect replay attacks on UMVs with less control cost. It is worth noting that while incorporating watermark signals into control systems of UMVs aids in replay attack detection, the approach also compromises control performance [25,26,27,28,29,30]. The advantages and limitations of different existing methods can be seen in

Table 1. Advantages and Limitations of Different Existing Methods.

Detection Method	Timestamp	Watermark	Vulnerable	Precise	Cost-Effective
Ref. [19]	✓		✓
Ref. [20]	✓		✓
Ref. [25]		✓		✓
Ref. [26]		✓		✓
Ref. [27]		✓		✓
Ref. [27]		✓		✓
Ref. [29]		✓		✓
Proposed Method		✓		✓	✓

. Therefore, it is interesting to study the schedule of watermarks to use control costs more effectively.

Inspired by the above-mentioned discussions, in this article, a periodic watermark signal method against replay attacks for UMV systems is investigated, which can precisely detect replay attacks and minimize the impact of the watermark signal on the control system’s performance. Significant contributions of this paper are highlighted as follows.

(1)

The proposed method utilizes the relative positions of watermark signals to detect replay attack, diverging from traditional watermarking methods in [30,31]. Compared with existing, this method significantly lowers control costs by reducing watermark signal variance.

(2)

A novel periodic watermarking-compensation mechanism is constructed based on a Gaussian signal, contrasting with binary watermark signals employed in [32]. Comparatively, this mechanism can effectively enhance the detection rate of replay attacks.

Throughout this article, the superscripts ‘T’ and ‘-’ denote the transposition and the inverse of a matrix.

2. Problem Formulation and Preliminaries

The considered system Structure is presented in Figure 1, which is comprised of UMVs, a Kalman filter, and an LQG controller for setpoint optimization. The following analysis will explore the Takagi–Sugeno (T-S) fuzzy modeling for UMVs, the replay attacks, and the applications of Kalman filters and LQG controllers.

2.1. UMVs System Modeling

It follows from [33,34,35] that the equations describing dynamics and kinematics of the UMVs shown in Figure 2 are depicted as follows:

$$\begin{array}{cc}\hfill & \mathbb{M}\dot{\tau }\left(t\right)+\mathbb{N}\tau \left(t\right)+\mathbb{O}\phi \left(t\right)=u\left(t\right)+\theta \left(t\right)\hfill \\ \hfill & \dot{\phi }\left(t\right)=\mathbb{Z}\left(\omega \left(t\right)\right)\tau \left(t\right)\hfill \end{array}$$

(1)

where $\tau \left(t\right)={\left[{\tau }_{surge}\left(t\right){\tau }_{sway}\left(t\right){\tau }_{yaw}\left(t\right)\right]}^T$ denotes the vector of body-fixed linear and angular velocities, where ${\tau }_{\mathrm{surge}}\left(t\right)$ stands for surge velocity, ${\tau }_{\mathrm{sway}}\left(t\right)$ for sway velocity, and ${\tau }_{\mathrm{yaw}}\left(t\right)$ for yaw velocity. $\phi \left(t\right)={\left[x\left(t\right)y\left(t\right)\omega \left(t\right)\right]}^T$ indicates the earth-fixed orientation vector, where $x\left(t\right)$ and $y\left(t\right)$ are positions, and $\omega \left(t\right)$ denotes the yaw angle. Control inputs are given by $u\left(t\right)={\left[u_1\left(t\right)u_2\left(t\right)u_3\left(t\right)\right]}^T$. $\theta \left(t\right)={[{\theta }_1\left(t\right),{\theta }_2\left(t\right),{\theta }_3\left(t\right)]}^T$ encapsulates wave-induced disturbances. $\mathbb{O}=\mathrm{diag}\{O_{11},O_{22},O_{33}\}$ stands for mooring forces.

Next, the UMVs system can be described as:

$$\dot{\tau }\left(t\right)=\mathbb{A}\phi \left(t\right)+\mathbb{B}\tau \left(t\right)+\mathbb{D}u\left(t\right)+\mathbb{D}\theta \left(t\right)$$

(2)

where $\mathbb{A}=-{\mathbb{M}}^{-1}\mathbb{O},\mathbb{B}=-{\mathbb{M}}^{-1}\mathbb{N},\mathbb{D}={\mathbb{M}}^{-1}$. In constructing the fuzzy dynamic model of the UMVs system, the yaw angle serves as the premise variable. In this case, Three pivotal yaw angle points $\omega \left(t\right)$ are selected at the origin and $\pm \pi /6$, respectively, and linearize the system at those three operating points, the fuzzy model of UMVs is given as follows:

$R^1:$ IF $\omega \left(t\right)$ is about 0 THEN

$$\left\{\begin{array}{cc}\hfill \dot{x}\left(t\right)& =A_{1}x\left(t\right)+Bu\left(t\right),\hfill \\ \hfill y\left(t\right)& =Cx\left(t\right).\hfill \end{array}\right.$$

(3)

$R^2:$ IF $\omega \left(t\right)$ is about $\pi /6$ THEN

$$\left\{\begin{array}{cc}\hfill \dot{x}\left(t\right)& =A_{2}x\left(t\right)+Bu\left(t\right),\hfill \\ \hfill y\left(t\right)& =Cx\left(t\right).\hfill \end{array}\right.$$

(4)

$R^3:$ IF $\omega \left(t\right)$ is about $-\pi /6$ THEN

$$\left\{\begin{array}{cc}\hfill \dot{x}\left(t\right)& =A_{3}x\left(t\right)+Bu\left(t\right),\hfill \\ \hfill y\left(t\right)& =Cx\left(t\right).\hfill \end{array}\right.$$

(5)

where

$$\begin{array}{c}\hfill A_p=\left[\begin{array}{cc}{0}_{3\times 3}& Q_i\\ \mathbb{B}& \mathbb{A}\end{array}\right],B=\left[\begin{array}{c}{0}_{3\times 3}\\ \mathbb{D}\end{array}\right],C=\left[\begin{array}{cc}{I}_{3\times 3}& 0_{3\times 3}\end{array}\right],Q_1=I_{3\times 3},Q_2=\left[\begin{array}{ccc}{\displaystyle \frac{\sqrt{3}}{2}}& -{\displaystyle \frac{1}{2}}& 0\\ {\displaystyle \frac{1}{2}}& {\displaystyle \frac{\sqrt{3}}{2}}& 0\\ 0& 0& 1\end{array}\right],Q_3=\left[\begin{array}{ccc}{\displaystyle \frac{\sqrt{3}}{2}}& -{\displaystyle \frac{1}{2}}& 0\\ -{\displaystyle \frac{1}{2}}& {\displaystyle \frac{\sqrt{3}}{2}}& 0\\ 0& 0& 1\end{array}\right].\end{array}$$

Given that the position and angle can be measured with relative ease, the output matrix C is selected accordingly. Furthermore, the membership functions for the three rules are illustrated in Figure 3.

The global T-S fuzzy model of the UMVs is derived using a singleton fuzzifier, product inference, and center average defuzzifier as described below:

$$\left\{\begin{array}{cc}\hfill \dot{x}\left(t\right)=& {\sum }_{p=1}^3{\hslash }_p\left(\varrho \left(t\right)\right)(A_{p}x\left(t\right)+D_{p}u\left(t\right)+\theta \left(t\right))\hfill \\ \hfill y\left(t\right)=& Cx\left(t\right)+v\left(t\right)\hfill \end{array}\right.$$

(6)

where

$$\begin{array}{c}{\hslash }_p\left(\varrho \left(t\right)\right)={\displaystyle \frac{{\upsilon }_p\left(\varrho \left(t\right)\right)}{{\sum }_{p=1}^3{\upsilon }_p\left(\varrho \left(t\right)\right)}},{\upsilon }_p\left(\varrho \left(t\right)\right)=M_{p1}\left({\varrho }_1\left(t\right)\right)M_{p1}\left({\varrho }_2\left(t\right)\right),{\hslash }_p\left(\varrho \left(t\right)\right)\ge 0,{\sum }_{p=1}^3{\hslash }_p\left(\varrho \left(t\right)\right)=1.\hfill \end{array}$$

The discrete-time fuzzy system results from the periodic sensor sampling with a period of $T_z$ and the application of the ZOH input mechanism:

$$\left\{\begin{array}{cc}\hfill x(k+1)=& {\sum }_{p=1}^3{\hslash }_p\left(\varrho \left(k\right)\right)(A_{dp}x\left(k\right)+B_{dp}u\left(k\right)+{\theta }_{dp}\left(k\right)),\hfill \\ \hfill y\left(k\right)=& Cx\left(k\right)+v\left(k\right)\hfill \end{array}\right.$$

(7)

where $A_{dp}=e^{A_{n}T}$, $B_{dp}={\int }_0^T{e}^{A_{n}T}{B}_{n}dt$, and C are defined as in Equation (6). The function ${\hslash }_p\left(\varrho \left(k\right)\right)$ represents the value of ${\hslash }_p\left(\varrho \left(t\right)\right)$ at the sampling instance $t=k$. The variables ${\theta }_{dp}\left(k\right)$ and $v\left(k\right)$ represent process noise and measurement noise, respectively. These noise components are mutually independent, with W and V serving as their respective covariance matrices.

In a UMVs system, optimal state estimation $\widehat{x}\left(k\right|k)$ can be achieved by implementing a Kalman filter. Sensor data are forwarded to an estimator that applies the Kalman filter for estimating the state. The system is presumed to initiate operations at $-\infty$, allowing for the convergence of the Kalman filter into a constant-gain linear estimator. The update process is outlined in Algorithm 1.

Algorithm 1 Kalman Filter Utilized in T-S UMVs Model

1: Step 1: Input $A_{dp},B_{dp},C,W,V$ 2: Step 2: $\widehat{x}(k+1|k)={\sum }_{p=1}^3{\hslash }_p\left(\varrho \left(k\right)\right)(A_{dp}\widehat{x}\left(k\right|k)+B_{dp}u\left(k\right))$ 3: Step 3: $\widehat{x}\left(k\right|k)=\widehat{x}\left(k\right|k-1)+Kz\left(k\right)$ 4: Step 4: $z\left(k\right)=y\left(k\right)-C\widehat{x}\left(k\right|k-1)$ 5: Step 5: $L=P{C}^T{(CP{C}^T+R)}^{-1}$ 6: Step 6: $P={\sum }_{p=1}^3(A_{dp}P{A}_{dp}^T+W)-A_{dp}P{C}^T{(CP{C}^T+R)}^{-1}CP{A}_{dp}^T$ 7: Step 7: Output $\widehat{x}\left(k\right|k)$

2.2. Linear Quadratic Gaussian Controller

The LQG controller is utilized with the goal of minimizing the LQG cost is assumed as:

$$J=min\underset{\ell \to \infty }{lim}{\displaystyle \frac{1}{\ell }}\mathbb{E}[\sum _{i=0}^{\ell -1}{\left(x\right(k\left)\right)}^{T}Qx\left(k\right)+{\left(u\left(k\right)\right)}^{T}Ru\left(k\right)]$$

(8)

where $Q^T=Q$ and $R^T=R$ denote the state and control variables, respectively. Consequently, the derived optimal feedback control law is:

$$\begin{array}{cc}\hfill u\left(k\right)=& -\sum _{p=1}^3{\hslash }_p\left(\varrho \left(k\right)\right)F\widehat{x}\left(k\right|k)\hfill \end{array}$$

(9)

The goal function given the optimal estimator and LQG controller is recast as:

$$J^{★}=tr\left(SW\right)+tr\left[\sum _{p=1}^3({A_{dp}}^{T}S{A}_{dp}+Q-S)(P-LCP)\right]$$

(10)

Controller enables the system to achieve a stable operational state. Consequently, parameters including $L,P,F$, and J can be computed offline.

2.3. Replay Attack Model

Mo et al. [31] initially defined a model for a replay attack. This model assumes that an attacker can capture a substantial amount of sensor data. During the attack, the real sensor measurement $y(t_1+i)$ is obstructed and replaced by $y(t_1+i)=y(t_0+i)$, which is forwarded to the estimator as demonstrated in Figure 4, with $d=t_1-t_0$ identified as the replay delay. The model is expressed as:

$$\begin{array}{c}\hfill y(t_1+i)=y(t_0+i),0\le i\le T.\end{array}$$

(11)

Remark 1.

In UMV systems, two prevalent types of replay attacks are commonly discussed: Stuxnet-like Zhu et al. [24] and time-delay attacks, which often target network-transmitted signals such as $y\left(k\right)$ and $r\left(k\right)$. Stuxnet-like attacks, which involve manipulated control inputs, present significant detection challenges due to their reliance on historical feedback signals. Time-delay attacks disrupt system performance by modifying the poles of the closed-loop system. This study specifically focuses on replay attacks that affect sensor signals $y\left(k\right)$. Future research will investigate the simultaneous tampering of sensor data and control actions.

Remark 2.

Replay attack threats to UMVs may come from state-sponsored groups, hacktivists, criminals, and insiders motivated by espionage, data theft, disruption, financial gain, and geopolitical leverage. The likelihood and success rate of replay attacks is contingent upon the security of the UMVs system’s transmission network—specifically, whether it employs advanced encryption technologies to protect its communications and data transmissions and the complexity of the attacker’s replay attack methods. When UMVs are fully deployed underwater, the inherent challenges of data transmission may reduce the success rate of such network attacks. Conversely, during docking operations or surface activities, UMV communications are more susceptible to the threat of replay attacks.

Assumption 1.

In this paper’s discussion of the replay attack scenario, the attack phase and the subsequent attack segment are temporally distinct with no overlap.

3. A Periodic Watermark-Based Detection of Replay Attacks

3.1. Periodic Watermarking Detection Mechanism

As shown in Figure 5, the attacker intercepts the actual sensor measurement $y\left(k\right)$ and forwards $y^r\left(k\right)$ to the estimator or controller regularly. Furthermore, it has been shown in Yang et al. [23] that under certain conditions, these attacks can evade detection.

Lemma 1.

In the scenario where a UMVs system faces a replay attack and equipping with a ${\chi }^2$ detector, ensuring $\epsilon =(A_{dp}+B_{dp}L)(I_n-LC)$ is Schur stable implies that the false alarm rate α and the positive detection rate ${\beta }_k$ at time k will equalize over time, confirmed by ${lim}_{k\to \infty }{\beta }_k=\alpha$.

According to Lemma 1, the detector’s alarm rate matches the false alarm rate, therefore complicating the confirmation of attacks upon alarms. To combat replay attacks, Mo et al. [31] introduced a technique of continuously adding Gaussian watermark signals $\Delta o\left(k\right)$ into the control inputs. It should be noted that, unless otherwise specified, the continuous watermark referenced hereafter is assumed to be a Gaussian signal. The actual control input can thus be described as:

$$\begin{array}{c}\hfill u^r\left(k\right)=u\left(k\right)+\Delta o\left(k\right)\end{array}$$

(12)

where $u^r\left(k\right)$ represents the actual control input of UMVs system added Gaussian watermark signals.

For UMV systems that are less responsive to watermarking, achieving an adequate detection rate might necessitate significantly increased watermark intensities Bian et al. [30]. To reduce the system control cost, a periodic watermarking mechanism is proposed in Figure 6.

Compared to the method in Bian et al. [30], the primary advantage of this periodic watermarking method lies in its reliance on comparing the relative magnitudes of the detection functions for compensation signals at various times within the same cycle rather than their absolute values. At time $k=t_0$, a periodic random watermark is introduced into the control variable, and a corresponding compensation signal is designed. As the system operates, this compensation signal develops periodicity that aligns with the watermark’s cycle. Furthermore, during each operational cycle of the UMV, there is a direct correlation between the compensation signal and the watermark.

3.2. Construction of Periodic Watermark Signal

In contrast to the approach presented in Fang et al. [32], which injects watermarks of varying sizes at periodic intervals, this study introduces a method that continuously injects a periodic Gaussian random signal into the control signal. The watermark signal’s period is denoted by T, where $T\in {\mathbb{N}}^r$. The series of periodic watermark signals is represented by $\Delta o_i=\left[\Delta o_0,\Delta o_1,\cdots ,\Delta o_{T-1}\right]$, with each $\Delta o_i(i=0,\dots ,T-1)$ being independently drawn from a Gaussian white noise signal characterized by a zero mean and a covariance matrix ${\Sigma }_o$. Assuming the system operates steadily at time $t_0$, a signal is continuously injected into the control variable. At any given time k, this injected watermark signal $\Delta o$ is defined as follows:

$$\Delta o\left(k\right)=\Delta o\left(i\right),i=(k-t_0)modT$$

(13)

where $(k-t_0)modT$ represents the remainder of k divided by T, and $0\le (kmodT)\le T-1$. After injecting the periodic watermark signal, the UMVs state equation can be expressed as:

$$\left\{\begin{array}{cc}\hfill x^r(k+1)=& {\sum }_{p=1}^3{\hslash }_p\left(\varrho \left(k\right)\right)(A_{dp}{x}^r\left(k\right)+B_{di}(u\left(k\right)+\Delta o\left(k\right))+{\theta }_{di}\left(k\right))\hfill \\ \hfill y^r\left(k\right)=& D{x}^r\left(k\right)+v\left(k\right)\hfill \end{array}\right.$$

(14)

3.3. Construction of Periodic Compensation Signal

To construct the auxiliary system for obtaining the measurement compensation signal $y^f\left(k\right)$ after watermark insertion ($k>t_0$):

$$\left\{\begin{array}{cc}\hfill x^f(k+1)& =\sum _{p=1}^3{\hslash }_p(\Delta o\left(k\right))\left(A_{dp}{x}^f\left(k\right)+B_{di}\Delta o\left(k\right)\right)\hfill \\ \hfill y^f\left(k\right)& =-C{x}^f\left(k\right)\hfill \end{array}\right.$$

(15)

where $x^f\left(t_0\right)=0.$

Theorem 1.

Assume $A_{dp}$ is stable, there exits an $n_0\in {\mathbb{N}}^r$ such that $A_{dp}^{(n_0-1)T}\approx 0$. By introducing watermark signals into the UMVs system over $n_0$ periods, the compensatory signal $y^f\left(k\right)$ achieves periodicity for $k\ge t_0+n_{0}T$, mirroring the periodic nature of the watermark. A series of these compensatory signals ${\mathbb{Y}}^f=[{\mathbb{Y}}^f\left(0\right),{\mathbb{Y}}^f\left(1\right),\dots ,{\mathbb{Y}}^f(T-1)]$. Therefore, each compensatory signal within a period corresponds directly to its associated watermark signal, as demonstrated by the relationship $o(i-1)\leftrightarrow {\mathbb{Y}}^f\left(i\right)$, where $i=(k-t_0)modT$.

Proof. 

To demonstrate that the compensation signal $y^f\left(k\right)$ becomes periodic for $k\ge t_0+n_{0}T$, it suffices to prove that the compensation state $x^f\left(k\right)$ becomes periodic for $k\ge t_0+n_{0}T$. When $k\in [t_0,t_0+T-1]$, the compensation signal state can be expressed as:

$$\begin{array}{cc}\hfill x^f\left(k\right)& ={\sum }_{p=1}^3{\hslash }_p\left(\varrho \left(k\right)\right)(A_{dp}{x}^f\left(k\right)+B_{dp}\Delta o\left(k\right))\hfill \\ \hfill & ={\sum }_{i=0}^{k-t_0}{\sum }_{p=1}^3{\hslash }_p\left(\varrho \left(k\right)\right)\left(A_{dp}{x}^f{\left(k\right)}^i{B}_{dp}\Delta o(k-t_0-i)\right)\hfill \end{array}$$

(16)

When $k=t_0+nT+i$, with $i\in [0,T-1]$ and $n\in {\mathbb{N}}^r$, the compensation signal state can be represented as:

$$\begin{array}{cc}\hfill x^f\left(k\right)=& x^f(t_0+nT+i)\hfill \\ \hfill =& {\sum }_{p=1}^3{\hslash }_p\varrho \left(k\right){\left(A_{dp}{x}^f(t_0+nT+i-1)\right)}^i{B}_{dp}\Delta o(t_0+nT+i-1))\hfill \\ \hfill =& {\sum }_{p=1}^3{\hslash }_p\varrho \left(k\right)\left(({A_{dp}}^2{\left(A_{dp}{x}^f(t_0+nT+i-3)\right)}^i{B}_{dp}\Delta o(i-3))\right)+A_{dp}{B}_{dp}\Delta o(i-2))+B_{dp}\Delta o(i-1)\hfill \\ \hfill =& {\sum }_{p=1}^3{\hslash }_p\varrho \left(k\right)({A_{dp}}^{(n-1)T}+x^f(t_0+T)+x^f(t_0+(l-1)T+i))\hfill \end{array}$$

(17)

Given the stability of the transition matrix $A_{dp}$, an integer $n_0\in {\mathbb{N}}^r$ is identified for which $A_{dp}^{(n_0-1)T}\approx 0$. This value of $n_0$ is determined solely by the matrix $A_{dp}$. Consequently, it is observed that for $n\ge n_0$, the state $x^f(t_0+nT+i)$ = $x^f(t_0+(n-1)T+i)$. We can achieve

$$x^f\left(k\right)=x^f(k-T)$$

(18)

It is indicated in (18) that the state values of the compensatory signal are deduced to become periodic after $n_0$ operational periods, reflecting the periodicity of the watermark signal. With the independence of watermark signals within each period, the state of the compensatory signal is confirmed to be distinct within any given period. Given that the compensatory signal $y^f\left(k\right)$ is defined as $y^f\left(k\right)=-C{x}^f\left(k\right)$ and output matrix C is constant, the compensatory signal is periodic, with each cycle exhibiting distinct values.

In accordance with (15), it is observed that for $k=t_0+n_{0}T+i$, where i spans from 0 to $T-1$, a precise correlation is established between the value of the compensatory signal ${\mathbb{Y}}^f\left(i\right)$ and the watermark signal value $\Delta o(k-1)=\Delta o\left(i\right)$ present in the measurement data. It has been confirmed that for any distinct i and j, $\mathrm{Prob}(\Delta o(i)=\Delta o(j\left)\right)=0$. Thus, a one-to-one correspondence between compensatory signal ${\mathbb{Y}}^f\left(i\right)$ and the watermark signal sequence $\Delta o(i-1)$ is established. □

3.4. Detection of Replay Attacks

At time k, the measurement $y^r\left(k\right)$ is received and augmented by adding the compensatory signal ${\mathbb{Y}}^f\left(m\right)$, resulting in a compensated measurement represented as $y^e(k,m)=y^r\left(k\right)+{\mathbb{Y}}^f\left(m\right)$. This process results in the modified measurement being defined as $y^e(k,m)=y^r\left(k\right)+{\mathbb{Y}}^f\left(m\right)$. The prediction state estimate from the Kalman filter at this point is designated as ${\widehat{x}}^e\left(k\right|k-1)$. Subsequently, the residual is computed as $r^e=y^e(k,m)-{Cx}^e\left(k\right|k-1)$. This residual is then employed to construct the chi-squared detection function $g(k,m)={\left(r^e(k,m)\right)}^T{\Sigma }_r^{-1}{r}^e(k,m)$, where ${\Sigma }_r$ is defined as ${CPC}^T+V$. The effectiveness of the proposed detection method is confirmed in this paper, and it is demonstrated that the subsequent theorem is applicable.

Theorem 2.

At time k, where $k=t_0+nT+i$, $i\in [0,T-1]$, and $n\ge n_0$, the control watermark signal is $\Delta o(i-1)$. Introducing different compensatory signals ${\mathbb{Y}}^f\left(m\right)$, $m\in [0,T-1]$, results in the chi-square detection function $g\left(k\right)$ satisfying $g(k,i)<g(k,j)$ for $j\in [0,T-1]$ and $j\ne i$. This indicates that assume compensatory signal ${\mathbb{Y}}^f\left(m\right)$ is injected with $m=i$, the detection function value $g(k,m)$ is minimized.

Proof. 

According to Theorem 1, the compensation signal at time $k=t_0+nT+i$ is denoted by ${\mathbb{Y}}^f\left(i\right)$. When $m=i$, ${\mathbb{Y}}^f\left(m\right)$ = ${\mathbb{Y}}^f\left(i\right)$. Thus, the measurement value $y^e(k,m)$ after adding the compensation signal ${\mathbb{Y}}^f\left(m\right)$ can be derived as follows:

$$\begin{array}{cc}\hfill y^e(k,m)=& y^r\left(k\right)+{\mathbb{Y}}^f\left(m\right)\hfill \\ \hfill =& y^r\left(k\right)+{\mathbb{Y}}^f(t_0+nT+i)C\sum _{p=1}^3{\hslash }_p\left(\varrho \left(k\right)\right)(A_{dp}{x}^r(k-1)+B_{dp}(u(k-1)\hfill \\ \hfill & +\Delta o(k-1))+{\theta }_{di}(k-1)-A_{dp}{x}^f(k-1)-B_{dp}\Delta o(k-1)+v\left(k\right)\hfill \\ \hfill =& C\sum _{p=1}^3{\hslash }_p\left(\varrho \left(k\right)\right)A_{dp}^{k}x\left(0\right)+\sum _{i=0}^{k-1}{A}_{dp}^i(B_{dp}u(k-1-i)+{\theta }_{di}(k-i-1))+v\left(k\right)\hfill \\ \hfill =& Cx\left(k\right)+v\left(k\right)\hfill \\ \hfill =& y\left(k\right)\hfill \end{array}$$

(19)

As established by (19), at time k, the measurement $y^e(k,m)$, which incorporates the compensatory signal ${\mathbb{Y}}^f\left(m\right)$, coincides with the system’s original measurement $y\left(k\right)$. This concurrence indicates that the disturbances induced by the watermark signal in the measurement are effectively neutralized by the compensatory signal.

The prediction state estimate by the Kalman filter post-compensation is represented as ${\widehat{x}}^e\left(k\right|k-1)$, and the original system’s state prediction estimate is denoted as $\widehat{x}\left(k\right|k-1)$. At the initial, it is affirmed that ${\widehat{x}}^e\left(0\right|-1)=\widehat{x}\left(0\right|-1)$. Following the integration with (19), it is derived that:

$$\begin{array}{cc}\hfill {\widehat{x}}^e\left(k\right|k-1)=& {\sum }_{n=1}^3{\hslash }_p\left(\varrho \left(k\right)\right)\left(A_{dp}{\widehat{x}}^e(k-1|k-1)+B_{dp}u(k-1)\right)\hfill \\ \hfill =& {\sum }_{n=1}^3{\hslash }_p\left(\varrho \left(k\right)\right)\left(A_{dp}+B_{dp}F\right){\widehat{x}}^e(k-1|k-1)\hfill \\ \hfill =& {\sum }_{n=1}^3{\hslash }_p\left(\varrho \left(k\right)\right)({\epsilon }^k\widehat{x}\left(0\right|1)+{\sum }_{i=0}^{k-1}{\epsilon }^i(A_{dp}+B_{dp}F)Ly(k-i-1))\hfill \\ \hfill =& {\sum }_{n=1}^3{\hslash }_p\left(\varrho \left(k\right)\right)\left(\widehat{x}(k-1|k-2)+(A_{dp}+B_{dp}F)Ly(k-1)\right)\hfill \\ \hfill =& \widehat{x}\left(k\right|k-1)\hfill \end{array}$$

(20)

Given the condition that $m=i$, it is found that the post-compensation state prediction estimate ${\widehat{x}}^e\left(k\right|k-1)$ aligns with the state prediction estimate $\widehat{x}\left(k\right|k-1)$ obtained when a control quantity watermark has not been injected. Hence, the residual $r^e(k,m)$ under the scheme presented in this paper is determined as:

$$\begin{array}{cc}\hfill r^e(k,m)=& y^e(k,m)-C{\widehat{x}}^e\left(k\right|k-1)\hfill \\ \hfill =& y^r\left(k\right)+{\mathbb{Y}}^f\left(m\right)-C{\widehat{x}}^e\left(k\right|k-1)\hfill \\ \hfill =& y\left(k\right)-C{\widehat{x}}^e\left(k\right|k-1)\hfill \\ \hfill =& y\left(k\right)-C\widehat{x}\left(k\right|k-1)\hfill \\ \hfill =& r\left(k\right)\hfill \end{array}$$

(21)

Consequently, the chi-squared detection function derived from the compensated residual is determined as follows:

$$\begin{array}{cc}\hfill g(k,m)& ={\left(r^e(k,m)\right)}^T{\Sigma }_r^{-1}{r}^e(k,m)\hfill \\ \hfill & ={\left(r\left(k\right)\right)}^T{\Sigma }_r^{-1}r\left(k\right)\hfill \\ \hfill & =g\left(k\right)\hfill \end{array}$$

(22)

Assume compensatory signal ${\mathbb{Y}}^f\left(m\right)$ is injected at time k. It has been established that the chi-squared detection function for compensated residual matches the $g\left(k\right)$ observed on the condition of no control quantity watermark is introduced. If at time $k-1$, a control quantity watermark signal $o(k-1)=o(i-1)$ was injected, and at time k, the compensatory signal ${\mathbb{Y}}^f\left(i\right)$ is introduced with $j\ne i$, then the measurement processed by the Kalman filter is as follows:

$$\begin{array}{cc}\hfill y^e(k,j)& =y^r\left(k\right)+{\mathbb{Y}}^f\left(j\right)\hfill \\ \hfill & =y^r\left(k\right)+{\mathbb{Y}}^f\left(k\right)+{\mathbb{Y}}^f\left(j\right)-{\mathbb{Y}}^f\left(k\right)\hfill \\ \hfill & =y^r\left(k\right)+{\mathbb{Y}}^f\left(i\right)+{\mathbb{Y}}^f\left(j\right)-{\mathbb{Y}}^f\left(i\right)\hfill \\ \hfill & =y\left(k\right)+{\mathbb{Y}}^f\left(j\right)-{\mathbb{Y}}^f\left(i\right)\hfill \end{array}$$

(23)

At this point, the residual $r^e(k,j)$ is as follows:

$$\begin{array}{cc}\hfill r^e(k,j)& =y^e(k,j)-C{\widehat{x}}^e\left(k\right|k-1)\hfill \\ \hfill & =y\left(k\right)+{\mathbb{Y}}^s\left(j\right)-{\mathbb{Y}}^s\left(i\right)-C{\widehat{x}}^e(k,j)\hfill \\ \hfill & =y\left(k\right)-C\widehat{x}\left(k\right|k-1)+{\mathbb{Y}}^s\left(j\right)-{\mathbb{Y}}^s\left(i\right)\hfill \\ \hfill & =r\left(k\right)+{\mathbb{Y}}^s\left(j\right)-{\mathbb{Y}}^s\left(i\right)\hfill \end{array}$$

(24)

The chi-squared detection signal based on the residual is defined as:

$$\begin{array}{cc}\hfill g(k,j)=& {\left(r^e(k,m)\right)}^T{\Sigma }_r^{-1}\left(r^e(k,j)\right)\hfill \\ \hfill =& {(r\left(k\right)+{\mathbb{Y}}^f\left(j\right)-{\mathbb{Y}}^f\left(i\right))}^T{\Sigma }_r^{-1}(r\left(k\right)+{\mathbb{Y}}^f\left(j\right)-{\mathbb{Y}}^f\left(i\right))\hfill \\ \hfill =& r^T\left(k\right){\Sigma }_r^{-1}r\left(k\right)+{\left({\mathbb{Y}}^f\left(j\right)-{\mathbb{Y}}^f\left(i\right)\right)}^T{\Sigma }_r^{-1}({\mathbb{Y}}^f\left(j\right)-{\mathbb{Y}}^f\left(i\right))\hfill \\ \hfill =& g\left(k\right)+\left|{\Sigma }_r^{-1}\right|{∥{\mathbb{Y}}^f\left(j\right)-{\mathbb{Y}}^f\left(i\right)∥}_2^2\hfill \end{array}$$

(25)

Given that $j\ne i$ and the compensatory signals within the same period are distinct, it invariably holds that ${∥{\mathbb{Y}}^f\left(j\right)-{\mathbb{Y}}^f\left(i\right)∥}_2^2>0$. Therefore, it is deduced that $g(k,j)>g\left(k\right)=g(k,i)$. □

Based on Theorem 2, the following replay attack detection method can be devised. Commencing from moment $t_0$, a periodic watermark signal is continuously injected into the control variable. At moment $k-1$, this watermark signal is given by $o(k-1)=o(i-1)$, where $i=(k-t_0)modT$. At moment k, where $k=t_0+nT+i$, and $i\in [0,T-1],n\ge n_0$, the minimum value of the chi-squared detection function after incorporating different compensatory signals is denoted by:

$$\begin{array}{c}\hfill a\left(k\right)=\underset{m}{min}g(k,m),m=0,\dots ,T-1\end{array}$$

(26)

A decision function is defined as:

$$\begin{array}{c}\hfill G\left(k\right)=a\left(k\right)-i\end{array}$$

(27)

In accordance with (27), the following hypothesis test is proposed: If $G\left(k\right)=0$, UMVs are considered to be operating normally. Otherwise, UMVs are presumed to be under a replay attack.

Theorem 3.

For a UMV system as depicted by (7), replay attacks can be detected to assume the delay of the replayed data is not an integer multiple of the watermark signal period, utilizing the approach described in this paper.

Proof. 

It is established from Theorem 2 that during normal system operation at moment $k=t_0+nT+i$, the compensatory signal $Y^f\left(m\right)$ that minimizes $g\left(k\right)$ satisfies $m=i$, with (26) yielding $a\left(k\right)=m$. Therefore, under the scheme proposed in this paper, the detection signal during normal operation is $G\left(k\right)=a\left(k\right)-i=m-i=0$.

When a replay attack occurs at time $k=t_0+nT+i$, where data from a normal state at time $k_1=t_0+n_{1}T+m_1$ is replayed, the corresponding control quantity watermark of the replayed measurement signal is $\Delta o(k_1-1)=\Delta o(m_1-1)$, with $m_1=(k_1-t_0)modT$. Given that the system operates normally at time $k-1$, it follows that $y^e(k-1)=y(k-1)$, and ${\widehat{x}}^e\left(k\right|k-1)=\widehat{x}\left(k\right|k-1)$. According to Theorem 2, for the detection function to achieve a minimal value, the injected compensatory signal should be $y^f\left(k_1\right)=Y^f\left(m_1\right)$, resulting in $a\left(k\right)=m_1$. If the delay of the replayed data $\Delta k=k-k_1=(n-n_1)T+i-m_1$ is not an integer multiple of the watermark signal period, i.e., $i\ne m_1$, as indicated by (27), the detection signal $G\left(k\right)=\alpha \left(k\right)-i=m_1-i\ne 0$ at this point, indicating the presence of a replay attack.    □

Remark 3.

Employing the method proposed in this paper, a replay attack can be detected whenever the condition $\left|{\Sigma }_r^{-1}\right|{∥{\mathbb{Y}}^f\left(j\right)-{\mathbb{Y}}^f\left(i\right)∥}_2^2$ is satisfied in (25). Therefore, in an ideal scenario, the detection rate of replay attacks using this scheme reaches 1. However, due to the presence of random noise disturbances and computational errors, especially assuming ${\Sigma }_o$ is small, there is a possibility of false detection occurring.

Remark 4.

To compromise system stability, an attacker must record and store system signals for an extended period prior to launching a replay attack. Conversely, control watermarks can be integrated into the control variables at the onset of system operations. It is established that $n_0$ is sufficient to ensure that $A_{di}^{(n_0-1)T}\approx 0$. Consequently, this paper assumes that no replay attack occurs within the initial $n_0$ periods.

Remark 5.

If the delay of the replayed data $\Delta k=k-k_1=(n-n_1)T+i-m_1$ is an integer multiple of the watermark signal period, i.e., $i=m_1$, as indicated by (27), the detection signal $G\left(k\right)=\alpha \left(k\right)-i=m_1-i=0$ at this point, which can result in a false positive. However, this method has certain limitations on the condition that the length of the chosen watermark period is sufficiently large that the occurrence of such cases can be minimized. Future work will focus on addressing this issue in more depth.

3.5. Analysis of System Performance Degradation

This paper focuses on analyzing the impact of this method on system control. Assuming no additional signals are present, the true state variables of the UMVs system are expressed by $x\left(k\right)$. The state error is then described as $ϵ\left(k\right)=x\left(k\right)-\widehat{x}\left(k\right)$. Employing the methodology presented herein, the state variables for the system during normal operation are given by ${\widehat{x}}^r\left(k\right)$, and the undisturbed steady-state error as defined within this section is $\widehat{ϵ}\left(k\right)$, thus yielding $\delta \left(k\right)={\widehat{x}}^r\left(k\right)-\widehat{x}\left(k\right)$. Upon constructing the matrix $\tilde{x}\left(k\right)={\left[x\left(k\right)ϵ\left(k\right)\delta \left(k\right)\right]}^T$, the following relation is established:

$$\begin{array}{cc}\hfill \tilde{x}(k+1)=& \sum _{p=1}^3\left[\begin{array}{ccc}{A}_{dp}+B_{dp}F& -B_{dp}F& 0\\ 0& A_{dp}-LC{A}_{dp}& 0\\ 0& 0& A_{dp}\end{array}\right] \tilde{x}\left(k\right)+\left[\begin{array}{c}w\left(k\right)\\ Hw\left(k\right)-Lv(k+1)\\ \sum _{p=1}^3{B}_{dp}u\left(k\right)\end{array}\right]\hfill \end{array}$$

(28)

where $H=I_n-LC$, $\mathrm{\Theta }$ and $\gamma \left(k\right)$ can be achieved as:

$$\begin{array}{ccc}\hfill \mathrm{\Theta }& =\sum _{p=1}^3\left[\begin{array}{ccc}{A}_{dp}+B_{dp}F& -B_{dp}F& 0\\ 0& A_{dp}-LC{A}_{dp}& 0\\ 0& 0& A_{dp}\end{array}\right],\gamma \left(k\right)\hfill & \hfill =\left[\begin{array}{c}w\left(k\right)\\ Hw\left(k\right)-Lv(k+1)\\ \sum _{p=1}^3{B}_{dp}u\left(k\right)\end{array}\right]\end{array}$$

Consequently, the covariance of the system is expressed as:

$$\begin{array}{c}\hfill cov\left(\tilde{x}\right)=\mathrm{\Theta }cov\left(\tilde{x}\right){\mathrm{\Theta }}^T+\iota \end{array}$$

(29)

Wherein, $\iota$ is characterized by:

$$\begin{array}{c}\hfill \iota =\left[\begin{array}{ccc}W& W{H}^T& 0\\ HW& HW{(I_n-LC)}^T& 0\\ LCW& 0& \sum _{p=1}^3\left(B_{dp}{\Sigma }_o{B}_{dp}^T\right)\end{array}\right]\end{array}$$

In the scheme proposed in this paper, the system control performance index can be represented as:

$$\begin{array}{cc}\hfill J=& \underset{\ell \to \infty }{lim}{\displaystyle \frac{1}{\ell +1}}\sum _{k=1}^{\ell }\mathbb{E}\left\{x^{*T}\left(k\right)Q{x}^{*}\left(k\right)+{(u\left(k\right)+\Delta o\left(k\right))}^T\right.\hfill \\ \hfill & \left.\times R\left(u\right(k)+\Delta o(k\left)\right)\right\}\hfill \\ \hfill =& \underset{\ell \to \infty }{lim}{\displaystyle \frac{1}{\ell +1}}\sum _{k=1}^{\ell }\mathbb{E}\left\{\tilde{x}\left(k\right)\mathrm{\Omega }{\tilde{x}}^T\left(k\right)+\Delta o^T\left(k\right)R\Delta o\left(k\right)\right\}\hfill \\ \hfill =& tr\left(cov\left(\tilde{x}\left(k\right)\right)\mathrm{\Omega }+{\Sigma }_{o}R\right)\hfill \end{array}$$

(30)

where $\mathrm{\Omega }=\left[\begin{array}{ccc}Q+F^{T}RF& Q& -F^{T}RF\\ -F^{T}RF& 0& F^{T}RF\\ Q& Q& 0\end{array}\right]$.

In the scheme proposed in this paper, the system performance degradation is expressed as:

$$\begin{array}{c}\hfill \Delta J=J-J^{★}\end{array}$$

(31)

wherein, the optimal system control performance index $J^{★}$ is derived using (10).

Remark 6.

According to (31), it can be observed that an increase in the covariance of the watermark signal ${\Sigma }_o$ results in a corresponding rise in the control performance function J. The methodology adopted in this paper, which utilizes the relative positions of watermark signals for attack detection, enables the application of smaller ${\Sigma }_o$ values, effectively minimizing control costs in UMV systems.

4. Numerical Simulation

This section validates the efficacy of replay attack detection. The UMVs model selected for this study, borrowed from Bian et al. [30] and Wang et al. [33], is presented below:

$$\begin{array}{c}\hfill \mathbb{M}=\left[\begin{array}{ccc}1.0852& 0& 0\\ 0& 2.0575& -0.4087\\ 0& -0.4087& 0.2153\end{array}\right],\mathbb{N}=\left[\begin{array}{ccc}0.0865& 0& 0\\ 0& 0.0762& 0.1510\\ 0& 0.1510& 0.0031\end{array}\right],\mathbb{O}=\left[\begin{array}{ccc}1.0389& 0& 0\\ 0& 0.0266& 0\\ 0& 0& 0\end{array}\right].\end{array}$$

For the UMVs to effectively track the reference system’s output through the yaw angle, consider the following matrix $C=\left[\begin{array}{cccccc}0& 0& 1& 0& 0& 0\end{array}\right]$. The disturbance $\theta \left(k\right)$ is presented as:

$$\left\{\begin{array}{l}{\theta }_1\left(k\right)=0.27F_1\left(s\right){\mathbb{N}}_1\left(k\right){\mathbb{N}}_2\left(k\right)\\ {\theta }_2\left(k\right)=-0.6cos\left(1.6k\right)e^{-0.12k}\\ {\theta }_3\left(k\right)=0.58F_2\left(s\right){\mathbb{N}}_3\left(k\right){\mathbb{N}}_4\left(s\right)\end{array}\right.$$

where $F_1\left(s\right)\stackrel{\Delta }{=}[K_{\theta 1}s/(s^2+2h_1{\varsigma }_{1}s+{\varsigma }_1^2)]$ and $F_2\left(s\right)\stackrel{\Delta }{=}[K_{\theta 2}s/(s^2+2h_2{\varsigma }_{2}s+{\varsigma }_2^2)]$. Designated wave strength coefficients are $K_{\theta 1}=0.26$ and $K_{\theta 2}=0.8$. The damping factors, $h_1=0.2$ and $h_2=1.7$. The frequencies of the waves are ${\varsigma }_1=1.3$ and ${\varsigma }_2=0.9$. Levels of white noise power are ${\mathbb{N}}_1\left(k\right)=2.69$ and ${\mathbb{N}}_3\left(k\right)=1.56$. Where

$$\begin{array}{c}{\mathbb{N}}_2\left(\mathrm{t}\right)=\left\{\begin{array}{c}1,t\in [0s,20s]\\ 0,t>20s\end{array}\right.\hfill \end{array},\begin{array}{c}{\mathbb{N}}_4\left(\mathrm{t}\right)=\left\{\begin{array}{c}1,t\in [0s,15s]\\ 0,t>15s\end{array}\right.\hfill \end{array}$$

At $t_0=50$ s, assuming the injected watermark signal has a periodicity of $T=10$ s with ${\Sigma }_o=0.5763$, there exist $A_{dp}^{10T}\approx 0$. After 10 periods of watermark signal injection, the injected watermark and compensation signals are illustrated in Figure 7. As depicted in the following figures, both the compensation and watermark signals exhibit periodic behavior with distinct values within each period.

In the following, an evaluation of watermark signal periods is conducted, detailing detection rates and false alarm rates under two types of replay attacks at various watermark intervals, as shown in

Table 2. Detection Rates of Replay Attacks under Different Watermark Periods.

T	${\mathbf{\Sigma }}_{\mathit{o}}$	$\mathit{\beta }$ of Continuous Replay Attacks	$\mathit{\beta }$ of Discontinuous Replay Attacks
6	0.5763	92.5%	87.3%
8	0.5763	92.7%	88.4%
10	0.5763	93.8%	90.1%
12	0.5763	93.3%	89.2%
14	0.5763	89.6%	87.5%

. For the same watermark signal value, ${\Sigma }_o=0.5763$, $\beta$ values for both continuous and discontinuous replay attacks exhibit an increase for $T<10$. Conversely, $\beta$ values decrease as the watermark period increases beyond $T>10$. Based on these observations, a period of $T=10$ has been selected.

In scenarios devoid of process and measurement noise interference, with ${\Sigma }_o=0.015$ and ${\Sigma }_o=0.4353$, the detection results under normal system operation are as illustrated in Figure 8. The values of the discriminant function are consistently zero, indicating the absence of false positives. With noise interference present in the system, the watermark signal variances are set to ${\Sigma }_o=0.015$ and ${\Sigma }_o=0.4353$, with the corresponding normal operational detection results displayed in Figure 9, respectively. The state noise and measurement noise in the system can lead to false negatives and false positives. As the variance ${\Sigma }_o$ of the watermark signal increases, the incidence of false positives decreases.

Case 1: The UMVs system is subjected to continuous replay attacks. Setting the watermark signal at ${\Sigma }_o=0.5763$, data are recorded by the attacker from 30 to 40 s. As illustrated in Figure 10 (left figure), the UMVs undergo replay attacks from 650 to 950 s. Similarly, as shown in Figure 10 (right figure), replay attacks occur from 750 to 1000 s. This case study aims to test the strategy’s effectiveness in detecting prolonged continuous replay attacks.

Case 2: The UMVs system is subjected to discontinuous replay attacks. Setting the watermark signal at ${\Sigma }_o=0.5763$, data are recorded by the attacker from 30 to 40 s. As illustrated in Figure 11 (left figure), the UMVs undergo replay attacks from 620 to 660 s, 700 to 740 s, 780 to 820 s, 860 to 900 s, and 940 to 980 s. Similarly, as shown in Figure 11 (right figure), replay attacks occur from 620 to 710 s, 750 to 840 s, and 880 to 970 s. This case study aims to test the strategy’s effectiveness in detecting prolonged continuous replay attacks.

Detection effectiveness of the UMVs system under replay attacks, as demonstrated for both continuous and discontinuous types of replay attacks using the method proposed, is showcased in Case 1 and Case 2. In Case 1, the UMVs underwent continuous replay attacks of two different durations during regular operations. As Figure 10 indicates, during normal system operation, $G\left(k\right)=0$. However, when the UMVs are undergoing a replay attack, significant changes occur in the values of the detection function. Between 650 and 750 s, $G\left(k\right)\ne 0$, indicating that the UMVs are experiencing a replay attack. In Case 2, the UMVs underwent several brief intermittent replay attacks. As displayed in Figure 11, the system faced five short, discontinuous attacks and two additional quick, intermittent ones. Similar to Case 1, with the application of the proposed strategy, the detection function maintains $G\left(k\right)=0$ during normal operations. Notable shifts in detection function values when under attack suggest a replay attack is occurring. After repeating experiments 1000 times, the detection rates for continuous and discontinuous replay attacks are $\beta =93.8\%$ and $\beta =90.1\%$, respectively, with false positive rates of $\alpha =4.5\%$ and $\alpha =6.7\%$. Thus, the method proposed in this article effectively detects the occurrence of continuous with discontinuous replay attacks of UMV systems.

As shown in Figure 12, in order to further validate the effectiveness of the proposed method, a comparative analysis is performed to compare the control performance of the proposed method with that of the method outlined in [30,36,37]. Simulations were conducted for 200 iterations on four approaches to obtain the average performance degradation and detection rates under replay attacks. It can be observed from Figure 12 that as the detection rate increases, the performance degradation of different methods gradually increases.

This analysis underscores the superiority of the proposed method by comparing various metrics, including the continuous replay attack detection rate ${\beta }_1$, discontinuous replay attack detection rate ${\beta }_2$, continuous replay attack false alarm rate ${\alpha }_1$, discontinuous replay attack false alarm rate ${\alpha }_2$, and the average control cost under multiple attacks $J_{ave}$, with those reported in previous studies [30,31,32,36,37]. The results of this comparative analysis are presented in

Table 3. Comparison of Detection Efficiency and Control Cost Between Methods in This Paper and Other Papers under the Same Detection Efficiency.

	${\mathit{J}}_{\mathit{ave}}$	${\mathit{\beta }}_1$	${\mathit{\alpha }}_1$	${\mathit{\beta }}_2$	${\mathit{\alpha }}_2$
Method in This Paper	24	$88–92\%$	$4–6\%$	$87–91\%$	$6–10\%$
Method in Bian et al. [30]	30	$88–92\%$	$4–6\%$	$87–91\%$	$6–10\%$
Method in Zhao et al. [37]	55	$88–92\%$	$4–6\%$	$87–91\%$	$6–10\%$
Method in Fang et al. [32]	93	$88–92\%$	$4–6\%$	$87–91\%$	$6–10\%$
Method in Mo et al. [31]	167	$88–92\%$	$4–6\%$	$87–91\%$	$6–10\%$
Method in Mo et al. [36]	175	$88–92\%$	$4–6\%$	$87–91\%$	$6–10\%$

and

Table 4. Comparison of Detection Efficiency and Control Cost Between Methods in This Paper and Other Papers under the Same Control Performance Loss.

	${\mathit{J}}_{\mathit{ave}}$	${\mathit{\beta }}_1$	${\mathit{\alpha }}_1$	${\mathit{\beta }}_2$	${\mathit{\alpha }}_2$
Method in This Paper	80	$95–96\%$	$1–2\%$	$95–96\%$	$1–2\%$
Method in Bian et al. [30]	80	$92–94\%$	$1–2\%$	$92–94\%$	$1–3\%$
Method in Zhao et al. [37]	80	$90–92\%$	$3–5\%$	$89–92\%$	$3–4\%$
Method in Fang et al. [32]	80	$82–84\%$	$9–12\%$	$79–80\%$	$11–13\%$
Method in Mo et al. [31]	80	$67–70\%$	$16–19\%$	$57–66\%$	$22–27\%$
Method in Mo et al. [36]	80	$65–70\%$	$17–23\%$	$53–62\%$	$28–35\%$

.

It is revealed in Table 3 that when the detection requirements are specified as follows: continuous replay attack detection rate ${\beta }_1$ between 88% and 92%, discontinuous replay attack detection rate ${\beta }_2$ between 87% and 91%, continuous replay attack false alarm rate ${\alpha }_1$ between 4% and 6%, and discontinuous replay attack false alarm rate ${\alpha }_2$ between 6% and 10%, the lowest control cost of 24 is achieved by the proposed method.

It is shown in Table 4 that when the control cost $J_{ave}$ for various methods is fixed at 80, other methods are surpassed by the proposed method in both continuous replay attack detection rate ${\beta }_1$ and discontinuous replay attack detection rate ${\beta }_2$. Additionally, the continuous replay attack false alarm rate ${\alpha }_1$ and discontinuous replay attack false alarm rate ${\alpha }_2$ are both lower than those reported in previously published methods.

The superiority of the proposed method is evident from the data and analysis presented. When the detection accuracy requirement is held constant, the proposed method results in reduced control costs. Conversely, when the control cost requirement is fixed, the proposed method achieves higher detection accuracy and a lower false alarm rate.

5. Conclusions

This research has presented a method for detecting replay attacks on UMVs by employing periodic watermark and compensatory signals that adhere to a Gaussian signal. Unlike conventional approaches that use fixed thresholds to compare residual detection function magnitudes, this method has improved detection efficiency by evaluating the relative position across compensatory signals. The technique has not only achieved high detection rates but also reduced the variance of watermark signals, thus preserving system performance. However, when the attack delay aligns with an integer multiple of the watermark signal period, the method may produce some false positives. In the method used by Bian et al. [30], although the method incurs a greater loss in control system performance compared to the paper, it does not produce false positives under these circumstances. Addressing this issue is planned for future research. Future research will also focus on designing new watermark signals to minimize their effect on system control performance further.