Dynamic probabilistic event-based fault detection for unmanned surface vehicles systems under hybrid attacks

Abstract

In this paper, the design of fault detection filter (FDF) for a class of unmanned surface vehicle (USV) systems based on dynamic probabilistic event-triggered transmission protocol under compound cyber attacks is studied. First, the dynamic probabilistic event-triggered mechanism and numerical quantization are used to boost system performance, reduce communication resource consumption, and prevent network congestion. Considering Denial-of-service (DoS) attack and spoofing attack, a compound attack model is established, which results in the filtering error system being converted into a stochastic switching system. And the sufficient conditions of the mean-square exponential stability for the USV system are obtained by using the Lyapunov functional and linear matrix inequality (LMI) technique. The gains of the designed filter and weighting matrix are obtained by optimizing the LMIs. Finally, a practical example is given to verify the feasibility of the proposed fault detection method.

Keywords

Unmanned surface vehicles fault detection filter compound cyber attacks dynamic probabilistic event-triggered mechanism linear matrix inequality

Introduction

With the development of Global Positioning System (GPS) and automation technology, unmanned surface vehicles (USVs) have offered innovative solutions for various waterborne tasks (Hao et al., 2019; Wang et al., 2019). These autonomous or remotely controlled vessels are equipped with cutting-edge technology, including sensors, cameras, and sophisticated navigation systems, allowing them to perform a multitude of operations efficiently and safely (Liu et al., 2017, 2016). The versatility of USVs makes them applicable in numerous fields, such as environmental monitoring, maritime security, search and rescue missions, and oceanographic research (Zhou et al., 2020). Due to the multiple advantages of USVs, the motion control issues have been widely explored by researchers. Key areas of focus include path planning and trajectory tracking (Liang et al., 2020), dynamic obstacle avoidance and environmental awareness (Woo and Kim, 2020), multi-vessel cooperative control (USV formations) (Chen et al., 2018; Qi et al., 2024), and dynamic positioning control (Ma et al., 2020; Zhang et al., 2021). Among these, the dynamic positioning control has attracted the greatest attention. For instance, Zhang et al. (2021) addressed the dynamic positioning problem of nonlinear USVs under communication interruptions and DoS attacks using a Takagi–Sugeno (T-S) fuzzy system. Through the design of a switching sliding mode controller, Ma et al. (2020) studied the positioning fault tolerance control of the USV whose membership function of the T-S fuzzy model is unknown.

As the USV operates in a humid and corrosive environment over extended periods, the systems are susceptible to damage from seawater corrosion and other adverse environmental factors, causing the deterioration and wear of electrical components, mechanical parts, and external structures. These environmental influences include salt spray, high humidity from seawater, and corrosive substances in the sea air, which can expedite the degradation of metal surfaces and harm critical components like sensors, power units, and communication systems, potentially leading to malfunctions (Wang and Han, 2016). Fault detection is essential for maintaining the reliability and safety of USVs, particularly in scenarios such as long-term autonomous missions, harsh marine environments, and network-based control systems (Abid et al., 2021; Liu et al., 2022). This has driven extensive research on fault detection and control strategies for USVs. Typically, the fault detection involves creating a residual signal and comparing it to a set threshold. If the residual signal surpasses this threshold, an alert is promptly generated. For instance, Li and Park (2024) investigated the design of an event-triggered fault detection filter (FDF) tailored for USV systems in networked environments.

In recent decades, in order to optimize the utilization of resources in the communication network and reduce redundant transmission, the choice of triggering mechanism has become particularly significant (Cheng et al., 2023; Wang et al., 2022). Based on the characteristics of the triggering mechanisms, they can generally be categorized into time-triggered and event-triggered approaches. Furthermore, event-triggered mechanisms (ETMs) can be subdivided into static event-triggered and dynamic event-triggered types. The core concept of event-triggered control is to reduce the frequency of computations or communications by defining a triggering threshold. The control law updates are initiated only when specific vectors, such as system states or errors, surpass this threshold (Chen et al., 2023). The dynamic positioning control of the USV system based on event-triggered has been widely studied. Ye et al. (2021) adopted a new event-triggered mechanism for intelligent positioning control of USV systems, which was based on a novel asynchronous advantage actor-critic (A3C) learning-based event-triggered approach. Li and Park (2024) studied the fault detection of the USV system under randomly occurring injection attacks based on an adaptive event-triggered mechanism.

Industrial communication networks play a crucial role in the functioning of contemporary industrial systems and are known for their open, interconnected structure. While this openness promotes seamless communication and data sharing, it also exposes the networks to a variety of cyber threats from unidentified attackers. Such cyber attacks can have devastating effects, such as reduced performance, compromised data integrity, unauthorized access, or even system failures. Consequently, securing these networks and ensuring their robustness has become a critical priority. In light of these vulnerabilities, researchers have extensively studied how to mitigate the negative impacts of cyber attacks, such as how to reduce the influence of DoS attacks (Ma et al., 2020; Qi et al., 2024; Ye et al., 2022, 2021; Zhang et al., 2021), distributed DoS attacks, deception attacks (injection attacks) (Chen et al., 2023; Li and Park, 2024), replay attacks (Liu et al., 2022; Shi et al., 2016), and hybrid attacks (Xing et al., 2025; Ye et al., 2021). Due to the limitations of the time-triggered mechanism in the presence of DoS attacks, a more elastic event-triggered approach was implemented to enhance fault detection and reduce the impact in Ma et al. (2020).

To sum up, we will complete the elastic fault detection of USV systems under hybrid cyber attacks. The main contributions of this paper are as follows: (1) by introducing the probabilistic event-triggered mechanism and considering the impact of data quantization on the system, the fault detection framework of USV system is established; (2) Considering aperiodic DoS attacks, the filtering error system is converted into a stochastic switching system; and (3) based on the Lyapunov functional and linear matrix inequality (LMI) technique, the sufficient conditions of the mean-square exponential stability for the filtering error system are obtained.

Notations

In this paper, $δ (t)$ denotes the rudder angle, which serves as the physical actuator input to control heading. $ω (t)$ represents the external disturbances such as wind and waves. The variables $r (t)$ , $p (t)$ , and $v (t)$ denote the yaw rate, roll rate, and sway velocity, respectively. The orientation of the USV is characterized by the heading angle $ψ (t)$ and the roll angle $φ (t)$ . System gain parameters are given by $K_{dp}$ , $K_{dr}$ , $K_{dv}$ , $K_{vp}$ , and $K_{vr}$ , which influence the system’s dynamic response. The pair $(ω_{n}, ζ)$ denotes the natural frequency and damping ratio, reflecting oscillatory behavior. The constants $T_{r}$ and $T_{v}$ denote time constants associated with yaw and sway dynamics. The event-triggered mechanism is characterized by the static threshold ρ and the time-varying threshold $ϵ (t)$ . $Φ$ is a positive definite weighting matrix used in evaluating the triggering condition, while $η (t)$ is an internal dynamic variable that evolves within the dynamic probabilistic event-triggered mechanism (DPETM). The Bernoulli variable $v_{p} (t)$ governs the random selection among threshold intervals, and ℓ denotes the number of such intervals. The sampling interval is denoted by h. Time delay due to network communication is represented by $τ (t)$ , with $τ_{M}$ as its upper bound. Spoofing attacks are modeled using the time-varying probability $α (t)$ , with mean and variance $\bar{α}$ and $\hat{α}$ , respectively. $F (y (t))$ denotes the spoofed (maliciously injected) sensor output. Denial-of-service (DoS) attacks are indicated by the Bernoulli variable $β (t)$ , with the off/on time durations bounded by $T_{off}^{\min}$ and $T_{on}^{\max}$ .

System description

USV modeling

In general, the USV dynamics comprise six degrees of freedom, including heave, pitch, roll, surge, sway, and yaw (Du et al., 2015). In order to facilitate the follow-up work, this paper only studies the motion of roll, sway, and yaw. The external disturbance is heave, surge, and pitch. The expression of the dynamics equation of USV system is obtained according to Wang and Han (2016):

{\begin{matrix} ṙ (t) = K_{vr} ∕ T_{r} v (t) + 1 ∕ T_{r} ω_{ψ} (t) + K_{dr} ∕ T_{r} δ (t) - 1 ∕ T_{r} r (t), \\ ṗ (t) = ω_{n}^{2} K_{dp} δ (t) - 2 ζ ω_{n} p (t) + ω_{n}^{2} ω_{ϕ} (t) - ω_{n}^{2} ϕ (t) + ω_{n}^{2} K_{vp} v (t), \\ \overset{\cdot}{v} (t) = K_{dv} ∕ T_{v} δ (t) - 1 ∕ T_{v} v (t), \\ \overset{\cdot}{ψ} (t) = r (t), \\ \overset{\cdot}{ϕ} (t) = p (t), \end{matrix}

(1)

where $r (t)$ denotes the yaw velocity, $υ (t)$ represents the sway velocity resulting from rudder movement, and $p (t)$ corresponds to the roll velocity. $ϕ (t)$ is used to describe the roll angle, $δ (t)$ indicates the rudder angle, while $ψ (t)$ refers to the heading angle. The external disturbances affecting the heading angle and roll angle due to wind and waves are denoted by $ω_{ψ} (t)$ and $ω_{ϕ} (t)$ , respectively. The parameters $K_{dp}$ , $K_{dr}$ , $K_{υp}$ , $K_{υr}$ , and $K_{dυ}$ are the predefined gains. In addition, $ω_{n}$ and ζ represent the undamped natural frequency and the damping ratio, while $T_{r}$ and $T_{υ}$ are the specified time constants.

Define $x (t) = {[υ (t), r (t), ψ (t), p (t), ϕ (t)]}^{T}, u (t) = δ (t)$ , $ω (t) = {[ω_{ψ} (t), ω_{ϕ} (t)]}^{T}$ with $x (t) \in ℝ^{n}$ , $ω (t) \in L_{2} [0, \infty)$ and $ω (t) \in ℝ^{m}$ , $u (t) \in$ $ℝ^{d} .$ The USV dynamic (1) can be restated as:

ẋ (t) = A_{0} x (t) + B_{0} u (t) + B_{ω} ω (t),

(2)

where

\begin{matrix} A_{0} & = [\begin{matrix} - 1 ∕ T_{v} & 0 & 0 & 0 & 0 \\ K_{vr} ∕ T_{r} & - 1 ∕ T_{r} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ ω_{n}^{2} K_{vp} & 0 & 0 & - 2 ζ ω_{n} & - ω_{n}^{2} \\ 0 & 0 & 0 & 1 & 0 \end{matrix}], \\ B_{0} & = {[\begin{matrix} K_{dv} ∕ T_{v} & K_{dr} ∕ T_{r} & 0 & ω_{n}^{2} K_{dp} & 0 \end{matrix}]}^{T}, \\ B_{ω} & = {[\begin{matrix} 0 & 1 ∕ T_{r} & 0 & 0 & 0 \\ 0 & 0 & 0 & ω_{n}^{2} & 0 \end{matrix}]}^{T} . \end{matrix}

Suppose there is a state feedback controller $u (t) = Kx (t)$ that stabilizes the USV system. The gain can be achieved by selecting matrix K such that $A = A_{0} + B_{0} K$ is simply Hurwitz. In addition, considering the system failure caused by factors such as metal rust, system (2) can be restated as:

{\begin{matrix} ẋ (t) = Ax (t) + B_{ω} ω (t) + B_{f} f (t), \\ y (t) = Cx (t), \end{matrix}

(3)

with $A = A_{0} + B_{0} K$ , the observable system output and potential system faults are denoted by $y (t) \in ℝ^{p}$ and $f (t) \in ℝ^{m}$ , respectively. Furthermore, $ω (t), f (t) \in L_{2} [0, \infty)$ . The system output matrix C and the fault allocation matrix $B_{f}$ for the USV system are given.

Fault weighting

In an effort to raise the power of the fault detection, in the framework of fault detection, a weighting fault function is usually contained, which can be referred to $\hat{f} (z) = D (z) f (z)$ . A appropriate weighting matrix $D (z)$ is selected to limit the frequency interval of interested fault signal, which can improve the performance of the fault detection. In this paper, the fault weighting system is expressed as:

{\begin{matrix} ẋ_{H} (t) = A_{H} x_{H} (t) + B_{H} f (t), \\ \hat{f} (t) = C_{H} x_{H} (t) + D_{H} f (t), \end{matrix}

(4)

where $x_{H} (t) \in ℝ^{H}$ and $\hat{f} (t) \in ℝ^{l}$ represent the state of the fault weighting system and the weighted fault of the system, respectively. The matrices $A_{H}$ , $B_{H}$ , $C_{H}$ , and $D_{H}$ are predefined and fully known.

Probabilistic event-triggered protocol

To conserve limited network energy, an event-triggered mechanism is proposed based on the following triggering condition:

\begin{matrix} t_{k + 1} h = in f_{k \in N^{+}} {k ∣ kh > t_{k} h, ρ G (kh) \geq η (kh), ρ > 0}, \\ G (t) = e^{T} (t) Φ e (t) - \sum_{p = 1}^{ℓ} v_{p} (t) ε_{p} (t) {[(y (t - τ (t)) - e (t))]}^{T} \\ Φ [(y (t - τ (t)) - e (t))], \end{matrix}

(5)

where $\sum_{p = 1}^{ι} v_{p} (t) ε_{p} (t)$ is the time-varying triggering threshold with $ε_{p} (t) \in {\vec{ε}}_{p} ≜ [ε_{p - 1}, ε_{p}), p = 1, 2, \dots, ℓ - 1$ , and $ε_{ℓ} (t) \in {\vec{ε}}_{ℓ} ≜ [ε_{ℓ - 1}, ε_{ℓ}]$ . $Φ$ is the weighting matrix to be solved.

We define the interval triggering threshold as $[ε_{\min}, ε_{\max}]$ , divide it into ℓ subintervals according to the method in Chen et al. (2021). $[ε_{\min}, ε_{\max}] = ⋃_{p = 1}^{ℓ} {\vec{ε}}_{p}, ε_{0} = ε_{\min}, ε_{ℓ} = ε_{\max}$ . $v_{p} (t)$ represents the introduced ℓ Bernoulli variables, which are used to determine the activity interval where the current trigger threshold is located. It satisfies $E {v_{p} (k)} = {\bar{v}}_{p} \in [0, 1], \sum_{p = 1}^{ℓ} v_{p} = 1 .$ Unlike sum-based ETMs (Yan et al., 2024), the proposed dynamic probabilistic event-triggered transmission protocol employs randomized threshold selection across multiple intervals, enhancing flexibility and robustness under hybrid attack scenarios.

In addition, the internal dynamic variables are given as follows:

\overset{\cdot}{η} (t) = - kη (t) - λ G (t), λ > 0, k > 0, η (0) = η_{0} > 0,

(6)

Inspired by Li et al. (2020), we convert the system into a delay system in order to better handle the measured output. So, we define:

\begin{matrix} [t_{k} h, t_{k + 1} h) = ⋃_{l_{k} = 0}^{B_{k}} H_{l_{k}}, \end{matrix}

(7)

where $H_{l_{k}} = [t_{k} h + l_{k} h, t_{k + 1} h + (l_{k} + 1) h)$ and $B_{k} = t_{k + 1}$ $- t_{k} - 1$ .

The piecewise functions defined as:

\begin{matrix} {\begin{matrix} τ (t) = t - t_{k} h - l_{k} h, \\ e (t) = y (t_{k} h + l_{k} h) - y (t_{k} h), \end{matrix} \end{matrix}

(8)

and we have:

\begin{matrix} y (t_{k} h) = y (t - τ (t)) - e (t) . \end{matrix}

(9)

Quantization

In scenarios with restricted communication, acquiring the full original measurement signal becomes challenging. Typically, signals must undergo quantization prior to transmission. The logarithmic quantizer is represented as:

Z_{j} (y_{j}) = {\begin{matrix} r_{i}^{l}, & \frac{1}{1 + C_{Z}} r_{i}^{l} < y_{j} < \frac{1}{1 - C_{Z}} r_{i}^{l}, y_{j} > 0, \\ 0, & y_{j} = 0, \\ - Z_{j} (- y_{j}), & y_{j} < 0, \end{matrix}

(10)

where the quantization density is $ϵ_{Z} = \frac{1 - P}{1 + P}, (0 < P < 1)$ , and the quantized levels are $R_{Z} = {\pm r_{i}^{x} : r_{i}^{x} = P_{i}^{x} r_{0}^{x}} \cup {0}, (l = 1, \pm 2, \dots) with r_{0}^{x} > 0, 0 < P_{i} < 1$ .

According to Wang et al. (2023), the quantization process can be represented by the following function:

\begin{matrix} Z_{j} (y_{j}) = (1 + Δ_{Z_{j}} y_{j}) y_{j} . \end{matrix}

(11)

Define the quantization uncertainty term is $Δ_{Z}$ . Combined with the above description of the dynamic probabilistic event-triggered transmission protocol and the quantizer, the quantized measurement output can be described as:

\begin{matrix} ỹ (t) & = Z_{j} (y (t_{k} h)) = (I + Δ_{Z}) y (t_{k} h), \\ ỹ (t) & = (I + Δ_{Z}) (Cx (t - τ (t) - e (t)) . \end{matrix}

(12)

Cyber attacks

When the information is transmitted, it constantly faces attacks from different attackers, which can lead to data loss. To overcome this obstacle, we propose a composite attack model that includes a class of power-limited DoS attacks and multi-channel spoofing attacks, which is described as follows:

ȳ (t) = β (t) [α (t) F (y (t)) + (1 - α (t)) ỹ (t)],

(13)

where $Y (t) = - y (t) + F (y (t))$ represents the false injection signals resulting from deception attacks, and $F (y (t))$ denotes a fully smooth and continuous mapping that satisfies $F (y (t)) : ℝ^{r} \times [0, T] \to ℝ^{r}$ and:

\begin{matrix} {[F (y (t)) - O_{1} y (t)]}^{T} [F (y (t)) - O_{2} y (t)] \leq 0, \end{matrix}

(14)

where

β (t) = {\begin{matrix} 1, t \in [T_{n}, T_{off} (n) + T_{n}] ≜ T_{1, n}, \\ 0, t \in [T_{off} (n) + T_{n}, T_{n + 1}] ≜ T_{2, n}, \end{matrix}

When $t \in [T_{n}, T_{off} (n) + T_{n}]$ , DoS attack is dormant stage, the data will be normally transmitted over a communication channel. When $t \in [T_{off} (n) + T_{n}, T_{n} + 1]$ , means the system is subjected to a DoS attack, the data cannot be normally transmitted. Here, the time interval $[T_{n}, T_{n} + 1]$ represents the nth DoS attack duration. $T_{off} (n)$ represents the inactive phase of the DoS attack.

$α (t), β (t) \in [0, 1]$ represent the occurrence probabilities of spoofing attacks and DoS attacks, respectively. $α (t)$ follow a Bernoulli distribution, it satisfies:

\begin{matrix} {\begin{matrix} \Pr {α (t) = 1} = E {α (t)} = \bar{α}, \\ \Pr {α (t) = 0} = 1 - E {α (t)} = 1 - \bar{α}, \\ E {{(α (t) - \bar{α})}^{2}} = \hat{α} . \end{matrix} \end{matrix}

(15)

Assumption 1. (Attack Duration Limitation) (Liu et al., 2021). Suppose that the sleeping period and active duration of the DoS attack are uniformly bounded by $T_{off}^{\min}$ and $T_{on}^{\max}$ , respectively, and meet the following conditions:

\begin{matrix} {\begin{matrix} in f_{n \in ℕ} {T_{off} (n - 1)} \geq T_{off}^{\min}, \\ su p_{n \in ℕ} {T_{n} - T_{n - 1} - T_{off} (n - 1)} \leq T_{on}^{\max}, \end{matrix} \end{matrix}

(16)

Assumption 2. (Attack Switch Times Limitation) (Liu et al., 2021). If the time interval between two consecutive DoS attacks is shorter than the minimum sampling period, it becomes challenging for any control system to ensure stability. Therefore, the frequency of DoS attacks must be appropriately restricted as:

\begin{matrix} s (t) \leq u + \frac{t}{m}, \end{matrix}

(17)

where $s (t)$ denotes the occurrence frequency of DoS attacks within the time interval $[0, t)$ , $u > 0, m > h$ are the known positive real numbers, and h is the average sampling period.

Fault filter

The main purpose of this paper is to detect the failure of the system by generating residual signal under the DPETM. Therefore, the FDF is designed as follows:

{\begin{matrix} ẋ_{F} (t) = A_{F} x_{F} (t) + B_{F} ȳ (t), \\ r (t) = C_{F} x_{F} (t) + D_{F} ȳ (t), \end{matrix}

(18)

where $x_{F} (t) \in ℝ^{n + H}, ȳ (t) \in ℝ^{p}$ , and $r (t) \in ℝ^{l}$ are respectively the state of filter, the input of filter, and the residual signal. The gains $A_{F}, B_{F}, C_{F}$ , and $D_{F}$ will be designed later.

Filtering error system

Define $\tilde{x} (t) = {[x_{H}^{T} (t), x^{T} (t), x_{F}^{T} (t)]}^{T}$ , $θ (t) = {[w^{T} (t), f^{T} (t)]}^{T}$ , $r_{e} (t) = r (t) - \hat{f} (t)$ , then the filtering error system can be established as the following form:

{\begin{matrix} \overset{\cdot}{\tilde{x}} (t) = A_{1} \tilde{x} (t) + (1 - α (t)) B_{1} \tilde{x} (t - τ (t)) + α (t) B_{2} F (y (t)) \\ + (1 - α (t)) B_{3} e (t) + B_{4} θ (t), t \in T_{1, n} \\ = A_{1} \tilde{x} (t) + B_{4} θ (t), t \in T_{2, n} \\ r_{e} (t) = C_{1} \tilde{x} (t) + D_{1} θ (t) + (1 - α (t)) C_{2} \tilde{x} (t - τ (t)) \\ + α (t) D_{F} F (y (t)) - (1 - α (t)) D_{2} e (t), t \in T_{1, n} \\ = C_{1} \tilde{x} (t) + D_{1} θ (t), t \in T_{2, n}, \end{matrix}

(19)

where

\begin{matrix} A_{1} = [\begin{matrix} A_{H} & 0 & 0 \\ 0 & A & 0 \\ 0 & 0 & A_{F} \end{matrix}], B_{1} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & B_{F} (I + Δ_{Z}) C & 0 \end{matrix}], \\ B_{2} = [\begin{matrix} 0 \\ 0 \\ B_{F} \end{matrix}], B_{3} = [\begin{matrix} 0 \\ 0 \\ - B_{F} (I + Δ_{Z}) \end{matrix}], \end{matrix}

\begin{matrix} B_{4} = [\begin{matrix} 0 & B_{H} \\ B_{ω} & B_{f} \\ 0 & 0 \end{matrix}], C_{1} = [\begin{matrix} - C_{H} & 0 & C_{F} \end{matrix}], \\ C_{2} = [\begin{matrix} 0 & D_{F} (I + Δ_{Z}) C & 0 \end{matrix}], D_{1} = [\begin{matrix} 0 & - D_{H} \end{matrix}], \\ D_{2} = D_{F} (I + Δ_{Z}) . \end{matrix}

And Figure 1 expresses the structure of the filtering error system.

For the convenience of subsequent research, we will introduce several key lemmas.

Lemma 1. (Liu et al., 2019). If there is the time delay $τ (t) \in [0, τ_{M}]$ , and any fitting $n \times n$ matrices $X > 0$ and $U > 0$ of appropriate dimensions satisfy $[\begin{matrix} X & * \\ U & X \end{matrix}] > 0$ , then it is satisfied as:

\begin{matrix} - \bar{τ} \int_{t - \bar{τ}}^{t} ẋ^{T} (t) Xẋ (t) dt \leq φ^{T} (t) Γ φ (t), \end{matrix}

(20)

where $φ (t) = [\begin{matrix} x (t) \\ x (t - τ (t)) \\ x (t - τ_{M}) \end{matrix}]$ , $Γ = [\begin{matrix} - X & * & * \\ X - U & - 2 X + U + U^{T} & * \\ U & X - U & - X \end{matrix}]$ .

Lemma 2. (Li et al., 2020). For a given real matrix with appropriate dimension $D, P, F$ and satisfy $F^{T} F \leq I$ . There exists $ϵ > 0$ that satisfies the following inequality:

\begin{matrix} DFP + P^{T} F^{T} D^{T} \leq ϵ^{- 1} D D^{T} + ϵ P^{T} P . \end{matrix}

(21)

Lemma 3. (Liu et al., 2020). For matrices $A > 0$ , P and the real numbers χ, it is satisfied as:

\begin{matrix} - P A^{- 1} P \leq χ^{2} A - 2 χP . \end{matrix}

(22)

Building upon the aforementioned operations, the fault detection problem for the USV system (3) can be recast as an auxiliary $H_{\infty}$ filtering problem for a time-delay system. Specifically, a filter in the form of equation (18) is to be designed such that both the filtering error and the residual signal are minimized. The objective is to determine a set of matrices $A_{F}, B_{F}, C_{F}$ , and $D_{F}$ that guarantee the mean-square stability of the fault detection dynamics described by equation (19) while simultaneously minimizing the infimum of the system performance index γ within the feasible solution set:

\begin{matrix} \sup_{θ (t) \neq 0, θ (t) \in L_{2} [0, + \infty)} \frac{∥ r_{e} (t) ∥^{2}}{∥ θ (t) ∥^{2}} < γ^{2}, γ > 0 . \end{matrix}

(23)

To effectively use the residual signal generated by the FDF for fault detection, we make a decision on fault occurrence by comparing the residual evaluation function $J (t) = {(\frac{1}{t} \int_{0}^{t} r^{T} (θ) r (θ) dθ)}^{1 ∕ 2}$ with the threshold $J_{th} = \sup_{w (t) \in L_{2}, f (t) = 0} J (t)$ :

\begin{matrix} {\begin{matrix} J (t) > J_{th} \Rightarrow fault occurrence, \\ J (t) \leq J_{th} \Rightarrow no fault . \end{matrix} \end{matrix}

(24)

In addition, the algorithm of fault detection process is given in Figure 2.

Main results

This section introduces the filter design method of USV using probabilistic event-triggered mechanism (PETM) scheme against multiple cyber attacks. First, the sufficient conditions for the exponentially mean-square stability are obtained by constructing Lyapunov functional in Theorem 1. Second, the term containing quantized nonlinear parameters is eliminated, and an equivalent inequality is obtained in Theorem 2. Finally, the design of filter gains and weighting matrix are obtained in Theorem 3.

Theorem 1. For given attack parameters $T_{off}^{\min}$ , $T_{on}^{\max}$ , u, m, the sampling interval $h < T_{off}^{\min}$ , triggering parameters $ρ, λ, k, ℓ, v_{p}, ϵ_{p}$ and scalars $β_{i}$ , $J_{i}$ , time delay parameters $τ_{M}$ , the filtering error system (23) is exponentially mean-square stable with $H_{\infty}$ performance γ, if there exist $P_{i} > 0$ , $Q_{i} > 0$ , $R_{i} > 0, Φ > 0$ , and matrices $M_{i}$ with suitable dimensions are defined to ensure that the following inequalities are satisfied:

\begin{matrix} Ξ^{i} < 0, \end{matrix}

(25)

\begin{matrix} P_{1} \leq J_{2} P_{2}, \end{matrix}

(26)

\begin{matrix} P_{2} \leq J_{1} e^{2 (β_{1} + β_{2}) h} P_{1}, \end{matrix}

(27)

\begin{matrix} Q_{i} \leq J_{3 - i} Q_{(3 - i)}, \end{matrix}

(28)

\begin{matrix} R_{i} \leq J_{3 - i} R_{(3 - i)}, \end{matrix}

(29)

\begin{matrix} [\begin{matrix} R_{i} & * \\ M_{i} & R_{i} \end{matrix}] > 0, \end{matrix}

(30)

\begin{matrix} \frac{2 β_{1} T_{off}^{\min} - 2 (β_{1} + β_{2}) h - 2 β_{2} T_{on}^{\max} - \ln (J_{1} J_{2})}{m} > 0, \end{matrix}

(31)

where

\begin{matrix} Ξ^{1} = [\begin{matrix} Ξ_{11}^{1} & Λ_{1}^{1} & Λ_{2}^{1} & Λ_{3}^{1} & Λ_{4}^{1} \\ * & - P_{1} R_{1}^{- 1} P_{1} & 0 & 0 & 0 \\ * & * & - P_{1} R_{1}^{- 1} P_{1} & 0 & 0 \\ * & * & * & - I & 0 \\ * & * & * & * & - I \end{matrix}], \\ X i_{11}^{1} = [\begin{matrix} γ_{11}^{1} & γ_{12}^{1} & γ_{13}^{1} & γ_{14}^{1} & γ_{15}^{1} & γ_{16}^{1} \\ * & γ_{22}^{1} & 0 & 0 & 0 & 0 \\ * & * & γ_{33}^{1} & γ_{34}^{1} & 0 & 0 \\ * & * & * & γ_{44}^{1} & 0 & 0 \\ * & * & * & * & - I & 0 \\ * & * & * & * & * & - γ^{2} I \end{matrix}], \\ Λ_{1}^{1} = [τ_{M} P_{1} A_{1} 0 τ_{M} (1 - \bar{α}) P_{1} B_{1} τ_{M} (1 - \bar{α}) P_{1} B_{3} \\ τ_{M} \bar{α} P_{1} B_{2} τ_{M} P_{1} B_{4}], \\ Λ_{2}^{1} = [\begin{matrix} 0 & 0 & τ_{M} \hat{α} P_{1} B_{1} & τ_{M} \hat{α} P_{1} B_{3} & - τ_{M} \hat{α} P_{1} B_{2} & 0 \end{matrix}], \\ Λ_{3}^{1} = [\begin{matrix} C_{1} & 0 & (1 - \bar{α}) C_{2} & - (1 - \bar{α}) D_{2} & \bar{α} D_{F} & D_{1} \end{matrix}], \\ Λ_{4}^{1} = [\begin{matrix} 0 & 0 & - \hat{α} C_{2} & \hat{α} D_{2} & \hat{α} D_{F} & 0 \end{matrix}], \\ γ_{11}^{1} = 2 β_{1} P_{1} + P_{1} A_{1} + A_{1}^{T} P_{1} + Q_{1} - {\bar{R}}_{1} - Ī^{T} C^{T} O_{1}^{T} O_{2} CĪ, \\ γ_{12}^{1} = M_{1}, γ_{13}^{1} = (1 - \bar{α}) P_{1} B_{1} + {\bar{R}}_{1} - M_{1}, γ_{14}^{1} = (1 - \bar{α}) P_{1} B_{3}, \\ γ_{15}^{1} = \bar{α} P_{1} B_{2} + \sqrt{O_{1}^{T} + O_{2}} Ī^{T} C^{T}, \\ γ_{16}^{1} = P_{1} B_{4}, γ_{22}^{1} = - e^{- 2 β_{1} τ_{M}} Q_{1} - {\bar{R}}_{1}, γ_{23}^{1} = {\bar{R}}_{1} - M_{1}, \end{matrix}

\begin{matrix} γ_{33}^{1} = (kλ + ρ) \sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) Ī^{T} C^{T} Φ CĪ - 2 {\bar{R}}_{1} + M_{1} + M_{1}^{T}, \end{matrix}

\begin{matrix} γ_{34}^{1} = (kλ + ρ) \sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) Ī^{T} C^{T} Φ, \end{matrix}

\begin{matrix} γ_{44}^{1} = (kλ + ρ) \sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) Φ - (kλ + ρ) Φ, \\ Ξ^{2} = [\begin{matrix} Ξ_{11}^{2} & Λ_{1}^{2} & Λ_{2}^{2} \\ * & - P_{2} R_{2}^{- 1} P_{2} & 0 \\ * & * & - I \end{matrix}], \end{matrix}

\begin{matrix} Ξ_{11}^{2} = [\begin{matrix} γ_{11}^{2} & γ_{12}^{2} & 0 & 0 & γ_{15}^{2} & γ_{16}^{2} \\ * & γ_{22}^{2} & 0 & 0 & 0 & 0 \\ * & * & γ_{33}^{2} & γ_{34}^{2} & 0 & 0 \\ * & * & * & γ_{44}^{2} & 0 & 0 \\ * & * & * & * & - I & 0 \\ * & * & * & * & * & - γ^{2} I \end{matrix}], \\ Λ_{1}^{2} = [\begin{matrix} τ_{M} P_{2} A_{1} & 0 & 0 & 0 & 0 & τ_{M} P_{2} B_{4} \end{matrix}], \\ Λ_{2}^{2} = [\begin{matrix} C_{1} & 0 & 0 & 0 & 0 & D_{1} \end{matrix}], \\ γ_{11}^{2} = - 2 β_{2} P_{2} + P_{2} A_{1} + A_{1}^{T} P_{2} + Q_{2} - {\bar{R}}_{2} - Ī^{T} C^{T} O_{1}^{T} O_{2} CĪ, \\ γ_{12}^{2} = M_{2}, γ_{13}^{2} = {\bar{R}}_{2} - M_{2}, \\ γ_{15}^{2} = \sqrt{O_{1}^{T} + O_{2}} Ī^{T} C^{T}, γ_{16}^{2} = P_{2} B_{4}, γ_{22}^{2} = - e^{2 β_{2} τ_{M}} Q_{2} - {\bar{R}}_{2}, \\ γ_{23}^{2} = {\bar{R}}_{2} - M_{2}, \\ γ_{33}^{2} = (kλ + ρ) \sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) Ī^{T} C^{T} Φ CĪ - 2 {\bar{R}}_{2} + M_{2} + M_{2}^{T}, \\ γ_{34}^{2} = (kλ + ρ) \sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) Ī^{T} C^{T} Φ, \\ γ_{44}^{2} = (kλ + ρ) \sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) Φ - (kλ + ρ) . \end{matrix}

Proof. To prove the stability of the filtering error systems (19), the Lyapunov–Krasovskii functional candidate is constructed as follows:

\begin{matrix} V_{i} (t) = V_{i 1} (t) + V_{i 2} (t) + V_{i 3} (t) + V_{i 4} (t), \end{matrix}

(32)

where

\begin{matrix} V_{i 1} (t) & = {\tilde{x}}^{T} (t) P_{i} \tilde{x} (t), \\ V_{i 2} (t) & = \int_{t - τ_{M}}^{t} e (\cdot) {\tilde{x}}^{T} (s) Q_{i} \tilde{x} (s) d s, \\ V_{i 3} (t) & = τ_{M} \int_{- τ_{M}}^{0} \int_{t + v}^{t} e (\cdot) {\overset{\cdot}{\tilde{x}}}^{T} (s) R_{i} \overset{\cdot}{\tilde{x}} (s) d s d v, \\ V_{i 4} (t) & = η (t), \end{matrix}

where $e (\cdot) = e^{2 {(- 1)}^{i} β_{i} (t - μ)}$ .

If $i = 1$ , differentiating $V_{i} (t)$ , we have:

\begin{matrix} L V_{11} = 2 {\tilde{x}}^{T} P_{1} [A_{1} \tilde{x} + B_{4} θ (t) + (1 - \bar{α}) B_{1} \tilde{x} (t - τ (t)) \\ + (1 - \bar{α}) B_{3} e (t) + \bar{α} B_{2} F (y (t)), \end{matrix}

(33)

\begin{matrix} L V_{12} = {\tilde{x}}^{T} (t) Q_{1} \tilde{x} (t) - e^{- 2 β_{1} τ_{M}} {\tilde{x}}^{T} (t - τ_{M}) Q_{1} \tilde{x} (t - τ_{M}), \end{matrix}

(34)

\begin{matrix} L V_{13} = E {{\overset{\cdot}{\tilde{x}}}^{T} (t) τ_{M}^{2} R_{1} \overset{\cdot}{\tilde{x}} (t)} \\ - τ_{M} \int_{t - τ_{M}}^{t} {\overset{\cdot}{\tilde{x}}}^{T} (s) e^{- 2 β_{1} τ_{M}} R_{1} \overset{\cdot}{\tilde{x}} (s) ds, \end{matrix}

(35)

\begin{matrix} L V_{14} = \overset{\cdot}{η} (t) = - kη (t) - ρ G (t) \\ \leq - (kλ + ρ) G (t) \\ = - (kλ + ρ) e {(t)}^{T} Φ e (t) \\ + (kλ + ρ) [\sum_{p = 1}^{ι} v_{p} (t) ϵ_{p} (t) {[(y (t - τ (t)) - e (t))]}^{T} \end{matrix}

(36)

\begin{matrix} Φ [(y (t - τ (t)) - e (t))]] . \end{matrix}

(37)

where

\begin{matrix} ξ (t) = {[\begin{matrix} {\tilde{x}}^{T} (t) & {\tilde{x}}^{T} (t - τ_{M}) & {\tilde{x}}^{T} (t - τ (t)) & e^{T} (t) & F^{T} (y (t)) \end{matrix}]}^{T} . \end{matrix}

The first term on the right side of equation (35) can be further expressed as:

\begin{matrix} E {{\overset{\cdot}{\tilde{x}}}^{T} (t) τ_{M}^{2} R_{1} \overset{\cdot}{\tilde{x}} (t)} = τ_{M}^{2} [\begin{matrix} ξ (t) \\ θ (t) \end{matrix}]^{T} [\begin{matrix} Ā_{1}^{T} \\ B_{4}^{T} \end{matrix}] R_{1} [\begin{matrix} Ā_{1}^{T} \\ B_{4}^{T} \end{matrix}]^{T} [\begin{matrix} ξ (t) \\ θ (t) \end{matrix}] \\ + {\hat{α}}^{2} [\begin{matrix} ξ (t) \\ θ (t) \end{matrix}]^{T} [\begin{matrix} {\bar{B}}_{1}^{T} \\ 0 \end{matrix}] R_{1} [\begin{matrix} {\bar{B}}_{1}^{T} \\ 0 \end{matrix}]^{T} [\begin{matrix} ξ (t) \\ θ (t) \end{matrix}], \end{matrix}

where

\begin{matrix} Ā_{1} = [\begin{matrix} A_{1} & 0 & (1 - \bar{α}) B_{1} & (1 - \bar{α}) B_{3} & \bar{α} B_{2} \end{matrix}], \\ {\bar{B}}_{1} = [\begin{matrix} 0 & 0 & B_{1} & B_{3} & B_{2} \end{matrix}] . \end{matrix}

According to Lemma 1, the second term on the right side of equation (35) results in:

\begin{matrix} - τ_{M} \int_{t - τ_{M}}^{t} {\overset{\cdot}{\tilde{x}}}^{T} (s) e^{- 2 β_{1} τ_{M}} R_{1} \overset{\cdot}{\tilde{x}} (s) ds \\ \leq [\begin{matrix} \tilde{x} (t) \\ \tilde{x} (t - τ (t)) \\ \tilde{x} (t - τ_{M}) \end{matrix}]^{T} [\begin{matrix} - {\bar{R}}_{1} & * & * \\ {\bar{R}}_{1} - M_{1} & - 2 {\bar{R}}_{1} + M_{1} + M_{1}^{T} & * \\ M_{1} & {\bar{R}}_{1} - M_{1} & - {\bar{R}}_{1} \end{matrix}] \\ [\begin{matrix} \tilde{x} (t) \\ \tilde{x} (t - τ (t)) \\ \tilde{x} (t - τ_{M}) \end{matrix}], \end{matrix}

where ${\bar{R}}_{1} = e^{- 2 β_{1} τ_{M}} R_{1}$ .

Combining equations (33)–(37), we have:

\begin{matrix} L V_{1} (t) + r_{e} {(t)}^{T} r_{e} (t) - γ^{2} θ {(t)}^{T} θ (t) \\ \leq - 2 β_{1} V_{1} (t) + 2 β_{1} V_{11} (t) + L V_{11} + L V_{12} + L V_{13} + L V_{14} \\ = - 2 β_{1} V_{1} (t) + 2 β_{1} V_{11} (t) + 2 \tilde{x} {(t)}^{T} P_{1} [A_{1} \tilde{x} (t) + B_{4} θ (t) \\ + (1 - \bar{α}) B_{1} \tilde{x} (t - τ (t)) + (1 - \bar{α}) B_{3} e (t) + \bar{α} B_{2} F (y (t)) \\ + {\tilde{x}}^{T} (t) Q_{1} \tilde{x} (t) - e^{- 2 β_{1} τ_{M}} {\tilde{x}}^{T} (t - τ_{M}) Q_{1} \tilde{x} (t - τ_{M}) \\ + τ_{M}^{2} {[\begin{matrix} ξ (t) \\ θ (t) \end{matrix}]}^{T} [\begin{matrix} Ā_{1}^{T} \\ B_{4}^{T} \end{matrix}] R_{1} {[\begin{matrix} Ā_{1}^{T} \\ B_{4}^{T} \end{matrix}]}^{T} [\begin{matrix} ξ (t) \\ θ (t) \end{matrix}] \\ + {\hat{α}}^{2} {[\begin{matrix} ξ (t) \\ θ (t) \end{matrix}]}^{T} [\begin{matrix} {\bar{B}}_{1}^{T} \\ 0 \end{matrix}] R_{1} {[\begin{matrix} {\bar{B}}_{1}^{T} \\ 0 \end{matrix}]}^{T} [\begin{matrix} ξ (t) \\ θ (t) \end{matrix}] \\ - {[\begin{matrix} \tilde{x} (t) \\ \tilde{x} (t - τ (t)) \\ \tilde{x} (t - τ_{M}) \end{matrix}]}^{T} [\begin{matrix} {\bar{R}}_{1} & * & * \\ {\bar{R}}_{1} - M_{1} & {\bar{R}}_{1} + M_{1} + M_{1}^{T} & * \\ M_{1} & {\bar{R}}_{1} - M_{1} & {\bar{R}}_{1} \end{matrix}] [\begin{matrix} \tilde{x} (t) \\ \tilde{x} (t - τ (t)) \\ \tilde{x} (t - τ_{M}) \end{matrix}] \\ - (kλ + ρ) e {(t)}^{T} Φ e (t) + (kλ + ρ) \\ [\sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) {[(y (t - τ (t)) - e (t))]}^{T} \\ Φ [(y (t - τ (t)) - e (t))]] \\ - {[F (y (t)) - O_{1} CĪ \tilde{x} (t)]}^{T} [F (y (t)) - O_{2} CĪ \tilde{x} (t)] \\ + r_{e} {(t)}^{T} r_{e} (t) - γ^{2} θ {(t)}^{T} θ (t) = - 2 β_{1} V_{1} (t) + \bar{ξ} {(t)}^{T} Ξ^{1} \bar{ξ} (t), \end{matrix}

(38)

where

\begin{matrix} \bar{ξ} = {[\begin{matrix} {\tilde{x}}^{T} (t) & {\tilde{x}}^{T} (t - τ_{M}) & {\tilde{x}}^{T} (t - τ (t)) & e^{T} (t) & F^{T} (y (t)) & θ^{T} (t) \end{matrix}]}^{T} . \end{matrix}

If $i = 2$ , by the same method, we have:

\begin{matrix} L V_{2} (t) + r_{e} {(t)}^{T} r_{e} (t) - γ^{2} θ {(t)}^{T} θ (t) \\ \leq 2 β_{1} V_{2} (t) - 2 β_{2} V_{21} (t) + L V_{21} + L V_{22} + L V_{23} + L V_{24} \\ = 2 β_{1} V_{2} (t) - 2 β_{2} V_{21} (t) + 2 \tilde{x} P_{2} [A_{1} \tilde{x} + B_{4} θ (t)] \\ + {\tilde{x}}^{T} (t) Q_{2} \tilde{x} (t) - e^{2 β_{2} τ_{M}} {\tilde{x}}^{T} (t - τ_{M}) Q_{2} \tilde{x} (t - τ_{M}) \\ + τ_{M}^{2} {[\begin{matrix} ξ (t) \\ θ (t) \end{matrix}]}^{T} [\begin{matrix} Ā_{1} \\ B_{4} \end{matrix}] R_{2} {[\begin{matrix} Ā_{1} \\ B_{4} \end{matrix}]}^{T} [\begin{matrix} ξ (t) \\ θ (t) \end{matrix}] \\ + {[\begin{matrix} \tilde{x} (t) \\ \tilde{x} (t - τ (t)) \\ \tilde{x} (t - τ_{M}) \end{matrix}]}^{T} [\begin{matrix} - {\bar{R}}_{2} & * & * \\ {\bar{R}}_{2} - M_{2} & - 2 {\bar{R}}_{2} + M_{2} + M_{2}^{T} & * \\ M_{2} & {\bar{R}}_{2} - M_{2} & - {\bar{R}}_{2} \end{matrix}] \\ [\begin{matrix} \tilde{x} (t) \\ \tilde{x} (t - τ (t)) \\ \tilde{x} (t - τ_{M}) \end{matrix}] \\ - (kλ + ρ) e {(t)}^{T} Φ e (t) + (kλ + ρ) \\ [\sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) {[(y (t - τ (t)) - e (t))]}^{T} Φ [(y (t - τ (t)) - e (t))]] \\ - {[F (y (t)) - O_{1} C \tilde{x} (t)]}^{T} [F (y (t)) - O_{2} C \tilde{x} (t)] + r_{e} {(t)}^{T} r_{e} (t) \\ - γ^{2} θ {(t)}^{T} θ (t) = 2 β_{2} V_{2} (t) + \bar{ξ} {(t)}^{T} Ξ^{2} \bar{ξ} (t) . \end{matrix}

(39)

Utilizing the Schur complement, we have $Ξ^{i} < 0$ . Then, we further rewrite equations (38) and (39) as:

\begin{matrix} {\begin{matrix} E {{\overset{\cdot}{V}}_{1} (t)} \leq - 2 β_{1} V_{1} (t) \\ E {{\overset{\cdot}{V}}_{2} (t)} \leq 2 β_{2} V_{2} (t) . \end{matrix} \end{matrix}

(40)

Next, according to i equals two cases described by different values, we have:

\begin{matrix} {\begin{matrix} E {V_{1} (t)} \leq e^{- 2 β_{1} (t - j_{n})} E {V_{1} (j_{n})}, \\ E {V_{2} (t)} \leq e^{2 β_{2} (t - l_{n})} E {V_{2} (l_{n})}, \end{matrix} \end{matrix}

(41)

where $j_{n} = T_{n}, l_{n} = T_{n} + T_{off} (n)$ .

If there exist constants $J_{i} > 0$ such that $P_{1} \leq J_{2} P_{2}$ , $P_{2} \leq J_{1} e^{2 (β_{1} + β_{2}) h}$ holds, we get:

\begin{matrix} {\begin{matrix} E {V_{1} (j_{n})} \leq J_{2} E {V_{2} (j_{n}^{-})}, \\ E {V_{2} (l_{n})} \leq J_{1} e^{2 (β_{1} + β_{2})} E {V_{1} (l_{n}^{-})} . \end{matrix} \end{matrix}

(42)

For $t \in T_{1, n}$ , combined equations (41) and (42) as described above, we have:

\begin{matrix} E {V_{1} (t)} & \leq J_{2} e^{- 2 β_{1} (t - j_{n})} E {V_{2} (j_{n}^{-})} \\ \leq J_{2} e^{- 2 β_{1} (t - j_{n})} e^{2 β_{2} (j_{n} - d_{n - 1})} E {V_{2} (l_{n - 1})} \\ \leq \cdot \cdot \leq e^{q_{1} (t)} E {V_{1} (0)}, \end{matrix}

(43)

where $q_{1} (t) = (u + \frac{t}{m}) (2 (β_{1} + β_{2}) h + 2 β_{2} T_{on}^{\max} - 2 β_{1} T_{off}^{\min} +$ $\ln (J_{1} J_{2}))$ . And then:

\begin{matrix} E {V_{1} (t)} \leq e^{q_{1}} e^{- nt} E {V_{1} (0)}, \end{matrix}

(44)

where

\begin{matrix} q_{1} & = u (2 (β_{1} + β_{2}) h + 2 β_{2} T_{on}^{\max} - 2 β_{1} T_{off}^{\min} + \ln (J_{1} J_{2})), \\ n & = \frac{β_{1} T_{off}^{\min} - (β_{1} + β_{2}) h - β_{2} T_{on}^{\max} - 0.5 \ln (J_{1} J_{2})}{m} . \end{matrix}

For $t \in T_{2, n}$ , applying the aforementioned methods, we obtain:

\begin{matrix} E {V_{2} (t)} \leq \frac{1}{J_{2}} e^{q_{2}} e^{- nt} E {V_{1} (0)}, \end{matrix}

(45)

where $q_{2} = (u + 1) (2 (β_{1} + β_{2}) h + 2 β_{2} T_{on}^{\max} - 2 β_{1} T_{off}^{\min} + \ln$ $(J_{1} J_{2}))$ .

Defining $U = \max {e^{q_{1}}, \frac{1}{J_{2}} e^{q_{2}}}, one obtains$ :

\begin{matrix} E {V (t)} \leq U e^{- nt} E {V_{1} (0)} . \end{matrix}

(46)

By equation (35), it follows that:

\begin{matrix} {\begin{matrix} E {V (t)} \geq μ_{1} ∥ \tilde{x} (t) ∥^{2}, \\ E {V_{1} (0)} \leq μ_{2} ∥ \bar{ζ} ∥_{h}^{2} . \end{matrix} \end{matrix}

(47)

where $μ_{1} = \min {J_{\min} (P_{i})}, μ_{2} = \max {J_{\max} (P_{i}) + h J_{\max} (Q_{i})$ $+ \frac{h^{2}}{2} J_{\max} (R_{i})}$ .

Finally, combining equations (46) and (47), we have:

\begin{matrix} ∥ \tilde{x} (t) ∥ \leq \sqrt{\frac{U μ_{2}}{μ_{1}}} e^{- \frac{n}{2} t} ∥ \bar{ζ} ∥_{h}^{2} . \end{matrix}

(48)

In conclusion, it means that the filtering error system is exponentially mean-square stable.

Theorem 1 gives the sufficient conditions of the mean-square exponential stability for the USV system with $H_{\infty}$ performance. However, it relates to a quantized uncertain parameter $Δ_{Z}$ . Consequently, in Theorem 2, we will eliminate the uncertain term using the approach in Lemma 2, resulting in an inequality equivalent. □

Theorem 2. For given attack parameters $T_{off}^{\min}$ , $T_{on}^{\max}$ ,u,m,the sampling interval $h < T_{off}^{\min}$ , triggering parameters $ρ, λ, k, ℓ, v_{p},$ $ϵ_{p}$ and scalars $ϵ_{1}, ϵ_{2}, ϵ_{3}, ϵ_{4}, ϵ_{Z}, β_{i}$ , $J_{i}$ , time delay parameters $τ_{M}$ , the filtering error system (23) is exponentially mean-square stable with $H_{\infty}$ performance γ, if there exist $P_{i} > 0$ , $Q_{i} > 0$ , $R_{i} > 0, Φ > 0$ , and matrices $M_{i}$ with suitable dimensions are defined to ensure that the following inequalities are satisfied:

\begin{matrix} {\bar{Ξ}}^{i} < 0, \end{matrix}

(49)

\begin{matrix} P_{1} \leq J_{2} P_{2}, \end{matrix}

(50)

\begin{matrix} P_{2} \leq J_{1} e^{2 (β_{1} + β_{2}) h} P_{1}, \end{matrix}

(51)

\begin{matrix} Q_{i} \leq J_{3 - i} Q_{(3 - i)}, \end{matrix}

(52)

\begin{matrix} R_{i} \leq J_{3 - i} R_{(3 - i)}, \end{matrix}

(53)

\begin{matrix} [\begin{matrix} R_{i} & * \\ M_{i} & R_{i} \end{matrix}] > 0, \\ \frac{2 β_{1} T_{off}^{\min} - 2 (β_{1} + β_{2}) h - 2 β_{2} T_{on}^{\max} - \ln (J_{1} J_{2})}{m} > 0, \end{matrix}

(54)

where

\begin{matrix} {\bar{Ξ}}^{1} & = [\begin{matrix} {\bar{Ξ}}_{1}^{1} & * \\ {\bar{Ξ}}_{2}^{1} & {\bar{Ξ}}_{3}^{1} \end{matrix}], \\ {\bar{Ξ}}_{1}^{1} & = [\begin{matrix} {\bar{Ξ}}_{11}^{1} & {\bar{Λ}}_{1}^{1} & {\bar{Λ}}_{2}^{1} & {\bar{Λ}}_{3}^{1} & {\bar{Λ}}_{4}^{1} \\ * & - 2 χ P_{1} + χ^{2} R_{1} & 0 & 0 & 0 \\ * & * & - 2 χ P_{1} + χ^{2} R_{1} & 0 & 0 \\ * & * & * & - I & 0 \\ * & * & * & * & - I \end{matrix}], \end{matrix}

\begin{matrix} {\bar{Ξ}}_{2}^{1} & = [\begin{matrix} (1 - \bar{α}) P_{1} {\bar{B}}_{F} & 0 & (1 - \bar{α}) P_{1} {\bar{B}}_{F} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \bar{C} & 0 & 0 & 0 & \bar{C} & 0 & 0 \\ 0 & 0 & 0 & - I & 0 & 0 & 0 & - I \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ (1 - \bar{α}) τ_{M} P_{1} {\bar{B}}_{F} & 0 & (1 - \bar{α}) τ_{M} P_{1} {\bar{B}}_{F} & 0 & 0 & 0 & 0 & 0 \\ \bar{α} τ_{M} P_{1} {\bar{B}}_{F} & 0 & \bar{α} τ_{M} P_{1} {\bar{B}}_{F} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & (1 - \bar{α}) D_{F} & 0 & (1 - \bar{α}) D_{F} & 0 \\ 0 & 0 & 0 & 0 & (1 - \bar{α}) D_{F} & 0 & (1 - \bar{α}) D_{F} & 0 \end{matrix}], \\ {\bar{Ξ}}_{3}^{1} & = diag {- \frac{ϵ_{1}}{ϵ_{Z}^{2}} I, - \frac{1}{ϵ_{1}} I, - \frac{ϵ_{2}}{Z^{2}} I, - \frac{1}{ϵ_{2}} I, - \frac{ϵ_{3}}{Z^{2}} I, - \frac{1}{ϵ_{3}} I, - \frac{ϵ_{4}}{Z^{2}} I, - \frac{1}{ϵ_{4}} I}, \end{matrix}

\begin{matrix} {\bar{Ξ}}_{11}^{1} & = [\begin{matrix} {\bar{γ}}_{11}^{1} & {\bar{γ}}_{12}^{1} & {\bar{γ}}_{13}^{1} & {\bar{γ}}_{14}^{1} & {\bar{γ}}_{15}^{1} & {\bar{γ}}_{16}^{1} \\ * & {\bar{γ}}_{22}^{1} & 0 & 0 & 0 & 0 \\ * & * & {\bar{γ}}_{33}^{1} & {\bar{γ}}_{34}^{1} & 0 & 0 \\ * & * & * & {\bar{γ}}_{44}^{1} & 0 & 0 \\ * & * & * & * & - I & 0 \\ * & * & * & * & * & - γ^{2} I \end{matrix}], \end{matrix}

\begin{matrix} {\bar{Λ}}_{1}^{1} & = [\begin{matrix} τ_{M} P_{1} A_{1} & 0 & τ_{M} (1 - \bar{α}) P_{1} {\bar{B}}_{1} & τ_{M} (1 - \bar{α}) P_{1} {\bar{B}}_{3} & τ_{M} \bar{α} P_{1} B_{2} & τ_{M} P_{1} B_{4} \end{matrix}], \\ {\bar{Λ}}_{2}^{1} & = [\begin{matrix} 0 & 0 & τ_{M} \hat{α} P_{1} {\bar{B}}_{1} & τ_{M} \hat{α} P_{1} {\bar{B}}_{3} & - τ_{M} \hat{α} P_{1} B_{2} & 0 \end{matrix}], \\ {\bar{Λ}}_{3}^{1} & = [\begin{matrix} C_{1} & 0 & (1 - \bar{α}) {\bar{C}}_{2} & - (1 - \bar{α}) D_{2} & \bar{α} D_{F} & D_{1} \end{matrix}], \\ {\bar{Λ}}_{4}^{1} & = [\begin{matrix} 0 & 0 & - \hat{α} {\bar{C}}_{2} & \hat{α} D_{2} & \hat{α} D_{F} & 0 \end{matrix}], \\ {\bar{γ}}_{11}^{1} & = 2 β_{1} P_{1} + P_{1} A_{1} + A_{1}^{T} P_{1} + Q_{1} - {\bar{R}}_{1} - Ī^{T} C^{T} O_{1}^{T} O_{2} CĪ, \\ {\bar{γ}}_{12}^{1} & = M_{1}, {\bar{γ}}_{13}^{1} = (1 - \bar{α}) P_{1} {\bar{B}}_{1} + {\bar{R}}_{1} - M_{1}, \\ {\bar{γ}}_{14}^{1} & = (1 - \bar{α}) P_{1} {\bar{B}}_{3}, {\bar{γ}}_{15}^{1} = \bar{α} P_{1} B_{2} + \sqrt{O_{1}^{T} + O_{2}} Ī^{T} C^{T} \\ {\bar{γ}}_{16}^{1} & = P_{1} B_{4}, {\bar{γ}}_{22}^{1} = - e^{- 2 β_{1} τ_{M}} Q_{1} - {\bar{R}}_{1}, {\bar{γ}}_{23}^{1} = {\bar{R}}_{1} - M_{1}, \\ {\bar{γ}}_{33}^{1} & = (kλ + ρ) \sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) Ī^{T} C^{T} Φ CĪ - 2 {\bar{R}}_{1} + M_{1} + M_{1}^{T}, \end{matrix}

\begin{matrix} {\bar{γ}}_{34}^{1} & = (kλ + ρ) \sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) Ī^{T} C^{T} Φ, \\ {\bar{γ}}_{44}^{1} & = (kλ + ρ) \sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) Φ - (kλ + ρ) Φ \\ {\bar{Ξ}}^{2} = [\begin{matrix} {\bar{Ξ}}_{11}^{2} & {\bar{Λ}}_{1}^{2} & {\bar{Λ}}_{2}^{2} \\ * & - 2 χ P_{2} + χ^{2} R_{2} & 0 \\ * & * & - I \end{matrix}], \\ {\bar{Λ}}_{1}^{2} = [\begin{matrix} τ_{M} P_{2} A_{1} & 0 & 0 & 0 & 0 & τ_{M} P_{2} B_{4} \end{matrix}], \\ {\bar{Λ}}_{2}^{2} = [\begin{matrix} C_{1} & 0 & 0 & 0 & 0 & D_{1} \end{matrix}], \end{matrix}

\begin{matrix} {\bar{γ}}_{12}^{2} = M_{2}, {\bar{γ}}_{13}^{2} = {\bar{R}}_{2} - M_{2}, {\bar{γ}}_{15}^{2} = \sqrt{O_{1}^{T} + O_{2}} Ī^{T} C^{T}, {\bar{γ}}_{16}^{2} = P_{2} B_{4}, \\ {\bar{γ}}_{22}^{2} = - e^{2 β_{2} τ_{M}} Q_{2} - {\bar{R}}_{2}, {\bar{γ}}_{23}^{2} = {\bar{R}}_{2} - M_{2}, \\ {\bar{γ}}_{33}^{2} = (kλ + ρ) \sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) Ī^{T} C^{T} Φ CĪ - 2 {\bar{R}}_{2} + M_{2} + M_{2}^{T}, \\ {\bar{γ}}_{34}^{2} = (kλ + ρ) \sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) Ī^{T} C^{T} Φ, {\bar{γ}}_{44}^{2} \\ = (kλ + ρ) \sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) Φ - (kλ + ρ), \\ {\bar{B}}_{1} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & B_{F} C & 0 \end{matrix}], {\bar{B}}_{3} = [\begin{matrix} 0 \\ 0 \\ - B_{F} \end{matrix}], {\bar{B}}_{F} = [\begin{matrix} 0 \\ 0 \\ B_{F} \end{matrix}], \\ \bar{C} = [\begin{matrix} 0 & C & 0 \end{matrix}], \\ {\bar{C}}_{2} = [\begin{matrix} 0 & D_{F} C & 0 \end{matrix}] . \end{matrix}

Other parameters are the same as Theorem 1.

Proof. From the description of $Ξ_{22}^{1}, Ξ_{33}^{1}, Ξ_{22}^{2}$ in Theorem 1 and based on Lemma 3, it can be concluded that:

\begin{matrix} {\bar{Ξ}}_{22}^{1} & = {\bar{Ξ}}_{33}^{1} \leq - 2 χ P_{1} + χ^{2} R_{1}, \end{matrix}

(55)

\begin{matrix} {\bar{Ξ}}_{22}^{2} & \leq - 2 χ P_{2} + χ^{2} R_{2} . \end{matrix}

(56)

By Lemma 2, there exist $ϵ_{1}, ϵ_{2}, ϵ_{3}, ϵ_{4}, Δ_{Z}^{2} \leq ϵ_{Z}^{2}$ such that:

\begin{matrix} {\bar{Ξ}}^{1} = {\bar{Ξ}}_{1}^{1} + K_{B 1} Δ_{Z} K_{C 1} + K_{C 1}^{T} Δ_{Z} K_{B 1}^{T} + K_{B 2} Δ_{Z} K_{C 2} \end{matrix}

(57)

\begin{matrix} + K_{C 2}^{T} Δ_{Z} K_{B 2}^{T} + K_{B 3} Δ_{Z} K_{C 3} \\ + K_{C 3}^{T} Δ_{Z} K_{B 3}^{T} + K_{B 4} Δ_{Z} K_{C 4} + K_{C 4}^{T} Δ_{Z} K_{B 4}^{T} \\ \leq {\bar{Ξ}}_{1}^{1} + ϵ_{1}^{- 1} K_{B 1} ϵ_{Z}^{2} K_{B 1}^{T} + ϵ_{1} K_{C 1} K_{C 1}^{T} + ϵ_{2}^{- 1} K_{B 2} ϵ_{Z}^{2} K_{B 2}^{T} \\ + ϵ_{2} K_{C 2} K_{C 2}^{T} + ϵ_{3}^{- 1} K_{B 3} ϵ_{Z}^{2} K_{B 3}^{T} + ϵ_{3} K_{C 3} K_{C 3}^{T} + ϵ_{4}^{- 1} K_{B 4} ϵ_{Z}^{2} K_{B 4}^{T} \\ + ϵ_{4} K_{C 4} K_{C 4}^{T}, \end{matrix}

(58)

where

\begin{matrix} K_{B 1} = K_{B 2} = [{\begin{matrix} (1 - \bar{α}) {\bar{B}}_{F}^{T} P_{1} & 0 & 0 & 0 & 0 & 0 & (1 - \bar{α}) τ_{M} {\bar{B}}_{F}^{T} P_{1} & \bar{α} τ_{M} {\bar{B}}_{F}^{T} P_{1} & 0 & 0 \end{matrix}}^{T}], \\ K_{C 1} = K_{C 3} = [\begin{matrix} 0 & 0 & \bar{C} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}], \\ K_{C 2} = K_{C 4} = [\begin{matrix} 0 & 0 & 0 & - I & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}], \\ K_{B 3} = K_{B 4} = [{\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & (1 - \bar{α}) D_{F}^{T} & - D_{F}^{T} \end{matrix}}^{T}] . \end{matrix}

This completes the proof. □

On the basis of the above theory, the gains of the filter are obtained by dividing the matrix P into blocks.

Theorem 3. For given attack parameters $T_{off}^{\min}$ , $T_{on}^{\max}$ ,u,m,the sampling interval $h < T_{off}^{\min}$ , triggering parameters $ρ, λ, k, v_{p}, ϵ_{p}$ and scalars $ϵ_{1}, ϵ_{2}, ϵ_{3}, ϵ_{4}, ϵ_{Z}, β_{i}$ , $J_{i}$ , time delay parameters $τ_{M}$ , the filtering error system (23) is exponentially mean-square stable with $H_{\infty}$ performance γ, if there exist $P_{i} = [\begin{matrix} P_{i 1} & P_{i 2} \\ P_{i 2} & P_{i 2} \end{matrix}] > 0$ , $Q_{i} > 0$ , $R_{i} > 0, Φ > 0, Â_{Fi}, {\hat{B}}_{F}$ , and matrices $M_{i}$ with appropriate dimensions such that the following inequalities hold:

\begin{matrix} {\hat{Ξ}}^{i} < 0, \end{matrix}

(59)

\begin{matrix} P_{1} \leq J_{2} P_{2}, \end{matrix}

(60)

\begin{matrix} P_{2} \leq J_{1} e^{2 (β_{1} + β_{2}) h} P_{1}, \end{matrix}

(61)

\begin{matrix} Q_{i} \leq J_{3 - i} Q_{(3 - i)}, \end{matrix}

(62)

\begin{matrix} R_{i} \leq J_{3 - i} R_{(3 - i)}, \end{matrix}

(63)

\begin{matrix} [\begin{matrix} R_{i} & * \\ M_{i} & R_{i} \end{matrix}] > 0, \end{matrix}

(64)

\begin{matrix} \frac{2 β_{1} T_{off}^{\min} - 2 (β_{1} + β_{2}) h - 2 β_{2} T_{on}^{\max} - \ln (J_{1} J_{2})}{m} > 0, \end{matrix}

(65)

where

\begin{matrix} {\hat{Ξ}}^{1} = [\begin{matrix} {\hat{Ξ}}_{1}^{1} & * \\ {\hat{Ξ}}_{2}^{1} & {\hat{Ξ}}_{3}^{1} \end{matrix}], {\hat{Ξ}}_{1}^{1} = [\begin{matrix} {\hat{Ξ}}_{11}^{1} & {\hat{Λ}}_{1}^{1} & {\hat{Λ}}_{2}^{1} & {\hat{Λ}}_{3}^{1} & {\hat{Λ}}_{4}^{1} \\ * & {\hat{Ξ}}_{22}^{1} & 0 & 0 & 0 \\ * & * & {\hat{Ξ}}_{33}^{1} & 0 & 0 \\ * & * & * & - I & 0 \\ * & * & * & * & - I \end{matrix}], \end{matrix}

\begin{matrix} {\bar{Ξ}}_{2}^{1} = [\begin{matrix} (1 - \bar{α}) {\hat{B}}_{F} & 0 & (1 - \bar{α}) {\hat{B}}_{F} & 0 & 0 & 0 & 0 & 0 \\ (1 - \bar{α}) {\hat{B}}_{F} & 0 & (1 - \bar{α}) {\hat{B}}_{F} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \bar{C} & 0 & 0 & 0 & \bar{C} & 0 & 0 \\ 0 & 0 & 0 & - I & 0 & 0 & 0 & - I \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ (1 - \bar{α}) τ_{M} {\hat{B}}_{F} & 0 & (1 - \bar{α}) τ_{M} {\hat{B}}_{F} & 0 & 0 & 0 & 0 & 0 \\ (1 - \bar{α}) τ_{M} {\hat{B}}_{F} & 0 & (1 - \bar{α}) τ_{M} {\hat{B}}_{F} & 0 & 0 & 0 & 0 & 0 \\ \bar{α} τ_{M} {\hat{B}}_{F} & 0 & \bar{α} τ_{M} {\hat{B}}_{F} & 0 & 0 & 0 & 0 & 0 \\ \bar{α} τ_{M} {\hat{B}}_{F} & 0 & \bar{α} τ_{M} {\hat{B}}_{F} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & (1 - \bar{α}) D_{F} & 0 & (1 - \bar{α}) D_{F} & 0 \\ 0 & 0 & 0 & 0 & (1 - \bar{α}) D_{F} & 0 & (1 - \bar{α}) D_{F} & 0 \end{matrix}], \end{matrix}

\begin{matrix} {\bar{Ξ}}_{3}^{1} = diag {- \frac{ϵ_{1}}{ϵ_{Z}^{2}} I, - \frac{1}{ϵ_{1}} I, - \frac{ϵ_{2}}{ϵ_{Z}^{2}} I, - \frac{1}{ϵ_{2}} I, - \frac{ϵ_{3}}{ϵ_{Z}^{2}} I, - \frac{1}{ϵ_{3}} I, - \frac{ϵ_{4}}{ϵ_{Z}^{2}} I, - \frac{1}{ϵ_{4}} I}, \\ {\hat{Ξ}}_{11}^{1} = [\begin{matrix} {\hat{γ}}_{11}^{1} & {\hat{γ}}_{12}^{1} & {\hat{γ}}_{13}^{1} & {\hat{γ}}_{14}^{1} & {\hat{γ}}_{15}^{1} & {\hat{γ}}_{16}^{1} \\ * & {\hat{γ}}_{22}^{1} & 0 & 0 & 0 & 0 \\ * & * & {\hat{γ}}_{33}^{1} & {\hat{γ}}_{34}^{1} & 0 & 0 \\ * & * & * & {\hat{γ}}_{44}^{1} & 0 & 0 \\ * & * & * & * & - I & 0 \\ * & * & * & * & * & - γ^{2} I \end{matrix}], {\hat{Ξ}}_{22}^{1} = {\hat{Ξ}}_{33}^{1} = - 2 χ [\begin{matrix} P_{11} & P_{12} \\ P_{12} & P_{12} \end{matrix}] + χ^{2} R_{1}, \end{matrix}

\begin{matrix} {\hat{Λ}}_{1}^{1} = τ_{M} [\begin{matrix} P_{11} Â & Â_{F 1} & 0 & 0 & (1 - \bar{α}) {\hat{B}}_{F} CĪ & 0 & (1 - \bar{α}) {\hat{B}}_{F} & \bar{α} {\hat{B}}_{F} & P_{11} {\bar{B}}_{ω} & P_{11} {\bar{B}}_{f} \\ P_{12} Â & Â_{F 1} & 0 & 0 & (1 - \bar{α}) {\hat{B}}_{F} CĪ & 0 & - (1 - \bar{α}) {\hat{B}}_{F} & \bar{α} {\hat{B}}_{F} & P_{12} {\bar{B}}_{ω} & P_{12} {\bar{B}}_{f} \end{matrix}], \\ {\hat{Λ}}_{2}^{1} = τ_{M} [\begin{matrix} 0 & 0 & 0 & 0 & \hat{α} {\hat{B}}_{F} CĪ & 0 & \hat{α} {\hat{B}}_{F} & - \hat{α} {\hat{B}}_{F} & 0 & 0 \\ 0 & 0 & 0 & 0 & \hat{α} {\hat{B}}_{F} CĪ & 0 & \hat{α} {\hat{B}}_{F} & - \hat{α} {\hat{B}}_{F} & 0 & 0 \end{matrix}], \\ {\hat{Λ}}_{3}^{1} = [\begin{matrix} C_{1} & 0 & (1 - \bar{α}) {\bar{C}}_{2} & - (1 - \bar{α}) D_{2} & \bar{α} D_{F} & D_{1} \end{matrix}], \\ {\hat{Λ}}_{4}^{1} = [\begin{matrix} 0 & 0 & - \hat{α} {\bar{C}}_{2} & \hat{α} D_{2} & \hat{α} D_{F} & 0 \end{matrix}], \\ {\hat{γ}}_{11}^{1} = 2 β_{1} [\begin{matrix} P_{11} & P_{12} \\ P_{12} & P_{12} \end{matrix}] + [\begin{matrix} P_{11} Â & Â_{F 1} \\ P_{12} Â & Â_{F 1} \end{matrix}] + [\begin{matrix} Â^{T} P_{11} & Â^{T} P_{12} \\ Â_{F 1}^{T} & Â_{F 1}^{T} \end{matrix}] \\ + Q_{1} - {\bar{R}}_{1} - Ī^{T} C^{T} O_{1}^{T} O_{2} CĪ, \\ {\hat{γ}}_{12}^{1} = M_{1}, {\hat{γ}}_{13}^{1} = (1 - \bar{α}) [\begin{matrix} {\hat{B}}_{F} CĪ & 0 \\ {\hat{B}}_{F} CĪ & 0 \end{matrix}] + {\bar{R}}_{1} - M_{1}, \\ {\hat{γ}}_{14}^{1} = [\begin{matrix} - (1 - \bar{α}) {\hat{B}}_{F} \\ - (1 - \bar{α}) {\hat{B}}_{F} \end{matrix}], \\ {\hat{γ}}_{15}^{1} = [\begin{matrix} \bar{α} {\hat{B}}_{F} \\ \bar{α} {\hat{B}}_{F} \end{matrix}] + \sqrt{O_{1}^{T} + O_{2}} Ī^{T} C^{T}, {\hat{γ}}_{16}^{1} = [\begin{matrix} P_{11} {\bar{B}}_{ω} & P_{11} {\bar{B}}_{f} \\ P_{12} {\bar{B}}_{ω} & P_{12} {\bar{B}}_{f} \end{matrix}], \\ {\hat{γ}}_{22}^{1} = - e^{- 2 β_{1} τ_{M}} Q_{1} - {\bar{R}}_{1}, \\ {\hat{γ}}_{23}^{1} = {\bar{R}}_{1} - M_{1}, {\hat{γ}}_{33}^{1} = (kλ + ρ) \\ \sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) Ī^{T} C^{T} Φ CĪ - 2 {\bar{R}}_{1} + M_{1} + M_{1}^{T}, \\ {\hat{γ}}_{34}^{1} = (kλ + ρ) \sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) Ī^{T} C^{T} Φ, \\ {\hat{γ}}_{44}^{1} = (kλ + ρ) \sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) Φ - (kλ + ρ) Φ, \end{matrix}

\begin{matrix} {\bar{Ξ}}^{2} = [\begin{matrix} {\bar{Ξ}}_{11}^{2} & {\bar{Λ}}_{1}^{2} & {\bar{Λ}}_{2}^{2} \\ * & {\hat{Ξ}}_{22}^{2} & 0 \\ * & * & - I \end{matrix}], \\ {\bar{Ξ}}_{11}^{2} = [\begin{matrix} {\bar{γ}}_{11}^{2} & {\bar{γ}}_{12}^{2} & 0 & 0 & {\bar{γ}}_{15}^{2} & {\bar{γ}}_{16}^{2} \\ * & {\bar{γ}}_{22}^{2} & 0 & 0 & 0 & 0 \\ * & * & {\bar{γ}}_{33}^{2} & {\bar{γ}}_{34}^{2} & 0 & 0 \\ * & * & * & {\bar{γ}}_{44}^{2} & 0 & 0 \\ * & * & * & * & - I & 0 \\ * & * & * & * & * & - γ^{2} I \end{matrix}], \\ {\hat{Ξ}}_{22}^{2} = - 2 χ [\begin{matrix} P_{21} & P_{22} \\ P_{22} & P_{22} \end{matrix}] + χ^{2} R_{2}, \\ {\hat{Λ}}_{1}^{2} = τ_{M} [\begin{matrix} P_{21} Â & Â_{F 2} & 0 & 0 & 0 & 0 & 0 & 0 & P_{21} {\bar{B}}_{ω} & P_{21} {\bar{B}}_{f} \\ P_{22} Â & Â_{F 2} & 0 & 0 & 0 & 0 & 0 & 0 & P_{22} {\bar{B}}_{ω} & P_{22} {\bar{B}}_{f} \end{matrix}], \\ {\hat{Λ}}_{2}^{2} = [\begin{matrix} C_{1} & 0 & 0 & 0 & 0 & D_{1} \end{matrix}], \\ {\hat{γ}}_{11}^{2} = - 2 β_{2} [\begin{matrix} P_{21} & P_{22} \\ P_{22} & P_{22} \end{matrix}] + [\begin{matrix} P_{21} Â & Â_{F 2} \\ P_{22} Â & Â_{F 2} \end{matrix}] + [\begin{matrix} Â^{T} P_{21} & Â^{T} P_{22} \\ Â_{F 2}^{T} & Â_{F 2}^{T} \end{matrix}] \\ + Q_{2} - {\bar{R}}_{2} - Ī^{T} C^{T} O_{1}^{T} O_{2} CĪ \\ {\hat{γ}}_{12}^{2} = M_{2}, {\hat{γ}}_{13}^{2} = {\bar{R}}_{2} - M_{2}, {\hat{γ}}_{15}^{2} = \sqrt{O_{1}^{T} + O_{2}} Ī^{T} C^{T}, \\ {\hat{γ}}_{16}^{2} = [\begin{matrix} P_{21} {\bar{B}}_{ω} & P_{21} {\bar{B}}_{f} \\ P_{22} {\bar{B}}_{ω} & P_{22} {\bar{B}}_{f} \end{matrix}], \\ {\hat{γ}}_{22}^{2} = - e^{2 β_{2} τ_{M}} Q_{2} - {\bar{R}}_{2}, {\hat{γ}}_{23}^{2} = {\bar{R}}_{2} - M_{2}, \\ {\hat{γ}}_{33}^{2} = (kλ + ρ) \sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) Ī^{T} C^{T} Φ CĪ - 2 {\bar{R}}_{2} + M_{2} + M_{2}^{T}, \\ {\hat{γ}}_{34}^{2} = (kλ + ρ) \sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) Ī^{T} C^{T} Φ, {\hat{γ}}_{44}^{2} = (kλ + ρ) \\ \sum_{p = 1}^{ℓ} v_{p} (t) ϵ_{p} (t) Φ - (kλ + ρ) . \end{matrix}

Proof. Define $P_{i} = [\begin{matrix} P_{i 1} & P_{i 2} \\ P_{i 2} & P_{i 2} \end{matrix}], Â = [\begin{matrix} A_{H} & 0 \\ 0 & A \end{matrix}]$ , by bring them into equations (49), we get:

\begin{matrix} P_{1} A_{1} = [\begin{matrix} P_{11} Â & P_{12} A_{F} \\ P_{12} Â & P_{12} A_{F} \end{matrix}], P_{1} {\bar{B}}_{1} = [\begin{matrix} P_{12} B_{F} CĪ & 0 \\ P_{12} B_{F} CĪ & 0 \end{matrix}], \\ P_{1} {\bar{B}}_{3} = [\begin{matrix} - (1 - \bar{α}) P_{12} B_{F} \\ - (1 - \bar{α}) P_{12} B_{F} \end{matrix}], \\ P_{1} B_{2} = [\begin{matrix} \bar{α} P_{12} B_{F} \\ \bar{α} P_{12} B_{F} \end{matrix}], P_{1} B_{4} = [\begin{matrix} P_{11} {\bar{B}}_{ω} & P_{11} {\bar{B}}_{f} \\ P_{12} {\bar{B}}_{ω} & P_{12} {\bar{B}}_{f} \end{matrix}], \\ P_{2} A_{1} = [\begin{matrix} P_{21} Â & P_{22} A_{F} \\ P_{22} Â & P_{22} A_{F} \end{matrix}] . \end{matrix}

And then, we define $Â_{Fi} = P_{i 2} A_{F}, {\hat{B}}_{F} = P_{12} B_{F}$ , so we can have:

\begin{matrix} A_{F} = P_{i 2}^{- 1} Â_{Fi}, B_{F} = P_{12}^{- 1} {\hat{B}}_{F}, C_{F} = C_{F}, D_{F} = D_{F} . \end{matrix}

(66)

This completes the proof. □

Remark 1. The proposed method relies on solving LMIs whose size is directly affected by the system dimension and the number of design variables. As these increase, the computational burden also grows. In particular, for high-dimensional USV systems or more complex triggering schemes, the size of the resulting LMIs may pose a significant challenge for standard convex optimization solvers, potentially resulting in long computation times or infeasibility.

Simulation results

In this section, the validity of the proposed method is demonstrated, and we have borrowed the actual parameters of USV from Wang and Han (2016):

\begin{matrix} K_{dp} = - 0.0202, K_{dr} = - 0.0103, K_{dv} = 0.0380, K_{vp} = 0.7980, \\ K_{vr} = - 0.4600, ω_{n} = 1.6300, ζ = 2.0840, T_{r} = 0.4211, T_{v} = 0.5263 . \end{matrix}

Borrowing the parameters from Wang and Han (2016), we can set $C = [\begin{matrix} 1 & 0.8 & 1 & - 1 & 0.6 \end{matrix}], B_{f} = {[\begin{matrix} 0.6 & - 1 & 2 & 0.8 & 1 \end{matrix}]}^{T}$ . The state feedback controller is pre-designed as $K = [\begin{matrix} 1.1118 & 3.4650 & 6.6327 & 1.5691 & 3.1662 \end{matrix}]$ . We have:

\begin{matrix} A = [\begin{matrix} - 1.8193 & 0.2502 & 0.4789 & - 0.1133 & 0.2286 \\ - 1.1197 & - 2.4595 & - 0.1622 & 0.0384 & - 0.0774 \\ 0 & 1 & 0 & 0 & 0 \\ 2.0602 & - 0.1860 & - 0.3560 & - 6.7096 & - 2.8268 \\ 0 & 0 & 0 & 1 & 0 \end{matrix}], \\ B_{ω} = [\begin{matrix} 0 & 0 \\ 2.3747 & 0 \\ 0 & 0 \\ 0 & 2.6569 \\ 0 & 0 \end{matrix}] . \end{matrix}

The parameters of the fault weighting system are selected as follows:

\begin{matrix} A_{H} = - 0.1, B_{H} = 0.25, C_{H} = 0.5, D_{H} = 0 . \end{matrix}

(67)

We suppose $ω_{ψ} (t)$ is $N (0, 1)$ , $ω_{ϕ} (t) = 0.5 \sin (0.3 t)$ . The initial conditions of the system and the filter are chosen as $x (0) = [\begin{matrix} 0.9781 & - 0.9781 & 0.2079 & - 0.2079 & 0 \end{matrix}]$ , $x_{F} (0) = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]$ , respectively. Contemplating the signal of fault as:

\begin{matrix} f (t) = {\begin{matrix} 1, t \in [6, 12] \\ 0, otherwise \end{matrix} \end{matrix}

(68)

We select the dynamic probabilistic event-triggered transmission protocol parameter $ρ = 0.3, k = 2, λ = 3, ℓ = 3, ε_{1} = 0.0115, ε_{2} = 0.02, ε_{3} = 0.025, v_{1} = v_{2} = 0.1, v_{3} = 0.8$ . Set thesystem parameters to $h = 0.1, \bar{α} = 0.1, O_{1} = 0.1, O_{2} = 0.2, J_{1} = J_{2} = 1.01, β_{1} = 0.085, β_{2} = 0.25$ , $τ_{M} = 0.1$ , quantization parameter $P = 0.8, ϵ_{Z} = 0.1111$ . The spoofing attack signal is represented as $F (y (t)) = - 0.2 \tanh (y (t)) + 0.2$ . And suppose that the DoS attack parameters $T_{off}^{\min} = 2.4, T_{on}^{\max} = 0.5, m = 2$ .

By solving the LMIs in Theorem 3, we have:

\begin{matrix} A_{F} = [\begin{matrix} 0.9059 & - 124.3752 & 17.6268 & 32.1438 & - 4.7281 & 25.7355 \\ 2.5354 & - 297.9292 & 40.3411 & 77.4703 & - 13.9718 & 60.1618 \\ - 2.5467 & 245.5719 & - 48.3860 & - 64.1502 & - 1.4067 & - 60.9909 \\ 5.8123 & - 672.0460 & 95.3334 & 172.9964 & - 29.8065 & 138.2266 \\ - 2.1253 & 241.2471 & - 36.0818 & - 62.3378 & 0.4228 & - 55.6803 \\ 2.5128 & - 300.1381 & 43.1949 & 77.4427 & - 11.1547 & 61.4406 \end{matrix}], \\ B_{F} = [\begin{matrix} - 0.5842 \\ - 1.2920 \\ 1.2339 \\ - 3.3167 \\ 0.9742 \\ - 1.5168 \end{matrix}], \\ C_{F} = [\begin{matrix} - 0.6176 & - 0.2239 & 0.8662 & 2.3573 & 0.3612 & - 3.4843 \end{matrix}], \\ D_{F} = 0.0888 . \end{matrix}

The information expressed in Figure 3 expresses that the state response of the system has been severely affected by a system failure, which can lead to irreparable consequences. Figure 4 represents spoofing attacks, which follow the Benoulli distribution. The triggering sequence can be clearly seen in Figure 5. Figure 6 shows the output signal $y (t)$ of the system and the input signal $ȳ (t)$ of the filter. The residual response is exhibited in Figure 7. The residual evaluation function is illustrated in Figure 8. Through 200 simulations, we can get the average threshold $J_{th} = 0.0987$ . Since the sampling period $h = 0.1$ seconds, the fault can be detected in 1.8 seconds after its occurrence since $J (7.5) = 0.0901 < J_{th} < J (7.6) = 0.0996$ .

Conclusion

In this paper, the fault detection problem for USV system under compound cyber attacks has been studied under the dynamic probability-based event-triggered mechanism. In order to further improve the system performance and reduce network channel congestion, a dynamic probability-based event-triggered mechanism has been proposed. Based on compound cyber attacks, the filtering error system is converted into a stochastic switching system. By establishing the Lyapunov functional and analyzing LMIs, the sufficient conditions of the exponentially mean-square stability with a given $H_{\infty}$ performance index have been proved. Finally, the effectiveness of the proposed fault detection method has been verified by simulation experiments. In the future, we will try to study the design of finite time elastic sliding mode control of the USV system under compound cyber attacks with actuator failure.

While the proposed fault detection method demonstrates robustness under compound cyber attacks, quantized signals, and probabilistic event-triggered communication, it also has certain limitations. First, the filter design depends on solving a set of LMIs, which may become computationally expensive for high-dimensional or large-scale systems. Second, the theoretical results rely on accurate knowledge of some attack-related parameters (e.g. probabilities, duration bounds), which may not always be fully available or measurable in real-world scenarios. Moreover, the current framework assumes fixed filter gains, which are not adaptively updated during runtime.

Looking ahead, several extensions are possible. Future work could explore adaptive or learning-based filter designs that can adjust to unknown or time-varying attack characteristics. The method could also be extended to multi-agent USV systems or heterogeneous autonomous platforms operating collaboratively under network constraints. Moreover, while this study—like most existing works—focuses primarily on communication-induced delays, another valuable direction would be to explicitly model and analyze the effects of inherent system delays arising from sensing, computation, or actuation processes in the USV dynamics. Incorporating such physical delays would further enhance the practical applicability of the approach in real-world settings. As for applications, the proposed strategy is particularly well suited to autonomous maritime systems involved in offshore surveillance, environmental monitoring, or long-duration missions in adversarial environments. It may also find relevance in broader cyber-physical systems such as industrial automation and smart sensor networks where communication is constrained and security threats are present.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partly supported by the National Natural Science Foundation of China under grants 62173174 and 62203248 and the Natural Science Foundation of Shandong Province under grants ZR2021MF087, ZR2022MF252, ZR2022MF297, and ZR2024MF081.

ORCID iDs

Yanqian Wang

Guangming Zhuang

Data availability statement

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study

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