Fixed-Time Event-Triggered Fault-Tolerant Formation Control for Autonomous Underwater Vehicle Swarms

Abstract

Autonomous Underwater Vehicle (AUV) swarms possess advantages such as efficiency, reliability, flexibility, and extensive coverage in underwater operations. However, their coordinated control is challenged by communication interruptions and actuator failures in complex marine environments. This paper proposes a fixed-time event-triggered fault-tolerant formation control method to address these challenges. First, the Prim algorithm and the Hungarian algorithm are employed to reconstruct the communication topology, mitigating AUV disconnections due to communication failures and ensuring formation stability. Second, a fixed-time extended state observer (ESO) is designed to estimate the lumped disturbance arising from model uncertainties, unknown ocean disturbances, and actuator failures. Finally, a performance function is introduced to reformulate error variables, and a fixed-time event-triggered formation control law is developed based on an auxiliary saturation system and an event-triggering mechanism. In addition, this paper demonstrates the stability of the entire closed-loop system, and no Zeno phenomenon will occur. Simulation experiments demonstrate the effectiveness and superiority of the proposed method in maintaining robust formation control of AUV systems under adverse conditions.

1. Introduction

Autonomous Underwater Vehicles (AUVs) are widely used in underwater operations due to their capabilities in detection, communication, intelligent decision-making, and control. However, AUVs rely on acoustic communication, which suffers from much lower bandwidth, higher latency, and more frequent interruptions due to the aquatic medium compared to terrestrial or aerial systems. In addition, the combined effect of actuator faults and strong, unpredictable ocean currents can rapidly escalate a single failure into a swarm-wide disruption. This critical interplay is a hallmark of the underwater environment. Furthermore, the absence of GPS and the inherent drift of underwater navigation make maintaining swarm cohesion especially challenging during faults [1]. Thus, achieving effective collaborative control of AUV swarms remains challenging due to factors such as communication interruptions, actuator failures, and the complexities of the underwater environment. Fault-tolerant control plays a critical role in enhancing the safety and accuracy of AUV operations, offering robust solutions to mitigate these challenges and ensure reliable swarm performance.

Recently, many studies on multi-vehicle cooperative formation have focused on the issue of communication interruptions under weak communication conditions. Yang et al. investigated the fault-tolerant formation maintenance problem for multi-UAV systems with a focus on minimal connected topology [2]. They proposed a topology design and reconstruction strategy to mitigate communication failures and analyzed system stability using switching system theory. Kamel et al. investigated the fault-tolerant cooperative control problem for multi-wheeled mobile vehicles in the event of severe actuator failures. They generated a new formation reconstruction when a vehicle in the formation experienced serious failure by solving the optimal allocation problem and proposed a hybrid method combining control parameterization, time discretization, and particle swarm optimization to solve the formation reconstruction issue [3]. Zúñiga and Mendoza proposed and demonstrated the effectiveness of Prim algorithm in solving the open vehicle routing problem (OVRP) [4]. In addressing leader failure issues in multi-AUV leader–follower formations, Juan Li et al. implemented minimum-cost formation reconfiguration using the Hungarian algorithm [5]. To address potential communication topology failures, Prim algorithm is commonly employed to efficiently reconstruct and optimize communication links. Meanwhile, the Hungarian algorithm is utilized to reassign target positions during formation reconfiguration following system failures. These methods collectively enhance the robustness and recovery capability of multi-robot systems.

While the aforementioned studies effectively addressed communication disruptions, they did not consider actuator failures in individual AUVs—an essential aspect of fault-tolerant control in AUV swarms. Tian et al. designed an adaptive law to estimate the actuator efficiency factor, enabling their fault-tolerant controller to effectively handle multiplicative actuator failures [6]. Additionally, methods that treat fault terms as unknown disturbance terms are also frequently employed. Li et al. achieved fault-tolerant control by designing an adaptive fixed-time integral sliding mode disturbance observer to estimate lumped disturbances that include fault terms [7,8]. These studies successfully solved fault-tolerant control for AUV actuator failures. However, in practical applications, the transient performance of the controller, particularly convergence rate and overshoot, can influence the overall performance of AUV fault-tolerant control.

Many studies have proposed methods for the transient performance of AUVs to enhance the system’s fault tolerance. Fischer et al. adopted a control strategy based on RISE (Robust Integral of the Sign of the Error) feedback technology to achieve asymptotic trajectory tracking for AUVs [9]. However, this work did not consider the transient performance of tracking errors and required the assumption that the upper bound of external disturbances is completely known, which greatly limits the applicability of the controller. Sun et al. employed preset performance control methods to constrain tracking errors within a given range [10], which effectively improved the transient performance of tracking errors, but only yielded results that ensured the system states are ultimately bounded. Moreover, prolonged peak outputs from AUV actuators can lead to failures, and input saturation is an essential consideration for fault-tolerant control. Yan et al. addressed distributed three-dimensional trajectory tracking for multiple AUVs with input saturation by employing hyperbolic tangent functions to mitigate actuator saturation risks [11]. Stability analysis based on the Lyapunov method demonstrated the convergence of estimation errors. Li et al. studied adaptive fixed-time fuzzy formation control for multi-AUV systems, considering time-varying tracking error constraints and asymmetric actuator saturation, employing smooth saturation functions and a new signal compensation mechanism to solve amplitude and rate-related input saturation issues [12]. Miao et al. explored the path tracking problem for multiple AUVs with uncertainties and input saturation, proposing an auxiliary dynamic system to compensate for input saturation [13]. Additionally, employing an event-triggered mechanism to reduce the communication frequency between controllers and actuators has also enhanced fault tolerance to some extent [14,15]. Chen et al. introduced a distributed event-triggered communication mechanism, allowing each AUV to communicate only when its own state is updated, thereby reducing communication frequency and improving stability [16]. These studies suggest that integrating prescribed performance control with auxiliary dynamic systems can effectively address error constraints and input saturation within a fixed time. Furthermore, incorporating event-triggered mechanisms to minimize resource consumption holds significant research value for improving the robustness and efficiency of AUV swarm control.

This paper proposes a fixed-time fault-tolerant control scheme for AUV formations that systematically addresses trajectory tracking under multiple practical constraints while maintaining computational feasibility. The proposed architecture achieves an optimal balance between performance and complexity through several key features: the integration of Prim and Hungarian algorithms provides effective solutions for topology reconstruction and position reassignment; the fixed-time extended state observer enables efficient disturbance estimation, while the performance function and auxiliary saturation system collectively handle error constraints while preserving stability. Furthermore, the event-triggered mechanism significantly reduces controller update frequency. These carefully integrated components ensure the scheme maintains practical implementability while achieving control objectives under communication interruptions, actuator failures, and system constraints.

(i)

A fault-tolerant control mechanism for multiple AUVs is proposed to address formation task failures caused by communication topology disruptions. The Prim algorithm is employed to reconstruct the communication topology after failures, while the Hungarian algorithm is used to solve the position assignment problem for formation transformation.

(ii)

To account for actuator failures within AUVs, a fixed-time extended state observer (ESO) is designed to accurately estimate velocity and observe lumped disturbances, including model uncertainties, unknown ocean disturbances, and actuator failures. This enhances the robustness against external and internal disturbances.

(iii)

Based on a performance function, the tracking error is reconstructed into error variables, forming a novel terminal sliding mode surface with these error variables. An auxiliary saturation system and event-triggered mechanism are integrated into the control design, leading to a fixed-time event-triggered formation control law and the stability of the closed-loop system is proved. This approach ensures compliance with error constraints, maintains system stability, and reduces the frequency of controller updates, thereby improving computational efficiency.

The specific research object model, along with some lemmas and assumptions, is presented in Section 2. A fixed-time fault-tolerant controller is designed in Section 3. Simulation results on multi-AUV are provided in Section 4. Finally, Section 5 provides the conclusion of this study.

2. Problem Formulation

2.1. Dynamic Models

The research object of this paper is a multi-AUV system consisting of one leader and N followers. The followers can obtain the state information of other AUVs through their own high-precision sensors. Based on this, the paper investigates the formation control problem of multi-AUV systems under the influence of actuator faults and communication topology faults. As shown in Figure 1, the dynamic model of the 6-DOF multi-AUV system is expressed as follows [17]:

$$\left\{\begin{array}{l}{\dot{\eta }}_i=J(\eta )v_i\\ M_i{\dot{v}}_i+C_i\left(v_i\right)v_i+D_i\left(v_i\right)v_i+g_i({\eta }_i)={\tau }_i+{\tau }_i^F+d_i(t)\end{array}\right.$$

(1)

where ${\eta }_i={[x_i,y_i,z_i,{\phi }_i,{\theta }_i,{\psi }_i]}^T$ denotes the position and attitude information of the AUV relative to the fixed frame, $v_i(t)={[u_i,v_i,w_i,p_i,q_i,r_i]}^T$ represents the linear and angular velocity in the motion frame, and $d_i(t)$ denotes the external disturbance.

In addition, $J(\eta )$ denotes the rotation matrix from the fixed frame to the motion frame, and it can be defined as follows:

$$J(\eta )=\left[\begin{array}{cc}R(\eta )& 0_{3\times 3}\\ 0_{3\times 3}& T(\eta )\end{array}\right]$$

(2)

$$R(\eta )=\left[\begin{array}{ccc}\cos \theta \cos \psi & \sin \phi \cdot \sin \theta \cdot \cos \psi -\cos \phi \cdot \sin \psi & \cos \phi \cdot \sin \theta \cdot \cos \psi +\sin \phi \cdot \sin \psi \\ \cos \theta \sin \psi & \sin \phi \cdot \sin \theta \cdot \sin \psi +\cos \phi \cdot \cos \psi & \cos \phi \cdot \sin \theta \cdot \sin \psi -\sin \phi \cdot \cos \psi \\ -\sin \theta & \sin \phi \cos \theta & \cos \phi \cos \theta \end{array}\right]$$

(3)

$$T(\eta )=\left[\begin{array}{ccc}1& \sin \phi \tan \theta & \cos \phi \tan \theta \\ 0& \cos \phi & -\sin \phi \\ 0& \sin \phi \sec \theta & \cos \phi \sec \theta \end{array}\right]$$

(4)

where $M_i$ is the inertial matrix, $C_i$ represents the matrix of centripetal and Coriolis, $D_i$ is the damping matrix composed of damping forces and moments and $g_i$ is the restoring forces and moments vector.

${\tau }_i^F$ denotes the impact caused by actuator faults, and its specific expression is given as follows:

$$\begin{gathered} {\tau }_i^F=B(t-T_f)((E-I)\mathrm{sat}({\tau }_i)+{\overline{\tau }}_i) \\ \mathrm{sat}({\tau }_i)=\left\{\begin{array}{c}\mathrm{sign}({\tau }_i){\tau }_{i\max },\left|{\tau }_i\right|\ge {\tau }_{i\max }\\ {\tau }_i,\left|{\tau }_i\right|<{\tau }_{i\max }\end{array},(i=1,2,3,4,5,6)\right. \end{gathered}$$

(5)

where ${\tau }_i={[{\tau }_{i1},{\tau }_{i2},{\tau }_{i3},{\tau }_{i4},{\tau }_{i5},{\tau }_{i6}]}^T$ denotes the control force and torque, and $\mathrm{sat}({\tau }_i)=\Delta {\tau }_i+{\tau }_i$ is saturation function, where $\Delta {\tau }_i$ represents the difference between the desired control output and the actual saturated output. ${\tau }_{i\max }$ represents the maximum force or torque output capability of each AUV’s actuator. To prevent the controller from sustaining peak output over extended periods and mitigate potential motor damage risks, it is imperative to implement a saturation function that incorporates ${\tau }_{i\max }$. $B$ denotes a constant matrix, $T_f$ denotes the time at which the fault occurs, and $I$ denotes the identity diagonal matrix. ${\overline{\tau }}_i={[{\overline{\tau }}_{i1},{\overline{\tau }}_{i2},{\overline{\tau }}_{i3},{\overline{\tau }}_{i4},{\overline{\tau }}_{i5},{\overline{\tau }}_{i6}]}^T$ represents faults. $B(t-T_f)=\mathrm{diag}(b_1(t-T_{f1}),b_2(t-T_{f2}),b_3(t-T_{f3}),b_4(t-T_{f4}),b_5(t-T_{f5}),b_6(t-T_{f6})]$ denotes the distribution of fault time. $E=\mathrm{diag}(e_{11},e_{12},e_{13},e_{14},e_{15},e_{16})$, $e_{ii}$ denotes the health status of actuator, and $0<e_{ii}<1$. The fault range distribution is expressed as follows:

$$b_i(t-T_{fi})=\left\{\begin{array}{cc}0& t<T_{fi}\\ 1-e^{-a_{fi}(t-T_{fi})}& t\ge T_{fi}\end{array}\right.$$

(6)

where $a_{fi}>0$ denotes the fault severity of the actuator.

2.2. Lemma and Assumption

Lemma 1 ([18] Fixed-time Stability).

If there exists a continuous radially unbounded function $V:{ℝ}^n\to {ℝ}_{+}\cup \left\{0\right\}$, such that:

(1)

$V(x)=0\iff x=0$;

(2)

Any solution $x_{\left(t\right)}$ satisfies the inequality: $V\dot{(}x)\le -r_1{V}^{\alpha }(x)-r_2{V}^{\beta }(x)$.

Then the system is globally fixed-time stable, and  $r_1$, $r_2$, $\alpha$, $\beta$ are constant. $0<\alpha <1$, $\beta >1$, and $T$ satisfies the inequality

$$T\le T_{\max }:={\displaystyle \frac{1}{r_1(1-\alpha )}}+{\displaystyle \frac{1}{r_2(\beta -1)}}$$

(7)

(3)

Any solution $x\left(t\right)$ satisfies the inequality:  $V\dot{(}x)\le -r_1{V}^{\alpha }(x)-r_2{V}^{\beta }(x)+\delta$

When  $\delta \in \left(0, \infty \right)$, $0<\theta <1$, the system is practically fixed-time stable, and  $T$ satisfies the inequality:

$$T\le T_{\max }:={\displaystyle \frac{1}{r_1\theta (1-\alpha )}}+{\displaystyle \frac{1}{r_2\theta (\beta -1)}}$$

(8)

Lemma 2  ([19] Young’s inequality).

For any vectors, the following inequality holds:

$$ab\le {\displaystyle \frac{k{a}^p}{p}}+{\displaystyle \frac{k^{-q/p}{b}^q}{q}}\le k{a}^p+k^{-q/p}{b}^q$$

(9)

where $a>0$, $b>0$, $p>1$, $q>1$, $k>1$ and ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$.

Specifically, when $p=q=2$, the above expression can be transformed into the following formula: $ab\le {\displaystyle \frac{k}{2}}{a}^2+{\displaystyle \frac{1}{2k}}{b}^2$.

Assumption 1.

The model fault term ${\tau }_i^F$ and disturbance $d_i(t)$ as well as their derivatives ${\dot{\tau }}_i^F$ and ${\dot{d}}_i(t)$ are all bounded. And ${\overline{\tau }}_i^F$, ${\overline{\tau }}_d^F$, $\overline{d}$, ${\overline{d}}_d$, all satisfy $‖{\tau }_i^F‖\le {\overline{\tau }}_i^F$, $‖{\dot{\tau }}_i^F‖\le {\overline{\tau }}_d^F$, $‖d_i(t)‖\le \overline{d}$, $‖{\dot{d}}_i(t)‖\le {\overline{d}}_d$.

Assumption 2.

The desired reference signal  ${\eta }_d$ is sufficiently smooth, and the derivative of  ${\eta }_d$ exists and is bounded.

Assumption 3.

There exists at least one directed path from the leader AUV to the follower AUV.

Assumption 4.

$G(x_{1i})\Delta {\tau }_i$ is bounded and satisfies $‖G(x_{1i})\Delta {\tau }_i‖\le \kappa$, where $\kappa$ is constant.

3. Fault-Tolerant Control of Multiple AUVs for Fixed-Time Formation

This section focuses on the design of a fixed-time fault-tolerant controller, with the overall framework illustrated in Figure 2, consisting of three modules, which will be introduced in three subsections. The first subsection completes the reconstruction of AUV formation under communication topology failures based on the Prim algorithm and the Hungarian algorithm. The second subsection designs a fixed-time ESO to estimate actuator failures, ocean current disturbances, and model uncertainties. The third subsection introduces a performance function and an auxiliary saturation system, reconstructs the error variables, and forms a novel terminal sliding mode surface using the error variables. Finally, this paper incorporates an event-triggered mechanism and proposes a fixed-time formation control law.

3.1. The Formation Reconfiguration Scheme Considering Communication Topology Failure

A fault-tolerant mechanism for multi-AUV systems is proposed to solve communication topology failure. The Prim algorithm is utilized to resolve the communication topology failure issue, while the Hungarian algorithm is employed to handle the position allocation problem during formation transformation after the multi-AUV formation is disrupted.

3.1.1. The Multi-AUV Communication Topology Reconstruction Method Based on the Prim Algorithm

When communication failures occur in an AUV system, the original communication topology can no longer maintain the formation structure. This section solved the problem of multi-AUV topology reconstruction based on the Prim algorithm. The Prim algorithm achieves a global optimal solution by iteratively finding local optimal solutions, calculating the edge subsets of all vertices in the graph, and ultimately generating a minimum spanning tree with the smallest total weight [20]. By utilizing the Prim algorithm, the reconstructed AUV formation minimizes communication costs, which are represented by a communication cost matrix. Algorithm 1 illustrates the fault-tolerant process of multi-AUV topology reconstruction based on the Prim algorithm:

Algorithm 1 The Multi-AUV Topology Reconstruction Process Based on Prim Algorithm

Input: $G=\left(V,E\right)$           $V$: the set of AUVs and E: the edges of graph           w: weight function for the edges of AUV communication distance           Output: Minimum spanning tree (T).                    T← ∅ Initialize the minimum spanning tree                    U ← {v0} Start from an arbitrary vertex v0                    while |U| < |V| do                            Select edge (u, v) ∈ E such that u ∈ U, v ∉ U and w(u, v) is minimum                            Add edge (u, v) to T                            Add vertex v to U                    end while            return T

The communication topology reconstruction of a multi-AUV system based on the Prim algorithm involves the following steps:

(1) Select the position of leader AUV as the vertex of the minimum spanning tree.

(2) Use the distances between multiple AUVs as the edge weights, identify all edges connected to the newly added vertex, and find the edge with the minimum weight to add it to the tree.

(3) Repeat step (2) until all AUV vertices are included, thereby constructing the minimum spanning tree.

3.1.2. The Formation Position Allocation Method Based on the Hungarian Algorithm

In the event of a communication topology failure in a multi-AUV system, the original formation can no longer perform its designated tasks. Under such circumstances, a formation transition is required. However, considering the limited energy carried by AUVs, the Hungarian algorithm is employed to achieve the optimal position allocation for the AUVs.

This study focuses on the problem of formation transition for follower AUVs after communication loss, assuming that the trajectory of the leader AUV is known. During the movement of multi-AUV formation, the distance transformation of follower AUVs relative to the leader AUV can be regarded as a transformation of relative distances. Therefore, in this section, the relative distances of follower AUVs before and after the two formation transitions are used to construct the coefficient matrix, and the assignment matrix for the follower formation transition is denoted as $E_{ij}$. The following diagram illustrates the formation transition position allocation process based on the Hungarian algorithm (Algorithm 2):

Algorithm 2 The Formation Transition Position Allocation Based on Hungarian Algorithm

Input: AUV matrix of size $n\times m$ Output: Optimal assignment matrix $E_{ij}$ Step 1: Subtract the row minimum from each row   for i from 1 to n: row_minimums = $E_{\min }^i$       for j from 1 to m: $E_{ij}-=E_{\min }^i$     for j from 1 to m: column_minimums = $E_{\min }^j$       for i from 1 to n: $E_{ij}-=E_{\min }^j$ Step 2: Cover all zeros with a minimum number of lines     while True://Find a zero in the matrix       for i from 1 to n:         for j from 1 to m:           if cost_matrix[i][j] = 0 and not cover_rows[i] = True and not           cover_cols[j] = True:             break         if zero is found: break       if all columns are covered: break Step 3: Create a new zero if necessary and find the smallest uncovered value       for i from 1 to n:         for j from 1 to m:           if not cover_rows[i] and not cover_cols[j]:             min_uncovered = min(min_uncovered, cost_matrix[i][j])       //Subtract the smallest uncovered value from all uncovered elements       for i from 1 to n:         for j from 1 to m:           if not cover_rows[i] and not cover_cols[j]: cost_matrix[i][j] −= min_uncovered           elif cover_rows[i] and cover_cols[j]: cost_matrix[i][j] + = min_uncovered Step 4: Construct the optimal assignment matrix $E_{ij}$     Back to step 2 until the number of circled zeros equals n     return $E_{ij}$

3.2. Fixed-Time Extended State Observer

This subsection estimates the lumped disturbance caused by model uncertainties, unknown ocean current, and actuator failures by fixed-time ESO, which is developed by integrating fixed-time control theory with observer theory.

The dynamic model of the AUV is reformulated as follows to facilitate the observer design:

$$\left\{\begin{array}{c}{x}_{1i}={\eta }_i\\ x_{2i}={\dot{\eta }}_i\end{array}\right.$$

(10)

where ${\eta }_i={[x_i,y_i,z_i,{\phi }_i,{\theta }_i,{\psi }_i]}^T$ denotes the position and attitude information of the AUV relative to the fixed frame, ${\dot{\eta }}_i=J(\eta )v_i$.

Substituting it into Equation (10), the following form can be obtained:

$$\left\{\begin{array}{c}{\dot{x}}_{1i}=x_{2i}\\ {\dot{x}}_{2i}=F(x_{1i},x_{2i})+G(x_{1i})\mathrm{sat}({\tau }_i)+D_i\end{array}\right.$$

(11)

where $F(x_{1i},x_{2i})=-M_{\eta i}^{-1}(C_{RB\eta i}{\dot{\eta }}_i+C_{A\eta i}{\dot{\eta }}_i+D_{\eta i}{\dot{\eta }}_i+g_{\eta i})$, $D_i=M_{\eta i}^{-1}({\tau }_i^F+d_i(t))$, $G(x_{1i})=M_{\eta i}^{-1}$.

Remark 1.

$D_i$ is continuously differentiable and bounded, including fault terms, model uncertainties, and ocean disturbances. According to Assumption 1, there exists a constant  $D_{\upsilon }$ such that $‖D_i‖\le D_n<D_{\upsilon }$,  $0<D_{\upsilon }<\infty$.

The fixed-time ESO can be designed as follows [21]:

$$\left\{\begin{array}{c}{\dot{\widehat{x}}}_{1i}=x_{2i}+c_1\mathrm{T}{\mathrm{sig}}^{{\alpha }_1}({\tilde{x}}_{1i})+c_1(1-\mathrm{T}){\mathrm{sig}}^{{\beta }_1}({\tilde{x}}_{1i})\\ \begin{array}{c}{\dot{\widehat{x}}}_{2i}=F(x_{1i},x_{2i})+G(x_{1i})\mathrm{sat}({\tau }_i)+{\widehat{D}}_i+c_2\mathrm{T}{\mathrm{sig}}^{{\alpha }_2}({\tilde{x}}_{1i})+c_2(1-\mathrm{T}){\mathrm{sig}}^{{\beta }_2}({\tilde{x}}_{1i})\\ {\dot{\widehat{D}}}_i=c_3\mathrm{T}{\mathrm{sig}}^{{\alpha }_3}({\tilde{x}}_{1i})+c_3(1-\mathrm{T}){\mathrm{sig}}^{{\beta }_3}({\tilde{x}}_{1i})+D_{\upsilon }\mathrm{sign}({\tilde{x}}_{1i})\end{array}\end{array}\right.$$

(12)

where ${\widehat{x}}_{1i}$ is the estimated value of $x_{1i}$, ${\tilde{x}}_{1i}=x_{1i}-{\widehat{x}}_{1i}$, $D_{\upsilon }$ are the observer gains, which satisfy $D_{\upsilon }>D_n$, ${\alpha }_1\in (1-\mathrm{{\rm Z}},1)$, $\mathrm{{\rm Z}}$ is a sufficiently small positive constant, and ${\alpha }_2=2{\alpha }_1-1$, ${\alpha }_3=3{\alpha }_1-2$. $\mathrm{sig}{(x)}^{\alpha }=|x{|}^{\alpha }\mathrm{sign}(x)$, where α > 0, while $\mathrm{sign}$ function returns values (−1, 0, or 1) solely based on the sign of the parameter. Similarly, ${\beta }_1\in (1,1+{\rm Z})$, ${\beta }_2=2{\beta }_1-1$, ${\beta }_3=3{\beta }_1-2$. The switching function $T:[0,\infty )\to \{0,1\}$ satisfies:

$$T=\left\{\begin{array}{c}0, \mathrm{if} t\le T_u\\ 1, \mathrm{if} t>T_u\end{array}\right.$$

(13)

where $T_u$ is the switching time, designed based on the observer gains. $c_1$, $c_2$, $c_3$ are constants to be designed. These constants are set to ensure that the coefficient matrix of the ESO is a Hurwitz matrix [21]. The coefficient matrix is as follows:

$$P=\left[\begin{array}{lll}-c_1\hfill & 1\hfill & 0\hfill \\ -c_2\hfill & 0\hfill & 1\hfill \\ -c_3\hfill & 0\hfill & 0\hfill \end{array}\right]$$

(14)

The observer is much more difficult to design in practical applications. However, by employing the observer gains $c_1$, $c_2$, $c_3$ and the switching time $T_u$, it is obvious that the proposed fixed time disturbance observer is more convenient for selecting the observer gains. Therefore, compared with the other fixed time observers in [14,15], the proposed extended state observer can estimate the exact state and the disturbances within a fixed time, and achieve a more concise structure and far fewer parameters.

Theorem 1.

Based on Assumption 1, the fixed-time ESO can estimate the state variables  $x_{1i}$  and  $x_{2i}$ within a fixed time $T_F$. In addition, the estimation error of the lumped disturbance approaches a small vicinity around the origin within $T_F$.

Proof of Theorem 1 is provided in Appendix A.1.

3.3. Controller Design

3.3.1. Controller Design Based on Prescribed Performance

First, define the following error variable:

$$z_{1i}={\displaystyle \sum _{j\in {{\rm N}}_i}{a}_{ij}(t)(x_{1i}-x_{1j}-x_{\mathit{ij}}(t))+a_{i0}(t)(x_{1i}-x_d-x_{\mathit{id}}(t))}$$

(15)

$${\widehat{z}}_{2i}={\displaystyle \sum _{j\in {{\rm N}}_i}{a}_{ij}(t)({\widehat{x}}_{2i}-{\widehat{x}}_{2j})+a_{i0}(t)({\widehat{x}}_{2i}-{\dot{x}}_d)}$$

(16)

where ${{\rm N}}_i$, $a_{ij}(t)$, and $a_{i0}(t)$ are elements in the graph theory matrix, $x_{\mathit{ij}}(t)$ is the desired distance between the AUV-i and the AUV-j, and $x_{id}(t)$ is the desired distance between the AUV-i and the leader AUV. $a_{ij}(t)$, $a_{i0}(t)$, $x_{\mathit{ij}}(t)$ and $x_{\mathrm{id}}(t)$ are time-varying constant vectors, and $a_{id}(t)={\displaystyle \sum _{j\in {\mathrm{N}}_i}{a}_{ij}(t)+a_{i0}(t)}$. ${\widehat{z}}_{2i}$ is the estimated value of $z_{2i}$, and $z_{2i}$ satisfies $z_{2i}={\displaystyle \sum _{j\in {{\rm N}}_i}{a}_{ij}(t)(x_{2i}-x_{2j})+a_{i0}(t)(x_{2i}-{\dot{x}}_d)}$.

Remark 2.

Based on the fixed-time ESO (12), at  $t\ge T_F$, satisfying ${\widehat{z}}_{2i}=z_{2i}$, (16) can be transformed into:

$${\widehat{z}}_{2i}=z_{2i}={\displaystyle \sum _{j\in {{\rm N}}_i}{a}_{ij}(t)(x_{2i}-x_{2j})+a_{i0}(t)(x_{2i}-{\dot{x}}_d)}$$

(17)

To achieve the purpose of error constraint, a time performance function is introduced, as shown below:

$${\rho }_i(t)=\left\{\begin{array}{c}({\rho }_{0i}-{\rho }_{\infty i}){(1-{\displaystyle \frac{t}{T_i}})}^{{\mu }_i}+{\rho }_{\infty i},0\le t<T_i\\ {\rho }_{\infty i},t\ge T_i\end{array}\right.$$

(18)

where $0<{\rho }_{\infty i}<\left|z_{1i}(0)\right|<{\rho }_{0i}$, $T_i>0$ and ${\mu }_i>2$ is a constant. The constant $T_i$ represents the maximum predefined time for ${\rho }_i(t)$ to converge from the largest initial value ${\rho }_{0i}$ to the maximum asymptotic value ${\rho }_{\infty i}$, and ${\dot{\rho }}_i(t)$ is the predefined minimum convergence rate. ${\rho }_{0i}$, as the initial value of the prescribed performance function ${\rho }_i(t)$, strictly defines the permissible initial range of the trajectory tracking error $z_{1i}$ for the AUV-i. These parameters directly govern transient performance and error constraint enforcement. The expression for ${\dot{\rho }}_i(t)$ can be obtained as: 

$${\dot{\rho }}_i(t)=\left\{\begin{array}{c}-{\displaystyle \frac{{\mu }_i}{T}}({\rho }_{0i}-{\rho }_{\infty i}){(1-{\displaystyle \frac{t}{T_i}})}^{{\mu }_i-1},0\le t<T_i\\ 0,t\ge T_i\end{array}\right.$$

(19)

According to (19), there exists ${\dot{\rho }}_i(t)<0$ at $0\le t<T_i$, ${\rho }_{0i}>{\rho }_{\infty i}$.

$$\underset{t\to T_i^{-}}{\lim }{\rho }_i(t)=({\rho }_{0i}-{\rho }_{\infty i}){(1-{\displaystyle \frac{t}{T_i}})}^{{\mu }_i}+{\rho }_{\infty i}={\rho }_{\infty i}=\underset{t\to T_i^{+}}{\lim }{\rho }_i(t)$$

(20)

$$\underset{t\to T_i^{-}}{\lim }{\dot{\rho }}_i(t)=-{\displaystyle \frac{{\mu }_i}{T_i}}({\rho }_{0i}-{\rho }_{\infty i}){(1-{\displaystyle \frac{t}{T_i}})}^{{\mu }_i-1}=0=\underset{t\to T_i^{+}}{\lim }{\dot{\rho }}_i(t)$$

(21)

Thus, ${\rho }_i(t)$ is a monotonically decreasing, bounded, smooth, and positive function. Since ${\rho }_i(0)={\rho }_{0i}$ and ${\rho }_i(\infty )={\rho }_{\infty i}$, we can obtain $0<{\rho }_{\infty i}\le {\rho }_i(t)\le {\rho }_{0i}$, and then $\underset{t\to T_i}{\lim }{\rho }_i(t)={\rho }_{\infty i}$.

To achieve the transient and steady-state performance of multi-AUV tracking and improve the operational safety of AUVs, the following prescribed performance is proposed based on the proposed time performance function (18):

$${\underset{\_}{c}}_i{\rho }_i(t)\le z_{1i}\le {\stackrel{\_}{c}}_i{\rho }_i(t)$$

(22)

$${\underset{\_}{c}}_i=\left\{\begin{array}{c}-{\nu }_i,z_{1i}(0)\ge 0\\ -1,z_{1i}(0)<0\end{array}\right.,{\stackrel{\_}{c}}_i=\left\{\begin{array}{c}1,z_{1i}(0)\ge 0\\ {\nu }_i,z_{1i}(0)<0\end{array}\right.$$

(23)

where $0<{\nu }_i\le 1$ is the overshoot exponential constant, $z_{1i}(0)$ is the initial position and attitude error of $z_{1i}$.

Formula (18) cannot be directly used in controller design. Therefore, an error transformation function $\xi (x)$ is defined to establish the relationship between the formation tracking error and the time performance function. The definition of $\xi (x)$ is as follows:

$$\xi (x)={\displaystyle \frac{{\underset{\_}{c}}_i{\overline{c}}_i(\exp (x)-1)}{{\underset{\_}{c}}_i\exp (x)-{\overline{c}}_i}}$$

(24)

Taking the derivative of the above equation:

$$\dot{\xi }(x)={\displaystyle \frac{-{\underset{\_}{c}}_i{\overline{c}}_i\exp (x)({\overline{c}}_i-{\underset{\_}{c}}_i)}{{({\underset{\_}{c}}_i\exp (x)-{\overline{c}}_i)}^2}}$$

(25)

From (23), ${\underset{\_}{c}}_i<0<{\overline{c}}_i$; thus it is known that $\dot{\xi }(x)>0$, and $\xi (x)$ is a monotonically increasing function, since $\underset{x\to \infty }{\xi (x)}={\overline{c}}_i$, $\underset{x\to -\infty }{\xi (x)}={\underset{\_}{c}}_i$; then we can obtain $\xi (x)\in ({\underset{\_}{c}}_i,{\overline{c}}_i)$. The relationship between the transformed error ${\epsilon }_{1i}$ and the formation tracking error $z_{1i}$ is defined as:

$$z_{1i}={\rho }_i\xi ({\epsilon }_{1i})$$

(26)

From the above equation, it can be concluded that $z_{1i}\in ({\underset{\_}{c}}_i{\rho }_i(t),{\overline{c}}_i{\rho }_i(t))$ satisfies the prescribed performance of (22). The inverse function of $\xi (x)$ is denoted as:

$${\xi }^{-1}(x)=\ln ({\displaystyle \frac{{\overline{c}}_i(x-{\underset{\_}{c}}_i)}{-{\underset{\_}{c}}_i(\overline{c}-x)}})$$

(27)

${\xi }^{-1}(x)$ is a smooth, monotonically increasing bijective function, satisfying $\underset{{\epsilon }_{1i}\to {\overline{c}}_i}{\lim }{\xi }^{-1}({\epsilon }_{1i})=\infty$, $\underset{{\epsilon }_{1i}\to {\underset{\_}{c}}_i}{\lim }{\xi }^{-1}({\epsilon }_{1i})=-\infty$. According to (31):

$${\epsilon }_{1i}={\xi }^{-1}({\displaystyle \frac{z_{1i}}{{\rho }_i}})$$

(28)

Differentiating the above equation yields:

$${\dot{\epsilon }}_{1i}=B_i({\dot{z}}_{1i}+A_i{z}_{1i})$$

(29)

where $B_i,A_i$ are defined as:

$$B_i=\vartheta ({\displaystyle \frac{z_{1i}}{{\rho }_i}})/{\rho }_i,A_i=-{\displaystyle \frac{{\dot{\rho }}_i}{{\rho }_i}}$$

(30)

$\vartheta (x)$ is the derivative of ${\xi }^{-1}(x)$, denoted as:

$$\vartheta (x)={\dot{\xi }}^{-1}(x)={\displaystyle \frac{1}{x-{\underset{\_}{c}}_i}}+{\displaystyle \frac{1}{{\overline{c}}_i-x}}$$

(31)

From (29), the error can be expressed as:

$${\dot{\epsilon }}_{1i}={\widehat{\epsilon }}_{2i}=B_i({\dot{z}}_{1i}+A_i{z}_{1i})$$

(32)

$${\dot{\widehat{\epsilon }}}_{2i}=B_i({\ddot{z}}_{1i}+{\dot{A}}_i{z}_{1i}+A_i{\dot{z}}_{1i})$$

(33)

According to reference [22], $A_i=\mathrm{diag}({\alpha }_1,\cdots ,{\alpha }_6)$ is a positive semi-definite diagonal matrix satisfying ${\alpha }_k\ge 0(k=1\cdots 6)$, and $B_i=\mathrm{diag}(b_1,\cdots ,b_6)$ is a positive definite diagonal matrix satisfying ${\beta }_k\ge 2{\rho }_{0i}^{-1}(k=1\cdots 6)$.

Based on the above error definition, a novel nonsingular terminal sliding mode surface is introduced as follows:

$$s_i={\mathrm{sig}}^{a_1}({\epsilon }_{1i})+{\displaystyle \frac{k_2{a}_2}{2a_2-1}}{\mathrm{sig}}^{2-{\displaystyle \frac{1}{a_2}}}({\widehat{\epsilon }}_{2i}+k_1{\mathrm{sig}}^{a_1}({\epsilon }_{1i}))$$

(34)

where $k_1>0$, $k_2>0$, $a_2>1$, $1<a_1<1-{\displaystyle \frac{1}{a_2}}$.

Theorem 2.

When $s_i=0$, both ${\epsilon }_{1i}$ and ${\epsilon }_{2i}$ can converge to zero within a fixed time $T_2$.

Proof of Theorem 2 is provided in Appendix A.2.

3.3.2. Design of Auxiliary Saturation Systems and Event-Triggered Mechanisms

The following auxiliary saturation system is introduced to solve the issue of input saturation [11]:

$${\dot{\varsigma }}_i=\left\{\begin{array}{c}0,\\ -l_1{\mathrm{sig}}^p({\varsigma }_i)-l_2{\mathrm{sig}}^q({\varsigma }_i)-h({\varsigma }_i,s_i,△{\tau }_i)+a_{id}(t)G(x_{1i})△{\tau }_i,\end{array}\begin{array}{c}‖{\varsigma }_i‖<{\zeta }_i\\ ‖{\varsigma }_i‖\ge {\zeta }_i\end{array}\right.$$

(35)

$$h({\varsigma }_i,s_i,\Delta {\tau }_i)={\displaystyle \frac{(‖s_i‖+‖{\varsigma }_i‖){\lambda }_{\max }(B_i)a_{id}(t)\kappa +k_{\varsigma }{\lambda }_{\max }({\mathrm{\Gamma }}_i)‖s_i‖‖{\varsigma }_i‖}{{‖{\varsigma }_i‖}^2}}{\varsigma }_i$$

(36)

where ${\varsigma }_i$ is the state vector of the auxiliary saturation system, $‖{\varsigma }_i(0)‖={\varsigma }_{0i}$, ${\varsigma }_{0i}$ is a position vector, $l_1>0$, $l_2>0$, $p>1$, $0<q<1$, ${\zeta }_i(i=1,2\cdots N)$ is a small positive constant, and $k_{\zeta }$ is a designed constant. When $‖{\varsigma }_i‖<{\zeta }_i$, no input saturation occurs, and the auxiliary variable is expressed as ${\dot{\varsigma }}_i=0$; when $‖{\varsigma }_i‖\ge {\zeta }_i$, input saturation occurs, and the auxiliary variable is expressed as follows:

$${\dot{\varsigma }}_i=-l_1{\mathrm{sig}}^p({\varsigma }_i)-l_2{\mathrm{sig}}^q({\varsigma }_i)-h({\varsigma }_i,s_i,\Delta {\tau }_i)+a_{id}(t)G(x_{1i})\Delta {\tau }_i$$

(37)

Considering input saturation, model uncertainties, and unknown ocean disturbances, the control law can be designed as:

$$u_i=u_{i1}+u_{i2}+u_{i3}$$

(38)

$$\begin{array}{ll}{u}_{i1}\hfill & =G^{-1}(x_{1i})(-F(x_{1i},{\hat{x}}_{2i}))+{\displaystyle \frac{G^{-1}(x_{1i})}{a_{id}(t)}}[{\displaystyle \sum _{j\in {{\rm N}}_i}{a}_{ij}(t){\dot{\widehat{x}}}_{2j}+a_{id}(t){\ddot{x}}_d}-{\dot{A}}_i{z}_{1i}-A{\dot{z}}_{1i}\hfill \\ \hfill & +B^{-1}(-{\gamma }_1{\mathrm{sig}}^p(s_i)-{\gamma }_2{\mathrm{sig}}^q(s_i)-\delta ({\sigma }_1\left|{\epsilon }_{1i}\right|+{\sigma }_2\left|{\widehat{\epsilon }}_{2i}\right|)\mathrm{sign}(s_i)\hfill \\ & -k_1{a}_1\mathrm{diag}{(\left|{\epsilon }_{1i}\right|)}^{a_1-1}({\widehat{\epsilon }}_{2i}+{\displaystyle \frac{{\mathrm{\Lambda }}_i}{k_1}}))]\hfill \end{array}$$

(39)

$$u_{i2}=-G^{-1}(x_{1i}){\widehat{D}}_i$$

(40)

$$u_{\mathrm{i}3}={\displaystyle \frac{G^{-1}(x_{1i})}{a_{id}(t)}}{B}^{-1}{k}_{{\varsigma }_i}{\varsigma }_{\mathrm{i}}$$

(41)

where the expression for ${\mathrm{\Lambda }}_i$ is as follows:

$${\mathrm{\Lambda }}_i={\displaystyle \frac{1}{k_2}}{\mathrm{sig}}^{{\displaystyle \frac{1}{a_2}}}[{\widehat{\epsilon }}_{2i}+k_1{\mathrm{sig}}^{a_1}({\epsilon }_{1i})]+{\displaystyle \frac{k_1{a}_1}{2a_2-1}}[{\widehat{\epsilon }}_{2i}+k_1{\mathrm{sig}}^{a_1}({\epsilon }_{1i})]$$

(42)

where ${\gamma }_1>0$, ${\gamma }_2>0$, $p>1$, $0<q<1$, $k_1$, $k_2$, $\delta$, ${\sigma }_1$, ${\sigma }_2$ are positive constants.

By introducing an event-triggered mechanism, we first define $t_0,t_1\cdots \cdots t_k$, which represents the time sequence of event triggers, where $t_k$ denotes the triggering instants. The measurement error $\Delta e_i(t)$ is defined as follows:

$$\Delta e_i(t)=u_i(t_k)-u_i(t)$$

(43)

Define the actual control input as:

$${\tau }_i(t)=u_i(t_k),t\in [t_k,t_{k+1})$$

(44)

The event-triggering condition is defined as:

$$t_{k+1}=\inf \{t>t_k:f_i(t)\ge 0;k\in {\rm N}\},t_0=0$$

(45)

where $f_i(t)$ is defined as:

$$\begin{array}{r}{f}_i(t)=0.5a_{id}{\lambda }_{\max }(G(x,t)){\lambda }_{\max }(B)({‖s_i‖}^2+{‖\Delta e_i(t)‖}^2)\\ -\delta ({\sigma }_1\left|{\epsilon }_{1i}\right|+{\sigma }_2\left|{ \epsilon ^}_{2i}\right|)\left|s_i\right|-{\sigma }_3{e}^{-\gamma t}\end{array}$$

(46)

3.3.3. Stability Analysis and the Exclusion of No Zeno Behavior for Controller

Zeno behavior [6] refers to the phenomenon in event-triggered control where an infinite number of triggering events occur within a finite time period, and the inter-event intervals converge to zero. This behavior can compromise system stability and is directly related to the design of the event-triggering condition. To prevent Zeno behavior, a strictly positive lower bound must be guaranteed for the inter-event intervals.

Theorem 3.

For the system (9) with model uncertainties, unknown ocean disturbances, actuator faults, and communication topology faults, under the fixed-time ESO (12), trajectory error constraints (22), and control law (38), combined with the event-triggered mechanisms (44)–(46), the entire AUV formation can achieve convergence within a fixed time. Moreover, the proposed scheme eliminates Zeno behavior while ensuring that the formation tracking error converges within fixed time, thus achieving fault-tolerant control for the multi-AUV formation system.

The proof of Theorem 3 is provided in Appendix A.3.

4. Simulation Results

In the simulation, one leader (labeled as 0) is selected, which typically represents the control signal issued by the command center in practical applications. Five followers (labeled as 1, 2, 3, 4, and 5) are included. The simulation environment models a multi-AUV system conducting resource exploration. During the process, one AUV suddenly loses communication. To improve the accuracy of resource exploration, a formation reconfiguration is performed, making the AUV formation more compact. The relative distances between AUVs are set as: $x_{10}={[3,4,0,0,0,0]}^T$, $x_{21}={[-6,0,0,0,0,0]}^T$, $x_{32}={[0,4,0,0,0,0]}^T$, $x_{53}={[3,4,0,0,0,0]}^T$, $x_{41}={[0,4,0,0,0,0]}^T$, the communication topology is shown as follows (Figure 3):

4.1. Simulation Settings

In this study, the Ocean Bottom Flying Node (OBFN), as a fully actuated AUV, is selected for simulation experiments. The specific model parameters are provided in

Table 1. AUV Inertia Parameters [23].

Mess (kg)	${\mathit{I}}_{\mathit{x}}$ (Nm·s2)	${\mathit{I}}_{\mathit{y}}$ (Nm·s2)	${\mathit{I}}_{\mathit{z}}$ (Nm·s2)	${\mathit{I}}_{\mathit{x}\mathit{y}}$ (Nm·s2)	${\mathit{I}}_{\mathit{y}\mathit{z}}$ (Nm·s2)	${\mathit{I}}_{\mathit{x}\mathit{z}}$ (Nm·s2)
92	49	84	79	0	0	0

and

Table 2. Hydrodynamic coefficients of AUVs [23].

Coefficients	Linear.Drag/kg·s−1	Quad.Drag/kg·m−1	Added Mass/kg
Surge	21.9	24.5	40.6
Sway	47.4	34.7	82.3
Heave	53.5	40.2	114.6
Heel	71.9	42.1	55.7
Trim	97.3	48.3	76.5
Yaw	86.9	39.3	59.1

, while the initial positions and velocities are listed in

Table 3. Initial positions (angles) of follower AUVs.

${\mathit{\eta }}_{\mathit{i}\mathit{j}}(0)$	$\mathit{i}=1$	$\mathit{i}=2$	$\mathit{i}=3$	$\mathit{i}=4$	$\mathit{i}=5$
$j=1(\mathrm{m})$	10	1	0	1	6
$j=2(\mathrm{m})$	8	5	2	1	2
$j=3(\mathrm{m})$	7	5	2	5	5
$j=4(\mathrm{rad})$	0.1	0	0.005	0.01	0.005
$j=5(\mathrm{rad})$	0.1	0	0.005	0.01	0.005
$j=6(\mathrm{rad})$	0.1	0	0.005	0.01	0.005

and

Table 4. Initial velocities (angle velocities) of follower AUVs.

${\mathit{v}}_{\mathit{i}\mathit{j}}(0)$	$\mathit{i}=1$	$\mathit{i}=2$	$\mathit{i}=3$	$\mathit{i}=4$	$\mathit{i}=5$
$j=1(\mathrm{m}/\mathrm{s})$	0.1	0.2	0.1	0.2	0.2
$j=2(\mathrm{m}/\mathrm{s})$	0.1	0.3	0.3	0.1	0.1
$j=3(\mathrm{m}/\mathrm{s})$	0.1	0	0.2	0.2	0.2
$j=4(\mathrm{rad}/\mathrm{s})$	0.1	0.1	0	0.1	0.1
$j=5(\mathrm{rad}/\mathrm{s})$	0.1	0.1	0	0.1	0.1
$j=6(\mathrm{rad}/\mathrm{s})$	0.1	0.1	0	0.1	0.1

.

In this study, the ocean current is simulated using a first-order Gauss–Markov process [23], as shown below:

$${\dot{V}}_c+\mu V_c=\omega$$

(47)

In the above equation, $V_c$ represents the magnitude of the ocean current velocity in the fixed coordinate system. $\omega$ is Gaussian white noise with a mean of 1 and a variance of 1, $\mu =3$.

The simulation selects a 30% model uncertainty to solve the model uncertainty of the AUV itself, meaning that the nominal value of the dynamic model in the controller is 70% of the actual value.

The position of the leader is specified as:

$$\left\{\begin{array}{l}{\eta }_1(t)=6\sin (0.1t)+5\hfill \\ {\eta }_2(t)=5\cos (0.1t)+1\hfill \\ {\eta }_3(t)=0.5144t\hfill \\ {\eta }_4(t)=0.2\hfill \\ {\eta }_5(t)=0\hfill \\ {\eta }_6(t)=0\hfill \end{array}\right.$$

(48)

To verify the effectiveness of the Prim algorithm for topology reconstruction, a simulation experiment is designed. The simulation time is set to 100 s, and at 20 s, Follower AUV-1 loses connection. At this moment, its communication cost matrix is as follows:

$$C_{ij}=\left[\begin{array}{lllll}0\hfill & 5\hfill & \sqrt{73}\hfill & \sqrt{73}\hfill & 12\hfill \\ 5\hfill & 0\hfill & 4\hfill & \sqrt{52}\hfill & \sqrt{52}\hfill \\ \sqrt{73}\hfill & 4\hfill & 0\hfill & 6\hfill & 5\hfill \\ \sqrt{73}\hfill & \sqrt{52}\hfill & 6\hfill & 0\hfill & 5\hfill \\ 12\hfill & \sqrt{52}\hfill & 5\hfill & 5\hfill & 0\hfill \end{array}\right]$$

(49)

After computation, the topology graph generated by the Prim algorithm is as follows:

To simulate formation reconfiguration following a topology failure, the transformation is triggered at t = 70 s in the simulation. The reconfigured formation is depicted in Figure 4, with the relative distances from the leader AUV defined as follows: $x_{1d}={[-3,4,0,0,0,0]}^T$, $x_{2d}={[0,4,0,0,0,0]}^T$, $x_{3d}={[3,4,0,0,0,0]}^T$, $x_{4d}={[0,8,0,0,0,0]}^T$. The relative distance position can be represented as follows (Figure 5):

The communication cost matrix of the Prim algorithm after the transformation is as follows:

$$C_{ij}=\left[\begin{array}{lllll}0\hfill & 5\hfill & 4\hfill & 5\hfill & 8\hfill \\ 5\hfill & 0\hfill & 3\hfill & 6\hfill & 5\hfill \\ 4\hfill & 3\hfill & 0\hfill & 3\hfill & 4\hfill \\ 5\hfill & 6\hfill & 3\hfill & 0\hfill & 5\hfill \\ 8\hfill & 5\hfill & 4\hfill & 5\hfill & 0\hfill \end{array}\right]$$

(50)

The communication topology graph calculated using the Prim algorithm is as follows (Figure 6) (①, ②, ③, and ④ represent the sequence in the topological diagram and do not refer to a specific AUV):

The Hungarian algorithm is employed for optimal position assignment during formation transformation, with its cost matrix constructed as follows:

$$E_{ij}=\left[\begin{array}{llll}0\hfill & 3\hfill & 6\hfill & 5\hfill \\ 4\hfill & 5\hfill & \sqrt{52}\hfill & 3\hfill \\ \sqrt{52}\hfill & 5\hfill & 4\hfill & 3\hfill \\ \sqrt{73}\hfill & 8\hfill & \sqrt{73}\hfill & 4\hfill \end{array}\right]$$

(51)

The formation transformation assignment results obtained using the Hungarian algorithm are summarized in Figure 7.

From the assignment results, positions ①, ②, ③, and ④ are assigned to AUV-2,3,4 and 5, respectively. Finally, the communication topology graph is shown in Figure 7.

The parameter selection is rigorously derived from fixed-time stability theory and tuned via numerical simulations to balance convergence speed, robustness, and implementation feasibility. All parameters are validated in both nominal and fault scenarios to ensure consistent performance. The control parameters are as follows, with the fixed-time ESO coefficients being $c_1=500$, $c_2=50$, $c_3=2000$, ${\alpha }_1=0.6$, and ${\beta }_1=1.4$. The parameters of the prescribed performance function are ${\rho }_{01}=10$, $T_1=8 \mathrm{s}$, ${\rho }_{02}=10$, $T_2=8 \mathrm{s}$, ${\rho }_{03}=14$, $T_3=10 \mathrm{s}$, ${\rho }_{04}=12$, $T_4=10 \mathrm{s}$, ${\rho }_{05}=14$, $T_5=11 \mathrm{s}$, ${\mu }_i=2.01(i=1~5)$, and ${\rho }_{\infty i}=0.1(i=1~5)$; the parameters of the fixed-time terminal sliding mode surface are $a_1=1.5$, $a_2=2$, $k_1=1.2$, and $k_2=1.2$; the parameters of the saturation auxiliary system are $l_1=1.2$, $l_2=1.2$, $p=1.1$, $q=0.7$, $k_{\zeta }=0.01$, and $\kappa =1000$; the initial parameters of ${\varsigma }_i$ are set as ${\varsigma }_i(0)={[0.01,0.01,0.01,0.01,0.01,0.01]}^T$, and ${\zeta }_i=0.001(i=1\cdots 6)$. The relevant parameters of the remaining controller are set as ${\tau }_{i\max }=300 \mathrm{N}(i=1\cdots 3)$, ${\tau }_{i\max }=300 \mathrm{N}\cdot \mathrm{m}(i=4\cdots 6)$, ${\gamma }_1=1.2$, ${\gamma }_2=1.2$, $\delta =0.01$, ${\sigma }_1=5$, ${\sigma }_2=5$, ${\sigma }_3=1.25$, and $\gamma =2$.

The fault model of follower AUVs: $a_{f1}={[1,1,1,1,1,1]}^T$, $T_{f1}=10 \mathrm{s}$, ${\overline{\tau }}_1={\left[4,4,4,4,4,4\right]}^T$, $E_1=\mathrm{diag}(0.5,0.5,0.5,0.7,0.7,0.7)$, $a_{f2}={[1,1,1,1,1,1]}^T$, $T_{f2}=10 \mathrm{s}$, ${\overline{\tau }}_2={\left[10,10,10,4,4,4\right]}^T$, $E_2=\mathrm{diag}(0.9,0.9,0.9,0.9,0.9,0.9)$, $a_{f4}={[1,1,1,1,1,1]}^T$, $T_{f4}=50 \mathrm{s}$, ${\overline{\tau }}_4={\left[4,4,4,4,4,4\right]}^T$, $E_4=\mathrm{diag}(0.7,0.7,0.7,0.7,0.7,0.7)$.

4.2. Results

Figure 3 and Figure 5 illustrate the communication topology reconstruction results using the Prim algorithm. Figure 3 shows that after a communication topology failure occurs for follower AUV-1 at 20 s, the Prim algorithm can reconstruct the communication topology with minimal communication cost. Figure 5 demonstrates that during the formation transformation at 70 s, the Prim algorithm calculates the minimal communication cost to reconstruct the multi-AUV communication topology, confirming the effectiveness of the Prim-based multi-AUV topology reconstruction. Figure 6 and Figure 7 present the results of formation transition position allocation using the Hungarian algorithm. The formation transition position allocation cost at this time is 13 m, which is the minimal position allocation cost, validating the effectiveness of the Hungarian algorithm for formation transition position allocation.

Figure 8 shows the trajectories of the multi-AUV formation, illustrating the formation shapes at 10 s, 40 s, and 85 s from bottom to top. During 0–20 s, the entire formation is in a hexagonal shape. At 20 s, follower AUV-1 loses connection, causing a communication topology failure. After reconstructing the topology, the formation changes to a pentagonal shape. Following a formation transformation after 70 s, the formation becomes quadrilateral. As illustrated in Figure 8, the entire AUV formation successfully transitions from a hexagonal configuration to a pentagonal one, and ultimately to a quadrilateral shape. This progression effectively demonstrates the algorithm’s capability in addressing communication topology failures and facilitating formation transformations.

Figure 9 and Figure 10 illustrate the displacement and angle variations of the multi-AUV system. At 20 s, when a topology failure occurs, there are no abrupt changes in the displacement or angle of each AUV, demonstrating the effectiveness of the Prim algorithm in reconstructing the topology and maintaining the formation. At 70 s, during the formation transformation, the displacement and angle of each AUV begin to change. At this point, the Prim algorithm continues to reconstruct the multi-AUV topology, while the Hungarian algorithm assigns positions for the formation transformation. Around 10 s later, the displacement and angle of each AUV stabilize at the new formation positions, completing the formation transformation with stable formation performance.

Taking follower AUV-2 as an example, Figure 11 shows the lumped disturbance force and disturbance torque values estimated by the fixed-time ESO. The red line represents the estimated value of the lumped disturbance by the fixed-time ESO of follower AUV-2, while the blue line represents the actual value. There are slight fluctuations at 20 s during the topology transformation and at 70 s during the formation transition, but these do not affect the subsequent control performance. Figure 12 demonstrates the accurate estimation of velocity states, effectively addressing the issue of measurement inaccuracies caused by rapid velocity changes in practical engineering applications.

We take follower AUV-2 as an example to illustrate the effectiveness of the error constraints discussed in this section. Figure 13 presents the error and its associated constraints for follower AUV-2. Figure 13 illustrates the effect of error constraints, where the blue line represents the boundary of the time performance function, and the green line represents the error of follower AUV-2. It can be observed that throughout the entire formation operation, the error of follower AUV-2 consistently remains within the boundaries of the specified time performance function. This error constraint enhances the control accuracy of the formation, thereby validating the effectiveness of the error constraint algorithm proposed in this section.

Figure 14 shows the control force and control torque diagrams. It can be observed that the force and torque are relatively large in the initial stage, and saturation occurs during the convergence process. Under the effect of the auxiliary saturation system, the control force is limited, and throughout this process, the control force and control torque consistently remain within the set saturation values, validating the effectiveness of the auxiliary saturation system.

Taking follower AUV-2 as an example, Figure 15 and Figure 16 show its triggering moments and triggering intervals from 20 s to 30 s. It can be observed that the triggering instants are non-uniform, and the maximum triggering interval exceeds 1.5 s.

Table 5. Event trigger counts of follower AUVs.

	AUV	Follower AUV-2	Follower AUV-3	Follower AUV-4	Follower AUV-5
Trigger Counts
${\tau }_1$		5092	5484	4158	2269
${\tau }_2$		3436	4918	2894	3862
${\tau }_3$		3510	5390	1762	4093
${\tau }_4$		3263	3968	4099	4229
${\tau }_5$		2473	3812	5287	2758
${\tau }_6$		4077	5051	4797	4670
No event triggered		10,000	10,000	10,000	10,000

presents the number of event triggers for multiple AUVs, with the non-triggering ratio reaching over 80%. It is evident that, compared to time-based control methods, the event-triggered approach reduces the number of transmissions between the controller and the actuator, conserving transmission resources and, to some extent, reducing actuator wear. Table 5 shows the number of event-triggered for each AUV.

To verify the superiority of the fixed-time sliding mode control with error constraints proposed in this section, a comparison is made with the fixed-time integral terminal sliding mode control (FTITSMC) presented in reference [24]. It was chosen for comparison because it represents an advanced fixed-time control solution in current literature and addresses a similar control challenge, though in a different application context. The selected fixed-time terminal sliding mode surface is as follows:

$$s_i=z_{i2}+{\displaystyle {\int }_0^t{k}_{i1}{G}_1(z_{i1}(}T))+k_{i2}{G}_2(z_{i2}(T))dT$$

(52)

The corresponding part of the control law can be transformed into:

$$\begin{array}{ll}{u}_{i1=}\hfill & G^{-1}(x_{1i})(-F(x_{1i},x_{2i}))+{\displaystyle \frac{G^{-1}(x_{1i})}{a_{id}}}({\displaystyle \sum _{j\in {{\rm N}}_i}{a}_{ij}{\dot{x}}_{2j}+a_{id}{\ddot{x}}_d}+k_1{G}_1(z_{i1}(T))\hfill \\ \hfill & +k_2{G}_2(z_{i2}(T))+{\gamma }_1{\mathrm{sig}}^{r_1}(s_i)+{\gamma }_2\mathrm{sign}(s_i))\hfill \end{array}$$

(53)

where $k_1=k_2=1.2$, ${\alpha }_1=0.95$, ${\beta }_1=1.05$, ${\gamma }_1={\gamma }_2=1.2$. The algorithm proposed in this paper is referred to as PPFNFTSMC, while the algorithm in reference [24] is referred to as FTITSMC. The error comparison diagram is shown in Figure 17.

The results show that, under the same parameter conditions, the error convergence speed of the algorithm proposed in this section is faster, and the error consistently remains within the boundary defined by the time performance function. In contrast, the algorithm from reference [24] exhibits a slower error convergence speed and some overshot. The simulations presented above validate the effectiveness of the control algorithm developed in this section.

5. Conclusions

In this research, issues such as communication interruptions, actuator failures, dynamic uncertainties, environmental disturbances, transient performance, and input saturation in fault-tolerant control of multi-AUV cooperative formation are considered. A formation reconstruction mechanism is designed based on the Prim algorithm and the Hungarian algorithm, allowing for the reconstruction of communication topology. In simulations, the AUV swarm is reformulated in response to AUV disconnection, achieving fault-tolerant control under weak communication conditions. A fixed-time ESO is introduced to accurately estimate the lumped disturbances in the system, which consist of model uncertainties, environmental disturbances, and actuator failures. By introducing performance functions and input saturation functions, a novel event-triggered terminal sliding mode controller is constructed, which constrains the error and actuator inputs within predefined limits while reducing the frequency of controller updates. This approach ensures system robustness and avoids potential damage to the actuators. The event-triggered fault-tolerant formation control for AUV swarms in this paper has been verified through theoretical proof and simulation research.