Fault-Tolerant Formation Control for Multiple Stochastic AUV System under Markovian Switching Topologies

Abstract

Formation control, which is a core problem in multi-autonomous underwater vehicle (AUV) systems, plays an important role in realizing safe and accurate cooperation of multi-AUV systems. This paper provides a study on fault-tolerant formation control for multiple stochastic AUV systems under Markovian switching topologies. Considering the effect of noise and Markovian switching communication topology, a novel leader-following group formation control protocol with an actuator fault for a multi-AUV system is developed under the influence of the two independent stochastic processes. The designed controller is proved by an infinitesimal generator with Lyapunov stability theory and can guarantee that the follower AUVs’ states will eventually converge to the leader’s state in each subgroup while forming the desired sub-formation. Finally, the effectiveness of the theoretical analysis is verified by a simulation experiment.

1. Introduction

In recent years, the formation control of multiple autonomous underwater vehicles (AUV) has received considerable attention due to various applications in the field of ocean exploration, deep sea resource development, submarine rescue, trajectory tracking and other fields of ocean engineering. AUV is a kind of self-propelled submersible vehicle that is utilized to perform the desired missions with no human control [1]. Compared with a single AUV, a multi-AUV system can complete the task well in complex applications such as investigating resources, target detection and task accuracy in harsh ocean environments. Further, multi-AUV formation cooperation has more advantages in improving work efficiency and operation scope. The fact that multiple AUVs transmit state information via the sonars in an acoustic channel ensures the coordination of a multi-AUV system. In reality, the AUV system is always affected by stochastic ocean noise. In that case, the multi-AUV system contains uncertain items. In addition, the interaction topology among AUVs may be dynamic or generate new information chains because of the complex marine environments. Stochastic noise and random communication always affect the stability of the multi-AUV system and even lead to the loss of the AUV. Thus, a reliable controller should be designed to guarantee coordination of the multi-AUVs system in the case of noise and random communication. It is still a great challenge because of the highly coupled nonlinearities in the AUV dynamics and the uncertainties in ocean environments.

As the complexity of underwater environment inspection missions increases, formation of multi-AUV can greatly improve the efficiency of exploration and complete more tasks in a limited time. The formation control of a multi-AUV system with a single leader has been investigated in [2,3,4,5,6]. However, in many engineering applications, the strategy with multiple subgroups and leaders of a multi-agent system has stronger practical significance due to the assignment of different geographical locations and tasks [7,8,9]. At the same time, various remarkable results have been dedicated to the formation control problem, and the study made certain progress. The development of formation controllers employs the leader–follower method [10,11,12], consensus method [13,14,15], Lyapunov’s direct method [16,17], model predictive control method [18,19] and back-stepping techniques [20], etc. Among the typical control methods described above, the leader–follower method is widely used in multi-agent formation control because of its good scalability. As a matter of fact, it is a significant theoretical formal method that allows all followers to track the state of leaders. The consensus method is the most basic method in the formation control problem, which is often applied in combination with the leader–follower method.

It is noteworthy that all the studies mentioned above focused on formation control problems with a deterministic switching model manner. In practical applications, the multi-agent systems often suffer from some disturbances, such as communication obstacles under uncertain environments [21,22]. Stochastic switching models could reflect the changes of the information structure, which can be governed by the Markovian switching model. There are many results on multi-agent systems with Markovian switching topologies [23,24,25]. Note that the multi-AUV systems equipped with actuators are at risk of actuator failure in each AUV due to encoding or decoding. Thus far, there have been many studies on the consistency strategy with actuator faults [26,27,28,29].

In the past few years, a rich body of results has been made on formation control for multi-agent systems with stochastic noises. In [30], three formation protocols were established by using a novel stochastic analysis approach for a nonlinear multi-agent systems with communication delays and noise disturbance. Considering the time-varying output and stochastic noise of formation systems, ref. [31] puts forward a fully adaptive practical time-varying output formation protocol. Further, under both measurement and communication noises, ref. [32] designed a robust distributed orientation estimate algorithm and formation control law. Ref. [33] dealt with formation control problems with time delays and multiplicative noises. In [34], a mean-square quasi-composite rotating formation problem with stochastic communication noises was considered by taking coordinate transformation. Up to now, scholars have developed some theories in the research of formation control for stochastic agent systems; however, there are few studies on formation control for multiple stochastic AUV systems with stochastic switching topologies and stochastic noises, which is proposed in our present investigation for the first time.

In previous studies of heterogeneous multi-agent system formation control problems, the heterogeneity dynamics can be divided into two cases: one is that there are different dynamic model equations [35]. For example, the agents’ orders may be first-order, second-order, or high-order in the same system [36,37]; some cross-dimensional formation issues [38], such as the cooperation of an unmanned aerial in three-dimension and surface vehicles in the two-dimensional plane. The other is that each agent has the same dynamic model but different heterogeneous dynamics because of different types, external environment disturbances and configurations of the control objects, as illustrated in [39,40]. Ref. [41] proposed a formation control scheme for multi-AUV systems with heterogenous nonlinear dynamics. Heterogeneous multi-agent systems can be used to solve more practical formation control problems and make the controller have high compatibility and adaptability. In this paper, the leader AUV is equipped with more sensors and constructed as a stable time-invariant system, which can steadily send instructions to the followers. However, the communication topology among all followers is switched randomly. Moreover, the leaders and the followers have different lengths and weights. Under these conditions, the paper focuses on solving the influences of the actuator fault, ocean noise and communication switching on formation control.

Motivated by the above discussions, this paper aims to discuss fault-tolerant formation control for multiple stochastic AUV systems under Markovian switching topologies with multiple leaders. Based on previous research, the contributions of the paper are as follows: (1) the works in [1,3] only considered formation control of a multi-AUV system with a single leader. The paper provides a discussion of the group formation problem in a multi-AUV system with multiple leaders. All the AUVs are divided into multiple subgroups, and each subgroup is controlled to track the average state of the leader while forming the desired sub-formation. (2) The previous research in [42] employed switching communication topologies based on Laplacian matrix theory. In the paper, the fault-tolerant formation control problem is analyzed for the multiple stochastic AUV system with Markov switching topologies, which considers the more random case that two independent stochastic processes exist simultaneously in the system. (3) The communication matrices driven by a Markov process are no longer of Laplacian type, $a_{ij}$ is permitted to be negative in accordance with the group competition and cooperation mechanism. The effect of switching topologies on group formation is determined by the union of topologies of the Markov process, which is more suitable for the limited and unreliable communication topology.

The rest of this paper is arranged as follows: Section 2 introduces some important lemmas, definitions, preliminaries and the AUV model description. Section 3 develops the formation control structure and controller design and presents the main results. Section 4 verifies the effectiveness of the proposed algorithms by simulation. The conclusion is offered in Section 5.

Notations:

Table 1. Notations.

Notations	Meanings
$\mathbb{E}\left(·\right)$	Mathematical expectation operator
$∥·∥$	Euclidean norm
$N_i\left(t\right)$	Neighbors set
$1_n$	n dimension column vector with elements 1
$P>0$	Matrix P is positive definite
⊗	Kronecker product
$diag\left\{·\right\}$	Block-diagonal matrix
$A={\left(a_{ij}\right)}_{N\times N}$	The weighted matrix
${\lambda }_{max}\left(·\right)$	The largest eigenvalue
${\lambda }_{min}\left(·\right)$	The smallest eigenvalue

lists some notations and their meanings.

2. Preliminaries

This section mainly introduces AUV dynamic models and some related concepts, assumptions and lemmas.

2.1. Problem Formulation

It is well known that the nonlinear characteristics of an AUV are strong when it moves to resist the ocean current in a deep sea with complex terrain. In this paper, the roll velocity of the AUV is ignored due to the fact that the roll motion has less influence on the lateral motion. The AUV is assumed to be under-actuated in the 5-DOF model with the body-fixed and earth-fixed coordination consisting of the surge, sway, heave, pitch and yaw in this paper, which is shown in Figure 1. The dynamic equations of AUV are given as [43].

$$\left\{\begin{array}{c}\dot{\mathit{\eta }}=\mathit{J}\left(\mathit{\eta }\right)\mathit{\upsilon },\hfill \\ \mathit{M}\dot{\mathit{\upsilon }}+\mathit{C}\left(\mathit{\upsilon }\right)\mathit{\upsilon }+\mathit{D}\left(\mathit{\upsilon }\right)\mathit{\upsilon }+\mathit{g}\left(\mathit{\eta }\right)=\mathit{T}+\tilde{\mathit{\omega }}\hfill \end{array}\right.$$

(1)

where the vector $\mathit{\eta }={\left[x,y,z,\theta ,\psi \right]}^T$ denotes the states of the position vector; $\mathit{\upsilon }={\left[u,v,w,q,r\right]}^T$ denotes the states of the velocity vector; the matrix $\mathit{J}\left(\mathit{\eta }\right)$ denotes the rotational transformation matrix; M denotes the inertia matrix including the added mass. The paper provides an assumption that the AUV has only a $yoz$ plane symmetry structure. $\mathit{C}\left(\mathit{\upsilon }\right)$ and $\mathit{D}\left(\mathit{\upsilon }\right)$ are the Coriolis and centripetal matrix and the damping matrix; $\mathit{g}\left(\mathit{\eta }\right)$ is the vector caused by the effect of gravity and buoyancy; $\mathit{T}$ denotes the vector of control input; $\tilde{\mathit{\omega }}$ represents the environmental disturbance forces due to ocean current.

In order to simplify the model, assume the AUV has zero buoyancy, and the parameters of the matrices $\mathit{M}$, $\mathit{C}\left(\mathit{\upsilon }\right)$, $\mathit{D}\left(\mathit{\upsilon }\right)$ and $\mathit{g}\left(\mathit{\eta }\right)$ are as follows:

$$\mathit{M}=\left[\begin{array}{ccccc}m-X_{\dot{u}}& 0& 0& 0& 0\\ 0& m-Y_{\dot{v}}& 0& 0& 0\\ 0& 0& m-Z_{\dot{w}}& 0& 0\\ 0& 0& 0& I_y-M_{\dot{q}}& 0\\ 0& 0& 0& 0& I_z-N_{\dot{r}}\end{array}\right]$$

(2)

$$\mathit{C}\left(\mathit{\upsilon }\right)=\left[\begin{array}{ccccc}0& 0& 0& mw-Z_{\dot{w}}w& -mv+Y_{\dot{v}}v\\ 0& 0& 0& 0& mu-X_{\dot{u}}u\\ 0& 0& 0& -mu+X_{\dot{u}}u& 0\\ -mw+Z_{\dot{w}}w& 0& mu-X_{\dot{u}}u& 0& 0\\ mv-Y_{\dot{v}}v& -mu+X_{\dot{u}}u& 0& 0& 0\end{array}\right]$$

(3)

$$\mathit{D}\left(\mathit{\upsilon }\right)=diag\left\{X_u,Y_v,Z_w,M_q,N_r\right\}+diag\left\{X_{\left|u\right|u}\left|u\right|,Y_{\left|v\right|v}\left|v\right|,Z_{\left|w\right|w}\left|w\right|,M_{\left|q\right|q}\left|q\right|,N_{\left|r\right|r}\left|r\right|\right\}$$

(4)

$$\mathit{g}\left(\mathit{\eta }\right)={\left[0,0,0,\rho gV\overline{G{M}_L}sin\theta ,0\right]}^T$$

(5)

where the details of the above parameters are presented in [43].

Let

$$\mathit{\varsigma }={\left[x,y,z,\theta ,\phi ,u,v,w,q,r\right]}^T$$

(6)

$$\mathit{N}\left(\mathit{\eta },\mathit{\upsilon }\right)=\mathit{C}\left(\mathit{\upsilon }\right)\mathit{\upsilon }+\mathit{D}\left(\mathit{\upsilon }\right)\mathit{\upsilon }+\mathit{g}\left(\mathit{\eta }\right)$$

(7)

$$\mathit{J}\left(\mathit{\eta }\right)\mathit{\upsilon }={\left[f_1\left(\varsigma \right),f_2\left(\varsigma \right),f_3\left(\varsigma \right),f_4\left(\varsigma \right),f_5\left(\varsigma \right)\right]}^T$$

(8)

$${\mathit{M}}^{-1}\mathit{N}\left(\mathit{\eta },\mathit{\upsilon }\right)={\left[f_6\left(\varsigma \right),f_7\left(\varsigma \right),f_8\left(\varsigma \right),f_9\left(\varsigma \right),f_{10}\left(\varsigma \right)\right]}^T$$

(9)

the model (1) of the AUV can be represented as [42]:

$$\dot{\mathit{\varsigma }}=\mathit{f}\left(\mathit{\varsigma }\right)+\mathit{g}\left(\mathit{\varsigma }\right)\tilde{\mathit{u}}$$

(10)

where

$$\mathit{f}\left(\mathit{\varsigma }\right)=\left[\begin{array}{cc}\mathit{I}& \mathit{0}\\ \mathit{0}& {\mathit{M}}^{-1}\end{array}\right]\left[\begin{array}{c}\mathit{J}\left(\mathit{\eta }\right)\mathit{\upsilon }\\ \mathit{N}\left(\mathit{\eta },\mathit{\upsilon }\right)\end{array}\right]+\left[\begin{array}{c}\mathit{0}\\ {\mathit{M}}^{-1}\end{array}\right]\tilde{\mathit{\omega }}$$

(11)

$$\mathit{g}\left(\mathit{\varsigma }\right)=\left[\begin{array}{c}\mathit{0}\\ {\mathit{M}}^{-1}\end{array}\right]{\mathit{g}}^{\prime }\left(\mathit{\varsigma }\right)$$

(12)

Define the output function $\mathit{p}=\mathit{h}\left(\mathit{\varsigma }\right)={\left[x,y,z,\theta ,\phi \right]}^T$. By using the differential manifold theory, the property of the Lie derivative, and relative order [44], we construct the transformation of the new vectors:

$$\mathit{p}=\left[h_1\left(\mathit{\varsigma }\right),h_2\left(\mathit{\varsigma }\right),\right.{\left.h_3\left(\mathit{\varsigma }\right),h_4\left(\mathit{\varsigma }\right),h_5\left(\mathit{\varsigma }\right)\right]}^T$$

(13)

$$\mathit{v}=\left[L_{\mathit{f}}{h}_1\left(\mathit{\varsigma }\right),L_{\mathit{f}}{h}_2\left(\mathit{\varsigma }\right),L_{\mathit{f}}{h}_3\left(\mathit{\varsigma }\right),\right.{\left.L_{\mathit{f}}{h}_4\left(\mathit{\varsigma }\right),L_{\mathit{f}}{h}_5\left(\mathit{\varsigma }\right)\right]}^T$$

(14)

Then, the control input $\tilde{\mathit{U}}$ and environmental disturbances $\tilde{F}$ in the new coordinate system satisfy:

$$\begin{array}{c}\tilde{\mathit{U}}=\mathit{\chi }\left(\mathit{\varsigma }\right)+L_{\mathit{g}}{L}_{\mathit{f}}\mathit{h}\left(\mathit{\varsigma }\right)\tilde{\mathit{u}}\hfill \\ \tilde{F}=\mathit{J}\left(\mathit{\eta }\right)\overline{\mathit{\omega }}=\mathit{J}\left(\mathit{\eta }\right){\mathit{M}}^{-1}\tilde{\mathit{\omega }}\hfill \end{array}$$

(15)

After the coordinate transformation (13) and (14), $\mu \left(\mathit{\varsigma }\right)$ is used instead of matrix $L_{\mathit{g}}{L}_{\mathit{f}}\mathit{h}\left(\mathit{\varsigma }\right)$. The dynamical equation of the AUV can be described as follows:

$$\begin{array}{c}\dot{\mathit{p}}=L_{\mathit{f}}\mathit{h}\left(\mathit{\varsigma }\right)=\mathit{J}\left(\mathit{\eta }\right)\mathit{\upsilon }=\mathit{v}\hfill \\ \dot{\mathit{v}}=L_{\mathit{g}}{L}_{\mathit{f}}\mathit{h}\left(\varsigma \right)\tilde{\mathit{u}}+{L_{\mathit{f}}}^{2}h\left(\mathit{\varsigma }\right)\hfill \\ =\mu \left(\mathit{\varsigma }\right)\tilde{\mathit{u}}+\mathit{\chi }\left(\mathit{\varsigma }\right)+\mathit{J}\left(\mathit{\eta }\right){\mathit{M}}^{-1}\tilde{\mathit{\omega }}\hfill \end{array}$$

(16)

In the transformed coordinate system, the vector of the control input is $\tilde{\mathit{U}}$, and $\tilde{F}$ denotes the environmental disturbance forces; therefore, $\tilde{\mathit{u}}={\mu }^{-1}\left(\mathit{\varsigma }\right)\left(\tilde{\mathit{U}}-\mathit{\chi }\left(\mathit{\varsigma }\right)\right)$, $\tilde{\mathit{\omega }}=\mathit{M}{\mathit{J}}^{-1}\left(\mathit{\varsigma }\right)\tilde{F}$.

Following the above steps, the control framework of the multi-AUV formation system, which consists of N AUVs, is shown in Figure 2. The control input $\mathit{U}={\left[{\tilde{\mathit{U}}}_1^T,{\tilde{\mathit{U}}}_2^T,\cdots {\tilde{\mathit{U}}}_N^T\right]}^T$ and the disturbance function $\mathit{F}={\left[{\tilde{F}}_1^T,{\tilde{F}}_2^T,\cdots {\tilde{F}}_N^T\right]}^T$ of the multi-AUV system in the formation are transformed into the control input $\mathit{u}={\left[{\tilde{\mathit{u}}}_1^T,{\tilde{\mathit{u}}}_2^T,\cdots {\tilde{\mathit{u}}}_N^T\right]}^T$ and current disturbance function $\mathit{\omega }={\left[{\tilde{\mathit{\omega }}}_1^T,{\tilde{\mathit{\omega }}}_2^T,\cdots {\tilde{\mathit{\omega }}}_N^T\right]}^T$ of the original multi-AUV model by inverse coordinate transformation. $\mathit{u}$ and $\mathit{\omega }$ are used to satisfy the formation coordination control. The output state $\mathit{\varsigma }$ of the multi-AUV is used as the control input state $(\mathit{p},\mathit{v})$ of formation control in the new coordinates.

In AUV system (16), no simplifications are made on system dynamics and disturbances. In this paper, the system (16) is inherently nonlinear and quite consistent with the actual situation, compared with the simplification of the AUV model in [1], which ignores the disturbances.

2.2. Preliminaries

Consider that a multi-AUV system is comprised of N AUVs and M leaders. In the multi-AUV system, the leader has no neighbor. Without loss of generality, we use $E=\left\{1,2,\cdots ,N\right\}$ to denote the AUV set and $O=\left\{N+1,N+2,\cdots ,N+M\right\}$ to denote the leader set. The matrix between the leader and the AUVs is defined by

$$D_k=\mathrm{diag}\left\{d_{1k},d_{2k},\cdots ,d_{Nk}\right\},k\in \left\{1,2,\cdots ,M\right\}$$

(17)

where $d_{ik}>0$ if AUV i can receive information from the leader k; otherwise, $d_{ik}=0$. It is important to note that an AUV can only follow one leader.

When a multi-AUV formation system performs an ocean survey mission, according to the different locations of the seafloor survey areas and the actual ocean environment, the multi-AUVs may need to arrive at the mission area by different paths, and then the multi-AUV system needs to be divided into several subgroups. In this paper, the multi-AUV system is decomposed into M sub-formations. Each sub-formation contains one leader, and the kth sub-formation contains $N_k$ AUVs. The first $N_1$ AUVs belong to the first sub-formation with the leader $N+1$; the $N_2$ AUVs belong to the second sub-formation with the leader $N+2$; the rest of the $N_M$ AUVs belong to the last sub-formation with the leader $N+M$. The number of AUVs in each sub-formation has to satisfy the following equation

$$\sum _{k=1}^M{N}_k=N.$$

Consider the formation control problem for a multiple stochastic AUV system with Markovian switching topologies and multiple leaders based on the AUV dynamical Equation (3), the dynamics of the AUV i are described by

$$\begin{array}{c}{\dot{\mathit{p}}}_i={\mathit{v}}_i,i\in E\hfill \\ {\dot{\mathit{v}}}_i={\tilde{U}}_i\left(t\right)+f\left({\mathit{p}}_i,{\mathit{v}}_i,t\right),i\in E\hfill \end{array}$$

(18)

where

$${\tilde{\mathit{U}}}_i=\mathit{\chi }\left(\mathit{\varsigma }\right)+\mu \left(\mathit{\varsigma }\right){\tilde{\mathit{u}}}_i$$

$$f\left({\mathit{p}}_i,{\mathit{v}}_i,t\right)={\tilde{F}}_i=\mathit{J}\left(\mathit{\eta }\right){\mathit{M}}^{-1}{\tilde{\mathit{\omega }}}_i$$

${\mathit{p}}_i={\left[x_i,y_i,z_i,{\theta }_i,{\phi }_i\right]}^T$ is the position state of the ith AUV, and ${\mathit{v}}_i={\left[{\overline{u}}_i,{\overline{v}}_i,{\overline{w}}_i,{\overline{q}}_i,{\overline{r}}_i\right]}^T$ is the velocity state after the coordinate transformation, respectively,

$$f\left({\mathit{p}}_i,{\mathit{v}}_i,t\right)=\left[f_1\left({\mathit{p}}_i,{\mathit{v}}_i,t\right)\right.,\cdots ,{\left.f_5\left({\mathit{p}}_i,{\mathit{v}}_i,t\right)\right]}^T$$

is a continuously differentiable nonlinear function.

The dynamics of the leaders can be described by

$$\begin{array}{c}{\dot{\mathit{p}}}_{lj}={\mathit{v}}_{lj},j\in O\hfill \\ {\dot{\mathit{v}}}_{lj}=f\left({\mathit{p}}_{lj},{\mathit{v}}_{lj},t\right),j\in O\hfill \end{array}$$

(19)

where

$$f\left({\mathit{p}}_{lj},{\mathit{v}}_{lj},t\right)=\mathit{J}\left(\mathit{\eta }\right){\mathit{M}}_{{}^{{}^{lj}}}^{-1}{\tilde{\mathit{\omega }}}_{lj}$$

is a differentiable nonlinear function. ${\mathit{p}}_{lj}={\left[x_{lj},y_{lj},z_{lj},{\theta }_{lj},{\phi }_{lj}\right]}^T$ is the position state of the jth leader, ${\mathit{v}}_{lj}=\left[{\overline{u}}_{lj},{\overline{v}}_{lj},{\overline{w}}_{lj},\right.{\left.{\overline{q}}_{lj},{\overline{r}}_{lj}\right]}^T$ is the velocity state after the coordinate transformation of the jth leader,

$$f\left({\mathit{p}}_{lj},{\mathit{v}}_{lj},t\right)={\left[f_1\left({\mathit{p}}_{lj},{\mathit{v}}_{lj},t\right),\cdots ,\left.f_5\left({\mathit{p}}_{lj},{\mathit{v}}_{lj},t\right)\right]\right.}^T.$$

It should be pointed out that the trajectories of the leaders are determined by the actual marine survey mission and the ocean environment, and the motions are assumed to be independent of that of other follower AUVs.

Next, we have the following definition:

Definition 1.

All AUVs in the multiple stochastic AUV system (18) and (19) converge to their respective leader’s trajectory if there exist fixed consensus gains $K_1,K_2$ such that for any given initial data ${\mathit{p}}_i\left(t_0\right),{\mathit{v}}_i\left(t_0\right),{\mathit{p}}_{li}\left(t_0\right),{\mathit{v}}_{li}\left(t_0\right)$ and initial distribution of ${\pi }_{ij}\left(t_0\right)$ and $r\left(t_0\right)$,

$$\begin{array}{c}\underset{t\to \infty }{lim}\mathbb{E}\left({∥{\mathit{p}}_i\left(t\right)-{\mathit{p}}_{lj}\left(t\right)-{{\mathit{p}}_i}^{\ast }∥}^2\right)=0\hfill \\ \underset{t\to \infty }{lim}\mathbb{E}\left({∥{\mathit{v}}_i\left(t\right)-{\mathit{v}}_{lj}\left(t\right)∥}^2\right)=0\hfill \end{array}$$

(20)

for $\forall i\in N_i,j=N+i$, where ${{\mathit{p}}_i}^{\ast }$ is the desired constant relative position vector between each AUV i and the leader $lj$.

To solve the above formation control problem, we give the following assumptions and lemmas.

Assumption 1.

The union of communication graphs is balanced and contains a directed spanning tree rooting at each leader.

Assumption 2.

For the follower AUV i, the sum of all communication coefficients of the other sub-formations that can exchange information with the AUV i is zero, i.e., ${\sum }_{j\in {\tilde{N}}_i}{a}_{ij}=0$,${\tilde{N}}_i$ represents the $ith$ AUV’s neighbor set in the other sub-formations.

The below Lipschitz condition [43,45,46,47,48,49] has been widely used to limit the unknown and bounded nonlinear functions.

Assumption 3.

The nonlinear continuous Lipschitz function $f\left(·,·,t\right)$ is assumed to satisfy the following inequality:

$$∥\tilde{f}\left({\mathit{p}}_i,{\mathit{v}}_i,t\right)-\tilde{f}\left({\mathit{p}}_j,{\mathit{v}}_j,t\right)∥\le l_1∥{\mathit{\epsilon }}_i-{\mathit{\epsilon }}_j∥$$

(21)

where $\tilde{f}\left({\mathit{p}}_i,{\mathit{v}}_i,t\right)={\left[\begin{array}{cc}\mathit{0}& f\left({\mathit{p}}_i,{\mathit{v}}_i,t\right)\end{array}\right]}^T$, $l_1>0$ is a constant.

Lemma 1

([50]). Let $S_1,S_2,S_3$ be given matrices such that $S_1>0$. Then

$$S=\left[\begin{array}{cc}{S}_1& S_2\\ S_2^T& S_3\end{array}\right]>0$$

(22)

if and only if $S_3-S_2^T{S}_1^{-1}{S}_2>0$.

Lemma 2

([51]). For M-matrix L, there exists a matrix $\mathrm{\Lambda }=diag\left\{e_1,e_2,\cdots ,e_N\right\}>0$ satisfying $L^T\mathrm{\Lambda }+\mathrm{\Lambda }L>0$, and ${\left[e_1,e_2,\cdots ,e_N\right]}^T={\left(L^T\right)}^{-1}{1}_N$.

Lemma 3

([52]). Suppose that $f\left(t\right)$ is $F-$ measurable and that $E\left(f\left(t\right)d\left(1_{\left\{r\left(t\right)=i\right\}}\right)\right)$ exists. Then for any $i\in S$,

$$E\left(f\left(t\right)d\left(1_{\left\{r\left(t\right)=i\right\}}\right)\right)=\sum _{j=1}^n{\pi }_{ji}E\left(f\left(t\right)1_{\left\{r\left(t\right)=i\right\}}d\left(t\right)\right)+o\left(d\left(t\right)\right)$$

Lemma 4

([53]). Consider a stochastic differential equation

$$d\epsilon \left(t\right)=f\left(t,\epsilon \left(t\right),r\left(t\right)\right)dt+g\left(t,\epsilon \left(t\right),r\left(t\right)\right)d\kappa \left(t\right)$$

(23)

where $r\left(t\right)$ obeys a Markov process, and $\kappa \left(t\right)$ is the standard Brownian motion, for a positive definite $V(t,\epsilon \left(t\right))$ satisfies $V_i\left(t,\epsilon \left(t\right)\right)1_{\left\{r\left(t\right)=i\right\}}=V\left(t,\epsilon \left(t\right)\right)$, the infinitesimal differential generator is defined as

$$\begin{array}{c}\mathcal{L}\left[\mathbb{E}{V}_i\left(t,\epsilon \left(t\right)\right)\right]=\mathbb{E}\left\{\left[\frac{\partial V\left(t,\epsilon \left(t\right)\right)}{\partial t}\right.\right.+\frac{\partial V\left(t,\epsilon \left(t\right)\right)}{\partial \epsilon }f\hfill \\ +\left.\left.\frac{1}{2}Trace\left[g^T\frac{{\partial }^{2}V\left(t,\epsilon \left(t\right)\right)}{\partial {\epsilon }^2}g\right]\right]\right\}{1}_{\left\{r\left(t\right)=i\right\}}\hfill \\ +{\displaystyle \sum _{j=1}^n}{\pi }_{ji}\mathbb{E}{V}_j\left(t\right)\hfill \end{array}$$

(24)

3. Main Results

In the complicated ocean environment, the AUV’s actuator may malfunction. For the ith follower AUV, the actuator fault model is established as follows [27]:

$$u_i^F\left(t\right)={\rho }_i\left(t\right){\tilde{U}}_i\left(t\right)$$

(25)

where random variable $0\le {\rho }_i\left(t\right)\le 1$ represents the unknown efficiency control input of the ith AUV with $\mathbb{E}\left[{\rho }_i\left(t\right)\right]\le \varpi$, $0<\varpi <1$, $\rho \left(t\right)=diag\left\{{\rho }_1\left(t\right),{\rho }_2\left(t\right),\cdots ,{\rho }_N\left(t\right)\right\}$. In this paper, we assume that the leader AUV in each sub-formation is fault-free.

Noting that the AUV propulsion system will be affected by the random ocean noise, and the position and velocity state information received by the AUV from its neighbors will be attenuated to a certain extent, it is important to take the impact of noise into account.

Based on the above analysis, the formation control consensus protocol can be designed as

$$\begin{array}{c}{\tilde{U}}_i\left(t\right)\hfill \\ =-\gamma K_1\left[{\displaystyle \sum _{j\in N_i^{}\left(t\right)}}{a}_{ij}\left(t\right)\left(\left({\mathit{p}}_i\left(t\right)-{\mathit{p}}_i^{\ast }\right)-\left({\mathit{p}}_j\left(t\right)-{\mathit{p}}_j^{\ast }\right)\right)\right.\hfill \\ \left.+d_{ik}\left(\left({\mathit{p}}_i\left(t\right)-{\mathit{p}}_i^{\ast }\right)-{\mathit{p}}_{lk}\left(t\right)\right)\right]\hfill \\ -\gamma K_2\left[{\displaystyle \sum _{j\in N_i^{}\left(t\right)}}{a}_{ij}\left(t\right)\left({\mathit{v}}_i\left(t\right)-{\mathit{v}}_j\left(t\right)\right)\right.\left.+d_{ik}\left({\mathit{v}}_i\left(t\right)-{\mathit{v}}_{lk}\left(t\right)\right)\right]\hfill \\ -{\sigma }_1\left[{\displaystyle \sum _{j\in N_i^{}\left(t\right)}}{\tilde{a}}_{ij}\left(t\right)\left(\left({\mathit{p}}_i\left(t\right)-{\mathit{p}}_i^{\ast }\right)-\left({\mathit{p}}_j\left(t\right)-{\mathit{p}}_j^{\ast }\right)\right)\right.\hfill \\ \left.+{\tilde{d}}_i\left(\left({\mathit{p}}_i\left(t\right)-{\mathit{p}}_i^{\ast }\right)-{\mathit{p}}_{lk}\left(t\right)\right)\right]\vartheta \left(t\right)\hfill \\ -{\sigma }_2\left[{\displaystyle \sum _{j\in N_i^{}\left(t\right)}}{\tilde{a}}_{ij}\left(t\right)\left({\mathit{v}}_i\left(t\right)-{\mathit{v}}_j\left(t\right)\right)\right.\left.+{\tilde{d}}_i\left({\mathit{v}}_i\left(t\right)-{\mathit{v}}_{lk}\left(t\right)\right)\right]\vartheta \left(t\right)\hfill \end{array}$$

(26)

where $i\in E,k\in O,\gamma >0,K_1,K_2\in {\mathbb{R}}^{5\times 5}$ are gain matrices to be designed, ${\sigma }_1$ and ${\sigma }_2$ represent noise intensity, ${\tilde{a}}_{ij}\left(t\right)$ represents the weight of the noise in the information channel of the state variable at time t from the information sending AUV i to the information receiving AUV j,

$$W=\left[{\stackrel{\widetilde{˜}}{a}}_{ij}\right],{\stackrel{\widetilde{˜}}{a}}_{ii}=\sum _{j=1,j\ne i}^n{\tilde{a}}_{ij};{\stackrel{\widetilde{˜}}{a}}_{ij}=-{\tilde{a}}_{ij},i\ne j$$

similarly, ${\tilde{d}}_i$ denotes the weight of the noise in the information channel of the leader AUV, $\tilde{D}=diag\left\{{\tilde{d}}_1,{\tilde{d}}_2,\cdots ,{\tilde{d}}_N\right\}$, $C=W+\tilde{D}$. $\vartheta \left(t\right)$ is the measurement noise satisfying

$${\int }_0^t\vartheta \left(\tau \right)d\tau =\kappa \left(t\right)$$

$\kappa \left(t\right)$ is a one-dimensional Brownian motion defined in total probability space $\left(\mathrm{\Omega },\mathcal{F},\mathbb{P}\right)$.

The communication topology among agents is described by a randomly switching graph: $\mathcal{G}=\left(\mathcal{V},\epsilon \left(t\right),\mathcal{A}\left(t\right)\right)$, which is governed by a Markovian chain $r\left(t\right)$ with the state space $S=\left\{1,2,\cdots ,n\right\}$, the transition probability matrix $\mathrm{\Pi }=\left({\pi }_{ij}\right)\in {\mathbb{R}}^{n\times n}$ is given by:

$$P\left(r\left(t+h\right)=j\left|r\left(t\right)=i\right.\right)=\left\{\begin{array}{c}{\pi }_{ij}h+o\left(h\right),i\ne j\hfill \\ 1+{\pi }_{ii}h+o\left(h\right),i=j\hfill \end{array}\right.$$

(27)

where $h>0,{\pi }_{ij}$ is the transition rate from i to j if $i\ne j$, ${\pi }_{ii}=-{\displaystyle \sum _{j\ne i}}{\pi }_{ij}$ [54].

This paper considers the Markov process $r\left(t\right)$ ergodic, and the invariant distribution is

$$\pi ={\left({\pi }_1,\cdots ,{\pi }_S\right)}^T$$

for all n possible communication graphs $\mathcal{G}\in \left({\mathcal{G}}^{\left(1\right)},{\mathcal{G}}^{\left(2\right)},\cdots ,{\mathcal{G}}^{\left(n\right)}\right)$ with ${\mathcal{G}}^{\left(i\right)}=\left(\mathcal{V},{\epsilon }^{\left(i\right)}\left(t\right),\right.\left.{\mathcal{A}}^{\left(i\right)}\left(t\right)\right)$, respectively, $N_i\left(t\right)$ represents the neighbor’s set of the ith AUV. The union communication graph is defined as

$$\overline{\mathcal{G}}={\bigcup }_{i=1}^n{\mathcal{G}}^{\left(i\right)}=\left(\mathcal{V},{\bigcup }_{i=1}^n{\epsilon }^{\left(i\right)}\left(t\right),{\bigcup }_{i=1}^n{\mathcal{A}}^{\left(i\right)}\left(t\right)\right)$$

(28)

and the Laplacian matrix of communication graph $\mathcal{G}\left(t\right)$ is defined as $L\left(t\right)$, we have

$$L\left(t\right)\in \left\{L_1,L_2,\cdots ,L_3\right\}$$

(29)

where $L_i$ is the Laplacian matrix of communication graph ${\mathcal{G}}^{\left(i\right)}$. The union of Laplacian matrix is denoted by $L={\displaystyle \sum _{k=1}^n}{L}_k$. The M sub networks of $\mathcal{G}$ on ${\mathcal{V}}_j=\left\{v_1,v_2,\cdots v_{N_j}\right\}$ is denoted by ${\mathcal{G}}_j$, $j=1,2,\cdots ,M$, and similar notations will be held for ${\mathcal{G}}_j^{\left(k\right)}$.

In this paper, we assume that the stochastic processes $r\left(t\right)$ and $\kappa \left(t\right)$ are independent of each other. According to the control protocol (26), the factors affected by noise in underwater acoustic communication are reflected in the form of state measurement error in the control protocol, which is beneficial to the effectiveness analysis of formation control.

Set

$$\begin{array}{c}\mathit{\xi }\left(t\right)={\left[{\left({\mathit{p}}_1\left(t\right)-{\mathit{p}}_1^{\ast }\right)}^T,{\left({\mathit{p}}_2\left(t\right)-{\mathit{p}}_2^{\ast }\right)}^T,...,{\left({\mathit{p}}_N\left(t\right)-{\mathit{p}}_N^{\ast }\right)}^T\right]}^T\hfill \\ \mathit{\zeta }\left(t\right)={\left[{\mathit{v}}_1^T\left(t\right),{\mathit{v}}_2^T\left(t\right),...,{\mathit{v}}_N^T\left(t\right)\right]}^T\hfill \end{array}$$

$$\begin{array}{c}{\mathit{\xi }}^l\left(t\right)=\left[{\mathit{p}}_{l,N+1}^T\left(t\right),\cdots ,{\mathit{p}}_{l,N+1}^T\left(t\right),{\mathit{p}}_{l,N+2}^T\left(t\right),\cdots ,{\mathit{p}}_{l,N+2}^T\left(t\right),\right.\hfill \\ {\left.\cdots ,{\mathit{p}}_{l,N+M}^T\left(t\right),\cdots ,{\mathit{p}}_{l,N+M}^T\left(t\right)\right]}^T\hfill \\ {\mathit{\zeta }}^l\left(t\right)=\left[{\mathit{v}}_{l,N+1}^T\left(t\right),\cdots ,{\mathit{v}}_{l,N+1}^T\left(t\right),{\mathit{v}}_{l,N+2}^T\left(t\right),\cdots ,{\mathit{v}}_{l,N+2}^T\left(t\right),\right.\hfill \\ {\left.\cdots ,{\mathit{v}}_{l,N+M}^T\left(t\right),\cdots ,{\mathit{v}}_{l,N+M}^T\left(t\right)\right]}^T\hfill \end{array}$$

$$\begin{array}{c}\stackrel{⌢}{\mathit{F}}\hfill \end{array}\left(\mathit{p}\left(t\right),\mathit{v}\left(t\right),t\right)=\left[\begin{array}{c}f\left({\mathit{p}}_1\left(t\right),{\mathit{v}}_1\left(t\right),t\right)\\ ⋮\\ f\left({\mathit{p}}_{N_1}\left(t\right),{\mathit{v}}_{N_1}\left(t\right),t\right)\\ f\left({\mathit{p}}_{N_1+1}\left(t\right),{\mathit{v}}_{N_1+1}\left(t\right),t\right)\\ ⋮\\ f\left({\mathit{p}}_{N_1+N_2}\left(t\right),{\mathit{v}}_{N_1+N_2}\left(t\right),t\right)\\ ⋮\\ f\left({\mathit{p}}_{N-N_M+1}\left(t\right),{\mathit{v}}_{N-N_M+1}\left(t\right),t\right)\\ ⋮\\ f\left({\mathit{p}}_N\left(t\right),{\mathit{v}}_N\left(t\right),t\right)\end{array}\right]$$

where ${\mathit{\xi }}^l\left(t\right) \in {\mathbb{R}}^N$, ${\mathit{\zeta }}^l\left(t\right) \in {\mathbb{R}}^N$, ${\mathit{p}}_{l,N+i}^T\left(t\right)$, i = 1, …, M in ${\mathit{\xi }}^l\left(t\right)$ is repeated Ni times, and ${\mathit{v}}_{l,N+i}^T\left(t\right)$, i = 1, …, M in ${\mathit{\zeta }}^l\left(t\right)$ is repeated Ni times, $\stackrel{⌢}{\mathit{F}}\left(p\left(t\right),v\left(t\right),t\right)$ is repeated Ni times too.

The system (18) and (19) can be expressed in a structured form as:

$$\begin{array}{c}\dot{\xi }\left(t\right)=\zeta \left(t\right)\hfill \\ \dot{\zeta }\left(t\right)=-\left(\left(L\left(t\right)+D_k\right)\otimes \rho \left(t\right)\gamma K_1\right)\xi \left(t\right)\hfill \\ -\left(C\otimes \rho \left(t\right){\sigma }_1{I}_5\right)\zeta \left(t\right)\xi \left(t\right)\hfill \\ -\left(\left(L\left(t\right)+D_k\right)\otimes \rho \left(t\right)\gamma K_2\right)\varsigma \left(t\right)\hfill \\ -\left(C\otimes \rho \left(t\right){\sigma }_2{I}_5\right)\zeta \left(t\right)\varsigma \left(t\right)\hfill \\ +\left(D_k\otimes \rho \left(t\right)K_1\right){\xi }^l\left(t\right)+\left(D_k\otimes \rho \left(t\right)K_2\right){\zeta }^l\left(t\right)\hfill \\ +\left(\tilde{D}\otimes \rho \left(t\right){\sigma }_1{I}_5\right)\zeta \left(t\right){\xi }^l\left(t\right)\hfill \\ +\left(\tilde{D}\otimes \rho \left(t\right){\sigma }_2{I}_5\right)\zeta \left(t\right){\zeta }^l\left(t\right)\hfill \\ +\stackrel{⌢}{F}\left(p\left(t\right),v\left(t\right),t\right)\hfill \end{array}$$

(30)

where $D_k=diag\left\{d_{1k},d_{2k},\cdots ,d_{Nk}\right\},k=N+1,\cdots ,N+M$.

Let ${\overline{\mathit{p}}}_{li}\left(t\right)={\mathit{p}}_i\left(t\right)-{\mathit{p}}_{lk}\left(t\right)-{\mathit{p}}_i^{\ast },{\overline{\mathit{v}}}_{li}\left(t\right)={\mathit{v}}_i\left(t\right)-{\mathit{v}}_{lk}\left(t\right),i\in E$, $k\in O$, and the system can be written in the following equations associated with the error dynamics:

$$\begin{array}{c}{\dot{\overline{\mathit{p}}}}_{li}\left(t\right)={\overline{\mathit{v}}}_{li}\left(t\right),\hfill \\ {\dot{\overline{\mathit{v}}}}_{li}\left(t\right)=-\rho \left(t\right)\gamma K_1{\displaystyle \sum _{j\in N_i^{}\left(t\right)}}{h}_{ij}{\overline{\mathit{p}}}_{li}\left(t\right)-\rho \left(t\right)\gamma K_2{\displaystyle \sum _{j\in N_i^{}\left(t\right)}}{h}_{ij}{\overline{\mathit{v}}}_{li}\left(t\right)\hfill \\ -\rho \left(t\right){\sigma }_1\vartheta \left(t\right){\displaystyle \sum _{j\in N_i^{}\left(t\right)}}{\tilde{h}}_{ij}{\overline{\mathit{p}}}_{li}\left(t\right)-\rho \left(t\right){\sigma }_2\vartheta \left(t\right){\displaystyle \sum _{j\in N_i^{}\left(t\right)}}{\tilde{h}}_{ij}{\overline{\mathit{v}}}_{li}\left(t\right)\hfill \\ +f\left({\mathit{p}}_i\left(t\right)-{\mathit{p}}_i^{\ast },{\mathit{v}}_i\left(t\right),t\right)-f\left({\mathit{p}}_{lj}\left(t\right),{\mathit{v}}_{lj}\left(t\right),t\right)\hfill \end{array}$$

(31)

where ${\left[h_{ij}\right]}_{N\times N}=H_k=L\left(t\right)+D_k$, ${\left[{\tilde{h}}_{ij}\right]}_{N\times N}=C=W+\tilde{D}$.

Set

$$\begin{array}{c}\mathit{\epsilon }\left(t\right)={\left[{\mathit{\epsilon }}_1^T\left(t\right),\cdots ,{\mathit{\epsilon }}_N^T\left(t\right)\right]}^T\hfill \\ ={\left[{\left[{\overline{\mathit{p}}}_{l,1}^T\left(t\right),{\overline{\mathit{v}}}_{l,1}^T\left(t\right)\right]}^T,\cdots ,{\left[{\overline{\mathit{p}}}_{l,N}^T\left(t\right),{\overline{\mathit{v}}}_{l,N}^T\left(t\right)\right]}^T\right]}^T\hfill \end{array}$$

The systems (18) and (19) can be recast in the following compact form:

$$\begin{array}{c}d\mathit{\epsilon }\left(t\right)=\left[\left(I_N\otimes A\right)\mathit{\epsilon }\left(t\right)-\left(H_k\otimes \gamma {\overline{K}}_1\right)\mathit{\epsilon }\left(t\right)\right.\hfill \\ \left.+\overline{F}\left(\mathit{p}\left(t\right),\mathit{v}\left(t\right),t\right)\right]dt-\left(C\otimes {\overline{K}}_2\right)\mathit{\epsilon }\left(t\right)d\kappa \left(t\right)\hfill \end{array}$$

(32)

where

$$A=\left[\begin{array}{cc}{0}_5& I_5\\ 0_5& 0_5\end{array}\right],B=\left[\begin{array}{cc}{0}_5& 0_5\\ 0_5& I_5\end{array}\right]$$

$${\overline{K}}_1=\left[\begin{array}{cc}{0}_5& 0_5\\ \rho \left(t\right)K_1& \rho \left(t\right)K_2\end{array}\right],{\overline{K}}_2=\left[\begin{array}{cc}{0}_5& 0_5\\ \rho \left(t\right){\sigma }_1{I}_5& \rho \left(t\right){\sigma }_2{I}_5\end{array}\right]$$

$$\begin{array}{c}\overline{\mathit{F}}=\overline{\mathit{F}}\left(p\left(t\right),v\left(t\right),t\right)=\hfill \\ \left[\begin{array}{c}{0}_5\\ f\left({\mathit{p}}_1^{}-{\mathit{p}}_1^{\ast },{\mathit{v}}_1^{},t\right)-f\left({\mathit{p}}_{l,N+1}^{},{\mathit{v}}_{l,N+1}^{},t\right)\\ ⋮\\ 0_5\\ f\left({\mathit{p}}_{N_1}^{}-{\mathit{p}}_{N_1}^{\ast },{\mathit{v}}_{N_1}^{},t\right)-f\left({\mathit{p}}_{l,N+1}^{},{\mathit{v}}_{l,N+1}^{},t\right)\\ 0_5\\ f\left({\mathit{p}}_{N_1+1}-{\mathit{p}}_{N_1+1}^{\ast },{\mathit{v}}_{N_1+1}^{},t\right)-f\left({\mathit{p}}_{l,N+2}^{},{\mathit{v}}_{l,N+2}^{},t\right)\\ ⋮\\ 0_5\\ f\left({\mathit{p}}_{N_1+N_2}^{}-{\mathit{p}}_{N_1+N_2}^{\ast },{\mathit{v}}_{N_1+N_2}^{},t\right)-f\left({\mathit{p}}_{l,N+2}^{},{\mathit{v}}_{l,N+2}^{},t\right)\\ ⋮\\ 0_5\\ f\left({\mathit{p}}_{N-N_M+1}^{}-{\mathit{p}}_{N-N_M+1}^{\ast },{\mathit{v}}_{N-N_M+1}^{},t\right)-f\left({\mathit{p}}_{l,N+M}^{},{\mathit{v}}_{l,N+M}^{},t\right)\\ ⋮\\ 0_5\\ f\left({\mathit{p}}_N^{}-{\mathit{p}}_N^{\ast },{\mathit{v}}_N^{},t\right)-f\left({\mathit{p}}_{l,N+M}^{},{\mathit{v}}_{l,N+M}^{},t\right)\end{array}\right]\hfill \end{array}$$

(33)

By applying the properties of the elementary matrix, the following relationship can be obtained

$$\widehat{E}·\overline{\mathit{F}}=\left(\begin{array}{c}{0}_{5N}\\ \tilde{\mathit{F}}\left(\mathit{p}\left(t\right),\mathit{v}\left(t\right),t\right)\end{array}\right)$$

(34)

where

$$\widehat{E}=\left(\begin{array}{cccccccccc}{I}_5& 0_5& 0_5& 0_5& \cdots & 0_5& 0_5& 0_5& 0_5& 0_5\\ 0_5& 0_5& 0_5& 0_5& \cdots & 0_5& 0_5& 0_5& I_5& 0_5\\ 0_5& 0_5& 0_5& 0_5& \cdots & 0_5& I_5& 0_5& 0_5& 0_5\\ & \cdots & & & \cdots & & & \cdots & & \\ 0_5& 0_5& 0_5& 0_5& \cdots & I_5& 0_5& 0_5& 0_5& 0_5\\ 0_5& 0_5& 0_5& 0_5& \cdots & 0_5& 0_5& I_5& 0_5& 0_5\\ 0_5& 0_5& 0_5& 0_5& \cdots & 0_5& 0_5& 0_5& 0_5& I_5\end{array}\right)$$

is an elementary matrix.

Lemma 5.

For communication expectation matrix $\mathbb{E}\left[L\left(t\right)\right.$$\left.+D_k\right]$, there exists a matrix $\mathrm{\Lambda }=diag\left\{e_1,e_2,\cdots ,e_N\right\}>0$ satisfying $\mathbb{E}\left[L^T\left(t\right)+D_k\right]\mathrm{\Lambda }+\mathrm{\Lambda }\mathbb{E}\left[L\left(t\right)+D_k\right]>0$, and ${\left[e_1,e_2,\cdots ,e_N\right]}^T$$={\left(\mathbb{E}\left[L\left(t\right)\right]\right)}^{-1}{1}_N$.

Proof. 

According to literature [55], when $\mathrm{{\rm Y}}={\left[e_1,e_2,\cdots ,e_N\right]}^T$ is the left eigenvector of $\mathbb{E}\left[L\left(t\right)\right]$ associated with eigenvalue $\lambda =0$, the generalized graph Laplacian potential can be defined as follows

$$\begin{array}{c}{V}_L={\displaystyle \sum _{i,j=1}^N}{e}_i{a}_{ij}{\left({\mathit{\epsilon }}_j-{\mathit{\epsilon }}_i\right)}^2\hfill \\ ={\mathit{\epsilon }}^T\left(\mathbb{E}\left[L^T\left(t\right)\right]\mathrm{\Lambda }+\mathrm{\Lambda }\mathbb{E}\left[L\left(t\right)\right]\right)\mathit{\epsilon }\hfill \end{array}$$

(35)

From Assumption 1, it is apparent that

$$\mathrm{\Lambda }=diag\left\{e_1,e_2,\cdots ,e_N\right\}>0,$$

(36)

although the matrix $\mathbb{E}\left[L\left(t\right)\right]$ does not satisfy the M-matrix condition. Not all $a_{ij}$ in the adjacency matrix are greater than zero when $i\ne j$, but according to Assumption 2, for element $a_{ij}$, which is less than zero, there must exist $a_{ik}$, $a_{ik}=-a_{ij}$ in matrix $E\left[L\left(t\right)\right]$ such that $e_i{a}_{ij}{\left({\mathit{\epsilon }}_j-{\mathit{\epsilon }}_i\right)}^2+e_i{a}_{ik}{\left({\mathit{\epsilon }}_j-{\mathit{\epsilon }}_i\right)}^2=0$; this yields that

$${\mathit{\epsilon }}^T\left(\mathbb{E}\left[L^T\left(t\right)\right]\mathrm{\Lambda }+\mathrm{\Lambda }\mathbb{E}\left[L\left(t\right)\right]\right)\mathit{\epsilon }>0$$

(37)

and according to the definition of $D_k$, we have

$${\mathit{\epsilon }}^T\left(D_k\mathrm{\Lambda }+\mathrm{\Lambda }{D}_k\right)\mathit{\epsilon }\ge 0,$$

(38)

thus, we deduce

$$\mathbb{E}\left[L^T\left(t\right)+D_k\right]\mathrm{\Lambda }+\mathrm{\Lambda }\mathbb{E}\left[L\left(t\right)+D_k\right]>0.$$

(39)

Our study is motivated by the M-matrix results in [55], where the M-matrix L satisfies Lemma 2. While the result in Lemma 5 is similar to the conclusion based on a non-M matrix, which generalizes the properties of the matrix.

Theorem 1.

Consider the multiple stochastic AUV formation system (32) consisting of M leaders and N followers, the switch topologies are ${\mathcal{G}}^{\left(1\right)},{\mathcal{G}}^{\left(2\right)},\cdots ,{\mathcal{G}}^{\left(n\right)}$. Under Assumptions 1, 2 and 3, the formation control problem of system (32) can achieve consensus if there exists real matrix P, such that

$$\left[\begin{array}{cc}AP+P{A}^T+a_{0}P-c_{0}PP-b_{0}B{B}^T& P\\ P& -\frac{1}{c_0}I\end{array}\right]<0$$

(40)

where $a_0=\varpi {\sigma }_2^2{\lambda }_{max}\left(C\right)+2l_1,b_0=\frac{\varpi \gamma {\lambda }_{min}\left(\mathbb{E}\left[H_k^T\right]\mathrm{\Lambda }+\mathrm{\Lambda }\mathbb{E}\left[H_k^{}\right]\right)}{max\left(e_j\right)}$, $c_0>0$.

Proof. 

For $i\in S$ and the Markov process $r\left(t\right)$, construct the function candidates

$$V_i\left(t\right)={\mathit{\epsilon }}^T\left(t\right)\left(\mathrm{\Lambda }\otimes P^{-1}\right)\mathit{\epsilon }\left(t\right)1_{\left\{r\left(t\right)=i\right\}}$$

(41)

then the Lyapunov functional candidate is

$$V\left(t\right)={\mathit{\epsilon }}^T\left(t\right)\left(\mathrm{\Lambda }\otimes P^{-1}\right)\mathit{\epsilon }\left(t\right)$$

(42)

Constructing the Lyapunov expectation equation [56]:

$$\mathbb{E}\left[V\left(t\right)\right]=\sum _{i=1}^{n}P\left\{r\left(t\right)=i\right\}\left[V\left(t\right)\left|r\left(t\right)=i\right.\right]$$

(43)

By Lemma 4, the infinitesimal differential generator along (32) is given as

$$\begin{array}{c}\mathcal{L}\left[\mathbb{E}V\left(t\right)\right]={\displaystyle \sum _{i=1}^n}P\left\{r\left(t\right)=i\right\}\mathbb{E}\left\{z^T\left(t\right)\right.\left(\mathrm{\Lambda }\otimes P^{-1}\right)\mathit{\epsilon }\left(t\right)+{\mathit{\epsilon }}^T\left(t\right)\left(\mathrm{\Lambda }\otimes P^{-1}\right)z\left(t\right)\hfill \\ +Trace\left.\left[{\mathit{\epsilon }}^T\left(t\right){\left(C\otimes B{\overline{K}}_2\right)}^T\left(\mathrm{\Lambda }\otimes P^{-1}\right)\left(C\otimes B{\overline{K}}_2\right)\mathit{\epsilon }\left(t\right)\right]\right\}\hfill \end{array}$$

(44)

where:

$$\mathit{z}\left(t\right)=\left(I_N\otimes A\right)\mathit{\epsilon }\left(t\right)-\left(H_k\otimes \gamma B{\overline{K}}_1\right)\mathit{\epsilon }\left(t\right)+\overline{\mathit{F}}\left(\mathit{p}\left(t\right),\mathit{v}\left(t\right),t\right)$$

and consider:

$$\begin{array}{c}\mathbb{E}\left[Trace\left[{\mathit{\epsilon }}^T\left(t\right){\left(C\otimes B{\overline{K}}_2\right)}^T\left(\mathrm{\Lambda }\otimes P^{-1}\right)\left(C\otimes B{\overline{K}}_2\right)\mathit{\epsilon }\left(t\right)\right]\right]\hfill \\ \le \varpi {\sigma }_2^2{\lambda }_{max}\left(C\right)\mathbb{E}\left[{\mathit{\epsilon }}^T\left(t\right)\left(\mathrm{\Lambda }\otimes P^{-1}\right)\mathit{\epsilon }\left(t\right)\right]\hfill \end{array}$$

(45)

Let

$$\left[\begin{array}{cc}{0}_5& 0_5\\ K_1& K_2\end{array}\right]=B^T{P}^{-1}$$

(46)

we can obtain

$$\begin{array}{c}\mathbb{E}\left\{z^T\left(t\right)\right.\left(\mathrm{\Lambda }\otimes P^{-1}\right)\epsilon \left(t\right)+{\epsilon }^T\left(t\right)\left(\mathrm{\Lambda }\otimes P^{-1}\right)\left.z\left(t\right)\right\}\hfill \\ =\mathbb{E}\left\{{\epsilon }^T\left(t\right)\right.\left[\mathrm{\Lambda }\otimes \left(P^{-1}A+A^T{P}^{-1}+\varpi {\sigma }_2^2{\lambda }_{max}\left(C\right)P^{-1}\right)\right]\epsilon \left(t\right)\hfill \\ -{\epsilon }^T\left(t\right)\left[\gamma \left(H_k^T\mathrm{\Lambda }+\mathrm{\Lambda }{H}_k\right)\otimes \varpi \left(P^{-1}B{B}^T{P}^{-1}\right)\right]\epsilon \left(t\right)\hfill \\ +2{\epsilon }^T\left(t\right)\left(\mathrm{\Lambda }\otimes P^{-1}\right)\left.\overline{F}\left(p\left(t\right),v\left(t\right),t\right)\right\}\hfill \end{array}$$

(47)

According to Assumption 3, when $\tilde{f}\left(·,·,t\right)$ satisfies Eq. (21), we have

$$2{\mathit{\epsilon }}^T\left(t\right)\left(\mathrm{\Lambda }\otimes P^{-1}\right)\overline{F}\left(\tilde{x}\left(t\right),\tilde{v}\left(t\right),t\right)\le 2l_1{\mathit{\epsilon }}^T\left(t\right)\left(\mathrm{\Lambda }\otimes P^{-1}\right)\mathit{\epsilon }\left(t\right)$$

(48)

From Lemma 5 and the properties of $\rho \left(t\right)$, we obtain

$$\begin{array}{c}\mathbb{E}\left[{\mathit{\epsilon }}^T\left(t\right)\left[\left(H_k^T\mathrm{\Lambda }+\mathrm{\Lambda }{H}_k\right)\otimes \varpi \left(P^{-1}B{B}^T{P}^{-1}\right)\right]\mathit{\epsilon }\left(t\right)\right]\hfill \\ =\varpi \mathbb{E}\left[{\mathit{\epsilon }}^T\left(t\right)\left[\gamma \left(H_k^T\mathrm{\Lambda }+\mathrm{\Lambda }{H}_k\right)\otimes \left(P^{-1}B{B}^T{P}^{-1}\right)\right]\mathit{\epsilon }\left(t\right)\right]\hfill \\ \ge \frac{\varpi \gamma {\lambda }_{min}\left(\mathbb{E}\left[H_k^T\right]\mathrm{\Lambda }+\mathrm{\Lambda }\mathbb{E}\left[H_k\right]\right)}{max\left(e_j\right)}\mathbb{E}\left[{\mathit{\epsilon }}^T\left(t\right)\left(\mathrm{\Lambda }\otimes P^{-1}B{B}^T{P}^{-1}\right)\mathit{\epsilon }\left(t\right)\right]\hfill \end{array}$$

(49)

From (45), (47), (48) and (49), we obtain

$$\mathcal{L}\left[\mathbb{E}V\left(t\right)\right]=\mathbb{E}\left\{{\mathit{\epsilon }}^T\left(t\right)\left\{\mathrm{\Lambda }\otimes \mathrm{\Theta }\right\}\mathit{\epsilon }\left(t\right)\right\}$$

(50)

where

$$\mathrm{\Theta }=P^{-1}A+A^T{P}^{-1}+a_0{P}^{-1}-b_0{P}^{-1}B{B}^T{P}^{-1}$$

$$\varpi {\sigma }_2^2{\lambda }_{max}\left(C\right)+2l_1=a_0,\frac{\varpi \gamma {\lambda }_{min}\left(\mathbb{E}\left[H_k^T\right]\mathrm{\Lambda }+\mathrm{\Lambda }\mathbb{E}\left[H_k^{}\right]\right)}{max\left(e_j\right)}=b_0$$

According to Lemma 1, it can be concluded that if matrix inequality (40) is achieved, then

$$\mathcal{L}\left[\mathbb{E}V\left(t\right)\right]<0$$

(51)

In summary, if the conditions in Theorem 1 hold together, the fault-tolerant multiple stochastic AUV system under Markovian switching topologies and multiple leaders attains formation control. The proof is completed.

According to Theorem 1, the design steps for the parameters in the control protocol (26) are as follows:

Step 1:

Determine the structure of switching networks and communication topology probability transition.

Step 2:

According to Lemma 5, solve matrix $\mathrm{\Lambda }=diag\left\{e_1,e_2,\cdots ,e_N\right\}$ by equation ${\left[e_1,e_2,\cdots ,e_N\right]}^T={\left(\mathbb{E}\left[L\left(t\right)\right]\right)}^{-1}{1}_N$.

Step 3:

Determine the noise and fault model, and choose the appropriate parameter $l_1$ satisfying $∥\tilde{f}\left(p_i,v_i,t\right)-\tilde{f}\left(p_j,v_j,t\right)∥\le l_1∥{\epsilon }_i-{\epsilon }_j∥$.

Step 4:

Choose parameters $\gamma$, $c_0$, calculate parameters $a_0$, $b_0$, solve matrix inequality (40), and compute the positive matrix P.

Step 5:

Compute the feedback matrices $K_1$, $K_2$ by equality (46).

In the control protocol (26), $\gamma$ is an adjustment parameter. In general, the bigger $\gamma$ is, the faster the system (32) converges [57]. The feedback matrices $K_1$ and $K_2$ depend on the union matrix L, which represents the global information of the communication. In [49], a fully distributed protocol was proposed under the directed graph. In the future, we will try to design a controller that only relies on local communication information to realize formation control under switching topology.

After comparison with previous references [11,44,47] on formation control in multi-AUV systems, the innovations of this paper include the following two points. Firstly, this paper studies a multiple stochastic AUV system, considering the general case that the dynamic equations of an AUV include a Brownian motion. Secondly, the topology switching satisfies a stochastic Markov process. Each AUV’s communication topology can be randomly chosen from several preset possible topological graphs according to the probability. Therefore, the paper focuses on the formation control problem where two independent stochastic processes exist in the multi-AUV system. However, there are few studies on the formation control problem of two stochastic processes coexisting in multi-AUV systems in previous literature.

Compared with the existing results [1,3,16] on multi-AUV systems, the group formation and the effects of actuator faults have been considered in this paper. A fault-tolerant control strategy is presented by designing the union communication topology and selecting a Lyapunov expectation equation.

It should be mentioned that the Lyapunov function in Theorem 1 is constructed by taking into account the above two stochastic processes, and the formation control stability is proved by the Lyapunov expectation equation and infinitesimal generator.

4. Simulation

In this section, the numerical example is given to test the effectiveness of the proposed algorithm, and the numerical results are compared with those obtained by the control method proposed in reference [58]. Consider a multi-AUV system with six AUVs and two leaders divided into two sub-formations. The parameters of the leader AUV model are adopted from [59], and the parameters of the follower AUVs model refers to [42]. In addition, we assume that there are 10% uncertainties in the physical parameters of the AUV; that is, the nominal model parameters are reduced by 10% compared to the real parameters. Suppose the constant current is $u_c=\left[0.2,0.2,0\right]$ m/s, there are no obstacles in the simulation environment, and the leader AUV in each sub-formation is fault–free.

The initial value of the leader in each sub-formation has been listed in

Table 2. The leaders’ initial value.

State	Value	State	Value
$x_{l1}\left(0\right)$	150 m	$u_{l1}\left(0\right)$	0.75 m·s${}^{-1}$
$y_{l1}\left(0\right)$	−250 m	$v_{l1}\left(0\right)$	0 m·s${}^{-1}$
$z_{l1}\left(0\right)$	10 m	$w_{l1}\left(0\right)$	0 m·s${}^{-1}$
${\theta }_{l1}\left(0\right)$	$-\pi /\phantom{\pi 36}36$	$q_{l1}\left(0\right)$	0 rad·s${}^{-1}$
${\psi }_{l1}\left(0\right)$	$\pi /\phantom{\pi 2}2$	$r_{l1}\left(0\right)$	0 rad·s${}^{-1}$
$x_{l2}\left(0\right)$	20 m	$u_{l2}\left(0\right)$	0.75 m·s${}^{-1}$
$y_{l2}\left(0\right)$	0 m	$v_{l2}\left(0\right)$	0 m·s${}^{-1}$
$z_{l2}\left(0\right)$	8 m	$w_{l2}\left(0\right)$	0 m·s${}^{-1}$
${\theta }_{l2}\left(0\right)$	$-\pi /\phantom{\pi 36}36$	$q_{l2}\left(0\right)$	0 rad·s${}^{-1}$
${\psi }_{l2}\left(0\right)$	$\pi /\phantom{\pi 2}2$	$r_{l2}\left(0\right)$	0 rad·s${}^{-1}$

. The initial locations of the follower AUVs are randomly distributed in the three-dimensional space $\left[-200,200\right]\times \left[-300,200\right]\times \left[0,60\right]$, and the initial value of velocity is zero. The initial value of pitch angles of the followers AUVs are set in the interval $[-0.4,0.4]$, yaw angles are in the interval $[0,4]$, and the simulation time is 2500 s. In order to simulate the motion of AUV more practically, the control input of each AUV is restricted. The thrust amplitude is 1000 N, and the rudder angle amplitude is ${35}^{\circ }$.

The interaction topologies randomly switch between ${\mathcal{G}}_{}^{\left(i\right)}$ following the Markovian chain $r\left(t\right)$ with the state space $S=\left\{1,2,3\right\}$, and the topologies are shown in Figure 3. $D_7=D_8=\mathrm{diag}\left\{1,1,1,1,1,1\right\},\tilde{D}=\mathrm{diag}\left\{1,0,1,0,1,0\right\}$, and the generator are chosen as

$$\mathrm{\Pi }=\left(\begin{array}{ccc}-2& 1& 1\\ 0& -1& 1\\ 1& 0& -1\end{array}\right)$$

(52)

the initial distribution of the Markov process is given by its invariant distribution $\pi ={\left(1/8,1/2,3/8\right)}^T$, and by calculating $\mathbb{E}\left[L\left(t\right)\right]=\frac{1}{8}{L}_1+\frac{1}{2}{L}_2+\frac{3}{8}{L}_3$, it gives

$${\lambda }_{min}\left(\mathbb{E}\left[H_k^T\right]\mathrm{\Lambda }\right.+\mathrm{\Lambda }\left.\mathbb{E}\left[H_k^{}\right]\right)=1.7225.$$

Consider the nonlinear functions:

$$\tilde{\mathit{F}}=\tilde{\mathit{F}}\left(\mathit{p}\left(t\right),\mathit{v}\left(t\right),t\right)=\left[\begin{array}{c}0.05tanh\left({\mathit{v}}_1\right)-0.05tanh\left({\mathit{v}}_{l1}\right)\\ 0.01cos\left({\mathit{p}}_2-{\mathit{p}}_2^{\ast }\right)-0.01cos\left({\mathit{p}}_{l1}\right)\\ 0.015sin\left({\mathit{p}}_3-{\mathit{p}}_3^{\ast }\right)+0.2tanh\left({\mathit{v}}_3\right)\\ -0.01cos\left({\mathit{p}}_4-{\mathit{p}}_4^{\ast }\right)+0.01cos\left({\mathit{p}}_{l2}\right)\\ -0.2tanh\left({\mathit{v}}_5\right)+0.2tanh\left({\mathit{v}}_{l2}\right)\\ 0.015cos\left({\mathit{p}}_6-{\mathit{p}}_6^{\ast }\right)-0.35tanh\left({\mathit{v}}_6\right)\end{array}\right]$$

(53)

Clearly, the function $\tilde{\mathit{F}}$ satisfies Assumption 3.

Set

$${\mathit{p}}_1^{\ast }={\left[0,0,0,0,0\right]}^T,{\mathit{p}}_2^{\ast }={\left[80,-80,0,0,0\right]}^T$$

$${\mathit{p}}_3^{\ast }={\left[80,80,0,0,0\right]}^T,{\mathit{p}}_4^{\ast }={\left[-80,-80,0,0,0\right]}^T$$

$${\mathit{p}}_5^{\ast }={\left[-80,80,0,0,0\right]}^T,{\mathit{p}}_6^{\ast }={\left[0,0,0,0,0\right]}^T$$

and take

$$\left[{\stackrel{\widetilde{˜}}{a}}_{ij}\right]=L_1+L_2+L_3,C=L_1+L_2+L_3+\tilde{D},l_1=0.1225$$

$${\sigma }_1=0.0523,{\sigma }_2=0.3586,\varpi =0.75,a_0=c_0=1.0295,b_0=4.8740,\gamma =1.8.$$

The actuator fault is considered as follows: the actuator of AUV 1 broke down at 1500 s–2500 s with ${\rho }_1\left(t\right)=0.6+0.4exp(1500-t)$, the actuator of AUV 5 broke down with ${\rho }_2\left(t\right)=0.85+0.15exp\left(t\right)$, and the actuators of other AUVs are normal. In view of Theorem 1, the consensus gains are solved as:

$$K_1=\left[\begin{array}{ccccc}0.0396& 0& 0& 0& 0\\ 0& 0.0396& 0& 0& 0\\ 0& 0& 0.0396& 0& 0\\ 0& 0& 0& 0.0396& 0\\ 0& 0& 0& 0& 0.0396\end{array}\right]$$

$$K_2=\left[\begin{array}{ccccc}0.8074& 0& 0& 0& 0\\ 0& 0.8074& 0& 0& 0\\ 0& 0& 0.8074& 0& 0\\ 0& 0& 0& 0.8074& 0\\ 0& 0& 0& 0& 0.8074\end{array}\right]$$

Table 3. The formation convergence time corresponding to different values of parameter $\gamma$.

Parameter $\mathit{\gamma }$	${\mathit{K}}_1$ and ${\mathit{K}}_2$	The Time of Forming Formations
		Formation 1	Formation 2
$\gamma =1.8$	$K_1=0.0396I_5,K_2=0.8074I_5$	About 3.7 min	About 7.8 min
$\gamma =1.2$	$K_1=0.0618I_5,K_2=0.7887I_5$	About 4.6 min	About 9.6 min
$\gamma =0.6$	$K_1=0.0857I_5,K_2=0.8360I_5$	About 5.3 min	About 12.4 min

shows that the larger the value of parameter $\gamma$, the faster the multiple stochastic AUV system converges, so we select the parameter $\gamma =1.8$.

The desired paths for the leaders are two helical curves and expressed as:

$$\left\{\begin{array}{c}{x}_{l7}=80cos\left(0.002\pi t\right)+200,0<t\le 2500\hfill \\ y_{l7}=80sin\left(0.002\pi t\right)-200,0<t\le 2500\hfill \\ z_{l7}=0.2t+30,0<t\le 2500\hfill \end{array}\right.$$

$$\left\{\begin{array}{c}{x}_{18}=\left\{\begin{array}{c}-t+50,0<t<50\hfill \\ 0,50\le t<200\hfill \\ 25cos\left(0.001\pi \right(t-200\left)\right),200\le t<1950\hfill \\ -25,1950\le t\le 2500\hfill \end{array}\right.\hfill \\ y_{18}=\left\{\begin{array}{c}t-100,0<t<100\hfill \\ 1.2t-120,100\le t<200\hfill \\ 25sin\left(0.001\pi \right(t-200\left)\right),200\le t\le 2500\hfill \end{array}\right.\hfill \\ z_{18}=0.2t+30,0<t\le 2500\hfill \end{array}\right.$$

Figure 4 shows the Markovian states of switching topology. Figure 5 and Figure 6 plot the stochastic noise, which is a Brownian motion $\kappa \left(t\right)$, and the distribution $\vartheta \left(t\right)$ is normal. Using the above-mentioned control parameters, Figure 7 presents the position states curves of a multi-AUV system. It can be observed that the follower AUVs 1, 2 and 3 could track the position states of leader 1 and asymptotically approach those of leader 1’s desired path. The follower AUVs 4, 5 and 6 could track the position states of leader 2 and asymptotically approach those of leader 2’s desired path. Figure 8 describes the trajectories of the multi-AUV system with two leaders in three-dimensional space, which shows that the AUV in the same sub-group can reach a consensus with switching topology and stochastic noise through control protocol (26). In Figure 9, one can obtain that the follower AUVs in the same sub-group can form and keep an equilateral triangle formation. In addition, the tracking errors between the leaders and the followers are given in Figure 10 and Figure 11. It can be seen from Figure 10 that the position tracking errors eventually converge to zero after about 500 s. In Figure 11, the velocity tracking error curves oscillate strongly before formation convergence, which is caused by the change in the motion state information. It is generated by the topological switching process, the influence of the external nonlinear disturbance, and the effect of random ocean noise.

In the study, we compared the proposed control method with the methods proposed in [58], which studied path tracking control for multi-AUVs considering sampled-data delays and packet losses, to show its effectiveness in dealing with random ocean noise and actuator faults. The sampled-data delays and packet losses are omitted, and the noise and the effect of the actuator fault are added into the path-tracking control method in [58] in order to facilitate the comparison. All other simulation conditions are kept consistent with those in this paper. Figure 12 shows the three-dimensional trajectory of the multi-AUV system. Figure 13 and Figure 14 show the formation errors of follower AUVs. It can be seen that there exist large deviations for the formation position and velocity tracking errors of follower AUVs, which means the leader AUVs of each sub-formation cannot be tracked well by the follower AUVs. Therefore, the above simulation results bring about the conclusion that the proposed control protocols in the paper are valid for multi-AUV system formation control.

5. Conclusions

This paper provides a study on the fault-tolerant formation control problem for a multiple stochastic AUV system with Markovian switching topology. Using the theory of coordinate transformation and differential manifold, the control input and external disturbance of the AUV system are transformed, which is more conducive to the design of the controller and the effectiveness of the formation. In addition, the control protocols with stochastic items and multiple leaders are proposed. The random processes are dominated by the communication Markov process and the Brownian motion of noise, which are independent of each other. The study employed the mathematical expectation function and infinitesimal generator, which contributes to the establishment of some consistent results for the formation control of multiple stochastic AUV systems. Finally, the simulation was performed to verify the effectiveness of the proposed control strategy. Future studies on the formation control for a semi-Markovian jump multiple stochastic AUV system with an event-triggering communication strategy will be conducted.