Distributed Event-Triggered Fixed-Time Leader–Follower Formation Tracking Control of Multiple Underwater Vehicles Based on an Adaptive Fixed-Time Observer

Abstract

This paper focuses on the fixed-time leader–follower formation control of multiple underwater vehicles (MUVs) in the presence of external disturbances. First, an adaptive fixed-time disturbance observer (AFxDO) is developed to deal with unknown time-varying environmental disturbances. The developed AFxDO guarantees the fixed-time convergence property of the disturbance observation error and no prior information on the external disturbances or their derivatives is required. Then, with the aid of the developed AFxDO, a distributed event-triggered fixed-time backstepping controller was developed to achieve the leader–follower formation tracking control of MUVs. To solve the “explosion of complexity” problem inherent in the conventional backstepping, a nonlinear filter is introduced to obtain the derivative of the virtual control law. Furthermore, to reduce the communication burden, the event-triggered mechanism is integrated into the formation tracking controller. The stability analysis shows that the closed-loop MUV system is practical fixed-time stable. Finally, simulation results demonstrate the effectiveness of the proposed scheme.

1. Introduction

Nowadays, multi-agent systems have garnered widespread attention due to their emerging applications in many fields [1,2]. As a typical multi-agent system, underwater vehicles have become important tools for ocean exploitation and are extensively used in oceanographic observations, hydrology research, target tracking, and so on [3,4,5,6]. To broaden the service coverage and enhance the system reliability, the cooperative control of multiple underwater vehicles (MUVs) has been increasingly deployed to perform complex and large missions. However, the control of MUVs is a challenging problem because of the effect of adverse factors such as strong nonlinearity and unknown time-varying external disturbances [7,8]. Various methods have been presented to achieve cooperative control of MUVs [9,10,11]. Among the existing cooperative control methods, the leader–follower formation control approach has been diffusely applied due to its reliability [12].

For leader–follower formation control of MUVs, coordination convergence is a critical performance index and has become a research hotspot. In [13], a distributed event-triggered control law was proposed for the formation tracking of MUVs, where an adaptive neural network observer was developed to estimate the local velocity information. In [14], a distributed Lyapunov-based model predictive control (MPC) law was presented for a team of underwater vehicles, where an extended state observer (ESO) was developed for handling environmental disturbances. However, the above control laws can only achieve the asymptotic stability of the formation control system, meaning that the settling time tends to infinity [15,16]. The finite-time control is extensively studied on account of its better performance on convergence rate and robustness than the asymptotic control. In [17], the finite-time consensus tracking of MUVs was investigated and an adaptive sliding mode control (SMC) method was presented to guarantee the finite-time stability. Unfortunately, the upper bound of the convergence time of the finite-time control is related to the initial conditions of the system [18,19,20]. Since the initial conditions are difficult to obtain in practical engineering scenarios, finite control is limited in its applications [21,22,23]. To handle these constraints, the fixed-time control is presented to ensure that the convergence time is upper-bounded by an a priori value, irrespective of the initial conditions, and only control parameters are associated [24]. In [25], a fixed-time sliding mode formation controller was proposed. In [26], a fast fixed-time SMC law was presented. However, the above control laws only consider horizontal plane formation control.

To handle the formation control of MUVs subject to external disturbances, different approaches have been proposed, such as adaptive parameter estimation and disturbance observers [27]. In [28], an adaptive finite-time formation control law was developed for MUVs with external disturbances. However, the reconstruction information about the external disturbances could not be obtained. The disturbance observer is an effective way to reconstruct external disturbances and it is becoming increasingly popular. In [29], a disturbance estimator was presented to deal with the unknown dynamic and disturbances. In [30], a finite-time ESO was proposed to estimate the system uncertainties. However, their observation errors can only achieve asymptotic or finite-time convergence. In [25,31], a fixed-time disturbance observer was developed, where the fixed-time convergence of the disturbance observation errors was guaranteed. Nevertheless, the upper bounds or the derivative upper bounds of the disturbances were assumed to be known in advance, which is a strict limitation for practical applications. To solve these problems, this paper presents an adaptive fixed-time disturbance observer (AFxDO). With the developed AFxDO, the rigorous restriction on prior information of disturbances is removed by designing the adaptive parameter estimation strategy, and the fixed-time stability of the observation errors is guaranteed at the same time.

Given the actual working conditions, the communication devices of underwater vehicles often face constraints [32]. Saving communication bandwidth and computing resources should be taken into consideration when designing the formation controller of MUVs [33]. In the literature, the event-triggered control has been studied by many researchers in formation control due to the efficient management of control signals [34]. In [35], the consensus problem of MUVs was studied, where an adaptive dynamic event-triggered mechanism was presented to reduce the transmission burden. In [36], a distributed adaptive event-triggered formation control law was developed for MUVs. Although the control schemes mentioned above adopt the event-triggering mechanism, most of them cannot guarantee the fixed-time stability of tracking errors.

Motivated by the aforementioned analyses, in this paper, an AFxDO-based distributed event-triggered fixed-time leader–follower formation trajectory tracking control methodology was developed for MUVs subject to unknown time-varying external disturbances under a directed network. The main contributions of this paper are as follows:

• A novel AFxDO combined with an adaptive parameter estimation strategy was developed to eliminate the adverse effects of unknown time-varying external disturbances. Unlike existing disturbance observers, the proposed AFxDO guarantees the fixed-time stability of the observation errors, and the strict limitation of the prior information of the bound value of external disturbances was removed through the adaptive parameter estimation strategy.
• Together with the developed AFxDO, a distributed event-triggered fixed-time backstepping control strategy is proposed to solve the leader–follower formation control problem for MUVs subject to external disturbances. A nonlinear first-order filter was designed to avoid the “explosion of complexity” problem in conventional backstepping approach. The proposed formation tracking control methodology ensures that the formation tracking errors converge to an arbitrarily small neighborhood around the origin within a fixed time, independent of initial conditions. Additionally, the bandwidth resources are saved and the communication burdens are reduced by introducing an event-triggered mechanism.

This paper is organized as follows. The model description and preliminaries are described in Section 2. The main results are given in Section 3, where the design and stability analysis of the AFxDO and the distributed event-triggered fixed-time backstepping controller are presented. The simulation results are provided in Section 4. The paper ends with concluding remarks in Section 5.

2. Model Description and Preliminaries

2.1. Modeling of Networked Underwater Vehicles

Consider a group of homogeneous underwater vehicles composed of followers (labeled as $1,\cdots ,n$) and a leader (labeled as 0) under a directed communication topology. In order to establish the six degrees of freedom (DOF) motion model of each underwater vehicle, we introduce two reference frames, namely, the inertial frame and body-fixed frame. As shown in Figure 1, the inertial frame is located on Earth, and the body-fixed frame is located in the center of mass of the vehicle. In the inertial frame, the position and attitude vector of the underwater vehicle is defined as $\mathit{\eta }={\left[x,y,z,\varphi ,\theta ,\psi \right]}^T$, while in the body-fixed frame, the linear and angular velocity vectors are defined as $\mathit{\nu }={\left[u,v,w,p,q,r\right]}^T$. Based on the defined frames and coordinate vectors, the 6 DOF kinematics and dynamics equations of the ith underwater vehicle can be described as [37]:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\mathit{\eta }}}_i={\mathit{J}}_i\left({\mathit{\eta }}_i\right){\mathit{\nu }}_i\hfill \\ {\mathit{M}}_i{\dot{\mathit{\nu }}}_i+{\mathit{C}}_i\left({\mathit{\nu }}_i\right){\mathit{\nu }}_i+{\mathit{D}}_i\left({\mathit{\nu }}_i\right){\mathit{\nu }}_i+{\mathit{g}}_i\left({\mathit{\eta }}_i\right)={\mathit{\tau }}_i+{\mathit{f}}_i\hfill \end{array}\right.,\end{array}$$

(1)

where ${\mathit{J}}_i\left({\mathit{\eta }}_i\right)$ is the Jacobian transformation matrix describing the relationship between the inertial frame and body-fixed frame; ${\mathit{M}}_i\in {\mathbb{R}}^{6\times 6}$ is the inertia matrix of underwater vehicles, including the rigidity mass and the additional mass; ${\mathit{C}}_i\left({\mathit{\nu }}_i\right)\in {\mathbb{R}}^{6\times 6}$ is the Coriolis and centrifugal matrix; ${\mathit{D}}_i\left({\mathit{\nu }}_i\right)\in {\mathbb{R}}^{6\times 6}$ is the hydrodynamic damping matrix; ${\mathit{g}}_i\left({\mathit{\eta }}_i\right)\in {\mathbb{R}}^6$ is the restoring forces (gravity and buoyancy) vector; ${\mathit{f}}_i\in {\mathbb{R}}^6$ is the unknown time-varying external disturbances vector caused by ocean currents and waves; ${\mathit{\tau }}_i\in {\mathbb{R}}^6$ is the vector of control forces and moments of propellants.

Assumption A1. 

The Jacobian transformation matrix ${\mathit{J}}_i\left({\mathit{\eta }}_i\right)$ is bounded, i.e., there exists known positive constants ${\overline{J}}_i$ such that $\parallel {\mathit{J}}_i\left({\mathit{\eta }}_i\right)\parallel \le {\overline{J}}_i$.

Assumption A2. 

The external disturbances ${\mathit{f}}_i$ are supposed to be unknown but bounded, such that $\parallel {\mathit{f}}_i\parallel \le \gamma$, where γ is the unknown upper bound of external disturbances.

2.2. Notation and Related Lemmas

In this paper, ${\mathit{x}}^T$ denotes the transpose of vector $\mathit{x}$, $\parallel \mathit{x}\parallel$ denotes the Euclidean norm of vector $\mathit{x}$, ${\sigma }_{min}(·)/{\sigma }_{max}(·)$ denotes the minimum/maximum eigenvalue of a matrix, ${\mathit{sig}}^a\left(\mathit{x}\right)={\left[|x_1{|}^a\mathrm{sign}\left(x_1\right),|x_2{|}^a\mathrm{sign}\left(x_2\right),\dots ,{\left|x_n\right|}^a\mathrm{sign}\left(x_n\right)\right]}^T$ where sign(·) is the sign function, and $tanh\left(\mathit{x}\right)={[tanh\left(x_1\right),tanh\left(x_2\right),\dots ,tanh\left(x_n\right)]}^T$.

Consider a nonlinear system

$$\begin{array}{c}\hfill \dot{\mathit{x}}\left(t\right)=\mathit{f}\left(\mathit{x}\left(t\right)\right),f\left(0\right)=0,\end{array}$$

(2)

where $\mathit{x}={[x_1,x_2,\dots ,x_n]}^T\in {\mathbb{R}}^n$ and $\mathit{f}\left(\mathit{x}\right):{\mathbb{R}}^n⟶{\mathbb{R}}^n$ is a nonlinear function.

Definition 1 

([25,39]). The equilibrium $x=0$ of system (2) is fixed-time stable if it is globally finite-time stable and the settling time function $T\left(x\right)$ is bounded by a positive constant $T_{max}$, i.e., $T\left(x\right)\le T_{max}$ for any $x\in {\mathbb{R}}^n$.

Lemma 1 

([25]). For positive constants $a,b,\alpha ,\beta$ with $0<\alpha <1,\beta >1$, if we define a Lyapunov candidate function $V\left(x\right)$ on a neighborhood $U\subset {\mathbb{R}}^n$, satisfying

$$\begin{array}{c}\hfill \dot{V}\left(x\right)\le -a{V}^{\alpha }\left(x\right)-b{V}^{\beta }\left(x\right),\end{array}$$

(3)

then the origin of (2) is fixed-time stable and the settling time T satisfies

$$\begin{array}{c}\hfill T\le T_{max}=\frac{1}{a}\frac{1}{1-\alpha }+\frac{1}{b}\frac{1}{\beta -1}.\end{array}$$

(4)

Remark 1. 

As reported in [24,25,26], the settling time of the finite-time stable system depends on the initial condition. Unlike the finite-time stable system, the convergence time of the fixed-time stable system in Lemma 1 is bounded by an a priori value, which is not determined by the initial condition but by the control parameters $a,b,\alpha ,\beta$. It implies that the settling time can be designed in advance even though the initial condition is unknown.

Lemma 2 

([25]). Suppose the Lyapunov function $V\left(x\right)$ in Lemma 1 satisfies

$$\begin{array}{c}\hfill \dot{V}\left(x\right)\le -a{V}^{\alpha }\left(x\right)-b{V}^{\beta }\left(x\right)+\Omega ,\end{array}$$

(5)

where $\Omega >0$, then the origin of (2) is practical fixed-time stable, and the residual set is given by

$$\begin{array}{c}\hfill \left\{\underset{t\to T_{max}}{lim}x|V\left(x\right)\le min\left\{{\left(\frac{\Omega }{a(1-\omega )}\right)}^{\frac{1}{\alpha }},{\left(\frac{\Omega }{b(1-\omega )}\right)}^{\frac{1}{\beta }}\right\}\right\},\end{array}$$

(6)

where $\omega \in (0,1]$, and T is bounded by

$$\begin{array}{c}\hfill T\le T_{max}=\frac{1}{a\omega (1-\alpha )}+\frac{1}{b\omega (\beta -1)}.\end{array}$$

(7)

Lemma 3 

([40]). For $\forall y\ge x\ge 0$ and $n>1$, we have

$$\begin{array}{c}\hfill x{(y-x)}^n\le \frac{n}{n+1}(y^{n+1}-x^{n+1}).\end{array}$$

(8)

Lemma 4 

([41]). For variables $x\in \mathbb{R},y\in \mathbb{R}$ and constants $a>0,b>0,c>0$, we have

$$\begin{array}{c}\hfill {\left|x\right|}^a{\left|y\right|}^b\le \frac{a}{a+b}{c\left|x\right|}^{a+b}+\frac{b}{a+b}{c}^{-\frac{a}{b}}{\left|y\right|}^{a+b}.\end{array}$$

(9)

Lemma 5 

([34]). For any variable x, the following inequalities hold:

$$\begin{array}{c}\hfill \left|x\right|-xtanh\left(\frac{x}{\varrho }\right)\le \iota \varrho ,\end{array}$$

(10)

where $\varrho >0$ is a constant and $\iota =\underset{t>0}{sup}\left(\frac{1}{1+e^t}\right)=0.2785$.

Lemma 6 

([39]). The following inequalities hold for any nonnegative real number $x_i$,

$$\begin{array}{c}\hfill {\left(\sum _{i=1}^n{x}_i\right)}^a\le \sum _{i=1}^n{x}_i^a,0<a<1,\end{array}$$

(11)

$$\begin{array}{c}\hfill {\left(\sum _{i=1}^n{x}_i\right)}^b\le n^{b-1}\sum _{i=1}^n{x}_i^b,b>1.\end{array}$$

(12)

2.3. Basic Graph Theory

The communication topology of the n underwater vehicle followers and a leader is described by a directed graph $\mathit{G}\triangleq \left\{\mathit{N},\mathit{\zeta },\mathit{A}\right\}$, where $N=\{0,1,2,\cdots ,n\}$ and $\mathit{\zeta }\subseteq \mathit{N}\times \mathit{N}$ are the sets of nodes and edges, respectively, and $\mathit{A}=\left[a_{ij}\right]\in {\mathbb{R}}^{n\times n}$ is the adjacency matrix. The information transfer from node $j\in \mathit{N}$ to node $i\in \mathit{N}$ is expressed as $\left({\zeta }_j,{\zeta }_i\right)\in \mathit{\zeta }$, $a_{ij}=1$, node j can be called the neighbor of node i, otherwise, $\left({\zeta }_j,{\zeta }_i\right)\notin \mathit{\zeta }$, $a_{ij}=0$. The in-degree $h_i$ of node i is defined as $h_i={\sum }_{j=0}^n{a}_{ij}$, and the in-degree matrix is defined as $\mathit{H}=\mathrm{diag}\{h_0,h_1,h_2,\cdots ,h_{n-1}\}$. The Laplacian matrix related to graph $\mathit{G}$ is defined as $\mathit{L}=\left[l_{ij}\right]\in {\mathbb{R}}^{n\times n}$ and $\mathit{L}=\mathit{H}-\mathit{A}$.

3. Main Results

3.1. Design of Adaptive Fixed-Time Disturbance Observer

In order to deal with the external disturbances, an AFxDO is developed for MUVs in this section. To facilitate the disturbance observer design, an auxiliary variable is firstly introduced

$$\begin{array}{c}\hfill {\mathit{e}}_i={\mathit{M}}_i{\mathit{\nu }}_i-{\mathit{\xi }}_i,\end{array}$$

(13)

where $\dot{{\mathit{\xi }}_i}$ is given by

$$\begin{array}{cc}\hfill {\dot{\mathit{\xi }}}_i=& -{\mathit{C}}_i\left({\mathit{\nu }}_i\right){\mathit{\nu }}_i-{\mathit{D}}_i\left({\mathit{\nu }}_i\right){\mathit{\nu }}_i-{\mathit{g}}_i\left({\mathit{\eta }}_i\right)+{\mathit{\tau }}_i\hfill \\ \hfill & +{\mathit{K}}_{10}\mathit{sig}{\left({\mathit{e}}_i\right)}^{2{\alpha }_0-1}+{\mathit{K}}_{20}\mathit{sig}{\left({\mathit{e}}_i\right)}^{2{\beta }_0-1}+\widehat{\gamma }\frac{{\mathit{e}}_i}{\parallel {\mathit{e}}_i\parallel },\hfill \end{array}$$

(14)

where ${\alpha }_0,{\beta }_0,$ are positive design parameters that satisfy $\frac{1}{2}<{\alpha }_0<1$, ${\beta }_0>1$, ${\mathit{K}}_{10}=\mathrm{diag}\left[k_{10,1},k_{10,2},\cdots ,k_{10,6}\right]$, ${\mathit{K}}_{20}=\mathrm{diag}\left[k_{20,1},k_{20,2},\cdots ,k_{20,6}\right]$ are positive definite design matrices and $\widehat{\gamma }>0$ is the estimation of $\gamma$.

Theorem 1. 

Considering the nonlinear uncertain dynamic model of underwater vehicle i given in (1) under Assumption 2, if the AFxDO and the adaptive parameter estimation strategy are designed as

$$\begin{array}{cc}\hfill {\widehat{\mathit{f}}}_i=& {\mathit{K}}_{10}\mathit{sig}{\left({\mathit{e}}_i\right)}^{2{\alpha }_0-1}+{\mathit{K}}_{20}\mathit{sig}{\left({\mathit{e}}_i\right)}^{2{\beta }_0-1}+\widehat{\gamma }\frac{{\mathit{e}}_i}{\parallel {\mathit{e}}_i\parallel },\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\dot{\widehat{\gamma }}}_i=& {\lambda }_0\parallel {\mathit{e}}_i\parallel -{\lambda }_1{\widehat{\gamma }}_i-{\lambda }_2{\widehat{\gamma }}_i^{2{\beta }_0-1},\hfill \end{array}$$

(16)

where ${\lambda }_0,{\lambda }_1,{\lambda }_2$ are positive constants, then the estimation ${\widehat{\mathit{f}}}_i$ will be stabilized to a small region around ${\mathit{f}}_i$ within a fixed time.

Proof. 

Choose a Lyapunov function candidate as

$$\begin{array}{c}\hfill V_0=\frac{1}{2}{\mathit{e}}_i^T{\mathit{e}}_i+\frac{1}{2{\lambda }_0}{\tilde{\gamma }}_i^2,\end{array}$$

(17)

where ${\tilde{\gamma }}_i={\gamma }_i-{\widehat{\gamma }}_i$ is the estimation error.

Differentiating $V_0$ with respect to time yields

$$\begin{array}{cc}\hfill {\dot{V}}_0=& {\mathit{e}}_i^T\left({\mathit{f}}_i-{\mathit{K}}_{10}\mathit{sig}{\left({\mathit{e}}_i\right)}^{2{\alpha }_0-1}-{\mathit{K}}_{20}\mathit{sig}{\left({\mathit{e}}_i\right)}^{2{\beta }_0-1}-\widehat{\gamma }\frac{{\mathit{e}}_i}{\parallel {\mathit{e}}_i\parallel }\right)-\frac{1}{{\lambda }_0}{\tilde{\gamma }}_i{\dot{\widehat{\gamma }}}_i\hfill \\ \hfill =& -\sum _{\rho =1}^6\left(k_{10,\rho }{e}_{i,\rho }^{2{\alpha }_0}+k_{20,\rho }{e}_{i,\rho }^{2{\beta }_0}\right)+{\mathit{e}}_i^T{\mathit{f}}_i-{\widehat{\gamma }}_i\parallel {\mathit{e}}_i\parallel -\frac{1}{{\lambda }_0}{\tilde{\gamma }}_i{\dot{\widehat{\gamma }}}_i\hfill \\ \hfill \le & -\sum _{\rho =1}^6\left(k_{10,\rho }{e}_{i,\rho }^{2{\alpha }_0}+k_{20,\rho }{e}_{i,\rho }^{2{\beta }_0}\right)+\frac{1}{{\lambda }_0}\tilde{\gamma }\left({\lambda }_0\parallel {\mathit{e}}_i\parallel -{\dot{\widehat{\gamma }}}_i\right).\hfill \end{array}$$

(18)

Substituting the adaptive law (16) into (18), we can obtain

$$\begin{array}{cc}\hfill {\dot{V}}_0\le & -\sum _{\rho =1}^6\left(k_{10,\rho }{e}_{i,\rho }^{2{\alpha }_0}+k_{20,\rho }{e}_{i,\rho }^{2{\beta }_0}\right)+\frac{{\lambda }_1}{{\lambda }_0}{\tilde{\gamma }}_i{\widehat{\gamma }}_i+\frac{{\lambda }_2}{{\lambda }_0}{\tilde{\gamma }}_i{\widehat{\gamma }}_i^{2{\beta }_0-1}.\hfill \end{array}$$

(19)

According to Lemma 3 and the definition of ${\tilde{\gamma }}_i$, we can obtain

$$\begin{array}{c}\hfill {\tilde{\gamma }}_i{\widehat{\gamma }}_i^{2{\beta }_0-1}={\tilde{\gamma }}_i{({\gamma }_i-{\tilde{\gamma }}_i)}^{2{\beta }_0-1}\le \frac{2{\beta }_0-1}{2{\beta }_0}\left({\gamma }_i^{2{\beta }_0}-{\tilde{\gamma }}_i^{2{\beta }_0}\right).\end{array}$$

(20)

By utilizing Lemma 4, let $x=\frac{{\tilde{\gamma }}_i^2}{2{\lambda }_0}$, $y=1$, $a={\alpha }_0$, $b=1-{\alpha }_0$ and $c=1/{\alpha }_0$, then we have

$$\begin{array}{c}\hfill {\left(\frac{{\tilde{\gamma }}_i^2}{2{\lambda }_0}\right)}^{{\alpha }_0}\le \frac{{\tilde{\gamma }}_i^2}{2{\lambda }_0}+(1-{\alpha }_0){\alpha }_0^{\frac{{\alpha }_0}{1-{\alpha }_0}}.\end{array}$$

(21)

Substituting (20) and (21) into (19), we have

$$\begin{array}{cc}\hfill {\dot{V}}_0\le & -\sum _{\rho =1}^6\left(k_{10,\rho }{e}_{i,\rho }^{2{\alpha }_0}+k_{20,\rho }{e}_{i,\rho }^{2{\beta }_0}\right)+\frac{{\lambda }_1}{{\lambda }_0}\left(-\frac{1}{2}{\tilde{\gamma }}_i^2+\frac{1}{2}{\gamma }_i^2\right)+\frac{{\lambda }_2(2{\beta }_0-1)}{2{\beta }_0{\lambda }_0}\left({\gamma }_i^{2{\beta }_0}-{\tilde{\gamma }}_i^{2{\beta }_0}\right)\hfill \\ \hfill \le & -\sum _{\rho =1}^6\left(k_{10,\rho }{e}_{i,\rho }^{2{\alpha }_0}+k_{20,\rho }{e}_{i,\rho }^{2{\beta }_0}\right)-\frac{2^{{\beta }_0}{\lambda }_0^{{\beta }_0-1}{\lambda }_2(2{\beta }_0-1)}{2{\beta }_0}{\left(\frac{{\tilde{\gamma }}_i^2}{2{\lambda }_0}\right)}^{{\beta }_0}\hfill \\ \hfill & -\frac{{\lambda }_1}{2{\lambda }_0}{\tilde{\gamma }}_i^2+\frac{{\lambda }_1}{2{\lambda }_0}{\gamma }_i^2+\frac{{\lambda }_2(2{\beta }_0-1)}{2{\beta }_0{\lambda }_0}{\gamma }_i^{2{\beta }_0}\hfill \\ \hfill \le & -{\sigma }_{min}\left({\mathit{K}}_{10}\right)\sum _{\rho =1}^6{\left(e_{i,\rho }^2\right)}^{{\alpha }_0}-{\sigma }_{min}\left({\mathit{K}}_{20}\right)\sum _{\rho =1}^6{\left(e_{i,\rho }^2\right)}^{{\beta }_0}-{\lambda }_1{\left(\frac{{\tilde{\gamma }}_i^2}{2{\lambda }_0}\right)}^{{\alpha }_0}+\frac{{\lambda }_1{\gamma }_i^2}{2{\lambda }_0}\hfill \\ \hfill & -\frac{{\left(2{\lambda }_0\right)}^{{\beta }_0-1}{\lambda }_2(2{\beta }_0-1)}{{\beta }_0}{\left(\frac{{\tilde{\gamma }}_i^2}{2{\lambda }_0}\right)}^{{\beta }_0}+\frac{{\lambda }_2(2{\beta }_0-1)}{2{\beta }_0{\lambda }_0}{\gamma }_i^{2{\beta }_0}+{\lambda }_1(1-{\alpha }_0){\alpha }_0^{\frac{{\alpha }_0}{1-{\alpha }_0}}\hfill \\ \hfill \le & -min\left\{{\sigma }_{min}\left({\mathit{K}}_{10}\right)2^{{\alpha }_0},{\lambda }_1\right\}\left({\left(\frac{{\mathit{e}}_i^T{\mathit{e}}_i}{2}\right)}^{{\alpha }_0}+{\left(\frac{{\tilde{\gamma }}_i^2}{2{\lambda }_0}\right)}^{{\alpha }_0}\right)\hfill \\ \hfill & -min\left\{2{\sigma }_{min}\left({\mathit{K}}_{20}\right)3^{1-{\beta }_0},\frac{{\left(2{\lambda }_0\right)}^{{\beta }_0-1}{\lambda }_2(2{\beta }_0-1)}{{\beta }_0}\right\}\left({\left(\frac{{\mathit{e}}_i^T{\mathit{e}}_i}{2}\right)}^{{\beta }_0}+{\left(\frac{{\tilde{\gamma }}_i^2}{2{\lambda }_0}\right)}^{{\beta }_0}\right)+{\Omega }_0\hfill \\ \hfill \le & -A_0{V}_0^{{\alpha }_0}-B_0{V}_0^{{\beta }_0}+{\Omega }_0,\hfill \end{array}$$

(22)

where

$$\begin{array}{cc}\hfill & A_0=min\left\{{\sigma }_{min}\left({\mathit{K}}_{10}\right)2^{{\alpha }_0},{\lambda }_1\right\},\hfill \\ \hfill & B_0=2^{1-{\beta }_0}min\left\{2{\sigma }_{min}\left({\mathit{K}}_{20}\right)3^{1-{\beta }_0},\frac{{\left(2{\lambda }_0\right)}^{{\beta }_0-1}{\lambda }_2(2{\beta }_0-1)}{{\beta }_0}\right\},\hfill \\ \hfill & {\Omega }_0=\frac{{\lambda }_1{\gamma }_i^2}{2{\lambda }_0}+\frac{{\lambda }_2(2{\beta }_0-1)}{2{\beta }_0{\lambda }_0}{\gamma }_i^{2{\beta }_0}+{\lambda }_1(1-{\alpha }_0){\alpha }_0^{\frac{{\alpha }_0}{1-{\alpha }_0}}.\hfill \end{array}$$

Since (22) satisfies Lemma 2, system (17) is, therefore, proven to be practical fixed-time stable, and the settling time can be expressed as

$$\begin{array}{c}\hfill t_0\le T_{\max }=\frac{1}{{\omega }_0{A}_0(1-{\alpha }_0)}+\frac{1}{{\omega }_0{B}_0({\beta }_0-1)},\end{array}$$

(23)

where $0<{\omega }_0<1$. If ${\tilde{\gamma }}_i=0$, then according to (17), we have

$$\begin{array}{c}\hfill {\dot{V}}_0={\mathit{e}}_i^T\dot{{\mathit{e}}_i}\le -\sum _{\rho =1}^6\left(k_{10,\rho }{e}_{i,\rho }^{2{\alpha }_0}+k_{20,\rho }{e}_{i,\rho }^{2{\beta }_0}\right).\end{array}$$

(24)

According to Lemma 1, the auxiliary variable ${\mathit{e}}_i$ will be stabilized to zero in a fixed time when ${\tilde{\gamma }}_i=0$. Thus, its derivative ${\dot{\mathit{e}}}_i$ will also converge to zero in fixed time. From (13)–(15), we have ${\tilde{\mathit{f}}}_i={\mathit{f}}_i-{\widehat{\mathit{f}}}_i={\mathit{M}}_i{\dot{\mathit{\nu }}}_i-{\dot{\xi }}_i={\dot{\mathit{e}}}_i$. Therefore, ${\mathit{f}}_i={\widehat{\mathit{f}}}_i$ when ${\tilde{\gamma }}_i=0$. For the case ${\tilde{\gamma }}_i\ne 0$, it is easy to obtain ${\tilde{\mathit{f}}}_i={\mathit{f}}_i-{\widehat{\mathit{f}}}_i=\left({\gamma }_i-{\widehat{\gamma }}_i\right)\frac{{\mathit{e}}_i}{\parallel {\mathit{e}}_i\parallel }={\tilde{\gamma }}_i\frac{{\mathit{e}}_i}{\parallel {\mathit{e}}_i\parallel }$. From (22), ${\tilde{\gamma }}_i$ will converge to the neighborhood of the origin when $t>t_0$. Thus, ${\widehat{\mathit{f}}}_i$ will converge to the neighborhood of ${\mathit{f}}_i$ when $t>t_0$. □

Remark 2. 

Different from the asymptotic and finite-time convergent disturbance observers, the developed AFxDO guarantees the fixed-time convergence property of observation errors, i.e., the convergence time is upper-bounded by an a priori value, irrespective of initial conditions. Additionally, the developed AFxDO eliminates the need for prior knowledge of the upper bound value of external disturbances, which is useful for practical applications.

3.2. Design of Distributed Event-Triggered Fixed-Time Formation Tracking Controller

The block diagram of the proposed control strategy is shown in Figure 2. As seen from Figure 2, a distributed event-triggered fixed-time backstepping controller is proposed for the formation tracking control of MUVs based on the developed AFxDO. The proposed formation tracking control strategy has strong robustness, a fast convergence rate, and reduces the communication burden. The details are as follows.

For underwater vehicle i, underwater vehicle j is supposed to be its neighbor. Define the formation tracking error ${\tilde{\mathit{\eta }}}_i\in {\mathbb{R}}^6$ of underwater vehicle i as

$$\begin{array}{cc}\hfill {\tilde{\mathit{\eta }}}_i=& \sum _{j=0}^n{a}_{ij}\left({\mathit{\eta }}_i-{\mathit{\eta }}_j-\Delta {\mathit{\eta }}_{ij}\right),\hfill \end{array}$$

(25)

where ${\tilde{\mathit{\eta }}}_i={\left[{\tilde{x}}_i,{\tilde{y}}_i,{\tilde{z}}_i,{\tilde{\varphi }}_i,{\tilde{\theta }}_i,{\tilde{\psi }}_i\right]}^T$ and $\Delta {\mathit{\eta }}_{ij}={\left[\Delta x_{ij},\Delta y_{ij},\Delta z_{ij},\Delta {\varphi }_{ij},\Delta {\theta }_{ij},\Delta {\psi }_{ij}\right]}^T$ is the desired formation pattern for i and j, $i\ne 0$.

The time derivative of (25) yields

$$\begin{array}{cc}\hfill {\dot{\tilde{\mathit{\eta }}}}_i=& \sum _{j=0}^n{a}_{ij}\left({\mathit{J}}_i\left({\mathit{\eta }}_i\right){\mathit{\nu }}_i-{\dot{\mathit{\eta }}}_j\right).\hfill \end{array}$$

(26)

To guarantee the fixed-time convergence of the formation tracking error ${\tilde{\mathit{\eta }}}_i$ of underwater vehicle i, the virtual control law ${\mathit{\chi }}_{c,i}$ can be designed as follows:

$$\begin{array}{cc}\hfill {\mathit{\chi }}_{c,i}=& \frac{1}{h_i}{\mathit{J}}_i^{-1}\left({\mathit{\eta }}_i\right)\left(\sum _{j=0}^n{a}_{ij}{\dot{\mathit{\eta }}}_j-{\mathit{K}}_{11}\mathit{sig}{\left({\tilde{\mathit{\eta }}}_i\right)}^{2{\alpha }_1-1}-{\mathit{K}}_{21}\mathit{sig}{\left({\tilde{\mathit{\eta }}}_i\right)}^{2{\beta }_1-1}\right),\hfill \end{array}$$

(27)

where ${\alpha }_1,{\beta }_1,$ are positive design parameters that satisfy $\frac{1}{2}<{\alpha }_1<1,{\beta }_1>1$, and ${\mathit{K}}_{11}=\mathrm{diag}\left[k_{11,1},k_{11,2},\cdots ,k_{11,6}\right]$, ${\mathit{K}}_{21}=\mathrm{diag}\left[k_{21,1},k_{21,2},\cdots ,k_{21,6}\right]$ are positive definite design matrices.

To deal with the “explosion of complexity” issue that is inherent in the traditional backstepping control, let ${\mathit{\chi }}_{c,i}$ pass through the following first-order nonlinear filter

$$\begin{array}{cc}\hfill \kappa {\dot{\mathit{\chi }}}_{d,i}=& -\mathit{sig}{({\mathit{\chi }}_{d,i}-{\mathit{\chi }}_{c,i})}^{2{\alpha }_1-1}-\mathit{sig}{({\mathit{\chi }}_{d,i}-{\mathit{\chi }}_{c,i})}^{2{\beta }_1-1},\hfill \end{array}$$

(28)

where ${\mathit{\chi }}_{d,i}$ is the filtered virtual control scheme and $\kappa >0$ is the designed parameter. According to (28), the differential term ${\dot{\mathit{\chi }}}_{d,i}$ is directly obtained to replace the complicated differential term ${\dot{\mathit{\chi }}}_{c,i}$ required in the traditional backstepping control. It is noteworthy that the filter employed here is nonlinear rather than linear since the linear filter cannot achieve the fixed-time convergence property of the overall system.

The velocity tracking error ${\tilde{\mathit{\nu }}}_i\in {\mathbb{R}}^6$ is defined as

$$\begin{array}{cc}\hfill {\tilde{\mathit{\nu }}}_i=& {\mathit{\nu }}_i-{\mathit{\chi }}_{d,i},\hfill \end{array}$$

(29)

Differentiating (29) with respect to time and substituting (1) into it, we have

$$\begin{array}{cc}\hfill {\dot{\tilde{\mathit{\nu }}}}_i=& {\mathit{M}}_i^{-1}\left(-{\mathit{C}}_i\left({\mathit{\nu }}_i\right){\mathit{\nu }}_i-{\mathit{D}}_i\left({\mathit{\nu }}_i\right){\mathit{\nu }}_i-{\mathit{g}}_i\left({\mathit{\eta }}_i\right)+{\mathit{\tau }}_i+{\mathit{f}}_i\right)-{\dot{\mathit{\chi }}}_{d,i},\hfill \end{array}$$

(30)

To achieve the fixed-time convergence of the velocity tracking error ${\tilde{\mathit{\nu }}}_i$ and enhance the system’s robustness against external disturbances, the AFxDO-based distributed event-triggered fixed-time formation tracking control scheme for the underwater vehicle i can be designed as

$$\begin{array}{cc}\hfill {\overline{\mathit{\tau }}}_i=& {\mathit{C}}_i\left({\mathit{\nu }}_i\right){\mathit{\nu }}_i+{\mathit{D}}_i\left({\mathit{\nu }}_i\right){\mathit{\nu }}_i+{\mathit{g}}_i\left({\mathit{\eta }}_i\right)-{\widehat{\mathit{f}}}_i+{\mathit{M}}_i{\dot{\mathit{\chi }}}_{d,i}\hfill \\ \hfill & -{\mathit{K}}_{12}\mathit{sig}{\left({\tilde{\mathit{\nu }}}_i\right)}^{2{\alpha }_1-1}-{\mathit{K}}_{22}\mathit{sig}{\left({\tilde{\mathit{\nu }}}_i\right)}^{2{\beta }_1-1}-{\mu }_{2}tanh\left(\frac{{\tilde{\mathit{\nu }}}_i}{{\varrho }_i}\right)-\frac{{\tilde{\mathit{\nu }}}_i{\parallel {\tilde{\mathit{\nu }}}_i\parallel }^2}{2},\hfill \end{array}$$

(31)

with the triggering event condition given as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathit{\tau }}_i\left(t\right)={\overline{\mathit{\tau }}}_i\left(t_{i,m}\right),t\in \left[t_{i,m},t_{i,m+1}\right),\hfill \\ {\mathit{t}}_{i,m+1}=inf\left\{t>t_{i,m}\left|\parallel {\tilde{\mathit{\tau }}}_i\left(t\right)\parallel \ge {\mu }_1\parallel {\tilde{\mathit{\nu }}}_i\left(t\right)\parallel +{\mu }_2\right.\right\}\hfill \end{array}\right.\end{array}$$

(32)

where ${\mu }_1,{\mu }_2$ are positive design parameters, ${\mathit{K}}_{12}=\mathrm{diag}\left[k_{12,1},k_{12,2},\cdots ,k_{12,6}\right]$, ${\mathit{K}}_{22}=\mathrm{diag}\left[k_{22,1},k_{22,2},\cdots ,k_{22,6}\right]$ are positive definite design matrices, $t_{i,m}$ ($m\in {\mathbb{Z}}^{+}$) is the triggering time of the control law for underwater vehicle i, and ${\tilde{\mathit{\tau }}}_i\left(t\right)={\overline{\mathit{\tau }}}_i\left(t\right)-{\mathit{\tau }}_i\left(t\right)$. Equation (32) indicates that when the triggering condition $\parallel {\tilde{\mathit{\tau }}}_i\parallel \ge {\mu }_1\parallel {\tilde{\mathit{\nu }}}_i\parallel +{\mu }_2$ is satisfied, the control input ${\mathit{\tau }}_i$ updates its value at $t_{i,m+1}$, and is then set to ${\overline{\mathit{\tau }}}_i$ during $t\in \left[t_{i,m},t_{i,m+1}\right)$.

Theorem 2. 

Considering the MUV system given in (1) under Assumptions 1 and 2, if the distributed fixed-time control scheme is designed as (31), based on the virtual control law (27), the first-order nonlinear filter (28), the event-triggering law (32), and the AFxDO (15), then the formation tracking control can be achieved for the MUV system, and all signals of the MUV system are practical fixed-time stable. Additionally, the Zeno behavior can be excluded.

Proof. 

The proof of Theorem 2 depends on the premise that the disturbance estimation error ${\tilde{\mathit{f}}}_i$ is stabilized toward a small neighborhood of the origin in fixed time. Actually, this premise can be achieved in Theorem 1, where the AFxDO is designed to estimate the external disturbances ${\mathit{f}}_i$ in a fixed time $t_0$. Therefore, we can assume that when $t>t_0$, ${\tilde{\mathit{f}}}_i$ converges to a small region of the origin, and there exists a small positive constant ${\delta }_i$, satisfying $\parallel {\tilde{\mathit{f}}}_i\parallel \le {\delta }_i$. Then, we consider the following Lyapunov function candidate

$$\begin{array}{c}\hfill V_1=\frac{1}{2}{\tilde{\mathit{\eta }}}_i^T{\tilde{\mathit{\eta }}}_i+\frac{1}{2}{\tilde{\mathit{\chi }}}_i^T{\tilde{\mathit{\chi }}}_i+\frac{1}{2}{\tilde{\mathit{\nu }}}_i^T{\mathit{M}}_i{\tilde{\mathit{\nu }}}_i.\end{array}$$

(33)

where ${\tilde{\mathit{\chi }}}_i={\mathit{\chi }}_{d,i}-{\mathit{\chi }}_{c,i}$ is the filtered error.

Differentiating $V_1$ with respective to time, we have

$$\begin{array}{c}\hfill {\dot{V}}_1={\tilde{\mathit{\eta }}}_i^T{\dot{\tilde{\mathit{\eta }}}}_i+{\tilde{\mathit{\chi }}}_i^T{\dot{\tilde{\mathit{\chi }}}}_i+{\tilde{\mathit{\nu }}}_i^T{\mathit{M}}_i{\dot{\tilde{\mathit{\nu }}}}_i.\end{array}$$

(34)

According to (26) and (29), we have

$$\begin{array}{cc}\hfill {\tilde{\mathit{\eta }}}_i^T{\dot{\tilde{\mathit{\eta }}}}_i=& {\tilde{\mathit{\eta }}}_i^T\left(\sum _{j=0}^n{a}_{ij}\left({\mathit{J}}_i\left({\mathit{\eta }}_i\right){\mathit{\nu }}_i-{\dot{\mathit{\eta }}}_j\right)\right)\hfill \\ \hfill =& {\tilde{\mathit{\eta }}}_i^T\left(h_i{\mathit{J}}_i\left({\mathit{\eta }}_i\right)\left({\tilde{\mathit{\chi }}}_i+{\tilde{\mathit{\nu }}}_i+{\mathit{\chi }}_{c,i}\right)-\sum _{j=0}^n{a}_{ij}{\dot{\mathit{\eta }}}_j\right)\hfill \\ \hfill =& -\sum _{\rho =1}^6\left(k_{11,\rho }{\tilde{\eta }}_{i,\rho }^{2{\alpha }_1}+k_{21,\rho }{\tilde{\eta }}_{i,\rho }^{2{\beta }_1}\right)+h_i{\tilde{\mathit{\eta }}}_i^T{\mathit{J}}_i\left({\mathit{\eta }}_i\right)\left({\tilde{\mathit{\chi }}}_i+{\tilde{\mathit{\nu }}}_i\right)\hfill \\ \hfill \le & -\sum _{\rho =1}^6\left(k_{11,\rho }{\tilde{\eta }}_{i,\rho }^{2{\alpha }_1}+k_{21,\rho }{\tilde{\eta }}_{i,\rho }^{2{\beta }_1}\right)+h_i{\overline{J}}_i\left({\tilde{\mathit{\eta }}}_i^T{\tilde{\mathit{\eta }}}_i+\frac{1}{2}{\tilde{\mathit{\chi }}}_i^T{\tilde{\mathit{\chi }}}_i+\frac{1}{2}{\tilde{\mathit{\nu }}}_i^T{\tilde{\mathit{\nu }}}_i\right).\hfill \end{array}$$

(35)

Based on the definition of ${\tilde{\mathit{\chi }}}_i$, its derivative can be expressed as

$$\begin{array}{c}\hfill {\dot{\tilde{\mathit{\chi }}}}_i=-\frac{1}{\kappa }\mathit{sig}{\left({\tilde{\mathit{\chi }}}_i\right)}^{2{\alpha }_1-1}-\frac{1}{\kappa }\mathit{sig}{\left({\tilde{\mathit{\chi }}}_i\right)}^{2{\beta }_1-1}-{\dot{\mathit{\chi }}}_{c,i}.\end{array}$$

(36)

In the light of (27), we can assume that ${\dot{\mathit{\chi }}}_{c,i}$ is bounded, i.e., $\parallel {\dot{\mathit{\chi }}}_{c,i}\parallel <{\varsigma }_i,{\varsigma }_i>0$. Then, we have

$$\begin{array}{cc}\hfill {\tilde{\mathit{\chi }}}_i^T{\dot{\tilde{\mathit{\chi }}}}_i=& -\frac{1}{\kappa }\sum _{\rho =1}^6\left({\tilde{\chi }}_{i,\rho }^{2{\alpha }_1}+{\tilde{\chi }}_{i,\rho }^{2{\beta }_1}\right)-{\tilde{\mathit{\chi }}}_i^T{\dot{\mathit{\chi }}}_{c,i}\hfill \\ \hfill \le & -\frac{1}{\kappa }\sum _{\rho =1}^6\left({\tilde{\chi }}_{i,\rho }^{2{\alpha }_1}+{\tilde{\chi }}_{i,\rho }^{2{\beta }_1}\right)+\parallel {\tilde{\mathit{\chi }}}_i\parallel \parallel {\dot{\mathit{\chi }}}_{c,i}\parallel \hfill \\ \hfill \le & -\frac{1}{\kappa }\sum _{\rho =1}^6\left({\tilde{\chi }}_{i,\rho }^{2{\alpha }_1}+{\tilde{\chi }}_{i,\rho }^{2{\beta }_1}\right)+\frac{1}{2}{\tilde{\mathit{\chi }}}_i^T{\tilde{\mathit{\chi }}}_i+\frac{1}{2}{\varsigma }_i^2.\hfill \end{array}$$

(37)

In view of (30)–(32), we have

$$\begin{array}{cc}\hfill {\tilde{\mathit{\nu }}}_i^T{\mathit{M}}_i{\dot{\tilde{\mathit{\nu }}}}_i=& {\tilde{\mathit{\nu }}}_i^T\left({\tilde{\mathit{f}}}_i-{\tilde{\mathit{\tau }}}_i-{\mathit{K}}_{12}\mathit{sig}{\left({\tilde{\mathit{\nu }}}_i\right)}^{2{\alpha }_1-1}-{\mathit{K}}_{22}\mathit{sig}{\left({\tilde{\mathit{\nu }}}_i\right)}^{2{\beta }_1-1}-{\mu }_{2}tanh\left(\frac{{\tilde{\mathit{\nu }}}_i}{{\varrho }_i}\right)-\frac{{\tilde{\mathit{\nu }}}_i{\parallel {\tilde{\mathit{\nu }}}_i\parallel }^2}{2}\right)\hfill \\ \hfill \le & -\sum _{\rho =1}^6\left(k_{12,\rho }{\tilde{\nu }}_{i,\rho }^{2{\alpha }_1}+k_{22,\rho }{\tilde{\nu }}_{i,\rho }^{2{\beta }_1}\right)+\frac{1}{2}{\tilde{\mathit{\nu }}}_i^T{\tilde{\mathit{\nu }}}_i+\frac{{\delta }_i^2}{2}\hfill \\ \hfill & +\parallel {\tilde{\mathit{\nu }}}_i\parallel \left({\mu }_1\parallel {\tilde{\mathit{\nu }}}_i\parallel +{\mu }_{i,2}\right)-{\mu }_2{\tilde{\mathit{\nu }}}_i^{T}tanh\left(\frac{{\tilde{\mathit{\nu }}}_i}{{\varrho }_i}\right)-\frac{\parallel {\tilde{\mathit{\nu }}}_i{\parallel }^4}{2}\hfill \\ \hfill \le & -\sum _{\rho =1}^6\left(k_{12,\rho }{\tilde{\nu }}_{i,\rho }^{2{\alpha }_1}+k_{22,\rho }{\tilde{\nu }}_{i,\rho }^{2{\beta }_1}\right)+\frac{1}{2}{\tilde{\mathit{\nu }}}_i^T{\tilde{\mathit{\nu }}}_i\hfill \\ \hfill & +{\mu }_2\left(\parallel {\tilde{\mathit{\nu }}}_i\parallel -{\tilde{\mathit{\nu }}}_i^{T}tanh\left(\frac{{\tilde{\mathit{\nu }}}_i}{{\varrho }_i}\right)\right)+\frac{{\delta }_i^2}{2}+\frac{{\mu }_1^2}{2}.\hfill \end{array}$$

(38)

From the definition of ${\tilde{\mathit{\nu }}}_i$ and $tanh\left(\frac{{\tilde{\mathit{\nu }}}_i}{{\varrho }_i}\right)$, and utilizing Lemma 5 yields

$$\begin{array}{c}\hfill \parallel {\tilde{\mathit{\nu }}}_i\parallel -{\tilde{\mathit{\nu }}}_i^{T}tanh\left(\frac{{\tilde{\mathit{\nu }}}_i}{{\varrho }_i}\right)\le \sum _{\rho =1}^6\left(|{\tilde{\nu }}_{i,\rho }|-{\tilde{\nu }}_{i,\rho }tanh\left(\frac{{\tilde{\nu }}_{i,\rho }}{{\varrho }_i}\right)\right)\le 1.671{\varrho }_i.\end{array}$$

(39)

Substituting (35)–(39) into (34), we have

$$\begin{array}{cc}\hfill {\dot{V}}_1\le & -\sum _{\rho =1}^6\left(k_{11,\rho }{\tilde{\eta }}_{i,\rho }^{2{\alpha }_1}+k_{21,\rho }{\tilde{\eta }}_{i,\rho }^{2{\beta }_1}\right)+h_i{\overline{J}}_i\left({\tilde{\mathit{\eta }}}_i^T{\tilde{\mathit{\eta }}}_i+\frac{1}{2}{\tilde{\mathit{\chi }}}_i^T{\tilde{\mathit{\chi }}}_i+\frac{1}{2}{\tilde{\mathit{\nu }}}_i^T{\tilde{\mathit{\nu }}}_i\right)-\frac{1}{\kappa }\sum _{\rho =1}^6\left({\tilde{\chi }}_{i,\rho }^{2{\alpha }_1}+{\tilde{\chi }}_{i,\rho }^{2{\beta }_1}\right)\hfill \\ \hfill & +\frac{1}{2}{\tilde{\mathit{\chi }}}_i^T{\tilde{\mathit{\chi }}}_i+\frac{1}{2}{\varsigma }_i^2-\sum _{\rho =1}^6\left(k_{12,\rho }{\tilde{\nu }}_{i,\rho }^{2{\alpha }_1}+k_{22,\rho }{\tilde{\nu }}_{i,\rho }^{2{\beta }_1}\right)+\frac{1}{2}{\tilde{\mathit{\nu }}}_i^T{\tilde{\mathit{\nu }}}_i+1.671{\mu }_2{\varrho }_i+\frac{{\delta }_i^2}{2}+\frac{{\mu }_1^2}{2}\hfill \\ \hfill & +\frac{1+h_i{\overline{J}}_i}{2}{\tilde{\mathit{\chi }}}_i^T{\tilde{\mathit{\chi }}}_i+\frac{1+h_i{\overline{J}}_i}{2}{\tilde{\mathit{\nu }}}_i^T{\tilde{\mathit{\nu }}}_i+1.671{\mu }_2{\varrho }_i+\frac{{\delta }_i^2}{2}+\frac{{\mu }_1^2}{2}+\frac{1}{2}{\varsigma }_i^2\hfill \\ \hfill \le & -\sum _{\rho =1}^6\left(\left(k_{11,\rho }-h_i{\overline{J}}_i\right){\tilde{\eta }}_{i,\rho }^{2{\alpha }_1}+\left(\frac{1}{\kappa }-\frac{1}{2}-\frac{h_i{\overline{J}}_i}{2}\right){\tilde{\chi }}_{i,\rho }^{2{\alpha }_1}+\left(k_{12,\rho }-\frac{1}{2}-\frac{h_i{\overline{J}}_i}{2}\right){\tilde{\nu }}_{i,\rho }^{2{\alpha }_1}\right)\hfill \\ \hfill & -\sum _{\rho =1}^6\left(\left(k_{21,\rho }-h_i{\overline{J}}_i\right){\tilde{\eta }}_{i,\rho }^{2{\beta }_1}+\left(\frac{1}{\kappa }-\frac{1}{2}-\frac{h_i{\overline{J}}_i}{2}\right){\tilde{\chi }}_{i,\rho }^{2{\beta }_1}+\left(k_{22,\rho }-\frac{1}{2}-\frac{h_i{\overline{J}}_i}{2}\right){\tilde{\nu }}_{i,\rho }^{2{\beta }_1}\right)\hfill \\ \hfill & +1.671{\mu }_2{\varrho }_i+\frac{{\delta }_i^2}{2}+\frac{{\mu }_1^2}{2}+\frac{1}{2}{\varsigma }_i^2\hfill \\ \hfill \le & -\left({\sigma }_{min}\left({\mathit{K}}_{11}\right)-h_i{\overline{J}}_i\right){\left({\tilde{\mathit{\eta }}}_i^T{\tilde{\mathit{\eta }}}_i\right)}^{{\alpha }_1}-\left(\frac{1}{\kappa }-\frac{1}{2}-\frac{h_i{\overline{J}}_i}{2}\right){\left({\tilde{\mathit{\chi }}}_i^T{\tilde{\mathit{\chi }}}_i\right)}^{{\alpha }_1}\hfill \\ \hfill & -\frac{{\sigma }_{min}\left({\mathit{K}}_{12}\right)-\frac{1}{2}-\frac{h_i{\overline{J}}_i}{2}}{{\sigma }_{max}^{{\alpha }_1}\left(\mathit{M}\right)}{\left({\tilde{\mathit{\eta }}}_i^T\mathit{M}{\tilde{\mathit{\eta }}}_i\right)}^{{\alpha }_1}+6^{1-{\beta }_1}\left(-\left({\sigma }_{min}\left({\mathit{K}}_{21}\right)-h_i{\overline{J}}_i\right){\left({\tilde{\mathit{\eta }}}_i^T{\tilde{\mathit{\eta }}}_i\right)}^{{\beta }_1}\right.\hfill \\ \hfill & \left.-\left(\frac{1}{\kappa }-\frac{1}{2}-\frac{h_i{\overline{J}}_i}{2}\right){\left({\tilde{\mathit{\chi }}}_i^T{\tilde{\mathit{\chi }}}_i\right)}^{{\beta }_1}-\frac{{\sigma }_{min}\left({\mathit{K}}_{22}\right)-\frac{1}{2}-\frac{h_i{\overline{J}}_i}{2}}{{\sigma }_{max}^{{\beta }_1}\left(\mathit{M}\right)}{\left({\tilde{\mathit{\eta }}}_i^T\mathit{M}{\tilde{\mathit{\eta }}}_i\right)}^{{\beta }_1}\right)\hfill \\ \hfill & +1.671{\mu }_2{\varrho }_i+\frac{{\delta }_i^2}{2}+\frac{{\mu }_1^2}{2}+\frac{1}{2}{\varsigma }_i^2\hfill \\ \hfill \le & -2^{{\alpha }_1}{\overline{A}}_1\left({\left(\frac{1}{2}{\tilde{\mathit{\eta }}}_i^T{\tilde{\mathit{\eta }}}_i\right)}^{{\alpha }_1}+{\left(\frac{1}{2}{\tilde{\mathit{\chi }}}_i^T{\tilde{\mathit{\chi }}}_i\right)}^{{\alpha }_1}+{\left(\frac{1}{2}{\tilde{\mathit{\nu }}}_i^T{\mathit{M}}_i{\tilde{\mathit{\nu }}}_i\right)}^{{\alpha }_1}\right)\hfill \\ \hfill & -6^{1-{\beta }_1}{\overline{B}}_1\left({\left(\frac{1}{2}{\tilde{\mathit{\eta }}}_i^T{\tilde{\mathit{\eta }}}_i\right)}^{{\beta }_1}+{\left(\frac{1}{2}{\tilde{\mathit{\chi }}}_i^T{\tilde{\mathit{\chi }}}_i\right)}^{{\beta }_1}+{\left(\frac{1}{2}{\tilde{\mathit{\nu }}}_i^T{\mathit{M}}_i{\tilde{\mathit{\nu }}}_i\right)}^{{\beta }_1}\right)+{\Omega }_1\hfill \\ \hfill =& -A_1{V}_1^{{\alpha }_1}-B_1{V}_1^{{\beta }_1}+{\Omega }_1,\hfill \end{array}$$

(40)

where $A_1=2^{{\alpha }_1}{\overline{A}}_1$, ${\overline{A}}_1=min\left\{{\sigma }_{min}\left({\mathit{K}}_{11}\right)-h_i{\overline{J}}_i,\frac{1}{\kappa }-\frac{1}{2}-\frac{h_i{\overline{J}}_i}{2},\frac{{\sigma }_{min}\left({\mathit{K}}_{12}\right)-\frac{1}{2}-\frac{h_i{\overline{J}}_i}{2}}{{\sigma }_{max}^{{\alpha }_1}\left(\mathit{M}\right)}\right\}$, $B_1={18}^{1-{\beta }_1}{\overline{B}}_1$, ${\overline{B}}_1=min\left\{{\sigma }_{min}\left({\mathit{K}}_{21}\right)-h_i{\overline{J}}_i,\frac{1}{\kappa }-\frac{1}{2}-\frac{h_i{\overline{J}}_i}{2},\frac{{\sigma }_{min}\left({\mathit{K}}_{22}\right)-\frac{1}{2}-\frac{h_i{\overline{J}}_i}{2}}{{\sigma }_{max}^{{\beta }_1}\left(\mathit{M}\right)}\right\}$, ${\Omega }_1=1.671{\mu }_2{\varrho }_i$$+\frac{{\delta }_i^2}{2}+\frac{{\mu }_1^2}{2}+\frac{1}{2}{\varsigma }_i^2$.

According to Lemma 2, system (33) is practical fixed stable, and the tracking errors of the MUV system converge to an arbitrarily small region of the origin within a fixed time, which is determined by

$$\begin{array}{c}\hfill t_1\le T_{\max }=\frac{1}{{\omega }_1{A}_1(1-{\alpha }_1)}+\frac{1}{{\omega }_1{B}_1({\beta }_1-1)}.\end{array}$$

(41)

To avoid the Zeno behavior, we show that minimal inter-event times $t_i^{*}$ exist, which satisfy $\left(t_{i,m+1}-t_{i,m}\right)>t_i^{*}$ for $m\in {\mathbb{Z}}^{+}$.

For $t\in [t_{i,m},t_{i,m+1})$, we have ${\dot{\mathit{\tau }}}_i\left(t\right)=0$, then taking the time derivative of the measurement error ${\tilde{\mathit{\tau }}}_i\left(t\right)$ yields

$$\begin{array}{c}\hfill \frac{d}{dt}\parallel {\tilde{\mathit{\tau }}}_i\left(t\right)\parallel =\frac{d}{dt}{\left[{\tilde{\mathit{\tau }}}_i^T\left(t\right){\tilde{\mathit{\tau }}}_i\left(t\right)\right]}^{\frac{1}{2}}=\frac{{\tilde{\mathit{\tau }}}_i^T\left(t\right){\dot{\tilde{\mathit{\tau }}}}_i\left(t\right)}{\parallel {\tilde{\mathit{\tau }}}_i\left(t\right)\parallel }\le ∥\frac{d}{dt}{\overline{\mathit{\tau }}}_i\left(t\right)∥.\end{array}$$

(42)

Since all the signals of the closed-loop MUV system are bounded, there is a positive constant ${\epsilon }_i$, satisfying $∥\frac{d}{dt}{\overline{\mathit{\tau }}}_i\left(t\right)∥<{\epsilon }_i$. Then, integrating (42) over $t\in [t_{i,m},t_{i,m+1})$ and employing the event-triggering condition given in (32), we have

$$\begin{array}{c}\hfill t_{i,m+1}-t_{i,m}\ge \frac{\parallel {\tilde{\mathit{\tau }}}_i\left(t\right)\parallel }{{\epsilon }_i}\ge \frac{{\mu }_1\parallel {\tilde{\mathit{\nu }}}_i\parallel +{\mu }_2}{{\epsilon }_i}\ge \frac{{\mu }_2}{{\epsilon }_i}=t_i^{*}.\end{array}$$

(43)

Therefore, it holds that $\left(t_{i,m+1}-t_{i,m}\right)>t_i^{*}$ with $t_i^{*}=\frac{{\mu }_2}{{\epsilon }_i}$. □

Remark 3. 

Different from the finite-time methods [17,19,20,21,22], in this paper, the designed distributed event-triggered fixed-time backstepping control scheme can make the formation tracking errors converge to the origin within fixed-time independent of initial conditions, while the signal transmission frequencies are effectively reduced.

Remark 4. 

Although the control design for MUVs subjected to external disturbances is investigated, the proposed control scheme can be easily extended to handle other engineering systems [42,43,44,45]. In addition, the developed control scheme is based on the model. The accurate underwater vehicle model will lead to better control performance. Therefore, some advanced parameter estimation algorithms [46,47,48,49] can be used to establish an accurate model.

Remark 5. 

The parameters of the proposed control scheme should be selected according to the following adjustment guidelines: (1) Controller parameters—increasing ${\mathit{K}}_{11},{\mathit{K}}_{21},{\mathit{K}}_{12},{\mathit{K}}_{22},{\beta }_1,{\mu }_1,{\mu }_2$ and decreasing ${\alpha }_1$ will improve the convergence rate, reduce the system tracking errors, and prolong the inter-event time. However, too large ${\mathit{K}}_{11},{\mathit{K}}_{10},{\mathit{K}}_{20},{\mathit{K}}_{22},{\beta }_1,{\mu }_1,{\mu }_2$, and too small ${\alpha }_1$ will result in severe input saturation and high-frequency changes in the vehicle states. (2) Observer parameters—increasing ${\mathit{K}}_{11},{\mathit{K}}_{21},{\lambda }_0,{\beta }_0$ and decreasing ${\lambda }_1,{\lambda }_2,{\alpha }_0$ will improve the convergence rate and enhance the estimation accuracy of the disturbance and adaptive parameters. However, being too large or too small will lead to an overshoot that is too large. A trade-off is required when tuning these parameters.

4. Simulation Results

In this section, numerical simulations are performed on MATLAB/Simulink using a 3.00 GHz Intel Core i7-9700 processor and some necessary discussions are provided to illustrate the effectiveness of the proposed formation control approach. The ODIN underwater vehicle developed at the University of Hawaii is considered in the simulations (see Figure 3a). Detailed model parameters of ODIN can be found in [7]. The propulsion system of ODIN is composed of four horizontal thrusters and four vertical thrusters. Each thruster is saturated within $\pm 150$ N, such as in [7]. Therefore, the control input ${\mathit{\tau }}_i$ can be produced jointly by individual thrusters. The relationship between the control input and thrusters can be described as

$$\begin{array}{c}\hfill {\mathit{\tau }}_i=\mathit{E}{\mathit{T}}_i,\end{array}$$

(44)

where $T_i\in {\mathbb{R}}^8$ denotes the vector of thrust forces generated by each thruster, and $\mathit{E}\in {\mathbb{R}}^{6\times 8}$ is the thrust force allocation matrix given by

$$\begin{array}{c}\hfill \mathit{E}=\left[\begin{array}{cccccccc}a& -a& -a& a& 0& 0& 0& 0\\ a& a& -a& -a& 0& 0& 0& 0\\ 0& 0& 0& 0& -1& -1& -1& -1\\ 0& 0& 0& 0& La& La& -La& -La\\ 0& 0& 0& 0& La& -La& -La& La\\ L_z& -L_z& L_z& -L_z& 0& 0& 0& 0\end{array}\right],\end{array}$$

(45)

where $a=\frac{\sqrt{2}}{2}$, $L=0.381$ m is the distance from the center of ODIN to the vertical thruster’s center, and $L_z=0.508$ m is the radial distance from the center of ODIN to the horizontal thruster’s center. The thruster layout of ODIN is plotted in Figure 3b.

In order to comprehensively illustrate the effectiveness of the proposed formation tracking control scheme, a group of ODIN underwater vehicles consisting of a leader and four followers described by (1) with different communication topologies is considered. The trajectory of the leader is described by two different trajectories, i.e., the spatial helical trajectory and the 3D Dubins trajectory. The details are as follows.

4.1. Scenario 1: Helical Trajectory Formation Tracking

In this scenario, four followers are commanded to track the trajectory of the leader described by the 3D helical trajectory. The trajectory of the leader in the inertial frame is given as

$$\begin{array}{c}\hfill {\mathit{\eta }}_0=\left[\begin{array}{c}{x}_0\hfill \\ y_0\hfill \\ z_0\hfill \\ {\varphi }_0\hfill \\ {\theta }_0\hfill \\ {\psi }_0\hfill \end{array}\right]=\left[\begin{array}{c}5sin\left(0.02\pi t\right)\mathrm{m}\\ 5cos\left(0.02\pi t\right)\mathrm{m}\\ 0.3t+5\mathrm{m}\\ 0\mathrm{rad}\\ 0\mathrm{rad}\\ 0.02\pi t\mathrm{rad}\end{array}\right].\end{array}$$

(46)

To verify the robustness of the proposed formation tracking control scheme against external disturbances, without loss of generality, the unknown time-varying environmental disturbances are selected as

$$\begin{array}{c}\hfill {\mathit{f}}_i=\left[\begin{array}{c}{f}_{i,u}\hfill \\ f_{i,v}\hfill \\ f_{i,w}\hfill \\ f_{i,p}\hfill \\ f_{i,q}\hfill \\ f_{i,r}\hfill \end{array}\right]=\left[\begin{array}{c}-4-3sin\left(0.3t\right)-2cos\left(0.2t\right)\mathrm{N}\hfill \\ -3-2sin\left(0.2t\right)-1.5sin\left(0.1t\right)\mathrm{N}\hfill \\ -2-3sin\left(0.1t\right)-2cos\left(0.2t\right)\mathrm{N}\hfill \\ -1-sin\left(0.2t\right)-cos\left(0.1t\right)\mathrm{N}·\mathrm{m}\hfill \\ -2-2sin\left(0.1t\right)-3cos\left(0.3t\right)\mathrm{N}·\mathrm{m}\hfill \\ -3-2cos\left(0.2t\right)-2cos\left(0.1t\right)\mathrm{N}·\mathrm{m}\hfill \end{array}\right].\end{array}$$

(47)

The communication topology of the MUV system is depicted in Figure 4, where node ‘L0’ represents the leader and nodes ‘F1’, ‘F2’, ‘F3’, and ‘F4’ stand for followers 1–4. The corresponding Laplacian matrix is given as

$$\begin{array}{c}\hfill \mathit{L}=\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ -1& 1& 0& 0& 0\\ -1& -1& 2& 0& 0\\ -1& 0& 0& 1& 0\\ -1& 0& 0& -1& 2\end{array}\right].\end{array}$$

(48)

The desired formation pattern of this MUV system is defined as

$$\begin{array}{cc}\hfill & \Delta {\mathit{\eta }}_{10}\left(0\right)={[-2\mathrm{m},2\mathrm{m},-2\mathrm{m},0\mathrm{rad},0\mathrm{rad},0\mathrm{rad}]}^T,\hfill \\ \hfill & \Delta {\mathit{\eta }}_{20}\left(0\right)={[-4\mathrm{m},4\mathrm{m},-4\mathrm{m},0\mathrm{rad},0\mathrm{rad},0\mathrm{rad}]}^T,\hfill \\ \hfill & \Delta {\mathit{\eta }}_{30}\left(0\right)={[-2\mathrm{m},2\mathrm{m},2\mathrm{m},0\mathrm{rad},0\mathrm{rad},0\mathrm{rad}]}^T,\hfill \\ \hfill & \Delta {\mathit{\eta }}_{40}\left(0\right)={[-4\mathrm{m},4\mathrm{m},4\mathrm{m},0\mathrm{rad},0\mathrm{rad},0\mathrm{rad}]}^T,\hfill \\ \hfill & \Delta {\mathit{\eta }}_{21}\left(0\right)={[-2\mathrm{m},2\mathrm{m},-2\mathrm{m},0\mathrm{rad},0\mathrm{rad},0\mathrm{rad}]}^T,\hfill \\ \hfill & \Delta {\mathit{\eta }}_{43}\left(0\right)={[-2\mathrm{m},2\mathrm{m},2\mathrm{m},0\mathrm{rad},0\mathrm{rad},0\mathrm{rad}]}^T.\hfill \end{array}$$

(49)

The initial positions of the leader and four followers are chosen as

$$\begin{array}{cc}\hfill & {\mathit{\eta }}_0\left(0\right)={[0\mathrm{m},5\mathrm{m},5\mathrm{m},0\mathrm{rad},0\mathrm{rad},0\mathrm{rad}]}^T,\hfill \\ \hfill & {\mathit{\eta }}_1\left(0\right)={[-1\mathrm{m},6\mathrm{m},4\mathrm{m},0.1\mathrm{rad},-0.1\mathrm{rad},0.1\mathrm{rad}]}^T,\hfill \\ \hfill & {\mathit{\eta }}_2\left(0\right)={[-2\mathrm{m},7\mathrm{m},3\mathrm{m},0.2\mathrm{rad},-0.2\mathrm{rad},-0.1\mathrm{rad}]}^T,\hfill \\ \hfill & {\mathit{\eta }}_3\left(0\right)={[-1\mathrm{m},6\mathrm{m},8\mathrm{m},-0.1\mathrm{rad},0.1\mathrm{rad},0.2\mathrm{rad}]}^T,\hfill \\ \hfill & {\mathit{\eta }}_4\left(0\right)={[-2\mathrm{m},7\mathrm{m},11\mathrm{m},-0.2\mathrm{rad},0.2\mathrm{rad},-0.2\mathrm{rad}]}^T.\hfill \end{array}$$

(50)

The initial velocities of four followers are set as ${\mathit{\nu }}_i\left(0\right)={[0\mathrm{m}/\mathrm{s},0\mathrm{m}/\mathrm{s},0\mathrm{m}/\mathrm{s},0\mathrm{rad}/\mathrm{s},0\mathrm{rad}/\mathrm{s},0\mathrm{rad}/\mathrm{s}]}^T$. The controller parameters are selected as ${\alpha }_1=0.9$, ${\beta }_1=1.1$, ${\mathit{K}}_{11}=1.5{\mathit{I}}_6$, ${\mathit{K}}_{21}=1.5{\mathit{I}}_6,{\mathit{K}}_{12}=10{\mathit{I}}_6,{\mathit{K}}_{22}=10{\mathit{I}}_6,\kappa =0.1,{\mu }_1=20,{\mu }_2=0.2$, and the observer parameters are chosen as ${\alpha }_0=0.9,{\beta }_0=1.1,{\mathit{K}}_{10}=5{\mathit{I}}_6,{\mathit{K}}_{20}=5{\mathit{I}}_6,{\lambda }_0=35,{\lambda }_1=0.1,{\lambda }_2=0.1$. Meanwhile, the step size is selected as 10 ms and the simulation time is set as 150 s. The simulation results of scenario 1 are shown in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11.

In order to demonstrate the superior performance of the proposed control strategy, a comparison with the nonlinear disturbance observer-based dynamic surface control (NDO-DSC) scheme, which is widely used for thee control of ocean vehicles [50], is performed. The comparison of the norm of the formation tracking error ${\tilde{\mathit{\eta }}}_i$ between the proposed control scheme and the NDO-DSC scheme is presented in Figure 12.

4.2. Scenario 2: Dubins Trajectory Formation Tracking

In this scenario, four followers are forced to track the trajectory of the leader described by a spatial Dubins trajectory. The trajectory of the leader in the inertial frame is expressed as

$$\begin{array}{c}\hfill x_0\left(t\right)=\left\{\begin{array}{cc}0.6t\mathrm{m},\hfill & 0\le t<25\hfill \\ 0.6(t-25)+15\mathrm{m},\hfill & 25\le t<50\hfill \\ 7sin\left(0.04\pi \right(t-50\left)\right)+30\mathrm{m},\hfill & 50\le t<75\hfill \\ -0.6(t-75)+30\mathrm{m},\hfill & 75\le t<100\hfill \\ -7sin\left(0.04\pi \right(t-100\left)\right)+15\mathrm{m},\hfill & 100\le t<125\hfill \\ 0.6(t-125)+15\mathrm{m},\hfill & 125\le t<150\hfill \end{array}\right.,\\ \hfill y_0\left(t\right)=\left\{\begin{array}{cc}1\mathrm{m},\hfill & 0\le t<25\hfill \\ 1\mathrm{m},\hfill & 25\le t<50\hfill \\ -7cos\left(0.04\pi \right(t-50\left)\right)+8\mathrm{m},\hfill & 50\le t<75\hfill \\ 15\mathrm{m},\hfill & 75\le t<100\hfill \\ -7cos\left(0.04\pi \right(t-100\left)\right)+22\mathrm{m},\hfill & 100\le t<125\hfill \\ 29\mathrm{m},\hfill & 125\le t<150\hfill \end{array}\right.,\end{array}$$

$$\begin{array}{c}\hfill {\psi }_0\left(t\right)=\left\{\begin{array}{cc}0\mathrm{rad},\hfill & 0\le t<25\hfill \\ 0\mathrm{rad},\hfill & 25\le t<50\hfill \\ 0.04\pi (t-50)\mathrm{rad},\hfill & 50\le t<75\hfill \\ \pi \mathrm{rad},\hfill & 75\le t<100\hfill \\ \pi -0.04\pi (t-100)\mathrm{rad},\hfill & 100\le t<125\hfill \\ 0\mathrm{rad},\hfill & 125\le t<150\hfill \end{array}\right.,\\ \hfill \begin{array}{c}{z}_0\left(t\right)=\left\{\begin{array}{cc}0.6t+4\mathrm{m},\hfill & 0\le t<25\hfill \\ 19\mathrm{m},\hfill & 25\le t<150\hfill \end{array}\right.,\hfill \\ \\ {\varphi }_0\left(t\right)=0\mathrm{rad},0\le t<150,\hfill \\ \\ {\theta }_0\left(t\right)=0\mathrm{rad},0\le t<150.\hfill \end{array}\end{array}$$

To further demonstrate the robustness of the proposed formation tracking control approach, the external disturbances are changed to $2\times {\mathit{f}}_i$. The communication topology of the MUV system is provided in Figure 13, and the corresponding Laplacian matrix is expressed as

$$\begin{array}{c}\hfill \mathit{L}=\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ -1& 1& 0& 0& 0\\ 0& -1& 2& -1& 0\\ -1& 0& 0& 1& 0\\ 0& -1& 0& -1& 2\end{array}\right].\end{array}$$

(51)

The desired formation pattern of this MUV system is defined as

$$\begin{array}{cc}\hfill & \Delta {\mathit{\eta }}_{10}\left(0\right)={[2\mathrm{m},0\mathrm{m},-2\mathrm{m},0\mathrm{rad},0\mathrm{rad},0\mathrm{rad}]}^T,\hfill \\ \hfill & \Delta {\mathit{\eta }}_{30}\left(0\right)={[-2\mathrm{m},0\mathrm{m},2\mathrm{m},0\mathrm{rad},0\mathrm{rad},0\mathrm{rad}]}^T,\hfill \\ \hfill & \Delta {\mathit{\eta }}_{21}\left(0\right)={[-2\mathrm{m},-2\mathrm{m},2\mathrm{m},0\mathrm{rad},0\mathrm{rad},0\mathrm{rad}]}^T,\hfill \\ \hfill & \Delta {\mathit{\eta }}_{23}\left(0\right)={[2\mathrm{m},-2\mathrm{m},-2\mathrm{m},0\mathrm{rad},0\mathrm{rad},0\mathrm{rad}]}^T,\hfill \\ \hfill & \Delta {\mathit{\eta }}_{41}\left(0\right)={[-2\mathrm{m},2\mathrm{m},2\mathrm{m},0\mathrm{rad},0\mathrm{rad},0\mathrm{rad}]}^T,\hfill \\ \hfill & \Delta {\mathit{\eta }}_{43}\left(0\right)={[2\mathrm{m},2\mathrm{m},-2\mathrm{m},0\mathrm{rad},0\mathrm{rad},0\mathrm{rad}]}^T.\hfill \end{array}$$

The initial positions of the leader and four followers are set as

$$\begin{array}{cc}\hfill & {\mathit{\eta }}_0\left(0\right)={[0\mathrm{m},1\mathrm{m},4\mathrm{m},0\mathrm{rad},0\mathrm{rad},0\mathrm{rad}]}^T,\hfill \\ \hfill & {\mathit{\eta }}_1\left(0\right)={[1\mathrm{m},1\mathrm{m},5\mathrm{m},0.1\mathrm{rad},-0.1\mathrm{rad},-0.1\mathrm{rad}]}^T,\hfill \\ \hfill & {\mathit{\eta }}_2\left(0\right)={[-1\mathrm{m},-1\mathrm{m},2\mathrm{m},0.3\mathrm{rad},-0.3\mathrm{rad},0.3\mathrm{rad}]}^T,\hfill \\ \hfill & {\mathit{\eta }}_3\left(0\right)={[-1\mathrm{m},2\mathrm{m},5\mathrm{m},-0.1\mathrm{rad},0.1\mathrm{rad},-0.1\mathrm{rad}]}^T,\hfill \\ \hfill & {\mathit{\eta }}_4\left(0\right)={[1\mathrm{m},3\mathrm{m},3\mathrm{m},-0.3\mathrm{rad},0.3\mathrm{rad},0.3\mathrm{rad}]}^T.\hfill \end{array}$$

The initial velocities of four followers are chosen as ${\mathit{\nu }}_i\left(0\right)={[0\mathrm{m}/\mathrm{s},0\mathrm{m}/\mathrm{s},0\mathrm{m}/\mathrm{s},0\mathrm{rad}/\mathrm{s},0\mathrm{rad}/\mathrm{s},0\mathrm{rad}/\mathrm{s}]}^T$. The controller parameters and the observer parameters are selected to be the same as scenario 1 except for ${\lambda }_0=50$. The simulation results of scenario 2 are presented in Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21.

4.3. Discussion

Figure 5a and Figure 14a depict the formation tracking results of the MUV system with respect to the helical and Dubins trajectories with different communication topologies under the action of the AFxDO-based distributed event-triggered fixed-time backstepping formation tracking control law. Figure 5b–d and Figure 14b–d show the formation tracking performances in x-y, x-z, and y-z planes. The corresponding position tracking errors ${\tilde{x}}_i,{\tilde{y}}_i,{\tilde{z}}_i$ and attitude tracking errors ${\tilde{\varphi }}_i,{\tilde{\theta }}_i,{\tilde{\psi }}_i$ are given in Figure 6 and Figure 15. The corresponding linear velocity tracking errors ${\tilde{u}}_i,{\tilde{v}}_i,{\tilde{w}}_i$ and angular velocity tracking errors ${\tilde{p}}_i,{\tilde{q}}_i,{\tilde{r}}_i$ are plotted in Figure 7 and Figure 16. From Figure 5, Figure 6 and Figure 7 and Figure 14, Figure 15 and Figure 16, we can observer that the four underwater vehicle followers under different initial conditions can track the leader underwater vehicle with the desired formation pattern, and the formation tracking errors between the leader and four followers are stabilized to a small region of the origin in fixed time, which demonstrates that the formation control of MUVs in the presence of external disturbances can be successfully achieved under the proposed AFxDO-based distributed event-triggered fixed-time backstepping formation tracking control scheme.

Figure 8 and Figure 17 present the actual and estimated external disturbances of follower 1 in both scenarios. The corresponding estimations of the upper bounds of external disturbances are shown in Figure 9 and Figure 18. From Figure 8, Figure 9, Figure 17, and Figure 18, we can clearly see that the external disturbances are estimated effectively and rapidly, and the designed adaptive law can track the changes of the upper bound values of disturbances, verifying the effectiveness of the designed AFxDO.

Figure 10a and Figure 19a present the inter-event intervals of four followers under the proposed event-triggered mechanism. Figure 10b and Figure 19b illustrate the comparison of triggering numbers between the time-triggered approach and the designed event-triggered approach. As shown in Figure 10a,b, the maximum inter-event intervals from follower 1 to follower 4 in scenario 1 are 0.94 s, 0.90 s, 0.78 s, and 0.76 s, respectively. The designed event-triggered mechanism can effectively reduce the signal transmission frequency. From Figure 19a,b, the same results can be obtained in scenario 2. In both scenarios, the designed event-triggered mechanism can save large amounts of communication resources with a maximum percentage of 80.5% and a minimum percentage of 76.5%.

Figure 11 and Figure 20 present the thrust force of each thruster of four followers under the action of the proposed control scheme. It is shown that the presented approach has a large initial thrust force, which is due to the presence of a large initial error and the need to increase the thrust force size to speed up its convergence. Additionally, in scenario 2, the pulses in the thrust forces with respective to the Dubins trajectory occur at moments 25 s, 50 s, 75 s, 100 s, and 125 s. The reason is that the Dubins trajectory is non-smooth at the above moments and the thrust forces are associated with the derivative of the reference trajectory.

From Figure 12 and Figure 21, we can conclude that under the action of the proposed control scheme, the convergence rate of the formation tracking error is faster and the steady state error is smaller than that of the NDO-DSC scheme, which means that the proposed control scheme can provide a faster convergence rate and stronger robustness.

5. Conclusions

External disturbances are inevitable in the MUV system. In this paper, we develop a robust leader–follower formation trajectory tracking control scheme for MUVs by designing the AFxDO and the distributed event-triggered fixed-time backstepping control approach. Under the designed control scheme, the occupation of bandwidth resources is reduced and the fixed-time convergence property of the disturbance estimation errors and the formation tracking errors is guaranteed, which is independent of the initial conditions of the system. Simulation results show that under different communication topologies and formation patterns, the developed control strategy is effective at achieving the formation tracking control of the MUV system.

This paper develops the leader–follower formation tracking control strategy under the assumption that the model parameters of the underwater vehicle are known. However, in practice, the model parameters are not available; one can obtain these parameters first through using parameter estimation algorithms [51,52,53,54,55] from observation data, such as gradient-based algorithms, least squares-based algorithms, Newton algorithms [56,57,58,59,60], and so on. Moreover, the faults of underwater vehicle have not been taken into consideration. In future works, we will investigate these factors.