Adaptive Second-Order Fast Terminal Sliding-Mode Formation Control for Unmanned Surface Vehicles

Abstract

The formation control of unmanned surface vehicles (USVs) while considering communication topology, dynamic model uncertainties, environmental disturbances, and a fast convergence rate is addressed in this paper. First, graph theory is introduced to describe the connective relationships and establish generalized formation errors among USVs. Then, a second-order fast nonsingular terminal sliding-mode control (SOFNTSMC) is designed to guarantee that the system converges quickly and without chatter. An adaptive update law is designed in order to estimate the model uncertainties and external disturbances without the requirement of the boundary information of the system uncertainties. With the application of the adaptive SOFNTSMC (ASOFNTSMC) and graph theory, a distributed control is developed for every USV to perform the desired formation pattern. Finally, the results of simulations and comparisons demonstrate the effectiveness of the proposed method.

1. Introduction

Nowadays, the control of multi-agent systems is a point of focus for many scientific communities [1,2,3,4,5,6] because such systems can save fuel and complete tasks more efficiently. A class of USV formation systems has received considerable attention [7,8]. So far, a number of schemes have been proposed in order to realize the formation control of USVs, such as the virtual structure method, behavior-based methods, artificial potential function schemes, and the leader–follower approach. The most popular strategy is the leader–follower approach due to its convenience and ease of implementation [9]. A distribution formation control that integrated a backstepping method was designed for USVs, and it guaranteed the formation pattern in the presence of actuator saturation [10]. The input–output linearization technology was used in combination with graph theory, the idea of consensus, and some nonlinear tools for the formation control of USVs. A state-feedback-based backstepping control was designed in [11] for a USV without considering the model uncertainties and disturbances.

It is worth noting that these studies of formation control for USVs did not consider model uncertainties and disturbances. It is impossible to accurately obtain USVs’ model parameters in practice due to their strongly nonlinear characteristics. Further, because the ocean environment is complex and always changes, it is necessary to consider it in order to achieve accurate control for rapid USVs. To deal with model uncertainties and external disturbances, an extended state observer was designed in [12,13], an extended state observer (ESO)-based robust dynamic surface control (DSC) method was developed for a triaxial MEMS gyroscope in [12], and a disturbance-rejection-based solution to the problem of robust output regulation was studied in [13]. An adaptive attitude tracking control with prescribed performance was developed in [14]. In the presence of unknown disturbances, an equivalent output injection sliding-mode observer and an output feedback sliding-mode control were designed for a nonlinear plant [15]. An output feedback terminal sliding-mode control was proposed in [16], and a sliding-mode observer was designed for a nonlinear second-order system that needed less information and could converge to zero in a finite time. A nonlinear adaptive fuzzy output feedback control was designed for the dynamic positioning of a ship in the presence of model uncertainties and external disturbances in [17]. Unfortunately, it should be pointed out that even though disturbances were considered in the aforementioned research, boundary values were necessary in the control law, but these are difficult to know in practice.

In addition, because of its strong robustness to model uncertainties and external environmental disturbances, sliding-mode control has attracted particular attention in the field of USV formation control [18]. A dynamic sliding-mode control combined with a backstepping method was proposed for trajectory tracking in un-actuated unmanned underwater vehicles, and globally uniform asymptotic stability was achieved in [19]. An adaptive sliding-mode control was developed for a simplified UAV model with two degrees of yaw and pitch while considering input nonlinearities and unknown disturbances [20], and this adaptive sliding-mode control was employed to cope with disturbances; the comparison of the results verified the utility of the proposed method. However, the system could only asymptotically convergence to zero with the linear sliding-mode manifold in these works because of the latter’s characteristics. For USV systems, due to their rapid response, fast convergence of the system is a priority.

Subsequently, a terminal sliding-mode control that used a nonlinear sliding-mode manifold was proposed to meet the requirements of finite-time convergence. Because of the faster convergence of terminal sliding-mode control, it has been applied to the control of the movements of marine surface vehicles, unmanned underwater vehicles, USVs, and other mechanical systems [21,22,23]. A sliding-mode control combined with fault tolerance was designed for spacecraft in [24]; a new sliding mode was developed into a control law to guarantee that the system converged to zero in a finite time, and the control was robust to uncertainties and disturbances. An event-trigger-based distribution control law that took the communication burden into account was designed for spacecraft in [25]; the control law was able to reject mass uncertainties and disturbances by adjusting the control parameters. Nevertheless, on one hand, the discontinuous sign function (DSF) was used in the switching term, and the switching term was directly introduced into the control law, which could lead to the problem of chatter in the control signal. On the other hand, though the convergence speed was faster than that of the linear sliding-mode manifold, boundary information for the disturbances was needed in the control law, which is impossible to obtain in practice.

A nonsingular terminal sliding-mode control method based on a disturbance observer for the stabilization of micro-electro-mechanical systems under lumped perturbation was designed in [26]. A robust H∞ integral control scheme for a class of nonlinear uncertainty was proposed in [27]. In these two studies, the LMI method was introduced in order to calculate the optimized coefficients of the sliding surface, thus reducing the workload when adjusting the coefficients. A novel non-singular terminal sliding-mode control combined with a barrier function was developed in [28], in which a super-twisting terminal sliding-mode control was designed for a quad-rotor UAV while considering input delay, model uncertainty, and wind disturbance. These two studies did not need the boundary information of the disturbances or model uncertainties, and the simulation results demonstrated the finite-time convergence of the tracking trajectory. The authors of [29] presented a lumped-perturbation-observer-based robust control method using multiple extended sliding surfaces for a system with matched and unmatched uncertainties. An adaptive non-singular fast terminal sliding-mode control with an integral surface for the finite-time tracking control of nonlinear systems with external disturbances was proposed in [30], and the effectiveness of the proposed approach was assessed by using both a simulation and an experimental study. A novel fast terminal sliding-mode control technique based on a disturbance observer was studied for the stabilization of underactuated robotic systems in [31]. The proposed controller was able to regulate the state trajectories of underactuated systems toward the origin within a finite time with the existence of external disturbances. A non-singular terminal sliding-mode control scheme was proposed for the purpose of the finite-time stability of a nonlinear switching surface in [32]. In addition, an adaptive technique for the barrier function was employed to estimate the unknown upper bounds of exterior disturbances. In [33], a robust control technique was investigated for the reference tracking of uncertain time-delayed systems in the presence of actuator saturation. The simulations illustrated the efficiency of the suggested control procedure. In most of these studies, integral FTSMC was used. However, PD-FTSMC is considered in this paper.

For a group of USVs moving in the sea, when they are working at a speed greater than 30 knots/s, if the converge speed of the formation error is very slow, this will affect the efficiency of the formation operation. In addition, it is impossible to build an accurate USV model, which is also a challenge for USV formation control. Last but not least, external disturbances, such as wind, wave, and sea currents, are inevitable in the ocean; moreover, their boundary information is impossible to obtain. Therefore, it is very important to design a USV formation control strategy that takes model uncertainty, convergence speed, a lack of chatter, external disturbances, and a lack of boundary information into account. Motivated by the above research and analysis, in this work, an ASOFNTSMC for USV formation is developed to guarantee that the system converges quickly and without chatter in the presence of model uncertainties and external disturbances.

The main advantages and contributions of the proposed algorithms can be summarized as follows:

• The SOFNTSMC consists of a PDSMM (proportional differential sliding-mode manifold) and an SOFNTSMM (second-order fast nonsingular terminal sliding-mode manifold), which is proposed to guarantee that the generalized formation error converges to zero faster than the SONTSMC.
• Unlike the traditional sliding-mode control, which directly uses a sign function, in this approach, the sign function is introduced into the derivative of the control input, which eliminates the problem of chatter.
• An adaptive update law is designed to estimate the model uncertainties and external environmental disturbances without requiring the boundary information.

This paper is organized as follows: Section 2 introduces some notations and describes the problem. Section 3 presents the design of the USV formation control law, while the simulation results are described to verify the effectiveness and advantages of the proposed method in Section 4. Conclusions are drawn in Section 5. Finally, a stability analysis is provided in Appendix A. Before proceeding, a list of abbreviations is given in

Table 1. Abbreviation list.

Abbreviations	Meanings
$USVs$	$\mathrm{unmanned}\mathrm{surface}\mathrm{vehicles}$
$SOFNTSMM$	$\sec \mathrm{ond}-\mathrm{order}\mathrm{fast}\mathrm{nonsingular}\mathrm{terminal}\mathrm{sliding}-\mathrm{mode}\mathrm{manifold}$
$PDSMM$	$\mathrm{proportional}\mathrm{differential}\mathrm{sliding}-\mathrm{mode}\mathrm{manifold}$
$FNTSMM$	$\mathrm{fast}\mathrm{nonsingular}\mathrm{terminal}\mathrm{sliding}-\mathrm{mode}\mathrm{manifold}$
$SONTSMC$	$\sec \mathrm{ond}-\mathrm{order}\mathrm{nonsingular}\mathrm{terminal}\mathrm{sliding}-\mathrm{mode}\mathrm{control}$
$SOFNTSMC$	$\sec \mathrm{ond}-\mathrm{order}\mathrm{fast}\mathrm{nonsingular}\mathrm{terminal}\mathrm{sliding}-\mathrm{mode}\mathrm{control}$
$DSC$	$\mathrm{dynamic}\mathrm{surface}\mathrm{control}$
$ASOFNTSMC$	$\mathrm{adaptive}\sec \mathrm{ond}-\mathrm{order}\mathrm{fast}\mathrm{nonsingular}\mathrm{terminal}\mathrm{sliding}-\mathrm{mode}\mathrm{control}$
$DSF$	$\mathrm{discontinuous}\mathrm{sign}\mathrm{function}$

.

2. Preliminaries and Problem Description

2.1. Preliminaries

In this section, some notations and assumptions involved in the communication topology are given.

Algebraic Graph Theory

A directed graph $G=\{V,\epsilon \}$ [7] is used to describe the topological interactions among the USVs in this work, where $V=\{1,\cdots ,N\}$ stands for a finite node set, the ith node represents the ith USV, and the finite edge set of $\epsilon$ indicates the information effectiveness among the USVs. If $(i,j)\in \epsilon$, this means that there exists a valid path from i to j for transfering the ith USV’s information to the jth USV. However, there is no need for feedback. The elements of the adjacency matrix $A=\left[a_{ij}\right]\in {\mathbb{R}}^{N\times N}$ are defined as follows: If $(i,j)\in \epsilon$, $a_{ij}=1$, and $a_{ij}=0$ for other cases. $D=\mathrm{diag}\{{\mathrm{d}}_1,\cdots ,{\mathrm{d}}_{\mathrm{N}}\}$ is a diagonal matrix with the diagonal elements $d_i={\sum }_{j=1}^N{a}_{ij}$. A Laplacian matrix related to the graph G is defined as $L=D-A$. The Laplacian matrix $L=\left(l_{ij}\right)\in {\mathbb{R}}^{N\times N}$ can be also written as $l_{ii}={\sum }_{j\ne i}{a}_{ij}$ and $l_{ij}=-a_{ij}$ for $i\ne j$.

Remark 1.

For the Laplacian matrix $L=D-A$, we define a new normalized directed Laplacian matrix ι as follows:

$$\iota ={\iota }_{ij}\in {\mathbb{R}}^{\mathbb{N}\times \mathbb{N}}$$

where

$$\begin{array}{c}\hfill {\iota }_{ij}=\left\{\begin{array}{c}0,l_{ii}=0\hfill \\ 1,l_{ii}\ne 0\mathrm{and}i=j\hfill \\ \frac{l_{ij}}{l_{ii}},l_{ii}\ne 0\mathrm{and}i\ne j\hfill \end{array}\right.\end{array}$$

According to the case studied in the simulation, the normalized directed Laplacian matrix $\iota$ can be written as

$$\begin{array}{c}\hfill \iota ={{\iota }_{ij}}_{(N+1)\times (N+1)}=\left[\begin{array}{cc}0& 0_{1\times N}\\ {\iota }_1& {\iota }_2\end{array}\right]\end{array}$$

(1)

where

$$\begin{array}{c}\hfill {\iota }_{ij}=\left\{\begin{array}{c}0,i=L\hfill \\ \frac{l_{ij}}{l_{ii}},i\ne L\mathrm{and}i\ne j\hfill \\ 1,i\ne L\mathrm{and}i=j\hfill \end{array}\right.\end{array}$$

with the block matrix ${\iota }_1={[{\iota }_{1L},{\iota }_{2L},...,{\iota }_{NL}]}^T$ and the block matrix ${\iota }_2=\left[\begin{array}{ccc}{\iota }_{11}& \cdots & {\iota }_{1N}\\ \cdots & \cdots & \\ {\iota }_{N1}& \cdots & {\iota }_{NN}\end{array}\right]$.

2.2. Problem Description

An inertial frame $x_I$-$y_I$ and a body-fixed frame $x_B$-$y_B$ are used to describe the motion control of USVs, as shown in Figure 1.

Consider $N+1$ USVs, including one leader and N followers, with the degrees of heave, roll, and pitch being ignored. The three degrees of the ith USV model are given as follows [34]:

$$\begin{array}{cc}\hfill {\dot{\eta }}_i\left(t\right)& =R\left({\eta }_i\left(t\right)\right){\nu }_i,\hfill \\ \hfill {\dot{\nu }}_i& =M_i^{-1}(-C_i\left({\nu }_i\right){\nu }_i-D_i\left({\nu }_i\right){\nu }_i+{\tau }_i+{\tau }_{wi}),i=L,1,2\cdots ,N,\hfill \end{array}$$

(2)

where L denotes the leader and $1,2,\cdots ,N$ represents the N followers. ${\eta }_i\left(t\right)={[x_i,y_i,{\psi }_i]}^T\in {\mathbb{R}}^{3\times 1}$ are the north and east positions and the yaw angle in the initial frame, ${\nu }_i={[u_i,v_i,r_i]}^T\in {\mathbb{R}}^{3\times 1}$ denotes the surge, sway, and yaw velocities of the ith USV in the body-fixed frame, ${\tau }_i\in {\mathbb{R}}^{3\times 1}$ is the control input, ${\tau }_{wi}\in {\mathbb{R}}^{3\times 1}$ represents the external disturbances, and $R\left({\eta }_i\left(t\right)\right)$ is a transformation matrix with the following format:

$$R\left({\eta }_i\left(t\right)\right)=\left[\begin{array}{ccc}cos{\psi }_i& -sin{\psi }_i& 0\\ sin{\psi }_i& cos{\psi }_i& 0\\ 0& 0& 1\end{array}\right],$$

(3)

where $R{\left({\eta }_i\left(t\right)\right)}^T=R{\left({\eta }_i\left(t\right)\right)}^{-1}$. $M_i\in {\mathbb{R}}^{3\times 1}$, $C_i\left({\nu }_i\right)\in {\mathbb{R}}^{3\times 1}$, and $D_i\left({\nu }_i\right)\in {\mathbb{R}}^{3\times 1}$ are the inertia matrix, Coriolis centrifugal matrix, and damping matrix, respectively.

$M_i=M_i^T=\left[\begin{array}{ccc}{m}_{11i}& 0& 0\\ 0& m_{22i}& m_{23i}\\ 0& m_{32i}& m_{33i}\end{array}\right],$$C_i\left({\nu }_i\right)=\left[\begin{array}{ccc}0& 0& c_{13i}\\ 0& 0& c_{23i}\\ -c_{13i}& c_{23i}& 0\end{array}\right],$

$D_i\left({\nu }_i\right)=\left[\begin{array}{ccc}{d}_{11i}& 0& 0\\ 0& d_{22i}& d_{23i}\\ 0& d_{32i}& d_{33i}\end{array}\right].$

Assumption 1.

The model parameters are $\parallel M_i\parallel \le {\lambda }_0,\parallel C_i\left({\nu }_i\right)\parallel \le {\lambda }_1\parallel {\nu }_1\parallel$ and

$\parallel D_i\left({\nu }_i\right)\parallel \le {\lambda }_2+{\lambda }_3\parallel {\nu }_i\parallel$, where ${\lambda }_i(i=1,2,3)$ are unknown positive constants.

Assumption 2.

The disturbances ${\tau }_{wi}$ is once differential in time, and there exist unknown positive constants ${\lambda }_4$ and ${\lambda }_5$ such that ${\tau }_{wi}\le {\lambda }_4,{\dot{\tau }}_{wi}\le {\lambda }_5$.

Using the kinematics model and dynamics model of (2), we can obtain the mathematical model a USV in the north–east frame, which is described as follows:

$$\begin{array}{c}\hfill M_i\left({\eta }_i\left(t\right)\right){\ddot{\eta }}_i\left(t\right)+C_i({\eta }_i\left(t\right),{\dot{\eta }}_i\left(t\right)){\dot{\eta }}_i\left(t\right)+D_i({\eta }_i\left(t\right),{\dot{\eta }}_i\left(t\right)){\dot{\eta }}_i\left(t\right)=R_i\left({\eta }_i\left(t\right)\right)({\tau }_i+{\tau }_{wi}),\end{array}$$

(4)

where

$$\begin{array}{cc}\hfill M_i\left({\eta }_i\left(t\right)\right)& =R_i\left({\eta }_i\left(t\right)\right)M_i{R}_i^{-1}\left({\eta }_i\left(t\right)\right),\hfill \\ \hfill C_i({\eta }_i\left(t\right),{\dot{\eta }}_i\left(t\right))& =R_i\left({\eta }_i\left(t\right)\right)[C_i\left({\nu }_i\right)-M_i{R}_i{\left({\eta }_i\left(t\right)\right)}^{-1}{\dot{R}}_i\left({\eta }_i\left(t\right)\right)]R_i^{-1}\left({\eta }_i\left(t\right)\right),\hfill \\ \hfill D_i({\eta }_i\left(t\right),{\dot{\eta }}_i\left(t\right))& =R_i\left({\eta }_i\left(t\right)\right)D_i\left({\nu }_i\right)R_i^{-1}\left({\eta }_i\left(t\right)\right).\hfill \end{array}$$

Uncertainties in the damping term of the USV model are considered in this work, and they can be described as $D_i\left({\nu }_i\right)={\overline{D}}_i\left({\nu }_i\right)+\mathsf{\Delta }{D}_i\left({\nu }_i\right)$, where ${\overline{D}}_i\left({\nu }_i\right)$ is the nominal value of the damping terms, and $\mathsf{\Delta }{D}_i\left({\nu }_i\right)$ is uncertain part of the damping matrix. Then, Equation (4) can be rewritten as

$$\begin{array}{c}\hfill M_i\left({\eta }_i\left(t\right)\right){\ddot{\eta }}_i\left(t\right)+C_i({\eta }_i\left(t\right),{\dot{\eta }}_i\left(t\right)){\dot{\eta }}_i\left(t\right)+{\overline{D}}_i({\eta }_i\left(t\right),{\dot{\eta }}_i\left(t\right)){\dot{\eta }}_i\left(t\right)={\overline{\tau }}_i+{\rho }_i\left(t\right),\end{array}$$

(5)

where ${\overline{\tau }}_i=R_i\left({\psi }_i\right){\tau }_i$ and ${\rho }_i\left(t\right)=R_i^{-1}\left({\eta }_i\left(t\right)\right)\mathsf{\Delta }{D}_i\left({\nu }_i\right)R_i{\left({\eta }_i\left(t\right)\right)}^{-1}\left({\eta }_i\left(t\right)\right)+R_i\left({\eta }_i\left(t\right)\right)\times {\tau }_{wi}$ are unknown functions that include the model uncertainties and environmental disturbances.

Assumption 3.

According to Assumptions 1 and 2, the following inequalities can hold:

$$\begin{array}{cc}\hfill M_i\left({\eta }_i\left(t\right)\right)=& R_i\left({\eta }_i\left(t\right)\right)M_i{R}_i^{-1}\left({\eta }_i\left(t\right)\right)\hfill \\ \hfill \le & \parallel R_i\left({\eta }_i\left(t\right)\right)\parallel \parallel M_i\parallel \parallel R_i^{-1}\left({\eta }_i\left(t\right)\right)\parallel \hfill \\ \hfill \le & {\lambda }_0,\hfill \\ \hfill C_i({\eta }_i\left(t\right),{\dot{\eta }}_i\left(t\right))=& R_i\left({\eta }_i\left(t\right)\right)[C_i\left({\nu }_i\right)-M_i{R}_i{\left({\eta }_i\left(t\right)\right)}^{-1}{\dot{R}}_i\left({\eta }_i\left(t\right)\right)]R_i^{-1}\left({\eta }_i\left(t\right)\right)\hfill \\ \hfill \le & {\lambda }_1{\nu }_1-{\lambda }_0\parallel {\dot{R}}_i\left({\eta }_i\left(t\right)\right)|,\hfill \\ \hfill D_i({\eta }_i\left(t\right),{\dot{\eta }}_i\left(t\right))=& R_i\left({\eta }_i\left(t\right)\right)D_i\left({\nu }_i\right)R_i^{-1}\left({\eta }_i\left(t\right)\right)\hfill \\ \hfill \le & {\lambda }_2+{\lambda }_3\parallel {\nu }_i\parallel .\hfill \end{array}$$

where $\parallel ·\parallel$ denotes the norm of ·.

In this paper, the leader–follower scheme is used to obtain the desired formation pattern of the USVs. Considering collisions, the desired position of the neighboring USV is represented by a known vector $E_{ij}\left(t\right)\in {\mathbb{R}}^{3\times 1},t\ge 0,i,j=L,1,...,N$, indicating the position and attitude errors between the ith and jth USVs, and L represents the leader.

The formation error without the consideration of the communication topology can be given as follows:

$$\begin{array}{c}\hfill e_{ij}={\eta }_i\left(t\right)-{\eta }_j\left(t\right)-E_{ij}\left(t\right),i,j=L,1,\cdots ,N,\end{array}$$

(6)

Moreover, the generalized error with the consideration of the communication topology can be denoted by

$$\begin{array}{cc}\hfill z_i={\eta }_i+\frac{l_{ij}}{l_{ii}}\sum ({\eta }_j+E_{ij}),i=1,\cdots ,N,j& =L,1,\cdots ,N,i\ne j\hfill \end{array}$$

(7)

Lemma 1

([35]). Consider a connectivity graph with the normalized directed Laplacian matrix $\iota \in {\mathbb{R}}^{(N+1)\times (N+1)}$ given in (1). We define $z\triangleq {[z_1^T,\cdots ,z_N^T]}^T$, where $z_i\in {\mathbb{R}}^{3\times 1}$ is given by (7) and $e_j\left(t\right)\triangleq [e_{1j}^T,\cdots ,e_{nj}^T],j=L,1,\cdots ,N$. Then, the following statements hold:

(1) 

If $z_i=0$, then $e_{ij}=0,i=1,\cdots ,N,j=L,1,\cdots ,N$.

(2) 

If $e_{ij}\left(t\right)=0,i=1,\cdots ,N,j=L,1,\cdots ,N,i\ne j,$ then $z_i=0$.

We can obtain the dynamics of the generalized error by taking the first derivative of (7):

$$\begin{array}{cc}\hfill {\dot{z}}_i& ={\dot{\eta }}_i+\frac{l_{ij}}{l_{ii}}\sum ({\dot{\eta }}_j\left(t\right)+{\dot{E}}_{ij})\hfill \\ \hfill & ={\dot{\eta }}_i+{\iota }_{ij}\sum ({\dot{\eta }}_j+{\dot{E}}_{ij}),i=1,\cdots ,N,j=L,1,\cdots ,N.\hfill \end{array}$$

(8)

Taking the second derivation of $z_i$, we can obtain

$$\begin{array}{c}\hfill {\ddot{z}}_i={\ddot{\eta }}_i+{\iota }_{ij}\sum ({\ddot{\eta }}_j+{\ddot{E}}_{ij}),i=1,\cdots ,N,j=L,1,\cdots ,N\end{array}$$

(9)

According to Lemma 1, the stabilization of the error in Equation (7) with the communication topology is equal to the stabilization of the normal errors given by (6).

Next, we present the design process of an ASOFNTSMC that ensures the formation of a multiple-USV system.

3. Control Design

The ASOFNTSMC is composed of PDSMM and FNTSMM. The block diagram of the scheme of the ASOFNTSMC for USV formation is shown in Figure 2.

The PDSMM $s_{1i}$ was first designed as follows:

$$\begin{array}{c}\hfill s_{1i}={\dot{z}}_i+K_{1i}{z}_i,i=1,\cdots ,N,\end{array}$$

(10)

where $K_{1i}=\mathrm{diag}({\mathrm{k}}_{11\mathrm{i}},{\mathrm{k}}_{12\mathrm{i}},{\mathrm{k}}_{13\mathrm{i}})$ represents a positive diagonal matrix, and $k_{11i},k_{12i}$, and $k_{13i}$ are positive constants.

The FNTSMM was chosen as follows:

$$s_{2i}=s_{1i}+K_{2i}{s}_{1i}^{g/h}+K_{3i}{\dot{s}}_{1i}^{a/b},$$

(11)

where $K_{2i}=\mathrm{diag}({\mathrm{k}}_{21\mathrm{i}},{\mathrm{k}}_{22\mathrm{i}},{\mathrm{k}}_{23\mathrm{i}})$ and $K_{3i}=\mathrm{diag}({\mathrm{k}}_{31\mathrm{i}},{\mathrm{k}}_{32\mathrm{i}},{\mathrm{k}}_{33\mathrm{i}})$ represent positive diagonal matrices. $k_{j1i},k_{j2i},$ and $k_{j3i}(j=1,2)$ are positive constants, and $a,b,g$, and h are positive odd integers that satisfy $1<a/b<2$ and $g/h>a/b$.

By differentiating both sides of (11), we can obtain

$${\dot{s}}_{2i}={\dot{s}}_1+\frac{g}{h}{K}_{2i}\mathrm{diag}\left({\mathrm{s}}_{1\mathrm{i}}^{\mathrm{g}/\mathrm{h}-1}\right){\dot{\mathrm{s}}}_{1\mathrm{i}}+\frac{\mathrm{a}}{\mathrm{b}}{K}_{3i}\mathrm{diag}\left({\dot{\mathrm{s}}}_{1\mathrm{i}}^{\mathrm{a}/\mathrm{b}-1}\right){\ddot{\mathrm{s}}}_{1\mathrm{i}}$$

(12)

By substituting (8), (9), and (10) into (12), we can obtain

$$\begin{array}{cc}\hfill {\dot{s}}_{2i}=& {\dot{s}}_1+\frac{g}{h}{K}_{2i}\mathrm{diag}\left({\mathrm{s}}_{1\mathrm{i}}^{\mathrm{g}/\mathrm{h}-1}\right){\dot{\mathrm{s}}}_{1\mathrm{i}}+\frac{\mathrm{a}}{\mathrm{b}}{K}_{3i}\mathrm{diag}\left({\dot{\mathrm{s}}}_{1\mathrm{i}}^{\mathrm{a}/\mathrm{b}-1}\right)\hfill \\ \hfill & \times [M_i{\left({\eta }_i\right)}^{-1}{\dot{\tau }}_i-\frac{d}{dt}(C_i({\eta }_i,{\dot{\eta }}_i){\dot{\eta }}_i+D_i({\eta }_i,{\dot{\eta }}_i){\dot{\eta }}_i)\hfill \\ \hfill & +\frac{d}{dt}\left(M_i\left({\eta }_i\right)\right)({\tau }_i-C_i({\eta }_i,{\dot{\eta }}_i){\dot{\eta }}_i+D_i({\eta }_i,{\dot{\eta }}_i){\dot{\eta }}_i)\hfill \\ \hfill & +{\iota }_{ij}\sum ({\stackrel{⃛}{\eta }}_i+{\stackrel{⃛}{E}}_{ij})+K_{1i}\left(M_i{\left({\eta }_i\right)}^{-1}\right({\tau }_i\hfill \\ \hfill & -C_i({\eta }_i,{\dot{\eta }}_i){\dot{\eta }}_i+D_i({\eta }_i,{\dot{\eta }}_i){\dot{\eta }}_i)+{\iota }_{ij}\sum ({\ddot{\eta }}_j+{\ddot{E}}_{ij}))\hfill \\ \hfill & +{\tau }_{F\eta }]\hfill \\ \hfill =& {\dot{s}}_1+\frac{g}{h}{K}_{2i}\mathrm{diag}\left({\mathrm{s}}_{1\mathrm{i}}^{\mathrm{g}/\mathrm{h}-1}\right){\dot{\mathrm{s}}}_{1\mathrm{i}}+\frac{\mathrm{a}}{\mathrm{b}}{K}_{3i}\mathrm{diag}\left({\dot{\mathrm{s}}}_{1\mathrm{i}}^{\mathrm{a}/\mathrm{b}-1}\right)\hfill \\ \hfill & \times [M_i{\left({\eta }_i\right)}^{-1}{\dot{\tau }}_i-\frac{d}{dt}(C_i({\eta }_i,{\dot{\eta }}_i){\dot{\eta }}_i+D_i({\eta }_i,{\dot{\eta }}_i){\dot{\eta }}_i)\hfill \\ \hfill & +(\dot{\overbrace{M_i{\left({\eta }_i\right)}^{-1}}}+K_{1i}{M}_i{\left({\eta }_i\right)}^{-1})({\tau }_i-C_i({\eta }_i,{\dot{\eta }}_i){\dot{\eta }}_i\hfill \\ \hfill & +D_i({\eta }_i,{\dot{\eta }}_i){\dot{\eta }}_i)+{\iota }_{ij}\sum ({\stackrel{⃛}{\eta }}_i+{\stackrel{⃛}{E}}_{ij})\hfill \\ \hfill & +K_{1i}{\iota }_{ij}\sum ({\ddot{\eta }}_j+{\ddot{E}}_{ij})+{\tau }_{F\eta }],\hfill \end{array}$$

(13)

where ${\tau }_{F\eta }$ represents the system uncertainties and environment disturbances, and it is defined as

$$\begin{array}{c}\hfill {\tau }_{F_{\eta }}=[{\dot{M}}_i\left({\eta }_i\right)+K_{1i}{M}_i{\left({\eta }_i\right)}^{-1}]{\rho }_i\left(t\right)+M_i{\left({\eta }_i\right)}^{-1}{\dot{\rho }}_i\left(t\right)\end{array}$$

(14)

Assumption 4.

Based on Assumptions 1–3, we can obtain the following results:

$$\begin{array}{c}\hfill {\tau }_{F_{\eta }}\le {\beta }_0+{\beta }_1\Vert {\nu }_i\Vert \Vert {\dot{\eta }}_i\Vert +{\beta }_2\parallel {\nu }_i\parallel \parallel {\eta }_i{\parallel }^2,\end{array}$$

(15)

where ${\beta }_i(i=0,1,2)$ are unknown positive constants.

Remark 2.

It should be noted that though the upper boundary information is assumed here, it is only used to analyze the stability of the system. It is not directly used in the control law and adaptive update law.

The formation control scheme for USVs based on the SOFNTSMC includes an equivalent control law and a switching control law. In [36,37,38], the switching terms were all designed by using a DSF and were directly introduced into the control law, which could lead to the problem of chatter in the system.

Remark 3.

Unlike in the normal sliding-mode control, a time derivative of the control input is designed in this paper. As a part of the derivation of the control law, a switching control term using a DSF is designed to make the control input continuous and chatter-free after integration.

The derivative of the control can be denoted as

$${\dot{\tau }}_i={\tau }_{eqi}+{\tau }_{swi},$$

(16)

where ${\tau }_{eqi}$ and ${\tau }_{swi}$ are the ith USV’s equivalent control law and switching control law, respectively.

We can obtain the equivalent control law from Equation (13) by setting ${\dot{s}}_{2i}=0$ and ${\tau }_{F\eta }=0$:

$$\begin{array}{cc}\hfill {\tau }_{eqi}=& -(M_i\left({\eta }_i\right)K_{1i}{M}_i{\left({\eta }_i\right)}^{-1}+M_i\left({\eta }_i\right)\dot{\overbrace{M_i{\left({\eta }_i\right)}^{-1}}})({\tau }_i\hfill \\ \hfill & -C_i({\eta }_i,{\dot{\eta }}_i){\dot{\eta }}_i-D_i({\eta }_i,{\dot{\eta }}_i){\dot{\eta }}_i-M_i\left({\eta }_i\right)K_{1i}{\iota }_{ij}\sum ({\ddot{\eta }}_j+{\ddot{E}}_{ij})\hfill \\ \hfill & -M_i\left({\eta }_i\right){\iota }_{ij}\sum ({\stackrel{⃛}{\eta }}_j+{\stackrel{⃛}{E}}_{ij})+\frac{d}{dt}(C_i({\eta }_i,{\dot{\eta }}_i){\dot{\eta }}_i+D_i({\eta }_i,{\dot{\eta }}_i){\dot{\eta }}_i)\hfill \\ \hfill & -M_i\left({\eta }_i\right)\frac{b}{a}{K}_{3i}^{-1}\mathrm{diag}\left(\right|{\dot{\mathrm{s}}}_{1\mathrm{i}}{|}^{2-\mathrm{a}/\mathrm{b}})\hfill \\ \hfill & -M_i\left({\eta }_i\right)\frac{bg}{ah}{K}_{3i}^{-1}{K}_{2i}\mathrm{diag}\left(\right|{\mathrm{s}}_{1\mathrm{i}}{|}^{\mathrm{g}/\mathrm{h}-1}\left)\right|{\dot{\mathrm{s}}}_{1\mathrm{i}}{|}^{2-\mathrm{a}/\mathrm{b}}.\hfill \end{array}$$

(17)

Next, ${\tau }_{swi}$ is designed to deal with the deviation that is introduced by model uncertainties and external disturbances. In this paper, a DSF is used to design ${\tau }_{swi}$. Note that though the parameters ${\beta }_i(i=0,1,2)$ in Equation (8) are bounded, we cannot obtain their precise values. So, an adaptive method is used to estimate these bounded parameters. The switching control law is designed as follows:

$$\begin{array}{c}\hfill {\tau }_{swi}=-M_i\left({\eta }_i\right)({\widehat{\beta }}_0+{\widehat{\beta }}_1\parallel {\nu }_i\parallel \parallel {\dot{\eta }}_i\parallel +{\widehat{\beta }}_2\parallel {\nu }_i\parallel \parallel {\dot{\eta }}_i{\parallel }^2)\mathrm{sgn}\left({\mathrm{s}}_{2\mathrm{i}}\right),\\ \hfill \end{array}$$

(18)

where ${\widehat{\beta }}_i(i=0,1,2)$ are the estimated values of ${\beta }_i(i=0,1,2)$. The adaptive update law is designed as follows:

$$\begin{array}{cc}\hfill {\dot{\widehat{\beta }}}_0& =\frac{a}{b}{\phi }_0\parallel s_{2i}^T{K}_{3i}\mathrm{diag}\left(\right|{\dot{\mathrm{s}}}_{1\mathrm{i}}{|}^{\mathrm{a}/\mathrm{b}-1})\parallel ,\hfill \end{array}$$

(19)

$$\begin{array}{c}\hfill {\dot{\widehat{\beta }}}_1=\frac{a}{b}{\phi }_1\parallel s_{2i}^T{K}_{3i}\mathrm{diag}\left(\right|{\dot{\mathrm{s}}}_{1\mathrm{i}}{|}^{\mathrm{a}/\mathrm{b}-1})\parallel \parallel {\nu }_i\parallel \parallel {\dot{\eta }}_i\parallel \end{array}$$

(20)

$${\dot{\widehat{\beta }}}_2=\frac{a}{b}{\phi }_2\parallel s_{2i}^T{K}_{3i}\mathrm{diag}\left(\right|{\dot{\mathrm{s}}}_{1\mathrm{i}}{|}^{\mathrm{a}/\mathrm{b}-1})\parallel \parallel {\nu }_i\parallel \parallel {\dot{\eta }}_i{\parallel }^2,$$

(21)

where ${\phi }_i>0$ represents the rate of the adaptation.

Remark 4.

Unlike in [15,16], we need to find the upper bound of the uncertainty terms by using the adaptation law in (19), (20), and (21) to deal with the model uncertainties and external disturbances; the boundary information ${\tau }_{F\eta }$ does not need to be known in this paper.

Remark 5.

The DSF is involved in the time derivative of the control input (16); it can eliminate the problem of chatter in the control input after integration.

4. Simulation Studies

4.1. Example 1

In this section, Matlab 2014a was used for the simulations. A system of three USVs was used to verify the effectiveness of the proposed method. The communication relationships are shown in Figure 3. The USV parameters were the following [34]:

$$\begin{array}{c}\hfill m_{11}=200\mathrm{kg},{\mathrm{m}}_{22}=250\mathrm{kg},{\mathrm{m}}_{33}=80\mathrm{kg},\\ \hfill d_{11}=70\mathrm{kg}/\mathrm{s},{\mathrm{d}}_{22}=100\mathrm{kg}/\mathrm{s},{\mathrm{d}}_{33}=50\mathrm{kg}/\mathrm{s}.\end{array}$$

The leader USV was moving counterclockwise along a circle with a radius of 3 and with $x_L=3sin\left(t\right),y_L=3cos\left(t\right)$. The initial states of the three USVs were as follows: ${\eta }_L\left(0\right)={[0,0,0]}^T,{\eta }_1\left(0\right)=[-3,3,5\pi /6],{\eta }_2\left(0\right)={[-1,5,\pi ]}^T$. The desired distances between the USVs were set to $E_{1L}={[-\sqrt{48},4,0]}^T,E_{2L}={[-\sqrt{48},-4,0]}^T,E_{12}={[0,8,0]}^T$. The control gain parameters were chosen as follows: $K_{1i}=diag\left[211\right],K_{2i}=diag\left[111\right]),K_{3i}=diag\left(\left[211\right]\right))$.

According to Figure 3, the corresponding adjacency matrix A and Laplacian matrix L can be written as follows:

$$A=\left[\begin{array}{ccc}0& 0& 0\\ 1& 0& 1\\ 1& 1& 0\end{array}\right]$$

(22)

$$L=\left[\begin{array}{ccc}0& 0& 0\\ -1& 2& -1\\ -1& -1& 2\end{array}\right]$$

(23)

We can obtain the normalized directed Laplacian matrix from Equation (1):

$$\iota =\left[\begin{array}{ccc}0& 0& 0\\ -\frac{1}{2}& 1& -\frac{1}{2}\\ -\frac{1}{2}& -\frac{1}{2}& 1\end{array}\right]$$

(24)

In order to reflect model uncertainties and disturbances, without losing generality, the external environmental disturbances and model uncertainties are expressed as follows:

${\tau }_{wi}=\left[\begin{array}{c}100+50sin\left(0.02t\right)cos\left(t\right)\\ 200+50sin\left(0.03t\right)cos\left(0.8t\right)\\ 50cos\left(0.01t\right)\end{array}\right]$, $\mathsf{\Delta }{D}_i=20\ast \left[\begin{array}{c}0.3u_i{v}_i^2+0.4v_i^2{r}_i\\ 0.1u_i{v}_i\\ 0.2u_i{r}_i^2+0.3u_i{r}_i{v}_i^3\end{array}\right]$.

In Figure 4, we can see the variation curve of the model uncertainties and external disturbances.

The north–east coordinates of the three USVs are shown in Figure 5. This shows that the actual formation pattern corresponds to the desired formation pattern shown in Figure 3. In Figure 6, it is obvious that even with the disturbances, the generalized error was able to converge to zero in a finite time with the ASOFNTSMC, which demonstrated the good robustness of the proposed method. The position $x_i,y_i$ and attitude ${\psi }_i$ are given in Figure 7. The surge, sway, and yaw velocities are given in Figure 8. Figure 9 shows the control forces of follower 1 and follower 2. This reveals that the control forces are smooth and without chatter; this is because the sign function was introduced into the switching term, which is a part of the derivative of the input rather than a part of the control input. Figure 10 illustrates the PDSMM $s_{1i}$. The ASONTSMC method was compared with the proposed ASOFNTSMC method. The simulation results are given in Figure 11, Figure 12 and Figure 13. Comparing Figure 6 with Figure 12, we can see that the error states of the proposed method converged faster than those of ASOTSMC. The PDSMM shown in Figure 10 also converged faster than that of the ASOTSMC shown in Figure 13, which reveals the reason for the faster convergence.

4.2. Example 2

In this part, we compare our method with another nonsingular fast terminal sliding-mode control method that was proposed in [39]. A sliding-mode switching surface function is designed as follows:

$$s=z_i+\frac{1}{\beta }{{\dot{z}}_i}^{\frac{p}{q}}$$

The derivation of s is calculated as:

$$\dot{s}={\dot{z}}_i+\frac{1}{\beta }\frac{p}{q}{{\dot{z}}_i}^{\frac{p}{q}}\ddot{{\dot{z}}_i}$$

The nonsingular fast terminal sliding-mode control law is designed as follows:

$${\tau }_i=\frac{\beta q}{p}{M}_i\left({\eta }_i\right){{\dot{z}}_i}^{-\frac{p}{q}}{\dot{z}}_i-C_i({\eta }_i,{\dot{\eta }}_i)\dot{\eta }+D_i({\eta }_i,{\dot{\eta }}_i)\dot{\eta }-M_i^{-1}\left({\eta }_i\right){\rho }_i\left(t\right)sgn\left(s\right)$$

Let $p=2.4,q=3,\beta =2$. The simulation results when using the method from [39] are shown in Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18. The formation patterns and relative positions are displayed in Figure 14. Comparing Figure 14 with Figure 5, we can see that both methods were able to achieve the desired formation pattern; however, as shown in Figure 14, the motion curve of follower 1, which is marked in red, reveals that its trajectory cannot coincide in a short time, and it rotates around the balance point. Looking back to Figure 5, we can see that the curve of follower 1 circles around a center point and that its trajectory is coincident, which demonstrates the fast convergence of the proposed method. On the other hand, comparing Figure 15 with Figure 6, subfigure 3 in Figure 15 shows the yaw error convergence time, which needs about 15 seconds to decrease to zero with the method from [39]. However, as shown in Figure 5, the decreasing time is less than 5 s, which shows that the error convergence time of the method used in [39] is longer than that of the method proposed in this paper, which also proves the advantages of the proposed method.

5. Conclusions

An ASOFNTSMC was designed for the formation control of USVs while considering the communication topology, dynamic model uncertainties, time-varying disturbances, and fast convergence performance. To improve the robustness and convergence speed of the system error, an ASOFNTSMC consisting of a PDSMM and an FNTSMM was used for the first time to achieve USV formation control. As shown in the comparison of the simulation results, the convergence speed of the proposed method was faster than that of the SONTSMM. The SOFNTSM caused the PDSMM to quickly converge to zero in a finite time. Then, the ASOFNTSMC ensured that the error moved along the sliding mode, even in presence of dynamic uncertainties and environmental disturbances. The updated law of adaptation does not require the boundaries of uncertainties. Furthermore, a sign function in the switching term is introduced into the derivative of the control law rather than the control law, which eliminates the problem of chatter. The simulation results indicate that the proposed method converges more quickly than the ASONTSMC, and it has a good robustness against disturbances. Our future research will concentrate on the formation control of USVs that are subjected to some constraints, such as input saturation, time delays, and unmeasured states. In addition, using some methods to adjust the control parameters in order to reduce the workload is also necessary in future work.