Containment control of autonomous surface vehicles: A nonlinear disturbance observer–based dynamic surface control design

Abstract

This study presents a solution to the problem of containment control of multiple fully actuated autonomous surface vehicles subject to environmental disturbances. Using finite-time stability theory, a nonlinear disturbance observer is constructed to provide an estimation of unknown disturbances online. By employing nonlinear disturbance observer to compensate disturbances, a robust containment controller is constructed by combining backstepping technique and dynamic surface control technique. It is proved that the proposed containment control scheme can force all the followers to move into the convex hull spanned by the leaders and guarantee that all the signals in the closed-loop system are globally uniformly ultimately bounded. Finally, four simulation examples are provided to demonstrate the effectiveness of the containment controller and its robustness to external disturbances.

Introduction

In the past few years, the field of the cooperative control for multiple autonomous surface vehicles (ASVs) is of increasing interest to the motion control community. This is due to the fact that a group of ships can be used to complete a specific task, where it contributes greater robustness, lower cost, and better performance. Some typical applications include environmental monitoring, rescue, and reconnaissance operations.^1–5 To achieve these objectives, an increasing number of studies have focused on the cooperative control of a group of ASVs.

Recently, the cooperative control problem of multiple ASVs has been studied by many researches. For instance, a design is presented for distributed maneuvering of multiple ASVs with the aid of neurodynamic optimization and fuzzy approximation in Peng et al. ⁶ The authors in the study of Fahimi ⁷ design a nonlinear model predictive formation control (NMPFC) law based on an underactuated model to stabilize the relative distances and orientations between the follower and the leader; in the study of Kyrkjebø et al., ⁸ a leader–follower (L-F) synchronization output feedback controller has been proposed for the ship replenishment problem using an nonlinear observer; in the study of Dong, ⁹ continuous time-varying cooperative control laws are designed to perform a geometric pattern using suitable transformations while assuming that the yaw velocity is nonzero; to cope with the uncertainties of the ASV model, a robust adaptive formation control law is developed that combines neural network (NN) with dynamic surface control (DSC) technique in the study of Peng et al.; ² in the study of Shojaei, ³ a L-F formation tracking controller has been proposed to force a group of ASVs to maintain the desired orientations and positions relative to one leading ship, considering the input saturation; in the study of Peng et al., ¹⁰ a robust cooperative control method has also been applied to form a desired geometric pattern with the aid of NN, backstepping (BP) method, and graph theory. A common trait in these designs is that only one leader exists in the group.

In practical applications, there might exist multiple leaders in the group, which can be called as the containment control problem. ¹¹ In this case, the control objective is containment of the followers in the convex hull of the leaders’ positions.^12,13 In the marine domain, a group of ASVs are guided by another group, and some ASVs are regarded as leaders, which are all equipped with sensors to detect the hazardous obstacles in the marine environment. ¹⁴ The dynamics considered in recent works correspond to single integrator systems, ¹⁵ double integrator systems, ¹⁶ strict-feedback form systems, ¹¹ general linear dynamics, ¹⁷ and Lagrange dynamics. ¹⁸ However, these proposed methods do not consider environment disturbance, which might affect the control performance. To cope with the uncertainties of the model, a neurodynamics-based output feedback scheme has been proposed for containment maneuvering of ASVs guided by multiple parameterized paths in Peng et al. ¹³ Accordingly, how to achieve the containment control of fully actuated ASVs in the presence of ocean external disturbances needs to be investigated further.

The control method based on nonlinear disturbance observer (NDO) can enhance system disturbance rejection and improve system robustness. At present, NDO is widely used in robotic manipulator, ¹⁹ missile, ²⁰ active suspension system, ²¹ ASV, ²² and other fields. The design and analysis of traditional disturbance observer (DO) are based on linearized model or linear system theory. ¹⁹ However, the model (such as ship) has a strong coupling nonlinearity, so the design of DO through linear system theory is inaccurate. Therefore, the authors in the study of Chen et al. ¹⁹ assume that disturbance time-varying rate is lower than the dynamic characteristics of DO and design the DO that could be applied to nonlinear model, where estimation error has exponential convergence properties. In the study of Kim, ²³ an NDO has been developed using the feature that fuzzy logic system could arbitrarily approach a strong nonlinear function. The authors in Zhu et al. ²⁴ design a radial basic function NN DO, which has an excellent approximation ability to monitor the uncertainties. By assuming that the upper bound of the disturbance change rate is known, an NDO is proposed which could make disturbance estimation error to converge to a small neighborhood of the equilibrium point exponentially in the study of Do. ²² The NDO proposed by the above literatures had common drawback, namely, convergence speed of estimate error is low and only asymptotic convergence can be obtained.

In recent years, BP method, as an effective method of nonlinear control has become a research hotspot, and it, together with dynamic sliding mode, ²⁵ the cascade system theory, ²⁶ and other methods, is applied in motion control of ship. BP method, however, involves the repeated derivatives of the virtual control signals in the design process, and the problem of “explosion of complexity”^27,28 in it increases the complexity of the control law. To overcome this drawback, the authors in the study of Swaroop et al. ²⁷ propose DSC technology, which can approximately estimate the first-order derivative of the virtual control law and reduce the complexity of control system.^29,30

Motivated by the above observations, this article considers the robust containment control problem of multiple fully actuated ASVs subject to unknown external disturbances. Compared with the existing results, the main features of this article can be summarized as follows:

1. In contract to the cooperative controllers proposed in the study of Peng et al., ² Shojaei, ³ Fahimi, ⁷ Kyrkjebø et al., ⁸ Dong, ⁹ and Peng et al., ¹⁰ where there exist a single leader in the group, this article considers the containment control problem of ships in the presence of multiple virtual leaders.

2. In contract to the traditional NDO,^19,23,24 a new NDO, with the aid of finite-time stability, is proposed. The NDO designed in this note is employed to estimate the unknown disturbances, which can quickly approach unknown disturbances within finite time.

3. For the first time, the NDO-based DSC technique is employed to solve the containment control problem of multiple fully actuated ASVs with unknown disturbances, which leads to a simpler control law than traditional BP-based design.

This article is organized as follows: section “Problem statement and preliminaries” introduces some preliminaries and formally states containment control problem. In section “Control design,” we will present our main result, which will be illustrated by a series of simulation experiments in section “Simulations.” Finally, we will close this article in section “Conclusion” with some concluding remarks.

Problem statement and preliminaries

Notations

λ_max( A ) and λ_min( A ) denote the maximum and the minimum eigenvalues of a square matrix A , respectively; |·| stands for the absolute value of a scalar; ||·|| denotes the Euclidean norm of a vector; let R ^n×m denote a set of Euclidean n × m matrices, and R ⁿ denote n-dimensional Euclidean space; diag(a₁, a₂, …, a_n) represents a diagonal matrix with entries a_i (i = 1, 2, …, n).

Concepts in graph theory

Suppose that a group of ASVs interacts with each other through a communication network. It is natural to model the communication topology among these ASV using results from graph theory. In this article, we assume that the individual ASV as a node, and the information interaction for ASVs can be represented by a directed graph G = {V, ε}, where V = {a₁, a₂, …, a_n} denotes a set of nodes and ε = {(n_i, n_j) ∈ V × V} denotes a set of edges with element (n_i, n_j), which implies that node n_j can receive information from node n_i. Here, we say that node n_i is a neighbor of node n_j, and the notation N_i is represented as the set of all neighbors of node i. Let the adjacency matrix A  = {a_ij} ∈  R ^N×N be defined such as a_ij = 1 if (n_i, n_j) ∈ ε, and a_ij = 0 otherwise. Note that we assume a_ii = 0 for all the nodes. The Laplacian matrix L  = {l_ij} of the directed graph G can be defined as

$$l_{ij}={\displaystyle \sum _{j=1}^N{a}_{ij}}\mathrm{if}j=i;l_{ij}=-a_{ij},\mathrm{otherwise}$$

Definition 1

For a group of ASVs, ship i is said to be a leader if a_ij = 0, and ASV i is said to be a follower if a_ij = 1, j = 1, 2, …, N.

Definition 2

The real set C ∈  R ⁿ is said to be convex if for real constants x, y ∈ C, there exists a point that satisfies (1−z)x + zy ∈ C for any z ∈ [0, 1]. The convex hull for a set of points X = {x₁, x₂, …, x_N} in C is the minimal convex set including all the points in X and let Co(X) be the convex hull of X. In particular, Co(X) can be represented as follows ³¹

$$\mathrm{Co}(X)=\{\sum _{i=1}^N{\lambda }_i{x}_i|x_i\in X,{\lambda }_i>0,\sum _{i=1}^N{\lambda }_i=1\}$$

(1)

where ${\lambda }_i(i=1,2,...,N)$ is a positive constant.

Ship modeling

Consider a group consisting of M followers, labeled as ASV 1 to M. In the horizontal plane, the 3-degree-of-freedom (DOF) ASV model can be formulated as ¹

$${\stackrel{·}{\mathit{\eta }}}_i=\mathit{R}({\theta }_i){\mathit{v}}_i$$

(2)

$${\mathit{M}}_i{\stackrel{·}{\mathit{v}}}_i+{\mathit{C}}_i({\mathit{v}}_{\mathit{i}}){\mathit{v}}_{\mathit{i}}+{\mathit{D}}_i({\mathit{v}}_{\mathit{i}}){\mathit{v}}_{\mathit{i}}={\mathbf{\tau }}_{\mathit{i}}+{\mathit{d}}_i$$

(3)

where

$$\mathit{R}({\theta }_i)=\left[\begin{array}{ccc}\cos {\theta }_i\hfill & -\sin {\theta }_i\hfill & 0\hfill \\ \multicolumn{1}{c}{\sin {\theta }_i}& \cos {\theta }_i\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \end{array}\right]$$

(4)

the vector η _i  = (x_i, y_i, θ_i) ^T  ∈  R ³ accounts for the position and orientation of the ASV i in the inertial reference frame {I}; v _i  = (u_i, v_i, r_i) ^T  ∈  R ³ is the velocity vector in the ship-fixed reference frame {B}; M _i , C _i ( v _i ), and D _i ( v _i ) account for the inertia matrix (including the added mass), the coriolis/centrifugal matrix, and the damping matrix; τ _i  ∈  R ³ accounts for the actual control signals of ASV i; d _i  = (d_ui, d_vi, d_ri) ^T  ∈  R ³ accounts for the disturbances from the ocean environment.

Then, suppose that there exist N-M virtual leaders in the group, labeled as ASV M + 1 to N. The trajectories of the virtual leaders are denoted ${\mathbf{\varphi }}_{\mathbf{i}}\in {\mathit{R}}^3$ by where i = M + 1, …, N. To describe the communication among the ships in this group, let L denote the Laplacian matrix. As each leader of this group has no neighbors, L can be represented by the following

$$\mathit{L}=\left[\begin{array}{cc}{\mathit{L}}_1\hfill & {\mathit{L}}_2\hfill \\ \multicolumn{1}{c}{{\mathit{0}}_{(N-M)\times M}}& {\mathit{0}}_{(N-M)\times (N-M)}\hfill \end{array}\right]$$

(5)

To achieve the control objective, the following assumptions is imposed.

Assumption 1

For the time-varying disturbances d_ui, d_vi, and d_ri, there exist positive constants ${\overline{d}}_{1i},{\overline{d}}_{2i},{\overline{d}}_{3i}$ , such that

$$|d_{\mathrm{ui}}|\le {\overline{d}}_{1i},|d_{\mathrm{vi}}|\le {\overline{d}}_{2i},|d_{\mathrm{ri}}|\le {\overline{d}}_{3i}$$

(6)

Assumption 2

For all t > 0, there exist constants N₁ and N₂ such that $||{\stackrel{·}{\varphi }}_i(t)||\le N_1$ , $||{\stackrel{··}{\varphi }}_i(t)||\le N_2$

Assumption 3

For every follower of the group, there exists at least one leader that has a directed path to it. ³¹

Remark 1

Assumption 2 guarantees that the desired trajectories of the virtual leaders ${\mathbf{\varphi }}_{\mathbf{i}}$ are smooth enough, avoiding discontinuous control caused by the mutations from tracking errors. Since marine environment has finite energy, Assumption 2 is reasonable in practice.

Problem statement

This control objective is to design a containment control scheme such that all the followers in the group move into the convex hull formed by the leaders,^18,32 that is

$$\underset{t\to \infty }{lim}d({\mathbf{\eta }}_{\mathbf{i}}-\mathrm{Co}({\mathit{X}}_L))=0$$

(7)

where ${\mathit{X}}_L:=[{\mathbf{\varphi }}_{\mathbf{M}\mathbf{+}1}^{\mathbf{T}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathbf{\varphi }}_{\mathbf{N}}^{\mathbf{T}}{\mathbf{]}}^{\mathbf{T}}$ ; d(x, M ) denotes the distance from a point x ∈  R ³ to a set M  ∈ R³, that is

$$d(x,\mathit{M});=\underset{y\in \mathit{M}}{inf}|x-y|$$

(8)

Lemma 1

Under Assumption 3, all the eigenvalues of L ₁ have positive real parts; furthermore, each of entry $-{\mathit{L}}_1^{-1}{\mathit{L}}_2$ is nonnegative, and all the row sums of this matrix are equal to 1. ³³

Remark 2

Let ${\mathit{X}}_L(t)=[{\mathbf{\varphi }}_{\mathbf{M}\mathbf{+}1}^{\mathbf{T}}\mathbf{(}\mathbf{t}\mathbf{)}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathbf{\varphi }}_{\mathbf{N}}^{\mathbf{T}}\mathbf{(}\mathbf{t}\mathbf{)}{\mathbf{]}}^{\mathbf{T}}$ be the positions of the dynamic leaders. From Definition 2 and Assumption 3, Lemma 1 implies that every point of $-({\mathit{L}}_1^{-1}{\mathit{L}}_2\otimes {\mathit{I}}_2){\mathit{X}}_L(t)$ is in the convex hull spanned by the leaders X _L (t).

In this article, our control objective is formally stated as follows.

Control objective

Consider the fully actuated ASV model given by equations (2) and (3) under Assumptions 1–3; the objective of this work is to design a robust cooperative controller for each ship on the basis of its local states and the information from neighbors and a portion of the leaders and such that

$$\underset{t\to \infty }{lim}||{\mathbf{\eta }}_{\mathbf{i}}(t)-{\mathbf{\eta }}_{\mathbf{di}}(t)||\le {\sigma }_i$$

(9)

where σ_i is a positive constant, η _di (t) ∈  R ³ denotes the desired position of follower i which satisfies

$${\mathit{P}}_d(t)=-({\mathit{L}}_1^{-1}{\mathit{L}}_2\otimes {\mathit{I}}_3){\mathit{X}}_L(t)$$

(10)

where ${\mathit{P}}_d(t)=[{\mathbf{\eta }}_{\mathbf{d}1}^{\mathbf{T}}\mathbf{(}\mathbf{t}\mathbf{)}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathbf{\eta }}_{\mathbf{di}}^{\mathbf{T}}\mathbf{(}\mathbf{t}\mathbf{)}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathbf{\eta }}_{\mathbf{dM}}^{\mathbf{T}}\mathbf{(}\mathbf{t}\mathbf{)}{\mathbf{]}}^{\mathbf{T}}$

Concepts of finite-time stability

Consider the following system

$$\stackrel{·}{\mathit{x}}=f(\mathit{x}),f(0)=0,x\in {\mathit{R}}^n$$

(11)

where f: D  →  R ⁿ is continuous with respect to the variable x on an open neighborhood D of the origin x = 0.

Definition 3

The zero solution of system (11) is locally finite-time stable (LFTS) if it is Lyapunov stable and finite-time convergent in a neighborhood D ₀ ⊂  D of the origin x = 0. The finite-time convergence implies that there exist a function T: D ₀\{0} → (0,+∞), such that for any x₀ ∈  D  ⊂  R ⁿ , the solution of equation (11) represented by x(t, x₀) with x₀ as the initial state and x(t, x₀) ∈  D ₀\{0} for [0, T(x₀)), and $\underset{t\to T(x_0)}{lim}x(t,x_0)=0$ . When D ₀ =  D  =  R ⁿ , the zero solution of system (11) is said to be globally finite-time stable (GFTS).

Control design

In this section, a robust containment control is proposed for the fully actuated ASVs with unknown external disturbances taken into account. The block diagram of the robust controller described is provided in Figure 1. The detailed design procedure will be given in the following section.

Figure 1.

Block diagram of the containment control system.

NDO design

The external disturbances from ocean may affect performance of control system and even make system unstable. Therefore, a new NDO is designed to estimate the external disturbances, and its estimation is fedback to control law. Thus, the real-time compensation of disturbances is realized.

Before designing the NDO, we shall recall the following lemma:

Lemma 2

Consider the system defined in Definition 3. Suppose that there is a continuous function V(x): D  → R defined on an open set D  ⊂  R ^m including the origin x = 0 such that the following conditions ³⁴

1. V(0) = 0 and V(x) is positive definite on D ;

2. There exist real numbers c > 0, l > 0, and α  ∈ (0, 1), and a open set D ₀ ⊂  D of the origin x = 0 such that

$$\stackrel{·}{\mathit{V}}(\mathit{X})+c{\mathit{V}}^{\alpha }(\mathit{X})+l\mathit{V}(\mathit{X})\le 0,\mathit{X}\in {\mathit{D}}_0\backslash \{0\}$$

(12)

Then, system (11) is LFTS, and the settling time can be estimated by

$$T({\mathit{X}}_0)\le \frac{\ln (1+\frac{l}{c}{V}^{1-\alpha }({\mathit{X}}_0))}{l(1-\alpha )},{\mathit{X}}_0\in {\mathit{D}}_0$$

(13)

If D ₀ =  D  =  R ^m , then system (11) is GFTS.

At first, we define the auxiliary variable p_i  =  M_iv_i  = [p _ui , p _vi , p _ri ] ^T  ∈  R ³ and its estimation ${\hat{\mathit{p}}}_i$ .Then, the estimated error ${\tilde{\mathit{p}}}_i$ can be described by

$${\tilde{\mathit{p}}}_i={\hat{\mathit{p}}}_i-{\mathit{p}}_i$$

(14)

The time derivative of ${\hat{\mathit{p}}}_i$ is designed as below

$$\begin{array}{l}{\stackrel{.}{\widehat{\mathit{p}}}}_{\mathit{i}}=-{\mathit{C}}_{\mathit{i}}{\mathit{v}}_{\mathit{i}}-{\mathit{D}}_{\mathit{i}}{\mathit{v}}_{\mathit{i}}+{\mathit{\tau }}_{\mathit{i}}-{\mathit{D}}_1{\tilde{\mathit{p}}}_{\mathit{i}}\\ -{\mathit{D}}_2|{\tilde{\mathit{p}}}_{\mathit{i}}{|}^{\gamma }\mathit{s}\mathit{i}\mathit{g}\mathit{n}({\tilde{\mathit{p}}}_{\mathit{i}})-{\overline{\mathit{d}}}_{\mathit{i}}\mathit{s}\mathit{i}\mathit{g}\mathit{n}({\tilde{\mathit{p}}}_{\mathit{i}})\end{array}$$

(15)

where D ₁ = diag{D₁₁, D₁₂, D₃₃} and D ₂ = diag{D₂₁, D₂₂, D₂₃} are diagonal matrices with their entries being positive constants; ${\overline{\mathit{d}}}_i=\mathrm{diag}\{{\overline{d}}_{1i},{\overline{d}}_{2i},{\overline{d}}_{3i}\}$ ; $|{\tilde{\mathit{p}}}_i{|}^{\gamma }\mathrm{sign}({\tilde{\mathit{p}}}_i)=[|{\tilde{p}}_{\mathrm{ui}}{|}^{\gamma }\mathrm{sign}({\tilde{p}}_{\mathrm{ui}});|{\tilde{p}}_{\mathrm{vi}}{|}^{\gamma }\times \mathrm{sign}({\tilde{p}}_{\mathrm{vi}});|{\tilde{p}}_{\mathrm{ri}}{|}^{\gamma }\mathrm{sign}({\tilde{p}}_{\mathrm{ri}})]$

From equation (15), we have the disturbance estimation

$$\begin{array}{c}\multicolumn{1}{c}{{\hat{\mathit{d}}}_i={\stackrel{·}{\hat{\mathit{p}}}}_i+{\mathit{C}}_i{\mathit{v}}_i+{\mathit{D}}_i{\mathit{v}}_i-{\mathbf{\tau }}_i}\\ =-{\mathit{D}}_1{\tilde{\mathit{p}}}_i-{\mathit{D}}_2|{\tilde{\mathit{p}}}_i|\mathrm{sign}({\tilde{\mathit{p}}}_i)-\overline{\mathit{d}}\mathrm{sign}({\tilde{\mathit{p}}}_i)\hfill \end{array}$$

(16)

Then, the main result can be summarized as Theorem 1:

Theorem 1

Consider the ASV model equations (2) and (3) in the presence of bounded disturbances d . If the DO is given by equations (14)–(16), then the global finite-time stability of the observer error ${\tilde{\mathit{d}}}_i={\mathit{d}}_i-{\hat{\mathit{d}}}_i$ can be achieved.

Proof

Consider the following Lyapunov function (LF)

$$V_0=\frac{1}{2}{\tilde{\mathit{p}}}_i^T{\tilde{\mathit{p}}}_i$$

(17)

whose derivative is

$${\stackrel{·}{V}}_0={\tilde{\mathit{p}}}_i^T{\stackrel{·}{\stackrel{~}{\mathit{p}}}}_i={\tilde{\mathit{p}}}^T({\stackrel{·}{\hat{\mathit{p}}}}_i-{\mathit{M}}_i{\stackrel{·}{\mathit{v}}}_i)$$

(18)

According to equations (3) and (15), we calculate the derivative of V₀ as follows

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_0={\tilde{\mathit{p}}}_i^T(-{\mathit{D}}_1{\tilde{\mathit{p}}}_i-{\mathit{D}}_2|{\tilde{\mathit{p}}}_i|\mathrm{sign}({\tilde{\mathit{p}}}_i)-\overline{\mathit{d}}\mathrm{sign}({\tilde{\mathit{p}}}_i)-{\mathit{d}}_i)}\\ \le -{\tilde{\mathit{p}}}_i^T{\mathit{D}}_1{\tilde{\mathit{p}}}_i-{\tilde{\mathit{p}}}_i^T{\mathit{D}}_2|{\tilde{\mathit{p}}}_i{|}^{\gamma }\mathrm{sign}({\tilde{\mathit{p}}}_i)\hfill \\ \le -2D_{1min}{V}_0-2^{\gamma +1}{D}_{1min}{V}_0^{(\gamma +1)/2}\hfill \end{array}$$

(19)

where D_1min = min{D₁₁, D₁₂, D₁₃} > 0, D_2min = min{D₂₁, D₂₂, D₂₃} > 0. By Lemma 2, it can be concluded that the observer error ${\tilde{\mathit{p}}}_i$ converges to zero in a finite time.

By substituting equation (16) into equation (14) can be rewritten as

$$\begin{array}{c}\multicolumn{1}{c}{{\tilde{\mathit{d}}}_i={\mathit{d}}_i-{\hat{\mathit{d}}}_i}\\ ={\mathit{D}}_1{\tilde{\mathit{p}}}_i+{\mathit{D}}_2|{\tilde{\mathit{p}}}_i{|}^{\gamma }\mathrm{sign}({\tilde{\mathit{p}}}_i)+\overline{\mathit{d}}\mathrm{sign}({\tilde{\mathit{p}}}_i)\hfill \\ +{\mathit{M}}_i{\stackrel{·}{\mathit{v}}}_i+{\mathit{C}}_i{\mathit{v}}_i+{\mathit{D}}_i{\mathit{v}}_i-{\tau }_i\hfill \\ ={\stackrel{·}{\mathit{p}}}_{\mathit{i}}-{\stackrel{·}{\hat{\mathit{p}}}}_{\mathit{i}}=-{\stackrel{·}{\stackrel{~}{\mathit{p}}}}_{\mathit{i}}\hfill \end{array}$$

(20)

Clearly, using the finite-time convergent observer equations (14)–(16), the observer ${\hat{\mathit{d}}}_i$ will converge to the unknown disturbances ${\mathit{d}}_i$ in a finite time.

Remark 3

NDO proposed by Chen et al. ¹⁹ guarantees that disturbance estimation converges to disturbance with exponential rate, but disturbance must be slowly time-varying (i.e. ${\stackrel{·}{\mathit{d}}}_i=0$ ); so, high-frequency disturbance cannot be accurately estimated. NDO applied in the study of Deshpande et al. ²¹ and Do ²² can accurately track high-frequency disturbance and make estimation error converge to a neighborhood of the equilibrium point with exponential rate, but the changing rate of disturbance should be bounded (i.e. $|{\stackrel{·}{d}}_{\mathrm{ji}}|\le C_d$ , j = u, v, r), which is difficult to get in practice. All of above NDO only obtains the asymptotic convergence of the tracking errors. On the basis of finite-time stability, NDO designed in the article can quickly approach unknown disturbances (including low-frequency and high-frequency disturbance) within settling time T(X₀).

Remark 4

Because ${\tilde{\mathit{p}}}_i$ converges to zero in a finite time, ${\tilde{\mathit{p}}}_i$ is bounded for all t > 0 (i.e. $|{\tilde{p}}_{\mathrm{ji}}|\le {\omega }_{\mathrm{ji}},j=u,v,r$ ); thus, ${\tilde{\mathit{d}}}_i$ is globally uniformly ultimately bounded (GUUB) and satisfy $|{\tilde{d}}_{\mathrm{ji}}|\le {\epsilon }_{\mathrm{ji}},j=u,v,r$ .

Controller design

In this section, the output of NDO can be viewed as a compensated signal. Thus, the actual control input ${\mathbf{\tau }}_{\mathit{i}}$ of ASV i is proposed as follows

$${\mathbf{\tau }}_{\mathit{i}}={\mathbf{\tau }}_{\mathrm{DSC},\mathit{i}}-{\hat{\mathit{d}}}_i$$

(21)

where ${\mathbf{\tau }}_{\mathrm{DSC},\mathit{i}}\in {\mathit{R}}^3$ is specified later.

By substituting equation (21) into equations (2) and (3), we have

$${\stackrel{·}{\mathit{\eta }}}_i=\mathit{R}{\mathit{v}}_i$$

(22)

$${\stackrel{·}{\mathit{v}}}_i={\mathit{M}}_i^{-1}(-{\mathit{C}}_i{\mathit{v}}_i-{\mathit{D}}_i{\mathit{v}}_i+{\mathbf{\tau }}_{\mathrm{DSC},i}+{\tilde{\mathit{d}}}_i)$$

(23)

Then, we design a robust containment control law for the ship combining the NDO equations (14)–(16) and DSC technique. The whole controller design procedure contains two steps as follows:

• Step 1. Kinematic loop design

Define the first dynamic surface error

$${\mathit{s}}_{1i}={\mathit{R}}^T({\theta }_i)[\sum _{j=1}^M{a}_{\mathrm{ij}}({\mathbf{\eta }}_{\mathbf{i}}-{\mathbf{\eta }}_{\mathbf{j}})+\sum _{j=M+1}^N{a}_{\mathrm{ij}}({\mathbf{\eta }}_{\mathbf{i}}-{\mathbf{\varphi }}_{\mathbf{j}})]$$

(24)

where a_ij is defined in section “Concepts in graph theory.”

Differentiating both sides of equation (24) along equation (22) results in

$$\begin{array}{l}{\stackrel{.}{s}}_{1i}=-r_{i}Q{s}_{1i}+d_i{v}_i-{\displaystyle \sum _{j=1}^M{a}_{ij}{R}^T({\theta }_i)R({\theta }_j)v_j}\\ -{\displaystyle \sum _{j=M+1}^N{a}_{ij}{R}^T({\theta }_i)\stackrel{.}{{\varphi }_j}}\end{array}$$

(25)

where

$$\mathit{Q}=\left[\begin{array}{ccc}0\hfill & -1\hfill & 0\hfill \\ \multicolumn{1}{c}{1}& 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \end{array}\right],d_i=\sum _{j=1}^N{a}_{\mathrm{ij}},i=1,2,...,N$$

Consider the first LF

$$V_{1i}=\frac{1}{2}{\mathit{s}}_{1i}^T{\mathit{s}}_{1i}$$

(26)

whose derivative is

$${\stackrel{.}{V}}_{1i}=s_{1i}^T{\stackrel{.}{s}}_{1i}=s_{1i}^T\left[d_i{\alpha }_{vi}-{\displaystyle \sum _{j=1}^M{a}_{ij}{R}^T({\theta }_i)R({\theta }_j)v_j}-{\displaystyle \sum _{j=M+1}^N{a}_{ij}{R}^T({\theta }_i)\stackrel{.}{{\varphi }_j}}+d_i{e}_{vi}\right]$$

(27)

where ${\mathbf{\alpha }}_{\mathbf{vi}}$ is the virtual controller; ${\mathit{e}}_{\mathrm{vi}}={\mathit{v}}_i-{\mathbf{\alpha }}_{\mathrm{vi}}$ .

To make $V_{1i}\to 0$ , ${\mathbf{\alpha }}_{\mathbf{vi}}$ can be chosen as

$${\alpha }_{vi}=\frac{1}{d_i}\left[-{\mathit{K}}_{i1}{\mathit{s}}_{i1}+{\displaystyle \sum _{j=1}^M{a}_{ij}{\mathit{R}}^{\mathit{T}}({\theta }_i)\mathit{R}({\mathit{\theta }}_{\mathit{j}}){\mathit{v}}_{\mathit{j}}}+{\displaystyle \sum _{\mathit{j}=\mathit{M}+1}^{\mathit{N}}{\mathit{a}}_{\mathit{i}\mathit{j}}{\mathit{R}}^{\mathit{T}}({\mathit{\theta }}_{\mathit{i}})\stackrel{.}{{\mathit{\varphi }}_{\mathit{j}}}}\right]$$

(28)

where K _i1 = diag{k_i1,1, k_i1,2, k_i1,3} is a positive definite design matrix.

By substituting equation (28) into equation (27) yields

$${\stackrel{·}{V}}_{1i}=-{\mathit{s}}_{1i}^T{\mathit{K}}_{i1}{\mathit{s}}_{1i}+d_i{\mathit{s}}_{1i}^T{\mathit{e}}_{\mathrm{vi}}$$

(29)

Before the next step, DSC technique can be applied here to solve the problem “explosion of complexity,” which occurs in a strict-feedback form system. Let ${\mathbf{\alpha }}_{\mathbf{vi}}$ pass through the following first-order filter (FOF)

$${\mathit{T}}_0{\stackrel{·}{\mathit{v}}}_{\mathrm{ri}}+{\mathit{v}}_{\mathrm{ri}}={\mathbf{\alpha }}_{\mathbf{vi}},{\mathit{v}}_{\mathrm{ri}}(0)={\mathbf{\alpha }}_{\mathbf{vi}}(0)$$

(30)

where ${\mathit{v}}_{\mathrm{ri}}\in {\mathit{R}}^3$ is the state vector of the FOF, and T₀ is a positive constant.

To make the presentation clearer, the following notations are used

$$\begin{array}{l}{L}_1^{-1}={[b_{ij}]}_{M\times M},L_1^{-1}{L}_2={[c_{ij}]}_{M\times (N-M)},\\ s_i(t)={\displaystyle \sum _{j=1}^M{a}_{ij}({\mathit{\eta }}_i-{\mathit{\eta }}_j)}+{\displaystyle \sum _{j=M+1}^N{a}_{ij}({\mathit{\eta }}_i-{\varphi }_j)}\end{array}$$

(31)

To prepare for the subsequent control design, the following lemma is presented.

Lemma 3

Consider the kinematic model described by equation (22) canceling the terms v _i − α _vi , with the virtual control law (equation (28)). If the proposed containment control system satisfies Assumptions 1–3 the following properties can be given:

1. The control objective (equation (9)) is achieved;

2. There exist positive constants ρ_1i and ρ_2i, such that $||{\mathbf{\alpha }}_{\mathrm{Vi}}||\le {\rho }_{1i}$ and $||{\stackrel{·}{\mathbf{\alpha }}}_{\mathrm{Vi}}||\le {\rho }_{2i}$ for all t ≥ t₀, respectively.

Proof

By substituting equation (28) into equation (22), canceling the nonlinear term v _i − α _vi and multiplying both sides of equation (22) by d_i, we have

$$\begin{array}{l}{\displaystyle \sum _{j=1}^N{a}_{ij}\stackrel{.}{{\mathit{\eta }}_i}}-{\displaystyle \sum _{j=1}^M{a}_{ij}\stackrel{.}{{\mathit{\eta }}_j}}-{\displaystyle \sum _{j=M+1}^N{a}_{ij}\stackrel{.}{{\varphi }_j}}\\ =-K_{i1}({\displaystyle \sum _{j=1}^N{a}_{ij}{\mathit{\eta }}_i}-{\displaystyle \sum _{j=1}^M{a}_{ij}{\mathit{\eta }}_j}-{\displaystyle \sum _{j=M+1}^N{a}_{ij}{\varphi }_j})\end{array}$$

(32)

Then, one obtains

$$\begin{array}{l}{\displaystyle \sum _{j=1}^M{a}_{ij}\left(\stackrel{.}{{\mathit{\eta }}_i}-\stackrel{.}{{\mathit{\eta }}_j}\right)}+{\displaystyle \sum _{j=M+1}^N{a}_{ij}\left(\stackrel{.}{{\mathit{\eta }}_i}-\stackrel{.}{{\varphi }_j}\right)}\\ =-K_{i1}[{\displaystyle \sum _{j=1}^M{a}_{ij}}\left({\mathit{\eta }}_i-{\mathit{\eta }}_j\right)+{\displaystyle \sum _{j=M+1}^N{a}_{ij}}\left({\mathit{\eta }}_i-{\varphi }_j\right)]\end{array}$$

(33)

Hence, defining $\mathit{P}(t)=[{\mathbf{\eta }}_1^{\mathbf{T}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathbf{\eta }}_{\mathbf{i}}^{\mathbf{T}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathbf{\eta }}_{\mathbf{M}}^{\mathbf{T}}{\mathbf{]}}^{\mathbf{T}}$ , equation (33) is rewritten as below

$$\begin{array}{l}\left({\mathit{L}}_1\otimes {\mathit{I}}_3\right)\stackrel{.}{\mathit{P}}(\mathit{t})+\left({\mathit{L}}_2\otimes {\mathit{I}}_3\right){\stackrel{.}{\mathit{X}}}_{\mathit{L}}(\mathit{t})\\ =-{\mathit{K}}_{\mathit{c}}[({\mathit{L}}_1\otimes {\mathit{I}}_3)\mathit{P}(\mathit{t})+\left({\mathit{L}}_2\otimes {\mathit{I}}_3\right){\mathit{X}}_{\mathit{L}}(\mathit{t})]\end{array}$$

(34)

where I ₂ = diag(1, 1, 1) and K _c  = diag( K _i1, K _i2, …, K _iM ). From equation (34), it can be seen that $({\mathit{L}}_1\otimes {\mathit{I}}_3)\mathit{P}(t)+({\mathit{L}}_2\otimes {\mathit{I}}_3){\mathit{X}}_L(t)$ exponentially converges to the origin as time approaches infinity. Because ${\mathit{L}}_1^{-1}$ exists, ³⁵ P (t) converges to $-({\mathit{L}}_1^{-1}{\mathit{L}}_2\otimes {\mathit{I}}_3){\mathit{X}}_L(t)$ as time approaches infinity, that is, the objective (equation (9)) is reached.

Multiplying both sides of equation (34) with ${\mathit{L}}_1^{-1}\otimes {\mathit{I}}_3$ results in

$$\begin{array}{l}\stackrel{.}{\mathit{P}}(\mathit{t})+\left({\mathit{L}}_1^{-1}{\mathit{L}}_2\otimes {\mathit{I}}_3\right){\stackrel{.}{\mathit{X}}}_{\mathit{L}}(\mathit{t})\\ =-{\mathit{K}}_{\mathit{c}}({\mathit{L}}_1^{-1}\otimes {\mathit{I}}_3)[({\mathit{L}}_1\otimes {\mathit{I}}_3)\mathit{P}(\mathit{t})+\left({\mathit{L}}_2\otimes {\mathit{I}}_3\right){\mathit{X}}_{\mathit{L}}(\mathit{t})]\end{array}$$

(35)

According to equations (31) and (35), we have

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\mathit{\eta }}}_{\mathit{i}}(\mathit{t})={\mathbf{\alpha }}_{\mathit{vi}}=-\sum _{\mathit{j}=\mathit{M}+1}^{\mathit{N}}{\mathit{c}}_{\mathit{ij}}{\stackrel{·}{\mathit{\varphi }}}_{\mathit{j}}(\mathit{t})-{\mathit{k}}_{\mathit{ci}}\sum _{\mathit{j}=1}^{\mathit{M}}{\mathit{b}}_{\mathit{ij}}{\mathit{s}}_{\mathit{j}}(\mathit{t})}\\ \multicolumn{1}{c}{=-\sum _{\mathit{j}=\mathit{M}+1}^{\mathit{N}}{\mathit{c}}_{\mathit{ij}}{\stackrel{·}{\mathit{\varphi }}}_{\mathit{j}}(\mathit{t})-{\mathit{k}}_{\mathit{ci}}\sum _{\mathit{j}=1}^{\mathit{M}}{\mathit{b}}_{\mathit{ij}}{\mathit{s}}_{\mathit{j}}({\mathit{t}}_0){\mathit{e}}^{-{\mathit{k}}_{\mathit{j}}\mathit{t}}}\end{array}$$

(36)

By the use of triangle inequality, one obtains the upper bounds of the virtual control α _vi and its derivative

$$||{\mathbf{\alpha }}_{\mathrm{vi}}||\le \sum _{j=M+1}^N|c_{\mathrm{ij}}|d_{2j}+k_{\mathrm{ci}}\sum _{j=1}^M|b_{\mathrm{ij}}|||{\mathit{s}}_j(t_0)||:={\rho }_{1i}$$

(37)

$$\Vert {\stackrel{·}{\mathbf{\alpha }}}_{\mathrm{Vi}}\Vert \le \sum _{j=M+1}^N{c}_{\mathrm{ij}}{b}_{\mathrm{ij}}+k_{\mathrm{ci}}\sum _{j=1}^M{k}_j\Vert {\mathit{s}}_j(t_0)\Vert :={\rho }_{2i}$$

(38)

where ρ_1i, ρ_2i are positive constants.

• Step 2: dynamic loop design

Define the second dynamic surface error

$${\mathit{s}}_{2i}={\mathit{v}}_i-{\mathit{v}}_{\mathrm{ri}}$$

(39)

Differentiating equation (39) along equation (23) yields

$${\stackrel{·}{\mathit{s}}}_{2i}={\mathit{M}}_i^{-1}(-{\mathit{C}}_i{\mathit{v}}_i-{\mathit{D}}_i{\mathit{v}}_i+{\mathbf{\tau }}_{\mathrm{DSC},i}+{\tilde{\mathit{d}}}_i)-{\stackrel{·}{\mathit{v}}}_{\mathrm{ri}}$$

(40)

To stabilize ${\mathit{s}}_{2i}$ , ${\mathbf{\tau }}_{\mathrm{DSC},i}$ is designed as

$${\mathbf{\tau }}_{\mathrm{DSC},i}=-{\mathit{M}}_i{\mathit{K}}_{i2}{\mathit{s}}_{2i}+{\mathit{C}}_i{\mathit{v}}_i+{\mathit{D}}_i{\mathit{v}}_i+{\mathit{M}}_i{\stackrel{·}{\mathit{v}}}_{\mathrm{ri}}$$

(41)

where K _i2 = diag{k_i2,1, k_i2,2, k_i2,3} is a positive definite design matrix.

Stability analysis

Define following the FOF estimated error

$${\mathit{e}}_{\mathrm{fi}}={\mathit{v}}_{\mathrm{ri}}-{\mathbf{\alpha }}_{\mathbf{vi}}$$

(42)

Substituting equation (28) into equation (42) yields

$$e_{fi}=v_{ri}+\frac{1}{d_i}\left[{\mathit{K}}_{\mathit{i}1}{\mathit{s}}_{\mathit{i}1}-{\displaystyle \sum _{\mathit{j}=1}^{\mathit{M}}{\mathit{a}}_{\mathit{i}\mathit{j}}{\mathit{R}}^{\mathit{T}}({\mathit{\theta }}_{\mathit{i}})\mathit{R}({\mathit{\theta }}_{\mathit{j}}){\mathit{v}}_{\mathit{j}}}-{\displaystyle \sum _{\mathit{j}=\mathit{M}+1}^{\mathit{N}}{\mathit{a}}_{\mathit{i}\mathit{j}}{\mathit{R}}^{\mathit{T}}({\mathit{\theta }}_{\mathit{i}})\stackrel{.}{{\mathit{\varphi }}_{\mathit{j}}}}\right]$$

(43)

The time derivative ${\stackrel{·}{\mathit{e}}}_{\mathrm{fi}}$ is derived

$${\stackrel{·}{\mathit{e}}}_{\mathrm{fi}}=-\frac{{\mathit{e}}_{\mathrm{fi}}}{T_0}-{\stackrel{·}{\mathit{\alpha }}}_{\mathrm{vi}}$$

(44)

With equations (24), (28), (39), and (41), we have

$${\stackrel{·}{\mathit{s}}}_{1i}=-{\mathit{K}}_{i1}{\mathit{s}}_{1i}+d_i({\mathit{s}}_{2i}+{\mathit{e}}_{\mathrm{fi}})$$

(45)

Then, by recalling equations (39) and (41), one obtains

$${\stackrel{·}{\mathit{s}}}_{2i}=-{\mathit{K}}_{i2}{\mathit{s}}_{2i}+{\mathit{M}}_i^{-1}{\tilde{d}}_i$$

(46)

From equations (14), (15), and (22), we have

$${\stackrel{·}{\stackrel{~}{\mathit{p}}}}_i=-{\mathit{D}}_1{\tilde{\mathit{p}}}_i-{\mathit{D}}_2|{\tilde{\mathit{p}}}_i{|}^{\gamma }\mathrm{sign}({\tilde{\mathit{p}}}_i)-\overline{\mathit{d}}\mathrm{sign}({\tilde{\mathit{p}}}_i)-{\mathit{d}}_i$$

(47)

Based on the control law design, the main result of this note can be summarized as the following theorem.

Theorem 2

Consider the follower model equations (2) and (3) with unknown external disturbances. If the proposed containment control system satisfies Assumptions 1–3, the NDO are chosen as equations (14)–(16). Then, the actual control inputs (equation (21)) guaranty that all the followers in the group move into the convex hull formed by the leaders, and all the signals of multiple ASVs in the closed-loop system are semi-global uniformly ultimately bounded (SGUUB) by appropriately choosing the design parameter K _i1, K _i2, T₀, D ₁, and D ₂.

Proof

Consider the following LF

$$V_2=\frac{1}{2}\sum _{i=1}^M[{\mathit{s}}_{1i}^T{\mathit{s}}_{1i}+{\mathit{s}}_{2i}^T{\mathit{M}}_i{\mathit{s}}_{2i}+{\mathit{e}}_{\mathrm{fi}}^T{\mathit{e}}_{\mathrm{fi}}+{\tilde{\mathit{p}}}_i^T{\tilde{\mathit{p}}}_i]$$

(48)

In terms of equations (44)–(47), the time derivative of V₂ is

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2=\sum _{i=1}^M[-{\mathit{s}}_{1i}^T{\mathit{K}}_{i1}{\mathit{s}}_{1i}+d_i{\mathit{s}}_{1i}^T({\mathit{s}}_{2i}+{\mathit{e}}_{\mathrm{fi}})-{\mathit{s}}_{2i}^T{\mathit{K}}_{i2}{\mathit{s}}_{2i}+{\mathit{s}}_2^T\tilde{\mathit{d}}-\frac{1}{T_0}{\mathit{e}}_{\mathrm{fi}}^T{\mathit{e}}_{\mathrm{fi}}-{\mathit{e}}_{\mathrm{fi}}^T{\stackrel{·}{\mathbf{\alpha }}}_{\mathrm{vi}}-{\tilde{\mathit{p}}}_i^T{\mathit{D}}_1{\tilde{\mathit{p}}}_i}\\ -D_{21}{\tilde{p}}_{\mathrm{ui}}^{\gamma +1}\mathrm{sign}({\tilde{p}}_{\mathrm{ui}})-D_{22}{\tilde{p}}_{\mathrm{vi}}^{\gamma +1}\mathrm{sign}({\tilde{p}}_{\mathrm{vi}})-D_{23}{\tilde{p}}_{\mathrm{ri}}^{\gamma +1}\mathrm{sign}({\tilde{p}}_{\mathrm{ri}})-{\tilde{d}}_{\mathrm{ui}}|{\tilde{p}}_{\mathrm{ui}}|-{\tilde{d}}_{\mathrm{vi}}|{\tilde{p}}_{\mathrm{vi}}|-{\tilde{d}}_{\mathrm{ri}}|{\tilde{p}}_{\mathrm{ri}}|-{\tilde{\mathit{p}}}_i^T{\mathit{d}}_i]\hfill \end{array}$$

(49)

From equations (44)–(47) and the Young equality $\mathrm{ab}\le a^2+0.25b^2$ , one obtains

$$\{\begin{array}{c}{d}_i{\mathit{s}}_{1i}^T({\mathit{s}}_{2i}+{\mathit{e}}_{\mathrm{fi}})\le 2d_i||{\mathit{s}}_{1i}|{|}^2+0.25d_i||{\mathit{s}}_{2i}|{|}^2+0.25d_i||{\mathit{e}}_{\mathrm{fi}}|{|}^2\hfill \\ {\mathit{s}}_{2i}^T{\tilde{\mathit{d}}}_i\le ||{\mathit{s}}_{2i}|{|}^2+0.25||{\tilde{\mathit{d}}}_i|{|}^2\hfill \\ {\mathit{e}}_{\mathrm{fi}}^T{\stackrel{·}{\mathbf{\alpha }}}_{\mathrm{vi}}\le ||{\mathit{e}}_{\mathrm{fi}}|{|}^2+0.25||{\stackrel{·}{\mathbf{\alpha }}}_{\mathrm{vi}}|{|}^2\hfill \\ {\tilde{\mathit{p}}}_i^T{\mathit{d}}_i\le ||{\tilde{\mathit{p}}}_i|{|}^2+0.25||{\overline{\mathit{d}}}_i|{|}^2\hfill \end{array}$$

(50)

Substituting equation (50) into equation (49) yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2=\sum _{i=1}^M[-({\lambda }_{min}({\mathit{K}}_{i1})-2d_i){\mathit{s}}_{1i}^T{\mathit{s}}_{1i}-({\lambda }_{min}({\mathit{K}}_{i2})}\\ -0.25d_i-1){\mathit{s}}_{2i}^T{\mathit{s}}_{2i}-(\frac{1}{T_0}-0.25d_i-1){\mathit{e}}_{\mathrm{fi}}^T{\mathit{e}}_{\mathrm{fi}}\hfill \\ -({\lambda }_{min}({\mathit{D}}_1)-1){\tilde{\mathit{p}}}_i^T{\tilde{\mathit{p}}}_i+Q_{0i}]\hfill \end{array}$$

(51)

where

$$Q_{0i}={\displaystyle \sum _{i=1}^M\left[\frac{1}{4}\left|\right|{\stackrel{.}{\alpha }}_{vi}|{|}^2+\frac{1}{4}\left({\epsilon }_{ui}^2+{\epsilon }_{vi}^2+{\epsilon }_{ri}^2\right)+D_{21}{\omega }_{ui}^{\gamma +1}+D_{22}{\omega }_{vi}^{\gamma +1}+D_{23}{\omega }_{ri}^{\gamma +1}+{\omega }_{ui}{\epsilon }_{ui}+{\omega }_{vi}{\epsilon }_{vi}+{\omega }_{ri}{\epsilon }_{ri}+\frac{1}{4}{\displaystyle \sum _{j=1}^3{\overline{d}}_{ji}}\right]}$$

Using Lemma 3, $Q_{0i}$ is bounded as

$$Q_{0i}\le Q_{1i}=\sum _{i=1}^M[\frac{1}{4}{\rho }_{2i}^2+\frac{1}{4}({\epsilon }_{\mathrm{ui}}^2+{\epsilon }_{\mathrm{vi}}^2+{\epsilon }_{\mathrm{ri}}^2)+D_{21}{\omega }_{\mathrm{ui}}^{\gamma +1}+D_{22}{\omega }_{\mathrm{vi}}^{\gamma +1}+D_{23}{\omega }_{\mathrm{ri}}^{\gamma +1}+{\omega }_{\mathrm{ui}}{\epsilon }_{\mathrm{ui}}+{\omega }_{\mathrm{vi}}{\epsilon }_{\mathrm{vi}}+{\omega }_{\mathrm{ri}}{\epsilon }_{\mathrm{ri}}+\frac{1}{4}\sum _{j=1}^3{\overline{d}}_{\mathrm{ji}}]$$

(52)

From equation (51), it follows that

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2\le -\sum _{i=1}^M[({\lambda }_{min}({\mathit{K}}_{i1})-2d_i){\mathit{s}}_{1i}^T{\mathit{s}}_{1i}+({\lambda }_{min}({\mathit{K}}_{i2})}\\ -0.25d_i-1){\mathit{s}}_{2i}^T{\mathit{s}}_{2i}+(\frac{1}{T_0}-0.25d_i-1){\mathit{e}}_{\mathrm{fi}}^T{\mathit{e}}_{\mathrm{fi}}\hfill \\ ({\lambda }_{min}({\mathit{D}}_1)-1){\tilde{\mathit{p}}}_i^T{\tilde{\mathit{p}}}_i]+\sum _{i=1}^M{Q}_{1i}\hfill \end{array}$$

(53)

Let

$$\{\begin{array}{c}{\lambda }_{min}({\mathit{K}}_{i1})-2d_i>0\hfill \\ \multicolumn{1}{c}{{\lambda }_{min}({\mathit{K}}_{i2})-0.25d_i-1>0}\\ \multicolumn{1}{c}{1/T_0-0.25d_i-1>0}\\ \multicolumn{1}{c}{{\lambda }_{min}({\mathit{D}}_1)-1>0}\end{array}$$

(54)

And then, equation (53) can be written as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2\le -\sum _{i=1}^M{\mathbf{\alpha }}_{\mathbf{i}}[{\mathit{s}}_{1i}^T{\mathit{s}}_{1i}+{\mathit{s}}_{2i}^T{\mathit{s}}_{2i}+{\mathit{e}}_{\mathrm{fi}}^T{\mathit{e}}_{\mathrm{fi}}+{\tilde{\mathit{p}}}_i^T{\tilde{\mathit{p}}}_i]+\sum _{i=1}^M{Q}_{1i}}\\ \le -2\mathbf{\alpha }{\mathbf{V}}_2\mathbf{+}{\mathbf{Q}}_1\hfill \end{array}$$

(55)

where ${\mathbf{\alpha }}_{\mathbf{i}}=min\{{\lambda }_{min}({\mathit{K}}_{i1})-2d_i,{\lambda }_{min}({\mathit{K}}_{i2})-0.25d_i-1,1/T_0-0.25d_i-1,{\lambda }_{min}({\mathit{D}}_1)-1\},\mathbf{\alpha }\mathbf{=}min\mathbf{\left\{}{\mathbf{\alpha }}_{\mathbf{i}}\mathbf{\right\}}$ , $Q_1=\sum _{i=1}^M{Q}_{1i}$ . By appropriately selecting the parameters K _i1, K _i2, T ₀, D ₁ and D ₂, and $\mathbf{\alpha }\mathbf{\ge }{\mathbf{Q}}_1\mathbf{/}2\mathbf{p}$ . The inequality (equation (55)) implies ${\stackrel{·}{V}}_2<0$ when $\mathbf{\alpha }\mathbf{\ge }{\mathbf{Q}}_1\mathbf{/}2\mathbf{p}$ . Therefore, the error signals in this closed-loop system, that is, ${\mathit{s}}_{1i},{\mathit{s}}_{2i},{\mathit{e}}_{\mathrm{fi}}$ , and ${\tilde{\mathit{p}}}_i$ , are SGUUB.

By integration of the inequality (equation (55)), it leads to

$$0\le V_2(t)\le \frac{Q_1}{2\mathbf{\alpha }}-(\frac{Q_1}{2\mathbf{\alpha }}-V_2(t_0))e^{-\mathbf{\alpha }\mathbf{(}\mathbf{t}\mathbf{-}{\mathbf{t}}_0\mathbf{)}}$$

(56)

Noting that $||{\mathit{S}}_1||\le 2V_2$ with ${\mathit{S}}_1=[{\mathit{s}}_{11},...,{\mathit{s}}_{1M}{]}^T$ and then the dynamic surface error vector ${\mathit{S}}_1$ is bounded and satisfies

$$\underset{t\to \infty }{lim}||{\mathit{S}}_1(t)||\le \frac{Q_1}{\mathbf{\alpha }}$$

(57)

that is, ${\mathit{S}}_1$ is ultimately bounded. Observing that

$${\mathit{S}}_1=({\mathit{L}}_1\otimes {\mathit{I}}_3)\mathbf{\eta }\mathbf{+}\mathbf{(}{\mathit{L}}_2\mathbf{\otimes }{\mathit{I}}_3\mathbf{)}{\mathit{X}}_{\mathbf{L}}$$

(58)

where ${\mathit{S}}_1=[{\mathit{s}}_{11}^T,...,{\mathit{s}}_{1M}^T{]}^T$ . Then, it can be concluded that there exists a positive constant s_max such that

$$||\mathbf{\eta }\mathbf{+}\mathbf{(}{\mathit{L}}_1^{\mathbf{-}1}{\mathit{L}}_2\mathbf{\otimes }{\mathit{I}}_3\mathbf{)}{\mathit{X}}_{\mathbf{L}}\mathbf{|}\mathbf{|}\mathbf{\le }{\mathbf{s}}_{max}$$

(59)

which means that the control objective (equation (9)) is achieved.

Remark 5

Since the auxiliary variable estimated error ${\tilde{\mathit{p}}}_i$ is SGUUB, the disturbance estimated error ${\tilde{\mathit{d}}}_i$ is SGUUB.

Remark 6

In kinematic loop design, the derivatives of the virtual control law ${\stackrel{·}{\mathbf{\alpha }}}_{\mathrm{vi}}$ can be expressed as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\mathbf{\alpha }}}_{\mathrm{vi}}=\frac{1}{d_i}[-{\mathit{K}}_{i1}{\stackrel{·}{\mathit{s}}}_{i1}+\sum _{j=1}^M{a}_{\mathrm{ij}}{\stackrel{·}{\mathit{R}}}^T({\mathbf{\theta }}_{\mathbf{i}})\mathit{R}({\mathbf{\theta }}_{\mathbf{j}}){\mathit{v}}_j+\sum _{j=1}^M{a}_{\mathrm{ij}}{\mathit{R}}^T({\mathbf{\theta }}_{\mathbf{i}})\stackrel{·}{\mathit{R}}({\mathbf{\theta }}_{\mathbf{j}}){\mathit{v}}_j+\sum _{j=1}^M{a}_{\mathrm{ij}}{\mathit{R}}^T({\mathbf{\theta }}_{\mathbf{i}})\mathit{R}({\mathbf{\theta }}_{\mathbf{j}}){\stackrel{·}{\mathit{v}}}_j}\\ +\sum _{j=M+1}^N{a}_{\mathrm{ij}}{\stackrel{·}{\mathit{R}}}^T({\mathbf{\theta }}_{\mathbf{i}}){\stackrel{·}{\varphi }}_j+\sum _{j=M+1}^N{a}_{\mathrm{ij}}{\mathit{R}}^T({\mathbf{\theta }}_{\mathbf{i}}){\stackrel{··}{\varphi }}_j]\hfill \end{array}$$

(60)

To simply this computation, FOF is introduced to estimate each virtual control law and its derivative.

Remark 7

In recent years, many papers have discussed the problem of containment and consensus maneuvering of fully actuated ASVs under external disturbances. In Peng et al., ¹³ an echo state network (ESN)-based observer using recorded input–output data is developed to identify unknown vessel dynamics. In Peng et al., ¹⁴ iterative updating laws using prediction errors are employed to identify the unknown dynamics of ASVs. However, the above two methods can only asymptotically approach to unknown disturbance. Different from the study of Peng et al.,^13,14 this article utilizes the finite-time stability theory to design a NDO with finite-time convergence.

Simulations

In this section, simulation studies are presented to validate the effectiveness of the proposed robust cooperative scheme to bounded environmental disturbances. We consider a group of ships with three virtual leaders and five followers. The communication topology among the ASVs is shown in Figure 2 with F = {1, 2, …, 5}, L = {5, 6, 7}. The design parameters for the proposed containment controllers are chosen as ${\overline{\mathit{d}}}_i=\mathrm{diag}\{2,2,2\},\gamma =0.6$ , T₀ = 0.02, D ₁ = diag{2, 2, 2}, D ₂ = diag{1, 1, 1}, K _i1 = diag{6, 6, 6}, K _i2 = diag{8, 8, 8}. Next, four simulation examples will be presented to illustrate the theoretical results.

Figure 2.

Communication topology.

Point stabilization with constant disturbances

In this section, the proposed containment control strategy in the presence of the constant disturbances is examined. Suppose that the positions of the virtual leaders are fixed point as shown in Figure 2. The disturbances due to ocean environment are set as ${\mathit{d}}_i=[1\mathrm{N};1\mathrm{N};1\mathrm{N}\mathrm{m}](i=1,...,5)$ . The initial follower states X _i  = (x_i; y_i; ψ_i; u_i; v_i; r_i) are X ₁ = (−12.5; −4; π/3; 0.1; 0.02; 0.02); X ₂ = (−7.5; −7; π/4; 0.05; 0.15; 0); X ₃ = (−1; −8; π/6; 0.04; 0.012; 0.05); X ₄ = (3.4; −4; π/3; 0.012; 0.012; 0.01); X ₅ = (10; −7; π/2.5; 0.01; 0.01; 0.01). The initial states of the DO are taken as ${\hat{\mathit{d}}}_1(0)=[2.5;2.5;2.5],{\hat{\mathit{d}}}_2(0)=[2;2;2],{\hat{\mathit{d}}}_3(0)=[1.5;1.5;1.5],{\hat{\mathit{d}}}_4(0)=[1;1;1],{\hat{\mathit{d}}}_5(0)=[0.5;0.5;0.5]$ .

The results are shown in Figures 3 and 4. In Figure 3, the red points represent the virtual leaders’ actual positions, the black dotted area indicates the area that the followers need to reach, and the other five color dashed lines indicate the actual trajectories of the followers. From this figure, we find that all the followers in this group ultimately converge to the stationary convex hull formed by the stationary leaders, and the final trajectories of the followers are constant. The estimated errors of the DOs are displayed in Figure 4. It can be observed that the proposed NDO provides the rapidly convergent estimation of external disturbances as proved in Theorem 1.

Figure 3.

Motion trace of the group.

Figure 4.

Disturbance estimation errors.

Point stabilization with time-variant disturbances

In this section, the proposed control strategy in the presence of the time-variant disturbances is also examined. The initial states of each follower are chosen as section “Point stabilization with constant disturbances.” The initial states of the DO are also taken as section “Point stabilization with constant disturbances.” The disturbances due to ocean environment are assumed to be

$$\{\begin{array}{c}{d}_{\mathrm{ui}}=1.3+2\sin (0.02t)+1.5\sin (0.1t)\hfill \\ \multicolumn{1}{c}{d_{\mathrm{vi}}=-0.9+2\sin (0.02t-\pi /6)+1.5\sin (0.3t)}\\ \multicolumn{1}{c}{d_{\mathrm{ri}}=-\sin (0.09t+\pi /3)-4\sin (0.01t)}\end{array}$$

(61)

The results are shown in Figures 5 and 6. Figure 5 shows the output trajectories of the five followers, from which it is evident that the followers in this group also converge to the convex hull formed by those of the moving leaders. The estimated errors of the DOs are displayed in Figure 6. From these figures, one can conclude that the proposed containment control strategy is also practical when each follower is exposed to time-variant disturbances. It is obvious that the designed control law is effective when each ASV is exposed to both unknown constant and time-variant external disturbances.

Figure 5.

Motion trace of the group.

Figure 6.

Disturbance estimation errors.

Curve tracking with constant disturbances

In this section, we assume that the predefined trajectories of the virtual leaders for this group are curved trajectories. The initial follower states are X ₁ = (−14.5; −5; π/3; 0.1; 0.02; 0.02); X ₂ = (−12; −7; π/4; 0.05; 0.015; 0); X ₃ = (−8; −6; π/6; 0.04; 0.012; 0.05); X ₄ = (−6; −7; π/3; 0.012; 0.012; 0.01); X ₅ = (−4; −5; π/2.5; 0.01; 0.01; 0.01), and the external disturbance is set as ${\mathit{d}}_i=[1.5\mathrm{N};1.5\mathrm{N};1.5\mathrm{N}\mathrm{m}](i=1,...,5)$ . The initial states of the DO are taken as ${\hat{\mathit{d}}}_1(0)=[3;3;3]$ , ${\hat{\mathit{d}}}_2(0)=[2.5;2.5;2.5]$ , ${\hat{\mathit{d}}}_3(0)=[2;2;2]$ , ${\hat{\mathit{d}}}_4(0)=[1.5;1.5;1.5]$ , ${\hat{\mathit{d}}}_5(0)=[1;1;1]$ . The leaders can be described by

$$\{\begin{array}{c}{\mathbf{\varphi }}_2=[-8\cos (\pi t/200);8\sin (\pi t/200)]\hfill \\ \multicolumn{1}{c}{{\mathbf{\varphi }}_{\mathbf{i}}={\mathbf{\varphi }}_2+\mathit{R}({\psi }_d){\mathit{l}}_i}\end{array}$$

(62)

where i = 1, 3; R (·) = [cos(·), −sin(·); sin(·), cos(·)]; ψ_d = atan2[π/25cos(πt/200), π/25sin(πt/200)]; l ₁ =[−4; 4], l ₂ = [−4; −4].

The results are shown in Figures 7 and 8. Similar with Figures 3 and 5, the red points indicate the leaders’ trajectories, the black dotted area represents the desired area of the leaders in the movement, while the other five color dotted lines indicate the actual trajectories of the followers. Figure 7 illustrates the motion traces of the five followers. As shown in Figure 7, the followers’ trajectories converge to the convex hull spanned by three leaders’ trajectories. The estimated errors of the NDOs are shown in Figure 8. It can be noted that the NDO provides the rapidly convergent estimation of external disturbances when the trajectories of the virtual leaders are curved trajectories.

Figure 7.

Motion trace of the group.

Figure 8.

Disturbance estimation errors.

Curve tracking with time-variant disturbances

In this section, we consider the curve tracking problem of followers in presence of time-variant disturbances. The initial states of the followers and the curved predefined trajectories of the virtual leaders are the same as the previous section. The disturbances are assumed to be

$$\{\begin{array}{c}{d}_{\mathrm{ui}}=1.21+0.17\sin (0.02t+\pi /4)-0.5\sin (0.1t+\pi /3)\hfill \\ \multicolumn{1}{c}{d_{\mathrm{vi}}=-0.45+2\sin (0.02t-\pi /6)+1.5\sin (0.3t)}\\ \multicolumn{1}{c}{d_{\mathrm{ri}}=0.78+\cos (0.13t-\pi /3)+4\sin (0.01t)}\end{array}$$

(63)

These simulation results are demonstrated in Figures 9 and 10. Figure 9 shows the output trajectories of the five followers, from which it is evident that the followers in this group also converge to the convex hull formed by those of the moving leaders. The estimated errors of the DOs are displayed in Figure 10.

Figure 9.

Motion trace of the group.

Figure 10.

Disturbance estimation errors.

In summary, from Figures 3–10, we can see that the proposed containment control law in this article provided acceptable results as proven in Theorem 2, and all the signals of multiple ships in the closed-loop system are SGUUB as time approaches infinity. In addition, the proposed NDO provides the rapidly convergent estimation of external disturbances as proved in Theorem 1.

Conclusion

This article presents an NDO-based DSC approach for containment control of multiple fully actuated ASVs in the presence of unknown external disturbances. Compared with existing results, the approach proposed in this article shows some advantages to handle the disturbances due to ocean environment and avoid the problem of “explosion of complexity” from the standard BP method. Based on Lyapunov stability theory, the proposed containment control law guarantees that all the signals of the closed-loop system are SGUUB. Simulation examples demonstrate the effectiveness of the containment controller and its robustness to external disturbances.

In the future, related works will be required to account for communication constraint among ships, such as switching communication topology as well as communication delays.