# Containment Control of Underactuated Ships with Environment Disturbances and Parameter Uncertainties

Authors: Qidan Zhu; Junda Ma; Zhilin Liu; Ke Liu
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

An implementable robust containment control algorithm is proposed for a group of underactuated ships in the presence of hydrodynamic parameter uncertainties and external disturbances. The control objective is to drive all the followers into the convex hull spanned by the virtual leaders, whose state information is available only to a subset of the followers. For this purpose, the ship model is primarily transformed to a strict-feedback form. In the kinematic design, a virtual containment controller, requiring the state information from its neighbors, is presented based on the results obtained from graph theory. In the dynamic design, a robust containment controller is developed through utilizing upper-to-up sliding mode control. In addition, in order to simplify the implementations of the control law, the command filtered backstepping (CFBP) method is introduced to prevent the analytic differentiations of the virtual law from each design step of the backstepping (BP) method. Subsequently, it is well proven that all the tracking errors could converge to and remain small neighborhoods of the equilibrium point. Finally, several simulation experiments are conducted to demonstrate the performance of the proposed control algorithm.


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## Body

## 1. Introduction

Over the past few years, a huge and rapidly growing body of research has focused on the cooperative control of the multiple vehicle systems within a motion-control community because of their broad applications, for example, wheeled mobile robot, manipulator, spacecraft, and ship [1–4]. Although an individual vehicle can be used to complete a specific task, some benefits including greater robustness, lower cost, and better performance can be achieved through a group of vehicles working cooperatively. In the marine domain, many applications using cooperative control include various tasks such as environmental monitoring as well as rescue and reconnaissance operations [4–7]. To perform these tasks, an increasing number of studies have focused on the cooperative control of multiple ships.These studies have included behavior-based method [8], virtual structure method [9–14], and leader-follower (L-F) method [4, 5, 15–19]. On the basis of null space methods, a formation control strategy has been developed to achieve multiple tasks in [8]. In [15], an output synchronization control strategy has been designed for the L-F control of ships. The authors in [9, 10] introduce passivity theory as a practical design tool for solving the formation problem, which includes group coordination and path following. A robust cooperative control method has also been applied to form a desired geometric pattern with the aid of a neural network (NN), a backstepping (BP) method, and graph theory [11]. In [8–11, 15], the vehicles are assumed to be fully actuated.In actuality, most ships are underactuated, meaning that they are not actuated in the sway axis [20]. Since the dynamics of each underactuated ship do not satisfy the necessary conditions of Brockett for stabilization [21], the cooperative control law for each ship cannot be smooth functions with theirs variables [16]. In [12], a decentralized formation control method has been presented to address such an issue by using the nonlinear cascaded system theory and line-of-sight guidance to ensure a straight-line path following for the formations of ships. On the basis of Lyapunov theory, the BP method, and the graph theory, a path-following coordination controller (PFCC) is proposed in [13] and its control strategy also considers time delayed communication among ships. The work of [14] is extended further in [13]. In [14], a dynamic surface control (DSC) technique is proposed to estimate the virtual control at each step of the BP method. An additional virtual control is adopted to solve the difficult problems when designing the formation controller for ships in [22]. The L-F method for ship has been also reported in [4, 5, 16–19]. In [16, 17], continuous time-varying cooperative control laws are designed to perform a geometric pattern by using suitable transformations. It is noted that, in [16, 17], the yaw velocity is assumed to be nonzero, which is referred to as the “persistent excitation” (PE) condition. Under this condition, a straight line cannot be tracked for underactuated ships. The authors in [18] address the design of the nonlinear model predictive formation controller, where the relative distances and orientations between the follower and the leader can be stabilized. To cope with the uncertainties of the model, two different robust L-F formation control laws have been designed that combine NN with the DSC technique in [4, 19]. More recently, a robust formation control algorithm is proposed to force ships to maintain the desired orientations and positions relative to one leading vessel, considering the limited magnitude of the control signal in [5]. In [4, 5, 19], the radial basis function NN and adaptive control method can approximate the nonlinear uncertainty systems, which increases the complexity of the online computing. In addition, in all the works on L-F formation control, a common trait is that only one leader exists in the group.In practical applications, for a group of vehicles multiple leaders might exist. In this case, the control objective is to drive the followers of the group into the convex hull spanned by the leaders, which is called the containment control problem [23]. Thus, containment control can be regarded as a special L-F formation control, and the study of containment control is motivated by its possible applications. For example, a group of ships are guided by another group, and some ships are regarded as leaders, which are all equipped with sensors to detect the hazardous obstacles in the marine environment [24]. As such, the leaders can form a safe area in which the followers can converge [25]. Recent works on the containment control of multiple vehicle systems focus on single integrator systems [25], double integrator systems [26], strict-feedback form systems [23], general linear dynamics [27], and Lagrange dynamics [28]. However, these proposed methods cannot be applied directly to an underactuated mechanical system, and the uncertainties from the model might affect the control performance. Accordingly, how to achieve the containment control objective in the presence of uncertain dynamics and external disturbances needs to be investigated further.For nonlinear systems, various methods have been adopted, and, among these, the BP method can be regarded as a major design tool. However, this method suffers from the “explosion of complexity” problem, which is caused by the repeated derivatives of the virtual control signals [29]. To overcome this drawback, the command filtered backstepping (CFBP) method, which introduces, at each design step of the BP method, a command filter to prevent the derivative of nonlinear functions, has also been applied to the strict-feedback form system in [30, 31]. The CFBP method is regarded as an improved version of standard BP method and is widely used in different fields [32–34]. However, these works cannot prove the boundedness of the tracking errors, which implies that the stability of the closed-loop system cannot be analyzed quantifiably. In addition, most current CFBP adopted second-order command filter (SOCF), and the first-order command filter (FOCF) with simpler structure is rarely mentioned.In conclusion, the containment control of underactuated ships involved the following main difficulties: (1) the underactuated surface ship lacks control input at side, so it belongs to a class of underactuated system; namely, the general nonlinear control theory could not be directly applied; (2) in complex sea cases, ship is often influenced by the external disturbances and parameter uncertainties (EDPU), which makes it complicated to accurately control the motion of ship; (3) the current research results on containment control are applied only in full-actuated system such as single integrator systems, double integrator systems, and Lagrange systems. However, it is not mentioned how to achieve containment control by combining the knowledge of graph theory and the underactuated system, which is one of the research difficulties. In this paper, a robust containment control strategy, inspired by the previous works, which is performed by using the standard BP method, the FOCF technique, the sliding mode control (SMC) method, the Lyapunov stability theory, and the results from graph theory, is developed for a group of ships in the presence of the EDPU. Compared with the existing results, the main features of this paper can be summarized as follows: (1) for the first time, this paper considers the containment problem of underactuated ships; (2) by incorporating the FOCF technique, the BP commands are simplified; (3) different from the previous conclusions about SOCF [30, 32–34], the stability of all the closed-loop systems is first analyzed quantifiably; (4) by introducing a polar coordinate, the model is transformed to a strict- feedback form system, and the PE condition from [16, 17] is avoided; (5) compared with the results in [4, 5, 19], the development, using upper-to-up SMC to design a robust containment controller, is more practical because the online computation of the uncertainties is avoided.


## 2. Control Problem Formulation

### 2.1. Notations

To prepare for the subsequent control design, some notations are standard as below:Rn×m denotes a set of Euclidean n×m matrices, and Rn denotes n-dimensional Euclidean space; diag⁡(a1,a2,…,an) denotes a diagonal matrix with entries ai(i=1,2,…,n); the maximum and the minimum eigenvalues of a square matrix are described as λmax(·) and λmin(·), respectively; let · be the absolute value of a scalar; let · be the Euclidean norm of a vector; let A⊗B be the Kronecker product of matrices A∈Rn×m and B∈Rp×q; C≔D denotes that D is defined as C.


### 2.2. Concepts in Graph Theory

Suppose that a group of underactuated ships interact with each other through a communication network to perform a containment control task. It is natural to model the communication topology among these ships by using graph theory. Without loss of generality, we assume that the individual ship is a node, and the interaction for ships can be described by a directed graphG={V,ε}, where V={a1,a2,…,an} denotes a set of nodes and ε={(ni,nj)∈V×V} denotes a set of edges with element (ni,nj), which implies that node nj can receive information from node ni. Here, we say that node ni is a neighbor of node nj, and the notation Ni is described as the set of all neighbors of node. Let the adjacency matrix A={aij}∈RN×N be defined such as aij=1 if (ni,nj)∈ε and aij=0 otherwise; Note that we assume aii=0 for all the nodes. The Laplacian matrix L={lij} of the directed graph G is defined such that(1)lij=∑j=1Naijifj=i;lij=-aij,otherwise.A directed path from ni to nj in the graph G is a set of edges: ni,ni+1,ni+1,ni+2,…,nj-2,nj-1,and(nj-1,nj), where all the nodes in this path are different [35].Definition 1.
For a group of ships, shipi is said to be a leader if aij=0, and ship i is said to be a follower if aij=1, j=1,2,…,N.Definition 2.
The real setC∈Rn is said to be convex if, ∀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,…,xN} 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) is defined as follows [36]:(2)CoX=∑i=1nλixi∣xi∈X,λi>0,∑i=1nλi=1.


### 2.3. Underactuated Ship Modeling

Suppose that there areM identical followers, labeled as ship 1 to M. According to [37], for the containment control task we can neglect the motions in heave, pitch, and roll; hence, for a ship, the 3-DOF mathematical model can be formulated as (see Figure 1)(3)x˙i=uicos⁡ψi-visin⁡ψi,y˙i=uicos⁡ψi+visin⁡ψi,ψ˙i=ri,u˙i=fuiui,vi,ri+m11i-1Fui+τwuit,v˙i=fviui,vi,ri+m22i-1τwvit,r˙i=friui,vi,ri+m33i-1Fri+τwrit,where(4)fui=m22im11i-1viri-duim11i-1ui-du2im11i-1uiui,fvi=-m11im22i-1uiri-dvim22i-1vi-dv2im22i-1vivi-dvrim22i-1ri,fri=m11i-m22im33i-1uivi-drim33i-1ri-dr2im33i-1riri-dvrim22i-1ri,where (xi, yi) and ψi denote the position and heading angle of the ship i in the inertial reference frame {I}; ui, vi, and ri denote surge, sway, and yaw velocities expressed in the vessel-fixed reference frame {B}, respectively; τwui, τwvi, and τwri describe ocean environment disturbances; Fui and Fri are used to describe actual control inputs; dui, du2i, dvi, dv2i, dvri, dri, dr2i, and drvi denote hydrodynamic damping parameters; m11i, m22i, and m33i denote ship inertia including added mass.Figure 1
Planar model of an underactuated ship.Because no control signal is introduced in the sway direction, this model is an underactuated system. Inspired by the work of [38], we define the following polar coordinates to solve this problem:(5)UTi=ui2+vi2,ψTi=ψi+βi,where βi=arctan⁡(vi,ui) is often recognized as side-angle and UTi and ψTi are recognized as the speed and the course angle of ship i. Then, substituting (5) into (3) yields(6)x˙i=UTicos⁡ψTi,y˙i=UTisin⁡ψTi,ψ˙Ti=ri+β˙i,(7)V˙Ti=cos⁡βifui+sin⁡βifvi+cos⁡βim11i-1τwuit+sin⁡βim22i-1τwvit+cos⁡βim11i-1Fui,r˙i=fri+m33i-1τwrit+m33i-1Frit.Then, suppose that the group containsN-M leaders in the group, labeled as leaders M+1 to N. To describe the communication among the followers and leaders in the group, let L denote the Laplacian matrix. As each leader has no neighbors, L can be described by the following:(8)L=L1L20N-M×M0N-M×N-M,where L1∈RM×M and L2∈RM×(N-M).Assumption 3.
The environment disturbances andτwui, τwvi, and τwri satisfy(9)τwui≤τ-wui,τwvi≤τ-wvi,τwri≤τ-wri,where τ-wui,τ-wvi,τ-wri>0 [38].Assumption 4.
For every follower of the group, there exists at least one leader that has a directed path to it [36].Assumption 5.
The trajectories of the virtual leadersϕi=xdi,ydiT(i=M+1,…,N) satisfy(10)0<b1i≤ϕ˙i≤b2i,ϕ¨i≤b3i,where b1i, b2i, and b3i are all positive constants.Remark 6.
(1) Assumption 3 means that the disturbances induced by winds, waves, and currents have finite energy; (2) Assumption 4 indicates that the proposed control law relies just on a directed communication topology, which reduces the complexity of the communication among ships; (3) the boundedness of the first derivatives and second derivatives of the leaders will be introduced to analyze the stability of the closed-loop system.Remark 7.
Because the surge controlFui in (7) is available only for the nonzero value of cos⁡βi, the condition UTi>0 can guarantee that the side-angle satisfies βi≠±0.5π; that is, cos⁡βi≠0. In the next part, we will give a certain initial condition for ship i such that UTi is positive at all times.Remark 8.
Using the polar coordinate transformation (5), the underactuated models (3) are transformed to a strict-feedback form of (6) and (7). Then, one can apply the standard BP method to design the control input signals Fui and Fri, which achieves the containment control objective.


### 2.4. Command Filter

To overcome the problem of “explosion of complexity” which occurs in a strict-feedback form [39], a FOCF is introduced to eliminate the analytic differentiation of the virtual control signal in the standard BP design procedure [30]. The standard state space form of FOCF is presented as(11)z˙=-ωnz-q,where the FOCF parameter ωn is a constant and satisfies ωn>0; z and z˙ are estimates of the input q and its derivative, respectively; the initial conditions of states z and z-q are λ3 and λ4; the following lemma proves the input-to-state stability property of FOCF which will be used in Section 3 to analyze the stability of the closed-loop system.Lemma 9.
Consider the FOCF described by (11). Assume that the positive constants λ1 and λ2 exist, such that |q(t)|≤λ1, |q˙(t)|≤λ2 for all t≥0. Then, the following properties of the FOCF can be given:(1)
z1 is bounded and satisfies(12)z≤λ1+λ3-λ4+λ1ωn.(2)
z˙1 is bounded and satisfies(13)z˙≤λ1+ωnλ3-λ4.Proof.
Define the tracking errorq~=z-q. By using (11), one can obtain(14)q~˙=-ωnq~-q˙.
Consider a Lyapunov function candidate (LFC)(15)Vq~=q~2.Then, the time derivative of (15) is(16)V˙q~≤-2ωnq~2+2q˙q~2≤-2ωnVq~+2λ2Vq~.Using inequality (16), one obtains(17)q~≤e-ωntq~0+λ21-e-ωntωn.Thus, for all t≥0,(18)q~≤λ3-λ4+λ2ωn.Because |q(t)|⩽λ1, it follows that z is bounded and satisfies(19)z=z-q+q≤z-q+z≤λ1+λ3-λ4+λ1ωn.From (11) and (18), one obtains(20)z˙=ωnz-q≤ωnλ3-λ4+λ1ωn=ωnλ3-λ4+λ1.That is, z1 and z˙1 are bounded for all t≥0.Remark 10.
For a given inputq, the larger value of the FOCF parameter ωn will decrease the estimated error z-q, which leads to better final estimate accuracy.Remark 11.
To address the problem of excessive estimation error in the initial time, we assume that the FOCF initial values arez(0)=q(0) in this paper. Combined with (11) and (18), the following inequalities hold for all t≥0:(21)z≤λ1ωn+λ1,z˙≤λ2.


### 2.5. Control Formulation

This article aims to design a containment control scheme such that all the followers in the group move into the convex hull formed by the leaders [28, 40]; that is,(22)limt→∞⁡dpi-CoXL=0,where pi=[xi,yi]T is the position of the follower i(i=1,2,…,N); XL≔[ϕM+1T,…,ϕNT]T; d(x,M) denotes the distance from a point x∈R2 to a set M∈R2; that is, (23)dx,M≔infy∈M⁡x-y.Lemma 12.
Under Assumption4, all the eigenvalues of L1 have positive real parts. Furthermore, each of -L1-1L2 is nonnegative, and all the row sums of this matrix are equal to 1 [41].Remark 13.
LetXL(t)≔[ϕM+1T(t),…,ϕNT(t)]T be the positions of the dynamic leaders. From Definition 2 and Assumption 4, Lemma 12 implies that every point of -(L1-1L2⊗I2)XL(t) is in the convex hull spanned by the leaders XL(t).With this previous notation, the control problem under study can be formulated as below.Control Objective. Consider the model given by (6) and (7) under Assumptions 3–5, and 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 such that(24)limt→∞⁡pit-pdit≤σi,where σi is a positive constant, pi(t)∈R2 denotes the actual position of follower i, and pdi(t)∈R2 denotes the desired position of follower i which satisfies(25)Pdt=-L1-1L2⊗I2XLt,where Pd(t)=[pd1T(t),…,pdiT(t),…,pdMT(t)]T.Remark 14.
Because of the movement of the ship affected by the ocean environment and the filtering error induced by FOCF, the tracking errorpi-pdi cannot converge to zero but can be uniformly ultimately bounded by σi, which will be discussed in detail in the next part.


## 2.1. Notations

To prepare for the subsequent control design, some notations are standard as below:Rn×m denotes a set of Euclidean n×m matrices, and Rn denotes n-dimensional Euclidean space; diag⁡(a1,a2,…,an) denotes a diagonal matrix with entries ai(i=1,2,…,n); the maximum and the minimum eigenvalues of a square matrix are described as λmax(·) and λmin(·), respectively; let · be the absolute value of a scalar; let · be the Euclidean norm of a vector; let A⊗B be the Kronecker product of matrices A∈Rn×m and B∈Rp×q; C≔D denotes that D is defined as C.


## 2.2. Concepts in Graph Theory

Suppose that a group of underactuated ships interact with each other through a communication network to perform a containment control task. It is natural to model the communication topology among these ships by using graph theory. Without loss of generality, we assume that the individual ship is a node, and the interaction for ships can be described by a directed graphG={V,ε}, where V={a1,a2,…,an} denotes a set of nodes and ε={(ni,nj)∈V×V} denotes a set of edges with element (ni,nj), which implies that node nj can receive information from node ni. Here, we say that node ni is a neighbor of node nj, and the notation Ni is described as the set of all neighbors of node. Let the adjacency matrix A={aij}∈RN×N be defined such as aij=1 if (ni,nj)∈ε and aij=0 otherwise; Note that we assume aii=0 for all the nodes. The Laplacian matrix L={lij} of the directed graph G is defined such that(1)lij=∑j=1Naijifj=i;lij=-aij,otherwise.A directed path from ni to nj in the graph G is a set of edges: ni,ni+1,ni+1,ni+2,…,nj-2,nj-1,and(nj-1,nj), where all the nodes in this path are different [35].Definition 1.
For a group of ships, shipi is said to be a leader if aij=0, and ship i is said to be a follower if aij=1, j=1,2,…,N.Definition 2.
The real setC∈Rn is said to be convex if, ∀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,…,xN} 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) is defined as follows [36]:(2)CoX=∑i=1nλixi∣xi∈X,λi>0,∑i=1nλi=1.


## 2.3. Underactuated Ship Modeling

Suppose that there areM identical followers, labeled as ship 1 to M. According to [37], for the containment control task we can neglect the motions in heave, pitch, and roll; hence, for a ship, the 3-DOF mathematical model can be formulated as (see Figure 1)(3)x˙i=uicos⁡ψi-visin⁡ψi,y˙i=uicos⁡ψi+visin⁡ψi,ψ˙i=ri,u˙i=fuiui,vi,ri+m11i-1Fui+τwuit,v˙i=fviui,vi,ri+m22i-1τwvit,r˙i=friui,vi,ri+m33i-1Fri+τwrit,where(4)fui=m22im11i-1viri-duim11i-1ui-du2im11i-1uiui,fvi=-m11im22i-1uiri-dvim22i-1vi-dv2im22i-1vivi-dvrim22i-1ri,fri=m11i-m22im33i-1uivi-drim33i-1ri-dr2im33i-1riri-dvrim22i-1ri,where (xi, yi) and ψi denote the position and heading angle of the ship i in the inertial reference frame {I}; ui, vi, and ri denote surge, sway, and yaw velocities expressed in the vessel-fixed reference frame {B}, respectively; τwui, τwvi, and τwri describe ocean environment disturbances; Fui and Fri are used to describe actual control inputs; dui, du2i, dvi, dv2i, dvri, dri, dr2i, and drvi denote hydrodynamic damping parameters; m11i, m22i, and m33i denote ship inertia including added mass.Figure 1
Planar model of an underactuated ship.Because no control signal is introduced in the sway direction, this model is an underactuated system. Inspired by the work of [38], we define the following polar coordinates to solve this problem:(5)UTi=ui2+vi2,ψTi=ψi+βi,where βi=arctan⁡(vi,ui) is often recognized as side-angle and UTi and ψTi are recognized as the speed and the course angle of ship i. Then, substituting (5) into (3) yields(6)x˙i=UTicos⁡ψTi,y˙i=UTisin⁡ψTi,ψ˙Ti=ri+β˙i,(7)V˙Ti=cos⁡βifui+sin⁡βifvi+cos⁡βim11i-1τwuit+sin⁡βim22i-1τwvit+cos⁡βim11i-1Fui,r˙i=fri+m33i-1τwrit+m33i-1Frit.Then, suppose that the group containsN-M leaders in the group, labeled as leaders M+1 to N. To describe the communication among the followers and leaders in the group, let L denote the Laplacian matrix. As each leader has no neighbors, L can be described by the following:(8)L=L1L20N-M×M0N-M×N-M,where L1∈RM×M and L2∈RM×(N-M).Assumption 3.
The environment disturbances andτwui, τwvi, and τwri satisfy(9)τwui≤τ-wui,τwvi≤τ-wvi,τwri≤τ-wri,where τ-wui,τ-wvi,τ-wri>0 [38].Assumption 4.
For every follower of the group, there exists at least one leader that has a directed path to it [36].Assumption 5.
The trajectories of the virtual leadersϕi=xdi,ydiT(i=M+1,…,N) satisfy(10)0<b1i≤ϕ˙i≤b2i,ϕ¨i≤b3i,where b1i, b2i, and b3i are all positive constants.Remark 6.
(1) Assumption 3 means that the disturbances induced by winds, waves, and currents have finite energy; (2) Assumption 4 indicates that the proposed control law relies just on a directed communication topology, which reduces the complexity of the communication among ships; (3) the boundedness of the first derivatives and second derivatives of the leaders will be introduced to analyze the stability of the closed-loop system.Remark 7.
Because the surge controlFui in (7) is available only for the nonzero value of cos⁡βi, the condition UTi>0 can guarantee that the side-angle satisfies βi≠±0.5π; that is, cos⁡βi≠0. In the next part, we will give a certain initial condition for ship i such that UTi is positive at all times.Remark 8.
Using the polar coordinate transformation (5), the underactuated models (3) are transformed to a strict-feedback form of (6) and (7). Then, one can apply the standard BP method to design the control input signals Fui and Fri, which achieves the containment control objective.


## 2.4. Command Filter

To overcome the problem of “explosion of complexity” which occurs in a strict-feedback form [39], a FOCF is introduced to eliminate the analytic differentiation of the virtual control signal in the standard BP design procedure [30]. The standard state space form of FOCF is presented as(11)z˙=-ωnz-q,where the FOCF parameter ωn is a constant and satisfies ωn>0; z and z˙ are estimates of the input q and its derivative, respectively; the initial conditions of states z and z-q are λ3 and λ4; the following lemma proves the input-to-state stability property of FOCF which will be used in Section 3 to analyze the stability of the closed-loop system.Lemma 9.
Consider the FOCF described by (11). Assume that the positive constants λ1 and λ2 exist, such that |q(t)|≤λ1, |q˙(t)|≤λ2 for all t≥0. Then, the following properties of the FOCF can be given:(1)
z1 is bounded and satisfies(12)z≤λ1+λ3-λ4+λ1ωn.(2)
z˙1 is bounded and satisfies(13)z˙≤λ1+ωnλ3-λ4.Proof.
Define the tracking errorq~=z-q. By using (11), one can obtain(14)q~˙=-ωnq~-q˙.
Consider a Lyapunov function candidate (LFC)(15)Vq~=q~2.Then, the time derivative of (15) is(16)V˙q~≤-2ωnq~2+2q˙q~2≤-2ωnVq~+2λ2Vq~.Using inequality (16), one obtains(17)q~≤e-ωntq~0+λ21-e-ωntωn.Thus, for all t≥0,(18)q~≤λ3-λ4+λ2ωn.Because |q(t)|⩽λ1, it follows that z is bounded and satisfies(19)z=z-q+q≤z-q+z≤λ1+λ3-λ4+λ1ωn.From (11) and (18), one obtains(20)z˙=ωnz-q≤ωnλ3-λ4+λ1ωn=ωnλ3-λ4+λ1.That is, z1 and z˙1 are bounded for all t≥0.Remark 10.
For a given inputq, the larger value of the FOCF parameter ωn will decrease the estimated error z-q, which leads to better final estimate accuracy.Remark 11.
To address the problem of excessive estimation error in the initial time, we assume that the FOCF initial values arez(0)=q(0) in this paper. Combined with (11) and (18), the following inequalities hold for all t≥0:(21)z≤λ1ωn+λ1,z˙≤λ2.


## 2.5. Control Formulation

This article aims to design a containment control scheme such that all the followers in the group move into the convex hull formed by the leaders [28, 40]; that is,(22)limt→∞⁡dpi-CoXL=0,where pi=[xi,yi]T is the position of the follower i(i=1,2,…,N); XL≔[ϕM+1T,…,ϕNT]T; d(x,M) denotes the distance from a point x∈R2 to a set M∈R2; that is, (23)dx,M≔infy∈M⁡x-y.Lemma 12.
Under Assumption4, all the eigenvalues of L1 have positive real parts. Furthermore, each of -L1-1L2 is nonnegative, and all the row sums of this matrix are equal to 1 [41].Remark 13.
LetXL(t)≔[ϕM+1T(t),…,ϕNT(t)]T be the positions of the dynamic leaders. From Definition 2 and Assumption 4, Lemma 12 implies that every point of -(L1-1L2⊗I2)XL(t) is in the convex hull spanned by the leaders XL(t).With this previous notation, the control problem under study can be formulated as below.Control Objective. Consider the model given by (6) and (7) under Assumptions 3–5, and 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 such that(24)limt→∞⁡pit-pdit≤σi,where σi is a positive constant, pi(t)∈R2 denotes the actual position of follower i, and pdi(t)∈R2 denotes the desired position of follower i which satisfies(25)Pdt=-L1-1L2⊗I2XLt,where Pd(t)=[pd1T(t),…,pdiT(t),…,pdMT(t)]T.Remark 14.
Because of the movement of the ship affected by the ocean environment and the filtering error induced by FOCF, the tracking errorpi-pdi cannot converge to zero but can be uniformly ultimately bounded by σi, which will be discussed in detail in the next part.


## 3. Control Design

### 3.1. Kinematic Loop Design

Step 1.
First, to prepare for the design, the following error variables are introduced:(26)eUi0=Ui-αUi,eψTi0=ψTi-αψTi,where αUi and αψTi denote the virtual control laws of UTi and ψTi, respectively. Then, it is convenient to expand (6) into(27)p˙i=Vi=αVi+Vi-αVi=αVi+Φ1ψTieUi0+Φ2αψTiΦ3eψTi0αUi,where(28)pi=xi,yiT,Vi=UTicos⁡ψTi,UTisin⁡ψTiT,αVi=αUicos⁡αψTi,αUisin⁡αψTiT,Φ1·=cos⁡·,sin⁡·T,Φ2·=cos⁡·-sin⁡·sin⁡·cos⁡·,Φ3·=cos⁡·-1,sin⁡·T.It is obvious that αVi is the virtual control law for (27) which will be specified later. To solve the control problem (24), αVi based on its local states and the information from neighbors and a portion of the leaders is designed as below:(29)αVi=1∑j=1Naij∑j=1Maijp˙j+∑j=M+1Naijϕ˙j-kci∑j=1Naijpj+kci∑j=1Maijpj+kci∑j=M+1Naijϕj,where kci>0 is a constant. To make the presentation clearer, the following notations are used:(30)L1-1=bijM×M,L1-1L2=cijM×N-M,di=∑j=1Naij,sit=∑j=1Maijpi-pj+∑j=M+1Naijpi-ϕj.Then, the following lemma is presented.Lemma 15.
Consider the model described by (27) canceling the terms Vi-αVi, with the virtual control law (29). If the proposed control system satisfies Assumptions 4 and 5, the following properties can be given: (1) the control objective (24) is achieved; that is, pi(t)-pdi(t), as t→∞, where pdi(t)∈R2 is defined as (25); (2) the positive constants ρ1i and ρ2i exist, such that αVi≤ρ1i and α˙Vi≤ρ2i for all t≥t0, respectively; (3) the norm αVi is strictly positive for all t≥t0 if the control parameter satisfies(31)kci<∑j=M+1Ncijd2j∑j=1Mbijsjt0,where i=1,2,…,M.Proof.
This proof is divided into three parts.
(1) By substituting (29) into (27), canceling the term Vi-αVi, and multiplying both sides of (27) by di, one obtains(32)∑j=1Naijp˙i-∑j=1Maijp˙j-∑j=M+1Naijϕ˙j=-kci∑j=1Naijpi-∑j=1Maijpj-∑j=M+1Naijϕj.From (32), it follows that(33)∑j=1Maijp˙i-p˙j+∑j=M+1Naijp˙i-ϕ˙j=-kci∑j=1Maijpi-pj+∑j=M+1Naijpi-ϕj.Hence, defining P(t)=[p1T,…,piT,…,pMT]T, (33) can be rewritten in a matrix form as below:(34)L1⊗I2P˙t+L2⊗I2X˙Lt=-KcL1⊗I2×Pt+L2⊗I2XLt,where I2=diag⁡(1,1) and Kc=diag⁡(k1c,k2c,…,kMc). From (34), it can be observed that (L1⊗I2)P(t)+(L2⊗I2)XL(t) exponentially converges to zero as t→∞. Because L1-1 exists [42], P(t) converges to -(L1-1L2⊗I2)XL(t) as t→∞; that is, the objective (24) is reached.
(2) Multiplying both sides of (34) with L1-1⊗I2 results in the following matrix form:(35)P˙t+L1-1L2⊗I2X˙Lt=-KcL1-1⊗I2L1⊗I2×Pt+L2⊗I2XLt.From (30) and (35), it follows that(36)p˙it=αVi=-∑j=M+1Ncijϕ˙jt-kci∑j=1Mbijsjt=-∑j=M+1Ncijϕ˙jt-kci∑j=1Mbijsjt0e-kjt.By use of the triangle inequality, one obtains the upper bounds of the virtual control and its derivative(37)αVi≤∑j=M+1Ncijd2j+kci∑j=1Mbijsjt0≔ρ1iα˙Vi≤∑j=M+1Ncijbij+kci∑j=1Mkjsjt0≔ρ2i,where ρ1i and ρ2i are positive constants. Thus, αVi and α˙Vi are bounded for all t≥0.
(3) Then, with the aid of the triangle inequality, the norm of αVi can be expressed as follows:(38)αVi≥∑j=M+1Ncijϕ˙ijt-kci∑j=1Mbijsjte-kjt≥∑j=M+1Ncijd1j-kci∑j=1Mbijsjt0≔ρ3i,where ρ3i is a constant. Clearly, if condition (31) holds, it would imply the strict positiveness of the norm of αVi for all t≥0. This calculation completes this proof.Remark 16.
To facilitate the analysis of the properties of the proposed containment control law (29), the tracking error variable Vi-αVi is omitted, which will be considered in the following BP design procedure.At this point, one should also notice that the design of this control system will utilize the virtual control ofαUi and αψTi in the following design steps. To facilitate the computation of the virtual control, let αVi in (29) be expressed as(39)αVi=ηxi,ηyiT,where(40)ηxi=1di∑j=1Maijx˙j+∑j=M+1Naijx˙dj-kci∑j=1Naijxj+kci∑j=1Maijxj+kci∑j=M+1Naijxdj,ηyi=1di∑j=1Maijy˙j+∑j=M+1Naijy˙dj-kci∑j=1Naijyj+kci∑j=1Maijyj+kci∑j=M+1Naijydj.Then, combing with (28), (29), and (39), the expressions of αUi and αψTi can be given by(41)αUi=ηxi2+ηyi2,αψTi=atan⁡2ηyi,ηxi.Lemma 17.
ForαUi and αψTi described by (41), if kci in (36) satisfies condition (31), then the positive constants B1i, D1i, B2i, and D2i exist, such that(42)αUi≤B1i,α˙Ui≤D1i,αψTi≤B2i,αψTi≤D2i.Proof.
From (28), αUi is bounded by(43)αUi=αVi≤ρ1i≔B1i.Differentiating both sides of the first equation of (41) gives(44)α˙Ui=αViTα˙Vi2αUi.By use of Lemma 9, one can obtain(45)α˙Ui≤0.251+α˙Vi≤0.251+ρ2i≕D1i.According to the expression of αψTi in (41), αψTi is bounded by(46)αψTi≤π≔B2i.As inequality (45), α˙ψTi satisfies the following:(47)α˙ψTi≤α˙ViαUi≤ρ2iρ3i≔D2i.This calculation completes this proof.Step 2.
In this work, the BP method is applied to the strict-feedback nonlinear system of (6) and (7). However, it uses the derivatives of the virtual control such as αUi and αψTi, which increases the complexity of calculations and needs the acceleration information from its neighbors. To avoid the problems described previously, FOCF is introduced to estimate each virtual control law and its derivative [30, 43].To facilitate the expressions of the variables involved in filtered BP method, a particular notation is introduced in this paper. For example, letαUi go through FOCF depicted in Figure 2 to obtain αUic and α˙Uic, which represent the estimated values of αUi and α˙Ui, respectively. Similarly, we can also define the variables αψTic, α˙ψTic, αric, and α˙ric. For the sake of the design work, the following tracking errors are defined as(48)eUi=Ui-αUic,eψTi=ψTi-αψTic,eri=ri-αric.Figure 2
Configuration of FOCF.Remark 18.
It is noted the boundedness ofαUi, α˙Ui, αψTi, and α˙ψTi ensures the boundedness of αUic, α˙Uicα˙Uic, and α˙ψTic by Lemma 9. According to Lemma 17, αUiα˙Ui, αψTi, and α˙ψTi are bounded by B1i, D1i, B2i, and D2i, respectively, and then the constants B3i, D3i, B4i, and B4i exist and satisfy(49)αUic≤B3i,α˙Uic≤D3i,αψTic≤B4i,α˙ψTic≤D4i.With (48) applied, the kinematic model (6) can be rewritten as(50)p˙i=Vi=αVi+Vi-Vic+Vic-αVi=αVi+Φ1ψTieUi+Φ2αψTicΦ3eψiαUic+Vic-αVi,where pi, Vi, αVi, Φ1(·), Φ2(·), and Φ3(·) are defined as (28), Vic=[αUiccos⁡αψTic,αUicsin⁡αψTic]T. Then, substituting (29) into (50) and multiplying both sides of (50) with di give(51)s˙i=-kcisi+∑j=1NaijΦ1ψTieUi+Φ2αψTicΦ3eψTi×αUic+∑j=1NaijVic-αVi.Clearly, if the last two terms converge to zero, si exponentially converges to zero. To prepare for the following step, define the compensating tracking error vector as(52)Esi=si-ξsi,EUi=eUi-ξUi,EψTi=eψTi-ξψTi,Eri=eri-ξri,where ξUi, ξψTi, and ξri denote the compensating signals of eUi, eψTi, and eri, respectively. In order to remove the effect of the nonlinear term Vic-αVi in (51), a compensating signal is selected as(53)ξ˙si=-kciξsi+∑j=1NaijΦ1ψTiξUi+Φ2αψTiΦ3eψTieψTi×αUiξψTi+∑j=1NaijVic-αVi,where the initial conditions are ξsi=[0;0] and ξUi=b0exp⁡(-kUit). From (51), (52), and (53), one obtains(54)E˙si=-kciEsi+∑j=1NaijΦ1ψTiEUi+Φ2αψTi×Φ3eψTieψTiαUiEψTi,noting that the last nonlinear term of (54) will be compensated in the next step. Then, considering the first LFC(55)V1=0.5∑i=1MEsiTEsi.Taking the time derivative of V1 along the solution of (54) gives(56)V˙1=-kci∑i=1MEsi2+∑i=1M∑j=1MaijEsiTΦ1ψTiEUi+∑i=1M∑j=1MaijEsiTΦ2αψTiΦ3eψTieψTiαUiEψTi.Step 3.
At this step,ri will be considered as a control input to stabilize the tracking error eUi. With notation (48), the kinematic model for ψTi can be rewritten as below:(57)ψ˙Ti=ri+β˙i=αri+eri+αric-αri+β˙iwhere αri is the virtual control for ri. To stabilize the tracking error eψTi in (48) and to compensate the last term for (56), the virtual control law αri is designed as(58)αri=α˙ψTic-kψTieψTi-∑j=1NaijEsiTΦ2αψTiΦ3eψTieψTi×αUi-β˙i,where kψTi>0 is a constant. Then, from (34) and (35), it follows that(59)e˙ψTi=-kψTieψTi-∑j=1NaijEsiTΦ2αψTiΦ3eψTieψTi×αUi+eri+αric-αri.Select the compensating signal for eψTi as(60)ξ˙ψTi=-kψTiξψTi+ξri+αric-αri,where ξψTi(0)=0. By the definition of EψTi in (52) and (59) and (60), the compensating tracking error EψTi can be expressed as(61)E˙ψTi=-kψTiEψTi-∑j=1NaijEsiTΦ2αψTiΦ3eψTieψTiαUi+Eri.Consider the second LFC:(62)V2=V1+0.5∑i=1MEψTi2whose time derivative along (56) and (61) is(63)V˙2=V˙1+∑i=1MEψTiE˙ψTi=-∑i=1MkiEsi2-∑i=1MkψTiEψTi2+∑i=1M∑j=1NaijEsiT×Φ1ψTiEUi-∑i=1MEψTiEri.It should be noted that the last two terms of (63) will be compensated in the dynamic design.


### 3.2. Dynamic Loop Design

This section aims to design the robust control inputsFui and Fri which make the compensated tracking errors Eui and Eri asymptotically stable.Step 1 (surge control design).
The definition ofEri in (52) is chosen as the sliding surface:(64)Eri=eri-ξri.Differentiating both sides of (64) along (7), one can obtain Eri dynamics as(65)E˙ri=e˙ri-ξ˙ri=r˙i-α˙ric-ξ˙ri=1m33im11iuivi-m22iuivi-driri-dr2iriri-dvriri+τwri+Fri-m33iα˙ric-m33iξ˙ri.To ensure that Eri is always equal to zero, the equivalent control input Freqi can be obtained by solving E˙ri=0 without the parameter uncertainties and external disturbances. Thus, Freqi can be chosen as(66)Freqi=-m^11iuivi+m^22iuivi+d^riri+d^r2iriri+d^vriri+m^33iα˙ric+m^33iξ˙ri,where “d^” denotes the estimated ship parameters. It could be easily found that the equivalent control input cannot guarantee Eri=0 when the EDPU are considered; to eliminate the effect of the uncertainties of the control system and to ensure the convergence of Eri to zero, an additional control input Fradi is introduced:(67)Fradi=-q2iEri-ηrisign⁡Eri,where q2i is a positive constant and ηui is the control parameter, determined from the following bounds of the ship parameters in (7):(68)m11i-m^11i≤M11,m22i-m^22i≤M22,dri-d^ri≤Dr,dr2i-d^r2i≤Dr2,dvri-d^vri≤Dvr.Moreover, to compensate for the last term of (63), a compensated control input Frcpi is set to be(69)Frcpi=-EψTi.Combined with the previously mentioned control input, the actual yaw control input Fri can be chosen as(70)Fri=Freqi+Fradi+Frcpi.To determine ηui, consider the LFC for Eri as(71)Vr=0.5∑i=1Mm33iEri2whose time derivative along (65) and (70) is(72)V˙r=∑i=1Mm33iEriE˙ri=∑i=1MErim11i-m^11iuivi+m^22i-m22iuivi+d^ri-driri+d^r2i-dr2iriri+d^vri-dvriri+τwri+m^33i-m33iα˙ric+ξ˙ri-ηrisign⁡Eri-q2iEri-EψTi≤∑i=1MEriM11+M22uivi+Drri+Dr2ri2+DvrriM33α˙ric+ξ˙ri+τ-wri-ηri-∑i=1Mq2iEri2+EriEψTi.Then, a reasonable choice for ηui can be given as(73)ηri=M11uivi+M22uivi+Drri+Dr2ri2+Dvrri+τ-wr+M33α˙ric+ξ˙ri.Considering the third LFC V3=V2+Vr and taking the time derivative of V3 along (63) give(74)V˙3=V˙2+V˙r≤-∑i=1MkiEsi2-∑i=1MkψTiEψTi2-∑i=1Mq2iEri2+∑i=1M∑j=1NaijEsiTΦ1ψTiEUi.Step 2 (yaw control design).
Similar to Step1, a sliding surface for EUi is introduced:(75)EUi=eUi-ξUi.Taking the time derivative of EUi along (7) results in(76)E˙Ui=e˙Ui-ξ˙Ui=U˙Ti-α˙Uic-ξ˙Ui=cos⁡βim11im22iviri-duiui-du2iuiui+cos⁡βim11iτwui+sin⁡βim22i-m11iuiri-dvivi-dv2ivivi-duriri+sin⁡βim22iτwvi+cos⁡βim11iFui-α˙Uic-ξ˙Ui.Similar to the previous step, the equivalent control input Fueqi can be designed as(77)Fueqi=-m^22iviri+d^uiui+d^u2iuiui+sin⁡βicos⁡βim^11i2m^22iuiri+m^11id^vim^22ivi+m^11id^v2im^22ivivi+m^11id^vrim^22iri+1cos⁡βim^11iα˙Uic+ξ˙Ui,where “d^” denotes the estimated ship parameters. To compensate for the last term of inequality (74) and reject the uncertainties, the actual surge control input can be chosen as(78)Fui=Fueqi+Fuadi,(79)Fuadi=-1cos⁡βi∑j=1NaijEsiTΦ1ψTi+q1iEui+ηuisign⁡Eui,where q1i>0 is a constant; ηui can be determined by the following bounds of the ship parameters in (7):(80)dui-d^ui≤Du,du2i-d^u2i≤Du2,m33i-m^33i≤M33,dvi-d^vi≤Dv,dv2i-d^v2i≤Dv2,m11im22i≤H0,m11i2m22i-m^11i2m^22i≤Hur,m11idvim22i-m^11id^vim^22i≤Hv,m11idv2im22i-m^11id^v2im^22i≤Hv2,m11idvrim22i-m^11id^vrim^22i≤Hr,and, to determine ηui, we consider the LFC for EUi as(81)VU=0.5∑i=1Mm11iEUi2,whose time derivative along (76) and (78) is(82)V˙U=∑i=1MEUicos⁡βim22i-m^22iviri+d^ui-duiui+d^u2i-du2iuiui+cos⁡βiτwui+sin⁡βim11im22iτwvi+sin⁡βim^11i2m^22i-m11i2m22iuiri+m^11id^vim^22i-m11idvim22ivi+m^11id^v2im^22i-m11idv2im22ivivi+m^11id^v2im^22i-m11idv2im22iri+m^11i-m11iα˙Ui+ξ˙Ui-ηuisign⁡Eui-∑i=1Mq1iEui2-∑i=1M∑j=1MaijEsiTΦ1ψTiEUi.According to (80) and (82), computing the first-order derivative of VU results in(83)V˙U≤∑i=1MEUiM11viri+Duui+Du2ui2+τ-wui+H0τ-wvi+Huruiri+Hvvi+Hv2vi2+Hrri+M11α˙Ui+ξ˙Ui-ηui-∑i=1Mq1iEUi2-∑i=1M∑j=1MaijEsiTΦ1ψTiEUi.To stabilize EUi at the origin, ηui can be defined as(84)ηui=M11viri+Duui+Du2ui2+Huruiri+Hvvi+Hv2vi2+Hrri+τ-wui+H0τ-wvi+M11α˙Ui+ξ˙Ui.Substituting (84) into (83), one obtains(85)V˙U≤-∑i=1Mq1iEUi2-∑i=1M∑j=1NaijEsiTΦ1ψTiEUi.Consider the overall LFC V4=V3+VU; take the time derivative of V4 along (74) and (85) gives(86)V˙4≤V˙3+V˙U≤-∑i=1MkiEsi2-∑i=1MkψTiEψTi2-∑i=1Mq2iEri2-∑i=1Mq1iEUi2.


### 3.3. Stability Analysis of Closed-Loop System

This section consists of four steps.Step 1 (boundedness ofEsi and EUi).
By using equalities (68) and (80), (86) can be derived as(87)V˙4≤-∑i=1MkciEsi2-∑i=1MkψTiEψi2-∑i=1Mq2im^33i+M33m33iEri2-∑i=1Mq1im^11i+M11m11iEUi2≤-ρV4,where ρ=min⁡2kci,2kψi,2q2i/m^33+M33,2q1i/m^11+M11.
Then, one can obtain(88)0.5∑i=1MEsi2≤V4t≤V40e-ρt.From equality (88), one can conclude that EU(t), Esi(t)≤χ (where χ=2V40) for all t≥0.Step 2 (boundedness ofβi).
The definition ofEUi in (52) and the definition of eUi in (48) can be rewritten as(89)EUi=Ui-αUic-ζUi.Then, the bounds of EUi, αUic, and ζUi give(90)Ui-B3i-b0≥-χ.It is obviously concluded that the strict positiveness of Ui can be guaranteed if ζUi satisfies the condition(91)b0>χ.Thus, βi<0.5π; that is, cos⁡βi≠0, for all t>0. Because the determinant of Fui in (78) is given by the term cos⁡βi, condition (91) can guarantee that Fui are well defined for all t>0.Remark 19.
Notice that, withβi being bounded by 0.5π, the constants l1i and l2i exist, such that β˙i≤l1i, β¨i≤l2i all the time.Step 3 (boundedness ofξsi).
Prior to analyzing the boundednessξsi the boundedness of αri, αric should be discussed first. From the definition of αri in (58) and the Young equality, one obtains(92)αri≤α˙ψTic+kψTieψTi+12∑j=1NaijEsi2+αUi2+β˙i≤D4i+kψTi0.5π+B4i+12∑j=1Naijχ2+B1i2+l1i2≕B5iwhich implies the constant D5i>0 exists which satisfies α˙ri≤D5i. Combining with Lemma 9, it is seen that αri-αric≤B6i, where B5i>0 is a constant. Then, consider the LFC as(93)V˙ξ=∑i=1MξsiTξ˙si+ξUiξ˙Ui+ξψTiξ˙ψTi≤-∑i=1Mkciξsi2+∑i=1M∑j=1Naij32ξsi2+∑i=1M∑j=1Naij12ξUi2+∑i=1M∑j=1Naij12αUi2ξψTi2+∑i=1M∑j=1Naij12Vic2+αVi2-∑i=1MkUiξUi2-∑i=1MkψTiξψTi2+∑i=1M12ξψTi2+αric-αri2≤-2KVξ+Q,where(94)K=min⁡kci-∑j=1N1.5aij,kUi-∑j=1N0.5aij,kψTi-0.5-∑j=1N0.5aijB1i2,Q=∑j=1N0.5aijB1i2+B3i2+∑j=1N0.5B6i2and kci, kUi, and kψTi satisfy(95)kci>∑j=1N1.5aij,kUi>∑j=1N0.5aij,kψTi>0.5+∑j=1N0.5aijB1i2.With equality (93), one obtains(96)Vξt≤Vξ0-Q2Ke-2Kt+Q2K≔κ.It could easily be found from equality (96) that ξsi≤2κ for all t≥0.Step 4 (implementation of the control objective).
By noting that, with the boundedness ofESi, ξSi, the signal si in (30) is bounded by χ+2κ, define the vector as(97)St=s1Tt,…,sMTtT=L1⊗I2Pt+L2⊗I2XLt.Then, the upper bound for the norm of S(t) can be obtained as below:(98)St≤∑i=1Msit≤Mχ+2κ.Because L1 is invertible, equality (98) means that(99)L1-1⊗I2St=Pt-L1-1L2⊗I2XLt≤λminL1-1Mχ+2κ≔Smax.Thus, pi-pdi is bounded by Smax for all t≥0, where pdi is defined in (25). Subsequently, the containment control objective (24) is solved.Theorem 20.
Consider a group of ships given by (6) and (7), and suppose Assumptions 3–5 hold. For each follower, consider the robust cooperative controllers (70) and (78) and FOCF (11); then, the tracking errors pi-pdi are globally uniformly ultimately bounded (GUUB), if the control parameters satisfy conditions (31), (91), and (95).Remark 21.
In [16, 17, 44], the yaw velocity is required to be nonzero, which means that a straight-line reference cannot be tracked for ships. However, the PE condition is very restrictive from a practical point of view. By using a polar coordinate transformation, the proposed control strategy for ships can track not only straight-line paths but curved-line paths as well.Remark 22.
It is important to note that the traditional DSC method estimates the value of the virtual control law and its derivative only through first-order filter and fails to adjust its estimate through feedback. By contrast, FOCF method introduces compensation tracking error signal on the basis of traditional DSC method, realizes the real-time correction of virtual control law, and greatly improves the accuracy of the estimation.Remark 23.
The authors in [45] also adopt CFBP method to design the trajectory coordinated control strategy and use SOCF to estimate the virtual control law in the BP design procedure online. However, SOCF belongs to a class of second-order systems, and it is more tedious to implement than FOCF. It should be mentioned that there are many deduction errors in [45]. In addition, the authors in [45] adopt the adaptive method to estimate the hydrodynamic parameters online in the dynamic design, but it is difficult to implement in practice. Therefore, in the paper, we refer to the design structure framework of literature [45], use first-order filter to estimate the virtual control laws, solve the containment control objective, and analyze the stability of the closed-loop system by using the input-output boundedness property of FOCF. Moreover, in the dynamic design, upper-to-up sliding mode method is used to design the actual control inputs of ship. Rather than online estimation of hydrodynamic parameters, the method only needs to use the boundedness of hydrodynamic parameters and external disturbances and has the advantages of simple structure and strong robustness.


### 3.4. Stability Analysis of Sway Motion

Since there is no available control input in the sway direction of the individual ship (3), it is necessary to analyze the boundedness of the sway velocity vi. Consider the following LFC:(100)V5=12m22ivi2.From (100), the time derivative of V5 can be obtained as(101)V˙5=m22iviv˙i≤-dvi+dv2ivi2-uiri+dvririvi+τ-wvivi.Becausem11i, dvri, ui, and vi are bounded as shown in Section 3.2, one can obtain(102)V˙5≤-dvivi2+φvi,where φ=max⁡m11iuiri+dvriri+τ-wvi. It could be easily found that if dvivi>φ, V˙5<0. That is to say, the sway velocity vi is uniformly ultimately bounded and will eventually converge to the invariant set Svi=vi∈R∣dvvi≤φ.


## 3.1. Kinematic Loop Design

Step 1.
First, to prepare for the design, the following error variables are introduced:(26)eUi0=Ui-αUi,eψTi0=ψTi-αψTi,where αUi and αψTi denote the virtual control laws of UTi and ψTi, respectively. Then, it is convenient to expand (6) into(27)p˙i=Vi=αVi+Vi-αVi=αVi+Φ1ψTieUi0+Φ2αψTiΦ3eψTi0αUi,where(28)pi=xi,yiT,Vi=UTicos⁡ψTi,UTisin⁡ψTiT,αVi=αUicos⁡αψTi,αUisin⁡αψTiT,Φ1·=cos⁡·,sin⁡·T,Φ2·=cos⁡·-sin⁡·sin⁡·cos⁡·,Φ3·=cos⁡·-1,sin⁡·T.It is obvious that αVi is the virtual control law for (27) which will be specified later. To solve the control problem (24), αVi based on its local states and the information from neighbors and a portion of the leaders is designed as below:(29)αVi=1∑j=1Naij∑j=1Maijp˙j+∑j=M+1Naijϕ˙j-kci∑j=1Naijpj+kci∑j=1Maijpj+kci∑j=M+1Naijϕj,where kci>0 is a constant. To make the presentation clearer, the following notations are used:(30)L1-1=bijM×M,L1-1L2=cijM×N-M,di=∑j=1Naij,sit=∑j=1Maijpi-pj+∑j=M+1Naijpi-ϕj.Then, the following lemma is presented.Lemma 15.
Consider the model described by (27) canceling the terms Vi-αVi, with the virtual control law (29). If the proposed control system satisfies Assumptions 4 and 5, the following properties can be given: (1) the control objective (24) is achieved; that is, pi(t)-pdi(t), as t→∞, where pdi(t)∈R2 is defined as (25); (2) the positive constants ρ1i and ρ2i exist, such that αVi≤ρ1i and α˙Vi≤ρ2i for all t≥t0, respectively; (3) the norm αVi is strictly positive for all t≥t0 if the control parameter satisfies(31)kci<∑j=M+1Ncijd2j∑j=1Mbijsjt0,where i=1,2,…,M.Proof.
This proof is divided into three parts.
(1) By substituting (29) into (27), canceling the term Vi-αVi, and multiplying both sides of (27) by di, one obtains(32)∑j=1Naijp˙i-∑j=1Maijp˙j-∑j=M+1Naijϕ˙j=-kci∑j=1Naijpi-∑j=1Maijpj-∑j=M+1Naijϕj.From (32), it follows that(33)∑j=1Maijp˙i-p˙j+∑j=M+1Naijp˙i-ϕ˙j=-kci∑j=1Maijpi-pj+∑j=M+1Naijpi-ϕj.Hence, defining P(t)=[p1T,…,piT,…,pMT]T, (33) can be rewritten in a matrix form as below:(34)L1⊗I2P˙t+L2⊗I2X˙Lt=-KcL1⊗I2×Pt+L2⊗I2XLt,where I2=diag⁡(1,1) and Kc=diag⁡(k1c,k2c,…,kMc). From (34), it can be observed that (L1⊗I2)P(t)+(L2⊗I2)XL(t) exponentially converges to zero as t→∞. Because L1-1 exists [42], P(t) converges to -(L1-1L2⊗I2)XL(t) as t→∞; that is, the objective (24) is reached.
(2) Multiplying both sides of (34) with L1-1⊗I2 results in the following matrix form:(35)P˙t+L1-1L2⊗I2X˙Lt=-KcL1-1⊗I2L1⊗I2×Pt+L2⊗I2XLt.From (30) and (35), it follows that(36)p˙it=αVi=-∑j=M+1Ncijϕ˙jt-kci∑j=1Mbijsjt=-∑j=M+1Ncijϕ˙jt-kci∑j=1Mbijsjt0e-kjt.By use of the triangle inequality, one obtains the upper bounds of the virtual control and its derivative(37)αVi≤∑j=M+1Ncijd2j+kci∑j=1Mbijsjt0≔ρ1iα˙Vi≤∑j=M+1Ncijbij+kci∑j=1Mkjsjt0≔ρ2i,where ρ1i and ρ2i are positive constants. Thus, αVi and α˙Vi are bounded for all t≥0.
(3) Then, with the aid of the triangle inequality, the norm of αVi can be expressed as follows:(38)αVi≥∑j=M+1Ncijϕ˙ijt-kci∑j=1Mbijsjte-kjt≥∑j=M+1Ncijd1j-kci∑j=1Mbijsjt0≔ρ3i,where ρ3i is a constant. Clearly, if condition (31) holds, it would imply the strict positiveness of the norm of αVi for all t≥0. This calculation completes this proof.Remark 16.
To facilitate the analysis of the properties of the proposed containment control law (29), the tracking error variable Vi-αVi is omitted, which will be considered in the following BP design procedure.At this point, one should also notice that the design of this control system will utilize the virtual control ofαUi and αψTi in the following design steps. To facilitate the computation of the virtual control, let αVi in (29) be expressed as(39)αVi=ηxi,ηyiT,where(40)ηxi=1di∑j=1Maijx˙j+∑j=M+1Naijx˙dj-kci∑j=1Naijxj+kci∑j=1Maijxj+kci∑j=M+1Naijxdj,ηyi=1di∑j=1Maijy˙j+∑j=M+1Naijy˙dj-kci∑j=1Naijyj+kci∑j=1Maijyj+kci∑j=M+1Naijydj.Then, combing with (28), (29), and (39), the expressions of αUi and αψTi can be given by(41)αUi=ηxi2+ηyi2,αψTi=atan⁡2ηyi,ηxi.Lemma 17.
ForαUi and αψTi described by (41), if kci in (36) satisfies condition (31), then the positive constants B1i, D1i, B2i, and D2i exist, such that(42)αUi≤B1i,α˙Ui≤D1i,αψTi≤B2i,αψTi≤D2i.Proof.
From (28), αUi is bounded by(43)αUi=αVi≤ρ1i≔B1i.Differentiating both sides of the first equation of (41) gives(44)α˙Ui=αViTα˙Vi2αUi.By use of Lemma 9, one can obtain(45)α˙Ui≤0.251+α˙Vi≤0.251+ρ2i≕D1i.According to the expression of αψTi in (41), αψTi is bounded by(46)αψTi≤π≔B2i.As inequality (45), α˙ψTi satisfies the following:(47)α˙ψTi≤α˙ViαUi≤ρ2iρ3i≔D2i.This calculation completes this proof.Step 2.
In this work, the BP method is applied to the strict-feedback nonlinear system of (6) and (7). However, it uses the derivatives of the virtual control such as αUi and αψTi, which increases the complexity of calculations and needs the acceleration information from its neighbors. To avoid the problems described previously, FOCF is introduced to estimate each virtual control law and its derivative [30, 43].To facilitate the expressions of the variables involved in filtered BP method, a particular notation is introduced in this paper. For example, letαUi go through FOCF depicted in Figure 2 to obtain αUic and α˙Uic, which represent the estimated values of αUi and α˙Ui, respectively. Similarly, we can also define the variables αψTic, α˙ψTic, αric, and α˙ric. For the sake of the design work, the following tracking errors are defined as(48)eUi=Ui-αUic,eψTi=ψTi-αψTic,eri=ri-αric.Figure 2
Configuration of FOCF.Remark 18.
It is noted the boundedness ofαUi, α˙Ui, αψTi, and α˙ψTi ensures the boundedness of αUic, α˙Uicα˙Uic, and α˙ψTic by Lemma 9. According to Lemma 17, αUiα˙Ui, αψTi, and α˙ψTi are bounded by B1i, D1i, B2i, and D2i, respectively, and then the constants B3i, D3i, B4i, and B4i exist and satisfy(49)αUic≤B3i,α˙Uic≤D3i,αψTic≤B4i,α˙ψTic≤D4i.With (48) applied, the kinematic model (6) can be rewritten as(50)p˙i=Vi=αVi+Vi-Vic+Vic-αVi=αVi+Φ1ψTieUi+Φ2αψTicΦ3eψiαUic+Vic-αVi,where pi, Vi, αVi, Φ1(·), Φ2(·), and Φ3(·) are defined as (28), Vic=[αUiccos⁡αψTic,αUicsin⁡αψTic]T. Then, substituting (29) into (50) and multiplying both sides of (50) with di give(51)s˙i=-kcisi+∑j=1NaijΦ1ψTieUi+Φ2αψTicΦ3eψTi×αUic+∑j=1NaijVic-αVi.Clearly, if the last two terms converge to zero, si exponentially converges to zero. To prepare for the following step, define the compensating tracking error vector as(52)Esi=si-ξsi,EUi=eUi-ξUi,EψTi=eψTi-ξψTi,Eri=eri-ξri,where ξUi, ξψTi, and ξri denote the compensating signals of eUi, eψTi, and eri, respectively. In order to remove the effect of the nonlinear term Vic-αVi in (51), a compensating signal is selected as(53)ξ˙si=-kciξsi+∑j=1NaijΦ1ψTiξUi+Φ2αψTiΦ3eψTieψTi×αUiξψTi+∑j=1NaijVic-αVi,where the initial conditions are ξsi=[0;0] and ξUi=b0exp⁡(-kUit). From (51), (52), and (53), one obtains(54)E˙si=-kciEsi+∑j=1NaijΦ1ψTiEUi+Φ2αψTi×Φ3eψTieψTiαUiEψTi,noting that the last nonlinear term of (54) will be compensated in the next step. Then, considering the first LFC(55)V1=0.5∑i=1MEsiTEsi.Taking the time derivative of V1 along the solution of (54) gives(56)V˙1=-kci∑i=1MEsi2+∑i=1M∑j=1MaijEsiTΦ1ψTiEUi+∑i=1M∑j=1MaijEsiTΦ2αψTiΦ3eψTieψTiαUiEψTi.Step 3.
At this step,ri will be considered as a control input to stabilize the tracking error eUi. With notation (48), the kinematic model for ψTi can be rewritten as below:(57)ψ˙Ti=ri+β˙i=αri+eri+αric-αri+β˙iwhere αri is the virtual control for ri. To stabilize the tracking error eψTi in (48) and to compensate the last term for (56), the virtual control law αri is designed as(58)αri=α˙ψTic-kψTieψTi-∑j=1NaijEsiTΦ2αψTiΦ3eψTieψTi×αUi-β˙i,where kψTi>0 is a constant. Then, from (34) and (35), it follows that(59)e˙ψTi=-kψTieψTi-∑j=1NaijEsiTΦ2αψTiΦ3eψTieψTi×αUi+eri+αric-αri.Select the compensating signal for eψTi as(60)ξ˙ψTi=-kψTiξψTi+ξri+αric-αri,where ξψTi(0)=0. By the definition of EψTi in (52) and (59) and (60), the compensating tracking error EψTi can be expressed as(61)E˙ψTi=-kψTiEψTi-∑j=1NaijEsiTΦ2αψTiΦ3eψTieψTiαUi+Eri.Consider the second LFC:(62)V2=V1+0.5∑i=1MEψTi2whose time derivative along (56) and (61) is(63)V˙2=V˙1+∑i=1MEψTiE˙ψTi=-∑i=1MkiEsi2-∑i=1MkψTiEψTi2+∑i=1M∑j=1NaijEsiT×Φ1ψTiEUi-∑i=1MEψTiEri.It should be noted that the last two terms of (63) will be compensated in the dynamic design.


## 3.2. Dynamic Loop Design

This section aims to design the robust control inputsFui and Fri which make the compensated tracking errors Eui and Eri asymptotically stable.Step 1 (surge control design).
The definition ofEri in (52) is chosen as the sliding surface:(64)Eri=eri-ξri.Differentiating both sides of (64) along (7), one can obtain Eri dynamics as(65)E˙ri=e˙ri-ξ˙ri=r˙i-α˙ric-ξ˙ri=1m33im11iuivi-m22iuivi-driri-dr2iriri-dvriri+τwri+Fri-m33iα˙ric-m33iξ˙ri.To ensure that Eri is always equal to zero, the equivalent control input Freqi can be obtained by solving E˙ri=0 without the parameter uncertainties and external disturbances. Thus, Freqi can be chosen as(66)Freqi=-m^11iuivi+m^22iuivi+d^riri+d^r2iriri+d^vriri+m^33iα˙ric+m^33iξ˙ri,where “d^” denotes the estimated ship parameters. It could be easily found that the equivalent control input cannot guarantee Eri=0 when the EDPU are considered; to eliminate the effect of the uncertainties of the control system and to ensure the convergence of Eri to zero, an additional control input Fradi is introduced:(67)Fradi=-q2iEri-ηrisign⁡Eri,where q2i is a positive constant and ηui is the control parameter, determined from the following bounds of the ship parameters in (7):(68)m11i-m^11i≤M11,m22i-m^22i≤M22,dri-d^ri≤Dr,dr2i-d^r2i≤Dr2,dvri-d^vri≤Dvr.Moreover, to compensate for the last term of (63), a compensated control input Frcpi is set to be(69)Frcpi=-EψTi.Combined with the previously mentioned control input, the actual yaw control input Fri can be chosen as(70)Fri=Freqi+Fradi+Frcpi.To determine ηui, consider the LFC for Eri as(71)Vr=0.5∑i=1Mm33iEri2whose time derivative along (65) and (70) is(72)V˙r=∑i=1Mm33iEriE˙ri=∑i=1MErim11i-m^11iuivi+m^22i-m22iuivi+d^ri-driri+d^r2i-dr2iriri+d^vri-dvriri+τwri+m^33i-m33iα˙ric+ξ˙ri-ηrisign⁡Eri-q2iEri-EψTi≤∑i=1MEriM11+M22uivi+Drri+Dr2ri2+DvrriM33α˙ric+ξ˙ri+τ-wri-ηri-∑i=1Mq2iEri2+EriEψTi.Then, a reasonable choice for ηui can be given as(73)ηri=M11uivi+M22uivi+Drri+Dr2ri2+Dvrri+τ-wr+M33α˙ric+ξ˙ri.Considering the third LFC V3=V2+Vr and taking the time derivative of V3 along (63) give(74)V˙3=V˙2+V˙r≤-∑i=1MkiEsi2-∑i=1MkψTiEψTi2-∑i=1Mq2iEri2+∑i=1M∑j=1NaijEsiTΦ1ψTiEUi.Step 2 (yaw control design).
Similar to Step1, a sliding surface for EUi is introduced:(75)EUi=eUi-ξUi.Taking the time derivative of EUi along (7) results in(76)E˙Ui=e˙Ui-ξ˙Ui=U˙Ti-α˙Uic-ξ˙Ui=cos⁡βim11im22iviri-duiui-du2iuiui+cos⁡βim11iτwui+sin⁡βim22i-m11iuiri-dvivi-dv2ivivi-duriri+sin⁡βim22iτwvi+cos⁡βim11iFui-α˙Uic-ξ˙Ui.Similar to the previous step, the equivalent control input Fueqi can be designed as(77)Fueqi=-m^22iviri+d^uiui+d^u2iuiui+sin⁡βicos⁡βim^11i2m^22iuiri+m^11id^vim^22ivi+m^11id^v2im^22ivivi+m^11id^vrim^22iri+1cos⁡βim^11iα˙Uic+ξ˙Ui,where “d^” denotes the estimated ship parameters. To compensate for the last term of inequality (74) and reject the uncertainties, the actual surge control input can be chosen as(78)Fui=Fueqi+Fuadi,(79)Fuadi=-1cos⁡βi∑j=1NaijEsiTΦ1ψTi+q1iEui+ηuisign⁡Eui,where q1i>0 is a constant; ηui can be determined by the following bounds of the ship parameters in (7):(80)dui-d^ui≤Du,du2i-d^u2i≤Du2,m33i-m^33i≤M33,dvi-d^vi≤Dv,dv2i-d^v2i≤Dv2,m11im22i≤H0,m11i2m22i-m^11i2m^22i≤Hur,m11idvim22i-m^11id^vim^22i≤Hv,m11idv2im22i-m^11id^v2im^22i≤Hv2,m11idvrim22i-m^11id^vrim^22i≤Hr,and, to determine ηui, we consider the LFC for EUi as(81)VU=0.5∑i=1Mm11iEUi2,whose time derivative along (76) and (78) is(82)V˙U=∑i=1MEUicos⁡βim22i-m^22iviri+d^ui-duiui+d^u2i-du2iuiui+cos⁡βiτwui+sin⁡βim11im22iτwvi+sin⁡βim^11i2m^22i-m11i2m22iuiri+m^11id^vim^22i-m11idvim22ivi+m^11id^v2im^22i-m11idv2im22ivivi+m^11id^v2im^22i-m11idv2im22iri+m^11i-m11iα˙Ui+ξ˙Ui-ηuisign⁡Eui-∑i=1Mq1iEui2-∑i=1M∑j=1MaijEsiTΦ1ψTiEUi.According to (80) and (82), computing the first-order derivative of VU results in(83)V˙U≤∑i=1MEUiM11viri+Duui+Du2ui2+τ-wui+H0τ-wvi+Huruiri+Hvvi+Hv2vi2+Hrri+M11α˙Ui+ξ˙Ui-ηui-∑i=1Mq1iEUi2-∑i=1M∑j=1MaijEsiTΦ1ψTiEUi.To stabilize EUi at the origin, ηui can be defined as(84)ηui=M11viri+Duui+Du2ui2+Huruiri+Hvvi+Hv2vi2+Hrri+τ-wui+H0τ-wvi+M11α˙Ui+ξ˙Ui.Substituting (84) into (83), one obtains(85)V˙U≤-∑i=1Mq1iEUi2-∑i=1M∑j=1NaijEsiTΦ1ψTiEUi.Consider the overall LFC V4=V3+VU; take the time derivative of V4 along (74) and (85) gives(86)V˙4≤V˙3+V˙U≤-∑i=1MkiEsi2-∑i=1MkψTiEψTi2-∑i=1Mq2iEri2-∑i=1Mq1iEUi2.


## 3.3. Stability Analysis of Closed-Loop System

This section consists of four steps.Step 1 (boundedness ofEsi and EUi).
By using equalities (68) and (80), (86) can be derived as(87)V˙4≤-∑i=1MkciEsi2-∑i=1MkψTiEψi2-∑i=1Mq2im^33i+M33m33iEri2-∑i=1Mq1im^11i+M11m11iEUi2≤-ρV4,where ρ=min⁡2kci,2kψi,2q2i/m^33+M33,2q1i/m^11+M11.
Then, one can obtain(88)0.5∑i=1MEsi2≤V4t≤V40e-ρt.From equality (88), one can conclude that EU(t), Esi(t)≤χ (where χ=2V40) for all t≥0.Step 2 (boundedness ofβi).
The definition ofEUi in (52) and the definition of eUi in (48) can be rewritten as(89)EUi=Ui-αUic-ζUi.Then, the bounds of EUi, αUic, and ζUi give(90)Ui-B3i-b0≥-χ.It is obviously concluded that the strict positiveness of Ui can be guaranteed if ζUi satisfies the condition(91)b0>χ.Thus, βi<0.5π; that is, cos⁡βi≠0, for all t>0. Because the determinant of Fui in (78) is given by the term cos⁡βi, condition (91) can guarantee that Fui are well defined for all t>0.Remark 19.
Notice that, withβi being bounded by 0.5π, the constants l1i and l2i exist, such that β˙i≤l1i, β¨i≤l2i all the time.Step 3 (boundedness ofξsi).
Prior to analyzing the boundednessξsi the boundedness of αri, αric should be discussed first. From the definition of αri in (58) and the Young equality, one obtains(92)αri≤α˙ψTic+kψTieψTi+12∑j=1NaijEsi2+αUi2+β˙i≤D4i+kψTi0.5π+B4i+12∑j=1Naijχ2+B1i2+l1i2≕B5iwhich implies the constant D5i>0 exists which satisfies α˙ri≤D5i. Combining with Lemma 9, it is seen that αri-αric≤B6i, where B5i>0 is a constant. Then, consider the LFC as(93)V˙ξ=∑i=1MξsiTξ˙si+ξUiξ˙Ui+ξψTiξ˙ψTi≤-∑i=1Mkciξsi2+∑i=1M∑j=1Naij32ξsi2+∑i=1M∑j=1Naij12ξUi2+∑i=1M∑j=1Naij12αUi2ξψTi2+∑i=1M∑j=1Naij12Vic2+αVi2-∑i=1MkUiξUi2-∑i=1MkψTiξψTi2+∑i=1M12ξψTi2+αric-αri2≤-2KVξ+Q,where(94)K=min⁡kci-∑j=1N1.5aij,kUi-∑j=1N0.5aij,kψTi-0.5-∑j=1N0.5aijB1i2,Q=∑j=1N0.5aijB1i2+B3i2+∑j=1N0.5B6i2and kci, kUi, and kψTi satisfy(95)kci>∑j=1N1.5aij,kUi>∑j=1N0.5aij,kψTi>0.5+∑j=1N0.5aijB1i2.With equality (93), one obtains(96)Vξt≤Vξ0-Q2Ke-2Kt+Q2K≔κ.It could easily be found from equality (96) that ξsi≤2κ for all t≥0.Step 4 (implementation of the control objective).
By noting that, with the boundedness ofESi, ξSi, the signal si in (30) is bounded by χ+2κ, define the vector as(97)St=s1Tt,…,sMTtT=L1⊗I2Pt+L2⊗I2XLt.Then, the upper bound for the norm of S(t) can be obtained as below:(98)St≤∑i=1Msit≤Mχ+2κ.Because L1 is invertible, equality (98) means that(99)L1-1⊗I2St=Pt-L1-1L2⊗I2XLt≤λminL1-1Mχ+2κ≔Smax.Thus, pi-pdi is bounded by Smax for all t≥0, where pdi is defined in (25). Subsequently, the containment control objective (24) is solved.Theorem 20.
Consider a group of ships given by (6) and (7), and suppose Assumptions 3–5 hold. For each follower, consider the robust cooperative controllers (70) and (78) and FOCF (11); then, the tracking errors pi-pdi are globally uniformly ultimately bounded (GUUB), if the control parameters satisfy conditions (31), (91), and (95).Remark 21.
In [16, 17, 44], the yaw velocity is required to be nonzero, which means that a straight-line reference cannot be tracked for ships. However, the PE condition is very restrictive from a practical point of view. By using a polar coordinate transformation, the proposed control strategy for ships can track not only straight-line paths but curved-line paths as well.Remark 22.
It is important to note that the traditional DSC method estimates the value of the virtual control law and its derivative only through first-order filter and fails to adjust its estimate through feedback. By contrast, FOCF method introduces compensation tracking error signal on the basis of traditional DSC method, realizes the real-time correction of virtual control law, and greatly improves the accuracy of the estimation.Remark 23.
The authors in [45] also adopt CFBP method to design the trajectory coordinated control strategy and use SOCF to estimate the virtual control law in the BP design procedure online. However, SOCF belongs to a class of second-order systems, and it is more tedious to implement than FOCF. It should be mentioned that there are many deduction errors in [45]. In addition, the authors in [45] adopt the adaptive method to estimate the hydrodynamic parameters online in the dynamic design, but it is difficult to implement in practice. Therefore, in the paper, we refer to the design structure framework of literature [45], use first-order filter to estimate the virtual control laws, solve the containment control objective, and analyze the stability of the closed-loop system by using the input-output boundedness property of FOCF. Moreover, in the dynamic design, upper-to-up sliding mode method is used to design the actual control inputs of ship. Rather than online estimation of hydrodynamic parameters, the method only needs to use the boundedness of hydrodynamic parameters and external disturbances and has the advantages of simple structure and strong robustness.


## 3.4. Stability Analysis of Sway Motion

Since there is no available control input in the sway direction of the individual ship (3), it is necessary to analyze the boundedness of the sway velocity vi. Consider the following LFC:(100)V5=12m22ivi2.From (100), the time derivative of V5 can be obtained as(101)V˙5=m22iviv˙i≤-dvi+dv2ivi2-uiri+dvririvi+τ-wvivi.Becausem11i, dvri, ui, and vi are bounded as shown in Section 3.2, one can obtain(102)V˙5≤-dvivi2+φvi,where φ=max⁡m11iuiri+dvriri+τ-wvi. It could be easily found that if dvivi>φ, V˙5<0. That is to say, the sway velocity vi is uniformly ultimately bounded and will eventually converge to the invariant set Svi=vi∈R∣dvvi≤φ.


## 4. Simulation Results

To verify the effectiveness of the proposed robust cooperative scheme, a networked group that consists of the following five ships with three virtual leaders is considered. The communication topology among the ships is shown in Figure3 with F={1,2,…,5} and L={6,7,8}. It can be noted from Figure 3 that the network topology has a directed spanning tree. Assume that all the following ships have the identical structure and that mathematical model parameters are presented in Table 1. To satisfy conditions (31), (91), and (96), the cooperative control parameters for each following ship are chosen as follows:(103)M11=m^11×0.02,M22=m^22×0.02,M33=m^33×0.02,Dv=d^v×0.02,Dv2=d^v2×0.02,Dr=d^r×0.02,Dv=d^v×0.02,Dv2=d^v2×0.02,Dr=d^r×0.02,Dr2=d^r2×0.02,Dvr=d^vr×0.02,H0=m^11×1.02m^22×0.98,Hur=m^112m^22×0.02,Hv=m^11d^vm^22×0.02,Hv2=m^11d^v2m^22×0.02,Hr=m^11d^vrm^22×0.02,kci=6,kUi=3,kψTi=5,q1i=0.1,q2i=0.1,τ-wui=0.6,τ-wvi=0.6,τ-wri=0.6,and the FOCF parameter is selected as ωn=10. Next, four experiments will be presented to verify the effectiveness of the proposed algorithms.Table 1
Parameters of ships.
Parameter
Value
Unit
m
^
11
25.8
kg
m
^
22
33.8
kg
m
^
33
2.76
kg⋅m2
d
^
u
12
kg/s
d
^
u
2
2.5
kg/m
d
^
v
17
kg/s
d
^
v
2
4.5
kg/m
d
^
v
r
0.2
kg
d
^
r
0.5
kg⋅m2/s
d
^
r
2
0.1
kg⋅m2
d
^
r
v
0.5
kgFigure 3
Communication topology.


### 4.1. Straight-Line Tracking without EDPU

In this section, an example validates the proposed containment control strategy without considering the disturbances for the followers. The predefined trajectories of the virtual leaders are chosen asϕ1=[0.08t;0.035t+12], ϕ2=[0.08t;0.035t], and ϕ3=[0.08t+12;0.035t]. The initial follower states Xi=(xi;yi;ψi;ui;vi;ri) are X1=(-7;-7.5;π/3;0.05;0.02;0); X2=(-4;5;π/4;0.02;0.012;0); X3=(-4;-1;π/3;0.04;0.03;0); X4=(2;-2.5;π/3;0.01;0.01;0); X5=(6;-7;π/2;0.024;0.015;0). It is noted that when the trajectories of virtual leaders are straight lines, the desired trajectories are straight lines. The simulation results are shown in Figures 4–7. Also, the total simulation time is 200 s. Figure 3 shows the motion traces of the followers. As shown in Figure 4, the five followers converge to the convex hull formed by those of the moving leaders. Figure 5 demonstrates that the norms of the tracking error for each follower converge asymptotically to zero as t→∞, which reveals that the containment control objective (24) is solved. Figure 6 shows the actual control inputs of each follower. Figure 7 shows the time evolution of the FOCF estimated errors:(104)e~Ui=αUi-αUic,e~ψTi=αψTi-αψTic,e~ri=αri-αric.It is obvious from Figure 7 that FOCF can accurately estimate virtual control laws.Figure 4
Motion trace of the group.Figure 5
Error norms of the tracking errorszi(i=1,…,5).Figure 6
Control inputs for each follower.Figure 7
Tracking errorse~Ui, e~ψTi, e~ri(i=1,…,5) for FOCF.


### 4.2. Straight-Line Tracking with EDPU

In this section, the robust containment control strategy in the presence of the disturbances is also examined. The trajectories of the virtual leaders are chosen as Section4.1. The EDPU for the followers in the inertial reference frame {I} are assumed to be(105)τwuit=0.28sin⁡0.016t+π3+0.02N,τwvit=0.014cos⁡0.01t+π4+0.04N,τwrit=0.042sin⁡0.008t+π4+0.008N·m,where i=1,2,…,5. The initial follower states are chosen as Section 4.1. The model parameters, used for simulation experiment, are given in the following, assuming that the model parameters are 1.5% parameter perturbation:(106)m11=m^111+0.015sin⁡0.1tkg,m22=m^221+0.015cos⁡0.1t+π3kg,m33=m^331+0.015sin⁡0.1t+π6kg·m2,du=d^u1-0.015sin⁡0.1tkg/s,du2=d^u21+0.015sin⁡0.1tkg/m,dv=d^v1+0.015sin⁡0.1tkg/s,dv2=d^v21+0.015cos⁡0.1t-π4kg/m,dvr=d^vr1+0.015cos⁡0.1t+π3kg,dr=d^r1+0.015sin⁡0.1t+π6kg·m2,dr2=d^r21-0.015sin⁡0.1tkg·m2/s,drv=d^rv1+0.015cos⁡0.1tkg.The simulation results are depicted in Figures 8–11, which exhibit almost the same control performance as those given in Section 4.1. Figure 8 shows the motion traces of the five followers, from which it is evident that the followers also converge to the convex hull formed by those of the moving leaders. The norms of the tracking errors are plotted in Figure 9, and it reveals that the tracking error for each follower are bounded to a small neighborhood of the origin as t→∞. It is clearly seen from Figures 5 and 9 that the proposed controller yields slightly worse performance than the previous section due to the EDPU. However, the tracking errors are bounded, which means that the tracking errors for each follower are GUUB as proved in Theorem 20. Figure 10 shows the actual control inputs applied to each follower under the EDPU, whereas Figure 11 shows the time evolution of the FOCF estimated errors. Therefore, from the simulation results in this section, it is obvious to conclude that the proposed containment control strategy is also practical when each follower is exposed to the EDPU. Also, it is clear that the proposed containment control strategy for each follower can track straight lines without assuming the PE condition.Figure 8
Motion trace of the group.Figure 9
Error norms of the tracking errorszii=1,…,5.Figure 10
Control inputs for each follower.Figure 11
Tracking errorseUi, eψTi, and eri(i=1,…,5) for FOCF.


### 4.3. Curve Tracking without EDPU

In this section, the predefined trajectories of the virtual leaders are curved trajectories. This experiment validates the control strategy without considering the EDPU. The initial follower statesXi=(xi;yi;ψi;ui;vi;ri) are X1=(-14.5;-5;π/3;0.01;0.02;0); X2=(-12;-7;π/4;0.02;0.012;0); X3=(-8;-6;π/6;0.02;0.03;0); X4=(-6;-7;π/3;0.01;0.01;0); X5=(-4;-5;π/2.5;0;0.015;0). In the following simulation, the curved trajectories of virtual leaders are chosen as(107)ϕ2=-8cos⁡πt200;8sin⁡πt200,ϕi=ϕ2+Rψdli,where i=1,3; R(·)=[cos⁡·,-sin⁡·;sin⁡·,cos⁡(·)]; ψd=atan⁡2[π/25cos⁡(πt/200),π/25sin⁡(πt/200)]; l1=[-4;4], l2=[-4;-4]. The simulation results are plotted in Figures 12–15. Figure 12 illustrates the motion traces of the five followers. It is obviously seen that the control strategy can also track the curved trajectories defined by the virtual leaders. The norms of the tracking errors are plotted in Figure 13. From this figure, we can observe that the tracking errors converge to a small neighborhood of the origin with fast convergence rate. The responses of the control inputs are displayed in Figure 14. In Figure 15, FOCF estimated errors converge to nearly zero.Figure 12
Motion trace of the group.Figure 13
Error norms of the tracking errorszii=1,…,5.Figure 14
Control inputs for each follower.Figure 15
Tracking errorseUi, eψTi, and eri(i=1,…,5) for FOCF.


### 4.4. Curve Tracking with EDPU

In this section, the simulation experiments are presented, which concern the robustness properties of the containment control law to the EDPU. The initial states of the followers and the curved predefined trajectories of leaders are the same as the previous section. These simulation results are demonstrated in Figures16–19. Figure 16 shows the actual trajectories of the followers. The norms of the tracking errors are plotted in Figure 17. As seen from Figures 16 and 17, the containment control laws can force the followers to converge into the triangular region in the presence of the EDPU, and the norms of the tracking errors are all less than 0.75 m. In Figure 18, the control forces Fui and Fri are shown. Figure 19 illustrates the responses of the FOCF estimated errors.Figure 16
Motion trace of the group.Figure 17
Error norms of the tracking errorszii=1,…,5.Figure 18
Control inputs for each follower.Figure 19
Tracking errorseUi, eψTi, and eri(i=1,…,5) for FOCF.In summary, from Figures4–19, we can see that the proposed containment control law in this paper provided acceptable results as proven in Theorem 20 and all the tracking errors of the closed-loop system converge to a small neighborhood of the origin as time approaches infinity.


## 4.1. Straight-Line Tracking without EDPU

In this section, an example validates the proposed containment control strategy without considering the disturbances for the followers. The predefined trajectories of the virtual leaders are chosen asϕ1=[0.08t;0.035t+12], ϕ2=[0.08t;0.035t], and ϕ3=[0.08t+12;0.035t]. The initial follower states Xi=(xi;yi;ψi;ui;vi;ri) are X1=(-7;-7.5;π/3;0.05;0.02;0); X2=(-4;5;π/4;0.02;0.012;0); X3=(-4;-1;π/3;0.04;0.03;0); X4=(2;-2.5;π/3;0.01;0.01;0); X5=(6;-7;π/2;0.024;0.015;0). It is noted that when the trajectories of virtual leaders are straight lines, the desired trajectories are straight lines. The simulation results are shown in Figures 4–7. Also, the total simulation time is 200 s. Figure 3 shows the motion traces of the followers. As shown in Figure 4, the five followers converge to the convex hull formed by those of the moving leaders. Figure 5 demonstrates that the norms of the tracking error for each follower converge asymptotically to zero as t→∞, which reveals that the containment control objective (24) is solved. Figure 6 shows the actual control inputs of each follower. Figure 7 shows the time evolution of the FOCF estimated errors:(104)e~Ui=αUi-αUic,e~ψTi=αψTi-αψTic,e~ri=αri-αric.It is obvious from Figure 7 that FOCF can accurately estimate virtual control laws.Figure 4
Motion trace of the group.Figure 5
Error norms of the tracking errorszi(i=1,…,5).Figure 6
Control inputs for each follower.Figure 7
Tracking errorse~Ui, e~ψTi, e~ri(i=1,…,5) for FOCF.


## 4.2. Straight-Line Tracking with EDPU

In this section, the robust containment control strategy in the presence of the disturbances is also examined. The trajectories of the virtual leaders are chosen as Section4.1. The EDPU for the followers in the inertial reference frame {I} are assumed to be(105)τwuit=0.28sin⁡0.016t+π3+0.02N,τwvit=0.014cos⁡0.01t+π4+0.04N,τwrit=0.042sin⁡0.008t+π4+0.008N·m,where i=1,2,…,5. The initial follower states are chosen as Section 4.1. The model parameters, used for simulation experiment, are given in the following, assuming that the model parameters are 1.5% parameter perturbation:(106)m11=m^111+0.015sin⁡0.1tkg,m22=m^221+0.015cos⁡0.1t+π3kg,m33=m^331+0.015sin⁡0.1t+π6kg·m2,du=d^u1-0.015sin⁡0.1tkg/s,du2=d^u21+0.015sin⁡0.1tkg/m,dv=d^v1+0.015sin⁡0.1tkg/s,dv2=d^v21+0.015cos⁡0.1t-π4kg/m,dvr=d^vr1+0.015cos⁡0.1t+π3kg,dr=d^r1+0.015sin⁡0.1t+π6kg·m2,dr2=d^r21-0.015sin⁡0.1tkg·m2/s,drv=d^rv1+0.015cos⁡0.1tkg.The simulation results are depicted in Figures 8–11, which exhibit almost the same control performance as those given in Section 4.1. Figure 8 shows the motion traces of the five followers, from which it is evident that the followers also converge to the convex hull formed by those of the moving leaders. The norms of the tracking errors are plotted in Figure 9, and it reveals that the tracking error for each follower are bounded to a small neighborhood of the origin as t→∞. It is clearly seen from Figures 5 and 9 that the proposed controller yields slightly worse performance than the previous section due to the EDPU. However, the tracking errors are bounded, which means that the tracking errors for each follower are GUUB as proved in Theorem 20. Figure 10 shows the actual control inputs applied to each follower under the EDPU, whereas Figure 11 shows the time evolution of the FOCF estimated errors. Therefore, from the simulation results in this section, it is obvious to conclude that the proposed containment control strategy is also practical when each follower is exposed to the EDPU. Also, it is clear that the proposed containment control strategy for each follower can track straight lines without assuming the PE condition.Figure 8
Motion trace of the group.Figure 9
Error norms of the tracking errorszii=1,…,5.Figure 10
Control inputs for each follower.Figure 11
Tracking errorseUi, eψTi, and eri(i=1,…,5) for FOCF.


## 4.3. Curve Tracking without EDPU

In this section, the predefined trajectories of the virtual leaders are curved trajectories. This experiment validates the control strategy without considering the EDPU. The initial follower statesXi=(xi;yi;ψi;ui;vi;ri) are X1=(-14.5;-5;π/3;0.01;0.02;0); X2=(-12;-7;π/4;0.02;0.012;0); X3=(-8;-6;π/6;0.02;0.03;0); X4=(-6;-7;π/3;0.01;0.01;0); X5=(-4;-5;π/2.5;0;0.015;0). In the following simulation, the curved trajectories of virtual leaders are chosen as(107)ϕ2=-8cos⁡πt200;8sin⁡πt200,ϕi=ϕ2+Rψdli,where i=1,3; R(·)=[cos⁡·,-sin⁡·;sin⁡·,cos⁡(·)]; ψd=atan⁡2[π/25cos⁡(πt/200),π/25sin⁡(πt/200)]; l1=[-4;4], l2=[-4;-4]. The simulation results are plotted in Figures 12–15. Figure 12 illustrates the motion traces of the five followers. It is obviously seen that the control strategy can also track the curved trajectories defined by the virtual leaders. The norms of the tracking errors are plotted in Figure 13. From this figure, we can observe that the tracking errors converge to a small neighborhood of the origin with fast convergence rate. The responses of the control inputs are displayed in Figure 14. In Figure 15, FOCF estimated errors converge to nearly zero.Figure 12
Motion trace of the group.Figure 13
Error norms of the tracking errorszii=1,…,5.Figure 14
Control inputs for each follower.Figure 15
Tracking errorseUi, eψTi, and eri(i=1,…,5) for FOCF.


## 4.4. Curve Tracking with EDPU

In this section, the simulation experiments are presented, which concern the robustness properties of the containment control law to the EDPU. The initial states of the followers and the curved predefined trajectories of leaders are the same as the previous section. These simulation results are demonstrated in Figures16–19. Figure 16 shows the actual trajectories of the followers. The norms of the tracking errors are plotted in Figure 17. As seen from Figures 16 and 17, the containment control laws can force the followers to converge into the triangular region in the presence of the EDPU, and the norms of the tracking errors are all less than 0.75 m. In Figure 18, the control forces Fui and Fri are shown. Figure 19 illustrates the responses of the FOCF estimated errors.Figure 16
Motion trace of the group.Figure 17
Error norms of the tracking errorszii=1,…,5.Figure 18
Control inputs for each follower.Figure 19
Tracking errorseUi, eψTi, and eri(i=1,…,5) for FOCF.In summary, from Figures4–19, we can see that the proposed containment control law in this paper provided acceptable results as proven in Theorem 20 and all the tracking errors of the closed-loop system converge to a small neighborhood of the origin as time approaches infinity.


## 5. Conclusions

(1) Because the sway force is unavailable, the underactuated model is transformed to a strict-feedback form by introducing a polar coordinate transformation. In the kinematic loop design, a virtual controller is designed on the basis of the results from graph theory so as to achieve the control objective.(2) To avoid the problem of “explosion of complexity” from the standard BP method, FOCF is introduced to estimate the virtual control laws and their derivatives. In addition, a compensating signal is introduced to compensate the nonlinear terms in BP design.(3) Under the consideration that each follower is exposed to the EDPU, the actual control inputs are presented by using the SMC method in the dynamic loop design, which can guarantee that all the signals of the closed-loop system are GUUB and all the tracking errors eventually converge to a small neighborhood of the equilibrium point.(4) Future works will be required to account for communication constraint among ships, such as packets dropouts and constant or time-varying communication delays as well as switching communication topology.



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*Source: 5086705-2017-04-02.xml*