Robust Coordinated Control Algorithm for Multiple Marine Vessels with External Disturbances

Abstract

The problem of coordinated control for multiple marine vessels in the presence of external disturbances is considered in this paper. A robust coordinated control algorithm is proposed for multiple marine vessels. The proposed robust coordinated control algorithm is divided into two parts. The first part develops an extended state observer to estimate the disturbances of marine vessels. The second part presents a robust coordinated control algorithm based on the output of the extended state observer. Furthermore, the robust coordinated control algorithm is designed using the dynamic surface control method. In light of the leader-follower strategy, the trajectory for each vessel is defined according to the desired trajectory of the assigned leader and the relative distance with respect to the leader. The effectiveness of the proposed coordination algorithm is demonstrated by the simulation results.

1. Introduction

In recent years, coordinated control of multiple vehicles has received increasing attention as an emerging technology [1]. Multiple vehicles can perform many complex tasks effectively with less time and lower cost than a single vehicle. And multiple vehicles can accomplish some tasks which cannot be executable by a single one. In order to perform these complicated practical tasks, it is necessary for these vehicles to move collectively as a whole formation. In practice, many relevant applications of coordinated control can be found on the land, in the sea, and in the air [2]. For instance, in the operations of underway replenishment by a fleet of surface vessels, it is required that the replenished vehicle should maintain a fixed relative position with respect to the replenishing one, in order to ensure the replenishment operation performed safely and effectively.

The problem of coordinated formation control has been reported in a large number of recent publications. Basic approaches of the coordinated control include leader-follower approach [3–5], behavioral approach [6, 7], and virtual structures approach [8, 9]. In the leader-follower approach, some agents are considered as the leaders, and the rest ones are considered as the followers. The followers will track the leaders, and the leaders will track the predefined desired trajectories. This method is easy to be manipulated and implemented. However, the main criticism of the leader-follower approach is that it depends heavily on the leader to achieve the goal of the formation task which may be undesirable [3–5]. In the behavioral approach, the collision avoidance/obstacle avoidance and the target tracking are prescribed for each agent, and the whole formation is achieved by calculating the weight of the relative importance of each behavior. However, it is difficult to analyze the stability of the group behavior using such approach [6, 7]. In the virtual-structure approach, each member in the formation is considered as a particle embedded in a rigid geometric structure, but the relative applications are limited when the formation structure is time-varying or needs to be frequently reconfigured [8, 9].

Some advanced approaches including graph theory [10], passivity-based control [11, 12], and hybrid control [13] are also used for coordinated control of multiple marine vessels. Most results about the coordinated control problem addressed in the earlier papers are on the assumption that marine vessels are free from environmental disturbances. However, coordinated control for multiple surface vessels encountering exogenous disturbances adds a new level of complexity to the problem. Other advanced methods are proposed to solve the robust coordinated formation control problem, for example, the Lagrangian approach [14], the nonlinear model predictive control [15], the adaptive control [16], and the sliding mode control [17]. In addition, the fault tolerant control and the fault diagnosis are studied in references [18–20]. In particular, the problem of coordinated path following multiple vessels has also been discussed in the related literature studies [21, 22]. The robustness to environmental disturbances is highly important when performing practical marine and offshore tasks for surface vessels, which is also the concerned issue in this paper. The core of the extended state observer is that the disturbances and the unknown dynamics can be considered as extend state, and then the detailed values can be estimated by designed observer. The correlative applications can be found in literature studies [23–26]. The stability of the extended state observer is analyzed in [27–29]. The robust coordination control algorithm for multiple surface vessels based on extend state observer and robust control technology is studied in this paper. The designed controller is useful for the practical marine operations.

In this paper, we consider the problem of coordinated formation control of multiple surface vessels in the presence of exogenous disturbances. The coordinated formation controller is proposed by combining the extended state observer and dynamic surface control using the leader-follower strategy. The extended state observer is developed to estimate the external disturbances of the surface vessels. The coordinated control algorithm is accomplished based on the output of the extended state observer. Furthermore, the trajectory of each vessel is defined using the desired trajectory of the assigned leader and the relative distance with respect to the leader. This paper is organized as follows. In Section 2, the vessel model is established. Section 3 contains a detailed algorithm of the coordination formation control for multiple vessels. Simulation is carried out in Section 4, and we draw conclusions in Section 5.

2. Preliminaries

The vessel model can be divided into two parts: the kinematics and nonlinear dynamics. Generally, only the motion in the horizontal plane is considered for the surface vessel. The elements corresponding to heave, roll, and pitch are neglected. The dynamic model for the ith surface vessel can be represented by the following 3 degrees of freedom (DOF) [30]: 𝜂̇_𝑖=𝐑_𝑖𝜓_𝑖𝐯_𝑖,𝐌_𝑖𝐯̇_𝑖+C_𝑖𝐯_𝑖𝐯_𝑖+𝐃_𝑖𝐯_𝑖𝐯_𝑖=𝜏_𝑖+𝜏_𝑑𝑖, (1)(2) where 𝜂_𝑖=[𝑛_𝑖,𝑒_𝑖,𝜓_𝑖]^𝑇 denotes the north position, east position, and orientation which are decomposed in the Earth-fixed reference frame, 𝐯_𝑖=[𝑢_𝑖,𝑣_𝑖,𝑟_𝑖]^𝑇 denotes the linear surge velocity, sway velocity, and angular velocity which are decomposed in the body-fixed reference frame. 𝐑_𝑖(𝜓_𝑖) is the transformations matrix from the body-fixed reference frame to the Earth-fixed reference frame, the form of which is as follows: 𝐑_𝑖𝜓_𝑖=cos𝜓_𝑖−sin𝜓_𝑖0sin𝜓_𝑖cos𝜓_𝑖0001. (3) The transformations matrix satisfies 𝐑_𝑖⁻¹(𝜓_𝑖)=𝐑^𝑇_𝑖(𝜓_𝑖), for all 𝜓_𝑖. 𝐌_𝑖 denotes the system inertia mass matrix including added mass which is positive definite. 𝐂_𝑖(𝐯_𝑖) and 𝐃_𝑖(𝐯_𝑖) denote the Coriolis-centripetal matrix and damping matrix, respectively. The detailed representation of the above three system matrices can be found in reference [30]. 𝜏_𝑖=[𝜏_𝑢𝑖𝜏_𝑣𝑖𝜏_𝑟𝑖]^𝑇 is the vector of forces and torques input from the thruster system. 𝜏_𝑑𝑖 is the vector of external environment forces and torques input which is generated by wind, wave, and current.

In order to design the backstepping sliding mode controller, we transform the vessel model as follows: 𝐯_𝑖=𝐑_𝑖⁻¹𝜓_𝑖𝜂̇_𝑖, (4) because 𝐑̇_𝑖(𝜓_𝑖)=𝐑_𝑖(𝜓_𝑖)𝑆, where 𝑆=0−𝑟_𝑖0𝑟_𝑖00000=−𝑆^𝑇 (5) with 𝑟_𝑖 being the angular velocity in the body-fixed reference frame. We can obtain that 𝐯̇_𝑖=𝐑̇_𝑖⁻¹𝜓_𝑖𝜂̇_𝑖+𝐑_𝑖⁻¹𝜓_𝑖𝜂̈_𝑖=−𝐑_𝑖⁻¹𝜓_𝑖𝐑̇_𝑖𝜓_𝑖𝐑_𝑖⁻¹𝜓_𝑖𝜂̇_𝑖+𝐑_𝑖⁻¹𝜓_𝑖𝜂̈_𝑖. (6) Taking (4) and (6) into the vessel dynamic model (2) yields 𝐌_𝑖𝐑_𝑖⁻¹𝜓_𝑖𝜂̈_𝑖−𝐌_𝑖𝐑_𝑖⁻¹𝜓_𝑖𝐑̇_𝑖𝜓_𝑖𝐑_𝑖⁻¹𝜓_𝑖𝜂̇_𝑖+𝐂_𝑖𝐯_𝑖𝐑_𝑖⁻¹𝜓_𝑖𝜂̇_𝑖+𝐃_𝑖𝐯_𝑖𝐑_𝑖⁻¹𝜓_𝑖𝜂̇_𝑖=𝜏_𝑖+𝜏_𝑑𝑖. (7) The above equation can be written as 𝐌_𝑛𝑖𝜂_𝑖𝜂̈_𝑖+𝐂_𝑛𝑖𝜂_𝑖,𝜂̇_𝑖𝜂̇_𝑖+𝐃_𝑛𝑖𝜂_𝑖,𝜂̇_𝑖𝜂̇_𝑖=𝜏_𝑖+𝜏_𝑑𝑖, (8) where 𝐌_𝑛𝑖𝜂_𝑖=𝐌_𝑖𝐑_𝑖⁻¹𝜓_𝑖;𝐂_𝑛𝑖𝜂_𝑖,𝜂̇_𝑖=𝐂_𝑖𝐯_𝑖−𝐌_𝑖𝐑_𝑖⁻¹𝜓_𝑖𝐑̇_𝑖𝜓_𝑖𝐑_𝑖⁻¹𝜓_𝑖;𝐃_𝑛𝑖𝜂_𝑖,𝜂̇_𝑖=𝐃_𝑖𝐯_𝑖𝐑_𝑖⁻¹𝜓_𝑖. (9)

3. Coordinated Formation Controller Design

In this section, the controller is designed from two inspects. One is the extended state observer design for each vessel, and the other is the coordinated controller for multiple vessels based on the output of the extended state observer.

3.1. Extended State Observer Design

In this section, we design the extended state observer for each vessel to estimate the disturbances.

Let 𝜂̇_𝑖=𝐯_𝑛𝑖, and then (8) can be written as 𝐌_𝑛𝑖𝜂_𝑖𝐯̇_𝑛𝑖+𝐂_𝑛𝑖𝜂_𝑖,𝐯_𝑛𝑖𝐯_𝑛𝑖+𝐃_𝑛𝑖𝜂_𝑖,𝐯_𝑛𝑖𝐯_𝑛𝑖=𝜏_𝑖+𝜏_𝑑𝑖. (10) Then we have 𝜂̇_𝑖=𝐯_𝑛𝑖,𝐯̇_𝑛𝑖=𝐌⁻¹_𝑛𝑖𝜂_𝑖𝜏_𝑖+𝐌⁻¹_𝑛𝑖𝜂_𝑖−𝐂_𝑛𝑖𝜂_𝑖,𝐯_𝑛𝑖𝐯_𝑛𝑖−𝐃_𝑛𝑖𝜂_𝑖,𝐯_𝑛𝑖𝐯_𝑛𝑖+𝜏_𝑑𝑖. (11)

Let 𝐊_𝑖=𝐌⁻¹_𝑛𝑖(𝜂_𝑖)(−𝐂_𝑛𝑖(𝜂_𝑖,𝐯_𝑛𝑖)𝐯_𝑛𝑖−𝐃_𝑛𝑖(𝜂_𝑖,𝐯_𝑛𝑖)𝐯_𝑛𝑖+𝜏_𝑑𝑖),𝐮_𝑖=𝐌⁻¹_𝑛𝑖(𝜂_𝑖)𝜏_𝑖. Then the above equation can be rewritten as 𝜂̇_𝑖=𝐯_𝑛𝑖,𝐯̇_𝑛𝑖=𝐊_𝑖+𝐮_𝑖. (12) Here 𝐊_𝑖 is assumed to be unknown. We assume that 𝐊_𝑖 is an extended state. However, 𝐊_𝑖 can be estimated using an extended state observer. Then the disturbances are observed and compensated by the designed controller.

The extended state observer is designed as 𝐳_1i=𝜂_𝑖−𝜂_𝑖,𝜂̇_𝑖=𝐯_𝑛𝑖+𝛽₁fal₁𝐳_1𝑖,𝐯̇,𝛼,𝛿_𝑛𝑖=𝐊_𝑖+𝛽₂fal₂𝐳_1𝑖,𝛼,𝛿+𝐮_𝑖,𝐊̇_𝑖=𝛽₃fal₃𝐳_1𝑖,𝛼,𝛿, (13) where fal𝐳_1𝑖=𝐳,𝛼,𝛿^1𝑖𝛼sign𝐳_1𝑖,𝐳_1𝑖𝐳>𝛿,_1𝑖𝛿^𝛼−1,𝐳_1𝑖≤𝛿. (14) And 𝛿>0, 0<𝛼<1.

Set 𝐳_1𝑖=𝜂_𝑖−𝜂_𝑖, 𝐳_2𝑖=𝐯_𝑛𝑖−𝐯_𝑛𝑖, 𝐳_3𝑖=𝐊_𝑖−𝐊_𝑖. 𝜂_𝑖,𝐯_𝑛𝑖,𝐊_𝑖 are the estimated values of 𝜂_𝑖,𝐯_𝑛𝑖,𝐊_𝑖, respectively. Taking the derivative of 𝐳_1𝑖,𝐳_2𝑖,𝐳_3𝑖, respectively, we can obtain that 𝐳̇_1𝑖=𝐯_𝑛𝑖+𝐳_2𝑖−𝐯_𝑛𝑖−𝛽₁fal₁𝐞_𝑖,𝛼,𝛿=𝐳_2𝑖−𝛽₁fal₁𝐳_1𝑖𝐳̇,𝛼,𝛿_2𝑖=𝐊_𝑖+𝐮_𝑖−𝐊_𝑖−𝛽₂fal₂𝐳_1𝑖,𝛼,𝛿−𝐮_𝑖=𝐳_3𝑖−𝛽₂fal₂𝐳_1𝑖𝐳̇,𝛼,𝛿_3𝑖=𝐊̇_𝑖−𝛽₃fal₃𝐳_1𝑖,𝛼,𝛿. (15)(16)(17)

The following assumptions are presumed.(1) The possibly unknown function 𝐊_𝑖 is continuously differentiable with respect to their variables. |𝐊̇_𝑖|≤𝑀 for all 𝑡>0, where 𝑀 is a positive constant.(2) Let 𝐳=[𝐳₁,𝐳₂,𝐳₃]^𝑇, and ‖•‖ denotes the Euclid norm of ℝ⁹. There exist positive definite constants 𝜆_𝑖(𝑖=1,2,3,4), 𝛼 and continuous differentiable functions 𝑉,𝑊∶ℝ⁹→ℝ such that(i) 𝜆₁‖𝐳‖²≤𝐕(𝐳)≤𝜆₂‖𝐳‖², 𝜆₃‖𝐳‖²≤𝐖(𝐳)≤𝜆₄‖𝐳‖²,(ii)(𝜕𝑉/𝜕𝐳₁)(𝐳₂−𝑓₁(𝐳₁))+(𝜕𝑉/𝜕𝐳₂)(𝐳₂−𝑓₂(𝐳₁))+(𝜕𝑉/𝜕𝐳₃)𝑓₃(𝐳₁)≤−𝑊(𝐳),(iii) ‖𝜕𝑉/𝜕𝐳₃‖≤𝛼‖𝐳‖.

The stability for the extended state observer is analyzed in [24]. Then we can obtain 𝜂_𝑖→𝜂_𝑖, 𝐯_𝑛𝑖→𝐯_𝑛𝑖, 𝐊_𝑖→𝐊_𝑖.

3.2. Coordinated Controller Design

3.2.1. Formation Setup

This paper considers a fleet of n vessels to perform the desired coordination formation task. Each vessel in the formation is identified by the index set 𝐼=[1,2,…,𝑛]. The desired formation is established using the leader-follower strategy as shown in Figure 1. Furthermore, the leader is a virtual vessel. If we assume the desired trajectory of the leader vessel is denoted as 𝜂_𝑑, where 𝜂_𝑑=[𝑛_𝑑(𝑡),𝑒_𝑑(𝑡),𝜓_𝑑(𝑡)]^𝑇, 𝑛_𝑑(𝑡), 𝑒_𝑑(𝑡) are sufficiently smooth functions, and 𝜓_𝑑(𝑡)=arctan(𝑒_𝑑̇(𝑡)/𝑛̇_𝑑(𝑡)). That means the vessel direction is chosen as the tangential vector of the respective desired trajectory. If we define the relative distance between the follower vessel and the leader vessel as 𝐥_𝑖=[𝑥_0𝑖,𝑦_0𝑖,𝜓_0𝑖]^𝑇, then the desired trajectory of the follower vessel is denoted as 𝜂_𝑑𝑖=𝜂_𝑑+𝐑(𝜓_𝑑)𝐥_𝑖.

Figure 1 

The formation of these vessels.

3.2.2. Controller Design

Define the first dynamic surface as 𝐒_1𝑖=𝜂_𝑖−𝜂_𝑑𝑖. (18) Taking the derivative of the first surface, with (13), we obtain 𝐒̇_1𝑖=𝜂̇_𝑖−𝜂̇_𝑑𝑖=𝐯_𝑛𝑖+𝛽₁fal₁𝐳_1𝑖,𝛼,𝛿−𝜂̇_𝑑𝑖. (19) Define a virtual velocity 𝐯_𝑟𝑖 as follows: 𝐯_𝑟𝑖=𝜂̇_𝑑𝑖−𝛽₁fal₁𝐳_1𝑖,𝛼,𝛿−Λ₁𝐒_1𝑖, (20) where Λ₁ is a positive definite matrix. With this definition, if 𝐯_𝑟𝑖=𝐯_𝑛𝑖, then 𝐒̇_1𝑖=−Λ₁𝐒_1𝑖. Then 𝐒_1𝑖→0 with a convergence rate determined by the choice of Λ₁. Because of the definition of 𝐒_1𝑖, this will also guarantee that 𝜂_𝑖→𝜂_𝑖→𝜂_𝑑𝑖.

𝐯_𝑟𝑖 is passed through a first order filter in order to avoid the problem existed in the backstepping scheme: 𝐓𝐯̇_𝑑𝑖+𝐯_𝑑𝑖=𝐯_𝑟𝑖, (21) where 𝐓 is a diagonal matrix of the filter time constants which are chosen to be as small as possible. Because 𝐯_𝑛𝑖→𝐯_𝑛𝑖, then we can define the second sliding surface as 𝐒_2𝑖=𝐯_𝑛𝑖−𝐯_𝑑𝑖, (22) where 𝐯_𝑑𝑖 is the estimated value of 𝐯_𝑟𝑖; take the derivative of 𝐯_𝑑𝑖, then we have 𝐯̇_𝑑𝑖=𝐓⁻¹𝐯_𝑟𝑖−𝐯_𝑑𝑖. (23) Taking the derivative of the second surface, with (13), we have 𝐒̇_2𝑖=𝐯̇_𝑛𝑖−𝐯̇_𝑑𝑖=𝐊_𝑖+𝛽₂fal₂𝐳_1𝑖,𝛼,𝛿+𝐮_𝑖−𝐯̇_𝑑𝑖. (24) We consider the following Lyapunov function candidate: 𝐕_1𝑖=12𝐒^𝑇_2𝑖𝐒_2𝑖. (25) We take the time derivative of (25): 𝐕̇_1𝑖=𝐒^𝑇_2𝑖𝐊_𝑖+𝛽₂fal₂𝐳_1𝑖,𝛼,𝛿+𝐮_𝑖−𝐯̇_𝑑𝑖. (26) So we choose the control input as 𝐮_𝑖=𝐯̇_𝑑𝑖−𝐊_𝑖−𝛽₂fal₂(𝐳_1𝑖,𝛼,𝛿)−𝐊_𝐷𝐒_2𝑖.

The control input of the vessel is 𝜏_𝑖=𝐌_𝑛𝑖(𝜂_𝑖)𝐮_𝑖. So the control force input 𝜏 is selected as 𝜏_𝑖=𝐌_𝑛𝑖𝜂_𝑖𝐯̇_𝑑𝑖−𝐊_𝑖−𝛽₂fal₂𝐳_1𝑖,𝛼,𝛿−𝐊_𝐷𝐒_2𝑖. (27)

Theorem 1. Consider the vessel with the nonlinear model as in (1), (2), and (8), with the control law (27), and then one can guarantee that the vessels approach the desired trajectory ultimately while holding the desired formation structure.

Proof. With the definition of the second surface, (19) can be rewrite as 𝐒̇_1𝑖=𝐒_2𝑖+𝐯_𝑑𝑖−𝛽₁fal₁𝐳_1𝑖,𝛼,𝛿−𝜂̇_𝑑𝑖. (28) Define the estimated error of the first order filter as 𝐒_3𝑖=−𝐓𝐯̇_𝑑𝑖=𝐯_𝑑𝑖−𝐯_𝑟𝑖. (29) Taking the derivative of 𝐒_3𝑖 yields 𝐒̇_3𝑖=𝐯̇_𝑑𝑖−𝐯_𝑟𝑖=𝐯̇_𝑑𝑖+𝛽₁𝑑fal₁𝐳_1𝑖,𝛼,𝛿𝑑𝑡+Λ₁𝐒̇_1𝑖−𝜂̈_𝑑𝑖𝐒=−^3𝑖𝐓+𝛽₁𝑑fal₁𝐳_1𝑖,𝛼,𝛿𝑑𝑡+Λ₁𝐒̇_1𝑖−𝜂̈_𝑑𝑖𝐒=−^3𝑖𝐓+𝑔𝛽₁,𝐳_1𝑖,𝛼,𝛿,𝐳̇_1𝑖,𝜂̈_𝑑𝑖,𝐒̇_1𝑖. (30) With (19) and (21), we obtain 𝐒̇_1𝑖=𝐒_2𝑖+𝐒_3𝑖−Λ_1𝑖𝐒_1𝑖. (31) Define the Lyapunov function as 𝐕_𝑖=12𝐒^𝑇_1𝑖𝐒_1𝑖+12𝐒^𝑇_2𝑖𝐒_2𝑖+12𝐒^𝑇_3𝑖𝐒_3𝑖. (32) Differentiating the above equation yields 𝐕̇_𝑖=𝐒^𝑇_1𝑖𝐒̇_1𝑖+𝐒^𝑇_2𝑖𝐒̇_2𝑖+𝐒^𝑇_3𝑖𝐒̇_3𝑖=𝐒^𝑇_1𝑖𝐒_2𝑖+𝐒_3𝑖−Λ₁𝐒_1𝑖−𝐒^𝑇_2𝑖𝐊_𝐷𝐒_2𝑖+𝐒^𝑇_3𝑖×−𝐒_3𝑖𝐓+𝑔𝛽₁,𝐳_1𝑖,𝛼,𝛿,𝐳̇_1𝑖,𝜂̈_𝑑𝑖,𝐒̇_1𝑖=−𝐒^𝑇_1𝑖Λ₁𝐒_1𝑖+𝐒^𝑇_1𝑖𝐒_2𝑖+𝐒^𝑇_1𝑖𝐒_3𝑖−𝐒^𝑇_2𝑖𝐊_𝐷𝐒_2𝑖−𝐒^𝑇_3𝑖𝐒_3𝑖𝐓+𝐒^𝑇_3𝑖𝑔𝛽₁,𝐳_1𝑖,𝛼,𝛿,𝐳̇_1𝑖,𝜂̈_𝑑𝑖,𝐒̇_1𝑖. (33) If we define the maximum of 𝑔(𝛽₁,𝐳_1𝑖,𝛼,𝛿,𝐳̇_1𝑖,𝜂̈_𝑑𝑖,𝐒̇_1𝑖) is 𝑔_max, we can know that 𝐒^𝑇_1𝑖𝐒_1𝑖+𝐒^𝑇_2𝑖𝐌𝐒_2𝑖+𝐒^𝑇_3𝑖𝐒_3𝑖≤2𝑝, 𝑝 is a positive constant. Then we can obtain that 𝑉_𝑖≤𝑝. Let Λ₁=𝐊_𝐷=𝛼₀+2𝐼₃, and the filter time constant can be chosen as 𝐓=(𝐼₃+(𝑔^𝑇_max𝑔_max/2𝜀𝐼₃)+𝛼₀)⁻¹; then we can obtain that 𝐕̇_𝑖=−𝐒^𝑇_1𝑖𝛼₀+2𝐼₃𝐒_1𝑖−𝐒^𝑇_2𝑖𝛼₀+2𝐼₃𝐒_2𝑖−𝐒^𝑇_3𝑖𝐼₃+𝑔²_max2𝜀𝐼₃+𝛼₀𝐒_3𝑖+𝐒^𝑇_1𝑖𝐒_2𝑖+𝐒^𝑇_1𝑖𝐒_3𝑖+𝐒^𝑇_3𝑖𝑔≤−𝐒^𝑇_1𝑖𝛼₀+2𝐼₃𝐒_1𝑖−𝐒^𝑇_2𝑖𝛼₀+2𝐼₃𝐒_2𝑖+2𝐒^𝑇_1𝑖𝐒_1𝑖+𝐒^𝑇_2𝑖𝐒_2𝑖+𝐒^𝑇_3𝑖𝐒_3𝑖2−𝐒^𝑇_3𝑖𝐒_3𝑖−𝐒^𝑇_3𝑖𝛼₀𝐒_3𝑖−𝐒^𝑇_3𝑖𝑔^𝑇_max𝑔_max2𝜀𝐼₃𝐒_3𝑖+𝑔²_max𝐒^𝑇_3𝑖𝐒_3𝑖2𝜀𝐼₃𝑔^𝑇𝑔𝑔²_max+𝜀2≤−2𝛼₀𝐕_𝑖+𝜀2. (34) If we choose 𝛼₀>𝜀/2𝑝, then 𝑉̇_𝑖<0. We can guarantee that 𝐒_2𝑖→0. This implies 𝐯_𝑛𝑖→𝐯_𝑛𝑖→𝐯_𝑑𝑖, in turn, 𝐒_1𝑖→0 and 𝜂_𝑖→𝜂_𝑖→𝜂_𝑑𝑖.

4. Simulation Results

In this section, experimental simulations are carried out to evaluate the effectiveness of the proposed coordinated formation control algorithm. The detailed parameters of the vessel are presented in the literature [11]. At the beginning, the proposed extended state observer of one vessel is evaluated by the simulation. Similarly, the performance of the extended state observer of other vessels is achieved. Compare with the existing literature studies, we let the initial position of the vessel is 𝜂=[0,0,0]^𝑇. The vessel moves in a beeline northward. assume that the vessels encounter the wind, wave, and current. The wind is assumed to be fixed direction and fixed velocity, and then the disturbance of wind is a constant; the wave and current are assumed to be the sine wave with a fixed frequency at one time. The external disturbances can be chosen as 𝜏_𝑑=10⁶∗[()(),]0.05sin𝜋𝑡/100+0.02,0.03sin𝜋𝑡/1000.01sin(𝜋𝑡/100)+0.01^𝑇(𝑁). (35) In the simulation, we assume that the external disturbances are unknown. The proposed observer parameters are selected as 𝛽₁=30, 𝛽₂=15, 𝛽₃=5, 𝛼=0.25, and 𝛿=0.1.

The simulation results are shown in Figures 2 to 5. Figure 2 shows the movements for the vessel in the plane. The practical value and estimated value of the north position, east position, and heading change curve of the vessel are shown in Figure 3. Figure 4 shows the practical value and estimated value of the surge velocity, sway velocity, and angular velocity of the vessel. The estimated value of the external disturbances are shown in Figure 5. In the simulation experiment, there is no measurement noise in the kinematics and nonlinear dynamics. So the practical value and estimated value of the position and velocity are consensus in a way. What’s more, the external disturbances can be estimated through introducing the extended state.

Figure 2 

The movement of the vessel in the plane.

Figure 3 

The practical value and estimated value of the positions.

Figure 4 

The practical value and estimated value of the velocities.

Figure 5 

The estimated value of the disturbances.

Then we evaluate the effectiveness of the proposed robust coordinated formation control algorithm. Three surface vessels are considered to perform the coordinated tracking task. The initial positions of the three vessels are 𝜂₁=[65782−𝜋/3]^𝑇,𝜂₂=[80831−7𝜋/30]^𝑇, and 𝜂₃=[75743−𝜋/4]^𝑇, respectively. In order to evaluate the performance of the coordinated tracking, the desired formation pattern of the coordinated formation controller is described by 𝐥₁=[000]^𝑇, 𝐥₂=[0−1000]^𝑇, and 𝐥₃=[01000]^𝑇. The desired trajectory for the assigned leader is chosen as 𝜂_𝑑(𝑡)=[𝑛_𝑑𝑒_𝑑𝜓_𝑑]^𝑇, and the detailed forms are 𝑛_𝑑=𝑡, 𝑒_𝑑=1000sin(𝑡/600), 𝜓_𝑑=arctan(𝑒_𝑑̇/𝑛̇_𝑑). The proposed controller parameters are selected as Λ₁=diag(0.05,0.05,0.05), 𝐓=diag(0.1,0.1,0.1), and 𝐊_𝐷=10⁴∗diag(6.5,6.5,6.5).

The simulation results are shown in Figures 6 to 10. Figure 6 shows the movements for these vessels in the plane. The heading change curve of each vessel is shown in Figure 7. Figures 8, 9, and 10 show the surge velocity, the sway velocity, and the angular velocity of each vessel during the coordinated control process, respectively. We can see that these vessels realized the coordinated tracking task from Figures 6 and 7. From Figures 8, 9, and 10, the velocities of these vessels achieve consensus as a whole, and the velocities cannot achieve consensus absolutely when the vessels move to the inflexion of the curves. With the analysis of the simulation results, we can conclude that these vessels can accomplish coordinated trajectory tracking task while keeping the desired formation. It means that the proposed coordination control algorithm is effective.

Figure 6 

Coordinated trajectory tracking of the vessels.

Figure 7 

Heading change of the vessels.

Figure 8 

Surge velocity change of the vessels.

Figure 9 

Sway velocity change of the vessels.

Figure 10 

Angular velocity change of the vessels.

5. Conclusion

This paper has proposed a new robust coordinated formation control algorithm for multiple surface vessels in the presence of external environmental disturbances. The proposed coordinated formation controller for these vessels is designed by combining the extended state observer and the dynamic surface control together. The extended state observer is designed to estimate the external disturbances of the surface vessels. The coordinated formation is realized based on the leader-follower strategy. The desired trajectory of each vessel is defined using the desired trajectory of the assigned leader and relative distance with respect to the leader. The controller is designed based on the output of the extended state observer and using the dynamic surface control method. The proposed coordinated controller is robust to the external disturbances. Finally, the effectiveness of the proposed robust coordination control algorithm is demonstrated by the simulation results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to acknowledge the support of the Natural Science Grant of Liaoning Province (2013020002).