# Robust Coordinated Control Algorithm for Multiple Marine Vessels with External Disturbances

Authors: Weixue Liu; Jianfang Jiao; Hamid Reza Karimi; Jian Jiao
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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 Section2, 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 theith surface vessel can be represented by the following 3 degrees of freedom (DOF) [30]:
(1)η˙i=Ri(ψi)vi,(2)Miv˙i+Ci(vi)vi+Di(vi)vi=τi+τdi,
where ηi=[ni,ei,ψi]T denotes the north position, east position, and orientation which are decomposed in the Earth-fixed reference frame, vi=[ui,vi,ri]T denotes the linear surge velocity, sway velocity, and angular velocity which are decomposed in the body-fixed reference frame. Ri(ψi) is the transformations matrix from the body-fixed reference frame to the Earth-fixed reference frame, the form of which is as follows:
(3)Ri(ψi)=[cos(ψi)-sin(ψi)0sin(ψi)cos(ψi)0001].
The transformations matrix satisfies Ri-1(ψi)=RiT(ψi), for all ψi. Mi denotes the system inertia mass matrix including added mass which is positive definite. Ci(vi) and Di(vi) denote the Coriolis-centripetal matrix and damping matrix, respectively. The detailed representation of the above three system matrices can be found in reference [30]. τi=[τuiτviτri]T is the vector of forces and torques input from the thruster system. τdi 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:(4)vi=Ri-1(ψi)η˙i,
because R˙i(ψi)=Ri(ψi)S, where
(5)S=[0-ri0ri00000]=-ST
with ri being the angular velocity in the body-fixed reference frame. We can obtain that
(6)v˙i=R˙i-1(ψi)η˙i+Ri-1(ψi)η¨i=-Ri-1(ψi)R˙i(ψi)Ri-1(ψi)η˙i+Ri-1(ψi)η¨i.
Taking (4) and (6) into the vessel dynamic model (2) yields
(7)MiRi-1(ψi)η¨i-MiRi-1(ψi)R˙i(ψi)Ri-1(ψi)η˙i+Ci(vi)Ri-1(ψi)η˙i+Di(vi)Ri-1(ψi)η˙i=τi+τdi.
The above equation can be written as
(8)Mni(ηi)η¨i+Cni(ηi,η˙i)η˙i+Dni(ηi,η˙i)η˙i=τi+τdi,
where
(9)Mni(ηi)=MiRi-1(ψi);Cni(ηi,η˙i)=[Ci(vi)-MiRi-1(ψi)R˙i(ψi)]Ri-1(ψi);Dni(ηi,η˙i)=Di(vi)Ri-1(ψi).


## 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.Letη˙i=vni, and then (8) can be written as
(10)Mni(ηi)v˙ni+Cni(ηi,vni)vni+Dni(ηi,vni)vni=τi+τdi.
Then we have
(11)η˙i=vni,v˙ni=Mni-1(ηi)τi+Mni-1(ηi)(-Cni(ηi,vni)vni-Dni(ηi,vni)vni+τdi).LetKi=Mni-1(ηi)(-Cni(ηi,vni)vni-Dni(ηi,vni)vni+τdi),ui=Mni-1(ηi)τi. Then the above equation can be rewritten as
(12)η˙i=vni,v˙ni=Ki+ui.
Here Ki is assumed to be unknown. We assume that Ki is an extended state. However, Ki 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(13)z1i=ηi-η^i,η^˙i=v^ni+β1fal1(z1i,α,δ),v^˙ni=K^i+β2fal2(z1i,α,δ)+ui,K^˙i=β3fal3(z1i,α,δ),
where
(14)fal(z1i,α,δ)={|z1i|αsign(z1i),|z1i|>δ,z1iδα-1,|z1i|≤δ.
And δ>0, 0<α<1.Setz1i=ηi-η^i, z2i=vni-v^ni, z3i=Ki-K^i. η^i,v^ni,K^i are the estimated values of ηi,vni,Ki, respectively. Taking the derivative of z1i,z2i,z3i, respectively, we can obtain that
(15)z˙1i=v^ni+z2i-v^ni-β1fal1(ei,α,δ)=z2i-β1fal1(z1i,α,δ)(16)z˙2i=Ki+ui-K^i-β2fal2(z1i,α,δ)-ui=z3i-β2fal2(z1i,α,δ)(17)z˙3i=K˙i-β3fal3(z1i,α,δ).The following assumptions are presumed.(1)
The possibly unknown functionKi is continuously differentiable with respect to their variables. |K˙i|≤M for all t>0, where M is a positive constant.(2)
Letz=[z1,z2,z3]T, and ∥•∥ denotes the Euclid norm of ℝ9. There exist positive definite constants λi(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→ηi, v^ni→vni, K^i→Ki.


### 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 setI=[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 ηd, where ηd=[nd(t),ed(t),ψd(t)]T, nd(t), ed(t) are sufficiently smooth functions, and ψd(t)=arctan(e˙d(t)/n˙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 li=[x0i,y0i,ψ0i]T, then the desired trajectory of the follower vessel is denoted as ηdi=ηd+R(ψd)li.Figure 1
The formation of these vessels.


#### 3.2.2. Controller Design

Define the first dynamic surface as(18)S1i=η^i-ηdi.
Taking the derivative of the first surface, with (13), we obtain
(19)S˙1i=η^˙i-η˙di=v^ni+β1fal1(z1i,α,δ)-η˙di.
Define a virtual velocity vri as follows:
(20)vri=η˙di-β1fal1(z1i,α,δ)-Λ1S1i,
where Λ1 is a positive definite matrix. With this definition, if vri=v^ni, then S˙1i=-Λ1S1i. Then S1i→0 with a convergence rate determined by the choice of Λ1. Because of the definition of S1i, this will also guarantee that η^i→ηi→ηdi.v
r
i is passed through a first order filter in order to avoid the problem existed in the backstepping scheme:
(21)Tv˙di+vdi=vri,
where T is a diagonal matrix of the filter time constants which are chosen to be as small as possible. Because v^ni→vni, then we can define the second sliding surface as
(22)S2i=v^ni-vdi,
where vdi is the estimated value of vri; take the derivative of vdi, then we have
(23)v˙di=T-1(vri-vdi).
Taking the derivative of the second surface, with (13), we have
(24)S˙2i=v^˙ni-v˙di=K^i+β2fal2(z1i,α,δ)+ui-v˙di.
We consider the following Lyapunov function candidate:
(25)V1i=12S2iTS2i.
We take the time derivative of (25):
(26)V˙1i=S2iT(K^i+β2fal2(z1i,α,δ)+ui-v˙di).
So we choose the control input as ui=v˙di-K^i-β2fal2(z1i,α,δ)-KDS2i.The control input of the vessel isτi=Mni(ηi)ui. So the control force input τ is selected as
(27)τi=Mni(ηi)(v˙di-K^i-β2fal2(z1i,α,δ)-KDS2i).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
(28)S˙1i=S2i+vdi-β1fal1(z1i,α,δ)-η˙di.
Define the estimated error of the first order filter as
(29)S3i=-Tv˙di=vdi-vri.
Taking the derivative of S3i yields
(30)S˙3i=v˙di-vri=v˙di+β1dfal1(z1i,α,δ)dt+Λ1S˙1i-η¨di=-S3iT+β1dfal1(z1i,α,δ)dt+Λ1S˙1i-η¨di=-S3iT+g(β1,z1i,α,δ,z˙1i,η¨di,S˙1i).
With (19) and (21), we obtain
(31)S˙1i=S2i+S3i-Λ1iS1i.
Define the Lyapunov function as
(32)Vi=12S1iTS1i+12S2iTS2i+12S3iTS3i.
Differentiating the above equation yields
(33)V˙i=S1iTS˙1i+S2iTS˙2i+S3iTS˙3i=S1iT(S2i+S3i-Λ1S1i)-S2iTKDS2i+S3iT×(-S3iT+g(β1,z1i,α,δ,z˙1i,η¨di,S˙1i))=-S1iTΛ1S1i+S1iTS2i+S1iTS3i-S2iTKDS2i-S3iTS3iT+S3iTg(β1,z1i,α,δ,z˙1i,η¨di,S˙1i).
If we define the maximum of g(β1,z1i,α,δ,z˙1i,η¨di,S˙1i) is gmax, we can know that S1iTS1i+S2iTMS2i+S3iTS3i≤2p, p is a positive constant. Then we can obtain that Vi≤p. Let Λ1=KD=α0+2I3, and the filter time constant can be chosen as T=(I3+(gmaxTgmax/2εI3)+α0)-1; then we can obtain that
(34)V˙i=-S1iT(α0+2I3)S1i-S2iT(α0+2I3)S2i-S3iT(I3+(gmax22εI3)+α0)S3i+S1iTS2i+S1iTS3i+S3iTg≤-S1iT(α0+2I3)S1i-S2iT(α0+2I3)S2i+2S1iTS1i+S2iTS2i+S3iTS3i2-S3iTS3i-S3iTα0S3i-S3iT(gmaxTgmax2εI3)S3i+gmax2S3iTS3i2εI3gTggmax2+ε2≤-2α0Vi+ε2.
If we choose α0>ε/2p, then V˙i<0. We can guarantee that S2i→0. This implies v^ni→vni→vdi, in turn, S1i→0 and η^i→ηi→ηdi.


## 3.1. Extended State Observer Design

In this section, we design the extended state observer for each vessel to estimate the disturbances.Letη˙i=vni, and then (8) can be written as
(10)Mni(ηi)v˙ni+Cni(ηi,vni)vni+Dni(ηi,vni)vni=τi+τdi.
Then we have
(11)η˙i=vni,v˙ni=Mni-1(ηi)τi+Mni-1(ηi)(-Cni(ηi,vni)vni-Dni(ηi,vni)vni+τdi).LetKi=Mni-1(ηi)(-Cni(ηi,vni)vni-Dni(ηi,vni)vni+τdi),ui=Mni-1(ηi)τi. Then the above equation can be rewritten as
(12)η˙i=vni,v˙ni=Ki+ui.
Here Ki is assumed to be unknown. We assume that Ki is an extended state. However, Ki 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(13)z1i=ηi-η^i,η^˙i=v^ni+β1fal1(z1i,α,δ),v^˙ni=K^i+β2fal2(z1i,α,δ)+ui,K^˙i=β3fal3(z1i,α,δ),
where
(14)fal(z1i,α,δ)={|z1i|αsign(z1i),|z1i|>δ,z1iδα-1,|z1i|≤δ.
And δ>0, 0<α<1.Setz1i=ηi-η^i, z2i=vni-v^ni, z3i=Ki-K^i. η^i,v^ni,K^i are the estimated values of ηi,vni,Ki, respectively. Taking the derivative of z1i,z2i,z3i, respectively, we can obtain that
(15)z˙1i=v^ni+z2i-v^ni-β1fal1(ei,α,δ)=z2i-β1fal1(z1i,α,δ)(16)z˙2i=Ki+ui-K^i-β2fal2(z1i,α,δ)-ui=z3i-β2fal2(z1i,α,δ)(17)z˙3i=K˙i-β3fal3(z1i,α,δ).The following assumptions are presumed.(1)
The possibly unknown functionKi is continuously differentiable with respect to their variables. |K˙i|≤M for all t>0, where M is a positive constant.(2)
Letz=[z1,z2,z3]T, and ∥•∥ denotes the Euclid norm of ℝ9. There exist positive definite constants λi(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→ηi, v^ni→vni, K^i→Ki.


## 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 setI=[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 ηd, where ηd=[nd(t),ed(t),ψd(t)]T, nd(t), ed(t) are sufficiently smooth functions, and ψd(t)=arctan(e˙d(t)/n˙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 li=[x0i,y0i,ψ0i]T, then the desired trajectory of the follower vessel is denoted as ηdi=ηd+R(ψd)li.Figure 1
The formation of these vessels.


### 3.2.2. Controller Design

Define the first dynamic surface as(18)S1i=η^i-ηdi.
Taking the derivative of the first surface, with (13), we obtain
(19)S˙1i=η^˙i-η˙di=v^ni+β1fal1(z1i,α,δ)-η˙di.
Define a virtual velocity vri as follows:
(20)vri=η˙di-β1fal1(z1i,α,δ)-Λ1S1i,
where Λ1 is a positive definite matrix. With this definition, if vri=v^ni, then S˙1i=-Λ1S1i. Then S1i→0 with a convergence rate determined by the choice of Λ1. Because of the definition of S1i, this will also guarantee that η^i→ηi→ηdi.v
r
i is passed through a first order filter in order to avoid the problem existed in the backstepping scheme:
(21)Tv˙di+vdi=vri,
where T is a diagonal matrix of the filter time constants which are chosen to be as small as possible. Because v^ni→vni, then we can define the second sliding surface as
(22)S2i=v^ni-vdi,
where vdi is the estimated value of vri; take the derivative of vdi, then we have
(23)v˙di=T-1(vri-vdi).
Taking the derivative of the second surface, with (13), we have
(24)S˙2i=v^˙ni-v˙di=K^i+β2fal2(z1i,α,δ)+ui-v˙di.
We consider the following Lyapunov function candidate:
(25)V1i=12S2iTS2i.
We take the time derivative of (25):
(26)V˙1i=S2iT(K^i+β2fal2(z1i,α,δ)+ui-v˙di).
So we choose the control input as ui=v˙di-K^i-β2fal2(z1i,α,δ)-KDS2i.The control input of the vessel isτi=Mni(ηi)ui. So the control force input τ is selected as
(27)τi=Mni(ηi)(v˙di-K^i-β2fal2(z1i,α,δ)-KDS2i).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
(28)S˙1i=S2i+vdi-β1fal1(z1i,α,δ)-η˙di.
Define the estimated error of the first order filter as
(29)S3i=-Tv˙di=vdi-vri.
Taking the derivative of S3i yields
(30)S˙3i=v˙di-vri=v˙di+β1dfal1(z1i,α,δ)dt+Λ1S˙1i-η¨di=-S3iT+β1dfal1(z1i,α,δ)dt+Λ1S˙1i-η¨di=-S3iT+g(β1,z1i,α,δ,z˙1i,η¨di,S˙1i).
With (19) and (21), we obtain
(31)S˙1i=S2i+S3i-Λ1iS1i.
Define the Lyapunov function as
(32)Vi=12S1iTS1i+12S2iTS2i+12S3iTS3i.
Differentiating the above equation yields
(33)V˙i=S1iTS˙1i+S2iTS˙2i+S3iTS˙3i=S1iT(S2i+S3i-Λ1S1i)-S2iTKDS2i+S3iT×(-S3iT+g(β1,z1i,α,δ,z˙1i,η¨di,S˙1i))=-S1iTΛ1S1i+S1iTS2i+S1iTS3i-S2iTKDS2i-S3iTS3iT+S3iTg(β1,z1i,α,δ,z˙1i,η¨di,S˙1i).
If we define the maximum of g(β1,z1i,α,δ,z˙1i,η¨di,S˙1i) is gmax, we can know that S1iTS1i+S2iTMS2i+S3iTS3i≤2p, p is a positive constant. Then we can obtain that Vi≤p. Let Λ1=KD=α0+2I3, and the filter time constant can be chosen as T=(I3+(gmaxTgmax/2εI3)+α0)-1; then we can obtain that
(34)V˙i=-S1iT(α0+2I3)S1i-S2iT(α0+2I3)S2i-S3iT(I3+(gmax22εI3)+α0)S3i+S1iTS2i+S1iTS3i+S3iTg≤-S1iT(α0+2I3)S1i-S2iT(α0+2I3)S2i+2S1iTS1i+S2iTS2i+S3iTS3i2-S3iTS3i-S3iTα0S3i-S3iT(gmaxTgmax2εI3)S3i+gmax2S3iTS3i2εI3gTggmax2+ε2≤-2α0Vi+ε2.
If we choose α0>ε/2p, then V˙i<0. We can guarantee that S2i→0. This implies v^ni→vni→vdi, in turn, S1i→0 and η^i→ηi→ηdi.


## 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 setI=[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 ηd, where ηd=[nd(t),ed(t),ψd(t)]T, nd(t), ed(t) are sufficiently smooth functions, and ψd(t)=arctan(e˙d(t)/n˙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 li=[x0i,y0i,ψ0i]T, then the desired trajectory of the follower vessel is denoted as ηdi=ηd+R(ψd)li.Figure 1
The formation of these vessels.


## 3.2.2. Controller Design

Define the first dynamic surface as(18)S1i=η^i-ηdi.
Taking the derivative of the first surface, with (13), we obtain
(19)S˙1i=η^˙i-η˙di=v^ni+β1fal1(z1i,α,δ)-η˙di.
Define a virtual velocity vri as follows:
(20)vri=η˙di-β1fal1(z1i,α,δ)-Λ1S1i,
where Λ1 is a positive definite matrix. With this definition, if vri=v^ni, then S˙1i=-Λ1S1i. Then S1i→0 with a convergence rate determined by the choice of Λ1. Because of the definition of S1i, this will also guarantee that η^i→ηi→ηdi.v
r
i is passed through a first order filter in order to avoid the problem existed in the backstepping scheme:
(21)Tv˙di+vdi=vri,
where T is a diagonal matrix of the filter time constants which are chosen to be as small as possible. Because v^ni→vni, then we can define the second sliding surface as
(22)S2i=v^ni-vdi,
where vdi is the estimated value of vri; take the derivative of vdi, then we have
(23)v˙di=T-1(vri-vdi).
Taking the derivative of the second surface, with (13), we have
(24)S˙2i=v^˙ni-v˙di=K^i+β2fal2(z1i,α,δ)+ui-v˙di.
We consider the following Lyapunov function candidate:
(25)V1i=12S2iTS2i.
We take the time derivative of (25):
(26)V˙1i=S2iT(K^i+β2fal2(z1i,α,δ)+ui-v˙di).
So we choose the control input as ui=v˙di-K^i-β2fal2(z1i,α,δ)-KDS2i.The control input of the vessel isτi=Mni(ηi)ui. So the control force input τ is selected as
(27)τi=Mni(ηi)(v˙di-K^i-β2fal2(z1i,α,δ)-KDS2i).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
(28)S˙1i=S2i+vdi-β1fal1(z1i,α,δ)-η˙di.
Define the estimated error of the first order filter as
(29)S3i=-Tv˙di=vdi-vri.
Taking the derivative of S3i yields
(30)S˙3i=v˙di-vri=v˙di+β1dfal1(z1i,α,δ)dt+Λ1S˙1i-η¨di=-S3iT+β1dfal1(z1i,α,δ)dt+Λ1S˙1i-η¨di=-S3iT+g(β1,z1i,α,δ,z˙1i,η¨di,S˙1i).
With (19) and (21), we obtain
(31)S˙1i=S2i+S3i-Λ1iS1i.
Define the Lyapunov function as
(32)Vi=12S1iTS1i+12S2iTS2i+12S3iTS3i.
Differentiating the above equation yields
(33)V˙i=S1iTS˙1i+S2iTS˙2i+S3iTS˙3i=S1iT(S2i+S3i-Λ1S1i)-S2iTKDS2i+S3iT×(-S3iT+g(β1,z1i,α,δ,z˙1i,η¨di,S˙1i))=-S1iTΛ1S1i+S1iTS2i+S1iTS3i-S2iTKDS2i-S3iTS3iT+S3iTg(β1,z1i,α,δ,z˙1i,η¨di,S˙1i).
If we define the maximum of g(β1,z1i,α,δ,z˙1i,η¨di,S˙1i) is gmax, we can know that S1iTS1i+S2iTMS2i+S3iTS3i≤2p, p is a positive constant. Then we can obtain that Vi≤p. Let Λ1=KD=α0+2I3, and the filter time constant can be chosen as T=(I3+(gmaxTgmax/2εI3)+α0)-1; then we can obtain that
(34)V˙i=-S1iT(α0+2I3)S1i-S2iT(α0+2I3)S2i-S3iT(I3+(gmax22εI3)+α0)S3i+S1iTS2i+S1iTS3i+S3iTg≤-S1iT(α0+2I3)S1i-S2iT(α0+2I3)S2i+2S1iTS1i+S2iTS2i+S3iTS3i2-S3iTS3i-S3iTα0S3i-S3iT(gmaxTgmax2εI3)S3i+gmax2S3iTS3i2εI3gTggmax2+ε2≤-2α0Vi+ε2.
If we choose α0>ε/2p, then V˙i<0. We can guarantee that S2i→0. This implies v^ni→vni→vdi, in turn, S1i→0 and η^i→ηi→ηdi.


## 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
(35)τd=106*[0.05sin(πt/100)+0.02,oooooo0.03sin(πt/100),oooooo0.01sin(πt/100)+0.01]T(N).
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 Figures2 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 areη1=[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 ηd(t)=[ndedψd]T, and the detailed forms are nd=t, ed=1000sin(t/600), ψd=arctan(e˙d/n˙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 KD=104*diag(6.5,6.5,6.5).The simulation results are shown in Figures6 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.



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*Source: 597195-2013-11-03.xml*