# Swarm Control for Multiple Unmanned Surface Vehicles with Unknown Time-Varying Environmental Disturbance and Input Saturation

Authors: Guoqing Xia; Xianxin Sun
Journal: Journal of Control Science and Engineering
 (2022)

---

## Abstract

This paper addresses the swarm tracking problem of multiple unmanned surface vehicles subjected to unknown time-varying environmental disturbance and input saturation. The main control objective of this paper is that USVs cluster to follow the virtual leader with the desired position and heading and are required to maintain a specified position separation relative to both neighbor vehicles. In order to achieve the design goal, we mainly focus on three aspects. Firstly, to estimate the external disturbance accurately and improve the convergence speed, a finite-time disturbance observer is designed. Secondly, an auxiliary dynamic system is introduced to solve the input saturation problem. Thirdly, an output feedback controller based on a finite-time disturbance observer and an auxiliary dynamic system is designed to achieve swarm control of multiple unmanned surface vehicles. The stability of the system is proved by the Lyapunov directly method. Finally, the simulation results show that the proposed control strategy is effective.


---

## Body

## 1. Introduction

In recent years, swarm control for multiple unmanned surface vehicles (USVs) has attracted increasing interest in many fields, such as search and track mission [1], rescue operations [2], and dynamic guarding [3, 4]. These have brought new challenges to USV cluster control, especially maintaining a desired position separation relative to what is often required when USVs swarm to perform the corresponding missions [5].These challenges can be divided into two aspects: unknown time-varying environmental disturbance and input saturation. The environmental disturbance is caused by the wind, waves, and ocean currents in surge, sway, and yaw, respectively. Due to the environmental disturbance being unknown and time-varying, it cannot be measured directly and accurately. Therefore, some control strategies based on estimating disturbance or compensating for disturbance are proposed in [6–11]. Through these proposed control strategies, the observation error only accurately estimates rather than converges to a small neighborhood of an equilibrium state as soon as possible. Furthermore, to improve the convergence speed, the finite-time control technology is adopted in [12–16], which ensures the consistency of all states in the closed-loop system in a finite time. Therefore, the combination of disturbance observer and finite-time control technology is an executable program.Except for environmental disturbance, another challenge to USVs’ cluster control is input saturation. Since the USV actuator cannot be infinite, there is a deviation between the expected input signal and the actual output of the actuator. Therefore, the amplitude of the control signal is usually limited to a certain range. In the process of designing the actuator, it is necessary to consider the physical constraints of the actuator, that is, input saturation. The existence of input saturation may lead to system oscillation and even system instability. In recent years, the input saturation problem of various systems has received the most extensive attention. In [17], an adaptive mechanism is devised to figure out the input saturation problem. In [18], a novel finite-time control approach is presented to overcome the effects of state constraints on system performance. The group consensus algorithms with input saturation are given in [19]. From the vessels’ work situation, it is very necessary to take input saturation into account in cluster control through the actual working conditions of the multiple USVs.Based on the above research background, for the first challenge, a finite-time disturbance observer is proposed to resolve the first challenge, which can not only measure the disturbance accurately, but also improve the converge speed. Secondly, by using an auxiliary dynamic system to solve the second problem of input saturation. Furthermore, to solve the above two challenges more perfectly, an output feedback controller is proposed, which is mainly composed of a finite time-disturbance observer, an auxiliary dynamic system, and other control technologies. Meanwhile, it is proved that all signals in the closed-loop system are bounded by the Lyapunov method.The main contributions of this paper are summarized as follows: firstly, different from existing disturbance observer approaches in [20], the unknown time-varying environmental disturbance can be measured accurately by the finite-time disturbance observer. Secondly, unlike the existing work ignoring the actuator constraints, an auxiliary dynamic system is introduced to resolve the input saturation problem. Thirdly, an output feedback controller is designed, which is mainly composed of a finite-time disturbance observer, an auxiliary dynamic system, and other control technologies.A summary of recent works is outlined in Table1, corresponding to different features and classification.Table 1
Recent works on consensus problems.
FeaturesClassificationReferencesAbout environmental disturbanceFormation control[6]Tracking control[7, 8]Output-feedback control[5, 9, 10]About time controlFixed-time control[11]Finite-time control[12–16]Time-varying control[21]ConstraintsInput saturation[17, 19, 22]Full state constraints[18]This paper is organized as follows: Section2 describes some necessary preliminaries and mathematical modeling of USVs. Section 3 depicts the finite-time disturbance observer design. Section 4 describes the output feedback controller design and stability analysis. Simulation results and comparison results are discussed in Section 5 and Section 6 concludes this paper.Notations: the following notations will be used throughout this paper.A/B denotes all elements that belong to A and not to B. ⋅ represents the absolute value of a scalar. ⋅ denotes the Euclidean norm. Rm×n represents the m×n dimensional Euclidean Space. diagai denotes a block-diagonal matrix with ai being the ith diagonal element. ⋅T and ⋅−1 represent the transpose and inverse of a matrix, respectively. ⊗ denotes the Kronecker product of matrix. λmin⋅ and λmax⋅ represent minimum and maximum of eigenvalues a matrix, respectively. sigδ⋅=⋅δsign⋅, sign represents the Symbolic function, i.e., signt=−1, ∀t<0; signt=0, t=0; signt=1, ∀t>0. In represents the n×n dimensional identity matrix. i is used to denote the subscript of USVs.


## 2. Preliminaries and Mathematical Modeling of USVs

### 2.1. Algebraic Graph Theory

Graph theory is used to describe the communication topology ofn follower USVs and a virtual leader vehicle (denoted by 0). A directed graph G=V,ε consists of a vertex set V=0,1,2,…,n and the set of edges ε⊆i,j∈V×V. A directed edge i,j is not only the incoming edge of node j but also the outgoing edge of node i. If i,j∈ε, node j is an adjacent node of node i. The set of all adjacent nodes of node i is represents by Ni=j∈V,i,j∈ε, see [21].Consider a directed graphG composed of n nodes, the adjacency matrix A=aijn×n is used to represent the link relationship between nodes, where aij=1, if i,j∈ε; aij=0, otherwise. If aij=aji, the graph is undirected; otherwise is directed. The Laplacian matrix ℒ associates with the graph G is defined as ℒ=D−A where D=diagd1,d2,…,dn with di=∑j=1naij. Laplacian matrix ℒ always has a right eigenvector of 1n=1,1,…,1T associated with eigen value λ1=0.In particular, a diagonal matrixA0=diagai0 is defined as a leader adjacency matrix, where ai0=1, if and only if the i th USV receives information from the virtual leader vehicle; ai0=0, otherwise. Finally, the information exchange matrix is defined as ℋ=ℒ+A0.Assump 1.
The graphG is directed, and there is at least one spanning tree from the root node to the leader node, i.e., the ℋ is a positive definite matrix.


### 2.2. Finite Time Stability

Lemma 1 (see [13]).
Consider the system of differential equations(1)y˙t=fyt,f0=0,where f:U⟶Rn is continuous on an open neighborhood U⊆Rn of the origin. A continuously differentiable function y:T⟶U is said to be a solution of formula (1) on the interval T⊂R if yt satisfies formula (1) for all t∈T.
Suppose there exists a continuous positive definite functionV:U⟶R, real numbers c1>0, c2∈0,1 and an open neighborhood U0⊆U of the origin such that(2)V˙x+c1Vxc2≤0,x∈U0∖0.Then the origin is a finite-time-stable equilibrium of (1).Moreover, ifU=Rn, Vx is proper, and V˙x takes negative values on Rn∖0, then the origin is a globally finite-time-stable equilibrium of (1). And T is called the convergence-time function, it satisfies(3)Tx≤1c11−c2Vx1−c2,x∈Rn∖0.


### 2.3. USVs Modeling

Consider a group of USVs swarm consisting a virtual leader vehicle (subscript is 0) andn follower USVs (subscript is 1,2,…,n), the 3 degrees of freedom (DOFs) kinematics and dynamics equations of the i th USV can be expressed in vector form as follows:[23](4)η˙i=Riψiυi,(5)Miυ˙i+Diυi=τi+di,where Riψi is the rotation matrix, it is given as follows:(6)Riψi=cosψi−sinψi0sinψicosψi0001,with properties: Riψi=1 and RiTRiψi=I3×3. ηi:=xi,yi,ψiT is the position and yaw angle in the earth-fixed frame XEOEYE (see Figure 1). υi:=ui,vi,riT is the velocity vector in the body-fixed frame XBOBYB. The system inertia matrix Mi∈R3×3 is positive definite and constant, where Mi=MiT and Mi.=0. The damping matrix Di∈R3×3 is also symmetric and positive define. τi:=τi1,τi2,τi3T is the control input, which is produced by the propellers. di:=di1,di2,di3T is unknown time-varying environmental disturbance, which caused by the wind, waves, and ocean currents in surge, sway, and yaw, respectively.Figure 1
Earth-fixed frame and body-fixed frame.In this paper, considering the input saturation constraint, the control forces and moment produced via the propellers are limited. The input saturation constraint can be described as follows:(7)τi=τi,max,ifτic>τi,max,τic,ifτi,min≤τic≤τi,max,τi,min,ifτic<τi,min,where τi,max∈R3 and τi,min∈R3 are the maximum and minimum control forces and moment of i th vehicle, respectively. τic=τic1,τic2,τic3T is calculated by the output feedback controller.The main goal of this paper is to design a control law so that the USVs can track the desired reference point while maintaining a fixed formation, i.e.,(8)limt⟶+∞ηi−ηd−μi≤σi,where ηd:=xd,yd,ψdT is the desired reference point. μi:=xiμ,yiμ,ψiμT represents the expected relative deviation position and heading between the i th USV and the desired reference point. In order to maintain fixed formation for USVs cluster, thus μi is a constant vector. σi>0 is a positive constant.Assump 2.
The reference signalηd is smooth and differentiable everywhere. Its first derivative η˙d and second derivative η¨d exist and are bounded.Assump 3.
The time-varying environmental disturbancesdi is unknown but bounded, and its first derivative di. exists and is bounded, i.e.,(9)di≤di,max,di.≤c3,where di,max>0 and c3>0 are positive constants.Assump 4.
The system inertia matrixMi and damping matrix Di are known.


### 2.4. Environmental Disturbances Modeling.

Unmodeled external forces and moment due to wind, ocean currents, and second waves are lumped together into an earth-fixed slowly-varying bias termdi∈R3 [24, 25]. A widely used bias model for USVs is the first-order Markov process [26–28]. In this paper, the environmental disturbances are modeled as follows:(10)di=−RiTb,where b represents the first-order Markov process, it is given as follows:(11)b˙=−T−1b+Ebϑb,where T∈R3×3 is the diagonal matrix of positive bias time constants, Eb∈R3×3 is the diagonal matrix scaling the amplitude of the ϑb, ϑb∈R3 is a vector of zero-mean Gaussian while noise.


## 2.1. Algebraic Graph Theory

Graph theory is used to describe the communication topology ofn follower USVs and a virtual leader vehicle (denoted by 0). A directed graph G=V,ε consists of a vertex set V=0,1,2,…,n and the set of edges ε⊆i,j∈V×V. A directed edge i,j is not only the incoming edge of node j but also the outgoing edge of node i. If i,j∈ε, node j is an adjacent node of node i. The set of all adjacent nodes of node i is represents by Ni=j∈V,i,j∈ε, see [21].Consider a directed graphG composed of n nodes, the adjacency matrix A=aijn×n is used to represent the link relationship between nodes, where aij=1, if i,j∈ε; aij=0, otherwise. If aij=aji, the graph is undirected; otherwise is directed. The Laplacian matrix ℒ associates with the graph G is defined as ℒ=D−A where D=diagd1,d2,…,dn with di=∑j=1naij. Laplacian matrix ℒ always has a right eigenvector of 1n=1,1,…,1T associated with eigen value λ1=0.In particular, a diagonal matrixA0=diagai0 is defined as a leader adjacency matrix, where ai0=1, if and only if the i th USV receives information from the virtual leader vehicle; ai0=0, otherwise. Finally, the information exchange matrix is defined as ℋ=ℒ+A0.Assump 1.
The graphG is directed, and there is at least one spanning tree from the root node to the leader node, i.e., the ℋ is a positive definite matrix.


## 2.2. Finite Time Stability

Lemma 1 (see [13]).
Consider the system of differential equations(1)y˙t=fyt,f0=0,where f:U⟶Rn is continuous on an open neighborhood U⊆Rn of the origin. A continuously differentiable function y:T⟶U is said to be a solution of formula (1) on the interval T⊂R if yt satisfies formula (1) for all t∈T.
Suppose there exists a continuous positive definite functionV:U⟶R, real numbers c1>0, c2∈0,1 and an open neighborhood U0⊆U of the origin such that(2)V˙x+c1Vxc2≤0,x∈U0∖0.Then the origin is a finite-time-stable equilibrium of (1).Moreover, ifU=Rn, Vx is proper, and V˙x takes negative values on Rn∖0, then the origin is a globally finite-time-stable equilibrium of (1). And T is called the convergence-time function, it satisfies(3)Tx≤1c11−c2Vx1−c2,x∈Rn∖0.


## 2.3. USVs Modeling

Consider a group of USVs swarm consisting a virtual leader vehicle (subscript is 0) andn follower USVs (subscript is 1,2,…,n), the 3 degrees of freedom (DOFs) kinematics and dynamics equations of the i th USV can be expressed in vector form as follows:[23](4)η˙i=Riψiυi,(5)Miυ˙i+Diυi=τi+di,where Riψi is the rotation matrix, it is given as follows:(6)Riψi=cosψi−sinψi0sinψicosψi0001,with properties: Riψi=1 and RiTRiψi=I3×3. ηi:=xi,yi,ψiT is the position and yaw angle in the earth-fixed frame XEOEYE (see Figure 1). υi:=ui,vi,riT is the velocity vector in the body-fixed frame XBOBYB. The system inertia matrix Mi∈R3×3 is positive definite and constant, where Mi=MiT and Mi.=0. The damping matrix Di∈R3×3 is also symmetric and positive define. τi:=τi1,τi2,τi3T is the control input, which is produced by the propellers. di:=di1,di2,di3T is unknown time-varying environmental disturbance, which caused by the wind, waves, and ocean currents in surge, sway, and yaw, respectively.Figure 1
Earth-fixed frame and body-fixed frame.In this paper, considering the input saturation constraint, the control forces and moment produced via the propellers are limited. The input saturation constraint can be described as follows:(7)τi=τi,max,ifτic>τi,max,τic,ifτi,min≤τic≤τi,max,τi,min,ifτic<τi,min,where τi,max∈R3 and τi,min∈R3 are the maximum and minimum control forces and moment of i th vehicle, respectively. τic=τic1,τic2,τic3T is calculated by the output feedback controller.The main goal of this paper is to design a control law so that the USVs can track the desired reference point while maintaining a fixed formation, i.e.,(8)limt⟶+∞ηi−ηd−μi≤σi,where ηd:=xd,yd,ψdT is the desired reference point. μi:=xiμ,yiμ,ψiμT represents the expected relative deviation position and heading between the i th USV and the desired reference point. In order to maintain fixed formation for USVs cluster, thus μi is a constant vector. σi>0 is a positive constant.Assump 2.
The reference signalηd is smooth and differentiable everywhere. Its first derivative η˙d and second derivative η¨d exist and are bounded.Assump 3.
The time-varying environmental disturbancesdi is unknown but bounded, and its first derivative di. exists and is bounded, i.e.,(9)di≤di,max,di.≤c3,where di,max>0 and c3>0 are positive constants.Assump 4.
The system inertia matrixMi and damping matrix Di are known.


## 2.4. Environmental Disturbances Modeling.

Unmodeled external forces and moment due to wind, ocean currents, and second waves are lumped together into an earth-fixed slowly-varying bias termdi∈R3 [24, 25]. A widely used bias model for USVs is the first-order Markov process [26–28]. In this paper, the environmental disturbances are modeled as follows:(10)di=−RiTb,where b represents the first-order Markov process, it is given as follows:(11)b˙=−T−1b+Ebϑb,where T∈R3×3 is the diagonal matrix of positive bias time constants, Eb∈R3×3 is the diagonal matrix scaling the amplitude of the ϑb, ϑb∈R3 is a vector of zero-mean Gaussian while noise.


## 3. Finite-Time Disturbance Observer Design

Owing the environmental disturbance is unknown and time-varying, the accurate value is difficult to receive directly. To solve this problem, a finite-time disturbance observer (FTDO) is designed as follows:(12)Miυ^˙i=−Diυi+τi+d^i,where d^i is the estimate of the unknown time-varying environmental disturbance di, υ^i is the estimate of υi.A new variable is defined as follows:(13)ωi=Miυi−Miυ^i,The update lawd^i of FTDO is obtained as follows:(14)d^i=Hi1sigδi1ωi+Hi2∫sigδi2ωidt,where Hi1∈R3×3 and Hi2∈R3×3 are positive define diagonal matrices. δi1>0 and δi2>0 are positive constants and satisfy 0.5≤δi1<1 and δi2=2δi1−1, respectively.Define the estimation error of disturbancesd˜i=di−d^i. The time derivative of (13) is given as follows:(15)ω˙i=di−d^i=d˜i=−Hi1sigδi1ωi−Hi2∫sigδi2ωidt+di.Define a new variable vectorρi=xi1T,xi2TT=ωiT,ςiTT, where ςi=−Hi2∫sigδi2ωidt+di. Thus, the formula (15) is rewritten as follows:(16)x˙i1=−Hi1sigδi1xi1+xi2,(17)x˙i2=−Hi2sigδi2xi1+d˙i.If a new closed-loop system is constructed by the formulas (16) and (17), then xi1 and xi2 can be regarded as the internal states of this closed-loop system. Under these circumstances, if xi1⟶0 and xi2⟶0 in finite time, then x˙i1⟶0 and d^i=di in finite time.Therefore, the following theorem holds.Theorem 1.
The disturbance observer composed by the formulas (12) and (14), the unknown time-varying environmental disturbance can be estimated in finite time, and the disturbance estimation error converges to a neighborhood of the equilibrium point in finite time.Proof.
The Lyapunov functionVi is chosen as follows:(18)Vi=12χiTPiχi,where the vector χi and positive define matrix Pi are designed as follows, respectively.(19)χi=sigδi1xi1xi2,Pi=2Hi2δi1+Hi12−Hi1−Hi12I3×3.
Note that Lyapunov functionVi is continuous and differentiable everywhere except ρi=xi1,xi2|xi1=03×1 and positive definite. Then, we get(20)12λminPiχi2≤Vi≤12λmaxPiχi2,where(21)χi2=sigδi1xi12+xi2Txi2=xi12δi1+xi2Txi2.
The time derivative of Lyapunov functionVi is given as follows:(22)V˙i=χiTδi1Hi1−Hi2−δi10Piχi+12χiTPi03×3d˙i+χiTPiδi1xi1δi1−1−Hi1sigδi1xi1+xi2−Hi2sigδi2xi1+12χiTPi03×3d˙i=χiTPixi1δi1−1δi1−Hi1sigδi1xi1+xi2−Hi2sigδi1−1xi1sigδi1xi1+χiTδi1Hi1−Hi2−δi10Piχi+χiTPi03×3d˙i=−xi1δi1−1χiTLiχi+χiTlid˙i≤−λminLixi1δi1−1χi2+c3χili,where(23)Li=Hi1Hi2+δi1Hi12−δi1Hi1−δi1Hi1δi1I3×3,li=−Hi12I3×3.
From formula (21), the following inequality holds:(24)χi2=xi12δi1+xi2Txi2⇒χi2≥xi12δi1⇒xi1δi1−1≥χiδi1−1/δi1.
Substituting the formula (24) into (22), the inequation (22) is written as follows:(25)V˙i≤−ci1χiδi1+δi2/δi1,where ci1=λminLi−c3li/χiδi2/δi1=λminLi−c3liχi−δi2/δi1.
To ensure the coefficientci1>0, the following inequality is holds(26)χi>c3liλminLiδi1/δi2.
From formulas (20) and (26), the term V˙i is bounded, i.e.,(27)V˙i≤−ci2Vici3,where(28)ci2=2ci1λmin−δi2/2δi1PiλminPi=2ci1λmin−ci3Pi>0,0<ci3=δi1+δi22δi1<1.
According to the Lemma1, the new closed-loop system composed of formulas (16) and (17) converges to a neighborhood of the equilibrium point in finite time.
This completes the proof.


## 4. Output Feedback Controller Design

In this chapter, inspired by references [29, 30], the output feedback controller is designed. The structure diagram of multiple unmanned surface vehicles swarm control is shown in Figure 2.Figure 2
Structure diagram of multiple unmanned surface vehicles swarm control.


### 4.1. Auxiliary Dynamic System Design

In this subsection, the auxiliary dynamic system (ADS) is introduced to settle the input saturation problem [22]. For ADS, as shown in Figure 2, the input of the system is the difference value Δτi between the control forces and moments with saturation τi and those without saturation constraint τic. The output of ADS is the compensation of position and heading tracking error βi1 and the compensation of velocity tracking error βi2 for i th USV. The purpose of designing ADS is to compensate for the specified variables in the closed-loop system.Among them,βi1 and βi2 are position and heading compensation and velocity compensation for zi1 and zi2, respectively. The ADS is designed as follows:(29)β˙i1=−Li1βi1+aidJiβi2,(30)β˙i2=−Li2βi2+Mi−1Δτi.where βi1∈R3×1 and βi2∈R3×1 are the output states of the ADS. Li1∈R3×3 and Li2∈R3×3 are diagonal positive matrices. aid=di+ai0. Δτi=τi−τic, Δτi≤Δτi,max, Δτi,max>0 is a positive constant.


### 4.2. Output Feedback Controller Design

In this subsection, the output feedback controller of multiple USVs is designed by using dynamic surface control technology [20]. The design process of output feedback controller is divided into the following steps.Step 1.
The first tracking error of thei th USV in earth-fixed frame is defined as follows:(31)zi1=∑j∈Niaijηi−μi−ηj−μj+ai0ηi−ηd−μi−βi1,where Ni, aij and ai0 are defined in Section 2.2. ηi, μi and ηd are explained in formula (8), ηj and μj have similar definitions.
The time derivative ofzi1, and using the formula (4) and (29), we obtain as follows:(32)z˙i1=∑j∈Niaijη˙i−∑j∈Niaijη˙j+ai0η˙i−ai0η˙d−β˙i1=∑j∈Niaij+ai0Riυi−∑j∈Niaijη˙j−ai0η˙d+Li1βi1−aidJiβi2=aidRiυi−∑j∈Niaijη˙j−ai0η˙d+Li1βi1−aidJiβi2.
Choosingυi as a virtual input in formula (32), by using dynamic surface control technology, the kinematic control law αi is designed as follows:(33)αi=RiTaid−Ki1zi1+∑j∈Niaijη˙j+ai0η˙d−Li1βi1,where Ki1∈R3×3 is the positive definite diagonal matrix.
From the kinematic control law obtained, it can be found that it is complex to calculate the time derivative of theαi. Therefore, a first-order low-poss filter is introduced to settle the matter.
Letαi pass through a first-order low-poss filter(34)ιiυ˙id=α˙i−υid,υid0=αi0,where ιi>0 is a time constant, υid∈R3×1 is the output vector of the first-order low-poss filter.Step 2.
According to the velocityυi of the i th USV, the output υid of the first-order low-pass filter, and the state βi2 of ADS, the second tracking error of the i th USV is defined as follows:(35)zi2=υi−υid−βi2.
By differentiatingzi2 and using the formula (5) and (30), we get the formula as follows:(36)Miz˙i2=−Diυi+τic+di−Miυ˙id−Li2βi2.
Furthermore, the termdi in formula (36) is replaced by the estimated value d^i of unknown time-varying environmental disturbance produced via the FTDO. Then, by using dynamic surface control method, the dynamic control law τic is designed as follows:(37)τic=−Ki2zi2+Diυi+Miυ˙id−Li2βi2−d^i,where Ki2∈R3×3 is the positive definite diagonal matrix.
By substituting formula (33) and (37) into (31) and (35), the error subsystem of the i th USV is obtained as follows:(38)z˙i1=−Ki1zi1+aidRizi2+qi,(39)Miz˙i2=−Ki2zi2−d˜i,where qi=υid−αi, d˜i is explained in formula (15).
According to the definition of vectorqi, by differentiating qi, and using the formula (34), we get the equation as follows:(40)q˙i=−qiιi−α˙i.
By integrating the two sides of formula (40), we have the following formula:(41)qit=qi0e−t/ιi−∫0te−t−τ/ιiα˙iυdυ.
Further calculation, the following inequality holds:(42)qit≤qi0e−t/ιi+supυ∈0,tα˙iυ∫0te−t−υ/ιidυ≤qi0e−t/ιi+ιiα˙i∞.
Owing to the control input of adjacent USVs are bounded, i.e.,∃αi,max>0:α˙i≤αi,max,∀t∈0,∞. In fact, since all systems are energy consuming system, the output of all USVs is bounded. Therefore, the following inequality holds:(43)qit≤qi0e−t/ιi+ιiαi,max.
Therefore,qit is bounded and qit≤qi,max.


### 4.3. Stability Analysis

Theorem 2.
A close-loop system is considered, which is composed ofn USVs cluster with input saturation, FTDO (12) and (14) with unknown time-varying environmental disturbance, ADS (29) and (30), output feedback controller (33) and (37). The swarm control scheme guarantees that all error signals in this close-loop system are bounded, and the tracking error converges to a neighborhood of the equilibrium point in finite time.Proof.
Consider the above closed-loop system, the Lyapunov functionV is chosen as follows:(44)V=∑i=1n12zi1Tzi1+12zi2TMizi2+12βi1Tβi1+12βi2Tβi2.
The time derivative ofV, we obtain the following equation:(45)V˙=∑i=1nzi1Tz˙i1+zi2TMiz˙i2+βi1Tβ˙i1+βi2Tβ˙i2.
From the formula (32), and using Young’s inequality, we get the following equation:(46)zi1Tz˙i1=zi1T−Ki1zi1+aidRizi2+qi=−zi1TKi1zi1+zi1TaidRizi2+zi1TaidRiqi≤−zi1TKi1zi1+aid2zi12+aid2zi22+aid2zi12+aid2qi2≤−zi1TKi1zi1+aidzi12+aid2zi22+aid2qi2.
Similarly, the following inequalities hold:(47)zi2TMiz˙i2≤−zi2TKi2zi2−12zi22+12d˜i2,βi1Tβ˙i1≤−βi1TLi1βi1+aid2βi12+aid2βi22,βi2Tβ˙i2≤−βi2TLi2βi2+12ΔτiTMi−1TMi−1Δτi+12βi22.
Substituting the formulas (46) and (47) into (45), then (45) is rewritten as follows:(48)V˙≤∑i=1nzi1Tz˙i1+zi2TMiz˙i2+βi1Tβ˙i1+βi2Tβ˙i2,≤∑i=1n−λminKi1−αidzi12−λminKi2−αid+12zi22−λminLi1−αid2βi12−λminLi2−αid+12βi22+12d˜i2+aid2qi2+12Mi−1ΔτiTMi−1Δτi.
From the FTDO, the termd˜i converges to zero in finite time. According to the formula (43), qit≤qi,max. For the i th USV, Mi is a known positive constant matrix. Δτi≤Δτi,max is defined in Section 4.1. Thus, we have the following equation:(49)c4=∑i=1n12d˜i2+aid2qi,max2+12λminMi−1TMi−1Δτi,max2.where c4 is bounded, i.e., 0<c4≤c4,max.
To ensure the stability of the closed-loop system, the parametersKi1, Ki2, Li1, and Li2 should meet the following conditions:(50)bi1=λminKi1−αid>0bi2=λminKi2−αid+12>0bi3=λminLi1−αid2>0bi4=λminLi2−αid+12>0where bi1, bi2, bi3, and bi4 are positive constants.
Substituting the formulas (49) and (50) into (48), then (48) is rewritten as follows:(51)V˙≤∑i=1n−bi1zi12−bi2zi22−bi3βi12−bi4βi22+c4,max≤−bV+c4,max.where b=2minbi1,bi2,bi3,bi4>0.
From the formula (51), either zi1>c4,max/bi1, or zi2>c4,max/bi2, or βi1>c4,max/bi3, or βi1>c4,max/bi3, then V˙≤0 holds. This implies that:(52)Vt≤V0−c4,maxbe−bt+c4,maxb.
It can be concluded that all signalszi1,zi2,βi1,βi2,d^i in the closed-loop system are bounded. Thus, the values of variables zi1 and βi1 are given as:(53)zi1≤c4,maxb,(54)βi1≤c4,maxb.
Next, it will be proved that the USV cluster can track the reference signal with the desired relative deviation.
The tracking error of thei th USV in the earth-fixed frame is defined as δi, it satisfies the following constraints:(55)δi=ηi−ηd−μi,
According to the formula (31), we obtain:(56)z1+β1=ℋ⊗I3δ,where z1=z11T,z21T,…,zn1TT, β1=β11T,β21T,…,βn1TT, and δ=δ1T,δ2T,…,δnTT. ℋ is defined in Section 2.2.
According to the Assumption1 of algebraic graph theory in the Section 2.2, it can be concluded that all eigenvalues of matrix ℋ have positive real parts. Furthermore, using the formulas (53) and (54), we get the following equation:(57)‖δ‖≤‖z1‖+‖β1‖λminℋ≤22c4,maxλminℋb.
According to the formulas (51) and (57), by adjusting the parameters of controller and disturbance observer, the upper bound of tracking error can be reduced and finally converges to a very small neighborhood close to zero.
This completes the proof.


## 4.1. Auxiliary Dynamic System Design

In this subsection, the auxiliary dynamic system (ADS) is introduced to settle the input saturation problem [22]. For ADS, as shown in Figure 2, the input of the system is the difference value Δτi between the control forces and moments with saturation τi and those without saturation constraint τic. The output of ADS is the compensation of position and heading tracking error βi1 and the compensation of velocity tracking error βi2 for i th USV. The purpose of designing ADS is to compensate for the specified variables in the closed-loop system.Among them,βi1 and βi2 are position and heading compensation and velocity compensation for zi1 and zi2, respectively. The ADS is designed as follows:(29)β˙i1=−Li1βi1+aidJiβi2,(30)β˙i2=−Li2βi2+Mi−1Δτi.where βi1∈R3×1 and βi2∈R3×1 are the output states of the ADS. Li1∈R3×3 and Li2∈R3×3 are diagonal positive matrices. aid=di+ai0. Δτi=τi−τic, Δτi≤Δτi,max, Δτi,max>0 is a positive constant.


## 4.2. Output Feedback Controller Design

In this subsection, the output feedback controller of multiple USVs is designed by using dynamic surface control technology [20]. The design process of output feedback controller is divided into the following steps.Step 1.
The first tracking error of thei th USV in earth-fixed frame is defined as follows:(31)zi1=∑j∈Niaijηi−μi−ηj−μj+ai0ηi−ηd−μi−βi1,where Ni, aij and ai0 are defined in Section 2.2. ηi, μi and ηd are explained in formula (8), ηj and μj have similar definitions.
The time derivative ofzi1, and using the formula (4) and (29), we obtain as follows:(32)z˙i1=∑j∈Niaijη˙i−∑j∈Niaijη˙j+ai0η˙i−ai0η˙d−β˙i1=∑j∈Niaij+ai0Riυi−∑j∈Niaijη˙j−ai0η˙d+Li1βi1−aidJiβi2=aidRiυi−∑j∈Niaijη˙j−ai0η˙d+Li1βi1−aidJiβi2.
Choosingυi as a virtual input in formula (32), by using dynamic surface control technology, the kinematic control law αi is designed as follows:(33)αi=RiTaid−Ki1zi1+∑j∈Niaijη˙j+ai0η˙d−Li1βi1,where Ki1∈R3×3 is the positive definite diagonal matrix.
From the kinematic control law obtained, it can be found that it is complex to calculate the time derivative of theαi. Therefore, a first-order low-poss filter is introduced to settle the matter.
Letαi pass through a first-order low-poss filter(34)ιiυ˙id=α˙i−υid,υid0=αi0,where ιi>0 is a time constant, υid∈R3×1 is the output vector of the first-order low-poss filter.Step 2.
According to the velocityυi of the i th USV, the output υid of the first-order low-pass filter, and the state βi2 of ADS, the second tracking error of the i th USV is defined as follows:(35)zi2=υi−υid−βi2.
By differentiatingzi2 and using the formula (5) and (30), we get the formula as follows:(36)Miz˙i2=−Diυi+τic+di−Miυ˙id−Li2βi2.
Furthermore, the termdi in formula (36) is replaced by the estimated value d^i of unknown time-varying environmental disturbance produced via the FTDO. Then, by using dynamic surface control method, the dynamic control law τic is designed as follows:(37)τic=−Ki2zi2+Diυi+Miυ˙id−Li2βi2−d^i,where Ki2∈R3×3 is the positive definite diagonal matrix.
By substituting formula (33) and (37) into (31) and (35), the error subsystem of the i th USV is obtained as follows:(38)z˙i1=−Ki1zi1+aidRizi2+qi,(39)Miz˙i2=−Ki2zi2−d˜i,where qi=υid−αi, d˜i is explained in formula (15).
According to the definition of vectorqi, by differentiating qi, and using the formula (34), we get the equation as follows:(40)q˙i=−qiιi−α˙i.
By integrating the two sides of formula (40), we have the following formula:(41)qit=qi0e−t/ιi−∫0te−t−τ/ιiα˙iυdυ.
Further calculation, the following inequality holds:(42)qit≤qi0e−t/ιi+supυ∈0,tα˙iυ∫0te−t−υ/ιidυ≤qi0e−t/ιi+ιiα˙i∞.
Owing to the control input of adjacent USVs are bounded, i.e.,∃αi,max>0:α˙i≤αi,max,∀t∈0,∞. In fact, since all systems are energy consuming system, the output of all USVs is bounded. Therefore, the following inequality holds:(43)qit≤qi0e−t/ιi+ιiαi,max.
Therefore,qit is bounded and qit≤qi,max.


## 4.3. Stability Analysis

Theorem 2.
A close-loop system is considered, which is composed ofn USVs cluster with input saturation, FTDO (12) and (14) with unknown time-varying environmental disturbance, ADS (29) and (30), output feedback controller (33) and (37). The swarm control scheme guarantees that all error signals in this close-loop system are bounded, and the tracking error converges to a neighborhood of the equilibrium point in finite time.Proof.
Consider the above closed-loop system, the Lyapunov functionV is chosen as follows:(44)V=∑i=1n12zi1Tzi1+12zi2TMizi2+12βi1Tβi1+12βi2Tβi2.
The time derivative ofV, we obtain the following equation:(45)V˙=∑i=1nzi1Tz˙i1+zi2TMiz˙i2+βi1Tβ˙i1+βi2Tβ˙i2.
From the formula (32), and using Young’s inequality, we get the following equation:(46)zi1Tz˙i1=zi1T−Ki1zi1+aidRizi2+qi=−zi1TKi1zi1+zi1TaidRizi2+zi1TaidRiqi≤−zi1TKi1zi1+aid2zi12+aid2zi22+aid2zi12+aid2qi2≤−zi1TKi1zi1+aidzi12+aid2zi22+aid2qi2.
Similarly, the following inequalities hold:(47)zi2TMiz˙i2≤−zi2TKi2zi2−12zi22+12d˜i2,βi1Tβ˙i1≤−βi1TLi1βi1+aid2βi12+aid2βi22,βi2Tβ˙i2≤−βi2TLi2βi2+12ΔτiTMi−1TMi−1Δτi+12βi22.
Substituting the formulas (46) and (47) into (45), then (45) is rewritten as follows:(48)V˙≤∑i=1nzi1Tz˙i1+zi2TMiz˙i2+βi1Tβ˙i1+βi2Tβ˙i2,≤∑i=1n−λminKi1−αidzi12−λminKi2−αid+12zi22−λminLi1−αid2βi12−λminLi2−αid+12βi22+12d˜i2+aid2qi2+12Mi−1ΔτiTMi−1Δτi.
From the FTDO, the termd˜i converges to zero in finite time. According to the formula (43), qit≤qi,max. For the i th USV, Mi is a known positive constant matrix. Δτi≤Δτi,max is defined in Section 4.1. Thus, we have the following equation:(49)c4=∑i=1n12d˜i2+aid2qi,max2+12λminMi−1TMi−1Δτi,max2.where c4 is bounded, i.e., 0<c4≤c4,max.
To ensure the stability of the closed-loop system, the parametersKi1, Ki2, Li1, and Li2 should meet the following conditions:(50)bi1=λminKi1−αid>0bi2=λminKi2−αid+12>0bi3=λminLi1−αid2>0bi4=λminLi2−αid+12>0where bi1, bi2, bi3, and bi4 are positive constants.
Substituting the formulas (49) and (50) into (48), then (48) is rewritten as follows:(51)V˙≤∑i=1n−bi1zi12−bi2zi22−bi3βi12−bi4βi22+c4,max≤−bV+c4,max.where b=2minbi1,bi2,bi3,bi4>0.
From the formula (51), either zi1>c4,max/bi1, or zi2>c4,max/bi2, or βi1>c4,max/bi3, or βi1>c4,max/bi3, then V˙≤0 holds. This implies that:(52)Vt≤V0−c4,maxbe−bt+c4,maxb.
It can be concluded that all signalszi1,zi2,βi1,βi2,d^i in the closed-loop system are bounded. Thus, the values of variables zi1 and βi1 are given as:(53)zi1≤c4,maxb,(54)βi1≤c4,maxb.
Next, it will be proved that the USV cluster can track the reference signal with the desired relative deviation.
The tracking error of thei th USV in the earth-fixed frame is defined as δi, it satisfies the following constraints:(55)δi=ηi−ηd−μi,
According to the formula (31), we obtain:(56)z1+β1=ℋ⊗I3δ,where z1=z11T,z21T,…,zn1TT, β1=β11T,β21T,…,βn1TT, and δ=δ1T,δ2T,…,δnTT. ℋ is defined in Section 2.2.
According to the Assumption1 of algebraic graph theory in the Section 2.2, it can be concluded that all eigenvalues of matrix ℋ have positive real parts. Furthermore, using the formulas (53) and (54), we get the following equation:(57)‖δ‖≤‖z1‖+‖β1‖λminℋ≤22c4,maxλminℋb.
According to the formulas (51) and (57), by adjusting the parameters of controller and disturbance observer, the upper bound of tracking error can be reduced and finally converges to a very small neighborhood close to zero.
This completes the proof.


## 5. Simulation Results

In this section, a USV swarm consisting of one virtual leader vehicle (indexed by 0) and six follower USVs (indexed by1,2,…,6) is considered to demonstrate the effectiveness of the proposed control strategy. The directed communication graph is shown in Figure 3.Figure 3
Directed communication topology.In simulations, the model of the surface ship Cybership II is used [31]. The time-varying environmental disturbances are modeled as first-order Markov processes [24]. The control forces and moment are limited as τi1,max=−τi1,min=2N, τi2,max=−τi2,min=2N and τi3,max=−τi3,min=1.5Nm. The desired reference point is set as ηd=15m,15m,45degT. Some parameters setting are shown in Table 2.Table 2
The parameters of six follower USVs.
μidata/m,m,degηidata/m,m,degμ1−2.8,0.3,0Tη1−2.5,0,0Tμ2−1.6,2.5,0Tη2−1.3,2.2,0Tμ3−1.6,1.9,0Tη3−1.3,−2.2,0Tμ41.0,2.5,0Tη41.3,2.2,0Tμ51.6,−2.5,0Tη51.3,−2.2,0Tμ62.2,−0.3,0Tη62.5,0,0T


### 5.1. Performance of Proposed Control Strategy

In this subsection, the simulation results are given to verify the performance of proposed control strategy.The parameters of environmental disturbance are selected asT=diag10−3,10−3,10−3, b0=0,0,0T, Eb=diag10−5,10−5,10−5, ϑb=0.5,0.5,0.5T. The parameters of observer are selected as Li1=diag10,10,10, Li2=diag10,10,10, δi1=0.75 and δi2=0.5. The design parameters of controller are chosen as ιi=0.001, Ki1=diag1.2,1.2,1.2 and Ki2=diag0.6,0.6,0.6.The simulation results are shown in Figures4–8. Figure 4 shows the trajectories of six USVs under the constraints of time-varying environmental disturbance and input saturation. It can be seen from Figure 4, the position and heading of six USVs can track the reference signal in the form of swarm motion and maintain the desired relative position with each other, although there is a deviation in the initial position of each USV. Figure 5 depicts the position and heading tracking errors of six USVs. It can be seen from the figure that the tracking errors of six USVs are convergent. Figure 6 shows the control forces and moments of six USVs with an auxiliary dynamic system. Through the analysis of Figure 6, the control inputs of six USVs are limited within the range of input constraints, and finally the constraint of input amplitude saturation is realized. Figure 7 depicts the true value of the time-varying environmental disturbance acting on the 1st USV and the estimated values obtained by methods FTDO and MDA, respectively. Figure 8 shows the disturbance error d˜i of six USVs between the actual value di and estimation value d^i via the FTDO. The specific values of simulation test and comparison test are shown in Table 3.Figure 4
Trajectories of six follower USVs.Figure 5
Tracking errors of six follower USVs.Figure 6
The control inputτi of six USVs with ADS.Figure 7
The time-varying environmental disturbance estimation of the 1th USV.Figure 8
The disturbance errord˜i of six USVs.Table 3
Specific values of simulation test and comparison test.
Performance parameterValuesConsider ADSConvergence time24.0 sNo ADSConvergence time31.2 sConsider input saturationτi1,max and τi2,max2 NNo input saturationτi1,max and τi2,max200 NConsider input saturationτi3,max1.5 NmNo input saturationτi3,max150 Nm


### 5.2. Comparison Group

The first comparison simulation is carried out by using a modular design approach (MDA) proposed in [20].Simulation results on the proposed predictor module for estimating the value and estimation error of the unknown ocean disturbances are shown in Figures7 and 9. Figure 7 shows the different time-varying environmental disturbance estimations of the 1st USV about actual value, MDA, and FTDO, respectively. Figure 9 shows the disturbance error of six USVs by using MDA. From Figures 7–9, the control inputs of two methods are both limited, but the more accurate estimation of the environmental disturbance is obtained by FTDO.Figure 9
The disturbance errord˜i of six USVs via MDA.The other simulation results of the proposed state feedback controller without the auxiliary dynamic system and input saturation are illustrated in Figures10 and 11, Figure 10 shows the control input of six follower USVs without considering the auxiliary dynamic system. Figure 11 shows the control input of six follower USVs without actuator constraint. Through comparing Figures 6, 10, and 11, the auxiliary dynamic system can effectively finish off the input saturation, and it is very necessary to take the physical constraints of the system into account when designing the controller.Figure 10
The control inputτi of six USVs with the input saturation without ADS.Figure 11
The control inputτi of six USVs without the actuator constraint.


## 5.1. Performance of Proposed Control Strategy

In this subsection, the simulation results are given to verify the performance of proposed control strategy.The parameters of environmental disturbance are selected asT=diag10−3,10−3,10−3, b0=0,0,0T, Eb=diag10−5,10−5,10−5, ϑb=0.5,0.5,0.5T. The parameters of observer are selected as Li1=diag10,10,10, Li2=diag10,10,10, δi1=0.75 and δi2=0.5. The design parameters of controller are chosen as ιi=0.001, Ki1=diag1.2,1.2,1.2 and Ki2=diag0.6,0.6,0.6.The simulation results are shown in Figures4–8. Figure 4 shows the trajectories of six USVs under the constraints of time-varying environmental disturbance and input saturation. It can be seen from Figure 4, the position and heading of six USVs can track the reference signal in the form of swarm motion and maintain the desired relative position with each other, although there is a deviation in the initial position of each USV. Figure 5 depicts the position and heading tracking errors of six USVs. It can be seen from the figure that the tracking errors of six USVs are convergent. Figure 6 shows the control forces and moments of six USVs with an auxiliary dynamic system. Through the analysis of Figure 6, the control inputs of six USVs are limited within the range of input constraints, and finally the constraint of input amplitude saturation is realized. Figure 7 depicts the true value of the time-varying environmental disturbance acting on the 1st USV and the estimated values obtained by methods FTDO and MDA, respectively. Figure 8 shows the disturbance error d˜i of six USVs between the actual value di and estimation value d^i via the FTDO. The specific values of simulation test and comparison test are shown in Table 3.Figure 4
Trajectories of six follower USVs.Figure 5
Tracking errors of six follower USVs.Figure 6
The control inputτi of six USVs with ADS.Figure 7
The time-varying environmental disturbance estimation of the 1th USV.Figure 8
The disturbance errord˜i of six USVs.Table 3
Specific values of simulation test and comparison test.
Performance parameterValuesConsider ADSConvergence time24.0 sNo ADSConvergence time31.2 sConsider input saturationτi1,max and τi2,max2 NNo input saturationτi1,max and τi2,max200 NConsider input saturationτi3,max1.5 NmNo input saturationτi3,max150 Nm


## 5.2. Comparison Group

The first comparison simulation is carried out by using a modular design approach (MDA) proposed in [20].Simulation results on the proposed predictor module for estimating the value and estimation error of the unknown ocean disturbances are shown in Figures7 and 9. Figure 7 shows the different time-varying environmental disturbance estimations of the 1st USV about actual value, MDA, and FTDO, respectively. Figure 9 shows the disturbance error of six USVs by using MDA. From Figures 7–9, the control inputs of two methods are both limited, but the more accurate estimation of the environmental disturbance is obtained by FTDO.Figure 9
The disturbance errord˜i of six USVs via MDA.The other simulation results of the proposed state feedback controller without the auxiliary dynamic system and input saturation are illustrated in Figures10 and 11, Figure 10 shows the control input of six follower USVs without considering the auxiliary dynamic system. Figure 11 shows the control input of six follower USVs without actuator constraint. Through comparing Figures 6, 10, and 11, the auxiliary dynamic system can effectively finish off the input saturation, and it is very necessary to take the physical constraints of the system into account when designing the controller.Figure 10
The control inputτi of six USVs with the input saturation without ADS.Figure 11
The control inputτi of six USVs without the actuator constraint.


## 6. Conclusion

This paper investigates swarm control for multiple USVs in the presence of an unknown time-varying environmental disturbance and input saturation. Firstly, an observer is designed to estimate disturbance by using finite time control technology. Secondly, an auxiliary dynamic system is introduced to finish off the input saturation problem. Thirdly, an output feedback controller is designed, which is mainly composed of a finite-time disturbance observer, an auxiliary dynamic system, and other control technologies. At the same time, the stability of the system is proved by the Lyapunov method. Finally, the effectiveness of the proposed control strategy is verified by simulation results.However, in the design process of the controller, only input amplitude saturation and time-varying environmental disturbance are considered: collision avoidance and obstacle avoidance are ignored. Safe, reliable, and collision-free track planning is an important research direction in the field of unmanned surface vehicles. This is the focus of our future research.



---

*Source: 8962970-2022-07-11.xml*