# Adaptive Synchronization Control of Multiple Vessels with Switching Communication Topologies and Time Delay

Authors: Fuguang Ding; Yanqin Ma; Yuanwei Zhou; Jiangjun Li
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

Recently, synchronization movement control of multiple vessels has been studied broadly. In most of the studies, the communication network among vessels is considered to be fixed and the time delay is often ignored. However, the communication network among vessels maybe vary because of switching of different tasks, and the time delay is necessary to be considered when the communication network is unreliable. In this paper, the synchronization movement of multiple vessels with switching connected communication topologies is studied, and an adaptive synchronization control algorithm that is based on backstepping sliding mode control is proposed. The control algorithm is achieved by defining cross coupling error which is combination of the trajectory tracking error and velocity tracking error. And an adaptive control term is used to estimate the external disturbances, so that the unknown external disturbances can be compensated. Furthermore, the robustness of the control law to time-varying time delay is also discussed. At last, some simulations are carried out to validate the effectiveness of the proposed synchronization control algorithm.


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## Body

## 1. Introduction

Recently, the applications of synchronization movement control of multiple vessels are increasing. For example, in underway replenishment, its need to control supply vessels maintains the same state with the receiving vessel to insure safety. And another example is a group of vessels performing the seabed mapping operations together, which can implement larger area in shorter time compared with a single vessel. Multiple vessels work together not only improving operation performance but also reducing difficulty. And this paper mainly discusses the synchronization movement control of multiple vessels.In recent years, synchronization control of multiple agents has been extensively studied in different fields, such as robot systems [1, 2], chaotic systems consensus [3, 4], and coordinated formation control of aircraft and spacecraft [5, 6]. As the development of synchronization control of multiple agents, there are several synchronization control approaches which are proposed. For example, distributed cooperative attitude synchronization control approach has been discussed in [7]. And adaptive control is a classical strategy used to address synchronization movement [8, 9]. Besides, using cascaded system theory and graph theory, a distributed attitude cooperative control scheme is studied in [10]. Compared with the earlier work in computational load and required states, Chung and Slotine proposes a simple synchronization framework to achieve cooperative movement of multiple agents [11]. In the presence of uncertainties of underactuated autonomous surface vehicles and ocean disturbances, a robust adaptive dynamic surface control law is proposed in [12]. There are some other synchronization movement control methods, such as backstepping control design [13, 14], artificial potential approach [15], leader-follower network strategy [16, 17], and methods that are based on graph theory [18, 19]. The main traits of the aforementioned articles are that they assume the communication topology is fixed without considering time delay.Switching communication topologies and time delay are two fundamental realities in the communication among vessels, but these factors are usually ignored in [20–22]. In recent years, switching communication topologies or time delay has been investigated [23–25], and time delay in the communication network is usually assumed to be a constant. However, in multiple-vessel network, because the bandwidth of communication is limited, communication topology among vessels may be varied when the marine task changes. And the time delay cannot be a constant because of the varying relative position among vessels and unknown external statistics. Thus, it is necessary to design a synchronization controller for multiple vessels with switching communication topologies and time delay. Moreover, for the surface vessels, which often encounter external disturbances, and the disturbances are usually difficult to model because of the disturbances varying with the complex ocean circumstance. So the synchronization movement control law should be robust to unknown disturbances and an adaptive control is useful to solve this problem [26–28].In view of the above reasons, the main innovation of this paper can be drawn as follows: an adaptive synchronization movement control law that is based on backstepping sliding mode control for multiple vessels with switching communication topologies is proposed. Different from the traditional controller design approach that maybe suffers difficulties to determine the Lyapunov function, the backstepping-based design method provides us with an appropriate Lyapunov function simply and ensures the stability of closed-loop system, and the sliding mode method shows robustness to external disturbances and system uncertainty; the combination of backstepping and sliding mode takes both advantages of the two methods, and an adaptive term is introduced to improve the synchronization control algorithm. Moreover, the directed strongly connected communication topology may be not balanced, which means one vessel can receive information from the neighbors and not necessarily share its own information with the neighbors, and compared to the existing studies, the requirement on the communication is relaxed. And cross coupling error using trajectory tracking error and velocity tracking error is defined, and the cross coupling error is introduced into the switched function. Furthermore, in the presence of time delay, the designed synchronization control strategy is improved, and the synchronization control method is robust to time delay.The arrangement of this paper is as follows. In Section2, the basic knowledge for graph theory and the mathematical model of vessels are given. The adaptive synchronization control approach with switching communication topologies and time delay is discussed in Section 3. In Section 4, some simulations are carried out to validate the proposed control algorithm. At last, conclusions and constructive prospects are drawn in Section 5.


## 2. Preliminaries

### 2.1. Graph Theory

For the multiple vessels, the communication topology and information exchange among vessels can be described by a graph. LetG={V,E} describe the information exchanges among vessels, which consists of nodes V(G)={v1,v2,…,vn} and edges E(G)⊆V×V, and E(G) composes edges where joint nodes come from V. A node in the graph represents a vessel. The edges represent the information exchange links among the vessels. Node j is a neighbor of node i, if (vi,vj)∈E(G). Let the set of the neighbors of node i denoted by Ni={vj∈V/(vi,vj)∈E}. Assume matrix A is the adjacency matrix of graph G, and aij∈A is defined as aij>0, if (vi,vj)∈E, otherwise,   aij=0, if (vi,vj)∉E. Define di=∑j≠iaij,D=diag⁡{di}; then the Laplacian matrix of the weighted graph G can be noted L=D-A. For all (vi,vj)∈E(G), if (vj,vi)∈E(G), then the graph is called undirected. Otherwise, it is directed.Define a set ofm communication topology graphs G={G1⋯Gm}. And the nodes of graph Gt are the same, but the edge sets are different. Therefore, the Laplacian matrix of graph Gt is denoted by Lt. It is necessary to satisfy that all the communication topologies are connected and Lt is a positive semidefinite matrix.If the communication topology is connected, then define∀x=x1T⋯xmTT is a column stack vector, and x satisfies xT(L⊗I)x=1/2∑i=1n∑j=1naij(xi-xj).


### 2.2. Mathematical Model of Vessel

The 3-DOF surface motion model of the vessel can be described as(1)η˙=Rψν,Mv˙+Cνν+Dνν=τv+τvd,
where η=n,e,ψT is the position and heading in the earth-fixed reference frame and v=u,υ,rT is the velocity with regard to the body-fixed reference frame. R(ψ) is a rotation matrix, which can be written as
(2)Rψ=cos⁡⁡ψ-sin⁡ψ0sin⁡ψcos⁡⁡ψ0001,
where M is a positive definite inertia mass matrix and M=MT. C(v) represents the hydrodynamic Coriolis and centripetal matrix. D(v) denotes damping matrix. τv illustrates the control forces and torques input. τvd are the external disturbances.The motion mathematical model of the vessel in the earth-fixed reference frame is(3)ν=RTψη˙,Mηη¨+Cη,η˙η˙+Dη,η˙η˙=τη+τd,
where the relation of the conversion yields
(4)Mη=R-TψMR-1ψ,Cη,η˙=R-T(ψ)C(v)R-1(ψ)-R-TψMR-1ψR˙-1ψ,Dη,η˙=R-TψDvR-1ψ,τη=R-Tψτv,τd=R-Tψτvd.The vessel model (3) holds the following properties.Property 1.
The inertia matrixM(η) is symmetric, positive definite, and bounded as
(5)λmin⁡M≤Mη≤λmax⁡M.Property 2.
The damping matrixD(η,η˙) satisfies
(6)ηTD(η,η˙)η=ηTDη,η˙+Dη,η˙Tη2≥0.


## 2.1. Graph Theory

For the multiple vessels, the communication topology and information exchange among vessels can be described by a graph. LetG={V,E} describe the information exchanges among vessels, which consists of nodes V(G)={v1,v2,…,vn} and edges E(G)⊆V×V, and E(G) composes edges where joint nodes come from V. A node in the graph represents a vessel. The edges represent the information exchange links among the vessels. Node j is a neighbor of node i, if (vi,vj)∈E(G). Let the set of the neighbors of node i denoted by Ni={vj∈V/(vi,vj)∈E}. Assume matrix A is the adjacency matrix of graph G, and aij∈A is defined as aij>0, if (vi,vj)∈E, otherwise,   aij=0, if (vi,vj)∉E. Define di=∑j≠iaij,D=diag⁡{di}; then the Laplacian matrix of the weighted graph G can be noted L=D-A. For all (vi,vj)∈E(G), if (vj,vi)∈E(G), then the graph is called undirected. Otherwise, it is directed.Define a set ofm communication topology graphs G={G1⋯Gm}. And the nodes of graph Gt are the same, but the edge sets are different. Therefore, the Laplacian matrix of graph Gt is denoted by Lt. It is necessary to satisfy that all the communication topologies are connected and Lt is a positive semidefinite matrix.If the communication topology is connected, then define∀x=x1T⋯xmTT is a column stack vector, and x satisfies xT(L⊗I)x=1/2∑i=1n∑j=1naij(xi-xj).


## 2.2. Mathematical Model of Vessel

The 3-DOF surface motion model of the vessel can be described as(1)η˙=Rψν,Mv˙+Cνν+Dνν=τv+τvd,
where η=n,e,ψT is the position and heading in the earth-fixed reference frame and v=u,υ,rT is the velocity with regard to the body-fixed reference frame. R(ψ) is a rotation matrix, which can be written as
(2)Rψ=cos⁡⁡ψ-sin⁡ψ0sin⁡ψcos⁡⁡ψ0001,
where M is a positive definite inertia mass matrix and M=MT. C(v) represents the hydrodynamic Coriolis and centripetal matrix. D(v) denotes damping matrix. τv illustrates the control forces and torques input. τvd are the external disturbances.The motion mathematical model of the vessel in the earth-fixed reference frame is(3)ν=RTψη˙,Mηη¨+Cη,η˙η˙+Dη,η˙η˙=τη+τd,
where the relation of the conversion yields
(4)Mη=R-TψMR-1ψ,Cη,η˙=R-T(ψ)C(v)R-1(ψ)-R-TψMR-1ψR˙-1ψ,Dη,η˙=R-TψDvR-1ψ,τη=R-Tψτv,τd=R-Tψτvd.The vessel model (3) holds the following properties.Property 1.
The inertia matrixM(η) is symmetric, positive definite, and bounded as
(5)λmin⁡M≤Mη≤λmax⁡M.Property 2.
The damping matrixD(η,η˙) satisfies
(6)ηTD(η,η˙)η=ηTDη,η˙+Dη,η˙Tη2≥0.


## 3. Adaptive Synchronization Control Design

### 3.1. Adaptive Synchronization Control with Switching Communication Topologies

The objective of this section is to develop a control method for achieving state synchronization of multiple vessels while tracking time-varying trajectoriesηid, and i=1,2,…,n, in the presence of switching communication topologies, and a new backstepping sliding mode synchronization controller is designed.The model of vesseli in a compact form yields
(7)vi=RiTψη˙i,η¨i=Mi-1τi+τid-Ci-Di.Define the trajectory tracking error of vesseli as
(8)η~i=ηi-ηid.Take the derivative ofη~i as
(9)η~˙i=η˙i-η˙id.Define the stabilizing function for vesseli as
(10)αi=λη~i,
where λ∈R3×3 is a diagonal positive definite matrix.Define the velocity tracking error of vesseli as
(11)v~i=η~˙i+λη~i.The switched function of vesseli is chosen as
(12)σi=η~i+v~i=η~˙i+Λη~i,
where Λ=λ+I3×3 is a diagonal positive definite matrix.Define a new parameterσij as
(13)σij=σi-σj.The vessels are said to synchronize if(14)lim⁡t→∞σij=0,∀i,j∈Ni.With vessel model (7), we can get
(15)Miσ˙i=Miη~¨i+Λη~˙i=Miη¨i-η¨id+MiΛη~˙i=τi+τid-Ciη˙i-Diη˙i-Miη¨id+MiΛη~˙i.In order to prove stability ofm switching communication topologies, define a common Lyapunov function for all m communication topologies.Define the first Lyapunov function as(16)V1=12∑i=1nη~iTη~i.The time derivative of (16) yields
(17)V˙1=∑i=1nη~iTη~˙i=∑i=1n(η~iTσi-η~iTΛη~i).Choose the second Lyapunov function as(18)V2=V1+12∑i=1nσiTσi.Therefore, the time derivative of (18) yields
(19)V˙2=V˙1+∑i=1nσiTσ˙i=∑i=1n(η~iTσi-η~iTΛη~i)+∑i=1nσiTMi-1τi+τid-Ciη˙i-Diη˙i-Miη¨id+MiΛη~˙i.Let the coupling control law for vesseli be
(20)fi=aij∑j=1n(σi-σj)=aij∑j=1nσij.Choose the synchronization control input as(21)τi=Ciη˙i+Diη˙i-τ^id+Miη¨id-MiΛη~˙i-Miη~i+σi+fi+sign⁡σi,
where τ^id is the estimate of external disturbances and aij is the element of weighted adjacency matrix A.Define the third Lyapunov function as(22)V3=V2+∑i=1n12κiτ~idTτ~id,
where τ~id=τid-τ^id and κi is a positive number.Assume the disturbancesτid are unknown in advance and τid vary slowly, which means τ˙id=0.The time derivative of (22) yields
(23)V˙3=V˙2-∑i=1n1κiτ~idTτ^˙id=∑i=1nη~iTσi-η~iTΛη~i-σiT(η~i+σi+sign⁡(σi))+∑i=1nσiTMi-1τ~id-σiTaij∑j=1nσij-1κiτ~idTτ^˙id=∑i=1n-η~iTΛη~i-σiTσi-σiTsign⁡(σi)+∑i=1nv~iTMi-1τ~id-σiTaij∑j=1nσij-1κiτ~idTτ^˙id=-∑i=1nτ~idTκiη~iTΛη~i+σiTσi+σiTsign⁡σikkkkkkk+τ~idTκi-κiMi-Tσi+τ^˙idτ~idTκi-12∑i=1n∑j=1nσijTaijσij,
where
(24)12∑i=1n∑j=1nσijTaijσij=12∑i=1n∑j=1nσiTaijσij-σjTaijσij=12∑i=1n∑j=1nσiTaijσij+12∑i=1n∑j=1nσjTaijσji=∑i=1n∑j=1nσiTaijσij.Then, we adopt an adaptive term to estimate the disturbances. Define the adaptive control law ofτ^id as
(25)τ^˙id=κiMi-Tσi,
where κi is a positive constant.With (25) and (23), we can obtain
(26)V˙3=-∑i=1nη~iTΛη~i+σiTσi+σiTsign⁡(σi)-12∑i=1n∑j=1nσijTaijσij=-∑i=1nη~iTΛη~i+σiTσi+σiTsign⁡(σi)-σTLt⊗Ikσ≤-∑i=1nη~iTΛη~i+σiTσi+σi-σTLt⊗Ikσ,
where t∈{1,…,m} represents the tth communication topology and σ=σ1T⋯σnTT.Consider the communication topologies are connected; then (26) is a negative definite function, and in view of the Lyapunov function (22) is a strict common function for the switched system. Therefore, we can get η~i→0, σi→0, and σij→0 as t→∞ with the arbitrary switching among the m communication topologies; with (8) and (12), we can get ηi→ηid, η˙i→η˙id, and ν~→0 as t→∞; the velocity tracking error and trajectory tracking error approached zero asymptotically.With (12) and (13), σij yields
(27)σij=σi-σj=η~˙i+Λη~i-η~˙j-Λη~j=η˙i-η˙j+l˙ij+Λ(ηi-ηj+lij)=η˙ij+l˙ij+Λ(ηij+lij)=μ˙ij+Λμij,
where ηij=ηi-ηj denotes the synchronization position error between vessel i and vessel j and lij=ηid-ηjd is the desired relative position between vessel i and vessel j. And note μij=ηij+lij. From (27), we can know that (27) represents a linear exponentially stable system with the input σij, as σij→0 and μij is bounded. Then, it can be obtained that lim⁡t→∞‖μij‖=0, which means ηi-ηj→lij, η˙i-η˙j→l˙ij. Therefore, the synchronization position error and synchronization velocity error approach zero asymptotically; that is, the vessels achieve state synchronization.Theorem 1.
Consider the model of vesseli described by (7) with the synchronization control law (21) and adaptive control law (25). If the communication topologies are connected, then the synchronization errors and the tracking errors are uniformly ultimately bounded in spite of switching communication topologies and unknown disturbances.


### 3.2. Adaptive Synchronization Control with Time Delay

In view of unreliable communication networks among multiple vessels, time-varying time delay is introduced to describe the condition. In this section,Tij denotes the time delay from vessel i to vessel j, and it is assumed that time delay is bounded and continuously differentiable, which means 0<Tij≤Tm<∞, T˙ij≤Tm<1, ∀i=1,…,n.In the presence of time delay, the vessels are considered to be delayed synchronization if(28)lim⁡t→∞σjt-Tji-σit=0,∀i,j∈Ni.To achieve delayed synchronization, define a positive constant gain as(29)dji2≤1-Tm.Similar to the front section, the procedure yields the following.Define the first Lyapunov function as(30)V1=12∑i=1nη~iTη~i.The time derivative of (30) yields
(31)V˙1=∑i=1nη~iTη~˙i=∑i=1nη~iTσi-η~iTΛη~i.Choose the second Lyapunov function as(32)V2=V1+12∑i=1nσiTσi+12∑i=1n∑j=1naij∫t-TjitσjTεσjεdε.Therefore, the time derivative of (32) yields
(33)V˙2≤V˙1+∑i=1nσiTσ˙i+12∑i=1n∑j=1naijσjTσj-(1-T˙ji)σjT(t-Tji)σj(t-Tji)≤∑i=1nη~iTσi-η~iTΛη~i+∑i=1nσiTMi-1(τi+τid-Ciη˙i-Diη˙i-Miη¨id+MiΛη~˙i)+12∑i=1n∑j=1naijσjTσj-1-T˙jiσjTt-Tjiσjt-Tji.Choose the delayed synchronization control input as(34)τi=Ciη˙i+Diη˙i-τ^id+Miη¨id-MiΛη~˙i-Mi(η~i+σi+sign⁡(σi))-Miaij∑j=1ndji2σjt-Tji-dji2+12σi,
where τ^id is the estimate of the disturbances.Define the third Lyapunov function as(35)V3=V2+∑i=1n12κiτ~idTτ~id,
where τ~id=τid-τ^id, and κi is a positive number.The time derivative of (35) yields
(36)V˙3=V˙2-∑i=1n1κiτ~idTτ^˙id≤∑i=1n(-η~iTΛη~i+σiTMi-1τ~id-σiTsign⁡(σi))+∑i=1nσiTaij∑j=1ndji2σjt-Tji-dji2+12σi-∑i=1n1κiτ~idTτ^˙ids+12∑i=1n∑j=1naijσjTσj-(1-T˙ji)σjT(t-Tji)σj(t-Tji)≤∑i=1n-η~iTΛη~i+σiTMi-1τ~id-σiTsign⁡(σi)-1κiτ~idTτ^˙id+aij2∑i=1n∑j=1n(2σiTdji2σj(t-Tji)-σiTdji2σi-σiTσi)+aij2∑i=1n∑j=1n(σjTσj-(1-T˙ji)σjT(t-Tji)σj(t-Tji))≤∑i=1n-η~iTΛη~i+σiTMi-1τ~id-σiTsign⁡(σi)-1κiτ~idTτ^˙id+aij2∑i=1n∑j=1n2σiTdji2σjt-Tji-σiTdji2σi-aij2∑i=1n∑j=1ndji2σjTt-Tjiσjt-Tji.Choose the adaptive control law ofτ^id is
(37)τ^˙id=κiMi-Tσi.Then, (36) yields
(38)V˙3≤-∑i=1nη~iTΛη~i+σiTsign⁡(σi)-aij2∑i=1n∑j=1ndji2σi-σjt-TjiT(σi-σj(t-Tji))≤-∑i=1nη~iTΛη~i+σi-aij2∑i=1n∑j=1ndji2σi-σjt-TjiTσi-σjt-Tji.Hence,η~i,σi,σi-σj(t-Tji) are bounded, and η~i→0, σi→0, (σi-σj(t-Tji))→0, as t→∞. The delayed synchronization can be further rewritten as
(39)σi-σjt-Tji=η˙i-η˙id+Λ(ηi-ηid)-η˙j(t-Tji)+η˙jd(t-Tji)-Λ(ηj(t-Tji)-ηjd(t-Tji))=η˙i-η˙jt-Tji+η˙jdt-Tji-η˙id+Ληi-ηjt-Tji+ηjdt-Tji-ηid=e˙ij+Λeij,
where eij=ηi-ηj(t-Tji)+ηjd(t-Tji)-ηid; similar to the preceding section, we can approve that the trajectory tracking error, velocity tracking error, and synchronization error are asymptotically stable.Theorem 2.
Consider a group ofn vessels described by (7) with communication delay in the communication network, using the delayed synchronization control law (34) and disturbances adaptive control law (37); the trajectory tracking error, velocity tracking error, and synchronization error are asymptotically stable, and the multiple vessels can realize synchronization movement.


## 3.1. Adaptive Synchronization Control with Switching Communication Topologies

The objective of this section is to develop a control method for achieving state synchronization of multiple vessels while tracking time-varying trajectoriesηid, and i=1,2,…,n, in the presence of switching communication topologies, and a new backstepping sliding mode synchronization controller is designed.The model of vesseli in a compact form yields
(7)vi=RiTψη˙i,η¨i=Mi-1τi+τid-Ci-Di.Define the trajectory tracking error of vesseli as
(8)η~i=ηi-ηid.Take the derivative ofη~i as
(9)η~˙i=η˙i-η˙id.Define the stabilizing function for vesseli as
(10)αi=λη~i,
where λ∈R3×3 is a diagonal positive definite matrix.Define the velocity tracking error of vesseli as
(11)v~i=η~˙i+λη~i.The switched function of vesseli is chosen as
(12)σi=η~i+v~i=η~˙i+Λη~i,
where Λ=λ+I3×3 is a diagonal positive definite matrix.Define a new parameterσij as
(13)σij=σi-σj.The vessels are said to synchronize if(14)lim⁡t→∞σij=0,∀i,j∈Ni.With vessel model (7), we can get
(15)Miσ˙i=Miη~¨i+Λη~˙i=Miη¨i-η¨id+MiΛη~˙i=τi+τid-Ciη˙i-Diη˙i-Miη¨id+MiΛη~˙i.In order to prove stability ofm switching communication topologies, define a common Lyapunov function for all m communication topologies.Define the first Lyapunov function as(16)V1=12∑i=1nη~iTη~i.The time derivative of (16) yields
(17)V˙1=∑i=1nη~iTη~˙i=∑i=1n(η~iTσi-η~iTΛη~i).Choose the second Lyapunov function as(18)V2=V1+12∑i=1nσiTσi.Therefore, the time derivative of (18) yields
(19)V˙2=V˙1+∑i=1nσiTσ˙i=∑i=1n(η~iTσi-η~iTΛη~i)+∑i=1nσiTMi-1τi+τid-Ciη˙i-Diη˙i-Miη¨id+MiΛη~˙i.Let the coupling control law for vesseli be
(20)fi=aij∑j=1n(σi-σj)=aij∑j=1nσij.Choose the synchronization control input as(21)τi=Ciη˙i+Diη˙i-τ^id+Miη¨id-MiΛη~˙i-Miη~i+σi+fi+sign⁡σi,
where τ^id is the estimate of external disturbances and aij is the element of weighted adjacency matrix A.Define the third Lyapunov function as(22)V3=V2+∑i=1n12κiτ~idTτ~id,
where τ~id=τid-τ^id and κi is a positive number.Assume the disturbancesτid are unknown in advance and τid vary slowly, which means τ˙id=0.The time derivative of (22) yields
(23)V˙3=V˙2-∑i=1n1κiτ~idTτ^˙id=∑i=1nη~iTσi-η~iTΛη~i-σiT(η~i+σi+sign⁡(σi))+∑i=1nσiTMi-1τ~id-σiTaij∑j=1nσij-1κiτ~idTτ^˙id=∑i=1n-η~iTΛη~i-σiTσi-σiTsign⁡(σi)+∑i=1nv~iTMi-1τ~id-σiTaij∑j=1nσij-1κiτ~idTτ^˙id=-∑i=1nτ~idTκiη~iTΛη~i+σiTσi+σiTsign⁡σikkkkkkk+τ~idTκi-κiMi-Tσi+τ^˙idτ~idTκi-12∑i=1n∑j=1nσijTaijσij,
where
(24)12∑i=1n∑j=1nσijTaijσij=12∑i=1n∑j=1nσiTaijσij-σjTaijσij=12∑i=1n∑j=1nσiTaijσij+12∑i=1n∑j=1nσjTaijσji=∑i=1n∑j=1nσiTaijσij.Then, we adopt an adaptive term to estimate the disturbances. Define the adaptive control law ofτ^id as
(25)τ^˙id=κiMi-Tσi,
where κi is a positive constant.With (25) and (23), we can obtain
(26)V˙3=-∑i=1nη~iTΛη~i+σiTσi+σiTsign⁡(σi)-12∑i=1n∑j=1nσijTaijσij=-∑i=1nη~iTΛη~i+σiTσi+σiTsign⁡(σi)-σTLt⊗Ikσ≤-∑i=1nη~iTΛη~i+σiTσi+σi-σTLt⊗Ikσ,
where t∈{1,…,m} represents the tth communication topology and σ=σ1T⋯σnTT.Consider the communication topologies are connected; then (26) is a negative definite function, and in view of the Lyapunov function (22) is a strict common function for the switched system. Therefore, we can get η~i→0, σi→0, and σij→0 as t→∞ with the arbitrary switching among the m communication topologies; with (8) and (12), we can get ηi→ηid, η˙i→η˙id, and ν~→0 as t→∞; the velocity tracking error and trajectory tracking error approached zero asymptotically.With (12) and (13), σij yields
(27)σij=σi-σj=η~˙i+Λη~i-η~˙j-Λη~j=η˙i-η˙j+l˙ij+Λ(ηi-ηj+lij)=η˙ij+l˙ij+Λ(ηij+lij)=μ˙ij+Λμij,
where ηij=ηi-ηj denotes the synchronization position error between vessel i and vessel j and lij=ηid-ηjd is the desired relative position between vessel i and vessel j. And note μij=ηij+lij. From (27), we can know that (27) represents a linear exponentially stable system with the input σij, as σij→0 and μij is bounded. Then, it can be obtained that lim⁡t→∞‖μij‖=0, which means ηi-ηj→lij, η˙i-η˙j→l˙ij. Therefore, the synchronization position error and synchronization velocity error approach zero asymptotically; that is, the vessels achieve state synchronization.Theorem 1.
Consider the model of vesseli described by (7) with the synchronization control law (21) and adaptive control law (25). If the communication topologies are connected, then the synchronization errors and the tracking errors are uniformly ultimately bounded in spite of switching communication topologies and unknown disturbances.


## 3.2. Adaptive Synchronization Control with Time Delay

In view of unreliable communication networks among multiple vessels, time-varying time delay is introduced to describe the condition. In this section,Tij denotes the time delay from vessel i to vessel j, and it is assumed that time delay is bounded and continuously differentiable, which means 0<Tij≤Tm<∞, T˙ij≤Tm<1, ∀i=1,…,n.In the presence of time delay, the vessels are considered to be delayed synchronization if(28)lim⁡t→∞σjt-Tji-σit=0,∀i,j∈Ni.To achieve delayed synchronization, define a positive constant gain as(29)dji2≤1-Tm.Similar to the front section, the procedure yields the following.Define the first Lyapunov function as(30)V1=12∑i=1nη~iTη~i.The time derivative of (30) yields
(31)V˙1=∑i=1nη~iTη~˙i=∑i=1nη~iTσi-η~iTΛη~i.Choose the second Lyapunov function as(32)V2=V1+12∑i=1nσiTσi+12∑i=1n∑j=1naij∫t-TjitσjTεσjεdε.Therefore, the time derivative of (32) yields
(33)V˙2≤V˙1+∑i=1nσiTσ˙i+12∑i=1n∑j=1naijσjTσj-(1-T˙ji)σjT(t-Tji)σj(t-Tji)≤∑i=1nη~iTσi-η~iTΛη~i+∑i=1nσiTMi-1(τi+τid-Ciη˙i-Diη˙i-Miη¨id+MiΛη~˙i)+12∑i=1n∑j=1naijσjTσj-1-T˙jiσjTt-Tjiσjt-Tji.Choose the delayed synchronization control input as(34)τi=Ciη˙i+Diη˙i-τ^id+Miη¨id-MiΛη~˙i-Mi(η~i+σi+sign⁡(σi))-Miaij∑j=1ndji2σjt-Tji-dji2+12σi,
where τ^id is the estimate of the disturbances.Define the third Lyapunov function as(35)V3=V2+∑i=1n12κiτ~idTτ~id,
where τ~id=τid-τ^id, and κi is a positive number.The time derivative of (35) yields
(36)V˙3=V˙2-∑i=1n1κiτ~idTτ^˙id≤∑i=1n(-η~iTΛη~i+σiTMi-1τ~id-σiTsign⁡(σi))+∑i=1nσiTaij∑j=1ndji2σjt-Tji-dji2+12σi-∑i=1n1κiτ~idTτ^˙ids+12∑i=1n∑j=1naijσjTσj-(1-T˙ji)σjT(t-Tji)σj(t-Tji)≤∑i=1n-η~iTΛη~i+σiTMi-1τ~id-σiTsign⁡(σi)-1κiτ~idTτ^˙id+aij2∑i=1n∑j=1n(2σiTdji2σj(t-Tji)-σiTdji2σi-σiTσi)+aij2∑i=1n∑j=1n(σjTσj-(1-T˙ji)σjT(t-Tji)σj(t-Tji))≤∑i=1n-η~iTΛη~i+σiTMi-1τ~id-σiTsign⁡(σi)-1κiτ~idTτ^˙id+aij2∑i=1n∑j=1n2σiTdji2σjt-Tji-σiTdji2σi-aij2∑i=1n∑j=1ndji2σjTt-Tjiσjt-Tji.Choose the adaptive control law ofτ^id is
(37)τ^˙id=κiMi-Tσi.Then, (36) yields
(38)V˙3≤-∑i=1nη~iTΛη~i+σiTsign⁡(σi)-aij2∑i=1n∑j=1ndji2σi-σjt-TjiT(σi-σj(t-Tji))≤-∑i=1nη~iTΛη~i+σi-aij2∑i=1n∑j=1ndji2σi-σjt-TjiTσi-σjt-Tji.Hence,η~i,σi,σi-σj(t-Tji) are bounded, and η~i→0, σi→0, (σi-σj(t-Tji))→0, as t→∞. The delayed synchronization can be further rewritten as
(39)σi-σjt-Tji=η˙i-η˙id+Λ(ηi-ηid)-η˙j(t-Tji)+η˙jd(t-Tji)-Λ(ηj(t-Tji)-ηjd(t-Tji))=η˙i-η˙jt-Tji+η˙jdt-Tji-η˙id+Ληi-ηjt-Tji+ηjdt-Tji-ηid=e˙ij+Λeij,
where eij=ηi-ηj(t-Tji)+ηjd(t-Tji)-ηid; similar to the preceding section, we can approve that the trajectory tracking error, velocity tracking error, and synchronization error are asymptotically stable.Theorem 2.
Consider a group ofn vessels described by (7) with communication delay in the communication network, using the delayed synchronization control law (34) and disturbances adaptive control law (37); the trajectory tracking error, velocity tracking error, and synchronization error are asymptotically stable, and the multiple vessels can realize synchronization movement.


## 4. Simulation Results

To verify the effectiveness of the proposed synchronization control algorithm, some simulations are carried out. Assume there are four vessels; the mode parameters are illustrated in [29]. The switching topologies are illustrated in Figure 1.The communication topologies of vessels.
(a)
(b)
(c)Therefore, the adjacency matrix and Laplacian matrix of the communication graphs are(40)Aa=0.5×0001100001000010,Ab=0.5×0001100001000110,Ac=0.5×0011100001000010,La=0.5×1-10001-10001-1-1001,Lb=0.5×1-10002-1-1001-1-1001,Lc=0.5×1-10001-10-102-1-1001.Assume the initial position of vessels as follows:(41)η1=0100T,η2=-4030πT,η3=-9050π6T,η4=50-10π3T.And the desired trajectory of each vessel is(42)ηid=nideidψidT,ψid=artann˙ide˙id,n1d=t,e1d=1000sint500,n2d=t,e2d=1000sint500+Rψ1d×0500T,n3d=t,e3d=1000sint500+Rψ1d×01000T,n4d=t,e4d=1000sint500+Rψ1d×0-500T.The external disturbances are chosen as(43)τid=2sint100sint150-1sint100+1T.The parameters of the controller are(44)κi=3,Λ=diag⁡10,15,8.The simulation results with switching topologies are shown in Figures2–6. In the first 1000 s, communication topology (a) is used, and from 1000 s to 2000 s, the communication topology (b) is employed. In the last 1000 s, the communication topology (c) is employed.Figure 2
Synchronization movement curves of vessels.Figure 3
The heading yaw curves of vessels.Figure 4
The surge velocity curves of vessels.Figure 5
The sway velocity curves of vessels.Figure 6
The yaw velocity curves of vessels.The synchronization movement curves of vessels are shown in Figure2. From the curves, it can be seen that the proposed control law can achieve synchronization movement for multiple vessels in the presence of switching communication topologies. The heading yaw curves of four vessels are shown in Figure 3. And in Figures 4, 5, and 6, the surge velocity, sway velocity, and yaw velocity in the synchronization process are shown.In the next section simulation, we discuss when there is time-varying time delay in the communication topology (a), and simulations are carried out. Assume the time delay as follows:T14=T21=0.1+0.3sint/10T43=T32=0.2sin(t/5),dji is assumed to be equal for the sake of simplicity which is selected as dji=0.4. The simulation results are shown in Figures 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and 18.Figure 7
Synchronization north position error with time delay.Figure 8
Synchronization north position error without time delay.Figure 9
Synchronization east position error with time delay.Figure 10
Synchronization east position error without time delay.Figure 11
Synchronization heading error with time delay.Figure 12
Synchronization heading error without time delay.Figure 13
Synchronization sway velocity error with time delay.Figure 14
Synchronization sway velocity error without time delay.Figure 15
Synchronization surge velocity error with time delay.Figure 16
Synchronization surge velocity error without time delay.Figure 17
Synchronization yaw velocity error with time delay.Figure 18
Synchronization yaw velocity error without time delay.From Figures7–18, it can be seen that comparing to the case without time delay in the communication network, the adjusting time may be longer. But the synchronization position error and velocity error among vessels asymptotically converge to zeros in the presence of time delay; that is, the vessels can achieve synchronization movement, and the designed control law is robust to time-varying time delay.


## 5. Conclusion

For the synchronization control of multiple vessels in the presence of switching communication topologies, an adaptive synchronization control algorithm is proposed. With the directed connected graph describing the communication topology among vessels, a new cross coupling error that includes trajectory tracking error and velocity tracking error is added into the backstepping sliding mode controller, and an adaptive term is introduced to estimate the unknown external disturbances. Furthermore, by introducing a new parameter, the improved synchronization control law is robust to time-varying time delay. Finally, the effectiveness of the synchronization control strategy is supported through several simulations. In the future work, it is desired to consider the mode uncertainty, and it would be useful considering that the velocities of vessels are unknown.



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*Source: 143597-2015-01-22.xml*