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

Recently, synchronizationmovement control ofmultiple vessels has been studied broadly. Inmost 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 vesselsmaybe vary because of switching of different tasks, and the time delay is necessary to be consideredwhen 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.


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][21][22].In recent years, switching communication topologies or time delay has been investigated [23][24][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][27][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 Section 2, 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.

Preliminaries
2.1.Graph Theory.For the multiple vessels, the communication topology and information exchange among vessels can be described by a graph.Let  = {,} describe the information exchanges among vessels, which consists of nodes () = {V 1 , V 2 , . . ., V  } and edges () ⊆  × , and () composes edges where joint nodes come from .
A node in the graph represents a vessel.The edges represent the information exchange links among the vessels.Node  is a neighbor of node , if (V  , V  ) ∈ ().Let the set of the neighbors of node  denoted by   = {V  ∈ /(V  , V  ) ∈ }.Assume matrix  is the adjacency matrix of graph G, and   ∈  is defined as Define a set of  communication topology graphs  = { 1 ⋅ ⋅ ⋅   }.And the nodes of graph   are the same, but the edge sets are different.Therefore, the Laplacian matrix of graph   is denoted by   .It is necessary to satisfy that all the communication topologies are connected and   is a positive semidefinite matrix.
If the communication topology is connected, then define is a column stack vector, and  satisfies

Mathematical Model of Vessel.
The 3-DOF surface motion model of the vessel can be described as where  = [, , ]  is the position and heading in the earthfixed reference frame and V = [, , ]  is the velocity with regard to the body-fixed reference frame.() is a rotation matrix, which can be written as where  is a positive definite inertia mass matrix and  =   .(V) represents the hydrodynamic Coriolis and centripetal matrix.(V) denotes damping matrix. V illustrates the control forces and torques input. V are the external disturbances.
The motion mathematical model of the vessel in the earth-fixed reference frame is where the relation of the conversion yields The vessel model (3) holds the following properties.
Define the trajectory tracking error of vessel  as Take the derivative of η as Define the stabilizing function for vessel  as where  ∈  3×3 is a diagonal positive definite matrix.Define the velocity tracking error of vessel  as The switched function of vessel  is chosen as where Λ =  +  3×3 is a diagonal positive definite matrix.
Define a new parameter   as The vessels are said to synchronize if lim With vessel model (7), we can get In order to prove stability of  switching communication topologies, define a common Lyapunov function for all  communication topologies.
Define the first Lyapunov function as The time derivative of ( 16) yields Choose the second Lyapunov function as Therefore, the time derivative of (18) yields Let the coupling control law for vessel  be Choose the synchronization control input as where τ is the estimate of external disturbances and   is the element of weighted adjacency matrix .
Define the third Lyapunov function as where τ =   − τ and   is a positive number.Assume the disturbances   are unknown in advance and   vary slowly, which means τ  = 0.The time derivative of (22) yields where Then, we adopt an adaptive term to estimate the disturbances.Define the adaptive control law of τ as where   is a positive constant.With ( 25) and ( 23), we can obtain where  ∈ {1, . . ., } represents the th communication topology and  = [  1 ⋅ ⋅ ⋅    ]  .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 → 0,   → 0, and   → 0 as  → ∞ with the arbitrary switching among the  communication topologies; with (8) and ( 12), we can get   →   , η  → η  , and ] → 0 as  → ∞; the velocity tracking error and trajectory tracking error approached zero asymptotically.
With ( 12) and ( 13),   yields where   =   −  denotes the synchronization position error between vessel  and vessel  and   =   −   is the desired relative position between vessel  and vessel .And note   =   +   .From ( 27), we can know that ( 27) represents a linear exponentially stable system with the input   , as   → 0 and   is bounded.Then, it can be obtained that lim  → ∞ ‖  ‖ = 0, which means   −   →   , η  − η  → l .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 vessel  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.

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,   denotes the time delay from vessel  to vessel , and it is assumed that time delay is bounded and continuously differentiable, which means 0 <   ≤   < ∞, Ṫ  ≤   < 1, ∀ = 1, . . ., .In the presence of time delay, the vessels are considered to be delayed synchronization if To achieve delayed synchronization, define a positive constant gain as Similar to the front section, the procedure yields the following.
Define the first Lyapunov function as The time derivative of (30) yields Choose the second Lyapunov function as Therefore, the time derivative of (32) yields Choose the delayed synchronization control input as where τ is the estimate of the disturbances.Define the third Lyapunov function as where τ =   − τ , and   is a positive number.The time derivative of (35) yields Choose the adaptive control law of τ is Then, (36) yields Hence, η ,   ,   −   ( −   ) are bounded, and η → 0,   → 0, (  −   ( −   )) → 0, as  → ∞.The delayed synchronization can be further rewritten as where   =   −   ( −   ) +   ( −   ) −   ; 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 of  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.

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.

Figure 3 :
Figure 3: The heading yaw curves of vessels.

Figure 4 :
Figure 4: The surge velocity curves of vessels.

Figure 5 :
Figure 5: The sway velocity curves of vessels.

Figure 6 :
Figure 6: The yaw velocity curves of vessels.

Figure 7 :
Figure 7: Synchronization north position error with time delay.