Distributed Formation–Containment Tracking Control for Multi-Hovercraft Systems with Compound Perturbations

: Aiming at the problem of hovercraft formation–containment control with compound perturbations including model uncertainties and ocean disturbances, a distributed control algorithm for underactuated hovercraft formation–containment is proposed by combining adaptive linear extended state observer (ALESO) and radial basis function neural network (RBFNN). Firstly, ALESO and RBFNN are designed to estimate the ocean disturbances and model uncertainties, respectively, for dynamic compensation in the controller. Then, the auxiliary variables are introduced into the formation error function, and the lateral and longitudinal error stabilization is transformed into the design of longitudinal force and rotational torque by using the skew-symmetric matrix transformation, which solves the lateral underactuated problem of the hovercraft. Finally, the uniform ultimate boundedness of formation–containment cooperative errors is proved by the Lyapunov stability theory. Digital simulation veriﬁes the eﬀectiveness of the proposed method.


Introduction
As a high-performance vessel, a hovercraft can travel at high speed on particular surfaces, such as ice, swamps, beaches, grass, etc., which cannot be achieved by conventional surface vessels [1,2].Currently, the research on hovercraft mainly focuses on the heading, rotation rate, and trajectory tracking control problem of a single hovercraft under constraint conditions [3][4][5][6][7].However, a single hovercraft may face issues such as insufficient personnel and carrying resources when the task is more complex.For specific complex tasks, there is a need for more intelligent multi-hovercraft systems that can encompass a broader range of practical engineering task requirements.The multi-hovercraft can complete the assigned task through cooperation and collaboration, which can significantly improve the efficiency of task completion.
With the in-depth study of multi-agent cooperative control problems, many related research branches have emerged, such as formation control [8][9][10][11], containment control [12][13][14], etc.Based on the leader-follower idea, the mathematical model of the formation is given in [15]; that is, all leaders will track the time-varying trajectory obtained by the decomposition of the desired formation configuration, and the followers are distributed to the convex hull formed by the leaders.Based on this model, this paper designs a formation-containment configuration model for multi-hovercraft systems to meet some specific mission requirements of multi-hovercraft systems, such as formation escort operations.The advantage of the multi-hovercraft formation-containment configuration is that some of the vessels in the multi-hovercraft system with be er performance and equipment can be considered as the leader vessels, and the rest as the follower vessels, and a control strategy can be adapted to converge the follower into the convex hull formed by the leaders.In the process of this cooperative control task, the followers are only responsible for the transportation task, while the leaders with be er performance are not only responsible for the transportation task but also responsible for monitoring the surrounding environment, eliminating potential security risks and other tasks while protecting the followers of the inner layer [16][17][18].
The hull structure and cushion navigation characteristics of the hovercraft are different from those of the conventional surface vessel, which causes it to have uncertainties such as parameter uncertainties, modeling errors, and other unknown ocean disturbances [19].At the same time, the hovercraft is a typical underactuated vessel due to the lack of a lateral propulsion mechanism.How to reduce the influence of uncertainties on hovercraft and improve the anti-interference ability of hovercraft is a difficult problem.For complex model modeling problems, scholars have proposed many solutions, such as extended state observer-based control [20,21], model-free adaptive control [22], neural network control [3,23], and data-driven control [24].The extended state observer (ESO) proposed by J.H. in his pioneering work [25] not only has the ability of state observation but also can estimate the generalized disturbances between the controlled object and the controlled object model in real time.In [20], an adaptive ESO is designed to estimate the unmeasurable linear velocity, angular velocity, and unknown external disturbances.The slowly varying environmental disturbances and model uncertainties are treated separately, and experiments prove the method's effectiveness.Since a neural network (NN) can approximate any nonlinear functions, the adaptive controller based on a neural network has been widely used in objects with model uncertainties and unknown disturbances [3].Reference [24] developed a model-free fuzzy control law based on a data-driven fuzzy predictor, which simultaneously learns unknown control gains and uncertain dynamics.Aiming at the underactuated problem of hovercraft, the literature [26] transformed the lateral thrust design into the yaw moment under underactuated by introducing an auxiliary variable, which solved the underactuated problem of a three-degree-of-freedom (3-DOF) surface vessel.In reference [17], the collision avoidance problem in formation-containment tracking control of multiple USVs with limited speed and driving force is studied, but the uncertainty of the model is not considered.In [16,18], the time-varying formation-containment problem of multiple unmanned underwater vehicle systems in three-dimensional space is studied and an extended state observer is designed to estimate the external disturbance and unknown nonlinearity in real time, but the underactuated problem of the system is not considered.
Inspired by the above scheme, this paper studies the formation-containment control problem of underactuated hovercraft with ocean disturbances and model uncertainties.The main contributions of this paper can be briefly summarized as follows: 1. Compared with the literature [27], this paper expands the formation-containment motion to the cooperative mission of 4-DOF hovercraft that considers compound uncertainties and underactuated problems.Furthermore, the asymptotic stability of the controller is proved by Lyapunov's method.This layered structure allows for more complex and flexible tasks and provides greater adaptability to the complex sea operations of hovercraft.2. By introducing an auxiliary variable in the cooperative error, the underactuated problem of the hovercraft is solved.Compared with the literature [7], multiple derivations of the virtual control law are avoided, thus reducing the excessive differential terms in the yaw direction control law.Compared with reference [18], a novel scheme is designed to solve the underactuated problem in the formation-containment controller design process.3. A novel ALESO is combined with RBFNN to estimate the unknown ocean disturbances and model uncertainties of hovercraft.Compared with reference [20], the ALESO relaxes the restriction on the change rate of disturbances.By adding a linear adaptive factor to the adaptive update law, it is easy to prove that all the closed-loop signals of the whole system are boundedly stable.
The remainder of this paper is organized as follows: Section 2 presents the mathematical models and preliminary knowledge.In Section 3, the formation-containment controller is proposed.Cooperative stability analysis is presented in Section 4. Furthermore, a simulation example is given in Section 5. Finally, Section 6 presents some conclusions.

Mathematical Models
Consider a multi-hovercraft system consisting of 1 R leaders,

R R
 followers, and a virtual leader, where . The virtual leader number is defined as 0, the leader number is defined as , and the follower number is defined . The reference system diagram of the hovercraft is shown in Figure 1.The 4-DOF kinematic model of the n th hovercraft is as follows where n     ,

 
, , , represents the position and a itude information of the n th hovercraft;

 
, , , represents the velocity information of the n th hov- . Suppose .   The resultant velocity of the n th hovercraft is defined as Then, (1) can be rewri en as where

Preliminary Knowledge
The directed topology graph   ,     is used to describe the communication rela- tionship between hovercrafts, where the node set represents the set of all hovercrafts, and the edge set      represents the set of edges of the communication relationship between hovercrafts.Edge   , n j  indicates that the n th hovercraft can ob- tain the information of the j th hovercraft, where the j th hovercraft refers to the adjacent hovercraft of the n th hovercraft.An adjacency matrix  is defined to describe the com- munication relationship between nodes, which is expressed as ; otherwise, 0 nj a  .In addition, the Laplacian matrix is defined as      , which can be expressed as in blocks, where   denotes the communication relationship between the leaders, and   denotes the communica- tion relationship between the followers and the leaders;   denotes the communication relationship between the followers.In addition, define   Assumption 2 [27].There is at least one path from the virtual leader to each leader.In addition, the leaders have at least one path to each follower.
Lemma 1 [27].Under the condition of Assumption 2, all eigenvalues of matrix  have positive real parts, and each element of

 
  is non-negative and the sum of each row of

Control Objective
In this paper, a distributed formation-containment control scheme for 4-DOF underactuated hovercraft under the condition of ocean disturbances and model uncertainties is proposed.The block diagram of the control system is shown in Figure 2. The control objectives are as follows: (1) In the formation layer, the control scheme is used to make the formula   is a small constant, and the leaders in the multi-hovercraft system realize for- mation tracking.
(2) In the containment layer, the control scheme is used to make

Formation-containment System Controller Design
This section proposes a formation tracking control scheme for the leaders and a containment control scheme for the followers in the backstepping framework.

Formation Layer Controller Design
In order to track the virtual leader, a formation control scheme is designed for the leader   i i  in the formation layer under the condition of ocean disturbances and model uncertainties.
Firstly, the distributed cooperative error of the formation layer is defined as follows where   cos sin sin cos represent the position information of the leader i , l , respectively.The time derivative of ( 4) can be wri en as where Considering that there is no thruster on the side of the hovercraft, an auxiliary variable i  is introduced to transform the lateral thrust design into the yaw moment design to solve the hovercraft's underactuated problem [28].The cooperative error is redefined as follows where   , 0  is a positive number.According to ( 5) and ( 6), the dynamic equation To stabilize the dynamic process of ( 7), the virtual control law of the formation layer is designed as follows   where   is a diagonal gain matrix.Substituting (8) into (7), we have where  is the output of the linear tracking differentiator, which will be introduced later; ie i ir      is the dynamic tracking error.
According to (3), we have   Taking the time derivative of ie  and combined with (10), we have where   is the output value of the differentiator.The total impact of i  and i d can be expressed as   The following linear differentiator is designed to estimate the differential value i where   is the diagonal gain matrix.Ac- cording to the convergence analysis of the linear differentiator in [29], it is known that the output error ie ir i  of the differentiator can converge to any small bounded domain.
Since the ocean disturbance term i d in ( 11) is unknown, an adaptive extended state observer (ALESO) is designed to estimate it.
The extended state vector i Q is defined as  , and unknown ocean disturbance i d is defined as an extended state variable.According to (11), we have where Inspired by [20], the ALESO in this paper is designed as follows where , where , is the observer estimation error.The observer gain i  is designed as , where     where [ , ] where Based on Assumption 1, it is easy to know that i  is bounded, so 1  has an unknown upper bound i  that satisfies the inequality relation where . Subtract ( 14) from (13), we can obtain where is Hurwi .There is a positive definite symmetric matrix i P satisfying 4 4 In order to prove the convergence of the observer estimation error, the following Lyapunov function is selected The time derivative of 1 i V can be wri en as Substituting ( 15), ( 16), ( 18) and ( 20) into ( 22), we have where . Solving (23), we have  , when t   , the estimation error of the observer can converge to a bounded domain, and the size of the bounded domain can be adjusted by reasonably selecting parameters.
Next, to deal with the model uncertainties i  in (11), the RBFNN is designed to ap- proximate i  , and the projection operator is introduced to limit the upper and lower bounds of the weights of the adaptive neural network.The input-output RBFNN algorithm is designed as follows where * * * ,  is the ideal weight matrix, which satisfies  is the network approximation error, basis function , where i c , i b respects the central position and width of the i th neuron, respectively.

 
, , , , where ˆˆ, . The adaptive weight matrix update law of the neural network is designed as follows: The definition and properties of the projection operator   , Proj   can be found in ref- erence [30].
Based on the output values of differentiator, ALESO and RBFNN, the distributed cooperative anti-interference control law of multi-hovercraft is designed as follows where whose time derivative alone ( 9), ( 23), ( 24), ( 25), ( 26) and ( 27), we have where According to (29), we have system error can converge to a bounded domain, and the size of the bounded domain can be adjusted by reasonably selecting the parameters.

Containment Layer Controller Design
Next, a control scheme is designed for the following hovercraft   k k  .The defini- tion of the containment layer error function is as follows where   cos sin sin cos represent the position information of the followers k , j and the leader l , respectively.
Taking the time derivative of ( 30) along ( 2), we have where . The auxiliary variable k  is introduced to solve the underactuated problem of the hovercraft, and the tracking error is redefined as follows where  is a positive constant whose time derivative (32) along (31) satisfies where In order to stabilize (33), the distributed virtual control law of the underactuated hovercraft with a containment layer is designed as follows   where   is the diagonal gain matrix.Substituting (34) into (33), one has where  is the output of the tracking differentiator, which will be intro- where   is the output value of the differentiator, and the design form of the differentiator is completely consistent with (12).
The total impact of k  and k d can be expressed as   Next, ALESO and RBFNN are introduced to estimate the unknown ocean disturbance k d and model uncertainties k  in (36), respectively.The design form of ALESO and RBFNN is completely consistent with the formation layer, and the design process is omitted here.
The distributed cooperative control law of the multi-hovercraft with a containment layer is designed as follows where  is a diagonal gain matrix.
Referring to the formation layer Lyapunov function form, the containment layer Lyapunov function can be finally organized into the following form where

Stability Analysis
Theorem 1.Consider a multi-hovercraft system consisting of 1 R leaders and 2 1 R R  followers and satisfying the relevant assumptions, as expressed in (3).Then, the control law ( 27), (37) and the virtual control law Equations ( 8) and (34) proposed in this paper can make the multi-hovercraft system achieve formation-containment control, and the formation tracking error and the containment tracking error can converge to a bounded domain.

Proof of Theorem 1. Consider the following Lyapunov function
The time derivative of (39) can be wri en as where According to (40), we have Then, we have Equation (41) implies that V is bounded, and the error signals , , , , , , , , , , , , , , , According to (42), we have where Substituting ( 6) into (33), we have The formation tracking error of the   i i  leaders can be defined as Then, the formation tracking error of all leaders can be defined as where     is the minimum singular value of matrix  . Define , , , , , , , , , , , ,

 
  is 1, and the containment error of all following hovercraft is wri en as   . According to (42), one has According to (30), we have Then, the containment error where      is the minimum singular value of matrix   .
By adjusting the parameters reasonably, the tracking error can be adjusted to the bounded domain, that is, . This completes the proof.□

Simulation
In this section, simulation results are provided to illustrate the efficacy of the proposed multi-hovercraft formation-containment control scheme.The communication topology between the multi-hovercraft system is shown in Figure 3, which includes the virtual leader (Hovercraft #0), three leaders (Hovercraft #1-Hovercraft #3) in the formation layer, and two follower (Hovercraft #4-Hovercraft#5) in the containment layer.For detailed information on the main parameters of the hovercraft used in the simulation, please refer to [3].The formation structure vector information is set as , , , , . The initial state of the ALESO, linear differentiator and RBFNN is set as zero.The initial state of the multi-hovercraft system is designed as   0 0 0, 0, 0, 90 The parameters of the control are designed as   4-7. Figure 4 illustrates the trajectories of three leaders (Hovercraft #1-Hovercraft #3) and two followers (Hovercraft #4-Hovercraft #5) under compound perturbation conditions.At the beginning of the simulation, Hovercraft #4 and Hovercraft #5 are located outside the convex envelope shaped by Hovercraft #1-Hovercraft #3.By using the proposed formation-containment scheme, Hovercraft #4 and Hovercraft #5 can be successfully guided into the convex envelope formed by the leaders.The multi-hovercraft system distributed cooperative errors, formation tracking error of Hovercraft #1-Hovercraft #3, and containment tracking error of Hovercraft #4 and Hovercraft #5 are plo ed in Figure 5 and Figure 6, respectively.From Figure 5 and Figure 6, it can be seen that the coordination error, formation tracking control error, and containment tracking control error can converge to the bounded domain.In order to show the superiority of the algorithm in this paper, the control algorithm provided in the literature [17] is compared with this paper in the simulation.The contrast effect diagram is shown in Figure 6.Method 1 represents the control scheme of this paper, and method 2 represents the control scheme of reference [17].It can be seen from Figure 6 that the method designed in this paper makes the followers have higher tracking accuracy.When 300 s t  and 500 s , the tracking errors fluctuate due to the change in the rotation rate, but the errors converge quickly.Figure 7 represents the total estimated effect of ALESO on the unknown ocean disturbances and RBFNN on the model uncertainties.From Figure 7, it can be observed that ALESO and RBFNN achieve a good estimation effect.The velocity curve of the hovercraft is shown in Figure 8.It is worth noting that the heeling velocity of the hovercraft fluctuates obviously.Due to the lack of control torque, the heeling angle and angular velocity cannot be limited.Future work should consider reducing the fluctuation of the heeling angle.The control force and torque of the hovercraft are shown in Figure 9.In the initial phase, the response curve of control force and torque fluctuates largely due to the large formation tracking control error and containment tracking control error at the beginning.When the multi-hovercraft system reaches a stable state, the control torque value fluctuates within a small range.

Conclusions
This paper studies the formation-containment control problem of multi-hovercraft systems under compound uncertainties.The designed ALESO and RBFNN effectively compensate for unknown ocean disturbances and model uncertainties of the multi-hovercraft system.The stability of the multi-hovercraft system is analyzed using Lyapunov's stability theory, and it is concluded that the tracking errors can converge to a bounded domain.Finally, simulation results show that the scheme proposed in this paper can achieve multi-hovercraft formation-containment control with small distributed cooperative errors under compound perturbations.In future research, we will focus on modelfree control schemes and connectivity maintenance of the multi-hovercraft system.

Figure 1 .
Figure 1.Reference frames of the hovercraft.

1 i
i th leader can obtain the information of the virtual leader, then 0 is the position information of the k th hovercraft, and k    is a small constant, and the followers in the multi-hovercraft system realize the containment control.

Figure 2 .
Figure 2. The framework of control system.
represents the expected position deviation of the leader i relative to the leader l .0 i  represents the expected position deviation of the leader i relative to the virtual leader.
and adaptive variable ˆi  will be introduced later.Assume that i  satisfies the following matching condition.

.
According to Lemma 1, the sum of each row of 1 
generated by a virtual leader consists of two straight lines and a circular arc[4].

Figure 5 .
Figure 5. Distributed cooperative errors of multi-hovercraft system.

Figure 6 .
Figure 6.Formation tracking errors and containment tracking errors.

Figure 9 .
Figure 9. Force and torque of multi-hovercraft system.
in which i p and 0 p represent the position infor- mation of the i th leader and the virtual leader, respectively.0 i  represents the expected position deviation of the leader hovercraft i relative to the virtual leader hovercraft, i  