Multi-Group Formation Tracking Control for Second-Order Nonlinear Multi-Agent Systems Using Adaptive Neural Networks

This paper investigates the multi-group formation tracking (MGFT) control problem for second-order nonlinear multi-agent systems (MASs) with unknown dynamics. The objective of the MGFT control is to divide all agents into several subgroups to form different desired sub-formations while following their respective leaders. Firstly, the neural network (NN) approximator is constructed to solve the problem of unknown dynamics. Then, the distributed adaptive control protocol is designed based on the NN approximator. According to the Lyapunov stability theory and algebraic graph theory, sufficient criteria are obtained to realize the MGFT control. The semi-globally uniformly ultimately boundedness of formation errors is proved in detail. Finally, a numerical simulation example is given to confirm the validity of our theoretical results.


I. INTRODUCTION
As one of the most active and attractive research topics in the coordinated control for MASs, formation control has received increasing attention in the last two decades because of its wide applications in practical systems [1][2][3][4][5][6][7][8][9]. MASs are composed of multiple interacting intelligent agents which generate complex cluster behaviors through local information interaction. For formation control, it is essential to design suitable control protocols that enable the agents to reach and maintain desired geometry to accomplish the task.
Several common formation control strategies of MASs are the virtual structure strategy [10], graph based approach [11,12], behavior based approach [13] and leader-follower strategy [14]. Among these methods, the leader-follower method has been widely used by many researchers due to its simplicity, reliability and scalability [15][16][17]. Han et al. [18] studied the formation tracking problem using a fast terminal sliding mode control approach under a linear MAS. Hashim et al. [19] proposed a distributed robust neuro-adaptive cooperative tracking controller for higher-order nonlinear MASs with prescribed performance, where the uncertain parameters and external disturbances were considered. Safaei [20] investigated consensus and formation-tracking problems of networked agents with completely unknown nonlinear dynamics, and proposed the distributed adaptive model-free control algorithm. Yang et al. [21] studied the formation control with collision, obstacle avoidance and connectivity maintenance problems for nonlinear MASs under external disturbances, and proposed a novel control protocol based on adaptive neural networks. Wen et al. [22] studied the leaderfollower formation tracking problem for nonlinear MASs on the basis of neural network techniques. The above studies merely considered the normal formation tracking problem of single target, where all the followers form the same desired geometry and track the leader. However, multi-objective searching and cooperative enclosing for multiple targets ubiquitously exist in reality, such as the surveillance operation of satellites cluster for multi-target, predators' formation in multi-prey hunting, and division of labor in society for different interests and conflicts, which implies that MASs can be decomposed into several subgroups, each subgroup can achieve different distributed tasks in each own desired sub-formation geometry. Therefore, it is vital to investigate the MGFT control problem.
Currently, owing to the aforementioned superiorities, a lot of effort has been made to investigate the formation problems for multiple groups, e.g., cluster formation control under communication delays and aperiodic sampling [23], timevarying multi-formation acquisition [24], multi-formation control for nonlinear MASs [25], multi-target tracking of heterogeneous collaborative-robots with external disturbances [26] and multi-formation tracking using impulsive control methods [27]. However, these works on the MGFT problem of MASs have not considered the impact of the unknown dynamics. In fact, on account of the influence coming from environmental disturbance and agent's own intrinsic dynamics, there are unknown nonlinear dynamics in practical engineering systems. By applying the adaptive learning and nonlinear continuous function approximation ability of NNs, the control strategy combining NNs with adaptive control technology can effectively solve the problem of unknown dynamics. Inspired by the above analysis, this paper investigates the MGFT control problem for second-order nonlinear MASs with unknown dynamics using adaptive NN control method.
This paper is organized as follows. In Section 2, the necessary preliminaries, related lemmas and assumptions are given. Section 3 designs the MGFT control protocol and proves the system's stability for second-order nonlinear MASs with unknown dynamics. In Section 4, a numerical simulation example is presented to illustrate the effectiveness of the proposed protocol. The conclusions are summarized in Section 5.
Notations: Let n  be the n-dimensional Euclidean space and m n   be the space of m n  real matrices. Give a matrix A , if matrix is positive definite (positive semi-definite). min ( ) A  and max ( ) A  denote the minimum and the maximum eigenvalue of the matrix A , respectively.  is the Kronecker product operator.  denotes the Euclidean norm for vectors.

A. GRAPH THEORY
To describe the communications among the N follower agents, we consider a weighted undirected graph denotes the set of nodes and      is the edge set. If an edge ( , ) i j   , then it implies that the information can be exchanged between node i and node j . The elements of the adjacency matrix  are positive if and only if ( , ) i j   , i.e., 0 (, )  as the set of M leaders (marked 0 ) and F  as the set of the N followers (marked 1, , N  ), respectively. Then, the leader adjacency matrix is defined as 1 2 ( , , , ) where if there exists a connection between the i th follower and the leader, then 0 represents the neuron number. m x     is the input vector of NNs. The basis function vector is denoted by is the center of the receptive field.
Based on the above fact, for any smooth continuous function ( ) f x , one has the optimal weight matrix then ( ) f x can be rewritten as In order to minimize the possible deviation between ( ) and ( ) f x , the optimal NN weight matrix * W must be obtained. However, since * W is just an "artificial" quantity for analysis, it is unavailable for the actual control design. Usually, the estimation of * W is employed to construct the actual control, which is obtained by online adaptive tuning.

C. SOME USEFUL LEMMAS AND ASSUMPTIONS
Lemma 1. [28] If there is at least one follower communicated to the leader in the connected undirected graph  , then    is a positive definite matrix. Lemma 2. [29] The matrix inequality that 1 3 symmetric matrices, is equivalent to any conditions as follows: 1) holds, where  and  are two positive constants, the following one can be obtained: Assumption 2. In the MGFT problem, the undirected subgraph l  for the followers in the l th subgroup is connected, and there is at least one follower connected to the leader 0 l , that is, Remark 1. Although the cooperation weight between different subgroups can be positive or negative, its sum is 0. Therefore, for one subgroup, Assumption 1 indicates that the total information coming from all other subgroups is 0, which enables each subgroup to accomplish the task as a whole.

A. PROBLEM FORMULATION
Consider a second-order nonlinear MAS containing M N  agents. The nonlinear dynamics of the i th follower is given by the following form  denote the position, velocity and control input of the i th follower, respectively.
is the unknown nonlinear function, which represents the inherent dynamics of the i th follower. The dynamics of the j th leader can be described as where 0 ( )  (1) and (2), MGFT is said to achieve if for bounded initial states, there exists a distributed control protocol ( ) i u t such that all agents in the same subgroup can form and maintain a desired formation while the geometric center tracks the leader's trajectory, i.e., for The Control Objective. Design a distributed adaptive formation control scheme to solve the MGFT problem of the second-order nonlinear MASs with unknown dynamics such that 1) the position and velocity errors can converge to the desired accuracy; 2) all formation errors are semi-globally uniformly ultimately bounded (SGUUB).

B. CONTROL PROTOCOL DESIGN
In order to realize the MGFT control for the nonlinear second-order MASs (1) and (2) According to the systems (1) and (2), the error dynamics can be described as For the l th subgroup, let The compact form of the error dynamics (6) can be obtained Since not all followers can directly connect with the leader in each subgroup, the formation tracking errors are defined as where ij a and i b are the elements of the adjacency matrix is the set of neighbors of agent i in the l th subgroup. On the basis of (5), then the formation tracking error dynamics can be rewritten as Based on the NN approximation (4), by applying the estimation ˆ( ) i W t of the optimal NN weight matrix i W  , the adaptive MGFT control protocol is proposed as where 0 p k > , 0 v k > denote the position damping gain and velocity damping gain, respectively. ˆ( ) where 0 i  > is a positive design constant, 1, , , where min ( ) l   is the minimum eigenvalue of matrix l l l      , then the MGFT control objectives can be achieved under the distributed adaptive control protocol (10) with the NN weight updating law (11).
Proof. For the l th subgroup, define the following Lyapunov function candidate for the system (7) as Due to Assumption 2 holds, the condition (12) and the Lemmas 1-2 are satisfied, which means ( ) l V t is positive definite. Then the time derivative of ( ) l V t along the trajectory of system (7) and (11) is Using the above equation (9), we can obtain

k e t e t e t t e t e t u t f p v a t
and the above equation (17), it follows that  T  T  l  p p  p  v v  v  i  i   n  n  T  T  T (18) According to Young's inequality and Cauchy-Schwarz inequality, it obtains It is easy to learn from Figs. 2-6 that the MGFT control of the MASs (1) and (2) is accomplished, which also confirms with the theoretical result obtained in Theorem 1. As presented in Fig. 2, all agents are divided into three subgroups in the desired sub-formation shapes, and the followers track their respective leaders. Fig. 3 depicts the evolutions of the norms of position errors between the followers and the respective leader along X, Y and Z axes, respectively. Fig. 4 depicts the evolutions of the norms of velocity errors between the followers and the respective leader along X, Y and Z axes, respectively. As presented in Figs. 3 and 4, the position and velocity errors can converge to the desired accuracy in three subgroups. Figs. 5 and 6 describes the evolutions of followers' position and velocity states along X, Y and Z axes, respectively. As presented in Figs. 5 and 6, the position and velocity states of followers can converge to the desired formation and maintain it in three subgroups.

V. CONCLUSION
In this paper, the multi-group formation tracking control problem for second-order nonlinear multi-agent systems with unknown dynamics was studied. The neural network approximator was used to compensate the unknown dynamics. Based on adaptive neural network control method, the multi-group formation tracking control protocol was proposed and analyzed in detail. Using the knowledge of algebraic graph theory and Lyapunov stability theory, sufficient conditions were obtained such that the multi-agent systems were proved to achieve the multi-group formation tracking control under the proposed control scheme, where all the followers can reach the desired sub-formation and the geometric center of the followers can track the corresponding leader in each subgroup. Finally, a numerical simulation example in 3D space has been provided to verify the effectiveness of the theoretical results.
In the future work, we will further investigate the multigroup formation tracking control problem with data packet loss and communication delays.