Formation Tracking for Nonaffine Nonlinear Multiagent Systems Using Neural Network Adaptive Control

Adaptive tracking control for distributed multiagent systems in nonaffine form is considered in this paper. Each follower agent is modeled by a nonlinear pure-feedback system with nonaffine form, and a nonlinear system is unknown functions rather than constants. Radial basis function neural networks (NNs) are employed to approximate the unknown nonlinear functions, and weights of NNs are updated by adaptive law in finite-time form. .en, the adaptive finite NN approach and backstepping technology are combined to construct the consensus tracking control protocol. Numerical simulation is presented to demonstrate the efficacy of suggested control proposal.


Introduction
Due to the limitations of a single agent in completing some special tasks and the needs of human society, the multiagent system has received extensive attention from the academic engineering community. In recent years, distributed information processing technology has been widely used, such as urban transportation [1], intelligent robots [2], flexible manufacturing [3], and coordinated expert systems [4] due to higher fault tolerance and reliability. It has attracted the attention of experts and scholars in different fields such as artificial intelligence and control engineering.
In the 1980s, multiagent systems were researched and applied, and in recent years, it has become one of the hot spots in the field of artificial intelligence. For multiagent systems, the consensus problem is the most basic control problem; see a large number of academic papers [5][6][7]. e key to consensus is to design a reasonable consensus protocol to ensure that the multiagent system is consistent. Scholars have further explored multiagent systems with different structures, such as directed topography and undirected topography [8][9][10]. However, it is difficult to avoid the interference factors of uncertainty, thus applying robust consensus control [11][12][13]. In recent years, due to the inevitable interference factors, the robust consensus problem has received widespread attention and received widespread attention. In [14], the robust consensus control of the Laplacian matrix uncertainty multiagent system is studied, and the conditions for achieving state consensus are proposed. In [15], the robust consensus of linear multiagents with random switching topology was studied. In [16], the second-order robustness of nonlinear multiagents with extended state observers is studied. In [17], scholars studied the distributed robust adaptive consensus control of uncertain nonlinear fractional multiagent systems. As we all know, finite-time convergence is a topic that has attracted much attention and is of great significance. erefore, some scholars have raised the problem of finite-time consensus [18][19][20]. Some scholars consider the issue of fixed-time consensus.
In recent years, consensus-tracking problem of distributed multiagent systems has broad applications in many areas, such as formation control, flocking, cooperative control of UAV, and distributed sensor networks [21,22]. Formation tracking control originated from various natural phenomena, such as flocking in birds, fish, and so on. ere exist some kinds of consensus problems of multiagent systems, because sometimes it is needed that all agents agree on some desired quantity of interest, sometimes not. In the leader follower multiagent systems, consensus means that all the follower agents reach the leader values in a finite time [23,24].
Recently, finite-time stability has received much attention due to its efficient performance in many areas [25,26]. Especial neural networks control is a powerful control method, because neural networks can approximate nonlinear system without model [27,28].
ough Lyapunov uniformly ultimately bounded (UUB) results solve some nonlinear system control problem, both bounded and exponential convergence speed cause confusion. It should be noted that the research on finite-time neural network control is still in a very beginning stage. e key issue is how to systematically obtain finite-time adaptive law of neural network weight from finite-time convergence of closed-loop systems. In regarding to such neural network-based adaptive control to the authors' best knowledge, there are a few results about finite-time adaptive neural network control because it is not easy to design the finite-time neural network adaptive controller, and there exists lack of relevant inequality skills to finish finite-time stability analysis.
ere has been any reference to show finite-time adaptive algorithm for weights of NNs having been expanded to solve the problems of finite control for pure-feedback nonaffine nonlinear systems. e remainder of this paper is organized as follows. e multiagent system is described in Section 2. e proposed algorithms for formation tracking, based on the finite-time adaptive control, are presented in Section 3. eoretical analysis of the model is given in Section 4. Simulation results are given in Section 5. Finally, conclusions are drawn in Section 6.

System Description.
Consider a class of the nonlinear pure-feedback multiagent system, composed of N follower agents (labeled from 1 to N), and a leader (labeled d). e communication topology of followers are described by a digraph G. e dynamic model of the ith (i � 1, 2, . . . , N) follower is . , x i,n i ] T ∈ R n i is the entire state variables of the ith agent, u i ∈ R, y i ∈ R, φ i (t) indicate the state, control, output, and initial condition, respectively, and, f i,k (·) are nonlinear smooth functions.

Algebraic Graph eory. A directed graph is represented by
meanwhile, it is also said that the agent j is one of agent i's neighbors, not vice versa. Hence, the agent i's neighbor set is When weight of edges is considered, the graph is said to be a weighted graph. A � [a ij ] ∈ R k×k (adjacency matrix) is often used to express the graphic topology. For the element a ij , it is When there exists at least one agent (root) in the digraph which can transmit information through direct path to all other agents, the direct graph is thus said to include a directed spanning tree. e local tracking error for agent i can be described as where i � 1, 2, . . . , N, the pinning gain b i ≥ 0, where b i > 0 denotes the weight between the ith agent and leader agent.

Radial Basis Function Neural Networks (RBFNNs) and
Function Approximation. In brief, the following radial basis function (RBF) NN is used to approximate the continuous function F(x): R n ⟶ R over a compact set where input x ∈ Ω ⊂ R n , weight vector W � [w 1 . . .
is being chosen as the commonly used Gaussian function as follows: T is the center of the receptive field and η i is the width of the Gaussian function. It has been proven that RBF NN can approximate any continuous function over a compact set Ω x ⊂ R n as where W * is the ideal NN weight and ε(x) is the NN approximation error: Notation: throughout this paper, W ∈ R m×n represents the matrix, W * , W, W indicate ideal weight, estimated weight, and error between ideal and estimated weight. roughout this paper, W ∈ R m×n represents the matrix, weight, and error between ideal and estimated weight.
, where x is a state vector and u is the input vector. e solution is practical en, the trajectory of the system _ x � f(x, u) is PFS.

Remark 1.
Based on Young's inequality, then the following inequalities hold:

Distributed Adaptive Tracking Controller Design
Consider system (1) and tracking error (2), and define where α i,j , (2 ≤ j ≤ n i ) is the virtual control; in the first step, consider the system z i,1 . en, it has Base on ideal virtual control law, and choose the NNs to approximate the nonlinear system N j�1 erefore, Choose the practical virtual control law Choose the adaptive law where Γ i,1 � Γ T i,1 > 0, and σ i,1 > 0 is positive constant design parameters.
en, based on (12) and (13), Let erefore, where Choose the Lyapunov candidate function en, Based on based inequalities, the following holds: Mathematical Problems in Engineering en, based on (21), it gives en, it has where e jth step 2 ≤ j ≤ n i : And x i,j+1 � u i , _ u i � v i , then choose the virtual control law, and choose the NN to approximate the nonlinear system _ x i,j − _ α i,j−1 : Based on the system, Choose the practical virtual control law Choose the adaptive law where Γ i,j � Γ T i,j > 0, and σ i,j > 0 is positive constant design parameters. en, Choose the Lyapunov candidate function en, Based on basic equation, the following inequalities hold: en, based on (33)，it yields where e nth step is the most important step. Based on the system,

(39)
Choose the NN to approximate the nonlinear system Based on the system, Choose practical virtual control law Choose adaptive law where Γ i,n i � Γ T i,n i > 0, and σ i,n i > 0 are positive constant design parameters; then, where From the following inequality, Choose the Lyapunov candidate function en, Based on Lemma, the following inequalities hold: en, it gives en, where en, the virtual control can be got as Choose Lyapunov candidate functions en, based on (24), (37), and (51), it has where η i � min η i,j , j � 1, 2, 3, . . . , n i + 1, Theorem 1. Consider the nonlinear system for which the model dynamics is approximated by neural networks (12), (27), and (40), the control law (53) with the virtual control (14), (29), and (42), and adaptive laws (15), (30), and (43); then, the following statements hold: (1) All the signals of the closed-loop system, including x i,j , α i,j , W i,j , remain bounded all the time.

Mathematical Problems in Engineering
(2) e closed-loop signal z i,j converge to a compact set defined by (i) where δ i , η i are constants related to the design parameters. (3) e finite-time T is given by

Simulation Example
In this section, simulation example shows the validity and feasibility of the proposed NN finite adaptive control design approach. Figure 1 shows the topology of communication graph of MAS with one leader and five followers. Consider the following MAS with adjacency matrix A and Laplacian matrix L as follows: e dynamic systems are described by y d � 5sin(t) is the state, control input, control output, and ideal output, respectively. e control objective of the proposed design method is to make the system output y follow the desired reference signal y d � 5sin(t). Select the controller _ u and virtual control α 1 , α 2 as follows:      Figure 2 shows the trajectory of output and desired reference trajectory of MAS. Figure 3 shows the error trajectory of five followers. Figure 4 shows the trajectory of controller. e method in this paper is more generally used, and both output and weight of NNs are convergence in finite time.

Conclusion
In this article, the adaptive tracking control method is proposed for distributed multiagent systems in nonaffine form. Based on finite neural network algorithm, a finite-time tracking result can be got. In addition, each follower agent is modeled by a nonlinear pure-feedback system with nonaffine form, and the nonlinear system is unknown functions rather than constants. Simulations and theoretical analysis are carried out to verify the feasibility and correctness of the proposed method.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper. Mathematical Problems in Engineering 7