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

Authors: Tongjuan Zhao; Jiuhe Wang; Jianhua Zhang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

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. Then, 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.


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## Body

## 1. 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 [5–7]. The 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 [8–10]. However, it is difficult to avoid the interference factors of uncertainty, thus applying robust consensus control [11–13]. 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. Therefore, some scholars have raised the problem of finite-time consensus [18–20]. 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. There 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]. Though 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. The 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. There 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.The remainder of this paper is organized as follows. The multiagent system is described in Section2. The proposed algorithms for formation tracking, based on the finite-time adaptive control, are presented in Section 3. Theoretical analysis of the model is given in Section 4. Simulation results are given in Section 5. Finally, conclusions are drawn in Section 6.


## 2. Problem Description

### 2.1. System Description

Consider a class of the nonlinear pure-feedback multiagent system, composed ofN follower agents (labeled from 1 to N), and a leader (labeled d). The communication topology of followers are described by a digraph G. The dynamic model of the ith i=1,2,…,N follower is(1)x˙i,k=fi,kx¯i,k+1,1≤k≤ni−1,x˙i,ni=fi,nix¯i,ni,ui,yi=xi,1,where x¯i,j=xi,1,xi,2,…,xi,jT∈ℜj, x¯i,ni=xi,1,xi,2,…,xi,niT∈ℜni is the entire state variables of the ith agent, ui∈ℜ, yi∈ℜ, φit indicate the state, control, output, and initial condition, respectively, and, fi,k⋅ are nonlinear smooth functions.


### 2.2. Algebraic Graph Theory

A directed graph is represented byG=V,E with E⊆V×V denoting edge set and V=v1,v2,…,vk is the node set. An edge eji=vj,vi∈E of the graph G means that i can get messages from j; meanwhile, it is also said that the agent j is one of agent i’s neighbors, not vice versa.Hence, the agenti’s neighbor set is Ni=vj|vj,vi∈E. When weight of edges is considered, the graph is said to be a weighted graph. A=aij∈ℜk×k (adjacency matrix) is often used to express the graphic topology. For the element aij, it is defined that aij>0 if eji=vj,vi∈E, otherwise, aij=0. Self-loop is not considered as usual, i.e., aii=0, and D=diagd1,d2,…,dk∈ℜk×k is defined as an in-degree matrix, where di=∑j=1kaij of D is in-degree for agent i. The row sum of Laplacian matrix L=D−A is 0 and L1=0. A directed graph is said to have direct path means that there exists an edge sequence in the form of vi,vr,vr,vs,…,vt,vj. 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. The local tracking error for agent i can be described as(2)zi,1=∑j=1Naijyi−yj+biyi−yd,where i=1,2,…,N, the pinning gain bi≥0, where bi>0 denotes the weight between the ith agent and leader agent.


### 2.3. Radial Basis Function Neural Networks (RBFNNs) and Function Approximation

In brief, the following radial basis function (RBF) NN is used to approximate the continuous functionFx:ℜn⟶ℜ over a compact set(3)FNNx,W=WTΨx,where input x∈Ω⊂ℜn, weight vector W=w1…wlT∈ℜl, and node vector Ψx=ψ1x…ψlxT, in which the element ψix is being chosen as the commonly used Gaussian function as follows:(4)ψix=exp−x−μiTx−μiηi2,i=1,2,…,l,where μi=μi1…μinT is the center of the receptive field and ηi is the width of the Gaussian function.It has been proven that RBF NN can approximate any continuous function over a compact setΩx⊂ℜn as(5)Fx=W∗TΨx+εx,where W∗ is the ideal NN weight and εx is the NN approximation error:(6)W∗=argminW∈ℜlsupFx−WTΨx.Notation: throughout this paper,W∈ℜm×n represents the matrix, W∗,W^,W˜ indicate ideal weight, estimated weight, and error between ideal and estimated weight. Throughout this paper, W∈ℜm×n represents the matrix, W=wijm×n, rational number η and matrix, matrix Wη denote element-by-element powers Wη=wijηm×n, and W˜ηT denote the transposition of matrix W˜η, such as W˜ηT=W˜ηT. W∗,W,W˜ indicate ideal weight, estimated weight, and error between ideal and estimated weight.Definition 1.
Consider the systemx˙=fx,u, where x is a state vector and u is the input vector. The solution is practical finite stable (PFS) if for all xt0=x0, there exists ε>0 and Tε,x0<∞, such that x<ε, for all t≥t0+T.Lemma 1.
Consider the systemx˙=fx,u, and suppose that there exist continuous function Vx, scalars λ>0,0<α<1, and 0<η<∞ such that(7)V˙x≤−λVαx+η.Then, the trajectory of the systemx˙=fx,u is PFS.Lemma 2.
Young’s inequality: for any constanta,b∈ℜ, the following inequality holds:(8)ab≤1pap+1qbq.where p>1, q>1, and 1/p+1/q=1.Remark 1.
Based on Young’s inequality, then the following inequalities hold:(9)zε≤34z4/3+14ε4,−W˜TW^1/3≤−12W˜2/3TW˜2/3+W∗2/3TW∗2/3.


## 2.1. System Description

Consider a class of the nonlinear pure-feedback multiagent system, composed ofN follower agents (labeled from 1 to N), and a leader (labeled d). The communication topology of followers are described by a digraph G. The dynamic model of the ith i=1,2,…,N follower is(1)x˙i,k=fi,kx¯i,k+1,1≤k≤ni−1,x˙i,ni=fi,nix¯i,ni,ui,yi=xi,1,where x¯i,j=xi,1,xi,2,…,xi,jT∈ℜj, x¯i,ni=xi,1,xi,2,…,xi,niT∈ℜni is the entire state variables of the ith agent, ui∈ℜ, yi∈ℜ, φit indicate the state, control, output, and initial condition, respectively, and, fi,k⋅ are nonlinear smooth functions.


## 2.2. Algebraic Graph Theory

A directed graph is represented byG=V,E with E⊆V×V denoting edge set and V=v1,v2,…,vk is the node set. An edge eji=vj,vi∈E of the graph G means that i can get messages from j; meanwhile, it is also said that the agent j is one of agent i’s neighbors, not vice versa.Hence, the agenti’s neighbor set is Ni=vj|vj,vi∈E. When weight of edges is considered, the graph is said to be a weighted graph. A=aij∈ℜk×k (adjacency matrix) is often used to express the graphic topology. For the element aij, it is defined that aij>0 if eji=vj,vi∈E, otherwise, aij=0. Self-loop is not considered as usual, i.e., aii=0, and D=diagd1,d2,…,dk∈ℜk×k is defined as an in-degree matrix, where di=∑j=1kaij of D is in-degree for agent i. The row sum of Laplacian matrix L=D−A is 0 and L1=0. A directed graph is said to have direct path means that there exists an edge sequence in the form of vi,vr,vr,vs,…,vt,vj. 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. The local tracking error for agent i can be described as(2)zi,1=∑j=1Naijyi−yj+biyi−yd,where i=1,2,…,N, the pinning gain bi≥0, where bi>0 denotes the weight between the ith agent and leader agent.


## 2.3. Radial Basis Function Neural Networks (RBFNNs) and Function Approximation

In brief, the following radial basis function (RBF) NN is used to approximate the continuous functionFx:ℜn⟶ℜ over a compact set(3)FNNx,W=WTΨx,where input x∈Ω⊂ℜn, weight vector W=w1…wlT∈ℜl, and node vector Ψx=ψ1x…ψlxT, in which the element ψix is being chosen as the commonly used Gaussian function as follows:(4)ψix=exp−x−μiTx−μiηi2,i=1,2,…,l,where μi=μi1…μinT is the center of the receptive field and ηi is the width of the Gaussian function.It has been proven that RBF NN can approximate any continuous function over a compact setΩx⊂ℜn as(5)Fx=W∗TΨx+εx,where W∗ is the ideal NN weight and εx is the NN approximation error:(6)W∗=argminW∈ℜlsupFx−WTΨx.Notation: throughout this paper,W∈ℜm×n represents the matrix, W∗,W^,W˜ indicate ideal weight, estimated weight, and error between ideal and estimated weight. Throughout this paper, W∈ℜm×n represents the matrix, W=wijm×n, rational number η and matrix, matrix Wη denote element-by-element powers Wη=wijηm×n, and W˜ηT denote the transposition of matrix W˜η, such as W˜ηT=W˜ηT. W∗,W,W˜ indicate ideal weight, estimated weight, and error between ideal and estimated weight.Definition 1.
Consider the systemx˙=fx,u, where x is a state vector and u is the input vector. The solution is practical finite stable (PFS) if for all xt0=x0, there exists ε>0 and Tε,x0<∞, such that x<ε, for all t≥t0+T.Lemma 1.
Consider the systemx˙=fx,u, and suppose that there exist continuous function Vx, scalars λ>0,0<α<1, and 0<η<∞ such that(7)V˙x≤−λVαx+η.Then, the trajectory of the systemx˙=fx,u is PFS.Lemma 2.
Young’s inequality: for any constanta,b∈ℜ, the following inequality holds:(8)ab≤1pap+1qbq.where p>1, q>1, and 1/p+1/q=1.Remark 1.
Based on Young’s inequality, then the following inequalities hold:(9)zε≤34z4/3+14ε4,−W˜TW^1/3≤−12W˜2/3TW˜2/3+W∗2/3TW∗2/3.


## 3. Distributed Adaptive Tracking Controller Design

Consider system (1) and tracking error (2), and define(10)zi,k=xi,k−αi,k−1,k=2,3,…,ni,zi,ni+1=ui−αi,ni,where αi,j,2≤j≤ni is the virtual control; in the first step, consider the system zi,1.Then, it has(11)z˙i,1=∑j=1Naij+bix˙i,1−∑j=1Naijx˙j,1−bifdxd,t=∑j=1Naij+bifi,1x¯i,2−∑j=1Naijfj,2x¯j,2−bifdxd,t.Base on ideal virtual control law, and choose the NNs to approximate the nonlinear system(12)∑j=1Naij+bifi,1x¯i,2−∑j=1Naijfj,2x¯j,2−bifdxd,t=Wi,1T∗Ψ1Z2+εi,1.Therefore,(13)z˙i,1=Wi,1∗TΨ1Z2+εi,1.Choose the practical virtual control law(14)αi,1=−k1zi,11/3+xi,2−W^1TΨ1Z2.Choose the adaptive law(15)W^.i,1=Γi,1zi,1Ψ1Z2−σi,1W^i,11/3,where Γi,1=Γi,1T>0, and σi,1>0 is positive constant design parameters.Then, based on (12) and (13),(16)z˙i,1=Wi,1∗TΨ1Z2+εi,1−ki,1zi,11/3+xi,2−αi,1−W^i,1TΨ1Z2.Let(17)zi,2=xi,2−αi,1.Therefore,(18)z˙i,1=−W˜i,1TΨ1Z2−ki,1zi,11/3+zi,2+εi,1,where(19)W˜i,1=W^i,1−Wi,1∗.Choose the Lyapunov candidate function(20)Vi,1=12zi,12+12W˜i,1TΓi,1−1W˜i,1.Then,(21)V˙i,1=zi,1z˙i,1+W˜i,1TΓi,1−1W˜.i,1=−ki,1zi,14/3+zi,1zi,2+zi,1εi,1−σi,1W˜i,1TW^i,11/3.Based on based inequalities, the following holds:(22)zi,1εi,1≤34zi,14/3+14εi,14−σi,1W^i,1TW^i,11/3≤−σ12W˜i,12/3TW˜i,12/3+σ1Wi,12/3∗TWi,12/3∗.Then, based on (21), it gives(23)V˙i,1=−ki,1−34zi,14/3+zi,1zi,2−σi,12W˜i,12/3TW˜i,12/3+14εi,14+σi,1Wi,12/3∗TWi,12/3∗.Then, it has(24)V˙i,1≤zi,1zi,2−ηi,1zi,1222/3−ηi,12W˜i,1TΓi,1−1W˜i,122/3+14εi,14+σi,1Wi,12/3∗TWi,12/3∗≤zi,1zi,2−ηi,1Vi,12/3+δi,1,where(25)ηi,1=min41/3ki,1−34,σi,121/3λmaxΓi,1−12/3,δi,1=14εi,14+σi,1Wi,12/3∗TWi,12/3∗.Thejth step 2≤j≤ni:(26)zi,j=xi,j−αi,j−1.Andxi,j+1=ui,u˙i=vi, then choose the virtual control law, and choose the NN to approximate the nonlinear system x˙i,j−α˙i,j−1:(27)fi,jx¯i,j+1−α˙i,j−1=Wi,j∗TΨjZj+1+εi,j.Based on the system,(28)z˙i,j=Wi,j∗TΨjZj+1+εi,j.Choose the practical virtual control law(29)αi,j=−zi,j−1−ki,jzi,j1/3+xi,j+1−W^i,jTΨjZj+1.Choose the adaptive law(30)W^.i,j=Γi,jzi,jΨjZj+1−σi,jW^i,j1/3,where Γi,j=Γi,jT>0, and σi,j>0 is positive constant design parameters. Then,(31)z˙i,j=−W˜i,jΨjZj+1−zi,j−1−ki,jzi,j1/3+zi,j+1+εi,j,where(32)W˜i,j=W^i,j−Wi,j∗.Choose the Lyapunov candidate function(33)Vi,j=12zi,j2+12W˜i,jTΓi,j−1W˜i,j.Then,(34)V˙i,j=zi,jz˙i,j+W˜i,jΓi,j−1W˜.i,jT=−zi,j−1zi,j−ki,jzi,j4/3+zi,jzi,j+1+zi,jεi,j−σi,jW˜i,jTW^i,j1/3.Based on basic equation, the following inequalities hold:(35)zi,jεi,j≤34zi,j4/3+14εi,j4−σi,jW^i,jTW^i,j1/3≤−σi,j2W^i,j2/3TW˜i,j2/3+σi,jWi,j∗2/3TWi,j∗2/3.Then, based on (33)，it yields(36)V˙i,j=zi,jz˙i,j+W˜i,jTΓi,j−1W˜.i,j=−zi,j−1zi,j−ki,j−34zi,j4/3+zi,jzi,j+1−σi,j2W˜i,j2/3TW˜i,j2/3+14εi,j4+σi,jWi,j2/3∗TWi,j2/3∗.Continue(37)V˙i,j≤−zi,j−1zi,j+zi,jzi,j+1−ηi,jzi,j222/3−ηi,jW˜i,jTΓi,j−1W˜i,j22/3+14εi,j4+σi,jWi,j2/3∗TWi,j2/3∗≤−zi,j−1zi,j+zi,jzi,j+1−ηi,jVi,j2/3+δi,j,where(38)ηi,j=min41/3ki,j−34,σi,j21/3λmaxΓi,j−12/3,δi,j=14εi,j4+σi,jWi,j2/3∗TWi,j2/3∗.Thenth step is the most important step. Based on the system,(39)zi,ni=xi,ni−αi,ni−1.Choose the NN to approximate the nonlinear system(40)fi,nix¯i,ni,ui−α˙i,ni−1=Wi,ni∗TΨniZni+1+εi,ni.Based on the system,(41)z˙i,ni=Wi,ni∗TΨniZni+1+εi,ni.Choose practical virtual control law(42)αi,ni=−zi,ni−1−ki,nizi,ni+ui−W^i,niTΨniZni+1.Choose adaptive law(43)W^.i,ni=Γi,nizi,niΨniTZni+1−σi,niW^ni1/3,where Γi,ni=Γi,niT>0, and σi,ni>0 are positive constant design parameters; then,(44)z˙i,ni=−W˜i,niTΨniZni+1t−zi,ni−1−ki,nizi,ni1/3+zi,ni+1+εi,ni,where(45)W˜i,ni=W^i,ni−Wi,ni∗.From the following inequality,(46)−σi,niW˜i,niTW^i,ni1/3≤−σi,ni2W˜i,ni2/3TW˜i,ni2/3+σi,niWi,ni2/3∗TWi,ni2/3∗.Choose the Lyapunov candidate function(47)Vi,ni=12zi,ni2+12W˜i,niΓi,ni−1W˜i,niT.Then,(48)V˙i,ni=zi,niz˙i,ni+W˜i,niΓi,ni−1W˜.i,niT=−zi,ni−1zi,ni−ki,nizi,ni4/3+zi,nizi,ni+1+zi,niεi,ni−σi,niW˜i,niTW^i,ni1/3.Based on Lemma, the following inequalities hold:(49)zi,jεi,ni≤34zi,ni4/3+14εi,ni4−σi,niW˜i,niTW^i,ni1/3≤−σi,ni2W˜i,ni2/3TW˜i,ni2/3+σi,niWi,ni∗2/3TWi,ni∗2/3.Then, it gives(50)V˙i,ni≤−zi,ni−1zi,ni−ki,ni−34zi,ni4/3+zi,nizi,ni+1−σi,ni2W˜i,ni2/3TW˜i,ni2/3+14εi,ni4+σi,niWi,ni2/3∗TWi,ni2/3∗.Then,(51)V˙i,ni≤−zi,ni−1zi,ni+zi,nizi,ni+1−ηi,nizi,ni222/3−ηi,niW˜i,niTΓi,ni−1W˜i,ni22/3+δi,ni≤−zi,ni−1zi,ni+zi,nizi,ni+1−ηi,niVi,ni2/3+δi,ni,where(52)ηi,ni=min41/3ki,ni−34,σi,ni21/3λmaxΓi,ni−12/3,δi,ni=14εi,ni4+σi,niWi,ni2/3∗TWi,ni2/3∗.Then, the virtual control can be got as(53)v=−zi,ni−ki,ni+1zi,ni+1−W^i,ni+1TΨi,ni+1Zi,ni+1t.Choose Lyapunov candidate functions(54)Vi=∑j=1ni+1Vi,j.Then, based on (24), (37), and (51), it has(55)V˙i≤−ηiVi2/3+δi,where(56)ηi=minηi,j,j=1,2,3,…,ni+1,δi=∑j=1ni+1δi.j,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, includingxi,j,αi,j,W^i,j, remain bounded all the time.(2)
The closed-loop signalzi,j converge to a compact set defined by(57)Ωi,s=zi|zi2≤δiηi,(i)
whereδi,ηi are constants related to the design parameters.(3)
The finite-timeT is given by(58)T≤3V1/3x0ηθ0,where η,θ0 are constants related to the design parameters and Vx0 is constant related to Lyapunov candidate functional.Proof.
considering Lyapunov function candidate (54) and (55), then Vi is bounded. Therefore, it can conclude that for all 1≤j≤ni+1, the error signals zi,j,W˜i,j are stable.


## 4. Simulation Example

In this section, simulation example shows the validity and feasibility of the proposed NN finite adaptive control design approach. Figure1 shows the topology of communication graph of MAS with one leader and five followers.Figure 1
Topology of communication graph.Consider the following MAS with adjacency matrixA and Laplacian matrix L as follows:(59)A=0000310000030000020000010,L=3000−3−110000−330000−220000−11.The dynamic systems are described by(60)x˙i,1=0.5sinxi,1−xi,1+sinxi,2+xi,2+5sint+5cost,x˙i,2=−sinxi,2−ixi,2+xi,2ui+ui+sinui,yi=xi,1.yd=5sint is the state, control input, control output, and ideal output, respectively. The control objective of the proposed design method is to make the system output y follow the desired reference signal yd=5sint. Select the controller u˙ and virtual control α1,α2 as follows:(61)u˙i=−zi,2−k3zi,31/3−W^i,3TΨ3x¯i,2,ui,αi,1=−k1zi,11/3+xi,2−W^i,1TΨ1x¯i,2,αi,2=−zi,1−k2zi,21/3+ui−W^i,2TΨ2x¯i,2,ui.The weight of NNs is updated by(62)W^.i,j=Γi,jzi,jΨix¯i,j+1−σi,jW^i,j+11/3,j=1,2,3,with zi,1=xi,1−yi,d, zi,2=xi,2−αi,1, and zi,3=ui−αi,2.Applying the finite-time NN control method, the simulation results are shown in Figures2–4. 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. The method in this paper is more generally used, and both output and weight of NNs are convergence in finite time.Figure 2
Output trajectory of five followers and leader.Figure 3
Error trajectory of five followers.Figure 4
Five control trajectory.


## 5. 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.



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