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

Zhao, Tongjuan; Wang, Jiuhe; Zhang, Jianhua

doi:https://doi.org/10.1155/2020/3697476

Mathematical Problems in Engineering

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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From Complexity to Simplicity in U-Model Enhanced Control System Design

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Research Article | Open Access

Volume 2020 | Article ID 3697476 | https://doi.org/10.1155/2020/3697476

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

Tongjuan Zhao,^1,2Jiuhe Wang,¹and Jianhua Zhang³

Academic Editor: Olfa Boubaker

Received08 May 2020

Revised29 Sept 2020

Accepted13 Oct 2020

Published26 Oct 2020

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.

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 Section 2. 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 of follower agents (labeled from 1 to ), and a leader (labeled ). The communication topology of followers are described by a digraph . The dynamic model of the th follower iswhere , is the entire state variables of the th agent, , , indicate the state, control, output, and initial condition, respectively, and, are nonlinear smooth functions.

2.2. Algebraic Graph Theory

A directed graph is represented by with denoting edge set and is the node set. An edge of the graph means that can get messages from ; meanwhile, it is also said that the agent is one of agent ’s neighbors, not vice versa.

Hence, the agent ’s neighbor set is . When weight of edges is considered, the graph is said to be a weighted graph. (adjacency matrix) is often used to express the graphic topology. For the element , it is defined that if , otherwise, . Self-loop is not considered as usual, i.e., , and is defined as an in-degree matrix, where of is in-degree for agent . The row sum of Laplacian matrix is 0 and . A directed graph is said to have direct path means that there exists an edge sequence in the form of . 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 can be described aswhere , the pinning gain , where denotes the weight between the th 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 function over a compact setwhere input , weight vector , and node vector , in which the element is being chosen as the commonly used Gaussian function as follows:where is the center of the receptive field and is the width of the Gaussian function.

It has been proven that RBF NN can approximate any continuous function over a compact set aswhere is the ideal NN weight and is the NN approximation error:

Notation: throughout this paper, represents the matrix, indicate ideal weight, estimated weight, and error between ideal and estimated weight. Throughout this paper, represents the matrix, , rational number and matrix, matrix denote element-by-element powers , and denote the transposition of matrix , such as . indicate ideal weight, estimated weight, and error between ideal and estimated weight.

Definition 1. Consider the system , where is a state vector and is the input vector. The solution is practical finite stable (PFS) if for all , there exists and , such that , for all .

Lemma 1. Consider the system , and suppose that there exist continuous function , scalars ,, and such that

Then, the trajectory of the system is PFS.

Lemma 2. Young’s inequality: for any constant , the following inequality holds:where , , and .

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

3. Distributed Adaptive Tracking Controller Design

Consider system (1) and tracking error (2), and definewhere is the virtual control; in the first step, consider the system .

Then, it has

Base on ideal virtual control law, and choose the NNs to approximate the nonlinear system

Therefore,

Choose the practical virtual control law

Choose the adaptive lawwhere , and is positive constant design parameters.

Then, based on (12) and (13),

Let

Therefore,where

Choose the Lyapunov candidate function

Then,

Based on based inequalities, the following holds:

Then, based on (21), it gives

Then, it haswhere

The th step :

And , then choose the virtual control law, and choose the NN to approximate the nonlinear system :

Based on the system,

Choose the practical virtual control law

Choose the adaptive lawwhere , and is positive constant design parameters. Then,where

Choose the Lyapunov candidate function

Then,

Based on basic equation, the following inequalities hold:

Then, based on (33)，it yields

Continuewhere

The th step is the most important step. Based on the system,

Choose the NN to approximate the nonlinear system

Based on the system,

Choose practical virtual control law

Choose adaptive lawwhere , and are positive constant design parameters; then,where

From the following inequality,

Choose the Lyapunov candidate function

Then,

Based on Lemma, the following inequalities hold:

Then, it gives

Then,where

Then, the virtual control can be got as

Choose Lyapunov candidate functions

Then, based on (24), (37), and (51), it haswhere

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 , remain bounded all the time.(2)The closed-loop signal converge to a compact set defined by(i)where are constants related to the design parameters.(3)The finite-time is given bywhere are constants related to the design parameters and is constant related to Lyapunov candidate functional.

Proof. considering Lyapunov function candidate (54) and (55), then is bounded. Therefore, it can conclude that for all , the error signals , are stable.

4. 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 and Laplacian matrix as follows:

The dynamic systems are described by

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 follow the desired reference signal . Select the controller and virtual control as follows:

The weight of NNs is updated bywith , , and .

Applying the finite-time NN control method, the simulation results are shown in Figures 2–4. 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.

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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This paper was supported by the 2018 Social and Science Fund (HB18GL075) and 2015 Hebei Province Natural Science Fund Project (G2015203378 and F2015208128).

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Copyright

Copyright © 2020 Tongjuan Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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