Elsevier

ISA Transactions

Volume 126, July 2022, Pages 160-170
ISA Transactions

Research article
Adaptive neural consensus of nonlinearly parameterized multi-agent systems with periodic disturbances

https://doi.org/10.1016/j.isatra.2021.07.024Get rights and content

Highlights

  • The unmeasurable periodic parameters are described based on fourier series.

  • A new neural network approximator is introduced.

  • This paper solves two kinds of consensus problems based on neural network.

Abstract

This article settles consensus of nonlinearly parameterized multi-agent systems with periodic disturbances by using matrix theory, adaptive control, neural networks and fourier series expansion. Firstly, uncertain nonlinear dynamics with unmeasurable periodic input disturbances are constructed and described by using fourier series expansion and neural networks. Secondly, a novel distributed control protocol based on adaptive control method and matrix theory is designed to make the second-order closed-loop systems asymptotically stable. Thirdly, another new distributed control protocol based on the above consensus protocol is designed to make the closed-loop system with unknown control directions asymptotically stable. Finally, the correctness of the two control protocols is verified by three simulation examples.

Introduction

The development of multi-agent systems (MASs) mainly comes from people’s observation and research on some interesting natural phenomena in the biological world, such as migratory birds, congregating ant colonies, swimming fish, etc. these weak individuals can always achieve relatively complex collective behavior through mutual communication and cooperation. Under the research of this kind of phenomenon by scholars in various fields, the coordinated control of MASs emerges as a new interdisciplinary subject, and has very important research value and application prospect in the fields of biology, artificial intelligence control and computer science. At present, it has become an important branch of control discipline.

As an important branch of distributed artificial intelligence, the consensus problem is an important part of the coordinated control of MASs, which is the basic problem of MASs [1], [2], [3]. It means that each agent and its neighbor agent share data and control according to their own coordination, so that the states of all agents finally reach the same state. This state is not fixed invariable, it can represent position consensus, velocity consensus, angle consensus, etc. At present, the problem of consensus of intelligent system has been widely used in multi robot cooperation, formation flying of satellites and so on [4], [5], [6], [7].

From the perspective of the dynamic system model of network nodes, the research on the consensus of MASs has also transited from the simple first-order, second-order and high-order linear MASs in the early stage to the more general nonlinear MASs now. Compared with the linear MASs, the existence of nonlinear term makes the nonlinear MASs more complex, and the related theory and processing method in linear system cannot be used. Therefore, it is a challenging and worthy research direction to extend the related achievements of linear MASs to nonlinear MASs. In recent ten years, researchers in various fields began to apply the nonlinear theory to the research of MASs [8], [9], [10], [11], [12], [13], where Duan et al. [8] and Zhan et al. [9] addressed synergy problem based on event-triggered, Chen et al. [10] achieved adaptive optimal output synchronization based on off-policy learning, Wang et al. [11] and Shi et al. [12] handled synergy problems of nonlinear MASs with unknown control gains based on adaptive control method (see, [14]), and Fu et al. [13] solved the globally asymptotic containment consensus problem based on the sliding mode control method.

From the current research results, there are still many problems in the study of nonlinear MASs, especially the uncertain nonlinear MASs. Uncertain nonlinear MASs can more generally describe the dynamic model of some actual systems affected by the external environment, so it is more valuable and more challenging to study it. According to the existing dynamic models of nonlinear systems, the research of uncertain nonlinear MASs can be reduced to two main categories: uncertain dynamic MASs and linear uncertain parameterized MASs.

The MASs with uncertain dynamics [15], [16], [17], [18], [19], [20] are usually studied by equivalent transformation method (i.e., the uncertain dynamic system is transformed into linear constant parameterized problem based on fuzzy logic systems (FLS) or neural networks (NN)), where the adaptive NN collaborative control for nonaffine nonlinear MASs was shown in [15], the distributed output consensus tracking control was investigated for stochastic nonlinear dynamic systems with unknown nonlinear dead-zone of the followers in [16], the global fuzzy consensus control for the first- and second-order unknown nonlinear dynamic systems with uncertain input disturbances was studied in [17], the distributed cooperative tracking control problems were studied for nonlinear MASs containing first- and second-order following agents in [18], [19], and an adaptive iterative learning control method for nonlinear MASs with heterogeneous dynamics was shown based on the NN in [20]. The linear uncertain parameterized MASs [21], [22], [23], [24], [25], [26], [27], [28] usually considers three kinds of system models, namely constant parameterized system model, periodic time-varying parameterized system model, and time-varying parameterized system model with finite time interval. These models are usually studied by means of adaptive control [21], [22], repetitive learning control [23], [24], [25], [26] and iterative learning control [27], [28]. However, most of the existing references mainly focus on the linear uncertain parameterized MASs.

With the gradual improvement of the consensus control technology of linear uncertain parameterized MASs, it is naturally expected that the uncertain nonlinearly parameterized MASs can also achieve the effect of consensus control. Because the existing linear uncertain parameterized system model cannot well describe the motion characteristics of uncertain nonlinearly parameterized system model, the control method based on linear uncertain parametric system model cannot achieve the expected goal. For the consensus control of the uncertain nonlinearly parameterized MASs, to the authors’ knowledge, there is no any related result.

Uncertain nonlinearly parameterized MASs is a class of system model which cannot be modeled linearly. Similar to the linear parameterized model, it can also be divided into three kinds of system models, namely constant parameterized system model (i.e., uncertain dynamic model), periodic time-varying parameterized system model, and time-varying parameterized system model with finite time interval. Up to now, as far as the authors know, these three kinds of uncertain nonlinearly parameterized control models have not been well solved, especially the consensus control of MASs with time-varying nonlinear parametric modeling.

Based on the above analysis, this paper will address consensus of nonlinearly parameterized MASs with periodic disturbances. From the theoretical point of view, this kind of control system has generality. For example, when the nonlinearly parameterized dynamics of the system can be linearized, the nonlinear parameterized periodically perturbed MASs will be transformed into a linear parameterized periodically perturbed MASs. From the perspective of engineering needs, this kind of control system has a wide range of research value, because periodic disturbance is an important part of the production process and exists in various production processes. Therefore, it is of great theoretical significance and practical value to study the distributed consensus control of nonlinearly parameterized MASs with periodic disturbances.

The objective of this paper is to solve this problem, the main contributions are as follows.

(1) Fourier series expansion and NN are used as the main approximation tools to describe an uncertain nonlinear dynamic with periodic disturbance, in which fourier series expansion is used to describe uncertain periodic disturbance, and then the estimated value is used as the input of NN to describe uncertain nonlinear dynamic. Based on fourier series expansion and NN approximator, the problem of describing uncertain nonlinear dynamic with unmeasurable periodic disturbance is solved.

(2) Unlike [15], [16], [17], [18], [19], [20] where the uncertain nonlinear dynamic is described only considering the state information of the system itself, the description of the uncertain nonlinear dynamic in this paper considers not only the state information but also the unknown periodic time-varying disturbance.

(3) Unlike the uncertain linear parameterized MASs [21], [22], [23], [24], [25], [26], [27], [28], this paper studies the consensus problem of uncertain nonlinearly parameterized MASs with periodic disturbances. From the description of nonlinear dynamic, the problems considered in this study are more complex and comprehensive.

Notations. A=ATA and A1=i=1n|ai|, where A=a1,anTRn; In=1,,1TRn.

Section snippets

Fundamentals of graph theory

In the study of distributed cooperative control of MASs, state information needs to be transmitted between agents, which is usually based on the relevant knowledge of graph theory. The information exchange between agents can be visually described by using topological graph, and some basic concepts of algebraic graph theory are briefly introduced here.

Suppose that the communication relationship between followers is denoted by undirected graph G=,Y, where =1,2,,n represents the set of

Main results

So far, the research on periodic disturbance has not formed a unified system, which is mainly determined by the essential characteristics of control objects in different fields. As for the consensus results of periodic disturbances [23], [24], [25], [26], the main concern is the consensus control of linearly parameterized MAS, while the research on the consensus control of nonlinearly parameterized MAS with periodic disturbances no any related result. To solve this problem, this paper considers

Simulations

In this section, the correctness of the two control protocols is verified by three simulation examples. The connection topology of the three simulation experiments is shown in Fig. 1, in which marks 1-4 represent followers and mark 0 represents leader. Let us take a look at these three simulation experiments.

Example 1

The dynamic systems of 4 followers and one leader are described as ẋ1i=x2iẋ2i=5sinx1iħi|cos(2πt)|0.95x2i+uiand ẋ10=x20ẋ20=sin(x10)x20where i=1,2,3,4, ħ1=0.1, ħ2=0.5, ħ3=0.2, ħ4

Conclusion

In this study, two new fully distributed consensus protocols are presented. Because the unmeasurable periodic disturbance is considered in the uncertain nonlinear MASs, the consensus algorithms presented are more practical and widely used than the existing protocol methods. To complete the control objective, the uncertain nonlinear dynamics are compensated by using NN, and the unmeasurable periodic disturbances are described by Fourier series expansion. A significant contribution of this study

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities under Grant No.20101217127.

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