Mathematical Problems in Engineering
Volume 2020 (2020), Article ID 2350431, 9 pages
https://doi.org/10.1155/2020/2350431

Research Article

Fully Distributed Event-Triggered Containment Control of Uncertain Multiagent Systems

Zhaodong Liu,¹ Zhi Liu,¹ Xuewu Qian,¹ Ancai Zhang,¹ and Zhenxing Li1,2

¹Department of Automation, Linyi University, Linyi 276000, China
²School of Mathematics, Southeast University, Nanjing 210096, China

Correspondence should be addressed to Zhenxing Li; zhxingli@gmail.com

Received 23 April 2020; Accepted 30 June 2020; Published 26 July 2020

Guest Editor: Rongwei Guo

Copyright © 2020 Zhaodong Liu 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.

Abstract.

This paper investigates the event-triggered containment control problem of a class of uncertain nonlinear multiagent systems (MASs). By employing the local relative information, we design an adaptive event-triggered containment algorithm. The proposed containment algorithm can cope with the unavailability of global topology information and uncertain dynamics of follower agents. Therefore, the presented containment algorithm is free of global topology information, i.e., the designed algorithm is fully distributed. In addition, it is proved that Zeno behavior will not occur. At last, a numerical example is given to verify our event-triggered containment algorithm.

1. Introduction

During past two decades, cooperative control of MASs has been a hot research field for the reason of broad engineering applications, such as formation control [1], attitude alignment [2], mobile sensor networks [3], multirobot systems [4], and so forth [5, 6]. In practice, MASs are composed of mobile units, such as UAVs and robots, which are equipped with embedded control chips and limited energy supply. To extend the working hours of MASs, researchers developed various cooperative algorithms. Event-triggered cooperative algorithms can effectively reduce communications among neighbor agents and have gained extensive attention.

MASs can be divided into three cases, i.e., leaderless MASs, leader-follower MASs, and multileader MASs. The control objectives of those three kinds of MASs are consensus control, tracking control, and containment control, respectively. Consensus control and tracking control have been widely studied, from linear MASs [1, 7, 8] to nonlinear MASs [9–11]. Event-triggered control technique is also employed to design consensus and tracking controllers for various kinds of MASs [12–16]. Unlike consensus and tracking control, the objective of containment control is that each follower agent enters into the convex hull spanned by leader agents. In papers [2, 17–19], containment control problems of different kinds of MASs were studied. Event-triggered containment problems are also considered by researchers [20–24]. Miao studied the event-triggered containment control for first/second-order MASs with constant time delay in [20]. Event-triggered containment control of second-order MASs with sampled position data and time-varying input delays are considered in [21, 22]. Rong and her coauthors studied the event-triggered containment control problem for general MIMO linear MASs in [23]. Xu studied the event-triggered containment problem of Euler–Lagrange MASs in [24].

Cooperative algorithms of MASs use the local information to achieve global control tasks. However, most of the existing control algorithms of MASs need the information of Laplacian which is a global information. Control algorithms that do not need any global information are said to be fully distributed. Many researchers devoted to develop the fully distributed algorithms for MASs. Papers [25–28] presented some typical works of fully distributed algorithms. Recently, some researchers studied the fully distributed control problems of MASs via event-triggered approach as well. Zhu and Cheng studied the fully distributed event-triggered control problems for linear MASs in [29, 30]. Li and his coauthors investigated both event-triggered consensus and tracking problems of second-order nonlinear MASs in [31]. Apart from Euler–Lagrange MASs in [24], event-triggered containment problem of multiple leaders MASs was seldom considered.

Inspired by the above papers, we study the fully distributed event-triggered containment problem of a class of uncertain MASs. Not only the Laplacian of MASs but also the dynamics of follower agents are unavailable to design the containment controllers. To deal with these two unknown data, adaptive techniques are used to design event-triggered containment controllers. The main contributions of this paper are listed as follows: (i) our containment algorithm is fully distributed and does not use any global information of MASs; (ii) our containment algorithm is robust and still works when the dynamics of follower agent is uncertain; and (iii) our presented event-triggered algorithm is free of Zeno behavior.

2. Problem Statement and Preliminaries

Notations. Let be the -dimensional identity matrix, be a vector with each entry being 1, be the Euclidean norm (2-norm), be the Kronecker product, and be a diagonal matrix. We use , to denote eigenvalues of a symmetric matrix . denotes the Frobenius norm of matrix .

2.1. Preliminaries

In this paper, there are agents in the considered MASs. The communication topology among those agents is depicted by a graph , where , , and are the agent set, edge set, and adjacency matrix, respectively. denotes a directed edge from agent to agent and the associated weighing factor ; otherwise, . If , then there is an undirected edge between agent and agent . We assume that there is no self-loop edge, i.e., . Edge sequence denotes a directed path from agent to agent . denotes the Laplacian matrix of graph , with and . Suppose that there are follower agents and leader agents in the considered MASs. For simplicity, we assume that agents indexed by are the followers, and agents indexed by are the leaders. In practice, the desired trajectories of followers are generated by leaders. Hence, we assume that each leader is not influenced by other agents, that is, . For brevity, we use and to denote the follower set and leader set, respectively.

For the topology of MASs, we have the following assumption.

Assumption 1. For every follower agent , there exists at least a leader agent that has a path to follower agent .

Assumption 2. The induced subgraph of follower agents is an undirected graph.
Since each leader agent is not influenced by other agents, the associated Laplacian matrix can be partitioned as follows:

Lemma 1 (see [19]). If Assumptions 1 and 2 hold, is positive definite, each entry of is nonnegative and the sum of each row of equals one.

Definition 1 (see [32]). A set is convex if , for any and any . The convex hull of a finite set of points is the minimal convex set containing all points in . That is, .

2.2. Problem Statement

In this paper, we study event-triggered containment control problem of uncertain MASs without using any global topology information.

Consider a network system of agents with the following dynamics:

where , and are the system matrix, input matrix, state vector, and control input of agent , respectively; input nonlinearity is an unknown smooth function.

Based on the fact that is smooth and Stone–Weierstrass approximation theory [33], the unknown nonlinear function can be approximated by

with being a known basis function, being the neural network (NN) weight matrix, being a compact set, and being the approximation error such that

Suppose that both and are bounded, i.e., there exist nonnegative numbers such that

For the control input of each leader, we assume that satisfies the following assumption, .

Assumption 3. is bounded by a nonnegative constant , that is, .
The control objective of MASs (2) and (3) is to design event-triggered controllers for follower agents such that the states of follower agents enter into the convex hull spanned by leaders’ states, that is, , where .

3. Adaptive Event-Triggered Containment Algorithm

In this section, we will give an adaptive event-triggered algorithm to solve the containment problem of MASs (2) and (3).

Let

From Lemma 1, we know that , is a convex combination of states of leader agents. If follower agent can track , then enters into , . We call the target state trajectory of follower agent .

Denote as the tracking error of follower agent and . Then, we have

where

Let be the local relative state of follower agent . denotes the sample value of at sample instant , and denotes sample error of , i.e.,

By using , we give the following dynamical containment controller for each follower agent:

where is the dynamical gain of with , is a positive constant, is a constant, is the dynamical estimation of with , and are the turning scalars, and is the unique solution of the following Riccati equation:

and is a discontinuous function such that, for ,

The trigger function for adaptive controller (11) is given as

where

Remark 1. Usually, the feedback gain of a consensus controller depends on the eigenvalues of the Laplacian matrix, which are the global topology information. Since adaptive gains can be adjusted by automatically, adaptive controller (11) does not rely on any global topology information.

Remark 2. Trigger function (14) is a hybrid trigger function. Unlike trigger functions in other papers, we use a positive constant to replace the exponential decay term. The exponential decay term only guarantees that there is no Zeno behavior for finite time . By choosing a relative small , a satisfactory tracking error and a lower bound of can be guaranteed.

Remark 3. In the design of dynamical controller (11), we use the -modification technique. Containment controller (11) and trigger function (14) only guarantee that tracking error is uniformly ultimately bounded. If we do not employ the -modification technique, the dynamical gains and estimation parameters may increase unboundedly. To ensure the boundedness of and , we use -modification technique to adjust the dynamical parameters.
Denote

Then, combining (4), (8), and (11), we have, for ,

Theorem 1. Suppose that Assumptions 1 and 2 hold, containment problem of MASs (2) and (3) will be solved by dynamical controller (11). The tracking errors and adaptive parameters and are uniformly ultimately bounded (UUB). Furthermore, controller (11) does not exist Zeno behavior.

Proof. Take Lyapunov function candidate as

where with being a constant to be determined later.
Along with (11 and 17), we compute as follows:

From the definition of , one obtains

Hence, we get

where are convex combinations of leaders’ inputs with . Since the sum of each row of is equal to 1, one has

Denote . Since , one gets

where .
For the third term of (21), one has

Note that

where .
For the fourth, fifth, and sixth terms, one obtains

Hence, we have

where is a real constant. From (11), we have

Substituting (23)–(31) into (21), we obtain

Note that

where . Since and is positive definite, we have the following inequalities:

where and are the smallest and largest eigenvalues of , respectively.
Combining (32)–(35), one has

where is a constant.
Choosing enough large such that

Then, Riccati equation (12) and trigger function (14) guarantee that

Since and , one gets

where is a constant.
Combining (38)–(40) yields

where . Since and , we get

where with being the largest eigenvalue of .
After simple calculations, we have

which indicates that tracking error , dynamical gains , and are UUB.
In the last step, we prove that event-triggered controller (11) does not exist Zeno behavior.
When , sample error will be set to zero. Dini derivative will be used to analyze , and we get

Since , for , we have

Denote

And, one gets

Notice that . Hence, for , satisfies following inequality:

Let . From trigger function (14), we know that the th event of agent will not be triggered unless exceeds , i.e.,

Because , one has

By (48) and (50), one obtains the interinterval satisfying

Due to the existence of positive real , is greater than zero for any given . That means is strictly positive.

4. Simulations

One numerical example is given in this section to verify the adaptive event-triggered containment controller (11) developed in this paper.

Consider a leader-follower multiagent system with four follower agents and two leader agents. The Laplacian matrix associated with topology is given as follows:

The dynamics of leaders is depicted by the linearized model of the longitudinal dynamics of an aircraft [34], described by (3) with

denotes the state of each agent with being the angle of attack, being the pitch rate, and being the elevator angle. From Stone–Weierstrass approximation theory, the approximation error can be arbitrarily selected. We set and . The initial conditions of follower agents are given as . We choose control inputs of leader agents as and initial conditions as .

We set the initial values of adaptive gains and as and . By solving Riccati equation (12), we get . According to controller design (11), we set , and .

The simulation results are presented in Figures 1–6. Figures 1 and 3 show the trajectories of follower agents and leader agents. The black lines denote the trajectories of leader agents which indicate that the states of follower agents enter into the convex hull spanned by leaders’ states. Figure 4 displays the triggering instants of each agent. Figure 5 displays the adaptive gains . Figure 6 presents the norms of estimation coefficients .

Figure 1: trajectories of each agent.

Figure 2: trajectories of each agent.

Figure 3: trajectories of each agent.

Figure 4: Triggering instants of each follower agent.

Figure 5: Adaptive gains .

Figure 6: Norms of .

5. Conclusions

In this paper, we have studied the event-triggered containment problem of a class of uncertain MASs. The global topology information is unavailable, and parameters of follower agents are uncertain. Based on adaptive control technique, we have proposed an event-triggered containment controller which does not rely on any global topology information and can estimate the uncertain parameters. Under this containment algorithm, follower agents enter into the convex hull of leader agents. Moreover, the proposed algorithm has no Zeno behavior. In the future, we will concentrate our study on event-triggered consensus problem of MAS with unknown control directions.

Data Availability

The simulation data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant nos. 61803193, 61803194, and 61773193, the Postdoctoral Scientific Foundation of China under Grant 2017M620183, and Jiangsu Planned Projects for Postdoctoral Research Funds under Grant 1701077B.

References

W. Ren and E. Atkins, “Distributed multi-vehicle coordinated control via local information exchange,” International Journal of Robust and Nonlinear Control, vol. 17, no. 10-11, pp. 1002–1033, 2007.

Z. Meng, W. Ren, and Z. You, “Distributed finite-time attitude containment control for multiple rigid bodies,” Automatica, vol. 46, no. 12, pp. 2092–2099, 2010.

H. Su, X. Chen, M. Z. Q. Chen, and L. Wang, “Distributed estimation and control for mobile sensor networks with coupling delays,” ISA Transactions, vol. 64, pp. 141–150, 2016.

L. Consolini, F. Morbidi, D. Prattichizzo, and M. Tosques, “Leader-follower formation control of nonholonomic mobile robots with input constraints,” Automatica, vol. 44, no. 5, pp. 1343–1349, 2008.

G. Zhao, Y. Wang, and H. Li, “A matrix approach to the modeling and analysis of networked evolutionary games with time delays,” IEEE/CAA Journal of Automatica Sinica, vol. 5, no. 4, pp. 56–64, 2018.

G. Zhao, H. Li, W. Sun, and F. E. Alsaadi, “Modelling and strategy consensus for a class of networked evolutionary games,” International Journal of Systems Science, vol. 49, no. 12, pp. 2548–2557, 2018.

R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1520–1533, 2004.

F. Borreli and T. Keviczky, “Distributed LQR design for identical dynamically decoupled systems,” IEEE Transactions on Automatic Control, vol. 53, no. 8, pp. 1901–1912, 2008.

A. Das and F. L. Lewis, “Distributed adaptive control for synchronization of unknown nonlinear networked systems,” Automatica, vol. 46, no. 12, pp. 2014–2021, 2010.

Z. Li and H. Ji, “Distributed consensus and tracking control of second-order time-varying nonlinear multi-agent systems,” International Journal of Robust and Nonlinear Control, vol. 27, pp. 3549–3563, 2017.

Z. Li and H. Ji, “Finite-time consensus and tracking control of a class of nonlinear multiagent systems,” IEEE Transactions on Automatic Control, vol. 63, no. 12, pp. 4413–4420, 2018.

D. V. Dimarogonas, E. Frazzoli, and K. H. Johansson, “Distributed event-triggered control for multi-agent systems,” IEEE Transactions on Automatic Control, vol. 57, no. 5, pp. 1291–1297, 2012.

M. Cao, F. Xiao, and L. Wang, “Event-based second-order consensus control for multi-agent systems via synchronous periodic event detection,” IEEE Transactions on Automatic Control, vol. 60, no. 9, pp. 2452–2457, 2015.

E. Garcia, Y. Cao, and D. W. Casbeer, “Decentralized event-triggered consensus with general linear dynamics,” Automatica, vol. 50, no. 10, pp. 2633–2640, 2014.

D. Liuzza, D. V. Dimarogonas, M. Di Bernardo, and K. H. Johansson, “Distributed model based event-triggered control for synchronization of multi-agent systems,” Automatica, vol. 73, pp. 1–7, 2016.

Z. Li, J. Yan, W. Yu, and J. Qiu, “Event-triggered control for a class of nonlinear multiagent systems with directed graph,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2020.

Y. Zheng and L. Wang, “Containment control of heterogeneous multi-agent systems,” International Journal of Control, vol. 87, no. 1, pp. 1–8, 2014.

S. J. Yoo, “Distributed adaptive containment control of uncertain nonlinear multi-agent systems in strict-feedback form,” Automatica, vol. 49, no. 7, pp. 2145–2153, 2013.

Z. Li and H. Ji, “Containment control of a class of heterogeneous nonlinear multi-agent systems,” International Journal of Control, vol. 90, no. 9, pp. 2020–2030, 2017.

G. Miao, J. Cao, A. Alsaedi, and F. E. Alsaadi, “Event-triggered containment control for multi-agent systems with constant time delays,” Journal of the Franklin Institute, vol. 354, no. 15, pp. 6956–6977, 2017.

H. Xia, W. X. Zheng, and J. Shao, “Event-triggered containment control for second-order multi-agent systems with sampled position data,” ISA Transactions, vol. 73, pp. 91–99, 2018.

T. Li, Z. Li, S. Fei, and Z. Ding, “Second-order event-triggered adaptive containment control for a class of multi-agent systems,” ISA Transactions, vol. 96, pp. 132–142, 2020.

L. Rong, Y. Hua, G. P. Jiang, and S. Xu, “Distributed event—triggered containment control for dynamical multiagent networks,” International Journal of Robust and Nonlinear Control, vol. 29, no. 7, pp. 2223–2235, 2019.

T. Xu, Y. Hao, and Z. Duan, “Fully distributed containment control for multiple Euler-Lagrange systems over directed graphs: an event-triggered approach,” IEEE Transactions on Circuits and Systems I, vol. 67, no. 6, pp. 2078–2090, 2019.

W. Yu, W. Ren, W. X. Zheng, G. Chen, and J. Lü, “Distributed control gains design for consensus in multi-agent systems with second-order nonlinear dynamics,” Automatica, vol. 49, no. 7, pp. 2107–2115, 2013.

C. Wang and H. Ji, “Leader-following consensus of multi-agent systems under directed communication topology via distributed adaptive nonlinear protocol,” Systems & Control Letters, vol. 70, pp. 23–29, 2014.

C. Wang, X. Wang, and H. Ji, “Leader-following consensus for a class of second-order nonlinear multi-agent systems,” Systems & Control Letters, vol. 89, pp. 61–65, 2016.

Z. Liu, A. Zhang, Z. Liu, and Z. Li, “Fully distributed tracking control of high-order nonlinear multi-agent systems,” IEEE Access, vol. 7, pp. 141594–141601, 2019.

W. Zhu, Q. Zhou, and D. Wang, “Consensus of linear multi-agent systems via adaptive event-based protocols,” Neurocomputing, vol. 318, pp. 175–181, 2018.

B. Cheng and Z. Li, “Fully distributed event-triggered protocols for linear multiagent networks,” IEEE Transactions on Automatic Control, vol. 64, no. 4, pp. 1655–1662, 2019.

Z. Li, J. Yan, W. Yu, and J. Qiu, “Adaptive event-triggered control for unknown second-order nonlinear multiagent systems,” IEEE Transactions on Cybernetics, 2020.

R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, USA, 1972.

M. H. Stone, “The generalized Weierstrass approximation theorem,” Mathematics Magazine, vol. 21, no. 5, pp. 237–254, 1948.

G. Wen, P. Wang, T. Huang, W. Yu, and J. Sun, “Robust neuro-adaptive containment of multileader multiagent systems with uncertain dynamics,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 49, no. 2, pp. 406–417, 2019.