Formation control for networked multiagent systems with a minimum energy constraint

Junlong LI, Jianxiang XI, Ming HE, Bing LI

a Department of Physics, Lu¨liang University, Lu¨liang 033001, China

b School of Missile Engineering, Rocket Force University of Engineering, Xi’an 710025, China

c First Military Representative Office of the Rocket Army Equipment Department in Xi’an, Xi’an 710025, China

KEYWORDS Energy constraint;Formation;Minimum energy consumption;Networked control systems;Switching networks

Abstract Minimum-energy formation achievement problems for networked multiagent systems are investigated, where information networks with leaderless and leader-follower structures are respectively addressed and information networks are randomly switching. The critical feature of this work is that the energy constraint is minimum in the sense of the linear matrix inequality,but limited-budget control and guaranteed-cost control cannot realize a minimum-energy formation.Firstly,the leaderless minimum-energy formation control problem is converted into an asymptotic stability one via a nonsingular transformation and state space decomposition, and based on linear matrix inequality techniques, sufficient conditions for analysis and design of leaderless minimum-energy formation achievement are proposed, respectively, which can be solved by the generalized eigenvalue method. Then, main results of minimum-energy formation achievement of leaderless networked multiagent systems are extended leader-follower networked multiagent systems, where the asymmetric property of the leader-follower information network is well dealt with by two nonsingular transformations.Finally,two simulation examples are shown to verify the main results for minimum-energy formation achievements of leaderless and leader-follower networked multiagent systems, respectively.

1. Introduction

With rapid development of microelectronics techniques, intelligent control, swarm intelligence, and communication technologies, many scientists paid their attentions to both theory research and technology exploration of formation control of networked multiagent systems, which have extensive applications in various fields including formation strike of unmanned aerial vehicle swarm, cooperative transport of multiple industrial robots, coordinative detection of multiple underwater detectors, etc., as discussed in Refs. 1-5. Actually, the formation behaviors of networked multiagent systems originate from the internal mechanism investigation of their flocking and consensus behaviors, as shown in Refs. 6-12.

The network structures of networked multiagent systems are usually classified into the leaderless network structure and the leader-follower network structure. In this case, the associated formation structures are called the leaderless formation and the leader-follower formation, respectively. Especially, the leader-follower formation is also regarded as formation tracking. For leaderless networked multiagent systems, the statuses of all the agents are identical, and they construct the required formation structure by interactive negotiation.In Refs.13-14,leaderless formation control problems were investigated by the consensus-based formation control strategy, which has inbuilt advantages in distributed realization and theoretical rigorism when compared with the behavior-based formation control strategy and the virtualstructure formation control strategy, as shown in Ref. 15.Liu et al.16proposed some new robust formation criteria for networked multiagent systems with disturbances and applied theoretical conclusions to a set of quad-copters. In Ref. 17, a new approach was proposed to investigate the robust formation design problem, where an extended state observer was constructed to deal with the impacts of external disturbances.For leader-follower networked multiagent systems, the leader plays a critical role, and it is required that following agents maintain a specific formation structure to track the coordinated states of the leader. Leader-follower formation criteria with one leader and multiple leaders were presented in Refs.18,19, respectively.

Based on the time-varying property of an information network,it can be classified as a fixed information network and a switching information network.Randomly switching information networks originate from electronic interference, equipment fault, active communication silence, etc. Especially, a network after switching usually maintains during some time intervals, which can be depicted by the minimum dwell time.For a switching information network, there exists a switching set consisting of all possible networks for networked multiagent systems to choose.Compared with fixed information networks,formation control with switching information networks is more complex and challenging.Dong and Hu20determined the impacts of switching information networks on timevarying formation of general high-order networked multiagent systems. In Ref. 21, intermittent communications were well modeled, and cooperative control conclusions for networked multiagent systems were presented, where the node dynamics was of a high order. Qin et al.22gave a novel formation protocol on the basis of an observer, where the influence mechanism of the time delay and the intermittent communication was determined.Yang et al.23introduced an adaptive strategy to realize completely distributed formation and applied theoretical results to cooperative teleoperators.For formation control of multiple quadrotors,leader-follower formation criteria were shown, and practical results of a flying test were presented in Ref. 24.

In most practical networked multiagent systems, it is well known that the energy or resource is usually limited. In this case, the energy constraint is essential for the formation achievement of networked multiagent systems. A reasonable formation controller of networked multiagent systems can be designed if the minimum value of the whole energy consumption can be determined. In Refs. 13-24, the energy consumption and/or the control performance for networked multiagent systems to achieve formation were not considered.When these factors are involved, cooperative control can be categorized into local optimization cooperation and global optimization cooperation. In Refs. 25,26, a local index function was assigned to each agent, and local optimization cooperation was realized. Cao and Ren27constructed a Linear Quadratic Regulator (LQR) global index function, and optimal cooperation was realized, where the information network is required to be complete, which means that all agents are connected with each other. In Refs. 28,29, guaranteed-cost optimization cooperation for networked multiagent systems was discussed,where different expressions of the upper bound of the guaranteed cost were determined.Yu et al.30dealt with the impacts of external disturbances on leader-follower guaranteed-cost formation and proposed associated sufficient conditions to determine the gain matrix of the formation protocol. In Ref. 31, limited-budget formation with switching information networks was analyzed, where the upper bound of the energy constraint was previously given, but this bound may be not minimum. Actually, guaranteed-cost control and limited-budget control are suboptimal and cannot guarantee that the whole energy constraint is minimum. To the best of our knowledge, formation control for networked multiagent systems with the minimum energy constraint is still open.

To minimize the energy constraint, the current paper focuses on minimum-energy formation control for networked multiagent systems with randomly switching information networks, and two types of network structures are considered,that is, the leaderless network structure and the leader-follower network structure. By the specific characteristics of the orthogonal transformation matrix, two different expressions of the whole energy consumption are introduced, and the interactive relationship between the energy consumption and coordinated states, formation functions, and switching movements is established.Based on this interactive relationship,sufficient conditions in the terms of the linear matrix inequality are presented to achieve leaderless minimum-energy formation, which can be checked by the generalized eigenvalue method in the linear matrix inequality toolbox. Furthermore,leaderless minimum-energy formation criteria are extended to leader-follower networked multiagent systems,where the main difficulty is that the leader-follower information network is asymmetric, which makes the orthogonal transformation applied in leaderless networked multiagent systems no longer valid. By dividing the whole information network into an information network between followers and the leader and an information network among followers, the impacts of the asymmetric information network are dealt with.

The arrangement of this work is shown as follows. In Section 2, the network structure and the problem description are presented, where the information network among agents is randomly switching and the energy constraint is required to be minimum. In Section 3, sufficient conditions for leaderless formation are proposed, which can guarantee that the energy constraint is minimum as formation is achieved asymptotically.Section 4 extends the main results from leaderless formation to leader-follower formation.In Section 5,two simulation examples are presented to check the main results. Section 6 summarizes the whole work.

Table 1 illustrates the key symbols applied in this work.2.Preliminaries and problem statements for leaderless minimumenergy formation

Table 1 Symbol indices.

In this section, some basic concepts and conclusions on graph theory are firstly summarized.Then,the switching information network is modeled.Finally,the problem description for leaderless minimum-energy formation is presented.

2.1. Basic concepts and conclusions about graph theory

Lemma 132. If the graph G is undirected and connected, then the Laplacian matrix L has a simple zero eigenvalue with the associated eigenvector being 1Mand M-1 positive eigenvalues.

2.2. Switching information network model

For a leaderless networked multiagent system with M homogeneous agents and randomly switching networks, the switching set ψ contains all possible information networks, all of which are undirected and connected. Let κ（t）: 0，+∞）［ →ψ denote the switching signal function and tj（j=0，1，···） be the j-th switching moment.For the j-th and j+1-th switching moments,it is required that tj+1-tj≥Tdwith Td＞0,which is called the minimum dwell time.In this case,the function κ（t）is piecewise continuous. For any moment t, Gκ（t）is applied to denote the information network of this networked multiagent system,which means that the information network is randomly switching according to the switching signal function.The mathematic property of an information network can be described by its Laplacian matrix as shown in Ref. 32. The Laplacian matrix associated with the information network Gκ（t）is denoted by Lκ（t）,whose row sum is zero,that is,Lκ（t）1=0.Especially,zero is its simple eigenvalue since it is assumed that each network in the set ψ is connected. It is assumed that the information network satisfies the following assumption.

Assumption 1. All information networks in the switching set are connected and the number of agents is known.

2.3. Problem description

The dynamics of this leaderless networked multiagent system is modeled as

Remark 1. Formation control strategies with the general energy constraint include two types: the guaranteed-cost formation control strategy and the limited-budget formation control strategy. For the guaranteed-cost formation control strategy discussed in Ref. 30, it is required to determine an expression of the upper bound of the guaranteed cost, but the design algorithm of the gain matrix of the formation protocol in the formation process is not related with the guaranteed cost. For the limited-budget formation control strategy discussed in Ref. 31, it is required to guarantee formation achievement under the condition that the upper bound of the energy consumption is given previously, and this upper bound impacts the design algorithm of the gain matrix in the formation process. However, this upper bound may be not minimum. It is more challenging to minimize the whole energy consumption via designing the gain matrix of the formation protocol in the formation process. To achieve minimum-energy formation control, two critical challenges are required to be dealt with. The first one is to design the interactive relationship between the whole energy consumption and coordinated states, formation functions, and switching information networks, that is, it is needed to introduce the whole energy consumption into networked multiagent systems.The second one is to minimize the energy constraint by constructing some optimization procedures. Moreover, it should also be pointed out that an arbitrary formation structure can be achieved by designing appropriate formation functions, but this formation structure may not be realized due to the physical structure constraints of each agent, which can be dealt with by introducing a formation feasible condition.

3. Design and analysis criteria for leaderless minimum-energy formation

This section firstly converts the leaderless formation control problem into an asymptotic stability one. Based on this transformation, sufficient conditions for the achievement of leaderless minimum-energy formation are respectively given via linear matrix inequality, where the minimum energy assumption problem can be solved by the generalized eigenvalue one.

Since it is assumed that randomly switching networks have a minimum dwell time,that is,Td＞0,and the switching signal function κ（t）and the Laplacian matrix Lκ（t）are piecewise continuous differentiable.In this case,the nonsingular matrix Tκ（t）is piecewise continuous differentiable, so one can convert the networked multiagent system in Eq. (5) intocr

Theorem 1. The networked multiagent system in Eq. (1) is leaderless minimum-energy formation achievable by the formation protocol in Eq. (2) if Afi（t）= ˙fi（t） i=1，2，···，M

（） and there exists an optimal solution for the following minimization problem:

Proof. We state the proofs by two steps. The first one is to design K so that the networked multiagent system in Eq. (1)achieves leaderless formation, and the second one is to add a specific condition so that the energy constraint is minimum.First of all, we design K so that the subsystems in Eq. (7) are asymptotically stable.Construct the following Lyapunov functional candidate:

By Eq. (25), the minimum energy consumption can be obtained by minimizing the parameter γ, and the matrix R in Eq.(26)can be regulated to minimize the energy consumption.In this case, the matrix inequality R-1＜γIdis introduced to realize the minimum energy consumption, which is equivalent to Id＜γR by left- and right-multiplying the matrix R. Due to x～（0）=x（0）-f（0）, by the convex feature of the linear matrix inequality, the conclusion of Theorem 1 is deduced. □

According to the generalized eigenvalue approach in Ref.33, the minimum energy constraint term R-1＜γIdis imposed to realize minimum-energy formation in the sense of the linear matrix inequality by minimizing the constant γ. In this case,the optimal solution of the matrix variable R can be obtained by minimizing the constant γ with the linear matrix inequality constraints Θi（i=1，2）,and the energy constraint is minimum if the matrix variable R has the optimal solution.Furthermore,it can be found that the conclusion in Theorem 1 has the scalability property, that is, the dimension of the variable of this conclusion is not associated with the number of agents.In this case, the consumption of the computing source does not become large as the number of agents increases.Furthermore,this conclusion is associated with the nonzero eigenvalues of the Laplacian matrix of the switching information networks,whose precise values are not required to be obtained and can be estimated by the approaches shown in Refs. 34,35, which have low computation complexity. Moreover, by the proofs of Theorem 1,one can deduce the following conclusion,which gives a linear matrix inequality sufficient condition for leaderless minimum-energy formation analysis for the networked multiagent system in Eq. (1) with the formation protocol in Eq. (2), where the gain matrix K is known.

Corollary 1. The networked multiagent system in Eq. (1) with the formation protocol in Eq. (2) achieves leaderless minimumenergy formation if Afi（t）= f˙i（t） （i=1，2，···，M） and there exists an optimal solution for the following minimization problem:

4. Design and analysis criteria for leader-follower minimumenergy formation

In this section, by the linear matrix inequality tool, sufficient conditions for design and analysis of leader-follower minimum-energy formation are given, respectively, where the impact of the asymmetric property of the Laplacian matrix of a leader-follower switching network on energy constraint minimization is well dealt with by introducing two nonsingular transformations.

Consider the networked multiagent system in Eq. (1) with leader-follower switching information networks, where the first agent is assigned as the leader and the other M-1 homogeneous agents as followers. The control input of the leader is set as zero,that is,u1（t）≡0,and the control input of each follower is determined by the formation protocol designed by the coordinated states of its neighbors. In this case, the dynamics of the networked multiagent system in Eq.(1)can be rewritten as

It is supposed that the information channels from the leader to the followers are unidirectional, but the information channels among followers are undirected, which can also be regarded as bidirectional information channels.Especially, only some followers can receive coordinated state information from the leader, and the information networks among all agents are switching, where there exists at least a path from the leader to each follower for all the information networks in the switching set ψ, which means that each information network contains a spanning tree defined in Ref. 32.

A leader-follower formation protocol with an energy consumption term and switching information networks is presented as follows:

From Eq.(34)to Eq.(35),according to the similar analysis to Theorem 1,the conclusion of Theorem 2 can be derived. □

When the gain matrix K of the formation protocol in Eq.(28) is given previously, the following sufficient condition for leader-follower minimum-energy formation analysis is proposed by the linear matrix inequality tool.

Corollary 2. The networked multiagent system in Eq. (27) with the formation protocol in Eq. (28) achieves leader-follower minimum-energy formation if Afi（t）= f˙i（t） （i=2，3，···，M）and there exists an optimal solution for the following minimization problem:

5. Numerical examples

This section gives two numerical simulation examples to illustrate the validness of main results for leaderless and leader-follower minimum-energy formations, respectively.

Example 1 ((Leaderless minimum-energy formation).). A group of six agents is considered, whose information topology is randomly switched from the switching topology set as depicted in Fig.1.The information strength wij=1 if agent i can receive information from agent j; otherwise, wij=0. The dynamics of each agent is described by （A，B） as follows:

Fig. 1 Leaderless switching topology set.

Fig. 2 Switching signal function κ（t）.

Fig. 3 State trajectories of all agents for leaderless case.

Fig. 4 Curves of Ellmin（t） and Ell（t）.

It can be found that the upper bound of the energy consumption with the minimum energy constrain is lower than that with the general energy constraint.

Example 2 ((Leader-follower minimum-energy formation).). A group of one leader and six followers is considered, whose switching topology set is shown in Fig. 5. The system matrices for each agent are modeled as

Fig. 5 Leader-follower switching topology set.

Fig. 6 State trajectories of all agents for leader-follower case.

Fig. 6 shows the state trajectories of all agents at t=0 s，8 s，9.5 s,and 10 s，where the leader and six followers are marketed by a hexagram and six different colored circles, respectively. From Fig. 6(a) to (d), it can be found that a desired formation is achieved.Meanwhile,the six followers are located at six vertices of the hexagon and continue to rotate around the leader. The trajectory of the leader within 10 s is shown in Fig. 7, where the trajectory starts at the blue hexagon and ends at the black hexagon.

Fig. 7 Trajectory of leader.

Fig. 8 Curves of Elfmin（t） and Elf（t）.

6. Conclusions

For networked multiagent systems with leaderless switching networks, a new formation protocol is shown to achieve formation control with a minimum energy constraint. By the common Lyapunov functional approach, sufficient conditions for design and analysis of leaderless minimum-energy formation are proposed,respectively,which realize the objective that the energy constraint in the sense of the linear matrix inequality is minimum and can be checked by the generalized eigenvalue method. Especially, those sufficient conditions are scalable, that is, the dimension of the variable is equal to that of each agent and is not associated with the number of agents.Moreover,sufficient conditions for leaderless minimum-energy formation control are extended to leader-follower networked multiagent systems, where the influence of the asymmetric structure of the leader-follower network is dealt with by combining a time-invariant nonsingular transformation with a specific structure and a time-varying orthogonal transformation. It should be pointed out that the key difference between the sufficient conditions for leaderless and leader-follower minimum-energy formations is that the relationship matrices have different structures, which introduce initial conditions and network information into the whole networked multiagent system.Further work will focus on the investigation of switching networks by relaxing some hypothesis, such as the information silence, the jointly connected switching mode, etc.Especially, a similar LQR regulation framework will be developed to actualize simultaneous optimization of the whole energy consumption and the control performance. Moreover,the associated results of the minimum-energy formation control will be extended to some practical multiagent systems,such as multiple-Unmanned Aerial Vehicle (UAV) systems,where each agent has specific dynamics associated with physical structures, and some numerical simulations and physical experiments will be implemented.

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 National Natural Science Foundation of China (Nos. 62176263, 62103434, 62003363,and 61703411), the Science Foundation for Distinguished Youth of Shaanxi Province, China (No. 2021JC-35), the Youth Science Foundation of Shaanxi Province, China (No.2021JQ-375), the China Postdoctoral Science Special Foundation (No. 2021T140790), and the China Postdoctoral Science Foundation (No. 271004).

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.cja.2022.01.015.