Elsevier

Neurocomputing

Volume 481, 7 April 2022, Pages 310-321
Neurocomputing

Minimum-energy synchronization for interconnected networks with non-periodical information silence

https://doi.org/10.1016/j.neucom.2022.01.083Get rights and content

Abstract

Minimum-energy synchronization control for interconnected networks is addressed, where network topologies contain leaderless and leader-follower structures and information transmission of the whole network is non-periodically silent. The key characteristic of the current work is that the total energy consumption is minimum in the sense of the linear matrix inequality, while both the guaranteed-cost synchronization and the limited-budget synchronization cannot make the total energy consumption be minimum. Firstly, the leaderless minimum-energy synchronization achievement problem is transformed into the asymptotic stabilization problem by the state decoupling strategy, and sufficient conditions of leaderless minimum-energy synchronization are presented by the Lyapunov-based method. Especially, those conditions can be solved by the generalized eigenvalue approach on the basis of the linear matrix inequality. Then, main results of leaderless minimum-energy synchronization are expanded to leader-follower interconnected networks, where the key challenge is that these networks are nonsymmetrical. Finally, two numerical examples are illustrated to verify main results.

Introduction

In recent years, a lot of researchers from different fields paid their attentions to interconnected networks, which promote significant developments for various practical applications, such as flocking [1], [2], synchronization analysis [3], [4], [5], [6], cooperative remote sensing [7], [8], [9] and formation control [10], [11], [12], etc. Synchronization for interconnected networks is a fundamental problem of cooperative control, which makes cooperative states or cooperative outputs of all agents achieve an agreement. Actually, synchronization and consensus own the same connotation in the literature. Accompanying the research on the internal mechanism of distributed interconnected networks, the extensive and in-depth investigation on the synchronization control acquires great achievements.

According to the structure features of the network topology, interconnected networks can be divided into the leaderless interconnected network and the leader-follower interconnected network. In this case, the related synchronization problems are called the leaderless synchronization and the leader-follower synchronization, respectively. In [13], synchronization strategies were applied to deal with cooperative control of multiple interconnected vehicles with leaderless topology structures. Nuno et al. [14] analyzed the synchronization behaviors of leaderless interconnected networks by the distributed synchronization protocol. By the impulsive control method, synchronization criteria for both leaderless and leader-follower interconnected networks were presented in [15]. Moreover, the information integrity is also critically important for synchronization control, which is mainly impacted by the information delay and the topology switching. Maghenem et al. [16] applied synchronization strategies to interconnected nonholonomic vehicles, where the information delay among agents is time-varying and non-differentiable. In [17], the double-integrator control was used to interconnected networks, where network topologies are switching and the dwell time is small.

In the above works on synchronization control for interconnected networks, the network topology among agents is connected and cooperative information is transmitted continuously. Since there may exist the electronic interference in the battlefield environment, information interactions among agents may be disappeared simultaneously and recover when the electronic interference is removed. Furthermore, an interconnected network may adopt the active anti-jamming strategy to save the energy or to avoid the detection of the electromagnetic spectrum feature by the hostile force. In this case, there exist some time intervals without any information interaction among all agents. Synchronization control was applied to achieve rendezvous of mobile networked agents in [18], where information interaction among agents is non-periodically silent. Based on the local observer with the non-periodical information silence, Zhang et al. [19] proposed a novel synchronization protocol to achieve the output synchronization, where the whole interconnected network is heterogeneous. In [20], a heterogeneous interconnected network was considered, where the network topology was modeled as the leader-follower structure and the influence mechanism of the non-periodic information silence was determined. Li et al. [21] modeled each agent as a second-order discrete-time dynamic system and applied synchronization control to realize containment, where the non-periodical information silence is asynchronous. It should be pointed out that the information silence and the intermittent control are essentially identical from the point of the information interaction of interconnected networks. However, from the point of the control approach, the intermittent control can also be caused by the event trigger control strategy.

Moreover, the energy consumption and/or the control performance are important factors in practical applications of interconnected networks, which can usually be modeled as optimization control problems. The global optimization synchronization contains the optimal one, the guaranteed-cost one and the limited-budget one. Cao and Ren [22] proposed optimal synchronization criteria under the condition that the network topology is complete, which requires that all agents interact with each other. For the guaranteed-cost synchronization control, a global optimization index was constructed by cooperative information of neighboring agents in [23], [24], [25], where different expressions of the upper bound of the guaranteed cost were determined, but those bounds may be not minimum. For the limited-budget synchronization control, it is required that the upper bound of the global optimization index is less than a previously given value, as discussed in [26], [27], but this value may be not minimum. For the interconnected network with the non-periodical information silence, it is new and challenging to minimize the whole energy consumption by regulating the gain matrix of the synchronization protocol. To the best of our knowledge, synchronization design and analysis problems for interconnected networks with the minimum energy constraint and the non-periodical information silence are still open.

The current paper focuses on minimum-energy synchronization for interconnected networks with the non-periodical information silence, where the leaderless network structure and the leader-follower network structure are unified into the same control framework. A new synchronization protocol is proposed by the local information interactive strategy, where an energy constraint term and the non-periodical information silence are introduced. Furthermore, sufficient conditions for leaderless minimum-energy synchronization achievement are presented in terms of the linear matrix inequality, where the minimum energy is realized by constructing the relationship between the energy constraint and the matrix variable. Especially, the impacts of the non-periodical information silence are constrained by imposing an inequality condition, which can make the convergent quantity during the non-silent component larger than the divergent quantity during the silent component. Moreover, a double transformation strategy is introduced to expand main results of the minimum-energy synchronization from leaderless interconnected networks to leader-follower interconnected networks, where the key challenge is to deal with the impacts of the nonsymmetrical structure.

Compared with the existing work about optimization synchronization control for interconnected networks, the essential contributions of the current paper include the following three aspects. Firstly, the current paper realizes the synchronization and the minimum-energy consumption simultaneously, but both the guaranteed-cost synchronization control and the limited-budget synchronization control in [23], [24], [25], [26], [27] cannot minimize the energy consumption of an interconnected network as a whole. Secondly, the impacts of the non-periodical information silence are constrained by an inequality associated with the maximum silent rate and the maximum time interval, where an exponential term is imposed to realize the minimum energy consumption. The impacts of the non-periodical information silence on the minimum energy consumption were not dealt with in [23], [24], [25], [26], [27]. Thirdly, the minimum-energy synchronization problems for the leaderless and leader-follower interconnected networks are unified into the same control framework by a double transformation strategy, where the main difficulty is that the leader-follower interconnected network is asymmetric. By separating the whole leader-follower interconnected network into the interconnected network among followers and the interconnected network between the leader and followers, the asymmetrical structure of the leader-follower interconnected network is well dealt with. The approaches in [23], [24], [25], [26], [27] cannot unify two types of interconnected networks with minimum energy constraints.

The statement of the current work is arranged as follows. In Section 2, the leaderless interconnected network with the non-periodical information silence is modeled, and the problem description is given, where the energy constraint is imposed to realize the minimum energy consumption in the sense of the linear matrix inequality. In Section 3, leaderless synchronization criteria in terms of the linear matrix inequality are proposed, which can ensure that the actual energy consumption is minimum when all agents achieve synchronization asymptotically. In Section 4, main synchronization results for leaderless interconnected networks are expanded to leader-follower interconnected networks. In Section 5, two simulation examples are presented to demonstrate main results for leaderless and leader-follower interconnected networks, respectively. Section 6 summarizes the whole work.

The following table illustrates key symbols used in this work. see Table 1.

Section snippets

Problem description for leaderless interconnected networks

Consider a leaderless interconnected network consisting of M identical agents, where each agent is modeled byẋi(t)=Axi(t)+Bui(t),where i=1,2,,M,ARd×d and BRd×m are system matrices, xi(t) is the cooperative state with a compatible dimension, and ui(t) is the synchronization protocol, which is needed to be determined on the basis of the cooperative states of neighboring agents of agent i; that is, the synchronization control is realized in a distributed manner.

The leaderless interconnected

Synchronization criteria for leaderless interconnected networks

In this section, by the state decoupling strategy, the leaderless synchronization achievement problem is converted into the asymptotic stabilization one. According to this decoupling, leaderless minimum-energy synchronization criteria are proposed in terms of the linear matrix inequality, where a special linear matrix inequality term is imposed to constrain the minimum energy consumption. These criteria can be solved by the generalized eigenvalue approach.

Let x(t)=x1T(t),x2T(t),,xMT(t)T, then

Synchronization criteria for leader-follower interconnected networks

This section expands main results of minimum-energy synchronization from leaderless interconnected networks to leader-follower interconnected networks and presents sufficient conditions for leader-follower minimum-energy synchronization achievement in terms of the linear matrix inequality. The critical challenge is to deal with the influence of the nonsymmetrical property of the leader-follower network topology on the minimum-energy synchronization.

For an interconnected network with M agents

Numerical simulations

This section presents two numerical examples to verify the validness of main results about leaderless minimum-energy synchronization and leader-follower minimum-energy synchronization, respectively.

Example 1

(Leaderless minimum-energy synchronization): Consider a leaderless interconnected network consisting of six agents with the network topology showing in Fig. 1. If agent i can receive the cooperative state of agent j directly, then wij=1 and wij=0 otherwise.

The system matrices A and B are designed as:A

Conclusions

For interconnected networks with leaderless topology structures, a new synchronization protocol with an energy constraint and the non-periodical information silence was proposed to realize the minimum-energy synchronization in the sense of the linear matrix inequality. Based on the state decoupling strategy and the Lyapunov-based approach, the leaderless minimum-energy synchronization achievement problem was converted into the asymptotic stabilization one and the associated synchronization

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.

Acknowledgements

This work was supported by National Natural Science Foundation of China (Nos. 62176263, 62103434, 62003363, and 61703411), Shaanxi Natural Science Foundation for Distinguished Young Scholars (No. 2021JC-35), Shaanxi Natural Science Foundation for Youths (No. 2021JQ-375), China Postdoctoral Science Special Foundation (No. 2021T140790), and China Postdoctoral Science Foundation (No. 271004).

Junlong Li received the B.S. and M.S. degrees from the North University of China, Taiyuan, China, in 2015 and 2018, respectively. He is currently pursuing the Ph.D. degree in control science and engineering from Rocket Force University of Engineering, Xi'an, China. His research interests include optimal control, swarm systems and interconnected networks.

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  • Junlong Li received the B.S. and M.S. degrees from the North University of China, Taiyuan, China, in 2015 and 2018, respectively. He is currently pursuing the Ph.D. degree in control science and engineering from Rocket Force University of Engineering, Xi'an, China. His research interests include optimal control, swarm systems and interconnected networks.

    Jianxiang Xi received the B.S. and M.S. degrees from High-Tech Institute of Xi'an, China in 2004 and 2007, respectively. He received the Ph.D. degree in control science and engineering from Rocket Force University of Engineering, Xi'an, China in 2012 by a coalition form with Tsinghua University. He is currently a Professor at the control science and engineering of Rocket Force University of Engineering, China. His research interests include complex systems control, switched systems and swarm systems.

    Le Wang received the B.S., M.S. and Ph.D. degrees from Rocket Force University of Engineering, Xi'an, China in 2014, 2016 and 2020, respectively. He is currently a Lecturer at the control science and engineering of Rocket Force University of Engineering, China. His research interests include optimal control, fault tolerant control, and multiagent systems.

    Donghao Qin received the B.S. degree in hot working technology and equipment from Taiyuan University of Science and Technology, Taiyuan, China, and M.S. degree in mechanical manufacture and automation from Northwestern Polytechnical University, Xi'an, China, in 2000 and 2010, respectively. He is currently pursuing the Ph.D. degree in control science and engineering from Rocket Force University of Engineering, Xi'an, China. His research interests include multi-agent systems, optimal control and formation control.

    Bing Li received the B.S. degree in missile test and control engineering from Rocket Force University of Engineering, Xi'an, China. He is currently an Engineer at First Military Representative Office of the Rocket Army Equipment Department in Xi'an, China. His research interests include optimal control, multiagent systems, and networked systems.

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