Eigenvalues Of Laplacian Matrix Research Articles

How do the eigenvalues of the Laplacian matrix affect route to synchronization patterns?

In cluster synchronization, network nodes are divided into synchronized groups before the whole network gets synchronized. This phenomenon is crucial in understanding the mechanism behind the synchronization of real-world and man-made complex networks like neuronal networks and power grids. The Laplacian matrix and its eigenvalues provide helpful information about the networks' synchronization, robustness, and controllability. Here, analyzing the relation between the Laplacian matrix eigenvalues and cluster synchronization demonstrates that the intensity of the eigenvalues has significant importance on the clusters' appearance. Results show that considering groups of equal eigenvalues yields the appearing of clusters in the network. So, this technique allows the ability to design networks with the desired number of clusters with defined cluster size. Synchronization of the clusters results in a plateau in the order parameter evolution. Furthermore, studying two different chaotic systems shows that this relationship depends on the systems' dynamics. Here, the eigenvector centrality tool is utilized to examine the existence of clusters besides the graph representation.

Dynamic Event-Triggered Formation Control for Heterogeneous Multiagent Systems With Nonautonomous Leader Agent.

In this article, the time-varying output formation issue of heterogeneous multiagent systems is investigated by the event-triggered control scheme. Only the outputs of all agents, including leader agent and follower agents, are measurable. The leader agent contains an unknown input signal to generate flexible reference trajectory. Also, only a subset of follower agents have the direct access to the leader agent. First, for each follower, the leader-state compensator is designed to estimate the state of leader. Two kinds of dynamic event-triggered (DET) mechanisms, i.e., node- and edge-based event-triggered schemes, can be equipped on the compensator to save the communication bandwidth of leader-follower and follower-follower interactions, respectively. Then, the distributed formation controller is built to drive each follower achieving formation tracking. The presented control protocol consisting of the DET state compensator and formation controller is fully distributed, which is independent of the global information of communication topology, such as the eigenvalues of Laplacian matrix of communication topology and amount of whole agents. Finally, the numerical experiments and comparison experiments are exhibited to verify the effectiveness of the presented control protocol.

Distributed adaptive time‐varying formation of multi‐UAV systems under undirected graph

AbstractIn this paper, a fully distributed control protocol is presented based on adaptive technology to solve the time‐varying formation problem of multi‐unmanned aerial vehicle (UAV) systems, where the coupling weights between neighbouring UAVs can be adjusted through adaptive method. Firstly, the proposed control protocol eliminates the need for global information on the interaction topology of multi‐UAV systems, the calculation of Laplacian matrix eigenvalues is avoided and reduces the computational complexity of the system. Secondly, an algorithm with two steps is proposed to determine the distributed adaptive control gain matrices, where the feasibility condition of time‐varying formation of multi‐UAV systems is given. Then, the closed‐loop stability of multi‐UAV systems is proved through Lyapunov theory. Finally, numerical simulation is conducted to verify the effectiveness of the theoretical results.

A fully distributed event-Triggered control approach to leader-following consensus of multiagent systems

This paper investigates the leader-following consensus problem for general linear multiagent systems under a fully distributed manner. To avoid using the global information of the communication network, two node-based adaptive parameters are designed and introduced to the fully distributed event-based control law. Such a protocol is more scalable and flexible for large-scaled and complex networked system, such as microgrids or multiple unmanned vehicles. A novel event-triggered rule is presented, and an internal dynamic variable is introduced in it to further cut down the triggering times and save the communication resources. A well-designed Lyapunov function candidate is constructed to not only ensure the convergence of the close-looped systems but also avoid using the eigenvalue of Laplacian matrix in the dynamic variable. Moreover, it is proved that the designed event-triggered conditions can avoid the Zeno behaviour. Finally, a simulation example based on a group of unmanned intelligent vehicle systems is provided to demonstrate the validity of the designed control method.

Consensus of fractional-order multi-agent systems with sampled position data

In this paper, the consensus of fractional-order multi-agent systems with the order α satisfying α∈(0,1] is investigated. A novel distributed control protocol is constructed, in which only the past sampled position data of neighbors are utilized. Then, based on Laplace transform, Mittag-Leffler function, and matrix theory, necessary and sufficient criteria depending on the order, coupling gains, sampling period, and communication topology, are established to attain consensus for the systems. Meanwhile, the intervals of coupling gains and sampling period associated with the maximum eigenvalue of Laplacian matrix over an undirected graph are presented to attain consensus. Finally, the availability of theoretical results is verified by numerical simulations.

Multiple-bipartite consensus for networked Lagrangian systems without using neighbours' velocity information in the directed graph

This paper investigates the multiple-bipartite consensus problem without considering neighbours' velocity information in networked Lagrangian systems (NLSs). A distributed adaptive control algorithm without using neighbours' velocity information is proposed, which facilitates the practical configuration deployments in the coopetition networks. By borrowing a subtle vector composed of the eigenvector components associated with zero eigenvalue of Laplacian matrix, a novel reference estimated vector is introduced to conduct the stability analysis step-by-step in the coopetition networks. Finally, simulations are provided to show the effectiveness of the proposed algorithm.

Energy of Graph

By given the adjacency matrix, laplacian matrix of a graph we can find the set of eigenvalues of graph in order to discussed about the energy of graph and laplacian energy of graph. (i.e. the sum of eigenvalues of adjacency matrix and laplacian matrix of a graph is called the energy of graph) and the laplacian energy of graph is greater or equal to zero for any graph and is greater than zero for every connected graph with more or two vertices (i.e. the last eigenvalues of laplacian matrix is zero), according to several theorems about the energy of graph and the laplacian energy of graph that are described in this work; I discussed about energy of graph, laplacian energy of graph and comparing them here.

A fully distributed adaptive event-triggered control for output regulation of multi-agent systems with directed network

This paper considers the cooperative output regulation problem for heterogeneous linear multi-agent systems via an event-triggered control strategy for the directed graph. First, to cancel the requirement of continuous update of the control rule, a novel distributed event-triggered control protocol such that each agent can determine when to update the control rule asynchronously, where the Zeno behavior is strictly ruled out based on the contradiction. Later, to make the multi-agent systems more flexible, based on the adaptive control technology, the cooperative output regulation problem of heterogeneous linear multi-agent systems can be solved in a fully distributed manner, where the global information, such as the number of agents or the eigenvalues of Laplacian matrix, is no longer needed. Furthermore, the considered communication network is a more general directed network, in which the minimum communication channel of the directed network is halved compared with the undirected case. Finally, the effectiveness of the proposed control strategy is verified on a group of heterogeneous agents.

Second-Order Hybrid Consensus of Multi-Agent Systems With Matrix-Weighted Networks

This article considers a hybrid consensus in the context of second-order multi-agent systems with matrix-weighted networks. A matrix-weighted hybrid consensus controller is built upon the sampled-data method in continuous-time subsystem. For the cases of undirected graph and directed graph, some sufficient and necessary conditions, which depend on the sampling period, coupling gains and the eigenvalues of Laplacian matrix, are derived with the assistance of stability theory and matrix theory. Finally, through simulation comparison, the validity of the obtained results is demonstrated.

Performance Analysis of Consensus Algorithms Over Prism Networks Using Laplacian Spectra

Prism networks belong to generalized petersen graph topologies with planar and polyhedral properties. These networks are extensively used to study the complex networks in the context of computer science, biological networks, brain networks, and social networks. In this letter, the performance analysis of consensus algorithms over prism networks is investigated in terms of convergence time, network coherence, and communication delay. Specifically, in this letter, we first derive the explicit expressions for eigenvalues of Laplacian matrix over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> -dimensional prism networks using spectral graph theory. Subsequently, the study of consensus metrics such as convergence time, maximum communication time delay, first order network coherence, and second order network coherence is performed. Our results indicate that the effect of noise and communication time-delay on the consensus dynamics in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> -dimensional prism networks is minimal. The obtained results illustrate that the scale-free topology of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> -dimensional prism networks along with loopy structure is responsible for strong robustness with respect to consensus dynamics in prism networks.

Identification of Stability Regions in Inverter-Based Microgrids

A new method for the stability assessment of inverter-based microgrids is presented in this paper. Directly determining stability boundaries by searching the multidimensional space of inverters’ droop gains is a computationally prohibitive task. Instead, we build a certified stability region by utilizing a generalized Laplacian matrix eigenvalues, which are a measure of proximity to stability boundary. We establish an upper threshold for the eigenvalues that determines the stability boundary of the entire system and demonstrate that this value depends only on the network’s R/X ratio but does not depend on the grid topology. We also provide a conservative upper threshold of the eigenvalues that are universal for any systems within a reasonable range of R/X ratios. We then construct approximate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">certified stability regions</i> representing convex sets in the multidimensional space of droop gains that could be utilized for gains optimization. We show how the certified stability region can be maximized by properly choosing droop gains, and we provide closed-form analytic expressions for the certified stability regions. The computational complexity of our method is almost independent of the number of inverters. The proposed methodology has been tested using IEEE 123 node test system with 10 inverters.

CONSENSUS ANALYSIS IN HIERARCHICAL NETWORKED SYSTEMS

This paper introduces evolving hierarchical networks based on hierarchical product. We discuss two important consensus behavior indexes in both un-weighted and weighted cases in those networked systems: (1) convergence speed in system without delay and noise; (2) delay robustness in system with communication time-delay. We analyze the approximative behaviors of these two consensus indexes by determining the second smallest and largest eigenvalue of Laplacian matrix, respectively, as well as how the scale of the weight factor affects the consensus performance. Moreover, the method applied in this paper is also applicable to many classic networks. Finally, two numerical simulations are given to verify the advantages of communication topology like hierarchical networks and the validity of the theoretical results.

Fully distributed observer-based tracking control for Lur’e systems with event-triggered communication

This paper investigates the problem of cooperative tracking for Lur’e systems under directed spanning tree topology. First, a control protocol is proposed to achieve cooperative tracking consensus by a distributed observer, which utilizes only the states of neighboring agents based on the event-triggering conditions with mixed node and edge. Then, an improved tracking protocol is developed by considering the case that only the outputs of neighbors can be obtained. With the aid of adaptive updating parameters, the two protocols do not utilize the minimum eigenvalue of Laplacian matrix, and can deal with the nonlinear dynamics of Lur’e systems in a fully distributed manner. Moreover, with the Lyapunov analysis framework, the tracking errors can be proved to converge to zero in both cases. Zeno behavior is excluded from the event-triggering conditions containing states and outputs of neighbors. Finally, the effectiveness of the proposed protocols is verified by two numerical simulations.

Graph Signal Processing Based Cross-Subject Mental Task Classification Using Multi-Channel EEG Signals

Classification of mental tasks from electroencephalogram (EEG) signals play a crucial role in designing various brain-computer interface (BCI) applications. Most of the current techniques consider each channel as independent, neglecting the functional connectivity of the brain during mental activity and are primarily subject specific. This paper proposes a graph signal representation to classify a pair of mental tasks using multi-channel EEG signals (MTMC-EEG) with cross subject classification within the database. Here, each channel of EEG signal corresponds to nodes of the task based graph whose EEG time series resides on the respective nodes. Functional connectivity of the brain between these nodes is obtained using smoothness constraint based Graph Signal Processing (GSP) technique. Graph spectral features namely, two-norm total variation of eigen vector (TNTV) corresponding to weighted adjacency matrix, graph Laplacian energy (GLE) using eigenvalues of Laplacian matrix and convex sum of TNTV and GLE in the form of joint total variation energy (JTVE) are proposed in this paper. The performance of the proposed methodology is evaluated on publicly available two different databases of MTMC EEG signals using benchmark classifiers and compared with the state of the art. Further, the superiority of the proposed metric obtained from the smoothened graph of GSP technique is validated by comparing it with Pearson correlation and Gaussian radial basis function (RBF) based functional connectivity in terms of accuracy, F-Score, and information transfer rate (ITR). The robustness of the proposed method is validated by adding white Gaussian noise (AWGN) to the EEG signals using different SNRs.

Critical parameters of the synchronisation's stability for coupled maps in regular graphs

Coupled Map Lattice (CML) models are particularly suitable to study spatially extended behaviours, such as wave-like patterns, spatio-temporal chaos, and synchronisation. Complete synchronisation in CMLs emerges when all maps have their state variables with equal magnitude, forming a spatially-uniform pattern that evolves in time. Here, we derive critical values for the parameters – coupling strength, maximum Lyapunov exponent, and link density – that control the synchronisation-manifold's linear stability of diffusively-coupled, identical, chaotic maps in generic regular graphs (i.e., graphs with uniform node degrees) and class-specific cyclic graphs (i.e., periodic lattices with cyclical node permutation symmetries). Our derivations are based on the Laplacian matrix eigenvalues, where we give closed-form expressions for the smallest non-zero eigenvalue and largest eigenvalue of regular graphs and show that these graphs can be classified into two sets according to a topological condition (derived from the stability analysis). We also make derivations for two classes of cyclic graph: k-cycles (i.e., regular lattices of even degree k, which can be embedded in Tk tori) and k-Möbius ladders, which we introduce here to generalise the Möbius ladder of degree k = 3. Our results highlight differences in the synchronisation manifold's stability of these graphs – even for identical node degrees – in the finite size and infinite size limit.

Matrix-weighted consensus for multi-agent systems with sampled-data

This paper researches the consensus issue for multi-agent systems on matrix-weighted directed fixed and undirected switching network topologies by sampled data control method which saves resources and is more practical. Using the sampled information, the distributed control laws are designed under two network topologies, respectively. Under directed fixed network topology, the consensus conditions based on the sampling period and the eigenvalues of Laplacian matrix are deduced by matrix theory and analysis theory. Under undirected switching network topology, by using Lyapunov stability theory, the consensus conditions based on the sampling period and switched network topologies are built. Lastly, two simulation examples are offered to verify the validity of the obtained results.

Distributed adaptive time-varying formation control for Lipschitz nonlinear multi-agent systems

In this paper, two kinds of distributed time-varying formation tracking problems for a class of Lipschitz nonlinear multi-agent systems are investigated, that is, the cases with leadless and leader-following formation. By devising a new type of edge-based adaptive algorithm, the controller of each agent can avoid the use of eigenvalues of Laplacian matrix in communication networks. The proposed algorithm only utilizes local information among neighbouring agents to form time-varying formation for nonlinear multi-agent systems with mismatched nonlinearities, and the formation tracking error is uniformly ultimately bounded. To accomplish the required time-varying formation characteristic, a predesigned formation compensation function is given. Finally, numerical simulation examples are presented to indicate the stability of the proposed algorithm, and a comparison is also given to show the superiority of the given algorithm.

The Role of Reverse Edges on Consensus Performance of Chain Networks

The minimal real part of all the nonzero eigenvalues of Laplacian matrix, also known as the dominant convergence rate, characterizes the consensus performance of multiagent systems on a directed graph. The effect of adding the second reverse edge to a directed chain graph on the dominant convergence rate is investigated. According to the relative positions of the two reverse edges, three cases are discussed, respectively. It is revealed that the dominant convergence rate of the whole network is determined only by the ranges and the relative positions of the reverse edges. Moreover, the consensus performance will not get better with the new reverse edge being added, and it decreases as the reverse range increases. Finally, the theoretical results are illustrated by some numerical examples.

Sampled-Based Consensus for Nonlinear Multiagent Systems With Deception Attacks: The Decoupled Method

In this paper, the sampled-based consensus problem is investigated for a class of nonlinear multiagent systems subjected to deception attacks. Due to network fluctuations and limited resources, deception attacks might destroy the sampled-data in communication networks. Additionally, the success of deception attacks greatly depends on some randomly fluctuated factors. This paper takes into account the deception attacks that are randomly launched at each sampling instant. The eigenvector of Laplacian matrix is utilized to construct a novel Lyapunov functional. Then, the decoupled criterion connected with eigenvalues of Laplacian matrix is obtained such that the addressed multiagent systems achieve the mean square consensus. Furthermore, the solutions of a set of matrix inequalities represent the controller gain matrix. Finally, a numerical example is provided to validate the effectiveness of the derived results.

Global Adaptive Leader-Following Consensus for Second-Order Nonlinear Multiagent Systems With Switching Topologies

In this note, the leader-following consensus problem is considered for a class of second-order nonlinear multiagent systems, where the communication topology is switching. In view of that the Lipschitz constants of nonlinear terms are unknown, adaptive control strategy is adopted. Using the back-stepping technique, a new adaptive protocol is proposed. It is noted that the global information, including the eigenvalues of Laplacian matrix, is not used in the protocol design. It is proven that the practical leader-following consensus can be reached by the given control scheme. Finally, we present a numerical example to show the effectiveness of the proposed protocol.