Algebraic Graph Research Articles

Quantized Multi-Tracking in the Multiagent System With Sampled Position Data via Intermittent Control

This paper studies the multi-tracking problem of multiagent systems with second-order dynamic. A pulse-modulation intermittent control protocol is proposed with only sampled position data of each agent. To deal with the bandwidth limitation, a stochastic quantization scheme is introduced. And the data of agents is quantified before transmission. Moreover, the proposed control protocol is improved under communication delay. Based on algebraic graph theory, stochastic quantization and stability theory, some necessary and sufficient conditions are obtained to ensure multi-tracking of the controlled system. Finally, simulation results are presented to demonstrate the effectiveness of the proposed control protocol. Agents in each subnetwork asymptotically converge to the same desired trajectory, while there is no consensus among different subnetworks.

Second-order consensus of multiagent systems with matrix-weighted network

This paper studies second-order consensus problem on the matrix-weighted undirected network, which is a generalization of second-order scalar-weighted network. The matrix coupling can describe the interdependencies of multi-dimensional states among agents. To highlight the influence of matrix coupling, we analyze the double integral model in matrix-weighted multiagent system network. A sufficient and necessary algebraic condition based on coupling strength, matrix theory and Lyapunov stability theory is established to achieve second-order consensus. With aid of the established property, a fairly straightforward algebraic graph condition is obtained, which ensures that the matrix-weighted multiagent system achieves second-order consensus. Moreover, owing to the existence of positive semi-definite connections, the clustering phenomena naturally exists, which shows that matrix coupling plays an important role in the convergence of the matrix-weighted multiagent system network. And an algebraic graph condition for finding all clusters is offered, which can guarantee the desired number of clusters via designing matrix-weights in practical applications. Finally, four simulation examples are given to illustrate the effectiveness of the obtained results.

Robust exponential stability of fractional-order coupled quaternion-valued neural networks with parametric uncertainties and impulsive effects

Abstract This paper investigates the robust exponential stability (RES) issue for fractional-order coupled quaternion-valued neural networks (FCQNNs) with parametric uncertainties and impulsive effects. According to the rules of quaternion algebra and its properties, a new fractional-order inequality is built, which greatly generalizes the existing fractional-order inequality in the real domain. On the basis of quaternion inequality technique, newly established inequality, together with algebraic graph theory and iterative method, several criteria for easy verification are presented, which depend on not only impulsive gain and maximum impulsive interval but also the scale of the controlled vertices. Furthermore, the convergence rate of the considered FCQNN is also estimated. Finally, numerical results are given to substantiate our theoretical criteria.

Distributed non-fragile containment control of nonlinear multi-agent systems with time-varying delays

ABSTRACT Distributed containment control for multi-agent systems with time delay is investigated, where the communication topology is directed. Different from the existing literature which mainly focuses on general linear dynamics with time delay, containment control of nonlinear dynamics is discussed with the help of a new follower-based time-varying delays control protocol. Furthermore, there always exist uncertainties in the controller and observer gain matrices, a novel class of non-fragile containment control with time delay is addressed for the first time, which can be more applicable and more significant to real physical systems. By making use of the convex analysis, algebraic graph theory, matrix transformation and Lyapunov theory, it is derived that the obtained results are cast as feasible linear matrix inequalities and the states of the followers finally converge to the convex hull formed by the leaders. Finally, two simulation examples of nine-agents topology are presented to verify the validity of obtained theoretical analysis.

Observer-based Event-triggered Bipartite Consensus of Linear Multi-agent Systems

The event-triggered bipartite consensus problem of linear multi-agent systems with a connected structurally balanced signed graph is considered in this paper. The state observer is set to estimate the actual states of the system. In the system structure, two event-triggered mechanisms are configured for each agent. The one is placed in the output side of the agent and the other in the controller. Thus the establishment of the state observer is based on the discrete output information sampled at the corresponding triggering time instants. And the inter-neighboring transmission of the observed states occurs at the moments when the corresponding triggering condition of the agent is satisfied. The triggering conditions designed do not rely on continuous inter-neighboring communications. With the help of the algebraic graph theory and Lyapunov stability theory, it is proved that the bipartite consensus can be achieved. Furthermore, Zeno behavior can be avoided in both sides. The effectiveness of the presented control strategy is demonstrated by a numerical simulation.

On solvable factors of almost simple groups

Let G be a finite almost simple group with socle G0. A (nontrivial) factorization of G is an expression of the form G=HK, where the factors H and K are core-free subgroups. There is an extensive literature on factorizations of almost simple groups, with important applications in permutation group theory and algebraic graph theory. In a recent paper, Xia and the second-named author describe the factorizations of almost simple groups with a solvable factor H. Several infinite families arise in the context of classical groups and in each case a solvable subgroup of G0 containing H∩G0 is identified. Building on this earlier work, in this paper we compute a sharp lower bound on the order of a solvable factor of every almost simple group and we determine the exact factorizations with a solvable factor. As an application, we describe the finite primitive permutation groups with a nilpotent regular subgroup, extending classical results of Burnside and Schur on cyclic regular subgroups, and more recent work of Li in the abelian case.

Moving Target Surrounding Control of Linear Multiagent Systems With Input Saturation

Formation control finds broad applications in numerous fields, such as cooperative detection, surveillance, transportation, and disaster rescue. In this article, aiming at hunting a moving target, a two-stage surrounding control algorithm is proposed for linear multiagent systems subject to input saturations. An adaptive distributed observer is developed for each agent to reconstruct the target’s position. With the assistance of the algebraic graph theory and low gain feedback technique, distributed controllers are designed to drive the agents to encircle the moving target with a fixed radius and evenly distributed phase angles. Finally, both numerical simulations and experiments are conducted to verify the effectiveness of the proposed control algorithm.

An Algebraic Graph Transformation Approach for RDF and SPARQL

We consider the recommendations of the World Wide Web Consortium (W3C) about RDF framework and its associated query language SPARQL. We propose a new formal framework based on category theory which provides clear and concise formal definitions of the main basic features of RDF and SPARQL. We define RDF graphs as well as SPARQL basic graph patterns as objects of some nested categories. This allows one to clarify, in particular, the role of blank nodes. Furthermore, we consider basic SPARQL CONSTRUCT and SELECT queries and formalize their operational semantics following a novel algebraic graph transformation approach called POIM.

Consensus Problems of Linear Multi-agent Systems involving Conformable Derivative

In this paper, we introduce linear multi-agent systems (MASs) composed of a conformable derivative. The protocol of each agent is designed by using its local information. Two sufficient and necessary conditions are derived for the consensus of the systems respectively under the topologies of undirected and directed networks based on algebraic graph theory and matrix theory. Meanwhile, the relationship between the agreement values and topology structures is discussed. Finally, we demonstrate the theoretical results and the advantage of the systems compared with traditional integer order systems by several numerical simulations.

Dissipativity-Based Consensus Tracking of Singular Multiagent Systems With Switching Topologies and Communication Delays.

This article, based on dissipativity theory, aims to tackle the consensus tracking issue for Lipschitz nonlinear singular multiagent systems (MASs) with switching topologies and communication delays. Rooted at the leader node, a directed spanning tree is assumed to be contained in the union of all possible interaction graphs. Within the framework of topology switching controlled by a Markov chain, communication delays encountered in the data transmission process are reasonably considered to be time-varying and dependent on Markovian jump modes. By using tools from the stochastic Lyapunov functional technique, algebraic graph theory, and strict (Q,S,R)-α -dissipativity analysis, the consensus controller collecting delayed in-neighboring agents' information is designed to ensure stochastic admissibility and strict dissipativity of the resulting consensus error system. The theoretical analysis is validated by numerical simulations.

A comparative look at two formation control approaches based on optimization and algebraic graph theory

This paper takes a novel look at formation control by comparing control setups based on two very different frameworks. These are applied to the distributed control of communicating omnidirectional mobile robots. One framework, which is possibly the most common approach to formation control, is based on algebraic graph theory, whereas the other, namely distributed model predictive control (DMPC), is based on distributed optimization, representing a rather uncommon view on the task. In this study, formation control is understood as the task of attaining and maintaining a specific relative positioning between robotic agents while moving the formation through the environment. While interesting on its own, formation control can serve as the basis for superordinate tasks like cooperative transportation. For an encompassing treatment of the task, two different control goals are considered, resulting in different setups for each control framework. One goal consists of moving the formation’s geometric center to a specific position, whereas the other aims at letting the whole formation move with the desired velocity. In both cases, the involved robots are subject to input constraints. Already during control design, some qualitative differences between the two frameworks become apparent, with the DMPC controller exhibiting characteristic beneficial qualities in exchange for its higher computational demand. Results from various simulation scenarios confirm these observations. Considerations on the practical implementation of the two schemes, as well as hardware experiments with tailor-made mobile robots, provide valuable insight for robotics practitioners, and highlight the applicability of the two frameworks.

Leader-Following Consensus of Non-linear Multi-agent Systems with Interval Time-Varying Delay via Impulsive Control

In this manuscript, the stability and consensus problem of the leader-following multi-agent system is studied. The state information of the leader is only available to the dynamics of all the followers, while the communication among the agents occurs at sampling instant. The innovative part of this paper is to reach the consensus and attain the stability for prescribed MASs with interval time varying delay using impulsive control in the presence of leader. The interaction between the agent is illustrated by an undirected graph. A class of distributed impulsive protocol relying on sampling information is suggested to reach the leader following consensus. The consensus is critically relying on the sampling period, control gains. By performing the similar procedures, the consensus problem of multi-agent system is converted to stability problem of the error system. By utilizing the stability theory of impulsive systems, algebraic graph theory, sufficient conditions are derived to guarantee the leader-following consensus of the multi-agent system. Terminally, a numerical example is given to show an efficacy of this presented approach and the sureness of theoretical analysis.

Consensus Tracking of Data-Sampled Nonlinear Multi-Agent Systems With Packet Loss and Communication Delay

This paper investigates the tracking consistency regulation of high-order multi-agent systems (MASs) subject to Lipschitz nonlinear perturbations. Rooting at the leader node, the interagent interaction is described by a directed graph that contains a directed spanning tree. On designing the tracking protocol, an aperiodic data sampling scheme is introduced such that in-neighbors' information can be aperiodically sampled and transmitted via broadcast channels, where packet loss and communication delay are unavoidably existent. Thanks to algebraic graph theory and Lyapunov-Krasovskii functional technique, the concerned consensus tracking issue is cast into asymptotically stabilizing a class of switched nonlinear systems which consist of delayed and non-delayed components. It is proved that, moreover, the asymptotic stabilization of such systems essentially depends on packet loss characteristics. By explicitly characterizing the interplay among sampling interval, communication delay and network structure, consensualization criteria are formulated within the linear matrix inequality (LMI) framework. The design method is validated by numerical simulations.

Graph product Khintchine inequalities and Hecke C⁎-algebras: Haagerup inequalities, (non)simplicity, nuclearity and exactness

Graph products of groups were introduced by Green in her thesis [32]. They have an operator algebraic counterpart introduced and explored in [14]. In this paper we prove Khintchine type inequalities for general C⁎-algebraic graph products which generalize results by Ricard and Xu [50] on free products of C⁎-algebras. We apply these inequalities in the context of (right-angled) Hecke C⁎-algebras, which are deformations of the group algebra of Coxeter groups (see [22]). For these we deduce a Haagerup inequality which generalizes results from [33]. We further use this to study the simplicity and trace uniqueness of (right-angled) Hecke C⁎-algebras. Lastly we characterize exactness and nuclearity of general Hecke C⁎-algebras.

Spectral properties of inverse sum indeg index of graphs

The inverse sum indeg (ISI) index is a vertex-degree-based topological index that was selected by Vukicevic and Gasperov in 2010 as a significant predictor of the total surface area of octane isomers. One of the main aims of algebraic graph theory is to determine how, or whether, properties of graphs are reflected in the algebraic properties of some matrices. The aim of this paper is to study the ISI index from an algebraic viewpoint. We introduce suitably modified versions of the classical adjacency matrix and the Laplacian matrix involving the degrees of the vertices of a graph. Moreover, we formulate the ISI index in terms of these matrices.

Projective Group Consensus of Multi-Agent Systems with Arbitrary Parameter

In this paper, the projective group consensus issue for second order multi-agent systems (MASs) in directed graphs with a dynamic leader is investigated. The proposed projective group consensus with arbitrary parameter includes traditional consensus, reverse group consensus and cluster consensus as its special cases. Novel distributed control protocols are designed to obtain projective group consensus without analyzing signed directed graph as in most current literatures on bipartite consensus problem. On the basis of Lyapunov stability property, algebraic graph and some necessary matrix theory, sufficient conditions for delay and delay-free cases are derived. Finally, simulations of nonlinear chaotic MASs are adopted to testify the theoretical results.

On Fundamental Solution for Autonomous Linear Retarded Functional Differential Equations

This document focuses attention on the fundamental solution of an autonomous linear retarded functional differential equation (RFDE) along with its supporting cast of actors: kernel matrix, characteristic matrix, resolvent matrix; and the Laplace transform. The fundamental solution is presented in the form of the convolutional powers of the kernel matrix in the manner of a convolutional exponential matrix function. The fundamental solution combined with a solution representation gives an exact expression in explicit form for the solution of an RFDE. Algebraic graph theory is applied to the RFDE in the form of a weighted loop-digraph to illuminate the system structure and system dynamics and to identify the strong and weak components. Examples are provided in the document to elucidate the behavior of the fundamental solution. The paper introduces fundamental solutions of other functional differential equations.

Distributed Multi-Area State Estimation for Power Systems With Switching Communication Graphs

We consider distributed multi-area state estimation algorithms for power systems with switching communication graphs. The power system is partitioned into multiple geographically non-overlapping areas and each area is assigned with an estimator to give a local estimate of the entire power system's state. The inter-area communication networks are assumed to switch among a finite set of digraphs. Each area runs a distributed estimation algorithm based on consensus+innovations strategies. By the binomial expansion of matrix products, time-varying system and algebraic graph theories, we prove that all area's local estimates converge to the global least square estimate with probability 1 if the measurement system is jointly observable and the communication graphs are balanced and jointly strongly connected. Finally, we demonstrate the theoretical results by an IEEE 118-bus system.

Formula omitted] delayed tracking protocol design of nonlinear singular multi-agent systems under Markovian switching topology

The consensus tracking of singular multi-agent systems (MASs) with Lipschitz-type nonlinearities and exogenous disturbances is researched in this paper. Governed by a Markov chain, the network interaction randomly switches in a directed graph set, where the directed spanning tree is not contained in each graph while exists in the union rooting at the leader node. By utilizing a collection of in-neighbors’ information that involves communication delay, the intention is to design a protocol such that the resultant consensus error system is stochastic admissible with an H∞ disturbance attenuation level. Based on algebraic graph theory, stochastic admissibility analysis and linear matrix inequality (LMI) technique, tracking consistency is first regulated in the concerned MAS by considering the case of completely known transition probabilities. Then, thanks to a group of free-connection weighting matrices, the obtained result is extended to the case that transition probabilities are partially known. Finally, the theoretical analysis is confirmed by some numerical examples.

H∞ group consensus of linear dynamical systems under directed switching topology

SummaryThis brief investigates the H∞ group consensus problem for linear dynamical systems with directed switching topology and external disturbance. The H∞ group consensus problem for multiagent systems (MASs) is complicated since the agents from different clusters may be cooperative or competitive. By using together algebraic graph theory and Lyapunov stability theory, we first analyze the stability of group consensus with desired H∞ performance, and then establish a sufficient condition to achieve the H∞ group consensus. This sufficient condition is derived in terms of the strengths of intra‐cluster couplings and the solution to an algebraic Riccati equation, whose solvability can be guaranteed if the MAS is state feedback stabilized with a bounded L2 gain. Moreover, the lower bound of the inter‐cluster coupling strength and dwell time are explicitly specified. Finally, the effectiveness of the theoretical results is verified by a numerical example.