Bounded consensus tracking of second-order multi-agent systems with sampling delay under directed networks

The bounded consensus tracking problems of second-order multi-agent systems under directed networks with sam- pling delay are addressed in this paper. When the sampling delay is more than a sampling period, new protocols based on sampled-data control are proposed so that each agent can track the time-varying reference state of the virtual leader. By using the delay decomposition approach, the augmented matrix method, and the frequency domain analysis, necessary and sufficient conditions are obtained, which guarantee that the bounded consensus tracking is realized. Furthermore, some numerical simulations are presented to demonstrate the effectiveness of the theoretical results.

Global Synchronization of Directed Networks with Fast Switching Topologies

Global synchronization of a class of directed dynamical networks with switching topologies is investigated.It is found that if there is a directed spanning tree in the fixed time-average of network topology and the time-averageis achieved sufficiently fast,then the network will reach global synchronization for sufficiently large coupling strength.

Spontaneous Recovery in Directed Dynamical Networks

Complex networked systems,which range from biological systems in the natural world to infrastructure systems in the human-made world,can exhibit spontaneous recovery after a failure;for example,a brain may spontaneously return to normal after a seizure,and traffic flow can become smooth again after a jam.Previous studies on the spontaneous recovery of dynamical networks have been limited to undirected networks.However,most real-world networks are directed.To fill this gap,we build a model in which nodes may alternately fail and recover,and we develop a theoretical tool to analyze the recovery properties of directed dynamical networks.We find that the tool can accurately predict the final fraction of active nodes,and the prediction accuracy decreases as the fraction of bidirectional links in the network increases,which emphasizes the importance of directionality in network dynamics.Due to different initial states,directed dynamical networks may show alternative stable states under the same control parameter,exhibiting hysteresis behavior.In addition,for networks with finite sizes,the fraction of active nodes may jump back and forth between high and low states,mimicking repetitive failure-recovery processes.These findings could help clarify the system recovery mechanism and enable better design of networked systems with high resilience.

Fixed-Time Cluster Consensus for Multi-Agent Systems with Objective Optimization on Directed Networks

This paper studies the cluster consensus of multi-agent systems(MASs)with objective optimization on directed and detail balanced networks,in which the global optimization objective function is a linear combination of local objective functions of all agents.Firstly,a directed and detail balanced network is constructed that depends on the weights of the global objective function,and two kinds of novel continuous-time optimization algorithms are proposed based on time-invariant and timevarying objective functions.Secondly,by using fixed-time stability theory and convex optimization theory,some sufficient conditions are obtained to ensure that all agents'states reach cluster consensus within a fixed-time,and asymptotically converge to the optimal solution of the global objective function.Finally,two examples are presented to show the efficacy of the theoretical results.

Percolation transitions in edge-coupled interdependent networks with directed dependency links

We propose a model of edge-coupled interdependent networks with directed dependency links(EINDDLs)and develop the theoretical analysis framework of this model based on the self-consistent probabilities method.The phase transition behaviors and parameter thresholds of this model under random attacks are analyzed theoretically on both random regular(RR)networks and Erd¨os-Renyi(ER)networks,and computer simulations are performed to verify the results.In this EINDDL model,a fractionβof connectivity links within network B depends on network A and a fraction(1-β)of connectivity links within network A depends on network B.It is found that randomly removing a fraction(1-p)of connectivity links in network A at the initial state,network A exhibits different types of phase transitions(first order,second order and hybrid).Network B is rarely affected by cascading failure whenβis small,and network B will gradually converge from the first-order to the second-order phase transition asβincreases.We present the critical values ofβfor the phase change process of networks A and B,and give the critical values of p andβfor network B at the critical point of collapse.Furthermore,a cascading prevention strategy is proposed.The findings are of great significance for understanding the robustness of EINDDLs.

Computing the SKT Reliability of Acyclic Directed Networks Using Factoring Method

This paper presents a factoring algorithm for computing source-to- K terminal (SKT) reliability, the probability that a source s can send message to a specified set of terminals K, in acyclic directed networks (AD-networks) in which both nodes and edges can fail. Based on Pivotal decomposition theorem, a new formula is derived for computing the SKT reliability of AD-networks. By establishing a topological property of AD-networks, it is shown that the SKT reliability of AD- networks can be computed by recursively applying this formula. Two new Reliability- Preserving Reductions are also introduced. The recursion tree generated by the presented algorithm has at most 2 leaf nodes, where V and K are the numbers of nodes and terminals, respectively, while C is the number of the nodes satisfying some specified conditions. The computation complexity of the new algorithm is O (E. V. 2) in the worst case, where E is the number of edges. For source-to-all-terminal (SAT) reliability, its computation complexity is O(E). Comparison of the new algorithm with the existing ones indicates that the new algorithm is more efficient for computing the SKT reliability of AD-networks.

ROBUST CONSENSUS FOR MULTI-AGENT SYSTEMS OVER UNBALANCED DIRECTED NETWORKS

In this paper,a distributed consensus protocol is proposed for discrete-time single-integer multi-agent systems with measurement noises under general fixed directed topologies.The time-varying control gains satisfying the stochastic approximation conditions are introduced to attenuate noises,thus the closed-loop multi-agent system is intrinsically a linear time-varying stochastic difference system.Then the mean square consensus convergence analysis is developed based on the Lyapunov technique,and the construction of the Lyapunov function especially does not require the typical balanced network topology condition assumed for the existence of quadratic Lyapunov function.Thus,the proposed consensus protocol can be applicable to more general networked multi-agent systems,particularly when the bidirectional and/or balanced information exchanges between agents are not required.Under the proposed protocol,it is proved that the state of each agent converges in mean square to a common random variable whose mathematical expectation is the weighted average of agents' initial state values;meanwhile,the random variable's variance is bounded.

Circular Formation Flight Control for Unmanned Aerial Vehicles With Directed Network and External Disturbance

This paper proposes a new distributed formation flight protocol for unmanned aerial vehicles(UAVs)to perform coordinated circular tracking around a set of circles on a target sphere.Different from the previous results limited in bidirectional networks and disturbance-free motions,this paper handles the circular formation flight control problem with both directed network and spatiotemporal disturbance with the knowledge of its upper bound.Distinguishing from the design of a common Lyapunov fiunction for bidirectional cases,we separately design the control for the circular tracking subsystem and the formation keeping subsystem with the circular tracking error as input.Then the whole control system is regarded as a cascade connection of these two subsystems,which is proved to be stable by input-tostate stability(ISS)theory.For the purpose of encountering the external disturbance,the backstepping technology is introduced to design the control inputs of each UAV pointing to North and Down along the special sphere(say,the circular tracking control algorithm)with the help of the switching function.Meanwhile,the distributed linear consensus protocol integrated with anther switching anti-interference item is developed to construct the control input of each UAV pointing to east along the special sphere(say,the formation keeping control law)for formation keeping.The validity of the proposed control law is proved both in the rigorous theory and through numerical simulations.

Distributed Fault-Tolerant Containment Control for Nonlinear Multi-Agent Systems Under Directed Network Topology via Hierarchical Approach

This paper investigates the distributed fault-tolerant containment control(FTCC)problem of nonlinear multi-agent systems(MASs)under a directed network topology.The proposed control framework which is independent on the global information about the communication topology consists of two layers.Different from most existing distributed fault-tolerant control(FTC)protocols where the fault in one agent may propagate over network,the developed control method can eliminate the phenomenon of fault propagation.Based on the hierarchical control strategy,the FTCC problem with a directed graph can be simplified to the distributed containment control of the upper layer and the fault-tolerant tracking control of the lower layer.Finally,simulation results are given to demonstrate the effectiveness of the proposed control protocol.

Flocking of multi-robot systems with connectivity maintenance on directed graphs

Analysis and design techniques for cooperative flocking of nonholonomic multi-robot systems with connectivity maintenance on directed graphs are presented. First, a set of bounded and smoothly distributed control protocols are devised via carefully designing a class of bounded artificial potential fields （APF） which could guarantee the connectivity maintenance, col ision avoidance and distance stabilization simultaneously during the system evolution. The connectivity of the underlying network can be preserved, and the desired stable flocking behavior can be achieved provided that the initial communication topology is strongly connected rather than undirected or balanced, which relaxes the constraints for group topology and extends the previous work to more generalized directed graphs. Furthermore, the proposed control algorithm is extended to solve the flocking problem with a virtual leader. In this case, it is shown that al robots can asymptotically move with the desired velocity and orientation even if there is only one informed robot in the team. Finally, nontrivial simulations and experiments are conducted to verify the effectiveness of the proposed algorithm.

Fully Distributed Nash Equilibrium Seeking for High-Order Players With Actuator Limitations

This paper explores the problem of distributed Nash equilibrium seeking in games, where players have limited knowledge on other players' actions. In particular, the involved players are considered to be high-order integrators with their control inputs constrained within a pre-specified region. A linear transformation for players' dynamics is firstly utilized to facilitate the design of bounded control inputs incorporating multiple saturation functions. By introducing consensus protocols with adaptive and time-varying gains, the unknown actions for players are distributively estimated. Then, a fully distributed Nash equilibrium seeking strategy is exploited, showcasing its remarkable properties: (1) ensuring the boundedness of control inputs;(2) avoiding any global information/parameters;and (3) allowing the graph to be directed. Based on Lyapunov stability analysis, it is theoretically proved that the proposed distributed control strategy can lead all the players' actions to the Nash equilibrium. Finally, an illustrative example is given to validate effectiveness of the proposed method.

Event-Triggered Zero-Gradient-Sum Distributed Algorithm for Convex Optimization with Time-Varying Communication Delays and Switching Directed Topologies

Nowadays, distributed optimization algorithms are widely used in various complex networks. In order to expand the theory of distributed optimization algorithms in the direction of directed graph, the distributed convex optimization problem with time-varying delays and switching topologies in the case of directed graph topology is studied. The event-triggered communication mechanism is adopted, that is, the communication between agents is determined by the trigger conditions, and the information exchange is carried out only when the conditions are met. Compared with continuous communication, this greatly saves network resources and reduces communication cost. Using Lyapunov-Krasovskii function method and inequality analysis, a new sufficient condition is proposed to ensure that the agent state finally reaches the optimal state. The upper bound of the maximum allowable delay is given. In addition, Zeno behavior will be proved not to exist during the operation of the algorithm. Finally, a simulation example is given to illustrate the correctness of the results in this paper.

MLDA:a multi-level k-degree anonymity scheme on directed social network graphs

With the emergence of network-centric data,social network graph publishing is conducive to data analysts to mine the value of social networks,analyze the social behavior of individuals or groups,implement personalized recommendations,and so on.However,published social network graphs are often subject to re-identification attacks from adversaries,which results in the leakage of users’privacy.The-anonymity technology is widely used in the field of graph publishing,which is quite effective to resist re-identification attacks.However,the current researches still exist some issues to be solved:the protection of directed graphs is less concerned than that of undirected graphs;the protection of graph structure is often ignored while achieving the protection of nodes’identities;the same protection is performed for different users,which doesn’t meet the different privacy requirements of users.Therefore,to address the above issues,a multi-level-degree anonymity(MLDA)scheme on directed social network graphs is proposed in this paper.First,node sets with different importance are divided by the firefly algorithm and constrained connectedness upper approximation,and they are performed different-degree anonymity protection to meet the different privacy requirements of users.Second,a new graph anonymity method is proposed,which achieves the addition and removal of edges with the help of fake nodes.In addition,to improve the utility of the anonymized graph,a new edge cost criterion is proposed,which is used to select the most appropriate edge to be removed.Third,to protect the community structure of the original graph as much as possible,fake nodes contained in a same community are merged prior to fake nodes contained in different communities.Experimental results on real datasets show that the newly proposed MLDA scheme is effective to balance the privacy and utility of the anonymized graph.

Theory and Experiments on Enclosing Control of Multi-Agent Systems

This paper proposes a control strategy called enclosing control.This strategy can be described as follows:the followers design their control inputs based on the state information of neighbor agents and move to specified positions.The convex hull formed by these followers contains the leaders.We use the single-integrator model to describe the dynamics of the agents and proposes a continuous-time control protocol and a sampled-data based protocol for multi-agent systems with stationary leaders with fixed network topology.Then the state differential equations are analyzed to obtain the parameter requirements for the system to achieve convergence.Moreover,the conditions achieving enclosing control are established for both protocols.A special enclosing control with no leader located on the convex hull boundary under the protocols is studied,which can effectively prevent enclosing control failures caused by errors in the system.Moreover,several simulations are proposed to validate theoretical results and compare the differences between the three control protocols.Finally,experimental results on the multi-robot platform are provided to verify the feasibility of the protocol in the physical system.

Distributed Containment Control of Networked Nonlinear Second-order Systems With Unknown Parameters

In this paper, we study the containment control problem for nonlinear second-order systems with unknown parameters and multiple stationary/dynamic leaders. The topologies that characterize the interaction among the leaders and the followers are directed graphs. Necessary and sufficient criteria which guarantee the control objectives are established for both stationary leaders(regulation case) and dynamic leaders(dynamic tracking case) based protocols. The final states of all the followers are exclusively determined by the initial values of the leaders and the topology structures. In the regulation case, all the followers converge into the convex hull spanned by the leaders,while in the dynamic tracking case, not only the positions of the followers converge into the convex hull but also the velocities of the followers converge into the velocity convex hull of the leaders.Finally, all the theoretical results are illustrated by numerical simulations.

A New Evaluation Method of Node Importance in Directed Weighted Complex Networks

Current researches on node importance evaluation mainly focus on undirected and unweighted networks, which fail to reflect the real world in a comprehensive and objective way. Based on directed weighted complex network models, the paper introduces the concept of in-weight intensity of nodes and thereby presents a new method to identify key nodes by using an importance evaluation matrix. The method not only considers the direction and weight of edges, but also takes into account the position importance of nodes and the importance contributions of adjacent nodes. Finally, the paper applies the algorithm to a microblog-forwarding network composed of 34 users, then compares the evaluation results with traditional methods. The experiment shows that the method proposed can effectively evaluate the node importance in directed weighted networks.

Linear and nonlinear generalized consensuses of multi-agent systems

Consensus in directed networks of multiple agents, as an important topic, has become an active research subject. Over the past several years, some types of consensus problems have been studied. In this paper, we propose a novel type of consensus, the generalized consensus （GC）, which includes the traditional consensus, the anti-consensus, and the cluster consensus as its special cases. Based on the Lyapunov＇s direct method and the graph theory, a simple control algorithm is designed to achieve the generalized consensus in a network of agents. Numerical simulations of linear and nonlinear GC are used to verify the effectiveness of the theoretical analysis.

DIRECT DISPLACEMENT OF PARALLEL MECHANISM WITH WAVELET NETWORK

A new method solution for the direct displacement of parallel mechanism, wavelet network method, is proposed. Comparing with the classical analytical and numerical methods, this method can be extended to any parallel mechanism with any selected degree of freedom and configuration. A wavelet network suiting to approach multi-input and multi-output system is constructed. The network is optimized by analyzing the sparseness of input data and selecting the fitting wavelets by orthogonalization method according to the output data. Then it is applied to solve the direct displace- ment of a general six-degree-of-freedom parallel mechanism as a numerical example. For comparison purposes, a BP neural network is also used for this problem. Simulation results show that the wavelet network performs better than BP neural network. In addition, the wavelet network learns much faster than BP network.

Empirical investigation of topological and weighted properties of a bus transport network from China

Many bus transport networks （BTNs） have evolved into directed networks. A new representation model for BTNs is proposed, called directed-space P. The bus transport network of Harbin （BTN-H） is described as a directed and weighted complex network by the proposed representation model and by giving each node weights. The topological and weighted properties are revealed in detail. In-degree and out-degree distributions, in-weight and out-weight distributions are presented as an exponential law, respectively. There is a strong relation between in-weight and in-degree （also between out-weight and out-degree）, which can be fitted by a power function. Degree-degree and weight-weight correlations are investigated to reveal that BTN-H has a disassortative behavior as the nodes have relatively high degree （or weight）. The disparity distributions of out-degree and in-degree follow an approximate power-law. Besides, the node degree shows a near linear increase with the number of routes that connect to the corresponding station. These properties revealed in this paper can help public transport planners to analyze the status quo of the BTN in nature.

Structural Reliability Analysis Based on Support Vector Machine and Dual Neural Network Direct Integration Method

Aiming at the reliability analysis of small sample data or implicit structural function,a novel structural reliability analysis model based on support vector machine(SVM)and neural network direct integration method(DNN)is proposed.Firstly,SVM with good small sample learning ability is used to train small sample data,fit structural performance functions and establish regular integration regions.Secondly,DNN is approximated the integral function to achieve multiple integration in the integration region.Finally,structural reliability was obtained by DNN.Numerical examples are investigated to demonstrate the effectiveness of the present method,which provides a feasible way for the structural reliability analysis.