Distributed Consensus of Stochastic Delayed Multi-agent Systems Under Asynchronous Switching

State Estimation of Switched Time-Delay Complex Networks With Strict Decreasing LKF

State estimation issue is investigated for a switched complex network (CN) with time delay and external disturbances. The considered model is general with a one-sided Lipschitz (OSL) nonlinear term, which is less conservative than Lipschitz one and has wide applications. Adaptive mode-dependent nonidentical event-triggered control (ETC) mechanisms for only partial nodes are proposed for state estimators, which are not only more practical and flexible but also reduce the conservatism of the results. By using dwell-time (DT) segmentation and convex combination methods, a novel discretized Lyapunov-Krasovskii functional (LKF) is developed such that the value of LKF at switching instants is strict monotone decreasing, which makes it easy for nonweighted L2 -gain analysis without additional conservative transformation. The main results are given in the form of linear matrix inequalities (LMIs), by which the control gains of the state estimator are designed. A numerical example is given to illustrate the advantages of the novel analytical method.

Input-to-State Stability for Time-Delay Systems With Large Delays

In this article, we consider the input-to-state stability (ISS) problem for a class of time-delay systems with intermittent large delays, which may cause the invalidation of traditional delay-dependent stability criteria. The topic of this article features that it proposes a novel kind of stability criterion for time-delay systems, which is delay dependent if the time delay is smaller than a prescribed allowable size. While if the time delay is larger than the allowable size, the ISS can be preserved as well provided that the large-delay periods satisfy the kind of duration condition. Different from existing results on similar topics, we present the main result based on a unified Lyapunov-Krasovskii function (LKF). In this way, the frequency restriction can be removed and the analysis complexity can be simplified. A numerical example is provided to verify the proposed results.

Topological Properties, Spectra Analysis, and Consensus Problems for a Class of Network Models Based on m-Fission Operation

The network model, especially the network model constructed by network operation, is an important tool to study complex networks. Many topological and dynamic properties of complex networks can be studied in this way. In this article, the m -fission operation is constructed based on the phenomenon of node splitting in the network, which is quite common in complex networks. Many network models, including dual Sierpinski gaskets, are built based on this operation, but it has never been systematically studied. Then, the topological and dynamic properties of the m -fission operation and the corresponding iterative fission network model are studied, and the influence of the operation on the network structure is revealed. Among them, the topological properties of the network include diameter, degree distribution, clustering coefficient, average distance, and modularity. By studying these properties, it can be concluded that the iterative fission network is a fractal homogeneous network with high clustering and high community characteristics. Since the dynamic properties are closely related to the spectrum of the Laplacian matrix corresponding to the network, the iterative relation of the spectrum in operation is studied, and the complete solution of the spectrum of the iterative fission network is obtained. Based on the above results, we calculate the analytical expressions of the characteristic quantities related to the dynamic properties on the network, including the Kirchhoff index and the average of hitting times. Finally, due to the close connection between the network model and the system, we further analyzed the consensus problems on the system corresponding to the network, including convergence rate, delay robustness, first-order noise coherence, and second-order noise coherence.

Coherence Scaling of Noisy Second-Order Scale-Free Consensus Networks

A striking discovery in the field of network science is that the majority of real networked systems have some universal structural properties. In general, they are simultaneously sparse, scale-free, small-world, and loopy. In this article, we investigate the second-order consensus of dynamic networks with such universal structures subject to white noise at vertices. We focus on the network coherence H_SO characterized in terms of the H₂ -norm of the vertex systems, which measures the mean deviation of vertex states from their average value. We first study numerically the coherence of some representative real-world networks. We find that their coherence H_SO scales sublinearly with the vertex number N . We then study analytically H_SO for a class of iteratively growing networks-pseudofractal scale-free webs (PSFWs), and obtain an exact solution to H_SO, which also increases sublinearly in N , with an exponent much smaller than 1. To explain the reasons for this sublinear behavior, we finally study H_SO for Sierpinśki gaskets, for which H_SO grows superlinearly in N , with a power exponent much larger than 1. Sierpinśki gaskets have the same number of vertices and edges as the PSFWs but do not display the scale-free and small-world properties. We thus conclude that the scale-free, small-world, and loopy topologies are jointly responsible for the observed sublinear scaling of H_SO.

Distributed Control of Time-Varying Signed Networks: Theories and Applications

Signed networks admitting antagonistic interactions among agents may polarize, cluster, or fluctuate in the presence of time-varying communication topologies. Whether and how signed networks can be stabilized regardless of their sign patterns is one of the fundamental problems in the network system control areas. To address this problem, this paper targets at presenting a self-appraisal mechanism in the protocol of each agent, for which a notion of diagonal dominance degree is proposed to represent the dominant role of agent's self-appraisal over external impacts from all other agents. Selection conditions on diagonal dominance degrees are explored such that signed networks in the presence of directed time-varying topologies can be ensured to achieve the uniform asymptotic stability despite any sign patterns. Further, the established stability results can be applied to achieve bipartite consensus tracking of time-varying signed networks and realize state-feedback stabilization of time-varying systems. Simulations are implemented to verify our uniform asymptotic stability results for directed time-varying signed networks.

Fast Approximation of Coherence for Second-Order Noisy Consensus Networks

It has been recently established that for second-order consensus dynamics with additive noise, the performance measures, including the vertex coherence and network coherence defined, respectively, as the steady-state variance of the deviation of each vertex state from the average and the average steady-state variance of the system, are closely related to the biharmonic distances. However, direct computation of biharmonic distances is computationally infeasible for huge networks with millions of vertices. In this article, leveraging the implicit fact that both vertex and network coherence can be expressed in terms of the diagonal entries of pseudoinverse L^2† of the square of graph Laplacian, we develop a nearly linear-time algorithm to approximate all diagonal entries of L^2† , which has a theoretically guaranteed error for each diagonal entry. The key ingredient of our approximation algorithm is an integration of the Johnson-Lindenstrauss lemma and Laplacian solvers. Extensive numerical experiments on real-life and model networks are presented, which indicate that our approximation algorithm is both efficient and accurate and is scalable to large-scale networks with millions of vertices.

Sampled-Data Consensus of Linear Time-Varying Multiagent Networks With Time-Varying Topologies

The main purpose of this article is to investigate the consensus of linear multiagent networks with time-varying characteristics under sampled-data communications, where the time-varying characteristics include both time-varying topologies and the node's linear time-varying dynamics. By using the decoupling method, we prove that the sampled-data consensus problem of multiagent networks is equal to the stability problem of sampled-data systems. Then, the globally asymptotical consensus is investigated for multiagent networks with time-varying characteristics by virtue of the Lyapunov function method. It should be noted that when the Lyapunov function method is utilized to investigate the stability problem of control systems, it is always assumed that the derivative of the constructed Lyapunov function is not more than zero. This assumption is removed here and as a replacement, the average value of the derivative of the Lyapunov function in a period to be negative is needed.

Consensus Control of Second-Order Time-Delayed Multiagent Systems in Noisy Environments Using Absolute Velocity and Relative Position Measurements

This article designs an effective consensus control protocol for continuous-time second-order time-delayed multiagent systems in a multiplicative noisy environment, using absolute velocity and relative position measurements. The nonlinear case and double-integrator case are studied, respectively. Due to the time delay and multiplicative noise in such models, the conventional methods for consensus analysis are not applicable. In this article, therefore, a degenerated Lyapunov functional is used to derive the conditions for mean-square consensus and almost-sure consensus, related to the Lipschitz constants of the nonlinear term, time delay, and noise intensity. In particular, for the double-integrator setting, it is shown that the mean-square consensus and the almost-sure consensus can be achieved by choosing appropriate control gains for any given time delay and noise intensity. To show the effectiveness of the proposed control protocol, some numerical simulations are demonstrated.

Formation Control of Multiagent Systems With Communication Noise: A Convex Analysis Approach

Under practical environments, some certain noises may arise from the communication of agents due to electromagnetic interference, sensor error, and other disturbances, which are usually modeled by additive noise in the relative position state between an agent and its neighbors. In this article, a formation control strategy is considered for this kind of relative position state with the additive noise, where the unmeasured velocity information is estimated by an observer-based strategy. On the other hand, the considered problem is transformed into a convergence problem of infinite products of a sequence of general stochastic matrices. The general stochastic matrix means that a matrix has row sum one but its entries do not necessarily require non-negative. This article develops the convex analysis to cope with such infinite products. Then, a sufficient condition is obtained to ensure that the specified formation shape in a noisy environment can be achieved. Subsequently, an example is presented to show the effectiveness of the result.

Necessary and Sufficient Conditions for Group Consensus of Fractional Multiagent Systems Under Fixed and Switching Topologies via Pinning Control

The group consensus problem for fractional-order multiagent systems is investigated in this paper. With the help of double-tree-form transformations, the group consensus problem of fractional-order multiagent systems is proved to be equivalent to the asymptotical stability problem of reduced-order error systems. A class of distributed control protocols and some simple LMI sufficient conditions as well as necessary and sufficient conditions are proposed in this paper to solve the group consensus problem for fractional multiagent systems. Moreover, pinning control strategy has been taken into consideration. It is shown that the system converges more rapidly when the designed pinning protocols are adopted. In addition, the case of fractional system with switching topologies is also discussed and some corresponding sufficient conditions are obtained. Finally, some simulation results are presented to illustrate the theoretical results.

Stability Analysis of Multi-Agent Tracking Systems with Quasi-Cyclic Switching Topologies

In this paper, the stability problem of a class of multi-agent tracking systems with quasi-cyclic switching topologies is investigated. The existing results of systems with switching topologies are usually achieved based on the assumption that the piecewise constant communication topologies are connected and the switchings are cyclic. The communication topologies are possible to be unconnected and it is difficult to guarantee the topologies switch circularly. The piecewise unconnected topology makes the interactive multi-agent tracking system to be an unstable subsystem over this time interval. In order to relax the assumption constraint, a quasi-cyclic method is proposed, which allows the topologies of multi-agent systems to switch in a less conservative way. Moreover, the stability of the tracking system with the existence of unstable subsystems is analyzed based on switched control theory. It is obtained that the convergence rate is affected by the maximum dwell time of unstable subsystems. Finally, a numerical example is provided to demonstrate the effectiveness of the theoretical results.

Synchronization of Time-Delayed Complex Networks With Switching Topology Via Hybrid Actuator Fault and Impulsive Effects Control

This article investigates global exponential synchronization almost surely (GES a.s.) of complex networks (CNs) with node delay and switching topology. By introducing transition probability (TP) and mode-dependent average dwell time (MDADT) to the switching signal, the considered model is more practical than the systems with average dwell-time (ADT) switching. Controllers with both impulsive effects and actuator fault feedback are considered. New analytical techniques are developed to obtain sufficient conditions to guarantee the GES a.s. Different from the existing results on the synchronization of switched systems, our results show that the GES a.s. can still be achieved even in the case that the upper bound of the dwell time (DT) of uncontrolled nodes is very large and the lower bound of the DT of controlled nodes is very small. Numerical examples demonstrate the effectiveness and the merits of the theoretical analysis.

Quasi-Consensus of Heterogeneous-Switched Nonlinear Multiagent Systems

In this paper, the quasi-consensus problem is investigated for a class of heterogeneous-switched nonlinear multiagent systems, in which both cooperation and competition interactions are considered simultaneously. By means of the Lyapunov function method, we show that quasi-consensus can be ensured for switched multiagent systems under the assumption that the activation time of cooperation interactions is sufficiently large. Moreover, a new Lyapunov function is considered to provide the lower and upper bounds of switching intervals explicitly. Thus, these bounds can be used to obtain less conservative stability results of switched systems. Furthermore, the established results are specialized to both the traditional consensus case and the stability of linear-switched systems. Finally, simulations are given to illustrate the theoretical results derived in this paper.

Scale-Free Loopy Structure is Resistant to Noise in Consensus Dynamics in Complex Networks

The vast majority of real-world networks are scale-free, loopy, and sparse, with a power-law degree distribution and a constant average degree. In this paper, we study first-order consensus dynamics in binary scale-free networks, where vertices are subject to white noise. We focus on the coherence of networks characterized in terms of the H ₂ -norm, which quantifies how closely the agents track the consensus value. We first provide a lower bound of coherence of a network in terms of its average degree, which is independent of the network order. We then study the coherence of some sparse, scale-free real-world networks, which approaches a constant. We also study numerically the coherence of Barabási-Albert networks and high-dimensional random Apollonian networks, which also converges to a constant when the networks grow. Finally, based on the connection of coherence and the Kirchhoff index, we study analytically the coherence of two deterministically growing sparse networks and obtain the exact expressions, which tend to small constants. Our results indicate that the effect of noise on the consensus dynamics in power-law networks is negligible. We argue that scale-free topology, together with loopy structure, is responsible for the strong robustness with respect to noisy consensus dynamics in power-law networks.

Distributed Optimization Based on a Multiagent System Disturbed by General Noise

A distributed optimization problem based on a continuous-time multiagent system (MAS) disturbed by general noise is considered in this paper. The general noise, under some relaxed assumptions, which may be a stationary process, is proposed to describe the disturbance among agents more accurately. The noise-to-state (NOS) stability of the concerned MAS is analyzed based on an improved theoretical result of random differential equations. Furthermore, the relative sufficient conditions in the form of linear matrix inequality are developed with less conservatism, from which the minimum estimation error between the optimal solution and the NOS stable state of the proposed MAS with general noise can be obtained by choosing some appropriate distributed optimization parameters. One example is used to verify the effectiveness of the proposed approach.

Nonfragile Near-Optimal Control of Stochastic Time-Varying Multiagent Systems With Control- and State-Dependent Noises

In this paper, the near-optimal nonfragile consensus control design problem is investigated for a class of discrete time-varying multiagent systems (MASs) with control- and state-dependent noises. A decentralized observer-based control protocol is proposed by using the relative output measurements. The gain perturbations/variations of the controller as well as the state- and control-dependent noises are simultaneously taken into consideration, which could better reflect the complexities in reality. The corresponding time-varying observer-based nonfragile near-optimal consensus protocol is designed for the underlying MASs over a finite horizon. To be specific, a certain upper bound is first derived for the associate cost function for the MASs. Then, such an upper bound is minimized by using the completing-the-square technique and Moore-Penrose pseudo inverse. The parameters of the time-varying observer/controller are obtained in terms of the solutions to the Riccati-like recursion. In virtue of the matrix partitioning technique, the explicit expressions of the control/observer parameters are presented. Finally, based on the derived consensus protocol, an upper bound of the associate cost function is provided as time goes to infinity. Some numerical simulations are conducted to demonstrate the validity of the proposed methodology.

Fuzzy Finite Time Control for Switched Systems via Adding a Barrier Power Integrator

This paper concentrates on the study of finite time control for nonlinear switched systems. Based on a newly introduced adding a barrier integrator technique, a novel adaptive fuzzy control strategy is proposed for a class of nonlinear switched systems. Compared with the existing adaptive control methods, the proposed method has several distinguishing features. Finite time control: the proposed adaptive control method can solve the exact finite time control problem for the stabilization and some types of tracking issues. Namely, the errors will converge to zero in finite time. For a general tracking problem, the practical finite time control can be achieved. More general systems: the proposed method is suitable for high order nonlinear switched systems with arbitrary switching and unknown control gains. Some strict assumptions on the system dynamics are relaxed. Full state constraints: the proposed method can be utilized to deal with the full state constraints problem. Simple controller structure: the "explosion of the complexity" in the backstepping design is avoided. Singularity free design: the singularity problem is carefully handled during the whole design procedure. Examples are presented to illustrate the effectiveness of the proposed method.

Adaptive Neural Tracking Control for Interconnected Switched Systems With Non-ISS Unmodeled Dynamics

The adaptive neural network tracking control problem is investigated for a class of interconnected switched systems. The considered systems are with unmodeled dynamics, some of which do not satisfy the input-to-state stable (ISS) condition. By utilizing the neural network to approximate the composite unknown nonlinear functions, the corresponding decentralized tracking controller is designed for each subsystem with the help of dynamic surface control method. Some subsystems are stable with the designed controller, while other subsystems may not be stable because of non-ISS unmodeled dynamics, but they have some special properties with the designed controller. Then, a novel switching signal scheme is established such that the interconnected switched system is stable in the sense of semi-global boundedness, and the tracking errors can converge to predefined residual sets with prescribed performance index. Moreover, the switching scheme allows the number of switches to grow faster than traditional average dwell time method. Finally, a numerical example is provided to demonstrate the effectiveness of the presented results.

Input Time Delay Margin in Event-Triggered Consensus of Multiagent Systems

In this paper, the event-triggered consensus problem of multiagent systems with input time delay is investigated. First, the normal event-triggered control scheme containing the input time delay is introduced to reduce the number of communication. Then the following results are achieved: 1) the procedure of setting parameters is carefully formulated for the event-triggered control scheme; 2) the precise input time delay margin is calculated for the event-triggered consensus of the multiagent systems; 3) a more general condition of constructing event-triggered functions is derived to exclude the Zeno behavior; 4) the self-triggered control scheme is further applied to avoid the continuous measurement; and 5) the observer-based control scheme is also utilized to tackle the problem of unmeasurable state. Finally, the correctness and the effectiveness of these results are demonstrated by numerical simulations.

Consensus in Self-Similar Hierarchical Graphs and Sierpiński Graphs: Convergence Speed, Delay Robustness, and Coherence

The hierarchical graphs and Sierpiński graphs are constructed iteratively, which have the same number of vertices and edges at any iteration, but exhibit quite different structural properties: the hierarchical graphs are nonfractal and small-world, while the Sierpiński graphs are fractal and "large-world." Both graphs have found broad applications. In this paper, we study consensus problems in hierarchical graphs and Sierpiński graphs, focusing on three important quantities of consensus problems, that is, convergence speed, delay robustness, and coherence for first-order (and second-order) dynamics, which are, respectively, determined by algebraic connectivity, maximum eigenvalue, and sum of reciprocal (and square of reciprocal) of each nonzero eigenvalue of Laplacian matrix. For both graphs, based on the explicit recursive relation of eigenvalues at two successive iterations, we evaluate the second smallest eigenvalue, as well as the largest eigenvalue, and obtain the closed-form solutions to the sum of reciprocals (and square of reciprocals) of all nonzero eigenvalues. We also compare our obtained results for consensus problems on both graphs and show that they differ in all quantities concerned, which is due to the marked difference of their topological structures.

Scaled Group Consensus in Multiagent Systems With First/Second-Order Continuous Dynamics

We investigate scaled group consensus problems of multiagent systems with first/second-order linear continuous dynamics. For a complex network consisting of two subnetworks with different physical quantities or task distributions, it is concerned with this case that the agents' states in one subnetwork converge to a consistent value asymptotically, while the states in the other subnetwork approach another value with a ratio of the former. For the case of the information exchange being directed, novel consensus protocols are designed for both first-order and second-order dynamics to solve the scaled group consensus problems. By utilizing algebra theory, graph theory, and Lyapunov stability theory, several necessary and sufficient conditions are established to guarantee the agents' states reaching the scaled group consensus asymptotically. Finally, several simulation results are presented to demonstrate the effectiveness of the theoretical results.

Finite-Horizon $H_\infty $ Consensus for Multiagent Systems With Redundant Channels via An Observer-Type Event-Triggered Scheme

This paper is concerned with the finite-horizon consensus problem for a class of discrete time-varying multiagent systems with external disturbances and missing measurements. To improve the communication reliability, redundant channels are introduced and the corresponding protocol is constructed for the information transmission over redundant channels. An event-triggered scheme is adopted to determine whether the information of agents should be transmitted to their neighbors. Subsequently, an observer-type event-triggered control protocol is proposed based on the latest received neighbors' information. The purpose of the addressed problem is to design a time-varying controller based on the observed information to achieve the consensus performance in a finite horizon. By utilizing a constrained recursive Riccati difference equation approach, some sufficient conditions are obtained to guarantee the consensus performance, and the controller parameters are also designed. Finally, a numerical example is provided to demonstrate the desired reliability of redundant channels and the effectiveness of the event-triggered control protocol.

Consensus of Heterogeneous Linear Multiagent Systems With Communication Time-Delays

This paper studies the consensus problem of heterogeneous linear multiagent systems with arbitrarily large constant, time-varying, or distributed communication delays. Novel distributed dynamic controllers are proposed for such multiagent systems with fixed and switching directed communication topologies, respectively. It is shown that the controlled heterogeneous linear multiagent system can reach consensus for arbitrarily large constant, time-varying, and distributed communication delays under some sufficient conditions. Simulation examples are provided to demonstrate the effectiveness of the proposed controllers.

Adaptive Leader-Following Consensus for Second-Order Time-Varying Nonlinear Multiagent Systems

The leader-following consensus problem is investigated for second-order time-varying nonlinear multiagent systems with unmodeled dynamics and unknown parameters over directed communication topology. Under the assumption that the unknown nonlinearities satisfy Lipschitz conditions with time-varying gains, a local adaptive law is introduced for the design of consensus protocol that enable all followers' state variables to consensus with that of leader asymptotically. The proposed protocols are independent of system parameters and only require the relative state information of its neighbors, and hence they are fully distributed. Simulation examples are given to illustrate the effectiveness of the theoretical results.

Distributed-observer-based cooperative control for synchronization of linear discrete-time multi-agent systems

This paper considers output synchronization of discrete-time multi-agent systems with directed communication topologies. The directed communication graph contains a spanning tree and the exosystem as its root. Distributed observer-based consensus protocols are proposed, based on the relative outputs of neighboring agents. A multi-step algorithm is presented to construct the observer-based protocols. In light of the discrete-time algebraic Riccati equation and internal model principle, synchronization problem is completed. At last, numerical simulation is provided to verify the effectiveness of the theoretical results.