Exhibit Zeno Behavior Research Articles

An event-based distributed least square linear equation solver employing network flow

In this article, we study a distributed, second-order, event-triggered network flow for solving linear algebraic equations described by the form z=Hy in the sense of least squares. An undirected graph is utilized to describe the distributed computation network. In the network, each node broadcasts its dynamic states to its neighbors in an asynchronous fashion, and updates the flow at the isolated event times. We construct a specific trigger function to determine the event times, in which an exponential decay function is introduced as a comparison term. We show that if the linear algebraic equation has a unique least squares solution, all nodes states converge to this unique solution exponentially fast, when the applied network is connected and the decay rate of the comparison term satisfies certain bounds determined jointly by the topology and the linear equations information. We obtain a strictly positive lower bound on the time interval between consecutive events for time t∈[0,∞), and therefore the event-triggered system does not exhibit Zeno behavior. We provide numerical examples to verify the effectiveness of the proposed event-based network flow. In addition, we illustrate the importance of the strictly positive inter-event time interval, in comparison with the case that the inter-event time interval may converge to zero when time t goes to infinity.

Model-independent event-based consensus of multiple Euler-Lagrange systems with input disturbances

This paper investigates leaderless consensus (LLC) and leader-follower consensus (LFC) issues of multiple Euler-Lagrange systems (MELSs) with uncertain system parameters and input disturbances. Firstly, by utilizing event-based control strategy, event-based consensus protocols are proposed to achieve LLC and LFC, respectively. Then, to confirm that no Euler-Lagrange (EL) system will exhibit Zeno behavior under the above event-based consensus protocols, we show that the low bound of event-triggering interval is strictly positive for any time instants. Distinct with the existing related works, the proposed event-triggered consensus protocols only rely on local sampled state, and are irrelevant to the system parameters. Finally, some simulation examples are presented to illustrate the feasibility of the proposed event-based consensus algorithms.

Distributed Nash equilibrium seeking with stochastic event-triggered mechanism

In this paper, we study the problem of consensus-based distributed Nash equilibrium (NE) seeking in a network of players represented as a directed graph, where each player aims to minimize their own local cost functions non-cooperatively. To address bandwidth constraints and limited energy, we propose a stochastic event-triggered algorithm that triggers individual players with a probability depending on certain events, thus enhancing communication efficiency through reduced continuous communication. We prove that our developed event-triggered algorithm achieves exponential convergence to the exact NE when the underlying communication graph is strongly connected. Furthermore, we establish that our proposed event-triggered communication scheme does not exhibit Zeno behavior. Finally, through numerical simulations of a spectrum access game and comparisons with existing event-triggered methods, we demonstrate the effectiveness of our proposed algorithm.

Distributed fixed-time event-triggered containment control for nonlinear multiagent systems with actuator faults

This paper is concerned with distributed fixed-time containment control for nonlinear multiagent systems with actuator faults via static and dynamic event-triggered strategies. A distributed fixed-time static event-triggered fault-tolerant controller is developed by employing the sign function, which eliminates the requirement for continuous control signal updates. To address the chattering phenomenon caused by the sign function, a distributed saturation controller is presented to enhance the control performance. To further reduce energy consumption and frequency of controller updates, a dynamic event-triggered strategy with an auxiliary parameter is proposed for each agent to dynamically regulate its threshold. The control scheme guarantees that the containment control objective is achieved within the settling time, and does not exhibit Zeno behavior. The effectiveness of the proposed control scheme is illustrated by simulation results.

Prescribed Settling Time Adaptive Neural Network Consensus Control of Multiagent Systems with Unknown Time-Varying Input Dead-Zone

For a class of multiagent systems with an unknown time-varying input dead-zone, a prescribed settling time adaptive neural network consensus control method is developed. In practical applications, some control signals are difficult to use effectively due to the extensive existence of an input dead-zone. Moreover, the time-varying input gains further seriously degrade the performance of the systems and even cause system instability. In addition, multiagent systems need frequent communication to ensure a system’s consistency. This may lead to communication congestion. To solve this problem, an event-triggered adaptive neural network control method is proposed. Further, combined with the prescribed settling time transform function, the developed consensus method greatly increases the convergence rate. It is demonstrated that all followers of multiagent systems can track the virtual leader within a prescribed time and not exhibit Zeno behavior. Finally, the theoretical analysis and simulation verify the effectiveness of the designed control method.

Fully distributed consensus control for second-order multi-agent systems based on adaptive dynamic clock communication

This paper proposes a novel event-triggered control strategy based on an adaptive dynamic clock to address the consensus problem of second-order multi-agent systems with undirected communication topology. The agent determines the triggering time according to its dynamic clock, adaptively resets the clock, and broadcasts its state information to neighbors at the triggering time. Each agent only obtains the status information of its neighbors at the triggering time and updates the clock and control signals based on its own state and the status of its neighbors at the triggering time. It does not require real-time status information from neighbors and does not rely on any global communication topology information. The designed control strategy is completely distributed and effectively avoids continuous communication. Using the Lyapunov stability analysis method and algebraic graph theory, it is proved that the proposed control strategy can ensure that the system is asymptotically stable and does not exhibit Zeno behavior. Finally, the effectiveness of the proposed dynamic event-triggered control strategy is demonstrated by a simulation example.

Adaptive Bipartite Output Tracking Consensus in Switching Networks of Heterogeneous Linear Multiagent Systems Based on Edge Events.

This article focuses on the problem of adaptive bipartite output tracking for a class of heterogeneous linear multiagent systems (MASs) by asynchronous edge-event-triggered communications under jointly connected signed topologies. By designing the observers to estimate the states of followers and the dynamic compensators to estimate the states of zero input and nonzero input leader, respectively, the fully distributed edge-event-triggered control protocol is presented. Moreover, it is proven that the bipartite output tracking problem is implemented, and the systems do not exhibit Zeno behavior under a fully distributed control strategy with edge-event-triggered mechanisms. Compared with the existing works, one of the highlights of this article is the design of triggering mechanisms, under which the leader avoids continuous information transmission and any pair of followers that make up the edge asynchronously transmit information through the edge. The methods greatly avoid unnecessary information transmission in the systems. Finally, several simulation examples are introduced to demonstrate the theoretical results obtained in this article.

Resilient Event-Triggered Control Strategies for Second-Order Consensus

This article investigates the second-order consensus issue for multiagent systems subject to both limited communication resources and replay attacks. First, an asynchronous dynamic edge event-triggered (DEET) communication scheme is developed to reduce the utilization of network resources in the absence of attacks. Then, we further consider the case of replay attacks launched by multiple adversaries, under which the transmitted information is maliciously replaced by a previous unnecessary message. To overcome the impact caused by replay attacks, a modified DEET scheme and an effective attack-resilient consensus protocol are well constructed, both of which successfully guarantee second-order consensus in the presence of replay attacks. In addition, internal dynamic variables are utilized in the proposed DEET schemes such that the triggering time sequence does not exhibit Zeno behavior. Finally, one numerical example is provided to illustrate our theoretical analysis.

Leader-Following Consensus for a Class of Nonlinear Multiagent Systems Under Event-Triggered and Edge-Event Triggered Mechanisms.

Considering that there are many systems with limited network bandwidth in practice, this article studies the leader-following consensus problem for a class of nonlinear multiagent systems (MASs). The purpose of this article is to reduce unnecessary information transmission between any pair of adjacent agents including the leader in the MASs through intermittent communication. The novel event-triggered and asynchronous edge-event triggered mechanisms are designed for the leader and all edges, respectively. The static and dynamic consensus protocols under these mechanisms are proposed to address the leader-following consensus problem for MASs with Lipschitz dynamics, and the systems will not exhibit Zeno behavior under these two control schemes. Note that the dynamic consensus protocol does not rely on any global values of MASs, it is a fully distributed way. Finally, a practice simulation example is introduced to illustrate the theoretical results obtained.

On Zeno Behavior in Event-Triggered Finite-Time Consensus of Multiagent Systems

This article studies Zeno behavior, where an infinite number of events occur in a finite-time interval, in the event-triggered multiagent systems that aim at achieving consensus in finite time. Two popular scenarios in the first-order multiagent systems with model-based event-triggered controllers are considered. One is that the event trigger in each agent samples the absolute information, and decides when to broadcast the information to its neighbors, and the other is that the update of control signals is only scheduled by the local event triggers via directly using the (combined) relative information. The events are triggered when the measurement error between the agent, and model states violates a given threshold function. Both cases are studied, where the threshold is generated by the agent state or by the model state. Then, sufficient conditions on the existence of Zeno behavior in event-triggered finite-time consensus of multiagent systems are provided. For the triggering conditions with the thresholds given by agent states, the system must exhibit Zeno behavior; while in the case of using model states, the existence of Zeno behavior is influenced by the properties of the communication topology, and the threshold functions. Finally, several simulations, and examples are provided to illustrate the effectiveness of the theoretical results.

Fully distributed event-triggered time-varying formation control of multi-agent systems subject to mode-switching denial-of-service attacks

In this study, the time-varying state formation control problem for general linear multi-agent systems (MASs) subject to mode-switching denial-of-service (MSDoS) attacks is considered. Based on only the sampled state information of itself and neighboring agents at event-triggered instants, a novel fully distributed event-triggered secure control strategy without continuous communication between agents is delicately designed. Note that the control strategy is implemented in a fully distributed manner, which means that it does not require any global network information. By using the multiple Lyapunov function approach based on edge-dependent average dwell time, this study presents the MASs subject to MSDoS attacks with limited attack frequency and attack width can achieve the specified time-varying state formation structure under the designed control strategy. Furthermore, this study presents that in any finite time, the control strategy does not exhibit Zeno behavior. Finally, the effectiveness and performance of the designed control strategy are validated by a numerical simulation.

Event‐triggered leader‐follower tracking control for interconnected systems

SummaryThis article considers the event‐triggered leader‐follower tracking control for interconnected systems with directed communication topology. Different from the existing work, here we consider systems which are physically coupled. We propose a model‐based event‐triggered tracking control strategy and an event triggering rule with a time‐dependent threshold which guarantees that the tracking errors in the systems asymptotically converge to zero. With the proposed event triggering rule, we prove that the system does not exhibit Zeno behavior. As a special case, the proposed event‐triggered tracking control strategy is also applied to decoupled multiagent systems. The efficacy of the proposed method is discussed using two simulation examples.

Event-triggered stabilization of nonlinear systems with time-varying sensing and actuation delay

This paper studies the problem of stabilization of a nonlinear system with time-varying delays in both sensing and actuation using event-triggered control. Our proposed strategy seeks to opportunistically minimize the number of control updates while guaranteeing stabilization and builds on predictor feedback to compensate for arbitrarily large known time-varying delays. We establish, using a Lyapunov approach, the global asymptotic stability of the closed-loop system as long as the open-loop system is globally input-to-state stabilizable in the absence of time delays and sampling. We further prove that the proposed event-triggered law has inter-event times that are uniformly lower bounded and hence does not exhibit Zeno behavior. For the particular case of a stabilizable linear system, we show global exponential stability of the closed-loop system and analyze the trade-off between the rate of exponential convergence and a bound on the sampling frequency. We illustrate these results in simulation and also examine the properties of the proposed event-triggered strategy beyond the class of systems for which stabilization can be guaranteed.

Towards event-triggered extended state observer for multi-agent systems

This paper deals with the event-triggered leader-follower consensus problem of linear multi-agent systems with disturbances over the directed graph. By using relative output information of the neighboring agents, we develop a novel class of distributed extended state observers. With the estimated values of tracking error and coupled states of the leader’s input and disturbances, event-triggered protocols are designed to solve the leader-follower consensus problem. The proposed protocols update intermittently and can suppress disturbances actively. Subsequently, we prove the systems don’t exhibit Zeno behavior. Furthermore, both the input of leader and disturbances are considered as unmodeled and not available to any follower. Finally, a simulation example is provided to demonstrate the theoretical results.

Adaptive Consensus Control of Linear Multiagent Systems With Dynamic Event-Triggered Strategies.

This paper is concerned with event-triggered consensus of general linear multiagent systems (MASs) in leaderless and leader-following networks, respectively, in the framework of adaptive control. A distributed dynamic event-triggered strategy is first proposed, in which an auxiliary parameter is introduced for each agent to regulate its threshold dynamically. The time-varying threshold ensures less triggering instants, compared with the traditional static one. Then under the proposed event-triggered strategy, a distributed adaptive consensus protocol is formed including the updating law of the coupling strength for each agent. Some criteria are derived to guarantee leaderless or leader-following consensus for MASs with general linear dynamics, respectively. Moreover, it is proved that the triggering time sequences do not exhibit Zeno behavior. Finally, the effectiveness of the proposed dynamic event-triggered control mechanism combined with adaptive control is validated by two examples.

Distributed Secure Cooperative Control Under Denial-of-Service Attacks From Multiple Adversaries.

This paper develops a fully distributed framework to investigate the cooperative behavior of multiagent systems in the presence of distributed denial-of-service (DoS) attacks launched by multiple adversaries. In such an insecure network environment, two kinds of communication schemes, that is, sample-data and event-triggered communication schemes, are discussed. Then, a fully distributed control protocol with strong robustness and high scalability is well designed. This protocol guarantees asymptotic consensus against distributed DoS attacks. In this paper, "fully" emphasizes that the eigenvalue information of the Laplacian matrix is not required in the design of both the control protocol and event conditions. For the event-triggered case, two effective dynamical event-triggered schemes are proposed, which are independent of any global information. Such event-triggered schemes do not exhibit Zeno behavior even in the insecure environment. Finally, a simulation example is provided to verify the effectiveness of theoretical analysis.

Event-triggered robust cooperative stabilization in nonlinearly interconnected multiagent systems

By way of the graph theoretic tools, we model closed-loop large-scale systems as two-layer multiagent systems with the agent layer describing nonlinearly interconnected dynamics and the control layer designed to stabilize the system by communications in agents’ neighborhoods. We consider a scenario in which the effect of dynamical interactions over the agent layer appears through some nonlinear functions which are in the range space of the control input matrix, yet the underlying agent layer interconnection topology and the associated nonlinearities are not known to the control layer designer. We develop a static linear cooperative algorithm and, having endowed each local controller with several buffers, we schedule the control layer information exchange events in a non-periodic non-synchronous manner. We combine graph theoretic and event triggering ideas with an optimal control formulation, and propose a design procedure to obtain the required robust control gain. We prove that all trajectories of the closed-loop multiagent system will exponentially converge to the origin in the presence of time-varying nonlinearly interconnected modeling uncertainties over agent layer. For the same cooperative stabilization algorithm, trading off with the exponential convergence to a neighborhood around the origin, we further show that a practical event triggering mechanism can be developed to better deal with the effect of modeling mismatch on the number of triggering events. We also prove the proposed event triggering mechanisms do not exhibit Zeno behavior to ensure the feasibility of the proposed ideas in the sense of reduced communication load and, thereby, energy consumption.

Dynamic Event-Triggered and Self-Triggered Control for Multi-agent Systems

We propose two novel dynamic event-triggered control laws to solve the average consensus problem for first-order continuous-time multiagent systems over undirected graphs. Compared with the most existing triggering laws, the proposed laws involve internal dynamic variables, which play an essential role in guaranteeing that the triggering time sequence does not exhibit Zeno behavior. Moreover, some existing triggering laws are special cases of ours. For the proposed self-triggered algorithm, continuous agent listening is avoided as each agent predicts its next triggering time and broadcasts it to its neighbors at the current triggering time. Thus, each agent only needs to sense and broadcast at its triggering times, and to listen to and receive incoming information from its neighbors at their triggering times. It is proved that the proposed triggering laws make the state of each agent converge exponentially to the average of the agents’ initial states if and only if the underlying graph is connected. Numerical simulations are provided to illustrate the effectiveness of the theoretical results.

Distributed Event-Triggered Adaptive Control for Consensus of Linear Multi-Agent Systems with External Disturbances.

This paper investigates the consensus problem of linear multi-agent systems subject to external disturbances via distributed event-triggered adaptive control. First, a distributed event-triggered adaptive output feedback control strategy is proposed for each agent. It is shown that under this control strategy, the consensus problem can be solved for any connected undirected communication graph in a fully distributed manner without using any global information. Then a distributed self-triggered adaptive output feedback control strategy is designed with which continuous monitoring of the measurement error is no longer needed. It is further shown that for the proposed event-triggered and self-triggered control strategies, no agent will exhibit Zeno behavior. Finally, the effectiveness of the proposed two control strategies is illustrated on a group of two-mass-spring systems.

Event-triggered time-varying formation control for general linear multi-agent systems

This paper investigates event-triggered formation control problems for general linear multi-agent systems. The time-varying formation this paper studied can be described by a bounded piecewise differentiable vector-valued function. Firstly, a time-varying formation control protocol based on event-triggered scheme is constructed by the states of the neighboring agents. Each agent broadcasts its state information to neighbor nodes if the triggering condition is satisfied, and the communication load is decreased significantly. Then, an algorithm consisting of three steps is proposed to design the event-triggered formation control protocol. Moreover, it is proven that under the designed event-triggered formation protocol, the multi-agent systems can achieve the desired time-varying formation which belongs to the feasible formation set with the bounded formation error and the closed systems do not exhibit Zeno behavior. Finally, simulation results are given to demonstrate the effectiveness of the theoretical analysis.