Second-order Consensus Algorithm Research Articles

Impulsive consensus algorithms for vector second-order Lipschitz nonlinear multi-agent systems using only velocity regulation

The existing impulsive consensus algorithms for second-order Lipschitz nonlinear multi-agent systems require to apply the impulsive control to both position and velocity vectors at the same time. Such a requirement cannot be met in most of the real-world applications. To overcome the limitations of these impulsive algorithms, two kinds of new second-order impulsive consensus algorithms using only velocity regulation are proposed. Through developing a weighted discontinuous Lyapunov function-based approach that is able to leverage the spectral property of Laplacian matrix, impulse-dwell-time-dependent sufficient conditions for solving second-order impulsive consensus are derived in the form of linear matrix inequalities. Further, it is shown that if the impulsively controlled velocity subsystems are globally exponentially stable, the impulsive static consensus algorithm is able to ensure that all agents tend to an agreed position. Based on the consensus conditions, two convex optimization problems are formulated, by which the impulsive gain matrices for ensuring a prescribed exponential convergence rate can be designed. Finally, the effectiveness of the proposed distributed impulsive consensus algorithms is certified through numerical simulations.

An enhanced second-order-consensus-based distributed secondary frequency controller of virtual synchronous generators for isolated AC microgrids

Virtual synchronous generator (VSG) control is a promising control method in microgrid (MG) for the inertia support characteristic. However, the VSG inherits the oscillation feature of synchronous generator (SG), thus the active power and frequency oscillation may be observed when VSG control is applied. In MG, the oscillation may cause the unreasonable dynamic active power sharing among VSGs. Subsequently, the VSGs are prone to overload and even unstable, and the existing works based on consensus control cannot suppress the oscillation effectively. In order to tackle this problem, this paper proposes an enhanced second-order-consensus-based distributed secondary frequency controller by introducing the active power second-order consensus control, and reveals the oscillation suppression capacity of the second-order consensus algorithm in depth. Since there is no specific control requirement for the second-order variables, i.e., the change rate of active power and frequency of MG in steady state, the proposed controller can be simplified by replacing the integral control of second-order variables with the proportional control of first-order variables, i.e., active power and frequency, thus the communication data volume will not increase comparing to the traditional first-order-consensus-based controller. This paper also provides a small-signal model of MG with the proposed controller to evaluate the system dynamic performance and design the parameters. Finally, the simulation results verify the effectiveness of the proposed controller.

Stability and Convergence of a Message-Loss-Tolerant Rendezvous Algorithm for Wireless Networked Robot Systems

This article considers the problem of swarm rendezvous when robots exchange local coordination messages through an unreliable wireless communication network (i.e., message-loss occurs). First, a discrete-time second-order consensus algorithm is provided to deal with the effects of unreliable channels on wireless networked robot (WNR) systems, and drive the robots to a given geometric formation. Then, necessary and sufficient conditions for the controlled WNR to converge to a desired formation, even in the presence of messages loss, are formulated and simulatively validated. The proposed conditions provide to the WNR designer a minimal topology connectivity requirement, and a relation among algorithm step-size and control gains to be fulfilled for guaranteeing swarm rendezvous.

UAV Group Formation Collision Avoidance Method Based on Second-Order Consensus Algorithm and Improved Artificial Potential Field

The problem of collision avoidance of an unmanned aerial vehicle (UAV) group is studied in this paper. A collision avoidance method of UAV group formation based on second-order consensus algorithm and improved artificial potential field is proposed. Based on the method, the UAV group can form a predetermined formation from any initial state and fly to the target position in normal flight, and can avoid collision according to the improved smooth artificial potential field method when encountering an obstacle. The UAV group adopts the “leader–follower” strategy, that is, the leader UAV is the controller and flies independently according to the mission requirements, while the follower UAV follows the leader UAV based on the second-order consensus algorithm and formations gradually form during the flight. Based on the second-order consensus algorithm, the UAV group can achieve formation maintenance easily and the Laplacian matrix used in the algorithm is symmetric for an undirected graph. In the process of obstacle avoidance, the improved artificial potential field method can solve the jitter problem that the traditional artificial potential field method causes for the UAV and avoids violent jitter. Finally, simulation experiments of two scenarios were designed to verify the collision avoidance effect and formation retention effect of static obstacles and dynamic obstacles while the two UAV groups fly in opposite symmetry in the dynamic obstacle scenario. The experimental results demonstrate the effectiveness of the proposed method.

Distributed optimal consensus algorithms in multi-agent systems

This paper investigates the distributed optimal consensus problem by designing the multi-agent communication systems, where each agent has local access to its own strongly convex utility function. Here, we design two distributed consensus-based optimization algorithms to tackle the optimization problem of multi-agent systems. One is the second-order optimization consensus algorithm, and another is the third-order algorithm. Moreover, the two algorithms can ensure that all agents achieve consensus at a common state which minimizes the sum of the objective function under the assumption that interaction graph is strongly connected and detailed balanced. For each of the proposed algorithms, it can be found that the equilibrium point of the algorithm is the optimal point and it is proved to be asymptotically stable under given conditions. Finally, the performance of the proposed algorithms are verified by two simulations based on an IEEE 14-bus test system and an IEEE 57-bus test system.

A Flexible Collision Avoidance Strategy for the Formation of Multiple Unmanned Aerial Vehicles

This paper develops a flexible collision avoidance strategy for the formation of multiple unmanned aerial vehicles (multi-UAV). Firstly, To make full use of the information flow from the communication of the formation, an improved artificial potential field (IAPF) function is constructed by adding the communication topology and communication weights. Based on the IAPF functions, both collision avoidance within the formation and obstacle avoidance outside the formation are well organized to ensure flight safety. Secondly, to fuse collision avoidance and formation keeping, we design the null space behavioral (NSB) approach to combine them into a single motion command, where the formation keeping algorithm is designed as a second-order consensus algorithm. In this way, the flexible collision avoidance strategy is proposed based on IAPF and consensus, handling the relationship between collision avoidance and formation keeping flexibly. Finally, three-dimensional multi-UAV flight simulation validates the effectiveness of the proposed strategy that the formation can both implement collision avoidance quickly and keep a good geometric formation configuration simultaneously.

Consensus-based three-dimensionalmulti-UAV formation control strategy with high precision

We propose a formation control strategy for multiple unmanned aerial vehicles (multi-UAV) based on second-order consensus, by introducing position and velocity coordination variables through neighbor-to-neighbor interaction to generate steering commands. A cooperative guidance algorithm and a cooperative control algorithm are proposed together to maintain a specified geometric configuration, managing the position and attitude respectively. With the whole system composed of the six-degree-of-freedom UAV model, the cooperative guidance algorithm, and the cooperative control algorithm, the formation control strategy is a closed-loop one and with full states. The cooperative guidance law is a second-order consensus algorithm, providing the desired acceleration, pitch rate, and heading rate. Longitudinal and lateral motions are jointly considered, and the cooperative control law is designed by deducing state equations. Closed-loop stability of the formation is analyzed, and a necessary and sufficient condition is provided. Measurement errors in position data are suppressed by synchronization technology to improve the control precision. In the simulation, three-dimensional formation flight demonstrates the feasibility and effectiveness of the formation control strategy.

Distributed attitude coordination tracking control for spacecraft formation with time-varying delays

To solve the problem of attitude synchronization for spacecraft formation with communication time-varying delays, this paper investigates a distributed attitude coordination tracking control strategy in which a directed graph contains a spanning tree with the leader as the root. Based on second-order consensus algorithm and graph theory, we propose a novel distributed attitude coordination control approach, which can regulate the attitude of spacecraft to a common time-varying reference states in case of time-varying delays. And the stability of the system is proved via Lyapunov Razumikhin theory. Simulation results demonstrate the effectiveness of the proposed control approach.

Convergence Analysis on Multi-AUV Systems With Leader-Follower Architecture

In multi-autonomous underwater vehicle (multi-AUV) systems, the convergence rate is characterized by the pace of consistency of the key state information for each member. The topology with leader–follower architecture is designed as a combination of an undirected graph between followers and a digraph between leaders and followers. An overview of influences on convergence rate of the second-order consensus algorithm is elaborated in three aspects, along with the main contributions in this paper. Specifically, the explicit expression of the maximum convergence rate is established based on the root locus method, and then, the effects of control parameters on the convergence rate are analyzed. Moreover, the influences of network topologies on the convergence rate are investigated from the view of adjusting the existing connectivity, changing the weights on links, and utilizing hierarchical structure. The combination of consensus and filtering algorithm is also an approach to enhance the capacity of multi-AUV systems. In order to eliminate the accumulated errors in the process of dead reckoning, a collaborative navigation model is presented, and then, a localization approach based on consensus-unscented particle filter algorithm is proposed. Simulations results are provided to verify location performance under the assumption of Gaussian white noise in the systems. In addition, the influences of the topologies on positioning accuracy are explored.

Robustness of First- and Second-Order Consensus Algorithms for a Noisy Scale-Free Small-World Koch Network

In this brief, we study first- and second-order consensus algorithms for the scale-free small-world Koch network, where vertices are subject to white noise. We focus on three cases of consensus schemes: 1) first-order leaderless algorithm; 2) first-order algorithm with a single leader; and 3) second-order leaderless algorithm. We are concerned with the coherence of the Koch network in the $H_{2}$ norm, which captures the level of agreement of vertices in face of stochastic disturbances. Based on the particular network construction, we derive explicit expressions of the coherence for all the three consensus algorithms, as well as their dependence on the network size. Particularly, for the first-order leader-follower model, we show that coherence relies on the shortest-path distance between the leader and the largest-degree vertices, as well as the degree of the leader. The asymptotic behaviors for coherence of the three consensus algorithms in Koch network behave differently from those associated with other networks lacking scale-free small-world features, indicating significant influences of the scale-free small-world topology on the performance of the consensus algorithms in noisy environments.

Three-dimensional multiple unmanned aerial vehicles formation control strategy based on second-order consensus

Considering three-dimensional formation control for multiple unmanned aerial vehicles, this paper proposes a second-order consensus strategy by utilizing the position and velocity coordinate variables. To maintain the specified geometric configuration, a cooperative guidance algorithm and a cooperative control algorithm are proposed together to manage the position and attitude, respectively. The cooperative guidance law, which is designed as a second-order consensus algorithm, provides the desired pitch rate, heading rate and acceleration. In addition, a synchronization technology is put forward to reduce the influence of the measurement errors for the cooperative guidance law. The cooperative control law, regarding the output of the cooperative guidance law as its input, is designed by deducing the state-space expression of both the longitudinal and lateral motions. The formation stability is analyzed to give a sufficient and necessary condition. Finally, the simulations for the three-dimensional formation control demonstrate the feasibility and effectiveness of the second-order consensus strategy.

Distributed time synchronization in wireless sensor networks via second-order consensus algorithms

This paper proposes a distributed second-order consensus time synchronization, which incorporates the second-order consensus algorithm into wireless sensor networks. Since local clocks may have different skews and offsets, the algorithm is designed to include offset compensation and skew compensation. The local clocks are not directly modified, thus the virtual clocks are built according to the local clocks via the compensation parameters. Each node achieves a virtual consensus clock by periodically updated compensation parameters. Finally, the effectiveness of the proposed algorithm is verified through a number of simulations in a mesh network. It is proved that the proposed algorithm has the advantage of being distributed, asymptotic convergence, and robust to new node joining.

Consensus and Coherence in Fractal Networks

We consider first- and second-order consensus algorithms in networks with stochastic disturbances. We quantify the deviation from consensus using the notion of network coherence, which can be expressed as an $H_{2}$ norm of the stochastic system. We use the setting of fractal networks to investigate the question of whether a purely topological measure, such as the fractal dimension, can capture the asymptotics of coherence in the large system size limit. Our analysis for first-order systems is facilitated by connections between first-order stochastic consensus and the global mean first passage time of random walks. We then show how to apply similar techniques to analyze second-order stochastic consensus systems. Our analysis reveals that two networks with the same fractal dimension can exhibit different asymptotic scalings for network coherence. Thus, this topological characterization of the network does not uniquely determine coherence behavior. The question of whether the performance of stochastic consensus algorithms in large networks can be captured by purely topological measures, such as the spatial dimension, remains open.

Disturbance Estimation, Compensation, and Dynamic Shaping: New Approach and an Application to Output Stabilization of Multiple Cooperative Control

This paper introduces a new disturbance estimation scheme, and a possible application to relative output stabilization of multiple systems. Using the proposed disturbance estimation scheme, total unknown external disturbance applied to a plant is estimated and compensated. Moreover, the model difference between an actual system and a desired system is also estimated and compensated. For the purpose of general use of the disturbance estimation scheme as an unknown input observer (UIO), a parameterized design method is given, even for the unstable and nonminimum phase systems. For the relative output stabilization of multiple systems, second-order consensus algorithm is additionally used. A case study, simulations, and experimental tests sequentially validate the proposed estimation and control methods.

Network Clock Synchronization Based on the Second-Order Linear Consensus Algorithm

In this paper a distributed algorithm for clock synchronization is proposed. This algorithm is based on an extension of the linear consensus algorithm which is able to synchronize a family of identical double integrators. Since in reality the various clocks may have different drifts, the algorithm needs to be designed in such a way that it works correctly also in case of heterogeneous double integrators. We start by reviewing an unrealistic synchronous implementation of the clock synchronization algorithm, that has been recently proposed in the context of noisy double integrators. The main contribution of this paper is to propose a realistic pseudo-synchronous implementation of this algorithm. This pseudo-synchronous algorithm is shown to be a perturbation of the synchronous one and so, through arguments related to the center manifold theorem, it is proved to be locally convergent under the assumption of absence of process noise, measurement noise and propagation delays. However, through numerical simulations, the performance of this algorithm is evaluated also in the case the communication delays are not negligible, the clock drifts are time-varying and the communication channels are unreliable. Moreover, again through numerical simulations, the strategy we propose in this paper is compared with other fully distributed strategies recently proposed in the literature. While being slightly slower to reach the asymptotic synchronization, our strategy seems to significantly outperform the other strategies in terms of robustness to process and measurement noises and time-varying clock drifts.

A Sufficient Condition for Convergence of Sampled-Data Consensus for Double-Integrator Dynamics With Nonuniform and Time-Varying Communication Delays

This technical note investigates a discrete-time second-order consensus algorithm for networks of agents with nonuniform and time-varying communication delays under dynamically changing communication topologies in a sampled-data setting. Some new proof techniques are proposed to perform the convergence analysis. It is finally shown that under certain assumptions upon the velocity damping gain and the sampling period, consensus is achieved for arbitrary bounded time-varying communication delays if the union of the associated digraphs of the interaction matrices in the presence of delays has a directed spanning tree frequently enough.

Second-Order Consensus of Multiple Agents with Bounded Control Inputs and Preserved Network Connectivity

This paper investigates second-order consensus of multi-agent systems with a virtual leader of varying velocity while preserving network connectivity. We propose a novel second-order consensus algorithm with bounded control inputs. Under the condition that the initial network is connected, the network will be connected all the time and all agents and the virtual leader can attain the same position and move with the same velocity. A simulation example is proposed to illustrate the effective of the proposed algorithm.

Adaptive second-order consensus of networked mobile agents with nonlinear dynamics

We investigate second-order consensus of multiple nonlinear dynamical mobile agents with a virtual leader in a dynamic proximity network. We assume that only a small fraction of agents in the group have access to the information about the position and velocity of the virtual leader through, for example, certain pre-designed communication mechanism such as wireless broadcasting or sensing. To avoid fragmentation, we propose a connectivity-preserving second-order consensus algorithm. Under the assumption that the initial network is connected, we introduce local adaptation strategies for both the weights on the velocity navigational feedback and the velocity coupling strengths that enable all agents to synchronize with the virtual leader even when only one agent is informed, without requiring any knowledge of the agent dynamics. We finally provide some convincing simulation results to illustrate the theoretical results.

Some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems

This paper studies some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems. First, basic theoretical analysis is carried out for the case where for each agent the second-order dynamics are governed by the position and velocity terms and the asymptotic velocity is constant. A necessary and sufficient condition is given to ensure second-order consensus and it is found that both the real and imaginary parts of the eigenvalues of the Laplacian matrix of the corresponding network play key roles in reaching consensus. Based on this result, a second-order consensus algorithm is derived for the multi-agent system facing communication delays. A necessary and sufficient condition is provided, which shows that consensus can be achieved in a multi-agent system whose network topology contains a directed spanning tree if and only if the time delay is less than a critical value. Finally, simulation examples are given to verify the theoretical analysis.

Consensus of Second-Order Delayed Multi-Agent Systems with Leader-Following

In this paper, a second-order consensus algorithm of multi-agent dynamical systems with leader-following is presented. With the hypothesis of a directed weighted connected graph composed of n agents together with a leader and the leader as a globally reachable node, the consensus algorithm with heterogeneous input delays is studied in the frequency domain. By applying Gershgorin disc theorem and curvature theory, decentralized consensus conditions for the multi-agent systems with asymmetric coupling weights are obtained. Finally, a simulation example is used to show the design of the parameters and the validity of the result.