Consensus analysis with large and multiple communication delays using spectral delay space concept

In this study, we consider the consensus problem for a group of second-order agents interacting under a fixed, undirected communication topology. Communication lines are affected by two rationally independent delays. The first delay is assumed to be in the position information channels, whereas the second delay is in the velocity information exchange. The delays are assumed to be large and uniform throughout the entire network. The stability analysis of such systems becomes quickly intractable as the number of agents increases and the delays enlarge. To resolve this dilemma, we first reduce the complexity of the problem dramatically, by decomposing the characteristic equation of the system into a set of second-order factors. Then, we assess the stability of the resulting subsystems exactly and exhaustively in the domain of the time delays using the cluster treatment of characteristic roots (CTCR) paradigm. CTCR requires the determination of all the potential stability switching loci in the domain of the delays. For this, a surrogate domain, called the ‘spectral delay space (SDS)’, is used. The result is a computationally efficient stability analysis of the given dynamics within the domain of the delays. Illustrative cases are provided to verify the analytical conclusions. On these examples, we also study the consensus speeds through eigenvalue analysis.     

Exhaustive stability analysis in a consensus system with time delay and irregular topologies

A consensus problem and its stability are studied for a group of agents with second-order dynamics and communication delays. The communication topologies are taken as irregular but always connected and undirected. The delays are assumed to be quasi-static and the same for all the interagent channels. A decentralised, PD-like control structure is proposed to create a consensus in the position and velocity of the agents. We present an interesting factorisation feature for the characteristic equation of the system which simplifies the stability analysis considerably from a prohibitively large dimensional problem to a manageable small scale. It facilitates a rare stability picture in the space of the control parameters and the delay, utilising a paradigm named cluster treatment of characteristic roots (CTCR). The influence of the individual factors on the absolute and relative stability of the system is studied. This leads to the introduction of two novel concepts: the most exigent eigenvalue, which refers to the one that defines the delay stability margin of the system, and the most critical eigenvalue, which is the one that dictates the consensus speed of the system. It is observed that the most exigent eigenvalue is not always the most critical, and this feature may be used as a design tool for the control logic. Case studies and simulations results are presented to verify these concepts.     

Finding the exact delay bound for consensus of linear multi-agent systems

This paper focuses on consensus problems for high-order, linear multi-agent systems. Undirected communication topologies along with a fixed and uniform communication time delay are taken into account. This class of problems has been widely studied in the literature, but there are still gaps concerning the exact stability bounds in the domain of the delays. The novelty of this paper lies in the determination of an exact and explicit delay bound for consensus. This is done in a very efficient manner by using the cluster treatment of characteristic roots (CTCR) paradigm. Before the stability analysis, a state transformation is performed to decouple the system and simplify the problem. CTCR is then deployed to the individual subsystems to obtain the stability margin in the domain of the delays without the conservatism introduced by other approaches more frequently found in the literature. Simulation results are presented to support the analytical claims.     

Group consensus of multi-agent systems in directed networks with noises and time delays

In this paper, group consensus problems in fixed directed networks of dynamic agents are investigated. Group consensus means that the agents in each group share a consistent value while there is no agreement between any two groups. Based on algebraic graph theory, sufficient conditions guaranteeing group consensus under the proposed control protocol in the presence of random noises and communication delays are derived. The analysis uses a stability result of Mao for stochastic differential delay equations, which ensures the consensus can be achieved almost surely and exponentially fast. Numerical examples are provided to demonstrate the availability of the obtained results as well as the effect of time delay/noise intensity.     

带有通信时滞的二阶多智能体系统一致问题

针对具有双向等时延的二阶无向通信拓扑系统,采用带有通信时滞的线性一致控制率协议,分析了使系统稳定的条件。由于系统的阶次较高,直接对其特征方程进行分析是比较困难的,提出了一种新的分析方法,把系统的特征方程分解为多个子系统的乘积,然后利用CTCR方法,求得每个子系统对应的时滞最大值,比较后得出使系统达到一致稳定的最大时滞,作出了控制率边界曲线图并标出了稳定区域。结果表明,在有向生成树的情况下,当时滞小于决策值时,系统能达到稳定。最后,数值仿真验证了所得结果的有效性。     

A stability study on first-order neutral systems with three rationally independent time delays

First-order linear time invariant and time-delayed dynamics of neutral type is taken into account with three rationally independent delays. There are two main contributions of this study. (a) It is the first complete treatment in the literature on the stability analysis of systems with three delays. We use a recent procedure, the cluster treatment of characteristic roots (CTCR), for this purpose. This procedure results in an exact and exhaustive stability tableau in the domain of the three delays. (b) It provides a proof of a complex concept called the delay-stabilisability (also known as strong stability) as a by-product of CTCR. Furthermore, we deploy a numerical method (infinitesimal generator approach) to approximate the dominant characteristic roots of this class of systems, which concur with the stability outlook generated by CTCR.     

Complete stability robustness of third-order LTI multiple time-delay systems

A unique procedure is presented in this paper, for a complete stability robustness of the third-order LTI multiple time-delay systems (LTI-MTDS). The uniqueness of the treatment is simply due to the fact that there is no comparable methodology, presently, in the literature. The end result of this procedure is an exhaustive and precise determination of the stable regions in the domain of time delays. The backbone of the method is a novel framework called “the cluster treatment of characteristic roots, (CTCR)”. CTCR is constructed over two fundamental propositions. The first proposition claims the existence of a bounded number of so-called “kernel curves”, where the only imaginary characteristic roots occur. The second proposition is on an interesting directional invariance property of the crossing tendencies of these imaginary roots. For simplicity of conveyance and without loss of generality, the number of time delays is taken as two in this document. The new methodology is expandable to higher-order dynamics with more time delays than two, as the authors intend to demonstrate in future publications.     

考虑时变时滞的多移动智能体分布式编队控制

考虑到多移动智能体编队控制中的时变时滞问题,在所设计的编队框架下,基于一致性算法设计了具有不对称时变时滞的分布式编队控制律.该控制律仅使用全局速度导引信息和邻居状态反馈信息;在固定通信拓扑条件下,推导了具有时变时滞的闭环系统状态方程,应用改进的自由权矩阵方法获得了保守性更小的系统稳定条件,并在时变通信拓扑条件下,将拓扑变化处理为系统结构的不确定性,同样获得了时变通信拓扑下的系统稳定条件;进行了6个智能体在平面内编队运动的仿真,实例证明,理论结果是正确的.     

Guaranteed Cost Consensus for High-dimensional Multi-agent Systems With Time-varying Delays

Guaranteed cost consensus analysis and design problems for high-dimensional multi-agent systems with timevarying delays are investigated. The idea of guaranteed cost control is introduced into consensus problems for high-dimensional multi-agent systems with time-varying delays, where a cost function is defined based on state errors among neighboring agents and control inputs of all the agents. By the state space decomposition approach and the linear matrix inequality (LMI), sufficient conditions for guaranteed cost consensus and consensualization are given. Moreover, a guaranteed cost upper bound of the cost function is determined. It should be mentioned that these LMI criteria are dependent on the change rate of time delays and the maximum time delay, the guaranteed cost upper bound is only dependent on the maximum time delay but independent of the Laplacian matrix. Finally, numerical simulations are given to demonstrate theoretical results.      

Weighted predictor‐feedback formation control in local frames under time‐varying delays and switching topology

This article presents a novel control strategy based on predictor‐feedback delay compensation for multiagent systems to reach a prescribed target formation under unknown but bounded communication delays and switching communication topology. Both communication delays and network topology can be subjected to arbitrarily‐fast time variations. The key idea is to implement predictor‐feedback strategies using only relative measurements between agents expressed in each local agent's frame, with the aim to counteract the negative effect of time delays. Nevertheless, due to the decentralized nature of the control, the presence of time‐varying delays and switching communication topology, only partial delay compensation is possible. Despite this, we show that better performance can be achieved with our proposal with respect to nonpredictor control schemes by introducing a weighting factor for predictor‐feedback terms in the control law. Sufficient conditions based on Linear Matrix Inequalities for robust stability are also provided, which allow to easily design the controller parameters in order to maximize the speed of convergence. Finally, simulation results are provided to show the effectiveness of the proposed approach.