The stabilization problem of distributed proportional-integral-derivative ( PID ) controllers for general first-order multi-agent systems with time delay is investigated in the paper. The closed-loop multi-input multi-output ( MIMO ) framework in frequency domain is firstly introduced for the multi-agent system. Based on the matrix theory, the whole system is decoupled into several subsystems with respect to the eigenvalues of the Laplacian matrix. Considering that the eigenvalues may be complex numbers, the consensus problem of the multi-agent system is transformed into the stabilizing problem of all the subsystems with complex coefficients. For each subsystem with complex coefficients, the range of admissible proportional gains is analytically determined. Then, the stabilizing region in the space of integral gain and derivative gain for a given proportional gain value is also obtained in an analytical form. The entire stabilizing set can be determined by sweeping proportional gain in the allowable range. The proposed method is conducted for general first-order multi-agent systems under arbitrary topology including undirected and directed graph topology. Besides, the results in the paper provide the basis for the design of distributed PID controllers satisfying different performance criteria. The simulation examples are presented to check the validity of the proposed control strategy.

中文翻译：

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具有时滞的一般一阶多智能体系统的分布式PID控制器的稳定参数区域

研究了具有时滞的通用一阶多智能系统的分布式比例-积分-微分（PID）控制器的稳定性问题。首先针对多智能体系统引入了频域的闭环多输入多输出（MIMO）框架。基于矩阵理论，相对于拉普拉斯矩阵的特征值，整个系统被分解为几个子系统。考虑到特征值可能是复数，将多智能体系统的共识问题转化为所有具有复杂系数的子系统的稳定问题。对于具有复杂系数的每个子系统，可以通过分析确定可接受的比例增益范围。然后，对于给定的比例增益值，在积分增益和微分增益空间中的稳定区域也可以通过解析形式获得。可以通过在允许范围内扫描比例增益来确定整个稳定设置。所提出的方法是在任意拓扑（包括无向图和有向图拓扑）下对通用的一阶多主体系统进行的。此外，本文的结果为满足不同性能指标的分布式PID控制器的设计提供了依据。通过仿真实例验证了所提出控制策略的有效性。所提出的方法是在任意拓扑（包括无向图和有向图拓扑）下对通用的一阶多主体系统进行的。此外，本文的结果为满足不同性能指标的分布式PID控制器的设计提供了依据。通过仿真实例验证了所提出控制策略的有效性。所提出的方法是在任意拓扑（包括无向图和有向图拓扑）下对通用的一阶多主体系统进行的。此外，本文的结果为满足不同性能指标的分布式PID控制器的设计提供了依据。通过仿真实例验证了所提出控制策略的有效性。