基于代数黎卡提方法的高阶网络一致性的收敛速度研究

The continuous-time algebraic Riccati equation and the discrete-time modified algebraic Riccati equation with a parameter multiplied in the quadratic term play an important role in the consensus problem for the high-order linear multi-agent networks. This project plans to study the solutions of the two types of quadratic matrix algebraic equations, and apply the solutions to the analysis of the convergence rate of the network consensus. Specifically, the research contents include: (1) search for the explicit solutions and the numerical solutions of the positive semi-definite solutions of the algebraic Riccati equations and inequalities, analyze the relationship between the solution matrix and the closed-loop system matrix, and study the uniqueness, the positive semi-definiteness, and the maximality of the stabilizing solution of the discrete-time modified algebraic Riccati equation; (2) apply the Riccati method to systematically study the convergence rate of the high-order linear network consensus, especially of the sampled consensus and the event-based consensus; and (3) study the output consensus problem for the heterogeneous networks with directed graphs, combine the advantages of the sampled control and the event-triggered control, design the distributed control law using the event-based asynchronous communication and sampled control methods, study the necessary and sufficient conditions required on the sampling period, the event-trigger threshold, the heterogeneous dynamics, and the directed graph, and analyze the convergence rate of the output consensus. The research plan is devoted to speeding up the convergence rate of the network consensus, reducing the frequency of communication and control updating, and facilitating the application of the consensus algorithms in practical engineering networks such as sensor networks and the formation of multiple unmanned aerial vehicles.

连续时间代数黎卡提方程和二次项含参数的离散时间代数黎卡提方程在高阶线性多智能体网络的一致性问题中发挥着重要作用。本项目将研究这两类二次矩阵代数方程的解，并应用于一致性的收敛速度分析，研究内容包括：（1）对黎卡提方程和不等式的半正定解进行解析和数值求解，分析解矩阵与闭环系统矩阵的关系，并研究含参数的离散时间代数黎卡提方程的镇定解的唯一性、半正定性和最大性；（2）运用黎卡提方法研究高阶网络一致性的收敛速度，并具体分析采样一致性和事件触发一致性的收敛速度；（3）研究有向图异质网络的输出一致性问题，结合采样和事件触发两种机制，设计基于异步时序通信的分布式采样事件触发控制算法，研究采样周期、事件触发阈值、系统矩阵、网络有向图需要满足的一致性充分必要条件，并分析输出一致性的收敛速度。此研究旨在加快网络一致性的收敛速度，降低通信和控制更新频率，促进一致性算法在传感器网络和无人机编队等工程网络中的应用。

本项目研究代数黎卡提方程的解和网络一致性收敛速度问题。首先，研究了二次项含参数的离散时间代数黎卡提方程MARE与不等式MARI，分析了其参数关键值与不等式正定解的存在性之间的联系。针对可控单输入系统，研究了非齐次MARE的显示解，并分析了解的镇定性以及稳定增益参数的取值范围。其次，运用黎卡提方法研究了滑模控制设计，然后针对无人机编队网络控制问题，提出了基于事件触发机制的滑模编队控制设计和基于离散积分滑模的编队跟踪控制设计。针对有向图网络的一致性问题，提出了基于黎卡提方程和网络图矩阵不等式的有向图网络李雅普诺夫方法，分析高阶网络一致性收敛速度与网络拓扑以及系统动力学之间的关系。同时，该方法可以应用到有向图网络的事件触发一致性、切换拓扑一致性、输出一致性等问题，并分析一致性的收敛速度。