[论文解读] Decentralized control of large vehicular formations: stability margin and sensitivity to external disturbances

He Hao, Prabir Barooah|arXiv (Cornell University)|Aug 5, 2011

Traffic control and management参考文献 30被引用 4

一句话总结

本文采用双积分器车辆模型，分析了基于有向树和无向信息图的大型车流队列的去中心化控制。研究发现，有向树结构具有与规模无关的稳定性裕度，但存在指数级误差增长；无向图的稳定性裕度衰减为$O(1/N)$，且$H_\infty$灵敏度为$O(N^3)$。非线性控制器可显著改善有向树结构的瞬态响应和扰动鲁棒性，但在无向图中效果甚微。

ABSTRACT

We study the stability and robustness of large-scale vehicular formations, in which each vehicle is modeled as a double-integrator. Two types of information graphs are considered: directed trees and undirected graphs. We prove stability of the formation with arbitrary number of vehicles for linear as well as a class of nonlinear controllers. In the case of linear control, we provide quantitative scaling laws of the stability margin and sensitivity to external disturbances (H-infinity norm) with respect to the number of vehicles $N$ in the formation. It is shown that the formation with directed tree graph achieves size-independent stability margin but suffers from high algebraic growth of initial errors. The stability margin in case of the undirected graph decays to 0 as at least $O(1/N)$. In addition, we show that the sensitivity to external disturbances in directed tree graphs is geometric in $k$ where $k <= N$ is the number of generations of the directed tree, while that of the undirected graph is only quadratic in $N$. In particular, for 1-D vehicular platoons, we obtain precise formulae for the H-infinity norm of the transfer function from the disturbances to the position errors. It is shown that the H-infinity norm scales as $O(α^N) (α>1)$ for predecessor-following architecture, but only as $O(N^3)$ for symmetric bidirectional architectures. For a class of nonlinear controllers, numerical simulations show that the transient response due to initial errors and sensitivity to external disturbances are improved considerably for the formation with directed tree graphs. However, by using the nonlinear controller considered, little improvement can be made for that with undirected graphs.

研究动机与目标

• 分析大规模车流队列在去中心化控制下的稳定性裕度与对外部扰动的敏感性。
• 比较有向树与无向信息图在稳定性与鲁棒性方面的性能差异。
• 评估非线性控制器在不同网络拓扑中对瞬态响应与扰动抑制的影响。
• 推导$H_\infty$范数与稳定性裕度随队列规模$N$变化的定量标度律。
• 探究非线性控制是否能够缓解线性控制在前向跟随与对称双向架构中的局限性。

提出的方法

• 将每辆车建模为基于相对位置与速度测量的双积分器系统，并采用去中心化控制。
• 利用李雅普诺夫函数分析稳定性，证明在扇区有界非线性条件下，一类非线性控制器具有全局指数稳定性。
• 推导一维队列中从扰动到位置误差的传递函数$H_\infty$范数的标度律。
• 利用谱分析与图论工具，表征有向树与无向图的稳定性裕度。
• 通过正弦与随机扰动的大量数值仿真，评估瞬态性能与鲁棒性。
• 应用瑞利-里茨定理与配方法，建立李雅普诺夫导数的上界，确保指数稳定性。

实验结果

研究问题

• RQ1 在有向树与无向图拓扑中，稳定性裕度如何随车辆数$N$变化？
• RQ2 前向跟随与对称双向架构的$H_\infty$范数（对扰动的敏感度）的定量标度关系为何？
• RQ3 非线性控制器能否改善去中心化车流队列的瞬态响应与扰动抑制性能？
• RQ4 在非线性控制下，无向图中的性能退化是否依然存在，或可得到缓解？
• RQ5 信息流不对称性（如有向树）相较于对称双向架构，如何影响稳定性与鲁棒性？

主要发现

• 有向树图的稳定性裕度与规模无关，而无向图的稳定性裕度至少以$O(1/N)$衰减。
• 前向跟随（有向树）架构中，扰动的$H_\infty$范数随$O(\alpha^N)$增长，其中$\alpha > 1$，表明存在指数级增长。
• 对于对称双向（无向）架构，$H_\infty$范数仅以$O(N^3)$增长，表明其鲁棒性显著优于前向跟随结构。
• 非线性控制器在有向树图中显著改善了瞬态响应与扰动敏感性，但在无向图中改善甚微。
• 有向树中初始误差放大呈代数增长，而随代数$k$呈几何增长。
• 无向图的$H_\infty$范数被限制在$O(N^3)$以内，证实有界度图在扰动下导致$O(N)$的误差增长。