PDE-Based Dynamic Density Estimation for Large-Scale Agent Systems
PDE-Based Dynamic Density Estimation for Large-Scale Agent Systems
复制标题
基于偏微分方程的大规模代理系统动态密度估计
DOI：
10.1109/lcsys.2020.3004417
复制
发表时间：
2020
影响因子：
3
通讯作者：
Hai Lin
中科院分区：
文献类型：
--
作者：
Tongjia Zheng;Qing Han;Hai Lin
大规模代理系统在不久的将来会有可预见的应用。估计其宏观密度对于许多基于密度的优化和控制任务（例如传感器部署和城市交通调度）至关重要。在这封信中，我们研究了估计其动态变化的概率密度的问题，考虑到代理的个体动态（可以是非线性和时变的）以及实时观察到的状态。密度演化被证明满足由主体动力学唯一确定的线性偏微分方程。我们提出了一种密度过滤器，它利用系统动力学来逐渐改进其估计，并且可扩展到代理群体。具体来说，我们使用核密度估计器（KDE）构建噪声测量，并表明，当代理群体很大时，测量噪声近似为“高斯”。凭借这一重要特性，无限维卡尔曼滤波器被用来设计密度滤波器。事实证明，测量噪声的协方差取决于真实密度。这种状态依赖性使得有必要近似相关算子 Riccati 方程中的协方差，从而使密度滤波器不是最优的。输入状态稳定性的概念用于证明次优密度滤波器的性能仍然接近最优密度滤波器。仿真结果表明，所提出的密度滤波器能够快速识别未知密度的潜在模式并自动忽略异常值，并且对 KDE 内核带宽的不同选择具有鲁棒性。
Large-scale agent systems have foreseeable applications in the near future. Estimating their macroscopic density is critical for many density-based optimization and control tasks, such as sensor deployment and city traffic scheduling. In this letter, we study the problem of estimating their dynamically varying probability density, given the agents’ individual dynamics (which can be nonlinear and time-varying) and their states observed in real-time. The density evolution is shown to satisfy a linear partial differential equation uniquely determined by the agents’ dynamics. We present a density filter which takes advantage of the system dynamics to gradually improve its estimation and is scalable to the agents’ population. Specifically, we use kernel density estimators (KDE) to construct a noisy measurement and show that, when the agents’ population is large, the measurement noise is approximately “Gaussian”. With this important property, infinite-dimensional Kalman filters are used to design density filters. It turns out that the covariance of measurement noise depends on the true density. This state-dependence makes it necessary to approximate the covariance in the associated operator Riccati equation, rendering the density filter suboptimal. The notion of input-to-state stability is used to prove that the performance of the suboptimal density filter remains close to the optimal one. Simulation results suggest that the proposed density filter is able to quickly recognize the underlying modes of the unknown density and automatically ignore outliers, and is robust to different choices of kernel bandwidth of KDE.