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PDE-based Dynamic Density Estimation for Large-scale Agent Systems

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Document pages: 6 pages

Abstract: Large-scale agent systems have foreseeable applications in the near future.Estimating their macroscopic density is critical for many density-basedoptimization and control tasks, such as sensor deployment and city trafficscheduling. In this paper, we study the problem of estimating their dynamicallyvarying probability density, given the agents individual dynamics (which canbe nonlinear and time-varying) and their states observed in real-time. Thedensity evolution is shown to satisfy a linear partial differential equationuniquely determined by the agents dynamics. We present a density filter whichtakes advantage of the system dynamics to gradually improve its estimation andis scalable to the agents population. Specifically, we use kernel densityestimators (KDE) to construct a noisy measurement and show that, when theagents population is large, the measurement noise is approximately``Gaussian . With this important property, infinite-dimensional Kalman filtersare used to design density filters. It turns out that the covariance ofmeasurement noise depends on the true density. This state-dependence makes itnecessary to approximate the covariance in the associated operator Riccatiequation, rendering the density filter suboptimal. The notion of input-to-statestability is used to prove that the performance of the suboptimal densityfilter remains close to the optimal one. Simulation results suggest that theproposed density filter is able to quickly recognize the underlying modes ofthe unknown density and automatically ignore outliers, and is robust todifferent choices of kernel bandwidth of KDE.

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