NSF-BSF: Real-Time Robust Estimation and Stochastic Control for Dynamic Systems with Additive Heavy-Tailed Uncertainties

钟形曲线，技术上称为高斯概率密度函数（pdf），一直是工程和金融算法的核心元素，处理数据并自动化所需的操作。然而，人们已经认识到，依赖高斯pdf可能过于简单化，因为工程、经济、生物、金融运动、地震、大气湍流等中的许多实际系统，高斯分布的概率密度函数描述得很差。结果表明，这些现象更好地描述了“重尾”的PDF。例如，在空中交通管制中，主动雷达测量动态环境中飞机的距离和方位。这些测量并不精确，其值存在不确定性或误差。这种不确定性不能用高斯概率密度函数很好地描述，因为钟形曲线远离峰值的部分，称为概率密度函数的尾部，远小于雷达数据所显示的;高斯钟形曲线被认为有一个轻的，快速（指数）衰减的尾部，而雷达数据被认为有一个重的尾部，与代数幂成反比衰减。目前，只有线性动态系统与加性高斯不确定性导致了递归和分析算法，允许听话的，实时的实施。工程文献中充满了这种算法的启发式变体。因此，需要一个新的严格的算法。我们新开发的递归和解析估计算法，基于一个非常重尾柯西pdf，是一个范式转变。由于柯西pdf尾部超出了其他现实的密度，估计器和控制器的基础上的柯西pdf假设是强大的未知的现实的物理密度。我们指的是统计意义上的鲁棒性，这意味着当面临离群值或无法解释的事件时，估计器实现了足够的性能，并且这些事件可能会出现大的测量误差，大的过程偏差，或由于动态模型的错误指定。数值实验证明了这种鲁棒性。由于极端数据被假设为可能的，柯西估计量在结构上是丰富的，因此在计算上比高斯估计量更密集。我们正在解决新的分析技术，使计算简化，并实现了通用图形处理单元上的柯西估计。我们的研究也集中在新的随机控制律。由于我们的估计是分析和递归的，新的随机成本标准可以制定，导致主机的新的随机控制器，一般来说，新的控制technology.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

The bell-shaped curve, known technically as the Gaussian probability density function (pdf), has been a central element in engineering and financial algorithms that process data and automate a desired operation. However, it has been well recognized that reliance on the Gaussian pdf can be overly simplistic, since many practical systems in engineering, economics, biology, financial movements, earthquakes, atmospheric turbulence, etc., are poorly described by Gaussian pdfs. It was demonstrated that those phenomena are better described by “heavy-tailed” pdfs. For example, in air traffic control, an active radar measures the distance and bearing of an aircraft in a dynamic environment. These measurements are not exact, having an uncertainty or error in their values. This uncertainty is not described well by the Gaussian pdf because the portion of the bell-shaped curve far from its peak, called the tail of the pdf, is far smaller than what the radar data would suggest; the Gaussian-shaped bell curve is known to have a light, rapidly (exponentially) decaying tail, while radar data is said to have a heavy tail, decaying inversely to an algebraic power. Currently, only linear dynamic systems with additive Gaussian uncertainties have resulted in a recursive and analytic algorithm that allows tractable, real-time implementations. The engineering literature is packed with heuristic variations of this algorithm. Hence, a new rigorous algorithm is needed.Our newly developed recursive and analytic estimation algorithm, based on a very heavy-tailed Cauchy pdf, is a paradigm shift. Since the Cauchy pdf tail over-bounds other realistic densities, estimators and controllers that are based on the Cauchy pdf are hypothesized to be robust to unknown realistic physical densities. We refer to robustness in the statistical sense, meaning that the estimator achieves adequate performance when faced with outliers or unexplained events, and where these events may arise either as large measurement errors, large process deviations, or due to misspecification of the dynamic model. Numerical experiments have demonstrated this robustness. Since extreme data is assumed likely, the Cauchy estimator is rich in structure and hence is computationally more intense than its Gaussian counterparts. We are addressing new analytic techniques to make the computation streamlined and have implemented this Cauchy estimator on general purpose graphical processing units. Our study also focuses on new stochastic control laws. Because our estimator is analytic and recursive, new stochastic cost criteria can be formulated, leading to a host of new stochastic controllers and, in general, new control technology.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.