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

钟形曲线，在技术上被称为高斯概率密度函数（pdf），一直是工程和金融算法处理数据和自动化所需操作的核心元素。然而，人们已经认识到，依赖高斯pdf可能过于简单化，因为在工程、经济学、生物学、金融运动、地震、大气湍流等许多实际系统中，高斯pdf都不能很好地描述。结果表明，用“重尾”pdf文件可以更好地描述这些现象。例如，在空中交通管制中，主动雷达测量飞机在动态环境中的距离和方位。这些测量并不精确，其值存在不确定性或误差。这种不确定性并没有被高斯pdf很好地描述，因为远离峰值的钟形曲线的一部分，称为pdf的尾部，远远小于雷达数据所显示的；众所周知，高斯形钟形曲线有一个轻的、迅速（指数）衰减的尾巴，而雷达数据据说有一个重的尾巴，以代数幂的反比衰减。目前，只有具有加性高斯不确定性的线性动态系统产生了可处理的、实时实现的递归和解析算法。工程文献中充斥着这种算法的启发式变体。因此，需要一种新的严格的算法。我们新开发的基于非常重尾柯西pdf的递归和分析估计算法是一种范式转变。由于柯西pdf尾部超过其他实际密度，基于柯西pdf的估计器和控制器被假设对未知的实际物理密度具有鲁棒性。我们指的是统计意义上的鲁棒性，这意味着当面对异常值或无法解释的事件时，以及这些事件可能由于大的测量误差、大的过程偏差或由于动态模型的错误说明而出现时，估计器实现了足够的性能。数值实验证明了这种鲁棒性。由于假设极端数据是可能的，柯西估计量结构丰富，因此计算强度比高斯估计量更大。我们正在研究新的分析技术，使计算流线化，并在通用图形处理单元上实现了柯西估计器。我们还研究了新的随机控制规律。由于我们的估计量是解析的和递归的，因此可以制定新的随机成本准则，从而产生许多新的随机控制器，总的来说，新的控制技术。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.