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)，一直是处理数据和自动化所需操作的工程和金融算法的核心元素。然而，人们已经很好地认识到，依赖高斯pdf可能过于简单，因为在工程、经济、生物、金融运动、地震、大气湍流等方面的许多实际系统，用高斯pdf描述得很差。事实证明，“重尾”pdf能更好地描述这些现象。例如，在空中交通管制中，有源雷达测量飞机在动态环境中的距离和方位。这些测量不准确，其值存在不确定性或误差。高斯pdf没有很好地描述这种不确定性，因为远离峰值的钟形曲线的部分，称为pdf的尾部，比雷达数据所显示的要小得多；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厚重的尾巴，与代数幂成反比衰减。目前，只有具有加性高斯不确定性的线性动态系统才产生了允许易于处理的实时实现的递归和解析算法。工程文献中充斥着该算法的启发式变体。因此，需要一种新的严格算法。我们新开发的基于重尾Cauchy pdf的递归和解析估计算法是一个范式的转变。由于Cauchy pdf尾部超出了其他真实密度的界限，因此假设基于Cauchy pdf的估计器和控制器对未知的真实物理密度是稳健的。我们指的是统计意义上的稳健性，这意味着估计器在面对异常值或无法解释的事件时，以及在这些事件可能作为大的测量误差、大的过程偏差或由于动态模型的错误指定而出现的情况下，获得足够的性能。数值实验证明了该方法的稳健性。由于假设可能存在极端数据，因此柯西估计器结构丰富，因此在计算上比高斯估计器更密集。我们正在解决新的分析技术，以使计算简化，并已在通用图形处理单元上实现了这个柯西估计器。我们的研究还集中在新的随机控制律上。由于我们的估计器是解析性和递归性的，因此可以制定新的随机成本标准，从而产生一系列新的随机控制器和一般的新控制技术。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.