Robust Estimation and Control of Dynamic Systems Experiencing Large Random Outliers

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。不幸的是，高斯函数有很大的局限性。例如，在空中交通管制中，在动态环境中与飞机的距离和方位是由主动雷达测量的。这种测量不精确，其值有不确定度或误差。这种不确定性没有被高斯概率分布曲线很好地描述，因为钟形曲线远离零的部分，称为概率分布曲线的尾部，远远小于雷达数据所显示的：高斯钟形曲线已知有一个轻的、快速衰减的尾部，而雷达数据据说有一个重的尾部。众所周知，依赖高斯pdf可能是危险的，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不能很好地描述，而用重尾pdf可以更好地描述。然而，目前大多数数据处理算法都基于高斯pdf假设，主要是因为它易于处理和实时实现。新提出的理论是一种范式转换，它提出了基于重尾柯西和拉普拉斯pdf的新的递归和解析算法。虽然没有物理过程是明确的柯西或拉普拉斯分布，因为它们的尾巴超过了其他实际密度，基于柯西或拉普拉斯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。这种鲁棒性尤其适用于尾部非常重的柯西pdf。鲁棒性在统计意义上意味着，当面对异常值或无法解释的事件时，估计器可以达到足够的性能，并且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。这些新的估计器有一个自适应的方面，而不是在目前常用的算法中发现的。由于极端数据是可能的，所以估计器结构丰富，因此计算强度比高斯估计器高。本研究的主要目标是通过揭示其基本属性和构建测量稳定性和鲁棒性的度量来确定鲁棒性、可实现性、实时性、估计器和随机控制器。因此，这些算法可以在诸如图形处理单元的计算硬件上实现。一类新的鲁棒的、可实现的、实时的、矢量状态的、估计器和随机控制器，用于具有加性重尾柯西过程和测量噪声的线性动态系统。该含加性柯西噪声的矢量-状态线性动力系统的估计方法仅通过对给定测量历史的状态的非归一化条件概率密度函数（ucpdf）的特征函数进行递推更新和传播来实现。在上一个授权期间，我们注意到类似的算法可以适用于具有加性拉普拉斯噪声的线性系统，其中ucpdf直接使用解析和递归关系确定。这两种算法由于其丰富的解析结构而具有显著的数值复杂性。实现实时矢量状态估计器和随机控制器的主要目标是确定将保留柯西函数和拉普拉斯函数的ucpdf特征函数的基本结构的近似，这些近似被证明是收敛的，性能误差可以忽略不计。这项研究是在两国科学基金会（BSF）资助下与以色列理工学院的一位同事一起完成的。这项国际合作将在NSF/ENG/ECCS-BSF和BSF的资助下继续进行。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. Unfortunately, the Gaussian is quite limiting. For example, in air traffic control the distance and bearing to an aircraft in a dynamic environment is measured by active radar. This measurement is not exact, having an uncertainty or error in its value. This uncertainty is not described well by the Gaussian pdf because the portion of the bell shaped curve far from zero, called the tail of the pdf, is far smaller than the radar data suggests: the Gaussian bell shaped curve is known to have a light, rapidly decaying tail, while radar data is said to have a heavy tail. It has been well recognized that reliance on the Gaussian pdf can be dangerous, since many practical systems in engineering, economics, biology, financial movements, earthquakes, atmospheric turbulence, etc., are poorly described by Gaussian pdfs and better described by heavy tailed ones. However, the majority of current data processing algorithms are based on the Gaussian pdf assumption mainly because it leads to tractable, real-time implementations. The newly proposed theory is a paradigm shift, which proposes new recursive and analytic algorithms based on the very heavy tailed Cauchy and Laplace pdfs. Although no physical process is explicitly Cauchy or Laplace distributed, since their tails over bound other realistic densities, estimators and controllers that are based on the Cauchy or Laplace pdfs are hypothesized to be robust to unknown realistic physical densities. This robustness is especially true for the Cauchy pdf which has a very heavy tail. Robustness is meant in the statistical sense, where 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. There is an adaptive aspect to these new estimators not found in the algorithms commonly used today. Since extreme data is likely, the estimator is rich in structure and hence is computationally more intense than its Gaussian counterparts. The primary goal of the proposed study is to determine robust, implementable, real-time, estimators and stochastic controllers by uncovering their fundamental properties and constructing metrics that measure stability and robustness. Thereby, these algorithms can be realized on computational hardware, such as graphic processing units. A new class of robust, implementable, real-time, vector-state, estimators and stochastic controllers for linear dynamic systems with additive heavy-tailed Cauchy process and measurement noises are to be further developed. The estimation methodology for this vector-state, linear dynamic system with additive Cauchy noises was realized only by developing a recursion for the analytic measurement update and propagation of the character function of the unnormalized conditional probability density function (ucpdf) of the state given the measurement history. Over the last grant period, we noticed that a similar algorithm could be adapted to linear systems with additive Laplace noises, in which the ucpdf is determined directly using analytic and recursive relations. Both of these algorithms entail significant numerical complexities due to their rich analytic structure. The primary goal for the implementation of real-time vector-state estimators and stochastic controllers is to determine approximations that will conserve the basic structure of the character function of the Cauchy and the ucpdf of the Laplacian, which are shown to be convergent with negligible performance error. This study was performed with a colleague from the Technion in Israel under a Bi-national Science Foundation (BSF) Grant. This international collaboration will continue under the NSF/ENG/ECCS-BSF and BSF grants.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.