NSF/ENG/ECCS-BSF: Vector-State Estimation and Control for Linear Systems with Additive Heavy-Tailed Distributions

NSF/ENG/ECCS-BSF：具有加性重尾分布的线性系统的矢量状态估计和控制

• 批准号: 1607502
• 负责人: Jason Speyer
• 金额: $ 36万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-15 至 2020-02-29
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1607502&HistoricalAwards=false
• 关键词: NSF; ENG; ECCS; BSF; Vector

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. For example, in air traffic control the distance and bearing to an aircraft in a dynamic environment is measured by an 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 (exponentially) 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 (reference: The Black Swan by Nessim Taleb). Many dynamic 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, 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 algorithms based on a heavy tailed pdf, known as the Cauchy pdf. The result is a more accurate and reliable automated system, which even for probabilistic models that are not Cauchy has demonstrated comparable performance to the Gaussian when the probabilistic environment is such. Since extreme data is likely, the estimator is rich in structure and hence is computationally more intense than its Gaussian counterparts.From a more technical viewpoint, a new class of 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 developed. The estimation methodology for this vector-state, linear dynamic system with additive Cauchy noises was addressed 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. Through a spectral transformation, the character function of the ucpdf is used explicitly in the development of stochastic controllers, based on a model predictive structure. These results entail significant analytical and numerical complexities due to the rich analytic form of the character function of the ucpdf, which produces a sum of terms that grows at each measurement update. The primary goal of the proposed study is to determine implementable real-time vector-state estimators and stochastic controllers by using simplifications that are due to the fundamental structure of the algorithms. Approximations that will conserve the basic structure of the character function are sought, and will be implemented on current computational hardware, such as graphic processing units. This work was performed with a colleague at the Technion under a Bi-national Science Foundation (BSF) Grant. This international collaboration will continue under the NSF/ENG/ECCS-BSF and BSF grants.

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。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有源雷达测量的。这种测量不精确，其值有不确定度或误差。这种不确定性不能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、迅速（指数）衰减的尾部，而雷达数据据说有一个重的尾部。人们已经充分认识到，依赖高斯pdf可能是危险的（参考：Nessim Taleb的《黑天鹅》）。在工程、经济、生物、金融运动、地震、大气湍流等领域的许多动态系统，用高斯pdf来描述是很差的，用重尾pdf来描述是更好的。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pdf的新算法，称为柯西pdf。结果是一个更准确和可靠的自动化系统，即使对于非柯西概率模型，在概率环境下也表现出与高斯模型相当的性能。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。从更技术性的角度来看，对于具有加性重尾柯西过程和测量噪声的线性动态系统，需要开发一类新的可实现的实时矢量状态估计器和随机控制器。通过发展一种递归的分析测量方法，对给定测量历史的状态的非归一化条件概率密度函数（ucpdf）的特征函数进行更新和传播，解决了具有加性柯西噪声的矢量状态线性动态系统的估计方法。通过谱变换，将ucpdf的特征函数明确地应用于基于模型预测结构的随机控制器的开发。由于ucpdf的特征函数的丰富解析形式，这些结果带来了显著的分析和数值复杂性，它产生了在每次测量更新时增长的项和。本研究的主要目标是通过简化算法的基本结构来确定可实现的实时矢量状态估计器和随机控制器。寻求将保留字符函数的基本结构的近似值，并将在当前的计算硬件（如图形处理单元）上实现。这项工作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。

项目成果

A Note on Hyper-Plane Arrangements in R^d

关于 R^d 中超平面排列的注释

• DOI: 10.47443/dml.2021.0048
• 发表时间: 2021
• 期刊: Discrete Mathematics Letters
• 影响因子: 0.8
• 作者: N. Duong, M. Idan

Properties of the Characteristic Function Generator of the Two-State Cauchy Estimator

二态柯西估计器特征函数发生器的性质

• DOI: 10.1137/18m1191580
• 发表时间: 2019
• 期刊: SIAM Journal on Control and Optimization
• 影响因子: 2.2
• 作者: Bai, Yu;Speyer, Jason L.;Idan, Moshe
• 通讯作者: Idan, Moshe