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。不幸的是，高斯是相当有限的。例如，在空中交通管制中，动态环境中飞机的距离和方位是由有源雷达测量的。这种测量是不准确的，其值有不确定度或误差。高斯pdf没有很好地描述这种不确定性，因为钟形曲线远离零的部分，称为pdf的尾部，比雷达数据显示的要小得多：众所周知，高斯钟形曲线有一个轻的、迅速衰减的尾巴，而雷达数据据说有一个厚重的尾巴。人们已经认识到，依赖高斯pdf可能是危险的，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用高斯pdf描述得很差，而用重尾pdf描述得更好。然而，目前的大多数数据处理算法都是基于高斯pdf假设的，这主要是因为它导致了易于处理的实时实现。新提出的理论是一种范式转换，它基于非常重尾的Cauchy和Laplace pdf提出了新的递归和分析算法。虽然没有明确的柯西或拉普拉斯分布的物理过程，但由于它们的尾部超过了其他实际密度的界，基于柯西或拉普拉斯分布的估计器和控制器被假设对未知的实际物理密度是稳健的。这种稳健性对于尾巴非常厚重的Cauchy pdf来说尤其如此。稳健性是指在统计意义上，估计器在面对离群值或无法解释的事件时获得足够的性能，并且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。这些新的估计器有一个自适应的方面，这在今天常用的算法中找不到。由于可能是极端数据，估计器的结构丰富，因此在计算上比高斯估计器更密集。所提出的研究的主要目标是通过揭示其基本性质并构建衡量稳定性和稳健性的度量来确定鲁棒的、可实现的、实时的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。具有加性重尾柯西过程和测量噪声的线性动态系统的一类新的鲁棒、可实现的实时矢量状态估值器和随机控制器有待进一步发展。对于这种具有加性柯西噪声的矢量状态线性动态系统的估计方法，只需开发一种递推方法，用于给定测量历史的状态的非归一化条件概率密度函数的特征函数的解析测量更新和传播。在上一个授权期内，我们注意到一个类似的算法可以适用于含有加性拉普拉斯噪声的线性系统，其中ucpdf直接使用解析和递推关系来确定。由于这两种算法都具有丰富的分析结构，因此需要大量的数值计算。实现实时向量状态估计器和随机控制器的主要目标是确定将保持柯西特征函数和拉普拉斯特征函数的基本结构的近似，它们被证明是收敛的，并且性能误差可以忽略不计。这项研究是与以色列理工学院的一位同事在两国科学基金会(BSF)的资助下进行的。这一国际合作将继续在NSF/ENG/ECCS-BSF和BSF的资助下进行。这一奖项反映了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. 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.