Robust Estimation and Control of Dynamic Systems Experiencing Large Random Outliers

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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.