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

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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. 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.