Publication: Control de ecuaciones diferenciales estocásticas
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2020-07-16
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Abstract
El objetivo de este trabajo es presentar de forma autocontenida el Filtro de Kalman-Bucy (FKB), que permite identificar la mejor aproximación del estado de un sistema dinámico estocástico lineal mediante una observación parcial, en presencia de perturbaciones aleatorias. Primero, se abordan dos problemas de control determinista que guardan relación con el FKB, a saber, el Problema de Seguimiento y el Filtro de Kalman Determinista. Después, y debido a la naturaleza estocástica del FKB, se realiza una formalización de las Ecuaciones Diferenciales Estocásticas, que comprende el estudio del Movimiento Browniano, la Integral y el Lema de Itô, y el Teorema de Existencia y Unicidad de Soluciones, entre otros conceptos. Por último, se define y se caracteriza el Estimador de Kalman-Bucy, consiguiendo la mejor aproximación en el sentido de mínimos cuadrados (i.e. de mínima varianza), y se estudia y se simula un ejemplo de aplicación para ilustrar su funcionamiento.
The aim of this work is to present the Kalman-Bucy Filter (KBF) in a self-contained way, which allows to identify the best approximation of the state of a linear stochastic dynamic system from a partial observation, in the presence of random disturbances. First, two deterministic control problems that are related to the KBF, namely, the Tracking Problem and the Deterministic Kalman Filter, are studied. Later, and due to the stochastic nature of the KBF, we do a formalization of the Stochastic Differential Equations, which includes the study of the Brownian Motion, the Itô’s Integral and Lemma, and the Existence and Uniqueness of Solutions Theorem, among other concepts. Finally, the Kalman-Bucy Estimator is defined and characterized, giving the best approximation in the minimum square sense (i.e. minimal variance), and an application example is studied and simulated to illustrate its functioning.
The aim of this work is to present the Kalman-Bucy Filter (KBF) in a self-contained way, which allows to identify the best approximation of the state of a linear stochastic dynamic system from a partial observation, in the presence of random disturbances. First, two deterministic control problems that are related to the KBF, namely, the Tracking Problem and the Deterministic Kalman Filter, are studied. Later, and due to the stochastic nature of the KBF, we do a formalization of the Stochastic Differential Equations, which includes the study of the Brownian Motion, the Itô’s Integral and Lemma, and the Existence and Uniqueness of Solutions Theorem, among other concepts. Finally, the Kalman-Bucy Estimator is defined and characterized, giving the best approximation in the minimum square sense (i.e. minimal variance), and an application example is studied and simulated to illustrate its functioning.
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