Thesis IDENTIFICACIÓN DE MODELOS DE ESTADO USANDO ALGORITMO EM CON RESTRICCIONES
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Date
2018
Authors
AVILA CARCAMO, FELIPE ELEAZAR
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Abstract
El objetivo principal de la Identificación de Sistemas es construir modelos matemáticosque representen sistemas reales a partir de datos medidos de ´este. La identificación de sistemasha sido ampliamente estudiada y existen distintos algoritmos que sirven para cumplirel objetivo de identificación en función de algún criterio de optimización.La presente Tesis tiene como objetivo estudiar la estimación de parámetros usando elalgoritmo EM que permite obtener el estimador de Máxima Verosimilitud para modelos delineales e invariantes en el tiempo cuando se imponen restricciones en la ubicación de lospolos del sistema. El estimador de máxima verosimilitud obtenido mediante el algoritmo EMes un estimador consistente y asintóticamente no sesgado, lo que implica que cuando existeuna cantidad limitada de datos (o baja relación señal a ruido) el modelo estimado puede noconservar características del sistema real, tales como estabilidad o comportamiento oscilatorio.Este problema es el que se intentar´a abordar mediante restricciones en la ubicaciónde lo autovalores de la matriz del sistema.Actualmente existen métodos de estimación con restricciones en las características delmodelo obtenido cuando se realiza estimación mediante métodos de subespacios. Estos métodosfueron estudiados y de ellos se destaca la idea del uso de regiones basadas en desigualdadeslineales matriciales. En esta Tesis se analiza el uso de estas desigualdades en el algoritmoEM.La implementación del algoritmo EM con las restricciones mencionadas presenta problemascomputacionales, por lo que en esta Tesis se presenta una sección donde se abordanestos problemas, mediante el uso del criterio de Sylvester para matrices reales y simétricasdefinidas positivas, y la aplicación de funciones de barrera logarítmicas para las restricciones,obteniendo así un algoritmo de rápida implementación.El algoritmo desarrollado se pone a prueba analizando estimaciones obtenidas y comparándolas con el algoritmo EM sin modificar para las mismas mediciones. Se analiza elimpacto de la cantidad de datos, relación señal a ruido a los mismos modelos con restriccionesaplicadas.
The main objective in System Identification is to obtain a mathematical model thatrepresents a real system from measured data. System Identification has been extensivelystudied. There are different algorithms that fulfills the objective based on different optimizationcriteria.The objective of this Thesis is to study parametric estimation using the EM algorithm,that obtains the Maximum Likelihood estimator for linear time invariant models whenlocation constraints are applied to the poles of the estimated model. The maximum likelihoodestimator obtained by the EM algorithm is consistent and asymptotically unbiased. Thisimplies that when a limited amount of data is available (or low signal-to-noise ratio) theestimated model may not preserve characteristics of the real system, such as stability oroscillatory behavior. This problem will be addressed by including constraints in the locationof the eigenvalues of the system matrix.There are estimation methods with constraints on the characteristics of the obtainedmodel using subspace methods. Some of this methods use the idea of regions based on linearmatrix inequalities. In this Thesis the application of constraints based on LMI region usingEM algorithm is analyzed.The implemention of the constrained EM algorithm presents computational problems, soin this Thesis we present this problem, and how to handle it, using the Sylvester Criterionfor real and symmetric positive defined matrices, and the application of log-barrier functionsfor the constraints.The proposed algorithm is tested by analyzing the obtained estimates and comparingthem with the unconstrained EM algorithm. The impact of the amount of data, and signalto-noise ratio is also studied.
The main objective in System Identification is to obtain a mathematical model thatrepresents a real system from measured data. System Identification has been extensivelystudied. There are different algorithms that fulfills the objective based on different optimizationcriteria.The objective of this Thesis is to study parametric estimation using the EM algorithm,that obtains the Maximum Likelihood estimator for linear time invariant models whenlocation constraints are applied to the poles of the estimated model. The maximum likelihoodestimator obtained by the EM algorithm is consistent and asymptotically unbiased. Thisimplies that when a limited amount of data is available (or low signal-to-noise ratio) theestimated model may not preserve characteristics of the real system, such as stability oroscillatory behavior. This problem will be addressed by including constraints in the locationof the eigenvalues of the system matrix.There are estimation methods with constraints on the characteristics of the obtainedmodel using subspace methods. Some of this methods use the idea of regions based on linearmatrix inequalities. In this Thesis the application of constraints based on LMI region usingEM algorithm is analyzed.The implemention of the constrained EM algorithm presents computational problems, soin this Thesis we present this problem, and how to handle it, using the Sylvester Criterionfor real and symmetric positive defined matrices, and the application of log-barrier functionsfor the constraints.The proposed algorithm is tested by analyzing the obtained estimates and comparingthem with the unconstrained EM algorithm. The impact of the amount of data, and signalto-noise ratio is also studied.
Description
Catalogado desde la version PDF de la tesis.
Keywords
ALGORITMO EM , IDENTIFICACION DE SISTEMAS , MAXIMA VEROSIMILITUD