Filtro de Kalman-Bucy
1. Introducción
Es de interés en el mundo del control automático poder modelizar un proceso y que en dicho modelo queden reflejadas, explícitamente todas las todas las variables que intervienen en su dinámica. La forma más popular que cumple con esta condición es la llamada representación en variables de estado. Otra inquietud que surge una vez obtenido este modelo es el poder obtener información de la dinámica del proceso sin necesidad de medir todas las variables, sino que, haciendo uso del conocimiento de su dinámica poder inferir u observar algunas de ellas. El tercer problema surge cuando intentamos utilizar este modelo con mediciones contaminado por algún tipo de incertidumbre. Estos tres conceptos se abordan con la llamada estimación estadística de señales y su versión más conocida es el Filtro de Kalman-Bucy que pretendemos abordar en este espacio.
El tratamiento de este tópico está ligado a dos aspectos del control: por un lado a la observación o estimación de estados internos de un sistema y por otro a esta misma observación cuando las mediciones están contaminadas por algún tipo de perturbación. Dividiremos el trabajo en tres partes: la primera será una revisión de la representación de un proceso mediante variables de estado y su observación. La segunda parte estará dedicada a mostrar la estimación de señales obtenidas con un error y por último presentaremos la versión clásica del filtrado estadístico.
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An Introduction to the Kalman Filter
Greg Welch, Gary Bishop
1. Introduction
The Kalman filter is a mathematical power tool that is playing an increasingly important role in computer graphics as we include sensing of the real world in our systems. The good news is you don’t have to be a mathematical genius to understand and effectively use Kalman filters. This tutorial is designed to provide developers of graphical systems with a basic understanding of this important mathematical tool.
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El filtro de Kalman
Alvaro Solera Ramirez
1. INTRODUCCIÓN
El propósito de este documento es proveer una introducción al filtro de Kalman y establecer la relación entre éste y la representación en la forma de estado-espacio. La importancia de estudiar el algoritmo de Kalman radica en que se constituye en el principal procedimiento para estimar sistemas dinámicos representados en la forma de estado-espacio (State-Space), los cuales tienen muchas aplicaciones econométricas de interés.
El filtro tiene su origen en el documento de Kalman (1960) donde describe una solución
recursiva para el problema del filtrado lineal de datos discretos. La derivación de Kalman fue dentro de un amplio contexto de modelos estado-espacio, en donde el núcleo es la estimación por medio de mínimos cuadrados recursivos. Desde ese momento, debido en gran parte al avance en el cálculo digital, el filtro de Kalman ha sido objeto de una extensiva investigación y aplicación, particularmente en el área de la navegación autónoma y asistida, en rastreo de misiles y en economía.
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Using nonlinear Kalman filtering to estimate signals
Dan Simon
The Kalman filter is a tool that estimates the variables of a wide range of processes. In
mathematical terms we'd say that a Kalman filter estimates the states of a linear system.
There are two reasons you might want to know the states of a system, whether linear or
nonlinear:
· First, you might need to estimate states in order to control the system. For example, the electrical engineer needs to estimate the winding currents of a motor in order to control its position. The aerospace engineer needs to estimate the velocity of a satellite in order to control its orbit. The biomedical engineer needs to estimate blood-sugar levels in order to regulate insulin injection rates.
· Second, you might need to estimate system states because they're interesting in their own right. For example, the electrical engineer needs to estimate powersystem parameters in order to predict failure probabilities. The aerospace engineer needs to estimate satellite position in order to intelligently schedule future satellite activities. The biomedical engineer needs to estimate bloodprotein levels to evaluate the health of a patient.
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An Adaptive Speed Estimator For Induction Motors Based On a Kalman Filter With Low Sample Time
J.L. Mora, A. Torralba, L.G. Franquelo
Nowadays, electrical drives play an important role as the electromechanical energy converters in transportation and most production processes. In the monitoring and control systems of rotary machines it is essential that rotor position and velocity measurements are known. Also, knowledge of these variables is indispensable in parameter estimation algorithms, electric or thermal modelling and in vector control techniques.
The classical way to known the speed of an induction motor is by measurement using an optical tachometer. The use of an incremental shaft encoder presents problems, especially in hostile environments, and, in addition to this, the encoder is a cost factor since special motor-shaft extensions and encoder mounting surfaces are obligatory.
One way to overcome these problems is to estimate the speed. Several speed estimators have been presented in the last years that determines speed by measuring stator voltages and currents. Between them, those based on speed adaptive flux observers and model reference adaptive systems [8], [9],[IO] have shown a good performance, but fail in the low speed range.
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Kalman filtering
DAN SIMON
Filtering is desirable in many situations in engineering and embedded systems. For example, radio communication signals are corrupted with noise. A good filtering algorithm can remove the noise from electromagnetic signals while retaining the useful information. Another example is power supply voltages. Uninterruptible power supplies are devices that filter line voltages in order to smooth out undesirable fluctuations that might otherwise shorten the lifespan of electrical devices such as computers and printers.
The Kalman filter is a tool that can estimate the variables of a wide range of processes. In mathematical terms we would say that a Kalman filter estimates the states of a linear system. The Kalman filter not only works well in
practice, but it is theoretically attractive because it can be shown that of all possible filters, it is the one that minimizes the variance of the estimation error. Kalman filters are often implemented in embedded control systems
because in order to control a process, you first need an accurate estimate of the process variables.
This article will tell you the basic concepts that you need to know to design and implement a Kalman filter. I will introduce the Kalman filter algorithm and we’ll look at the use of this filter to solve a vehicle navigation problem. In order to control the position of an automated vehicle, we first must have a reliable estimate of the vehicle’s present position. Kalman filtering provides a tool for obtaining that reliable estimate.
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A Tutorial on Dynamic Simulation of DC Motor and Implementation of Kalman Filter on a Floating Point DSP
Padmakumar S., Vivek Agarwal, and Kallol Roy
1. Intrroduction
THIS paper presents a tutorial on dynamic simulation of DC motor on a Digital Signal Processor (DSP). The 32 bit floating point DSP processors are readily available in the market and have high computing power and performance. This has helped in implementing some of the computationally complex observer algorithms such as Kalman filter which offers excellent estimates of states even in the presence of system and measurement noise. Sensor-less estimation of induction motor speed and GPS tracking are just a few examples, where these observer algorithms have been .
The objective of this paper is to acquaint a beginner with what all is involved in the design and implementation of such systems. As an example, a self excited fractional hp DC motor is modeled using Matlab® and the simulation results are presented for open loop time response of the state variables.
The measurements are then corrupted by super imposing a noise, which follow a Gaussian distribution. A Kalman filter is implemented in Matlab® to observe the states from the corrupted measurement signal. A comparison of the noisy measurement signal and the estimated states through Kalman Filter is presented. The same model is implemented in TI® floating point DSP C6713 and the results including the Kalman Filter responses are also presented. The Matlab® code for all the simulations and the Code Composer Studio® project
file including the required header files are available for download, from a link provided in Appendix-I.
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