Artículo

An Identification Genetic Algorithm for a Family of Duffing's System

Un Algoritmo Genético de Identificación para la Familia del Sistema de Duffing

Fortunato Flores–Ando¹, Carlos Aguilar–Ibáñez¹, Miguel S. Suárez–Castañón¹ and Francisco Cuevas de la Rosa²

¹ Centro de Investigación en Computación del IPN Av. Juan de Dios Bátiz s/n Esq. con Manuel Othón de Mendizabal Col. San Pedro Zacatenco, A.P. 75476 07700 México, D.F., México phone: 52–5–7296000, ext–56568

² Centro de Investigación en Óptica, León, Guanajuato

Abstract

This paper shows a simple way to recover the whole unknown parameters set of the Duffing's oscillator by using a genetic algorithm. The fact that the system is observable and constructible with respect to a suitable output helps in obtaining an integral parameterization of the output. Subsequently an integral parameterization of the output which depends upon the unknown parameters, and, a random estimation of the output is proposed, assuming that the set of unknown parameters are contained into a bounded set. This random estimation is chosen provided that the error between the actual output and the estimated output minimizes the errors of a quadratic function. The minimization problem and the random estimations of the output are formulated directly in terms of a genetic algorithm. A population of chromosomes is codified with the parameters of the Duffing's oscillator system. A fitness function is established to evaluate the chromosomes, in such a way that it minimizes the errors of a quadratic function. The chromosomes' population evolves till a fitness average threshold is obtained. This method is numerically possible and easy to implement in a digital computer.

Keywords: Mechanical Oscillator, Chaos, Genetic Algorithms, Reconstruction.

Resumen

En este artículo se presenta una forma sencilla para estimar los parámetros desconocidos del oscilador de Duffing mediante el empleo de un algoritmo genético. El hecho de que el sistema es observable y construible con respecto a una salida disponible, ayuda a obtener una parametrización integral de la salida. A partir de esta parametrización se propone un estimador aleatorio de la salida, asumiendo que los parámetros desconocidos están contenidos en un conjunto acotado. El estimador aleatorio es propuesto de tal forma que el error entre la salida real y la salida estimada minimiza una función cuadrática. Así, el problema de minimización y del estimador aleatorio son resueltos mediante un algoritmo genético. La población de cromosomas es codificada con los parámetros del oscilador de Duffing. La función de adaptabilidad es establecida para evaluar los cromosomas, de tal forma que se minimice el error de la función cuadrática. Los cromosomas de la población evolucionan hasta que un umbral promedio de adaptabilidad es alcanzado. Este método es numéricamente posible y fácil de implantar en una computadora digital.

Palabras Clave: Oscilador Mecánico, Caos, Algoritmos Genéticos, Reconstrucción.

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Acknowledgments

This research was sponsored by CIC–IPN, and by the Coordinación de Posgrado e Investigación (CGPI del IPN), under Research Grant 20020247 and the Consejo Nacional de Ciencias y Tecnologia de Mexico. Also, the authors want to thank Dr. Humberto Sossa A.

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