The Haar wavelets operational matrix of integration

Abstract

The Haar wavelets operational matrix of integration P is derived, which is similar to those previously derived for other types of orthogonal functions such as Walsh, block-pulse, Laguerre, Legendre and Chebyshev. A general procedure of forming this matrix P is summarized. This matrix P can be used to solve problems such as identification, analysis and optimal control, like that of the other orthogonal functions. However, in the process of solving practical problems, the different resolution bases can be selected on the local time domain but on the global time domain, in order to understand more clearly system behaviour on the local time domain. This advantage results from the property of Haar wavelet localization, and the other orthogonal functions do not have this property. A numerical example will be used to demonstrate this point.