Abstract
This research presents the integration, product, delay and inverse time operational matrices of Legendre wavelets with an arbitrary scaling parameter and illustrates how to design this parameter in order to improve their accuracy and capability in handling optimal control and analysis of time-delay systems. Using the presented Legendre wavelets, the piecewise delay operational matrix is derived to develop the applicability of Legendre wavelets in systems with piecewise constant time-delays or time-varying delays. With the aid of these matrices, the new Legendre wavelets method is applied on linear time-delay systems. The reliability and efficiency of the method are demonstrated by some numerical experiments.
Highlights
Wavelets as mathematical functions [1], when applied to time-delay systems, have advantages over orthogonal functions
We have shown the advantages of Chebyshev wavelets with scaling over orthogonal functions [5] in problems arising in such systems
With low order of approximation, the significant accuracy obtained and in applying arbitrary scaled Legendre wavelets (ASLWs) we have three options to increase the accuracy of the solution
Summary
Wavelets as mathematical functions [1], when applied to time-delay systems, have advantages over orthogonal functions. Their definition cannot be used to get other operational matrices like the piecewise delay operational matrix These are some practical limitations of CLW method in time-delay systems. To eliminate the source of error and to increase the applicability of Legendre wavelets, we use a flexible definition of Legendre wavelets and introduce the useful matrices which are required for the analysis and optimal control of general linear systems with delays and piecewise constant time delays.
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