Synthesis and Research of Orthonormal Functions Based on Chebyshev–Legendre Polynomials

Abstract

The special characteristics of dynamic systems put forward new requirements for the creation of stable algorithms for describing the dynamic characteristics of these systems. Spectral methods for dynamic characteristics analyzing make it possible to create models that can be used to solve problems of identification and diagnostics of technical systems, for example, data channels, power lines, electric or hydraulic drives. Impulse response functions, correlation and autocorrelation functions are used as dynamic characteristics of systems. Spectral models are determined on the basis of the well-known Fourier integral in the basis of functions, the justification of which is also very important. The phased implementation of the transformation procedures, the normalization of the Chebyshev–Legendre polynomials made it possible to synthesize the transformed generalized orthonormal Chebyshev–Legendre functions that retain their properties on the argument interval [0, ∞]. These functions can be used to approximate the impulse response functions of dynamic systems. The research of the properties of the synthesized orthonormal functions made it possible to establish their recurrence formulas, which form the basis of computational procedures in spectral mathematical models. The obtained results allow ensuring the uniqueness of mathematical models, their connection with other operator models (for example, Laplace), stability in determining the parameters of models, the implementation of computational procedures and create universal algorithms for identification and diagnostics.