The Function Space to Describe the Dynamics of Linear Systems

V N Tibabishev

doi:10.5923/j.ajis.20120205.01

Abstract

There are two models of a random stationary process with a continuous spectrum of Wiener and the discrete spectrum of E. Slutsky. The first model of Wiener is often used. It is shown that this assumption leads to the absurdity of the equation Wiener-Hopf. The contradictions disappear in the theory of random stationary processes in the transition to a different model of a random process with discrete spectrum. Random ergodic stationary processes have on the mean and do not have the ergodicity of the variance and correlation functions. The struggle is based on the correlation functions of random processes with interference on the Wiener. Correct correlation functions cannot be obtained. Therefore, frequency analysis of random processes is proposed instead of the correlation analysis. The frequency analysis results in a more effective method of combating additive noise. The new method was used to identify the dynamic characteristics of the Airbus on the data obtained during the automatic landing.

Highlights

Restrict ourselves to the simp lest case, when a one-dimensional dynamical system is linear and stationary.Authors of many studies suggest that the observed input and output of the system such signals are stationary ergodic random processes[1]
A simple test of time-averaging, for examp le, to model the input signal x i (t) = C + xi (t), we find that a stationary random process with discrete spectrum and a non-zero expectation of C has an ergodic properties of the first order of expectation and centered stochastic process (7) does not have the ergodic property of second-order dispersion and correlation functions
When solving applied problems of correlation functions are found by averaging the realizations of the time

Summary

Introduction

Restrict ourselves to the simp lest case, when a one-dimensional dynamical system is linear and stationary. Authors of many studies suggest that the observed input and output of the system such signals are stationary ergodic random processes[1] In this cross-correlation function Ryx (τ) , the autocorrelation function Rxx (τ) the input signal x(t), output signal y(t) and the weight function k(t) are interconnected by an integral operator of convolution type defined in the Hilbert space of Lebesgue. Using (3) in (2), we find that well-known algorith m for solving applied problems for rando m stationary ergodic processes leads to the trivial case when the operator equation (1) Rxx (τ) = 0 and Ryx (τ) = 0 on the whole line Such a paradoxical situation arises for the reason that in dealing with the problem of representing signals in a linear t ime-invariant system have been used incorrect assumptions. It follows fro m the fact that the noise n(t) does not generate involuntary movements in the control system and the noise m(t) is not part of the forced motion

The Space of Functions for Linear Time-Invariant Operators

Frequency Analysis of Signals with Discrete Spectrum

Examples of Applied Problems

Conclusions