The Analytical Model of Six-Dimensional Linear Dynamic Systems with Arbitrary Piecewise-Constant Parameters

Abstract

In this paper the 6 × 6 fundamental matrix of a linear six - dimensional system with arbitrary piecewise-constant parameters is obtained in the analytical form in elementary functions for the first time. To write this matrix the new sign-function Fq(i,j) is introduced and described here. This function determines the order of interaction of eigen-oscillations of intervals with constant parameters in a resulting equivalent oscillation. The fundamentally new concept of equivalent oscillations for six-dimensional systems is introduced for the first time too. The stability problem of these systems is considered and the stability conditions are obtained in the analytical form in original parameters of a linear six-dimensional system. The results obtained allow solving a number of problems in mechanics, electrical engineering, communication systems, optoelectronics, automatic control theory and information theory. It can be, for example, the problems of determining the stability of dynamical systems, the problems of synthesizing dynamical systems of various nature, etc.