INVESTIGATING THE SYMMETRIC SELF-OSCILLATIONS IN NONLINEAR TWO-DIMENSIONAL CONTROL SYSTEMS

Abstract

The task of investigation of self-oscillations in multivariable, i.e. having several inputs and several outputs, nonlinear feedback control systems is a complex task having at present no general solution. In engineering practice of development of nonlinear systems, the approximate method of harmonic linearization is used most often, which is based on taking into account only the first harmonic of periodic oscillations at the outputs of nonlinear elements.  The necessary conditions of self-oscillations’ existence in that case are written as a condition of being one of the characteristic systems of harmonically linearized multivariable system on the stability verge, and as a so-called collinearity condition of the vector of self-oscillations amplitudes and the corresponding axis of the canonical basis of the system. At that, the complexity of numerical calculations is growing sharply with an increase of the system dimension, that is with an increase of the number of the system channels.  A simple method of investigation based on the harmonic linearization and the characteristic transfer functions methods, of symmetric one-frequency self-oscillations in nonlinear two-dimensional control systems with identical transfer functions of separate channels and rigid cross-connections is proposed in the article. As opposed to the general approach, in which the determination of all vectors of self-oscillations is required, with the subsequent examination of the harmonic balance condition for the one-dimensional characteristic systems, the approach presented in the article is based on the determination of those values of amplitudes and frequencies of oscillations for which one of the two characteristic systems is on the stability verge and on the simple graphic test of the collinearity condition. At that, the procedure of investigation of self-oscillations is brought maximally close to the classic procedure of investigation of nonlinear systems with a single input and single output. A numerical example of self-oscillations’ investigation is given, as well as the results of dynamic simulation in the Simulink package environment that verify the calculated data are presented.