Elements of physical oscillation and control theory

Abstract

Parametric oscillation conditions for periodically timevariant system and stability failure of periodic motion of nonlinear dynamic system are derived within the monofrequency approximation. Graphical illustration and physically based explanation for oscillation conditions are given. Conditions for two-frequency combined oscillation of s u m and difference resonance phenomenon and high frequency synchronization are obtained. A problem of nonlinear system robust control avoiding one-frequency parametric resonances is set up.

Parametric and Coordinate Control of Oscillating Systems: Physics-Based Oscillation Feedback Design

“…The characteristics of this system are discussed in [19]. Similar results for second order systems can be found in the articles [20] and [21] and were received independently from [16][17][18].…”

“…The results of [9] are rather experimental and do not have generalization to higher order systems. The series of articles starting from [17] show several extensions for higher order systems. The parametrically controlled system with periodical time-varying parameters can be analyzed as linear time invariant system after some trigonometric approximations.…”

Frequency analysis of parametrically controlled oscillating systems

“…According to Chechurin and Chechurin (2003), when the time-varying gain in the feedback is sinusoidal, the controlled parameter may be approximated by linear gain and the timedelay element with the following transfer function ,…”

“…where is the frequency response function of the LTI oscillating object, is the frequency response approximation for the time-varying element (or the circumference of the first parametric resonance), is the amplitude of the time-varying parameter, and is its phase shift (see Chechurin, 2003 for details). The conditions of type (2) can define the instability region in coordinates "parameter varying magnitude" and "parameter varying frequency" for the first, second, etc.…”

“…However, the (stabilizing) properties of systems with a time-varying parameter and time-delay element in the same feedback are still left unattended. Here we suggest a method of analysis known as stationarization (Chechurin and Chechurin, 2003), which is based on harmonic analysis of the output of the time-varying parameter and the assumption that the higher frequencies are filtered out by the linear part of the system.…”