Kepler’s equation and limit cycles in a class of PWM feedback control systems

Abstract

The aim of this paper is to point out some new results concerning the ripple instability in the closed-loop control system using pulse width modulators (PWM), with natural sampling, as power amplifier. The presented analysis, based on the dual-input describing function method and the theoretical framework of Kepler's problem, shows an equivalence between the computation of switching instants of the PWM and the eccentric anomaly of the planet orbit around the sun, giving a simple stability criterion and a sufficient condition for the absence of solutions of the harmonic balance equation and, therefore, the probable absence of limit cycles of a period of a multiple of that characteristic of the modulator. The derived stability criterion, by using the describing function method, is successively compared with the local stability of the closed-loop PWM system for first-and second-order plants. In the first case it has been formally proved that the proposed criterion ensures the local stability of an equilibrium point, while in the second one a Monte Carlo simulation has confirmed that the selection of the modulator parameters, according to the proposed criterion, gives an effective method to avoid limit cycles and to ensure the local stability.

LVDT Conditioning on the LHC Collimators

“…An exponential model has been used to fit the standard deviations on the SNR: (12) Then the values of the parameters and have been fitted on the number of samples with the following model:…”

“…10. We applied square wave noise at 9 kHz and 18 kHz, since the LHC motor drivers work at a chopping frequency of 18 kHz and sub-harmonics may be present due to the PWM control; the main sub-harmonic involved is the half order [12]. The square waves have been generated with a SRS DS-360 frequency generator; to add the "noise" to the useful channel a Single Ended-Differential mixer has been built, since the output of the DS-360 is single ended while the LVDT's signals are differential.…”

On an integral representation of a class of Kapteyn (Fourier–Bessel) series: Kepler’s equation, radiation problems and Meissel’s expansion

“…Kepler's equation is probably the most famous transcendental equation in all branches of science and, although it has a simple form, for a long time an analytical solution was thought impossible to find. The relevance of Kapteyn series leads to a very large set of applications for which it is often helpful in obtaining explicit solutions: modern theory of electromagnetic radiation [3,4], Markov chains and queuing theory [5,6], atmospheric phenomena [7,8], spectral analysis of power conversion systems [9] and the analysis of ripple stability in Pulse-Width-modulated systems [10], just to name a few. It is well known that Kapteyn series often suffer from convergence problem.…”