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.
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Eisinberg, A., Fedele, G. & Frascino, D. Kepler’s equation and limit cycles in a class of PWM feedback control systems. Nonlinear Dyn 62, 215–227 (2010). https://doi.org/10.1007/s11071-010-9712-8
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DOI: https://doi.org/10.1007/s11071-010-9712-8