Extension of harmonic balance principle and its application to analysis of convergence rate of second-order sliding mode control algorithms

The harmonic balance condition is revisited. The existing methodology of analysis of periodic motions is extended to analysis of vanishing oscillations of variable frequency. Transient processes in the systems controlled by second-order sliding mode (SOSM) algorithms are analyzed. A simple criterion of the existence of finite-time convergence is proposed. It is shown that the convergence rate depends on the angle between the high-frequency asymptote of the Nyquist plot of the plant and the low-amplitude asymptote of the negative reciprocal of the describing function of the controller, which is named the phase deficit.