On frequency-domain criterion of finite-time convergence of second-order sliding mode control algorithms

Transient processes in the systems controlled by the twisting second-order sliding mode (SOSM) control algorithm and certain generic SOSM given by the describing function are analyzed in the frequency domain. The analysis is based on the approximate describing function method. The relationship between the frequency response (Nyquist plot) of the plant, the shape of the negative reciprocal describing function of the controller, and the transient process convergence rate is investigated. A simple criterion of the existence of finite-time convergence is proposed. It is shown that the convergence rate in a system controlled by a SOSM controller 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.