On Convergence Rate of Second-Order Sliding Mode Control Algorithms

Transient processes in the systems controlled by second-order sliding mode (SOSM) algorithms are analyzed in the frequency domain. A methodology of the frequency-domain analysis of transient processes that feature a decaying oscillation of variable frequency is developed. 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 relay or 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 called the phase deficit. Examples of analysis are given.