Chattering analysis for Lipschitz continuous sliding-mode controllers

In this article, an analysis of chattering in systems driven by Lipschitz continuous sliding-mode controllers (LCSMC) is performed using the describing function approach. Two kinds of LCSMC are considered: the first one is based on a linear sliding variable (LSV) and the second one on a terminal sliding variable (TSV). Predictions of amplitude, frequency, and average power of self-excited oscillations, are used to compare such LCSMC respect to the supertwisting controller (STC) in systems with fast-actuators. Theoretical predictions and simulations allow the following conclusions: (i) LCSMC still may induce fast-oscillations (chattering) of smaller amplitude and average power, than ones caused by the STC in the absence of the measurement noises. (ii) The level of chattering with LSV-LCSMC could be smaller than one produced by TSV-LCSMC. (iii) The zero (sliding) dynamics of the LSV-LCSMC cannot be arbitrarily fast or the closed-loop system may lose even practical stability, unlike the TSV-LCSMC whose trajectories are finally bounded.