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Summary Professor Utkin proposed an example showing that the amplitude of chattering caused by the presence of parasitic dynamics (stable actuators) in some systems governed by the First‐Order Sliding‐Mode Controller is lower than that produced by the Super‐Twisting Algorithm. This example served to motivate this paper reconsidering the problem of comparison of chattering in systems with stable actuators, and driven by Discontinuous Sliding‐Mode Controllers (DSMCs) and Continuous Sliding‐Mode Controllers (CSMCs). Comparison of chattering produced by DSMC and CSMC taking into account their amplitudes, frequencies, and average power (AP) needed to maintain the system into real‐sliding modes, allowing to conclude the following: (i) for systems with slow actuators, the amplitude of oscillations and AP produced by DSMC be smaller than those caused by CSMC; (ii) for bounded disturbances with fixed Lipschitz constant, there exist sufficiently fast actuators for which the amplitude of oscillations and AP produced by CSMC be smaller than those caused by DSMC.

A describing function‐based design for the continuous sliding mode controller in the proportional integral derivative (PID) form (PID‐CSMC) is provided. Two sets of the PID‐CSMC gains are suggested. The Harmonic Balance is used to predict the amplitude and frequency of the main harmonic of chattering caused by the presence of fast‐parasitic dynamics in the closed‐loop. Predicted values of amplitude and frequency allow to compute the average power needed to maintain the trajectories of the system into real sliding modes. The methodology for selection of suboptimal PID‐CSMC gains consists of amplitude of chattering and average power minimization, taking into account the presence of a critically damped actuator parameterizing the effects of parasitic dynamics. A novel homogeneous Lyapunov function proves that suggested sets of the PID‐CSMC gains ensure, in theory, finite‐time stability of the perturbed double integrator without parasitic dynamics, and exact compensation of Lipschitz perturbations. On the other hand, the Loeb's criterion allows to ensure that the PID‐CSMC with suggested gains generates orbitally asymptotically stable fast‐oscillations in the presence of fast‐actuators.

Summary 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.

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