  Lipschitz continuous sliding-mode controllers (LCSMC) are developed as the
integral of discontinuous SMC, producing control signals of finite slope.
Nevertheless, LCSMC still generate chattering in the presence of fast parasitic
dynamics. In this paper, an analysis of chattering in systems driven by LCSMC
is performed using the Harmonic Balance (HB) 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 switching variable (TSV). Predictions of the amplitude
and frequency of self-excited oscillations allowed to compute the average power
consumed by the controller, in order to maintain the trajectories into the real
sliding mode. A comparison of LCSMC with the Super-Twisting controller (STC),
which produce a continuous control signal with infinite slope, is performed.
Theoretical predictions and simulation results confirm that LCSMC may induce
fast-oscillations (chattering) of smaller amplitude and average power than
those ones caused by the STC. But, surprisingly, the chattering generated by
LSV-LCSMC could be smaller than that one caused by TSV-LCSMC, when the
actuators are fast enough. On the other hand, it tuns that if the sliding
dynamics of the LSV-LCSMC closed-loop is of similar speed as the actuators
dynamics, the system can loose even practical stability.
