Sliding mode control of nonlinear uncertain systems with relative degree two and uncertain control direction

This note consider the problem of controlling a class of single-input nonlinear uncertain systems with relative degree two and unknown sign of the high-frequency gain. It is suggested a simple modified version of a second-order sliding mode control algorithm, with the modification being consisting of the introduction of an adaptive gain switching between the values +1 and -1. It is shown that after a finite number of switchings the adaptive gain settles to the correct value guaranteeing the finite time convergence of the system to the origin. The proposed methology can be applied to other 2-SMC algorithms, such as the twisting algorithm. Simulation results are provided to show the effectiveness of the proposed method.