Quasi-continuous high-order sliding-mode controllers

A universal finite-time-convergent controller is developed capable to control the output a of any uncertain SISO system of a known permanent relative degree r. The mode sigma equivalent to 0 (r-sliding mode) is established by means of a control dependent only on sigma, sigma, ..., sigma((r-1)) and continuous everywhere except the set sigma = sigma = ... = sigma((r-1)) = 0. An output-feedback controller version is also developed. With 6 being the tracking deviation, the exact finite-time-convergent output tracking is provided in the absence of output noises, otherwise the tracking accuracy is proportional to the magnitude of the noise. In the latter case sigma equivalent to 0 is not attained producing an unswitched control continuous in time. The resulting performance is significantly improved compared with known r-sliding controllers.