High-order sliding mode controller design subject to lower-triangular nonlinearity and its application to robotic system

A common feature of the traditional high-order sliding mode (HOSM) control method is that the matched uncertainty of HOSM dynamics is enlarged by taking the high-order derivatives on the pre-selected sliding variable directly, which could cause much burden to the controller design. This deficiency is removed in this paper via introducing some nonlinear terms in the mismatched channels. On this basis, a new arbitrary-order HOSM dynamics with lower-triangular nonlinearities will be obtained. Then, a novel finite-time HOSM control approach an be developed by the aid of the mathematical tool named as adding a power integrator (API). Under the proposed HOSM algorithm, the mismatched uncertainties existing in the new HOSM dynamics are not required to be sufficiently smooth, but are needed to comply with a few homogeneous growth conditions. Moreover, the common constant boundary for the matched uncertainty in the majority of the available HOSM methods can be extended to a time-varying one. The rigorous proof is given based on the Lyapunov theory to show the finite-time stability of the considered closed-loop system. At last, an example of the robotic tracking system is adopted to validate the effectiveness of the developed finite-time HOSM control method.