Frequency-shaped sliding modes: analysis and experiments

A recent improvement over a motion control algorithm, sliding mode control with perturbation estimation, is considered in this paper. It enforces the nonlinear system to follow a desired trajectory despite the presence of modeling uncertainties and disturbances. This robustness feature is introduced by a control which may excite resonances of the system, especially those of high frequency. A complementary procedure, frequency shaping, is utilized in order to attenuate these controller introduced excitations. The combined control algorithm offers a couple of attractive features, such as suppression of high-frequency dynamics (including those which are unmodeled), as well as maintaining a desired level of tracking ability for the original uncertain system. The highlight of this work is to bring an equivalent conventional sliding surface to the frequency-shaped one. It is shown that such a strategy exists and can be found as a function of the initial conditions of the dynamics. This optimal conventional sliding surface is found by minimizing a quadratic cost over the trajectory tracking errors. The results of this study are experimentally verified using a single degree of freedom robot with a flexible appendage which represents the unmodeled dynamics.