Describing-function-based analysis to tune parameters of chattering reducing approximations of Sliding Mode controllers

Sliding Mode Control (SMC) is a powerful feedback control design technique leading to insensitivity to certain types of disturbances under ideal conditions. Among these conditions, there are discontinuity in the generated control signal and absence of unmodeled dynamics. In many cases, the former maybe not implementable while the latter is often not holdable. Such non-ideal conditions typically excite high-frequency oscillations, known as chattering, not only in the input to the plant but also in the output, leading to a degradation of performance and losing the attractive insensitivity property. One of well-known and widely used approaches to achieve chattering reduction, caused by delays, parasitic dynamics, unmatched perturbations, etc., is implicitly or explicitly approximating the discontinuity. Theoretically, removing the discontinuities in the input to the plant, as well as in the internal controller dynamics, leads to loosing the insensitivity property under ideal conditions, but may insure better performance under more realistic conditions. This paper presents a Describing Function analysis in frequency domain of some SMC algorithms under presence of unmodeled (parasitic) dynamics, in the case when the discontinuous sign function is replaced or approximated by a saturation function. We are to concentrate here on a systematic procedure to tune parameters of the approximation.