A new sliding mode control approach is introduced in this work with the dedicated mathematical tools. A time-delay modification/approximation of sign function is proposed, and it is shown that by substituting this new "sign" realization in the conventional sliding mode algorithms the main advantages of the sliding mode tools are preserved (like rejection of matched disturbances and hyper-exponential convergence, i.e. the rate of convergence to the origin is much faster than any exponential (Polyakov, Efimov, Perruquetti, & Richard, 2014)), while the chattering is reduced. These results are illustrated and confirmed by numerical simulations for the first order sliding mode control and the super-twisting algorithm. (C) 2015 Elsevier Ltd. All rights reserved.