On functional approximation of the equivalent control using learning variable structure control

A learning variable structure control (LVSC) approach is originated to obtain the equivalent control of a general class of multiple-input-multiple-output (MIMO) variable structure systems under repeatable control tasks. LVSC synthesizes variable structure control (VSC) as the robust part which stabilizes the system, and learning control (LC) as the "plug-in" intelligent part which completely nullifies the effects of the matched uncertainties on tracking error. Rigorous proof based on energy function and functional analysis shows. that the tracking error sequence converges uniformly to zero, and that the bounded LC sequence converges to the equivalent control almost everywhere.