Minimax sliding mode control with minimal-time reaching phase

An original linear time varying system with unmatched disturbances and uncertainties is replaced by a finite set of dynamic models, such that each one describes a particular uncertain case including exact realizations of possible dynamic equations as well as external bounded disturbances. Such a trade-off between an original uncertain linear time varying dynamic system and a corresponding higher order multi-model system with a complete knowledge leads to a linear multi-model system with known bounded disturbances. It is shown that for linear multi-model systems the optimal sliding surface is a hyperplane. During the pre-sliding motion the design of the control, minimizing the worst reaching time, is also obtained. The joint optimization problem is resolved using the suggested successive approximation technique based on the contraction mapping application.