Decomposition of the mini-max multimodel optimal problem via integral sliding mode control

An original linear time varying system with matched and 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 unmatched bounded disturbances. Such a trade-off between an original uncertain linear time varying dynamic system and a corresponding higher order multi model system containing only matched uncertainties leads to a linear multi-model system with known unmatched bounded disturbances and unknown matched disturbances as well. Each model from a given finite set is characterized by a quadratic performance index. The concept of integral sliding mode (ISM) permit to robustify the designed minimax control law starting from the beginning of the process. On the other hand, the equations for ISM dynamics has the same dimension that the dimension of the initial system equations. In order to reduce the dimension of the minimax control design the following steps revising ISM concept are made: the algorithm for correction of ISM dynamics; the correction of the LQ-index corresponding with the correction of the ISM dynamics. It allows to reduce the dimension of the minimax control design problem, to ensure the robustness of system trajectory with respect to matched uncertainties, to solve the minimax control design problem into the space of unmatched uncertainties only. Ilustrative numerical example concludes this study.