State decomposition principle in set invariance theory for linear discreate time systems

A state decomposition principle in set invariance theory for linear discrete time systems is analyzed. The proposed approach allows for an appropriate use of the superposition principle, for linear time invariant discrete time systems, in the set invariance theory. It is shown how to obtain a characterization of a novel family of robust control/positively invariant sets, given a collection of robust positively invariant sets for the system with respect to relaxed state and control constraints. The corresponding feedback control strategy is generally a selection from an appropriately defined set valued map. It is shown how to define a point-valued selection that, is in general case, non-linear function of state. The potential benefits of the proposed method are illustrated by a simple and illuminating example.