Minkowski algebra and Banach Contraction Principle in set invariance for linear discrete time systems

This paper re-examines the minimality of invariant sets for linear discrete time systems subject to bounded, additive, uncertainty in view of the theoretical framework reported by Artstein and Rakovic. The existence and uniqueness of the minimal invariant set are proven directly, under standard assumptions, by using the Banach fixed point theorem. The fundamentals of the Minkowski algebra and the Banach contraction principle are then utilized to provide characterization of a novel family of invariant sets. Members of this family can be, under modest assumptions, computed fairly efficiently. The unique feature of the characterized family is that it contains invariant sets that are characterized explicitly for any non-negative integer in strong contrast to the existing results in the literature. The corresponding computational considerations and a potential application are outlined. An illuminating example is also provided.