Linear Polytopic Positive Invariance

This article revisits verification of positive invariance for polytopic sets with respect to linear dynamics. The existing necessary and sufficient conditions are applicable only to fixed candidate polytopic sets. This article derives alternative necessary and sufficient conditions for two main representations of polytopic sets. These novel necessary and sufficient conditions are derived by utilizing Minkowski function in primal setting and support function in dual setting. Developed results enable direct positive invariance verification of fixed candidate polytopic sets, as well as enhanced positive invariance verification allowing for a structurally constrained redesign of candidate polytopic sets. The established necessary and sufficient conditions admit algebraic reformulations as feasibility checks of numerically appealing systems of affine inequalities and equalities indicating strongly their high numerical potential.