A simple LP formulation of the continuous-time positive invariance condition and its application to the constrained regulator problem

A system is said to be positively invariant with respect to a set S if each trajectory starting from S stays in S. A novel, brief proof of the algebraic positive invariance (PI) condition is given using the discussion in the dual space. A linear continuous-time time-invariant system with the initial state set being a convex polytope is considered. It is shown that the conventional PI condition has a lot of redundancy in general. The authors derive a reduced-order LP without redundancy and apply this result to solving the linear constrained regulator problem. An algorithm for computing an optimal feedback gain is proposed using an explicit LP condition of a feedback gain satisfying the design criteria