On invariant polyhedra of continuous-time systems subject to additive disturbances

This paper presents new necessary and sufficient algebraic conditions on the existence of positively D-invariant polyhedra of continuous-time linear systems subject to additive disturbances. In particular, for a convex unbounded polyhedron containing the origin in its interior, it is also shown that the positive D-invariance conditions can be split into two lower-dimensional sets of algebraic relations: the first corresponds to disturbance decoupling conditions and the second to positive D-invariance conditions for bounded polyhedra of a reduced-order system. The stability of the overall system is discussed as well. By exploring the results obtained, an LP approach is proposed for solution of a state-constrained regulator problem in the presence of additive disturbances.