Linear programming approach to constrained feedback control

This paper deals with the problem of stabilizing linear discrete-time systems under state and control linear constraints using linear programming techniques. Linear state constraints describe a polyhedron in the state space so that the problem considered is to make such a polyhedron positively invariant while the control does not violate its constraints. For this, necessary and sufficient conditions are given for the existence of a solution of the problem in terms of polyhedron's vertices and directions. These conditions are described by a set of linear constraints and, following the approach introduced by Vassilaki et al., they can be solved using linear programming techniques. The objective function proposed here turns out to be a natural one when describing the constraints in terms of polyhedron's vertices and directions.