Quadratically saturated regulator for constrained unstable linear systems

The problem of stabilization and null-controllability of open loop unstable discrete time multi-input systems with constraints on the states and the controls is addressed in this paper. In earlier work the authors have analyzed such problems by using a Frobenius matrix norm. These results were quite conservative. This paper attacks the problem using the more appropriate Euclidean norm. First necessary and sufficient conditions for solvability of the problem are derived. They guarantee the existence of a linear controller leaving the state constraint set for the closed loop system positively invariant. An optimal control law is computed and the admissible set of initial conditions is given such that along trajectories of the closed loop system the state and input constraints are satisfied. Then the domain of feasible initial conditions is enlarged using a saturating control if such is feasible (as decidable with the necessary and sufficient conditions)