Regulation of discrete-time linear systems with positive state and control constraints and bounded disturbances

A variety of control problems require the control action and/or state to be positive. Typical applications include situations where the operating point maximizes (steady state) efficiency so that the steady state control and/or the steady state itself lie on the boundaries of their respective constraint sets. Any deviation of the control and/or state from its steady state value must therefore be directed to the interior of its constraint set. We characterize a novel family of the robustly controlled invariant sets for linear systems under positivity constraints. The existence of a constraint admissible member of this family can be checked by solving a single linear or quadratic programming problem. The solution of this optimization problem yields the corresponding controller. These results are then used to devise a robust time-optimal controller.