Relatively optimal control: the static solution

A relatively optimal control is a stabilizing controller that, without initialization nor feedforwarding and tracking the optimal trajectory, produces the optimal (constrained) behavior for the nominal initial condition of the plant. In a previous work, a linear dynamic relatively optimal control, for discrete--time linear systems, was presented. Here a static solution is shown, namely a dead--beat piecewise affine state--feedback controller based on a suitable partition of the state space into polyhedral sets. The vertices of the polyhedrons are the states of the optimal trajectory, hence a bound for the complexity of the controller is known in advance. It is also shown how to obtain a controller that is not dead--beat by removing the zero terminal constraint while guaranteeing stability. Finally, the proposed static compensator is compared with the dynamic one.