Relatively optimal control: A static piecewise-affine 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, for discrete-time linear systems, we presented a linear dynamic relatively optimal control. Here we provide a static solution, namely a deadbeat piecewise-affine state-feedback controller based on a suitable partition of the state space into polyhedral sets. The vertices of the polyhedra are the states of the optimal trajectory; hence a bound for the complexity of the controller is known in advance. We also show how to obtain a controller that is not deadbeat by removing the zero terminal constraint while guaranteeing stability. Finally, we compare the proposed static compensator with the dynamic one.