Relatively optimal control with characteristic polynomial assignment and output feedback

A relatively optimal control is a stabilizing controller such that, if initialized at its zero state, produces the optimal (constrained) behavior for the nominal initial condition of the plant (without feedforwarding and tracking the optimal trajectory). In this paper, we prove that a relatively optimal control can be obtained under quite general constraints and objective function, in particular without imposing 0-terminal constraints as previously done. The main result is that stability of the closed-loop system can be achieved by assigning an arbitrary closed-loop characteristic stable polynomial to the plant. An explicit solution is provided. We also show how to choose the characteristic polynomial in such a way that the constraints (which are enforced on a finite horizon) can be globally or ultimately satisfied (i.e., satisfied from a certain time on). We provide conditions to achieve strong stabilization (stabilization by means of a stable compensator) precisely, we show how to assign both compensator and closed-loop poles. We consider the output feedback problem, and we show that it can be successfully solved by means of a proper observer initialization (based on output measurements only). We discuss several applications of the technique and provide experimental results on a cart-pendulum system.