On State Reconstruction for Linear Reduced-Order Output Feedback Optimal Control

The solution of optimal control problems is fundamental to numerous concepts, such as model predictive control. Despite recent advancements, solving these problems remains a challenge, particularly with complex systems involving multiple states. Employing reduced order models is a strategy to simplify these problems, but accurately estimating system states from output measurements continues to be difficult. We address the challenge of reconstructing the state of a linear dynamical system using measured input and output data, intending to use this reconstructed state in an optimal control framework, reducing the problem size. Our objective is to minimize the discrepancy between the optimal solution derived from the true system state and that from the reconstructed state. We approach the reconstruction problem through the lens of observability within a finite horizon, which allows us to confine our search to a subspace of the original state space containing only observable components. This confinement effectively yields a reduced-order system representation. We delineate conditions under which the reconstruction problem can be solved and demonstrate the practicality of our approach with a case study.