Optimal feedback control: Foundations, examples, and experimental results for a new approach

Typical optimal feedback controls are nonsmooth functions. Nonsmooth controls raise fundamental theoretical problems on the existence and uniqueness of state trajectories. Many of these problems are frequently addressed in control applications through the concept of a Filippov solution. In recent years, the simpler concept of a pi solution has emerged as a practical and powerful means to address these theoretical issues. In this paper, we advance the notion of Caratheodory-pi solutions that stem from the equivalence between closed-loop and feedback trajectories. In recognizing that feedback controls are not necessarily closed-form expressions, we develop a sampling theorem that indicates that the Lipschitz constant of the dynamics is a fundamental sampling frequency. These ideas lead to a new set of foundations for achieving feedback wherein optimality principles are interwoven to achieve stability and system performance, whereas the computation of optimal controls is at the level of first principles. We demonstrate these principles by way of pseudospectral methods because these techniques can generate Caratheodory-pi solutions at a sufficiently fast sampling rate even when implemented in a MATLAB (R) environment running on legacy computer hardware. To facilitate an exposition of the proposed ideas to a wide audience, we introduce the core principles only and relegate the intricate details to numerous recent references. These principles are then applied to generate pseudospectral feedback controls for the slew maneuvering of NPSAT1, a spacecraft conceived, designed, and built at the Naval Postgraduate School and scheduled to be launched in fall 2007.