(162d) Bridging the Gap Between Multigrid, Hierarchical, and Receding-Horizon Control

We analyze the structure of the Euler-Lagrange conditions of a lifted long-horizon optimal control problem. Here, lifting partitions the horizon into stages that are coupled using linear transition constraints (as is done in multiple shooting). The analysis reveals that the Euler-Lagrange conditions can be solved by using block Gauss-Seidel schemes and we prove that such schemes can be implemented by solving sequences of short-horizon optimal control problems. The analysis also reveals that a receding-horizon control scheme is equivalent to performing a single Gauss-Seidel sweep. Moreover, the analysis reveals that having optimal adjoint information at the transition constraints is sufficient for the receding-horizon scheme to recover the solution of the long-horizon problem. We use this observation to derive a strategy that computes approximate adjoints from a coarse long-horizon problem and we use this to correct the receding-horizon scheme. We observe that this strategy can be interpreted as a hierarchical control architecture in which a high-level controller transfers long-horizon information to a low-level, short-horizon controller. The information transfered in this hierarchy, however, is dual (adjoints) as opposed to primal (states); consequently, the proposed scheme differs from existing hierarchical control schemes. We demonstrate that the hierarchical scheme provides high quality approximate solutions of the original long-horizon optimal control problem. Moreover, this scheme can be fully implemented using off-the-shelf optimization solvers. We discuss how these new concepts are tightly connected to multigrid concepts arising in the control of distributed systems. We demonstrate our developments using optimal control problems arising in storage management for microgrids.