Linear Quadratic Path-Following via Online Trajectory Speed Optimization

Consider a linear system subject to stochastic disturbances and a path to be followed by a system's output. The path-following problem is posed here as choosing both the control input and the speed along the path to minimize the expected value of a quadratic function of the control input and of the error between the output and the resulting trajectory. The optimal control input policy for the deterministic version (no stochastic disturbances) is first provided and shown to be the sum of linear state feedback and path-dependent components, as for the twin linear quadratic trajectory-tracking problem. This policy is proven to also be optimal for the original stochastic problem when the path is a straight line. For general paths, it acts as a certainty equivalent policy that is shown to improve the cost of the optimal trajectory-tracking policy for any given trajectory, both when it can be exactly computed and when proposed approximate methods are used otherwise.