| This thesis presents a new model-based algorithm that computes predictive optimal controls on-line and in closed loop for broad classes of traditionally challenging nonlinear systems, including hybrid impulsive, underactuated, and state and control constrained systems. Rather than iteratively optimize finite horizon control sequences to minimize an objective, this thesis derives a closed-form expression for individual control actions (i.e., control values that can be applied for short duration) at each instant that optimally improve a tracking objective over a long time horizon. Under mild assumptions, the expression for optimal actions becomes a linear state feedback control law near equilibria that permits stability analysis and performance based parameter selection. Optimal actions are shown to reduce to LQ regulators with defined local stability properties as a special case. However, globally, derivations prove the optimal actions are solutions to a well-studied class of Tikhonov regularization problems and so inherit guarantees for existence and uniqueness.;By sequencing optimal actions on-line, the proposed control algorithm forms a (piecewise) continuous min-max constrained response to state that avoids the overhead typically required to impose control saturation constraints. Benchmark examples included in this thesis show that the approach can outperform both traditional optimal controllers and recently published, case-specific methods from literature in terms of tracking performance, and at computational speeds many orders of magnitude faster than traditionally achievable for nonlinear closed-loop predictive control. These results do not require seed trajectories and show that control synthesis based on optimal actions, which improve rather than directly minimize trajectory cost, can avoid local minima issues that can cause nonlinear trajectory optimization to converge to undesirable solutions. Further examples show the same approach controls both a high-dimensional continuous nonlinear system (a 56 dimensional, state and control constrained marionette model) and a hybrid model for dynamic locomotion (the spring-loaded inverted pendulum), based only on high-level models (e.g., a ROS URDF) and trajectory goals. These examples show the proposed algorithm automates control policy generation for very different systems and suggest it is likely these methods apply to a wide variety of robotics needs. |
