Optimal Control of Differentially Flat Systems is Surprisingly Easy
This addresses the problem of real-time optimal control for complex cyber-physical systems, offering a novel method that is incremental in improving efficiency for specific applications.
The paper tackles the complexity of solving optimal control problems for nonlinear cyber-physical systems by exploiting differential flatness to simplify Euler-Lagrange equations, eliminating numerical instabilities and enabling real-time trajectory generation. It demonstrates performance by generating a constrained optimal trajectory for a planar manipulator in 4.5 ms while respecting constraints.
As we move to increasingly complex cyber-physical systems (CPS), new approaches are needed to plan efficient state trajectories in real-time. In this paper, we propose an approach to significantly reduce the complexity of solving optimal control problems for a class of CPS with nonlinear dynamics. We exploit the property of differential flatness to simplify the Euler-Lagrange equations that arise during optimization, and this simplification eliminates the numerical instabilities that plague optimal control in general. We also present an explicit differential equation that describes the evolution of the optimal state trajectory, and we extend our results to consider both the unconstrained and constrained cases. Furthermore, we demonstrate the performance of our approach by generating the optimal trajectory for a planar manipulator with two revolute joints. We show in simulation that our approach is able to generate the constrained optimal trajectory in $4.5$ ms while respecting workspace constraints and switching between a `left' and `right' bend in the elbow joint.