Balanced active subspaces improve control of complex systems
Balanced Active Subspaces for Optimal Control via Trajectory Optimization
Machine Learning
Summary
Controlling complex systems with many variables is hard because traditional methods get too slow and complicated. The authors introduce Balanced Active Subspaces for Optimal Control (BASOC), a way to focus only on the most important directions that affect how well the system performs. They use data from optimal trajectories to simplify the problem, making it easier to design control strategies. Their method works better than some common techniques on tricky nonlinear systems.
What this means in practice
- •For aerospace engineers: Design real-time feedback controllers for complex flight conditions with reduced computational demands using BASOC-derived reduced models.
- •For fluid dynamicists: Simplify control of turbulent flow systems by focusing on key dynamics identified through balanced active subspaces for improved feedback strategies.
Authors
Emmanuel S. Ameh, Samuel E. Otto
Abstract
The curse of dimensionality associated with solving the Hamilton-Jacobi-Bellman equation for the value function remains a key obstacle for real-time optimal feedback control of high-dimensional nonlinear dynamical systems. This is especially severe in aerospace applications such as relaminarizing unsteady turbulent flows and recovering from aerodynamic stall where nonlinear dynamics play a significant role. We introduce Balanced Active Subspaces for Optimal Control (BASOC), a model reduction framework which seeks to find directions in state space along which the value function is most sensitive, and the distribution of states exhibits maximum variance along optimal trajectories. BASOC leverages trajectory optimization and the Pontryagin minimum principle to obtain state and co-state (value function gradient) data along locally optimal trajectories for constructing state and gradient covariance matrices. Balancing these matrices provides an oblique projection for approximating the value function, which can also be used to construct reduced-order models and surrogate feedback control laws. We show that for linear-quadratic optimal control problems, BASOC becomes equivalent to a snapshot method for initial condition balancing of the optimally controlled closed-loop realization. The framework is demonstrated on a nonlinear three-dimensional system where POD and balanced truncation are known to perform poorly, and on the nonlinear complex Ginzburg-Landau equation to demonstrate feasibility for larger-scale systems.