Blending Neural Control Density Functions for Stabilization and Safety

Recent work on Neural Network-based methods for nonlinear control use Lyapunov Functions to obtain controllers with guarantees of stability. However, Lyapunov-based methods are fundamentally limited: they cannot be used for smooth blending with formal Region of Attraction (RoA) expansion guarantees, and also fail to certify stability when unstable equilibria or saddle points are present. Density functions provide an alternate stability certificate, and address these limitations by certifying almost everywhere stability, and enable smooth blending of controllers. Learning valid density certificates is challenging due to integrability constraints, and the effect of density-based blending controllers on RoAs is not well understood. In this work, we provide the first guarantee that controllers blended with density functions yield RoAs containing the union of the RoAs achieved by the constituent controllers. Then, we propose a novel exponential characterization of density functions that provably satisfies the integrability condition, and introduce Neural Control Density Functions (NCDFs), that leverage this new parameterization. We also extend NCDFs for synthesizing safe-stable controllers by combining NCDFs with control barrier functions (NCDF-CBFs). Our experiments show that blended controllers obtain superior RoAs to state-of-the-art methods like Neural Lyapunov Control and Sum-of-Squares based techniques.

Robots and other automated systems need controllers that reliably guide them back to a safe, steady state - like keeping a drone hovering or a vehicle on its path. Typically, we prove a controller works using "Lyapunov functions," but these have a crucial limitation: two "good" controllers can't be blended together into a stronger one. They also do not work when a system has multiple stationary points. In this work, we use a different mathematical tool called a density function, which tracks where trajectories tend to flow rather than measuring energy. Density functions can be smoothly mixed, so several controllers can be combined into one. We provide the first mathematical proof that this blended controller works at least as well as all the originals put together, and we built neural networks that automatically learn both the controller and its safety guarantee. This potentially gives engineers a reliable way to combine controllers and expand the range of conditions/set points a system can achieve, while also keeping it away from unsafe states. In our tests, blended controllers handled a wider set of situations than today's leading methods.