Approximate Relative Entropy Constraints for Nonlinear Covariance Steering Under Distribution Ambiguity
RoboticsInformation Theory
Summary
The authors address the problem of controlling systems that behave unpredictably, especially when those systems are nonlinear. Traditional methods approximate these systems as simpler, Gaussian ones, which can lead to errors in estimating risks like collision chances. To improve this, the authors create a method that considers uncertainty about how well the Gaussian approximation matches reality, using a measure called Kullback-Leibler divergence. They develop mathematical ways to keep the true system behavior close to the approximation while safely controlling risk, and they test this on a tricky spacecraft navigation problem.
Authors
Trevor N. Wolf, Jay W. McMahon
Abstract
Covariance steering provides an efficient framework for designing linear stochastic feedback policies, but its extension to nonlinear systems relies on a Gaussian surrogate obtained through local linearization. Because this surrogate may differ substantially from the true nonlinear state distribution, risk-sensitive quantities such as collision probability and mean-squared error may be inaccurately estimated. This work develops a distributionally robust covariance-steering framework based on the relative entropy, also known as the Kullback-Leibler divergence (KLD), to account for ambiguity in the propagated probability density function. Using a variational representation of exponential integrals, we derive computable upper bounds on risk-sensitive quantities over a KLD ambiguity set. We then formulate an upper bound on the time rate of change of the KLD between the true nonlinear distribution and a Gaussian reference surrogate. Under some assumptions, this bound is controlled by decision variables within a covariance-steering formulation. The resulting constraints are incorporated into a sequential convex programming algorithm to design stochastic guidance policies that keep the true distribution close to its Gaussian surrogate while enforcing bounds on risk-sensitive performance measures. The proposed approach is demonstrated on a challenging nonlinear spacecraft transfer between two near-rectilinear halo orbits.