⚡ electrical engineering

Approximate Relative Entropy Constraints for Nonlinear Covariance Steering Under Distribution Ambiguity

This paper proposes a distributionally robust covariance-steering framework that utilizes relative entropy constraints and computable upper bounds on risk-sensitive quantities to design stochastic guidance policies for nonlinear systems, ensuring the true state distribution remains close to a Gaussian surrogate while mitigating estimation errors caused by distribution ambiguity.

Original authors: Trevor N. Wolf, Jay W. McMahon

Published 2026-07-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Trevor N. Wolf, Jay W. McMahon

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to steer a spaceship through a chaotic, swirling storm of invisible particles. In the world of space travel, nothing is ever perfectly predictable. There are tiny errors in your engines, bumps from space dust, and the gravitational pull of planets that can tug your ship in unexpected ways. Because of this, engineers don't just plan a single, straight line for a rocket to follow; they plan for a "cloud" of possibilities. Think of this cloud as a fuzzy, shifting balloon representing where the ship might be at any given moment.

For simple, straight-line journeys, scientists have a handy trick: they assume this fuzzy balloon always stays round and smooth, like a perfect sphere. This is called a "Gaussian" distribution. It's a mathematical shortcut that makes the heavy lifting of steering calculations much easier. However, space is rarely simple. When a ship swings close to a moon or a planet, the gravity gets messy and non-linear, stretching that perfect sphere into weird, lopsided shapes. If you keep pretending the balloon is still a perfect sphere when it's actually a squashed potato, your calculations for safety might be wrong. You might think you're safe from a collision, but your "safe zone" is actually too small, and the real ship could wander outside the predicted area. This paper tackles the problem of how to steer a spaceship when that "perfect sphere" assumption breaks down, ensuring the ship stays safe even when the math gets messy.

The authors of this paper, Trevor N. Wolf and Jay W. McMahon, propose a new way to handle this uncertainty. They introduce a method called "distributionally robust covariance steering." Instead of blindly trusting the simplified, round balloon model, they create a safety net that accounts for how much the real, messy cloud of possibilities might differ from the perfect model. They use a mathematical tool called "relative entropy" (or Kullback–Leibler divergence) to measure the "distance" between the real, messy cloud and the perfect, round model.

Think of it like this: Imagine you are drawing a map of a city. Your "Gaussian" model is a perfect, grid-like map where every street is straight. But the real city has winding, curvy roads and dead ends. If you try to drive using only the grid map, you might get lost. The authors' method is like having a sensor that constantly checks how far off your grid map is from the real, curvy streets. They don't just guess how big the error might be; they calculate a strict upper limit on how much the real path can deviate from their simple model. They call this the "ambiguity set." By keeping this "distance" small, they can provide strong guarantees that even if the real world is messy, the ship won't wander too far from the plan.

The paper develops a framework to steer the spaceship while actively controlling this "distance." They show that by using a specific type of optimization algorithm (called Successive Convexification, or SCvx), they can adjust the ship's controls to keep the real, messy cloud of possibilities close to their simple, round model. They do this by creating new rules, or "constraints," that the computer must follow. These rules ensure that the risk of a collision or the error in the ship's final position stays within safe limits, even when the simple model isn't perfect.

To test their idea, the authors simulated a challenging journey: transferring a spacecraft between two complex orbits called Near-Rectilinear Halo Orbits (NRHOs) around the Earth and Moon. These orbits are notoriously tricky because the spacecraft swings very close to the Moon, where gravity is intense and non-linear. In their simulations, they compared their new method against the standard approach (which just assumes the cloud stays round).

The results showed that the standard approach struggled to keep the ship within its predicted safety zones. When the ship got close to the Moon, the real cloud of possibilities stretched out wildly, and many simulated ships ended up outside the 99% confidence bounds predicted by the simple model. However, when the authors applied their new relative entropy constraints, the guidance system successfully kept the real cloud of possibilities tight and close to the plan. In one specific test, they set a strict limit on how much the real cloud could differ from the model (a relative entropy bound of 1.0). Under this constraint, the ship's actual error stayed well below the safety threshold, whereas the standard method let the error grow too large.

The paper suggests that this approach offers a more principled way to design space missions. Instead of guessing how much "wiggle room" to give the safety margins, the math tells you exactly how to keep the real world in check. By controlling the rate at which the real distribution drifts away from the simple model, the system can provide upper bounds on risk-sensitive quantities—like the chance of hitting the Moon or missing the target—ensuring they remain within acceptable limits. While these results come from computer simulations of a specific Earth-Moon transfer, the authors argue that this framework provides a robust, mathematically sound way to navigate the messy, uncertain reality of space travel.

Technical Summary: Approximate Relative Entropy Constraints for Nonlinear Covariance Steering Under Distribution Ambiguity

Problem Statement
Covariance steering offers an efficient framework for designing linear stochastic feedback policies, particularly for linear Gaussian systems where the problem can be formulated as a semidefinite program. However, extending this framework to nonlinear systems typically relies on local linearization around a nominal trajectory, assuming the state distribution remains Gaussian. This Gaussian surrogate often diverges substantially from the true nonlinear state distribution, leading to inaccurate estimations of risk-sensitive quantities such as collision probability and mean-squared error (MSE). While distributionally robust control techniques exist to handle uncertainty in the probability density function (PDF) itself, a principled method for selecting the radius of the ambiguity set (the bound on statistical distance) remains a challenge. This work addresses the need for a robust guidance framework that accounts for distribution ambiguity in nonlinear spacecraft dynamics without relying on heuristic ambiguity-set radii.

Methodology
The authors propose a distributionally robust covariance-steering framework based on the relative entropy, or Kullback–Leibler Divergence (KLD), to quantify the mismatch between the true state distribution ( $\pi_t$ ) and a Gaussian reference surrogate ( $\hat{\pi}_t$ ). The methodology proceeds through three primary analytical and algorithmic steps:

Variational Bounds on Risk-Sensitive Quantities:
Utilizing the Donsker-Varadhan variational formula for exponential integrals, the authors derive computable upper bounds for expected quantities of interest under distribution ambiguity. Specifically, they establish that for any bounded function $g(x)$ , the expectation under the true distribution $\pi$ is bounded by an expression involving the expectation under the surrogate $\hat{\pi}$ and the relative entropy $R(\pi \parallel \hat{\pi})$ . This allows for the derivation of explicit upper bounds for:
- Chance Constraints (Collision Risk): By defining the admissible state set as a half-space, the probability of collision is bounded using the surrogate's cumulative distribution function (CDF) and the KLD.
- Mean Squared Error (MSE): An upper bound for the delivery MSE is derived, which converges to the exact MSE of the Gaussian surrogate as the KLD approaches zero.
Bounding the Time Rate of Change of KLD:
To make the KLD constraint tractable within a steering formulation, the authors derive an approximate upper bound on the time rate of change of the relative entropy, $\frac{d}{dt}R(\pi_t \parallel \hat{\pi}_t)$ . Under the assumption that the reference surrogate is Gaussian and that model errors arise primarily from structural linearization (second-order truncation errors), this rate is shown to be a function of the covariance matrix $P_t$ and the Hessian of the nonlinear dynamics. Crucially, this bound is controllable via the decision variables in a covariance-steering problem.
Sequential Convex Programming (SCvx) Implementation:
The derived bounds are incorporated into a Sequential Convexification (SCvx) algorithm. The relative entropy upper bound and the variational MSE upper bound are treated as decision variables. The nonlinear dynamics of these bounds are linearized around the solution of the previous convex iteration, and slack variables are introduced to ensure dynamical consistency. The resulting subproblems are convex semidefinite programs that simultaneously optimize the feedback control policy and regulate the growth of the distributional mismatch.

Key Contributions

Distributionally Robust Framework: The paper introduces a covariance-steering framework that explicitly accounts for the ambiguity between the true nonlinear distribution and its Gaussian surrogate using KLD.
Principled Ambiguity Set Radius: Instead of selecting ambiguity-set radii heuristically, the work provides a method to determine appropriate bounds by linking them directly to the nonlinear dynamical evolution and the control policy.
Computable Risk Bounds: The derivation of upper bounds for collision risk and MSE that rely only on the known surrogate distribution and the KLD bound, eliminating the need for exact knowledge of the true distribution.
Controllable KLD Dynamics: The demonstration that the time rate of change of the KLD can be bounded and controlled within a standard covariance-steering formulation, enabling the enforcement of distributional consistency constraints.

Results
The proposed approach was validated on a challenging nonlinear spacecraft transfer problem between two Near-Rectilinear Halo Orbits (NRHOs) in the Earth-Moon Circular Restricted Three-Body Problem (CRTBP). This scenario features significant nonlinearities during close lunar encounters.

Relative Entropy Targeting: In Test Case #1, the algorithm successfully constrained the relative entropy upper bound to prescribed thresholds (ranging from 100 down to 0.1). Monte Carlo analysis (2,500 trials) demonstrated that while standard covariance steering resulted in a significant number of terminal states falling outside the 99% confidence bounds, the entropy-constrained policies maintained consistency with the designed terminal uncertainty. The results showed that tighter entropy constraints led to larger control variances but significantly reduced the divergence between the true distribution and the Gaussian surrogate.
MSE Variational Constraints: In Test Case #2, the algorithm enforced a variational upper bound on the MSE. The SCvx iterations successfully converged to an optimal Lagrange multiplier ( $\lambda^*$ ), and the resulting guidance policy ensured that the empirical MSE remained below the prescribed constraint, whereas the standard approach violated the bound.

Significance and Claims
The authors claim that this work offers a principled alternative to heuristic methods for selecting ambiguity-set radii in distributionally robust control. By connecting the ambiguity radius directly to the nonlinear dynamical evolution via the KLD rate bound, the framework provides a systematic way to design stochastic guidance policies that remain robust to distributional mismatch. The numerical results suggest that the proposed method can provide additional robustness beyond conventional Gaussian covariance steering, particularly in highly nonlinear regimes where linearization errors are significant. The paper concludes that the framework effectively regulates distributional mismatch while enforcing conservative bounds on risk-sensitive performance measures.

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