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Relative Entropy-Bounded Ambiguous Chance Constraints for Robust Planning in Nonlinear Systems

This paper proposes a robust planning framework for nonlinear stochastic systems that bounds risk under distributional ambiguity by defining a relative entropy-bounded set around a nominal Gaussian distribution, with the ambiguity size determined by covariance evolution and dynamical truncation errors, as demonstrated in a spacecraft guidance application.

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 guide a spaceship through a chaotic, swirling storm of space dust and gravitational tugs. In the perfect world of high school physics, we often pretend the universe is calm and predictable: if you know where the ship is and how fast it's going, you can calculate exactly where it will be tomorrow. But in the messy reality of engineering, things are never that neat. The ship might get hit by a stray particle, or the math describing gravity might be slightly off. This is where "stochastic control" comes in—a fancy term for steering things when you can't be 100% sure what's happening.

To keep the ship safe, engineers use "chance constraints." Think of this as a safety rule that says, "We must be 99% sure the ship won't crash into the Moon." Usually, to make this math work, engineers assume the ship's uncertainty follows a perfect "bell curve" (a Gaussian distribution). It's like assuming that if you throw a ball, it will land in a neat, symmetrical pile of sand. But in the real world, especially with complex systems like a spacecraft orbiting a planet, the wind and gravity can twist that pile of sand into a weird, lopsided blob. If you assume it's a perfect bell curve when it's actually a blob, your safety calculations might be dangerously wrong. This paper tackles the problem of how to stay safe when you know your "perfect bell curve" assumption is probably wrong, but you don't know exactly what the "blob" looks like.

The authors, Trevor N. Wolf and Jay W. McMahon, propose a new way to handle this uncertainty, which they call "distributional ambiguity." Instead of pretending we know the exact shape of the ship's uncertainty, they admit we don't. They create a "safety bubble" around their best guess (the Gaussian bell curve). Inside this bubble, the true uncertainty could be anything, as long as it isn't too different from the guess. They measure "too different" using a concept called "relative entropy," which is like a ruler that measures how much the real, messy reality has drifted away from the clean, perfect math model.

The core of their discovery is a mathematical trick that lets them calculate the worst-case risk inside this safety bubble. Imagine you are a captain trying to avoid a collision. You have a map that says, "There's a 1% chance we hit the rock." But you know your map might be slightly wrong. This paper gives you a formula to say, "Even if the map is wrong by this specific amount, the real chance of hitting the rock is definitely no more than X%." They found a way to make this "X%" calculation tight and useful, rather than just guessing a huge, useless number.

Crucially, the paper argues against the idea that we can just stick with the simple, perfect bell curve assumption when dealing with highly nonlinear systems (systems where small changes cause big, unpredictable effects). They show that for these complex systems, the bell curve assumption breaks down quickly. Instead of trying to find the exact, impossible-to-know shape of the real uncertainty, they use a variational expression (a fancy type of math formula) to bound the risk. This means they don't need to know the truth; they just need to know how far the truth could be from their guess.

The authors tested this idea using a simulation of a spacecraft navigating the Earth-Moon system. They set up a scenario where the ship had to fly near a "chaser" spacecraft without crashing, while dealing with random pushes from space noise. They ran thousands of computer trials (Monte Carlo simulations) to see what actually happened. The results showed that their new method works: the "safety bubble" they calculated successfully contained the real, messy risk. In fact, when they looked at the end of the journey, the old, simple method (assuming a perfect bell curve) underestimated the danger, while their new, robust method correctly warned that the risk was higher.

One of the most clever parts of their work is figuring out how big to make that "safety bubble." Instead of just picking a random size, they derived a rule that links the size of the bubble to how much the ship's path is curving and how much the math model is approximating the real physics. It's like having a rubber band that automatically stretches wider when the road gets bumpier and tighter when the road is smooth. They proved mathematically that as their "rubber band" shrinks to zero (meaning the model is perfect), their complex risk formula turns back into the simple, standard risk formula we already know.

In their specific simulation, they used a diffusion magnitude (a measure of how much random noise is pushing the ship) of $5 \times 10^{-6}$ km/s $^{3/2}$ and $1 \times 10^{-6}$ km/s $^{3/2}$ . They found that with higher noise, the gap between the "perfect guess" and the "messy reality" grew larger, and their method correctly adjusted the safety margin to account for it. They also showed that the "tuning knob" (a variable called $\lambda$ ) they use to tighten the math changes over time, adapting to the specific moment in the flight.

Ultimately, this paper doesn't claim to have solved the problem of perfect prediction. Instead, it offers a robust framework for engineers to say, "We don't know the exact truth, but we know how wrong we could be, and we can still guarantee safety within those limits." It turns the scary unknown of nonlinear chaos into a manageable, calculable risk, providing a new toolkit for guiding spacecraft through the unpredictable dance of the cosmos.

Problem Statement
The paper addresses the challenge of designing robust guidance and control policies for nonlinear stochastic systems under distributional ambiguity. Traditional chance-constrained control methods, particularly within the framework of covariance steering (CS), typically assume that the true state distribution is known exactly and follows a Gaussian distribution. These assumptions are often invalid for real-world engineering applications involving highly nonlinear dynamics (e.g., spacecraft guidance in the Earth-Moon system), where system dynamics may be only approximately modeled or where nonlinearities rapidly degrade Gaussianity. Consequently, standard methods may fail to accurately quantify the risk of constraint violations, such as collisions, when the true probability density function (PDF) deviates from the nominal Gaussian reference.

Methodology
The authors propose a framework for "distributionally robust chance constraints" using a relative entropy (Kullback-Leibler divergence, KLD) bounded ambiguity set. The methodology proceeds through several key steps:

Variational Upper Bound: The paper utilizes the Donsker–Varadhan variational formula to derive an upper bound on the expected risk (probability of constraint violation) under an unknown true distribution $\pi_t$ . This bound is expressed in terms of a nominal Gaussian reference distribution $\hat{\pi}_t$ and the relative entropy $R(\pi_t \| \hat{\pi}_t)$ . The bound recovers the exact risk value of the reference distribution as the relative entropy approaches zero.
Bounding Relative Entropy Growth: A critical contribution is the derivation of a time-varying upper bound for the relative entropy itself, without requiring explicit knowledge of the true distribution. By analyzing the Fokker-Planck equations for both the true and reference densities, the authors derive a differential inequality for the growth rate of the relative entropy. This rate is bounded by the expected squared error between the true nonlinear drift and the linearized reference drift, weighted by the process noise covariance.
Handling Nonlinearity: To make the bound computable, the authors approximate the drift error using a second-order Taylor expansion (quadratic remainder) centered at the reference mean. They leverage properties of Gaussian moments (specifically the expectation of quadratic forms) to express the bound in terms of the reference covariance matrix and the Hessian of the nonlinear drift.
Numerical Implementation: The derived relative entropy growth rate is integrated numerically using a first-order Euler method within a convex optimization framework. The authors employ a convex relaxation of the covariance steering constraints (using Schur complements) to solve for the optimal feedback control gains.

Key Contributions

Risk Bound under Ambiguity: The paper establishes a rigorous upper bound for chance constraints that accounts for distributional ambiguity, bridging the gap between nominal Gaussian risk and the worst-case risk within a relative entropy ball.
Dynamic Ambiguity Set Sizing: Unlike previous works that often treat the ambiguity set size as a fixed parameter, this work introduces a principled method to determine the relative entropy upper bound as a function of the reference covariance evolution and second-order dynamical truncation errors. This allows the ambiguity set size to be controlled dynamically based on the system's nonlinearity and noise characteristics.
Framework for Nonlinear CS: The work provides a foundational framework for incorporating distributional ambiguity into nonlinear covariance steering problems, moving beyond the limitations of linear time-invariant (LTI) assumptions or fixed Gaussian mixtures.

Results
The proposed method is evaluated using a nonlinear stochastic spacecraft guidance problem in the Earth-Moon circular restricted three-body problem (CRTBP) region.

Simulation Setup: The scenario involves steering a spacecraft's distribution over one orbital period while avoiding a collision with a "chaser" spacecraft. Two different diffusion magnitudes (process noise levels) were tested.
Validation: The authors compared their derived relative entropy upper bound against a "truth-surrogate" model, which computes the KLD between the reference Gaussian and a Gaussian reduction of a 4,000-trial Monte Carlo simulation.
Findings:
- The derived upper bound successfully tracked the evolution of the true relative entropy, remaining valid (conservative) throughout the trajectory for various tuning parameters ( $\gamma$ ).
- The results demonstrated that the nominal risk (calculated under the Gaussian assumption) significantly underestimates the true risk, particularly in the latter half of the trajectory where nonlinear errors accumulate.
- The proposed risk upper bound effectively constrained both the nominal and true risk probabilities, providing a robust safety margin.
- The optimal tuning parameter $\lambda^*$ for the variational bound was shown to be time-varying, dependent on the relative entropy bound.

Significance
The paper claims to provide the fundamental tools necessary for handling distributional ambiguity in nonlinear covariance steering problems. Its primary significance lies in offering a method to quantify and bound risk without assuming perfect knowledge of the system's probability distribution. By linking the size of the ambiguity set directly to the system's dynamical errors and covariance evolution, the approach offers a systematic way to design control policies that are robust to model inaccuracies and nonlinear effects. The authors position this work as a step toward making distributionally robust chance constraints operational in convex optimization procedures for complex engineering systems.

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