⚡ electrical engineering

Autonomous Satellite Rendezvous via Hybrid Feedback Optimization

This paper proposes a hybrid feedback optimization approach for autonomous satellite rendezvous that integrates a stabilizing controller with discrete-time gradient descent to handle measurement uncertainties and limited onboard processing, demonstrating exponential convergence to a bounded rendezvous region and a significant reduction in disturbance magnitude through simulations.

Original authors: Oscar Jed R. Chuy, Matthew T. Hale, Vignesh Sivaramakrishnan, Sean Phillips, Ricardo G. Sanfelice

Published 2026-02-26

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CC BY 4.0

Original authors: Oscar Jed R. Chuy, Matthew T. Hale, Vignesh Sivaramakrishnan, Sean Phillips, Ricardo G. Sanfelice

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 park a car in a very tight spot in a parking lot, but there are three major problems:

The GPS is glitchy: You can't perfectly predict where the car will be in the next second because the sensors are noisy.
The brain is slow: The car's computer is an old, slow model that can't do complex math quickly.
The wind is unpredictable: Random gusts keep pushing the car off course.

Most traditional methods try to solve this by predicting the future. They say, "Okay, if I turn the wheel now, and the wind blows this way, I will be in the perfect spot in 5 seconds." But if your GPS is glitchy or the wind changes unexpectedly, that prediction fails, and you crash.

This paper proposes a completely different approach called Hybrid Feedback Optimization. Instead of guessing the future, the car just constantly checks where it is right now and makes tiny adjustments.

Here is how the paper's solution works, broken down into simple concepts:

1. The "Slow Brain" Problem (The Hardware)

Satellites are like that old car. They have limited computing power. They can't run a super-complex simulation to figure out the perfect path to a target satellite.

The Old Way: "Calculate the perfect path, then execute it." (Too slow, too much math).
The New Way: "Take a step, check where you are, take another step." (Fast, simple math).

2. The "Glitchy GPS" Problem (The Uncertainty)

In space, you can't always know exactly where you are or what the future holds.

The Old Way: Relies on a perfect model of physics. If the model is wrong, the mission fails.
The New Way: It ignores the "what ifs" and focuses on the "what is." It measures the satellite's current position, feeds that data into a simple math loop, and asks, "How do I move right now to get closer to the target?"

3. The "Hybrid" Engine (The Secret Sauce)

The paper calls this a Hybrid System. Think of it like a driver who is constantly switching between two modes:

Mode A (Driving): The satellite is flying smoothly through space (Continuous time).
Mode B (Thinking): Every few seconds, the satellite pauses its "thinking" to run a quick calculation. It looks at where it is, runs a simple "gradient descent" (which is just a fancy way of saying "rolling downhill to find the lowest point"), and updates its steering command.

Because the satellite's computer is slow, it doesn't wait to find the perfect answer. It just runs the math for a few seconds, gets a "good enough" answer, and applies it immediately. Then it repeats the process.

4. The "Stabilizer" (The Training Wheels)

The physics of satellites (described by the Clohessy-Wiltshire equations) are naturally unstable. Imagine trying to balance a broomstick on your hand; without help, it falls over.

The authors first built a "training wheel" system (a stabilizing controller) that keeps the satellite from drifting away wildly.
Once the satellite is stable, they let the "Feedback Optimization" take over to gently nudge it toward the target.

5. The Results: A Magic 98% Improvement

The researchers ran simulations with this new method. They introduced random "gusts of wind" (disturbances) to see if the satellite would crash.

The Result: The new method reduced the effect of these disturbances by 98.4%.
The Analogy: If a gust of wind would normally push your car 100 meters off course, this new system only lets it drift 1.6 meters. It's incredibly robust.

Why This Matters

This is the first time this specific "check-and-adjust" method has been applied to satellite docking. It proves that you don't need a supercomputer to dock two satellites. You just need a smart, simple loop that constantly checks reality and adjusts, rather than trying to predict a perfect future that might never happen.

In a nutshell: Instead of trying to be a psychic who predicts the future, this system is a very observant driver who constantly checks the rearview mirror and makes small, quick corrections to stay on the road, even when the road is bumpy and the GPS is broken.

1. Problem Statement

The paper addresses the Autonomous Rendezvous, Proximity Operations, and Docking (ARPOD) problem, specifically focusing on a "chaser" satellite attempting to rendezvous with an uncontrolled "target" satellite in Low Earth Orbit (LEO).

Key Challenges:

Computational Constraints: Onboard satellite processors are significantly slower than terrestrial counterparts, limiting the ability to solve complex optimization problems in real-time.
Uncertainty and Disturbances: Space environments introduce modeling errors, measurement noise, and unmodeled disturbances (e.g., drag, solar pressure). Predicting future states based on imperfect models is unreliable.
System Stability: The relative dynamics of the satellites, modeled by the Clohessy-Wiltshire (CW) equations, are marginally stable (neutrally stable), meaning they do not naturally converge to a desired state without active control.
Limitations of Existing Methods: Traditional Model Predictive Control (MPC) requires predicting future states and solving optimization problems offline or in a feedforward manner, which is computationally expensive and sensitive to model inaccuracies.

Objective:
Develop a control strategy that:

Does not rely on predicting future states.
Rejects disturbances as they arise.
Accommodates slow onboard processors by allowing sub-optimal, iterative computations.
Ensures the chaser converges to a rendezvous point despite uncertainties.

2. Methodology

The authors propose a Hybrid Feedback Optimization framework. This approach integrates continuous-time physical dynamics with discrete-time optimization algorithms running in a feedback loop.

A. System Modeling

Dynamics: The chaser's relative motion is modeled using the linearized Clohessy-Wiltshire (CW) equations.
Stabilization: Since CW equations are marginally stable, the authors first design a state-feedback controller ($v = -Ky + u$) to asymptotically stabilize the system. They derive closed-form expressions for the gain matrix $K$ that place the closed-loop eigenvalues at desired negative real values, ensuring stability.
Hybrid System Formulation: The system is modeled as a Hybrid System (following the framework of Goebel, Sanfelice, and Teel):
- Continuous Flow ( $\dot{\zeta} \in F(\zeta)$ ): The satellite state $x$ evolves continuously according to the stabilized dynamics. Timers ( $\tau_c, \tau_g, \tau_d$ ) count down or up.
- Discrete Jumps ( $\zeta^+ \in G(\zeta)$ ):
  - Optimization Jumps: When a gradient descent timer ( $\tau_g$ ) reaches zero, the optimization variable $z$ is updated using a projected gradient descent step.
  - Control Jumps: When a control timer ( $\tau_c$ ) reaches zero, the current optimization iterate $z$ is applied as the control input $u$ , and a new output sample $y_s$ is taken.

B. Feedback Optimization Loop

Instead of solving for the optimal input offline, the system runs a projected gradient descent algorithm in the loop:

Objective Function: Minimizes a cost function $\Phi(u, y)$ that penalizes control effort and deviation from a nominal rendezvous point $\hat{y}$ .
Real-time Adaptation: The algorithm uses the actual sampled output $y_s$ (which includes disturbances) rather than a predicted model output.
Iterative Computation: The system does not wait for the gradient descent to converge to the exact optimum. It performs a limited number of iterations ( $\ell$ ) between control updates, making it suitable for slow processors.

3. Key Contributions

Stabilizing Controller Design: The authors design a stabilizing feedback controller for the marginally stable CW equations and provide closed-form analytical expressions for the gain matrix $K$ based on desired closed-loop eigenvalues (Theorem 1).
Hybrid System Modeling: They formulate the first hybrid feedback optimization model for ARPOD, rigorously proving that the system is well-posed (satisfies hybrid basic conditions) and that all maximal solutions are complete and non-Zeno (Proposition 1).
Convergence Analysis:
- They prove that the chaser satellite converges exponentially fast to a ball around a time-varying "chosen rendezvous point" $\tilde{x}(t)$ .
- They derive an explicit bound on the steady-state error radius in terms of system parameters, disturbance bounds, and processor speed (Theorem 2).
Robustness Guarantees: The paper establishes that the system is robust to perturbations. Bounded errors in the timing of computations and control updates (perturbations to the hybrid model) result in bounded deviations in the satellite's trajectory (Corollary 1).
Simulation Validation: Extensive simulations demonstrate the approach's efficacy, showing a 98.4% reduction in the magnitude of disturbances compared to the raw disturbance input.

4. Results

Nominal Performance: Simulations show the chaser converges to within approximately 0.082 meters of the chosen rendezvous point, despite a disturbance magnitude of 5 units.
Disturbance Rejection: The feedback optimization mechanism effectively rejects disturbances, reducing their impact on the rendezvous error by 98.4%.
Robustness to Timing Errors: The system was tested with perturbations in the timers ( $\tau_c$ and $\tau_g$ ) representing computational delays or timing jitter. Even with significant perturbations (up to $\rho=0.9$ ), the asymptotic error remained bounded (e.g., ~4.23 meters), demonstrating resilience.
Initial Condition Independence: Tests with 20 different random initial conditions confirmed that the system reliably converges to the rendezvous point regardless of the starting state.

5. Significance

This work represents a paradigm shift in autonomous space operations by moving away from prediction-based control (like MPC) toward measurement-based feedback optimization.

Computational Efficiency: It allows satellites with limited processing power to perform complex tasks by trading exact optimality for iterative, real-time adaptability.
Robustness: By avoiding state prediction, the method is inherently robust to modeling errors and unmodeled dynamics, which are critical in the unpredictable space environment.
Theoretical Rigor: The use of hybrid systems theory provides a mathematically rigorous framework to guarantee stability, convergence, and robustness, filling a gap in the literature where feedback optimization was previously applied mostly to continuous-time systems without discrete implementation constraints.
Practical Application: The results suggest that this approach is viable for real-world ARPOD missions, enabling safer and more autonomous satellite servicing and docking operations.