# Physics-informed Gaussian Processes as Linear Model Predictive Controller

arXiv: https://arxiv.org/abs/2412.04502 · HTML: https://mlsanity.com/paper/2412.04502/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 36 references that are in the map.
- First posted on arXiv 2 Dec 2024.

## Builds on (most cited first, 11 of 11)

- [Reinforcement Learning and Control as Probabilistic Inference: Tutorial and Review](https://mlsanity.com/paper/1805.00909/) (2018)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Linearly constrained Gaussian processes](https://mlsanity.com/paper/1703.00787/) (2017)
- [Algorithmic Linearly Constrained Gaussian Processes](https://mlsanity.com/paper/1801.09197/) (2018)
- [Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients](https://mlsanity.com/paper/2212.14319/) (2022)
- [Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equations](https://mlsanity.com/paper/2208.12515/) (2022)
- [Cautious NMPC with Gaussian Process Dynamics for Autonomous Miniature Race Cars](https://mlsanity.com/paper/1711.06586/) (2017)
- [Thomas Decomposition and Nonlinear Control Systems](https://mlsanity.com/paper/2001.08424/) (2020)
- [Efficiently Computable Safety Bounds for Gaussian Processes in Active Learning](https://mlsanity.com/paper/2402.18260/) (2024)
- [Implementation of soft-constrained MPC for Tracking using its semi-banded problem structure](https://mlsanity.com/paper/2403.04601/) (2024)
- [Model predictive control for tracking using artificial references: Fundamentals, recent results and practical implementation](https://mlsanity.com/paper/2406.06157/) (2024)

## Abstract (arXiv)

We introduce a novel algorithm for controlling linear time invariant systems in a tracking problem. The controller is based on a Gaussian Process (GP) whose realizations satisfy a system of linear ordinary differential equations with constant coefficients. Control inputs for tracking are determined by conditioning the prior GP on the setpoints, i.e. control as inference. The resulting Model Predictive Control scheme incorporates pointwise soft constraints by introducing virtual setpoints to the posterior Gaussian process. We show theoretically that our controller satisfies open-loop stability for the optimal control problem by leveraging general results from Bayesian inference and demonstrate this result in a numerical example.