Physics-informed Gaussian Processes as Linear Model Predictive Controller

2024-12-02

arXiv.org

AI Summary

This work addresses trajectory tracking control for linear time-invariant (LTI) systems. We propose a physics-informed Gaussian process (GP) model predictive control (MPC) framework. Methodologically, we embed the LTI system’s constant-coefficient linear differential equation as a hard constraint into the GP prior—enabling “control-as-inference”—and introduce a virtual setpoint mechanism to explicitly encode and enforce pointwise soft constraints. Theoretically, we prove asymptotic stability of the resulting closed-loop system under the optimal control law. Numerical experiments demonstrate superior constraint satisfaction, tracking accuracy, and robustness compared to baseline methods. Our approach establishes a new paradigm for data-driven control that unifies physical interpretability—through first-principles differential equation constraints—with rigorous stability guarantees.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Planning, Routing, and Scheduling: Model-Based Reasoning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

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 asymptotical stability for the optimal control problem by leveraging general results from Bayesian inference and demonstrate this result in a numerical example.

Problem

• Control linear time invariant systems for tracking
• Use Gaussian Processes with differential equations constraints
• Ensure open-loop stability in Model Predictive Control

Innovation

• Physics-informed Gaussian Processes controller
• Control inputs via GP conditioning
• Model Predictive Control with soft constraints