[Paper Review] A Tutorial on Gaussian Process Learning-based Model Predictive Control

Jie Wang, Youmin Zhang|arXiv (Cornell University)|Apr 2, 2024

Advanced Control Systems Optimization4 citations

TL;DR

This tutorial presents a systematic mathematical formulation of Gaussian process learning-based model predictive control (GP-MPC), integrating probabilistic GP regression with MPC to enhance predictive accuracy and uncertainty quantification in complex systems. It derives approximations for multi-step mean and variance propagation in GP predictions, enabling robust, constrained optimization in robotics applications like path-following and platooning with demonstrated improvements in tracking and stability.

ABSTRACT

This tutorial provides a systematic introduction to Gaussian process learning-based model predictive control (GP-MPC), an advanced approach integrating Gaussian process (GP) with model predictive control (MPC) for enhanced control in complex systems. It begins with GP regression fundamentals, illustrating how it enriches MPC with enhanced predictive accuracy and robust handling of uncertainties. A central contribution of this tutorial is the first detailed, systematic mathematical formulation of GP-MPC in literature, focusing on deriving the approximation of means and variances propagation for GP multi-step predictions. Practical applications in robotics control, such as path-following for mobile robots in challenging terrains and mixed-vehicle platooning, are discussed to demonstrate the real-world effectiveness and adaptability of GP-MPC. This tutorial aims to make GP-MPC accessible to researchers and practitioners, enriching the learning-based control field with in-depth theoretical and practical insights and fostering further innovations in complex system control.

Motivation & Objective

• Address the lack of comprehensive mathematical foundations for integrating Gaussian processes (GPs) into model predictive control (MPC) frameworks, particularly for multi-step predictions.
• Provide a systematic derivation of mean and variance propagation in GP-based multi-step predictions to enable uncertainty-aware MPC.
• Bridge the gap between theoretical GP learning and practical MPC implementation for researchers and practitioners in robotics and control systems.
• Demonstrate the effectiveness of GP-MPC in real-world robotic applications, including path-following and mixed-vehicle platooning, under dynamic and uncertain conditions.
• Offer a self-contained, accessible tutorial that consolidates theoretical principles and practical implementation insights for GP-MPC.

Proposed method

• Formulate GP regression as a non-parametric Bayesian surrogate model to capture system dynamics with uncertainty quantification.
• Derive first-order Taylor approximations for the expectation and variance of GP predictions over time, enabling tractable multi-step forecasting in MPC.
• Integrate the approximated GP mean and variance into the MPC optimization cost function to account for predictive uncertainty.
• Use linearization techniques to approximate non-Gaussian predictive distributions, ensuring computational tractability in the MPC framework.
• Implement a sequential quadratic programming (SQP)-like approach for solving the resulting non-convex optimization problem with gradient-based methods.
• Apply analytical derivatives of the cost function using matrix calculus, leveraging symmetry of Q and R matrices to simplify optimization.

Experimental results

Research questions

• RQ1 How can Gaussian process regression be systematically integrated into model predictive control to improve predictive accuracy and uncertainty handling?
• RQ2 What mathematical approximations are required to propagate GP mean and variance predictions over multiple time steps in an MPC framework?
• RQ3 How does incorporating predictive uncertainty into the MPC cost function affect control performance in dynamic, uncertain environments?
• RQ4 What are the practical benefits of GP-MPC in real-world robotic control tasks such as path-following and platooning?
• RQ5 How can the computational complexity of GP-MPC be managed while preserving robustness and accuracy in real-time control?

Key findings

• The paper provides the first systematic mathematical derivation of multi-step GP mean and variance propagation in MPC, enabling uncertainty-aware prediction.
• The derived approximation for the predictive variance ensures that uncertainty is properly propagated through the prediction horizon, improving robustness.
• In path-following tasks, GP-MPC demonstrated superior tracking performance and stability on uneven terrain compared to standard MPC, due to better modeling of system uncertainty.
• For mixed-vehicle platooning, GP-MPC achieved improved string stability and reduced inter-vehicle spacing errors by accounting for uncertain dynamics in leader-follower interactions.
• The optimization framework successfully computes control inputs that balance tracking performance and control effort, with the derived gradient expression enabling efficient numerical solution.
• The inclusion of predictive uncertainty in the cost function leads to more conservative and reliable control actions, especially in regions of high model uncertainty.