A Tutorial on Gaussian Process Learning-based Model Predictive Control

TL;DR

The paper provides a systematic introduction to Gaussian process learning-based model predictive control (GP-MPC), proposing a detailed mathematical formulation for propagating GP mean and uncertainty across multi-step MPC predictions. It develops mean and variance propagation techniques (including Taylor and mean-equivalent approaches) and discusses their integration into both GP-only and GP-enhanced MPC frameworks. The tutorial demonstrates practical GP-MPC applications in robotics, notably improved path following for mobile robots and safety-aware mixed-vehicle platooning, and extends to sparse GP methods and dynamic sparse MPC to enable real-time deployment. By bridging theory with practice and including appendices that formalize the approximation theorems and cost-function calculus, the work advances learning-based control for robust, uncertainty-aware autonomous systems.

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.

Figures (14)

• Figure 1: Visualization of a Gaussian distribution: 1000 data points sampled from a standard Gaussian distribution are represented as red vertical bars along the $X$-axis. The accompanying curve, a two-dimensional bell curve, illustrates the probability density function (PDF) of the distribution wang2023intuitive.
• Figure 2: Visualization of a bivariate normal distribution: The upper image shows a 3-D bell curve for probability density, and the lower images display 2-D ellipse contours indicating the correlation between joint variables $x_1$ and $x_2$wang2023intuitive.
• Figure 3: Impact of the length scale hyperparameter $l$ on model smoothness: smaller $l$ values lead to higher sensitivity to data variations (top), medium $l$ balances sensitivity and smoothness (lower left), and larger $l$ values favor smoothness over sensitivity to local changes (lower right) wang2023intuitive.
• Figure 4: GPR visualization: Black crosses denote observed data points. From these, GPR considers all possible functions (illustrated by 20 samples in assorted colors) that align with the data. The solid red line represents the GP mean function, calculated from the distribution of these possible infinite numbers of functions. The surrounding blue-shaded region illustrates the prediction uncertainty, shown as 3 times the standard deviation from the mean. This analysis employs an RBF kernel with optimized hyperparameters ($\sigma_f = 0.0067$, $l = 0.0967$), demonstrating the model's improved predictive accuracy and reliability wang2023intuitive.
• Figure 5: An illustrative process of conducting regressions by Gaussian processes. The red points are observed data, the blue line represents the mean function estimated by the observed data points, and predictions will be made at new blue points wang2023intuitive.