A Tutorial on Gaussian Process Learning-based Model Predictive Control

This tutorial provides an accessible and implementation-oriented introduction to Gaussian process learning-based model predictive control (GP-MPC), which combines probabilistic residual modeling with receding-horizon control. Its central tutorial contribution is a detailed, step-by-step derivation of multi-step mean and covariance propagation for GP-augmented prediction models. The derivation shows how commonly used propagation formulas follow from the laws of total expectation and total covariance, and clarifies the roles of the GP posterior mean, GP posterior covariance, uncertain inputs, and query–output cross-covariances. Building on this foundation, the paper distinguishes mean-only unconstrained GP predictive control from uncertainty-aware constrained GP-MPC, clarifies regulation and output-tracking formulations, and provides concise implementation and computational-complexity guidance, including practical software-tool references and discussion of the scope of closed-loop guarantees. Mobile-robot path-following examples illustrate mean-only unconstrained implementations, whereas mixed-vehicle platooning illustrates uncertainty-aware constrained GP-MPC with uncertainty-dependent safety-distance tightening. The tutorial is intended to help researchers and practitioners understand, implement, and critically evaluate GP-MPC designs for robotic and autonomous systems operating under modeling uncertainty.