Model Predictive Control using Efficient Gaussian Processes for Unknown Disturbance Inputs

Gaussian Processes (GPs) are a versatile tool to model unknown disturbances affecting a physical system. Combining the model of a physical system with a nonparametric GP prior for the input disturbances results in a model structure referred to as latent force model (LFM). Recently, it was (re-)discovered that using spectral factorization, GPs can be represented by equivalent/approximate state-space models driven by Gaussian white-noise with a given spectral density. The classical GP regression problem can in turn be solved efficiently using Kalman filters and smoothers. In this paper, we exploit such state-space formulation of LFMs for designing model predictive control algorithms to regulate the physical system in the presence of unknown input disturbances. The GP model is used for propagating the system disturbances over the MPC prediction horizon to counteract their effects. The probabilistic representation of the LFM model allows for considering probabilistic constraints on the state variables of the physical system, which yields a nominal MPC problem subject to tightened state constraints using the predicted covariance matrix of the LFM state. For recursive feasibility, a slack variable is included to soften the incorporated state constraints. Two numerical examples are given for illustration.