Provably Safe and Robust Learning-Based Model Predictive Control

Controller design for systems typically faces a trade-off between robustness and performance, and the reliability of linear controllers has caused many control practitioners to focus on the former. However, there is a renewed interest in improving system performance to deal with growing energy and pollution constraints. This paper describes a learning-based model predictive control (MPC) scheme. The MPC provides deterministic guarantees on robustness and safety, and the learning is used to identify richer models of the system to improve controller performance. Our scheme uses a linear model with bounds on its uncertainty to construct invariant sets which help to provide the guarantees, and it can be generalized to other classes of models and to pseudo-spectral methods. This framework allows us to handle state and input constraints and optimize system performance with respect to a cost function. The learning occurs through the use of an oracle which returns the value and gradient of unmodeled dynamics at discrete points, and the oracle could be parametric or non-parametric statistical identification tools. Moreover, we show convergence of the control action determined by the MPC with oracle to the control action of an MPC that knows the unmodeled dynamics. To integrate learning where there is no prior knowledge about unmodeled dynamics with control, we construct a new non-parametric estimator with properties that make it amenable to use with numerical optimization algorithms. Deterministic and probabilistic properties of this estimator are proven.