Learning-based adaptive-scenario-tree model predictive control with improved probabilistic safety using robust Bayesian neural networks

Scenario-based model predictive control (MPC) methods can mitigate the conservativeness inherent to open-loop robust MPC. Yet, the scenarios are often generated offline based on worst-case uncertainty descriptions obtained a priori, which can in turn limit the improvements in the robust control performance. To this end, this paper presents a learning-based, adaptive-scenario-tree model predictive control approach for uncertain nonlinear systems with time-varying and/or hard-to-model dynamics. Bayesian neural networks (BNNs) are used to learn a state- and input-dependent description of model uncertainty, namely the mismatch between a nominal (physics-based or data-driven) model of a system and its actual dynamics. We first present a new approach for training robust BNNs (RBNNs) using probabilistic Lipschitz bounds to provide a less conservative uncertainty quantification. Then, we present an approach to evaluate the credible intervals of RBNN predictions and determine the number of samples required for estimating the credible intervals given a credible level. The performance of RBNNs is evaluated with respect to that of standard BNNs and Gaussian process (GP) as a basis of comparison. The RBNN description of plant-model mismatch with verified accurate credible intervals is employed to generate adaptive scenarios online for scenario-based MPC (sMPC). The proposed sMPC approach with adaptive scenario tree can improve the robust control performance with respect to sMPC with a fixed, worst-case scenario tree and with respect to an adaptive-scenario-based MPC (asMPC) using GP regression on a cold atmospheric plasma system. Furthermore, closed-loop simulation results illustrate that robust model uncertainty learning via RBNNs can enhance the probability of constraint satisfaction of asMPC.