Learning-based Adaptive-Scenario-Tree Model Predictive Control with Probabilistic Safety Guarantees Using Bayesian Neural Networks

Abstract

Scenario-based optimization and control has proven to be an efficient approach to account for system uncertainty. In particular, the performance of scenario-based model predictive control (MPC) schemes depends on the accuracy of uncertainty quantification. However, current learning-and scenario-based MPC (sMPC) approaches employ a single timeinvariant probabilistic model (learned offline), which may not accurately describe time-varying uncertainties. Instead, this paper presents a model-agnostic meta-learning (MAML) of Bayesian neural networks (BNN) for adaptive uncertainty quantification that would be subsequently used for adaptive-scenario-tree model predictive control design of nonlinear systems with unknown dynamics to enhance control performance. In particular, the proposed approach learns both a global BNN model and an updating law to refine the BNN model. At each time step, the updating law transforms the global BNN model into more precise local BNN models in real time. The adapted local model is then used to generate scenarios for sMPC design at each time step. A probabilistic safety certificate is incorporated in the scenario generation to ensure that the trajectories of the generated scenarios contain the real trajectory of the system and that all the scenarios adhere to the constraints with a high probability. Experiments using closed-loop simulations of a numerical example demonstrate that the proposed approach can improve the performance of scenario-based MPC compared to using only one BNN model learned offline for all time steps.

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

“…Furthermore, in Figure 6, we compare our strategies with BNNs to the worst‐case sMPC strategy (in dotted red) and a published kernel‐based method (using Gaussian process (GP) regression in dashed orange) to compute the plant‐model mismatch, as described in the work 14 . As found in the works, 14,17 the worst‐case sMPC strategy fails to achieve the setpoint within a desired amount of time, here $$$ \le 25 $$$ s, because of the over‐conservative uncertainty quantification. The kernel‐based method (in orange) can achieve the desired CEM within the allotted time, but the BNN (in blue) and RBNN (in green) methods offer improved performance in delivering the desired CEM dose.…”

“…RBNN architecture and training: Similar to our recent work, 17 a DenseVariational layer with linear activation connected to a four‐layer fully‐connected neural network with the Exponential Linear Unit activation functions was used to represent the plant‐model mismatch $$$ y $$$. Each of the three hidden layers had 32 units.…”

“…However, GPR suffers from a number of shortcomings, including: (i) a cubic complexity because of the inversion and determinant of its $$$ {N}_s\times {N}_s $$$ kernel matrix, $$$ {N}_s $$$ being the data size, although a variety of scalable GPR methods have been presented, 16 and (ii) a reliance on the assumption that the model uncertainty can be described by a joint Gaussian distribution. Alternatively, the authors have recently demonstrated the use of Bayesian neural networks (BNNs) for describing state‐ and input‐dependent model uncertainty in learning‐ and scenario‐based MPC 17 . BNNs treat the weights of a deterministic neural network (NN) as random variables with given prior distributions and provide an estimation of the posterior distributions conditioned on data using variational inference 18 .…”

“…While BNNs can provide robust predictions using model averaging, 20 uncertainties quantified by BNNs may exhibit limitations in adversarial settings, 21 especially when minor perturbations to input samples result in large prediction errors due to process noise. In Reference 17, we observed that BNN estimations of model uncertainty for certain states and inputs could even be worse than the worst‐case estimates of uncertainty bounds, resulting in too conservative control performance. Lipschitz constant has recently been used as a measure of robustness of NN models, 22 wherein the robustness of neural networks (NNs) can be enhanced via regularizing their training loss function by the Lipschitz constant 23,24 .…”

“…Therefore, the probability of ensuring safety is related to the probability with which the real states are bounded by the states of scenarios along the trajectory. By quantifying tighter bounds for model uncertainty via RBNN, the proposed adaptive‐scenario‐tree MPC increases the probability of safety, as compared to adaptive‐scenario‐tree MPC based on BNNs 17 …”