Nonlinear sparse variational Bayesian learning based model predictive control with application to PEMFC temperature control

TL;DR

This work tackles the challenge of reliable MPC for unknown nonlinear systems by learning a predictive model from data with nonlinear sparse variational Bayesian learning (NSVB). NSVB-MPC uses a NARX model with sparsity via automatic relevance determination (ARD) and variational inference to produce a predictive distribution $p(y_{k+1}|\cdot)$, enabling uncertainty-aware MPC with input-to-state stability (ISS) guarantees and an invariant terminal region $\Omega_\vartheta$ without terminal constraints. The learning phase yields closed-form variational updates for $Q_{\boldsymbol{\omega}}$, $Q_{\boldsymbol{\alpha}}$, and $Q_{\beta}$, promoting sparsity and robust parameter estimation, while the control phase integrates the predictive model into a finite-horizon MPC with a barrier term in the stage cost and a terminal cost $V_f$ computed via LQR around the reference. The method is validated on PEMFC temperature control, showing improved predictive reliability and reduced computation time compared to GP-based MPC, with robust performance under noise and with limited training data.

Abstract

The accuracy of the underlying model predictions is crucial for the success of model predictive control (MPC) applications. If the model is unable to accurately analyze the dynamics of the controlled system, the performance and stability guarantees provided by MPC may not be achieved. Learning-based MPC can learn models from data, improving the applicability and reliability of MPC. This study develops a nonlinear sparse variational Bayesian learning based MPC (NSVB-MPC) for nonlinear systems, where the model is learned by the developed NSVB method. Variational inference is used by NSVB-MPC to assess the predictive accuracy and make the necessary corrections to quantify system uncertainty. The suggested approach ensures input-to-state (ISS) and the feasibility of recursive constraints in accordance with the concept of an invariant terminal region. Finally, a PEMFC temperature control model experiment confirms the effectiveness of the NSVB-MPC method.

Figures (10)

• Figure 1: Cooling system schematic.
• Figure 2: GP validation results with limited training data.
• Figure 3: NSVB validation results with limited training data.
• Figure 4: GP validation results with noise.
• Figure 5: NSVB validation results with noise.

Theorems & Definitions (24)

• Definition 1
• Remark 1
• Theorem 1
• proof
• Theorem 2
• proof
• Corollary 1
• proof
• Corollary 2
• proof