Safe and High-Performance Learning of Model Predicitve Control using Kernel-Based Interpolation

We present a method, which allows efficient and safe approximation of model predictive controllers using kernel interpolation. Since the computational complexity of the approximating function scales linearly with the number of data points, we propose to use a scoring function which chooses the most promising data. To further reduce the complexity of the approximation, we restrict our considerations to the set of closed-loop reachable states. That is, the approximating function only has to be accurate within this set. This makes our method especially suited for systems, where the set of initial conditions is small. In order to guarantee safety and high performance of the designed approximated controller, we use reachability analysis based on Monte Carlo methods.

Safe Learning-Based Optimization of Model Predictive Control: Application to Battery Fast-Charging

Model predictive control (MPC) is a powerful tool for controlling complex nonlinear systems under constraints, but often struggles with model uncertainties and the design of suitable cost functions. To address these challenges, we discuss an approach that integrates MPC with safe Bayesian optimization to optimize long-term closed-loop performance despite significant model-plant mismatches. By parameterizing the MPC stage cost function using a radial basis function network, we employ Bayesian optimization as a multi-episode learning strategy to tune the controller without relying on precise system models. This method mitigates conservativeness introduced by overly cautious soft constraints in the MPC cost function and provides probabilistic safety guarantees during learning, ensuring that safety-critical constraints are met with high probability. As a practical application, we apply our approach to fast charging of lithium-ion batteries, a challenging task due to the complicated battery dynamics and strict safety requirements, subject to the requirement to be implementable in real time. Simulation results demonstrate that, in the context of model-plant mismatch, our method reduces charging times compared to traditional MPC methods while maintaining safety. This work extends previous research by emphasizing closed-loop constraint satisfaction and offers a promising solution for enhancing performance in systems where model uncertainties and safety are critical concerns.

Safe and Stable Closed-Loop Learning for Neural-Network-Supported Model Predictive Control

Safe learning of control policies remains challenging, both in optimal control and reinforcement learning. In this article, we consider safe learning of parametrized predictive controllers that operate with incomplete information about the underlying process. To this end, we employ Bayesian optimization for learning the best parameters from closed-loop data. Our method focuses on the system's overall long-term performance in closed-loop while keeping it safe and stable. Specifically, we parametrize the stage cost function of an MPC using a feedforward neural network. This allows for a high degree of flexibility, enabling the system to achieve a better closed-loop performance with respect to a superordinate measure. However, this flexibility also necessitates safety measures, especially with respect to closed-loop stability. To this end, we explicitly incorporated stability information in the Bayesian-optimization-based learning procedure, thereby achieving rigorous probabilistic safety guarantees. The proposed approach is illustrated using a numeric example.

Neural Horizon Model Predictive Control -- Increasing Computational Efficiency with Neural Networks

The expansion in automation of increasingly fast applications and low-power edge devices poses a particular challenge for optimization based control algorithms, like model predictive control. Our proposed machine-learning supported approach addresses this by utilizing a feed-forward neural network to reduce the computation load of the online-optimization. We propose approximating part of the problem horizon, while maintaining safety guarantees -- constraint satisfaction -- via the remaining optimization part of the controller. The approach is validated in simulation, demonstrating an improvement in computational efficiency, while maintaining guarantees and near-optimal performance. The proposed MPC scheme can be applied to a wide range of applications, including those requiring a rapid control response, such as robotics and embedded applications with limited computational resources.

Lithium-Ion Battery System Health Monitoring and Fault Analysis from Field Data Using Gaussian Processes

Health monitoring, fault analysis, and detection are critical for the safe and sustainable operation of battery systems. We apply Gaussian process resistance models on lithium iron phosphate battery field data to effectively separate the time-dependent and operating point-dependent resistance. The data set contains 29 battery systems returned to the manufacturer for warranty, each with eight cells in series, totaling 232 cells and 131 million data rows. We develop probabilistic fault detection rules using recursive spatiotemporal Gaussian processes. These processes allow the quick processing of over a million data points, enabling advanced online monitoring and furthering the understanding of battery pack failure in the field. The analysis underlines that often, only a single cell shows abnormal behavior or a knee point, consistent with weakest-link failure for cells connected in series, amplified by local resistive heating. The results further the understanding of how batteries degrade and fail in the field and demonstrate the potential of efficient online monitoring based on data. We open-source the code and publish the large data set upon completion of the review of this article.

Stability-informed Bayesian Optimization for MPC Cost Function Learning

Designing predictive controllers towards optimal closed-loop performance while maintaining safety and stability is challenging. This work explores closed-loop learning for predictive control parameters under imperfect information while considering closed-loop stability. We employ constrained Bayesian optimization to learn a model predictive controller's (MPC) cost function parametrized as a feedforward neural network, optimizing closed-loop behavior as well as minimizing model-plant mismatch. Doing so offers a high degree of freedom and, thus, the opportunity for efficient and global optimization towards the desired and optimal closed-loop behavior. We extend this framework by stability constraints on the learned controller parameters, exploiting the optimal value function of the underlying MPC as a Lyapunov candidate. The effectiveness of the proposed approach is underlined in simulations, highlighting its performance and safety capabilities.

Learning Model Predictive Control Parameters via Bayesian Optimization for Battery Fast Charging

Tuning parameters in model predictive control (MPC) presents significant challenges, particularly when there is a notable discrepancy between the controller's predictions and the actual behavior of the closed-loop plant. This mismatch may stem from factors like substantial model-plant differences, limited prediction horizons that do not cover the entire time of interest, or unforeseen system disturbances. Such mismatches can jeopardize both performance and safety, including constraint satisfaction. Traditional methods address this issue by modifying the finite horizon cost function to better reflect the overall operational cost, learning parts of the prediction model from data, or implementing robust MPC strategies, which might be either computationally intensive or overly cautious. As an alternative, directly optimizing or learning the controller parameters to enhance closed-loop performance has been proposed. We apply Bayesian optimization for efficient learning of unknown model parameters and parameterized constraint backoff terms, aiming to improve closed-loop performance of battery fast charging. This approach establishes a hierarchical control framework where Bayesian optimization directly fine-tunes closed-loop behavior towards a global and long-term objective, while MPC handles lower-level, short-term control tasks. For lithium-ion battery fast charging, we show that the learning approach not only ensures safe operation but also maximizes closed-loop performance. This includes maintaining the battery's operation below its maximum terminal voltage and reducing charging times, all achieved using a standard nominal MPC model with a short horizon and notable initial model-plant mismatch.

Cycle Life Prediction for Lithium-ion Batteries: Machine Learning and More

Batteries are dynamic systems with complicated nonlinear aging, highly dependent on cell design, chemistry, manufacturing, and operational conditions. Prediction of battery cycle life and estimation of aging states is important to accelerate battery R&D, testing, and to further the understanding of how batteries degrade. Beyond testing, battery management systems rely on real-time models and onboard diagnostics and prognostics for safe operation. Estimating the state of health and remaining useful life of a battery is important to optimize performance and use resources optimally. This tutorial begins with an overview of first-principles, machine learning, and hybrid battery models. Then, a typical pipeline for the development of interpretable machine learning models is explained and showcased for cycle life prediction from laboratory testing data. We highlight the challenges of machine learning models, motivating the incorporation of physics in hybrid modeling approaches, which are needed to decipher the aging trajectory of batteries but require more data and further work on the physics of battery degradation. The tutorial closes with a discussion on generalization and further research directions.

Interpretation of High-Dimensional Linear Regression: Effects of Nullspace and Regularization Demonstrated on Battery Data

High-dimensional linear regression is important in many scientific fields. This article considers discrete measured data of underlying smooth latent processes, as is often obtained from chemical or biological systems. Interpretation in high dimensions is challenging because the nullspace and its interplay with regularization shapes regression coefficients. The data's nullspace contains all coefficients that satisfy $\mathbf{Xw}=\mathbf{0}$, thus allowing very different coefficients to yield identical predictions. We developed an optimization formulation to compare regression coefficients and coefficients obtained by physical engineering knowledge to understand which part of the coefficient differences are close to the nullspace. This nullspace method is tested on a synthetic example and lithium-ion battery data. The case studies show that regularization and z-scoring are design choices that, if chosen corresponding to prior physical knowledge, lead to interpretable regression results. Otherwise, the combination of the nullspace and regularization hinders interpretability and can make it impossible to obtain regression coefficients close to the true coefficients when there is a true underlying linear model. Furthermore, we demonstrate that regression methods that do not produce coefficients orthogonal to the nullspace, such as fused lasso, can improve interpretability. In conclusion, the insights gained from the nullspace perspective help to make informed design choices for building regression models on high-dimensional data and reasoning about potential underlying linear models, which are important for system optimization and improving scientific understanding.

LMI-based Data-Driven Robust Model Predictive Control

Predictive control, which is based on a model of the system to compute the applied input optimizing the future system behavior, is by now widely used. If the nominal models are not given or are very uncertain, data-driven model predictive control approaches can be employed, where the system model or input is directly obtained from past measured trajectories. Using a data informativity framework and Finsler's lemma, we propose a data-driven robust linear matrix inequality-based model predictive control scheme that considers input and state constraints. Using these data, we formulate the problem as a semi-definite optimization problem, whose solution provides the matrix gain for the linear feedback, while the decisive variables are independent of the length of the measurement data. The designed controller stabilizes the closed-loop system asymptotically and guarantees constraint satisfaction. Numerical examples are conducted to illustrate the method.