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.

Model Predictive Control with Gaussian-Process-Supported Dynamical Constraints for Autonomous Vehicles

We propose a model predictive control approach for autonomous vehicles that exploits learned Gaussian processes for predicting human driving behavior. The proposed approach employs the uncertainty about the GP's prediction to achieve safety. A multi-mode predictive control approach considers the possible intentions of the human drivers. While the intentions are represented by different Gaussian processes, their probabilities foreseen in the observed behaviors are determined by a suitable online classification. Intentions below a certain probability threshold are neglected to improve performance. The proposed multi-mode model predictive control approach with Gaussian process regression support enables repeated feasibility and probabilistic constraint satisfaction with high probability. The approach is underlined in simulation, considering real-world measurements for training the Gaussian processes.

Safe Machine-Learning-supported Model Predictive Force and Motion Control in Robotics

Many robotic tasks, such as human-robot interactions or the handling of fragile objects, require tight control and limitation of appearing forces and moments alongside sensible motion control to achieve safe yet high-performance operation. We propose a learning-supported model predictive force and motion control scheme that provides stochastic safety guarantees while adapting to changing situations. Gaussian processes are used to learn the uncertain relations that map the robot's states to the forces and moments. The model predictive controller uses these Gaussian process models to achieve precise motion and force control under stochastic constraint satisfaction. As the uncertainty only occurs in the static model parts -- the output equations -- a computationally efficient stochastic MPC formulation is used. Analysis of recursive feasibility of the optimal control problem and convergence of the closed loop system for the static uncertainty case are given. Chance constraint formulation and back-offs are constructed based on the variance of the Gaussian process to guarantee safe operation. The approach is illustrated on a lightweight robot in simulations and experiments.

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.