Richard D. Braatz

Joachim Schaeffer, Paul Gasper, Esteban Garcia-Tamayo et al.

Analysis of Electrochemical Impedance Spectroscopy (EIS) data for electrochemical systems often consists of defining an Equivalent Circuit Model (ECM) using expert knowledge and then optimizing the model parameters to deconvolute various resistance, capacitive, inductive, or diffusion responses. For small data sets, this procedure can be conducted manually; however, it is not feasible to manually define a proper ECM for extensive data sets with a wide range of EIS responses. Automatic identification of an ECM would substantially accelerate the analysis of large sets of EIS data. We showcase machine learning methods to classify the ECMs of 9,300 impedance spectra provided by QuantumScape for the BatteryDEV hackathon. The best-performing approach is a gradient-boosted tree model utilizing a library to automatically generate features, followed by a random forest model using the raw spectral data. A convolutional neural network using boolean images of Nyquist representations is presented as an alternative, although it achieves a lower accuracy. We publish the data and open source the associated code. The approaches described in this article can serve as benchmarks for further studies. A key remaining challenge is the identifiability of the labels, underlined by the model performances and the comparison of misclassified spectra.

Liang Wu, Richard D. Braatz

Providing an execution time certificate is a pressing requirement when deploying Model Predictive Control (MPC) in real-time embedded systems such as microcontrollers. Real-time MPC requires that its worst-case (maximum) execution time must be theoretically guaranteed to be smaller than the sampling time in closed-loop. This technical note considers input-constrained MPC problems and exploits the structure of the resulting box-constrained QPs. Then, we propose a \textit{cost-free} and \textit{data-independent} initialization strategy, which enables us, for the first time, to remove the initialization assumption of feasible full-Newton interior-point algorithms. We prove that the number of iterations of our proposed algorithm is \textit{only dimension-dependent} (\textit{data-independent}), \textit{simple-calculated}, and \textit{exact} (not \textit{worst-case}) with the value $\left\lceil\frac{\log(\frac{2n}ε)}{-2\log(\frac{\sqrt{2n}}{\sqrt{2n}+\sqrt{2}-1})}\right\rceil \!+ 1$, where $n$ denotes the problem dimension and $ε$ denotes the constant stopping tolerance. These features enable our algorithm to trivially certify the execution time of nonlinear MPC (via online linearized schemes) or adaptive MPC problems. The execution-time-certified capability of our algorithm is theoretically and numerically validated through an open-loop unstable AFTI-16 example.

Leilei Cui, Richard D. Braatz

A gradient-based method is proposed for solving the linear quadratic regulator (LQR) problem for linear systems with nonlinear dependence on time-invariant probabilistic parametric uncertainties. The approach explicitly accounts for model uncertainty and ensures robust performance. By leveraging polynomial chaos theory (PCT) in conjunction with policy optimization techniques, the original stochastic system is lifted into a high-dimensional linear time-invariant (LTI) system with structured state-feedback control. A first-order gradient descent algorithm is then developed to directly optimize the structured feedback gain and iteratively minimize the LQR cost. We rigorously establish linear convergence of the gradient descent algorithm and show that the PCT-based approximation error decays algebraically at a rate $O(N^{-p})$ for any positive integer $p$, where $N$ denotes the order of the polynomials. Numerical examples demonstrate that the proposed method achieves significantly higher computational efficiency than conventional bilinear matrix inequality (BMI)-based approaches.

Yingjie Ma, Jing Guo, Richard D. Braatz

Efficient implementation of nonlinear model predictive control (NMPC) for bioprocesses remains challenging because large nonlinear models are difficult to organize, simulate, and embed within optimization and control workflows. This difficulty is particularly pronounced for large-scale and multiscale systems that require hierarchical model construction and customized simulation strategies. To address this issue, we present GlycoPy, a CasADi-based Python framework for hierarchical modeling, optimization, and control of bioprocesses. GlycoPy combines an equation-oriented, object-oriented modeling architecture with CasADi's symbolic and differentiable computational capabilities, enabling hierarchical model composition, numerical and symbolic simulation, parameter estimation, dynamic optimization, and NMPC within a unified workflow. A key feature of the framework is its support for customized differentiable simulation algorithms that can be embedded directly in gradient-based optimization and control. GlycoPy is demonstrated on a multiscale monoclonal antibody glycosylation process in Chinese hamster ovary cell culture, where it is used for hierarchical model construction, quasi-steady-state simulation, and adaptive NMPC. The results show that GlycoPy provides a practical and reusable framework for applying advanced optimization and control methods to computationally demanding bioprocesses.

Pedro Seber, Richard D. Braatz

The LASSO-Clip-EN (LCEN) algorithm was previously introduced for nonlinear, interpretable feature selection and machine learning. However, its design and use was limited to regression tasks. In this work, we create a modified version of the LCEN algorithm that is suitable for classification tasks and maintains its desirable properties, such as interpretability. This modified LCEN algorithm is evaluated on four widely used binary and multiclass classification datasets. In these experiments, LCEN is compared against 10 other model types and consistently reaches high test-set macro F$_1$ score and Matthews correlation coefficient (MCC) metrics, higher than that of the majority of investigated models. LCEN models for classification remain sparse, eliminating an average of 56% of all input features in the experiments performed. Furthermore, LCEN-selected features are used to retrain all models using the same data, leading to statistically significant performance improvements in three of the experiments and insignificant differences in the fourth when compared to using all features or other feature selection methods. Simultaneously, the weighted focal differentiable MCC (diffMCC) loss function is evaluated on the same datasets. Models trained with the diffMCC loss function are always the best-performing methods in these experiments, and reach test-set macro F$_1$ scores that are, on average, 4.9% higher and MCCs that are 8.5% higher than those obtained by models trained with the weighted cross-entropy loss. These results highlight the performance of LCEN as a feature selection and machine learning algorithm also for classification tasks, and how the diffMCC loss function can train very accurate models, surpassing the weighted cross-entropy loss in the tasks investigated.

Joachim Schaeffer, Eric Lenz, Duncan Gulla et al.

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.

Joachim Schaeffer, Giacomo Galuppini, Jinwook Rhyu et al.

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.

Sebastian Hirt, Andreas Höhl, Joachim Schaeffer et al.

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.

Pedro Seber, Richard D. Braatz

Interpretable architectures can have advantages over black-box architectures, and interpretability is essential for the application of machine learning in critical settings, such as aviation or medicine. However, the simplest, most commonly used interpretable architectures, such as LASSO or elastic net (EN), are limited to linear predictions and have poor feature selection capabilities. In this work, we introduce the LASSO-Clip-EN (LCEN) algorithm for the creation of nonlinear, interpretable machine learning models. LCEN is tested on a wide variety of artificial and empirical datasets, frequently creating more accurate, sparser models than other architectures, including those for building sparse, nonlinear models. LCEN is robust against many issues typically present in datasets and modeling, including noise, multicollinearity, data scarcity, and hyperparameter variance. LCEN is also able to rediscover multiple physical laws from empirical data and, for processes with no known physical laws, LCEN achieves better results than many other dense and sparse methods -- including using 10.8-fold fewer features than dense methods and 8.1-fold fewer features than EN on one dataset, and is comparable to or better than ANNs on multiple datasets.

Yunhong Che, Vivek N. Lam, Jinwook Rhyu et al.

Diverse usage patterns induce complex and variable aging behaviors in lithium-ion batteries, complicating accurate health diagnosis and prognosis. Separate diagnostic cycles are often used to untangle the battery's current state of health from prior complex aging patterns. However, these same diagnostic cycles alter the battery's degradation trajectory, are time-intensive, and cannot be practically performed in onboard applications. In this work, we leverage portions of operational measurements in combination with an interpretable machine learning model to enable rapid, onboard battery health diagnostics and prognostics without offline diagnostic testing and the requirement of historical data. We integrate mechanistic constraints within an encoder-decoder architecture to extract electrode states in a physically interpretable latent space and enable improved reconstruction of the degradation path. The health diagnosis model framework can be flexibly applied across diverse application interests with slight fine-tuning. We demonstrate the versatility of this model framework by applying it to three battery-cycling datasets consisting of 422 cells under different operating conditions, highlighting the utility of an interpretable diagnostic-free, onboard battery diagnosis and prognosis model.

Liang Wu, Yunhong Che, Wallace Gian Yion Tan et al.

Lasso problems arise in many areas, including signal processing, machine learning, and control, and are closely connected to sparse coding mechanisms observed in neuroscience. A continuous-time ordinary differential equation (ODE) representation of the Lasso problem not only enables its solution on analog computers but also provides a framework for interpreting neurophysiological phenomena. This article proposes a fixed-time-stable ODE representation of the Lasso problem by first transforming it into a smooth nonnegative quadratic program (QP) and then designing a projection-free Newton-based ODE representation of the Lasso problem by first transforming it into a smooth nonnegative quadratic program (QP) and then designing a projection-free Newton-based fixed-time-stable ODE system for solving the corresponding Karush-Kuhn-Tucker (KKT) conditions. Moreover, the settling time of the ODE is independent of the problem data and can be arbitrarily prescribed. Numerical experiments verify that the trajectory reaches the optimal solution within the prescribed time.

Efstathios Iliakis, Wallace Gian Yion Tan, Liang Wu et al.

This paper develops a parametric Koopman operator framework for Stochastic Model Predictive Control (SMPC), where the Koopman operator is parametrized by Polynomial Chaos Expansions (PCEs). The model is learned from data using the Extended Dynamic Mode Decomposition -- Dictionary Learning (EDMD-DL) method, which preserves the convex least-squares structure for the PCE coefficients of the EDMD matrix. Unlike conventional stochastic Galerkin projection approaches, we derive a condensed deterministic reformulation of the SMPC problem whose dimension scales only with the control horizon and input dimension, and is independent of both the lifted state dimension and the number of retained PCE terms. Our framework, therefore, enables efficient nonlinear SMPC problems with expectation and second-order moment constraints with standard convex optimization solvers. Numerical examples demonstrate the efficacy of our framework for uncertainty-aware SMPC of nonlinear systems.

Mohammed Aatif Shahab, Francesco Destro, Richard D. Braatz

The biopharmaceutical industry is increasingly developing digital twins to digitalize and automate the manufacturing process in response to the growing market demands. However, this shift presents significant challenges for human operators, as the complexity and volume of information can overwhelm their ability to manage the process effectively. These issues are compounded when digital twins are designed without considering interaction and collaboration with operators, who are responsible for monitoring processes and assessing situations, particularly during abnormalities. Our review of current trends in biopharma digital twin development reveals a predominant focus on technology and often overlooks the critical role of human operators. To bridge this gap, this article proposes a collaborative intelligence framework that emphasizes the integration of operators with digital twins. Approaches to system design that can enhance operator trust and human-machine interface usability are presented. Moreover, innovative training programs for preparing operators to understand and utilize digital twins are discussed. The framework outlined in this article aims to enhance collaboration between operators and digital twins effectively by using their full capabilities to boost resilience and productivity in biopharmaceutical manufacturing.

Joachim Schaeffer, Eric Lenz, William C. Chueh et al.

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.

Weike Sun, Antonio R. C. Paiva, Peng Xu et al.

In processing and manufacturing industries, there has been a large push to produce higher quality products and ensure maximum efficiency of processes. This requires approaches to effectively detect and resolve disturbances to ensure optimal operations. While the control system can compensate for many types of disturbances, there are changes to the process which it still cannot handle adequately. It is therefore important to further develop monitoring systems to effectively detect and identify those faults such that they can be quickly resolved by operators. In this paper, a novel probabilistic fault detection and identification method is proposed which adopts a newly developed deep learning approach using Bayesian recurrent neural networks~(BRNNs) with variational dropout. The BRNN model is general and can model complex nonlinear dynamics. Moreover, compared to traditional statistic-based data-driven fault detection and identification methods, the proposed BRNN-based method yields uncertainty estimates which allow for simultaneous fault detection of chemical processes, direct fault identification, and fault propagation analysis. The outstanding performance of this method is demonstrated and contrasted to (dynamic) principal component analysis, which are widely applied in the industry, in the benchmark Tennessee Eastman process~(TEP) and a real chemical manufacturing dataset.