arXiv subjects

Richard D. Braatz

Publications and source records attributed to Richard D. Braatz.

At least 37 records · Page 2Linked to original sources

arXiv2025

A Time-certified Predictor-corrector IPM Algorithm for Box-QP

Minimizing both the worst-case and average execution times of optimization algorithms is equally critical in real-time optimization-based control applications such as model predictive control (MPC). Most MPC solvers have to trade off between certified worst-case and practical average execution times. For example, our previous work [1] proposed a full-Newton path-following interior-point method (IPM) with data-independent, simple-calculated, and exact $O(\sqrt{n})$ iteration complexity, but not as efficient as the heuristic Mehrotra predictor-corrector IPM algorithm (which sacrifices global convergence). This letter proposes a new predictor-corrector IPM algorithm that preserves the same certified $O(\sqrt{n})$ iteration complexity while achieving a $5\times$ speedup over [1]. Numerical experiments and codes that validate these results are provided.

math.OC↗

arXiv2025

Radial Variations in Residence Time Distribution for Pipe Flows

Suspensions of low-diffusing particles in pipe flows exhibit a difference in age at different radial positions. Particles near the channel walls have higher residence times than the cross-sectional average. We quantify this effect using Monte-Carlo simulations, and show the existence of two different regimes: a "transitional" regime where delay compounds with channel length, and a "far-field" regime where diffusion counterbalances advection. The results presented therein can be used to quantify residence time distributions near the walls of the tube. This effect is important to consider in experiments involving the kinetics of nanometer-scale particles using modern inline analytical tools. This work also provide a radially resolved extension of classical Taylor dispersion results.

physics.flu-dyn↗

arXiv2025

Bang-Ride Optimal Control: Monotonicity, External Positivity, and Fast Battery Charging

This work studies a class of optimal control problems with scalar inputs and general constraints, whose solutions follow a bang-ride pattern that always activates a constraint and enables efficient numerical computation. As a motivating example, fast battery charging leads to computationally demanding optimal control problems when detailed electrochemical models are used. Recently proposed optimization-free heuristics reduce this computational cost while producing input profiles observed in practice, following a bang-ride pattern and applying the maximum feasible input. We investigate when such heuristics satisfy necessary optimality conditions. By leveraging Pontryagin's maximum principle, we unify and formalize existing insights on the bang-ride structure and on the optimal control attaining the maximum feasible input under monotonicity. We further establish a novel connection between the structured optimal control and the external positivity of the costate dynamics. These results provide a rigorous theoretical foundation for heuristic charging strategies and explain the efficiency of optimization-free algorithms.

math.OC↗

arXiv2025

Highly Efficient Optimal Control for Lyophilization via Simulation of Discrete/Continuous Mixed-index Differential-algebraic Equations

This article presents a highly efficient optimal control algorithm and policies for lyophilization (also known as freeze drying). The optimal solutions and control policies are derived using an extended version of the simulation-based algorithm, which reformulates the optimal control problem as a hybrid discrete/continuous system of mixed-index differential-algebraic equations and subsequently calculates the optimal control vector via simulation of the resulting DAEs. Our algorithm and control policies are demonstrated via a number of case studies that encompass various lyophilization and optimal control strategies. All the case studies can be solved within less than a second on a normal laptop, regardless of their complexity. The method is several orders of magnitude faster than the traditional optimization-based techniques while giving similar/better accuracy. The proposed algorithm offers an efficient and reliable framework for optimal control of lyophilization, which can also be extended to other similar systems with phase transitions.

math.OC↗

arXiv2025

A Quadratic Programming Algorithm with $O(n^3)$ Time Complexity

Solving linear systems and quadratic programming (QP) problems are both ubiquitous tasks in the engineering and computing fields. Direct methods for solving systems, such as Cholesky, LU, and QR factorizations, exhibit data-independent time complexity of $O(n^3)$. This raises a natural question: could there exist algorithms for solving QPs that also achieve \textit{data-independent} time complexity of $O(n^3)$? This raises a natural question: could there exist algorithms for solving QPs that also achieve data-independent time complexity of $O(n^3)$? This is critical for offering an execution time certificate for real-time optimization-based applications such as model predictive control. This article first demonstrates that solving real-time strictly convex QPs, Lasso problems, and support vector machine problems can be turned into solving box-constrained QPs (Box-QPs), which support a cost-free initialization strategy for feasible interior-point methods (IPMs). Next, focusing on solving Box-QPs, this article replaces the exact Newton step with an approximated Newton step (substituting the matrix-inversion operation with multiple rank-1 updates) within feasible IPMs. For the first time, this article proposes an implementable feasible IPM algorithm with $O(n^3)$ time complexity, by proving the number of iterations is exact $O(\sqrt{n})$ and the number of rank-1 updates is bounded by $O(n)$. Numerical validations/applications and codes are provided.

math.OC↗

arXiv2025

Perturbed Gradient Descent Algorithms are Small-Disturbance Input-to-State Stable

This article investigates the robustness of gradient descent algorithms under perturbations. The concept of small-disturbance input-to-state stability (ISS) for discrete-time nonlinear dynamical systems is introduced, along with its Lyapunov characterization. The conventional linear Polyak-Lojasiewicz (PL) condition is then extended to a nonlinear version, and it is shown that the gradient descent algorithm is small-disturbance ISS provided the objective function satisfies the generalized nonlinear PL condition. This small-disturbance ISS property guarantees that the gradient descent algorithm converges to a small neighborhood of the optimum under sufficiently small perturbations. As a direct application of the developed framework, we demonstrate that the LQR cost satisfies the generalized nonlinear PL condition, thereby establishing that the policy gradient algorithm for LQR is small-disturbance ISS. Additionally, other popular policy gradient algorithms, including natural policy gradient and Gauss-Newton method, are also proven to be small-disturbance ISS.

math.OC↗

arXiv2025

Line Tension Reshapes Nucleation at Surface Edges: A Generalized Theory for Nanopore Activation

Heterogeneous nucleation at surface edges is pervasive across nature and industry, yet the role of line tension, arising from asymmetric capillary interactions at geometric singularities, remains poorly understood. Herein we develop a generalized nucleation theory that explicitly incorporates line tension induced by edge pinning, thereby extending classical frameworks to account for nanoscale confinement and interfacial asymmetry. Through analytical treatment of droplet formation within geometrically defined nanopores, we derive a closed-form expression for the edge-pinned line tension as a function of Laplace pressure, pore geometry, and wettability. This formulation reveals that line tension can significantly reshape the nucleation energy landscape, introducing nontrivial dependencies on contact angle and pore morphology. Our results uncover a tunable, geometry-mediated mechanism for controlling nucleation barriers, offering predictive insight into phase transitions in confined environments and suggesting new strategies for design in applications ranging from nanofluidics to crystallization control.

cond-mat.soft↗

arXiv2025

Simulation-based Approach for Fast Optimal Control of a Stefan Problem with Application to Cell Therapy

This article describes a new, efficient way of finding control and state trajectories in optimal control problems by reformulation as a system of differential-algebraic equations (DAEs). The optimal control and state vectors can be obtained via simulation of the resulting DAE system with the selected DAE solver, eliminating the need for an optimization solver. Our simulation-based approach is demonstrated and benchmarked against various optimization-based algorithms via four case studies associated with the optimization and control of a Stefan problem for cell therapy. The simulation-based approach is faster than every optimization-based method by more than an order of magnitude while giving similar/better accuracy in all cases. The solution obtained from the simulation-based approach is guaranteed to be optimal provided that at least one constraint or algebraic equation resulting from the reformulation remains active at all times. The proposed technique offers an efficient and reliable framework for optimal control, serving as a promising alternative to the traditional techniques in applications where speed is crucial, e.g., real-time online model predictive control.

eess.SY↗

arXiv2025

Data-Driven Nonlinear Regulation: Gaussian Process Learning

This article addresses the output regulation problem for a class of nonlinear systems using a data-driven approach. An output feedback controller is proposed that integrates a traditional control component with a data-driven learning algorithm based on Gaussian Process (GP) regression to learn the nonlinear internal model. Specifically, a data-driven technique is employed to directly approximate the unknown internal model steady-state map from observed input-output data online. Our method does not rely on model-based observers utilized in previous studies, making it robust and suitable for systems with modelling errors and model uncertainties. Finally, we demonstrate through numerical examples and detailed stability analysis that, under suitable conditions, the closed-loop system remains bounded and converges to a compact set, with the size of this set decreasing as the accuracy of the data-driven model improves over time.

eess.SY↗

arXiv2025

Mechanistic Modeling of Lipid Nanoparticle Formation for the Delivery of Nucleic Acid Therapeutics

Nucleic acids such as mRNA have emerged as a promising therapeutic modality with the capability of addressing a wide range of diseases. Lipid nanoparticles (LNPs) as a delivery platform for nucleic acids were used in the COVID-19 vaccines and have received much attention. While modern manufacturing processes which involve rapidly mixing an organic stream containing the lipids with an aqueous stream containing the nucleic acids are conceptually straightforward, detailed understanding of LNP formation and structure is still limited and scale-up can be challenging. Mathematical and computational methods are a promising avenue for deepening scientific understanding of the LNP formation process and facilitating improved process development and control. This article describes strategies for the mechanistic modeling of LNP formation, starting with strategies to estimate and predict important physicochemical properties of the various species such as diffusivities and solubilities. Subsequently, a framework is outlined for constructing mechanistic models of reactor- and particle-scale processes. Insights gained from the various models are mapped back to product quality attributes and process insights. Lastly, the use of the models to guide development of advanced process control and optimization strategies is discussed.

cond-mat.soft↗

arXiv2025

Bayesian Analysis of Interpretable Aging across Thousands of Lithium-ion Battery Cycles

The Doyle-Fuller-Newman (DFN) model is a common mechanistic model for lithium-ion batteries. The reaction rate constant and diffusivity within the DFN model are key parameters that directly affect the movement of lithium ions, thereby offering explanations for cell aging. This work investigates the ability to uniquely estimate each electrode's diffusion coefficients and reaction rate constants of 95 Tesla Model 3 cells with a nickel cobalt aluminum oxide (NCA) cathode and silicon oxide--graphite (LiC$_\text{6}$--SiO$_{\text{x}}$) anode. The parameters are estimated at intermittent diagnostic cycles over the lifetime of each cell. The four parameters are estimated using Markov chain Monte Carlo (MCMC) for uncertainty quantification (UQ) for a total of 7776 cycles at discharge C-rates of C/5, 1C, and 2C. While one or more anode parameters are uniquely identifiable over every cell's lifetime, cathode parameters become identifiable at mid- to end-of-life, indicating measurable resistive growth in the cathode. The contribution of key parameters to the state of health (SOH) is expressed as a power law. This model for SOH shows a high consistency with the MCMC results performed over the overall lifespan of each cell. Our approach suggests that effective diagnosis of aging can be achieved by predicting the trajectories of the parameters contributing to cell aging. As such, extending our analysis with more physically accurate models building on DFN may lead to more identifiable parameters and further improved aging predictions.

eess.SY↗

arXiv2025

Mechanistic Modeling of Lipid Nanoparticle (LNP) Precipitation via Population Balance Equations (PBEs)

Lipid nanoparticles (LNPs) are precisely engineered drug delivery carriers commonly produced through controlled mixing processes, such as nanoprecipitation. Since their delivery efficacy greatly depends on particle size, numerous studies have proposed experimental and theoretical approaches for tuning LNP size. However, the mechanistic model for LNP fabrication has rarely been established alongside experiments, limiting a profound understanding of the kinetic processes governing LNP self-assembly. Thus, we present a population balance equation (PBE)-based model that captures the evolution of the particle size distribution (PSD) during LNP fabrication, to provide mechanistic insight into how kinetic processes control LNP size. The model showed strong agreement with experimentally observed trends in the PSD. In addition to identifying the role of each kinetic process in shaping the PSD, we analyzed the underlying mechanisms of three key operational strategies: manipulation of (1) lipid concentration, (2) flow rate ratio (FRR), and (3) mixing rate. We identified that the key to producing precisely controlled particle size lies in controlling super-saturation and lipid dilution to regulate the balance between nucleation and growth. Our findings provide mechanistic understanding that is essential in further developing strategies for tuning LNP size.

physics.chem-ph↗

arXiv2025

Emergent kinetics of in vitro transcription from interactions of T7 RNA polymerase and DNA

The in vitro transcription reaction (IVT) is of growing importance for the manufacture of RNA vaccines and therapeutics. While the kinetics of the microscopic steps of this reaction (promoter binding, initiation, and elongation) are well studied, the rate law of overall RNA synthesis that emerges from this system is unclear. In this work, we show that a model that incorporates both initiation and elongation steps is essential for describing trends in IVT kinetics in conditions relevant to RNA manufacturing. In contrast to previous reports, we find that the IVT reaction can be either initiation- or elongation-limited depending on solution conditions. This initiation-elongation model is also essential for describing the effect of salts, which disrupt polymerase-promoter binding, on transcription rates. Polymerase-polymerase interactions during elongation are incorporated into our modeling framework and found to have nonzero but unidentifiable effects on macroscopic transcription rates. Finally, we develop an extension of our modeling approach to quantitatively describe and experimentally evaluate RNA- and DNA-templated mechanisms for the formation of double-stranded RNA (dsRNA) impurities. We show experimental results that indicate that an RNA-templated mechanism is not appropriate for describing macroscopic dsRNA formation in the context of RNA manufacturing.

q-bio.MN↗

arXiv2025

Digital Twins in Biopharmaceutical Manufacturing: Review and Perspective on Human-Machine Collaborative Intelligence

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.

cs.HC↗

arXiv2025

Diagnostic-free onboard battery health assessment

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.

eess.SY↗

arXiv2025

Probabilistically Robust Uncertainty Analysis and Optimal Control of Continuous Lyophilization via Polynomial Chaos Theory

Lyophilization, aka freeze drying, is a process commonly used to increase the stability of various drug products in biotherapeutics manufacturing, e.g., mRNA vaccines, allowing for higher storage temperature. While the current trends in the industry are moving towards continuous manufacturing, the majority of industrial lyophilization processes are still being operated in a batch mode. This article presents a framework that accounts for the probabilistic uncertainty during the primary and secondary drying steps in continuous lyophilization. The probabilistic uncertainty is incorporated into the mechanistic model via polynomial chaos theory (PCT). The resulting PCT-based model is able to accurately and efficiently quantify the effects of uncertainty on several critical process variables, including the temperature, sublimation front, and concentration of bound water. The integration of the PCT-based model into stochastic optimization and control is demonstrated. The proposed framework and case studies can be used to guide the design and control of continuous lyophilization while accounting for probabilistic uncertainty.

cs.CE↗

arXiv2025

EIQP: Execution-time-certified and Infeasibility-detecting QP Solver

Solving real-time quadratic programming (QP) is a ubiquitous task in control engineering, such as in model predictive control and control barrier function-based QP. In such real-time scenarios, certifying that the employed QP algorithm can either return a solution within a predefined level of optimality or detect QP infeasibility before the predefined sampling time is a pressing requirement. This article considers convex QP (including linear programming) and adopts its homogeneous formulation to achieve infeasibility detection. Exploiting this homogeneous formulation, this article proposes a novel infeasible interior-point method (IPM) algorithm with the best theoretical $O(\sqrt{n})$ iteration complexity that feasible IPM algorithms enjoy. The iteration complexity is proved to be \textit{exact} (rather than an upper bound), \textit{simple to calculate}, and \textit{data independent}, with the value $\left\lceil\frac{\log(\frac{n+1}ε)}{-\log(1-\frac{0.414213}{\sqrt{n+1}})}\right\rceil$ (where $n$ and $ε$ denote the number of constraints and the predefined optimality level, respectively), making it appealing to certify the execution time of online time-varying convex QPs. The proposed algorithm is simple to implement without requiring a line search procedure (uses the full Newton step), and its C-code implementation (offering MATLAB, Julia, and Python interfaces) and numerical examples are publicly available at https://github.com/liangwu2019/EIQP.

eess.SY↗

arXiv2025

A Phase Diagram for Crystallization of a Complex Macromolecular Assembly

Crystallization of biological molecules has high potential to solve some challenges in drug manufacturing. Thus, understanding the process is critical to efficiently adapting crystallization to biopharmaceutical manufacturing. This article describes phase behavior for the solution crystallization of recombinant adeno-associated virus (rAAV) capsids of serotypes 5, 8, and 9 as model biological macromolecular assemblies. Hanging-drop vapor diffusion experiments are used to determine the combined effects of pH and polyethylene glycol (PEG) and sodium chloride concentrations in which full and empty capsids nucleate and grow. Full and empty capsids show different crystallization behavior although they possess similar capsid structure and similar outer morphology with icosahedral symmetry and 2-fold, 3-fold, and 5-fold symmetry. The differential charge environment surrounding full and empty capsids is found to influence capsid crystallization. The crystal growth rate is found to be affected by the mass of the macromolecular assembly rather than the structure/shape of the macromolecular assembly. The regions of precipitant concentrations and pH in which crystallization occurs are found to be different for different rAAV serotypes and for full and empty capsids for each serotype. Depending on the precipitant concentrations and the rAAV serotype, a variety of complex crystal morphologies are formed and a variety of non-crystallization outcomes such as unidentified dense solid-phase/opaque crystals and an oil/dense phase is observed. The well-defined dense phase/oil is found to be converted into a solid phase over a long period of time. Trends in the crystallization of full and empty capsids between serotypes is observed to be altered by the extent of post-translational modifications (PTMS) associated with the massive macromolecular proteinaceous assembly.

cond-mat.soft↗