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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 487 records · Page 27
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Data-driven modeling and control of dynamical systems using Koopman and Perron-Frobenius operators

This dissertation studies the data-driven modeling and control problem of nonlinear systems by exploiting the linear operator theoretic framework involving Koopman and Perro-Frobenius operator. A systematic linear-operator based controller design procedure has been established, which can be used to solve a variety of nonlinear control problems, including feedback stabilization using control Lyapunov functions, optimal quadratic regulation using Koopman eigenfunctions and convex optimization formulation of optimal control problem using P-F and Koopman operator approximation. As the core of data-driven modeling, we first propose a new algorithm for the finite-dimensional approximation of the linear transfer Koopman and Perron-Frobenius operator from time-series data. We argue that the existing approach for the finite-dimensional approximation of these transfer operators such as Dynamic Mode Decomposition (DMD) and Extended Dynamic Mode Decomposition (EDMD) do not capture two important properties of these operators, namely positivity and Markov property. The algorithm we propose preserves these two properties. We call the proposed algorithm as naturally structured DMD (NSDMD) since it retains the inherent properties of these operators. Naturally structured DMD algorithm leads to a better approximation of the steady-state dynamics of the system regarding computing Koopman and Perron- Frobenius operator eigenfunctions and eigenvalues. However, preserving positivity property is critical for capturing the real transient dynamics of the system. This positivity property of the transfer operators and it's finite-dimensional approximation play an important role for controller and estimator design of nonlinear systems. To solve the feedback stabilization problem for nonlinear control systems, we tried to take advantage of the Koopman operator framework. The Koopman operator approach provides a linear representation for a nonlinear dynamical system and a bilinear representation for a nonlinear control system. The problem of feedback stabilization of a nonlinear control system is then transformed to the stabilization of a bilinear control system. We propose a control Lyapunov function (CLF)-based approach for the design of stabilizing feedback controllers for the bilinear system. The search for finding a CLF for the bilinear control system is formulated as a convex optimization problem. This leads to a schematic procedure for designing CLF-based stabilizing feedback controllers for the bilinear system and hence the original nonlinear system. Another advantage of the proposed controller design approach outlined in this dissertation is that it does not require explicit knowledge of system dynamics. In particular, the bilinear representation of a nonlinear control system in the Koopman eigenfunction space can be obtained from time-series data. Next, we study the optimal quadratic regulation problem for nonlinear systems. The linear operator theoretic framework involving the Koopman operator is used to lift the dynamics of nonlinear control system to an infinite-dimensional bilinear system. The optimal quadratic regulation problem for nonlinear system is formulated in terms of the finite-dimensional approximation of the bilinear system. A convex optimization-based approach is proposed for solving the quadratic regulator problem for bilinear system. We applied a variety of examples and compared the simulation results between our framework and conventional LQR control using linearized model. For more general optimal control problems, we provide a density-function based convex formulation for the optimal control problem of the nonlinear system. The convex formulation relies on the duality result in the stability theory of a dynamical system involving density function and Perron-Frobenius operator. The optimal control problem is formulated as an infinite-dimensional convex optimization program. The finite-dimensional approximation of the optimization problem relies on the recent advances made in the data-driven computation of the Koopman operator, which is dual to the Perron-Frobenius operator. Simulation results are presented to demonstrate the application of the developed framework.

Huang, Bowen↗
NASA NTRS1994

The Local Supercluster as a test of cosmological models

The Local Supercluster kinematic properties (the Local Group infall toward the Virgo Cluster and galaxy density distribution about the Virgo Cluster) in various cosmological models are examined utilizing large-scale N-body (PM) simulations 500(exp 3) cells, 250(exp 3) particles, and box size of 400 h(exp -1) Mpc) and are compared to observations. Five models are investigated: (1) the standard, Cosmic Background Explorer Satellite (COBE)-normalized cold dark matter (CDM) model with omega = 1, h = 0.5, and sigma(sub 8) = 1.05; (2) the standard Hot Dark Matter (HDM) model with omega = 1, h = 0.75, and sigma(sub 8) = 1; (3) the tilted CDM model with omega = 1, h = 0.5, n = 0.7, and sigma(sub 8) = 0.5; (4) a CDM + lambda model with omega = 0.3, lambda = 0.7, h = 2/3, and sigma(sub 8) = 2/3; (5) the PBI model with omega = 0.2, h = 0.8, x = 0.1, m = -0.5, and sigma(sub 8) = 0.9. Comparison of the five models with the presently available observational measurements v(sub LG) = 85 - 305 km/s (with mean of 250 km/s), delta(n(sub g))/(n(sub g)-bar) = 1.40 + or - 0.35) suggests that an open universe with omega approximately 0.5 (with or without lambda) and sigma(sub 8) approximately 0.8 is preferred, with omega = 0.3-1.0 (with or without lambda) and sigma(sub 8) = 0.7-1.0 being the acceptable range. At variance with some previous claims based on either direct N-body or spherical nonlinear approaches, we find that a flat model with sigma(sub 8) approximately 0.7-1.0 seems to be reasonably consistent with observations. However, if one favors the low limit of v(sub LG) = 85 km/s, then an omega approximately 0.2-0.3 universe seems to provide a better fit, and flat (omega = 1) models are ruled out at approximately 95% confidence level. On the other hand, if the high limit of v(sub LG) = 350 km/s is closer to the truth, then it appears that omega approximately 0.7-0.8 is more consistent. This test is insensitive to the shape of the power spectrum, but rather sensitive to the normalization of the perturbation amplitude on the relevant scale (e.g., sigma(sub 8)) and omega. We find that neither linear nor nonlinear relations (with spherical symmetry) are good approximations for the relation between radial peculiar velocity and density perturbation, i.e., nonspherical effects and gravitational tidal field are important. The derived omega using either of the two relations is underestimated. In some cases, this error is as large as a factor of 2-4.

Cen, Renyue↗
DOE OSTI2024

A windowed mean trajectory approximation for condensed phase dynamics

We propose a trajectory-based quasi-classical method for approximating dynamics in condensed phase systems. Building upon the previously developed optimized mean trajectory approximation that has been used to compute linear and nonlinear spectra, we borrow some ideas from filtering trajectory methods to obtain a novel semiclassical method for the dynamical propagation of density matrices. This new approximation is tested rigorously against standard multistate electronic models, spin-boson models, and models of the Fenna–Matthews–Olson complex. For dissipative systems, the current method is significantly better or as good as many other semiclassical methods available, especially at low temperatures and for off-diagonal density matrix elements, whereas for scattering models, the current method bears similar limitations as mean-field propagation schemes. All results are tested against the numerically exact hierarchical equations of motion method. In conclusion, the new method shows excellent agreement across various parameter regimes with numerically exact results, highlighting the robustness and accuracy of our approach.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

A data-driven sensor placement approach for detecting voltage violations in distribution systems

Stochastic fluctuations in power injections from distributed energy resources (DERs) combined with load variability can cause constraint violations (e.g., exceeded voltage limits) in electric distribution systems. To monitor grid operations, sensors are placed to measure important quantities such as the voltage magnitudes. Here, in this paper, we consider a sensor placement problem which seeks to identify locations for installing sensors that can capture all possible violations of voltage magnitude limits. We formulate a bilevel optimization problem that minimizes the number of sensors and avoids false sensor alarms in the upper level while ensuring detection of any voltage violations in the lower level. This problem is challenging due to the nonlinearity of the power flow equations and the presence of binary variables. Accordingly, we employ recently developed conservative linear approximations of the power flow equations that overestimate or underestimate the voltage magnitudes. By replacing the nonlinear power flow equations with conservative linear approximations, we can ensure that the resulting sensor locations and thresholds are sufficient to identify any constraint violations. Additionally, we apply various problem reformulations to significantly improve computational tractability while simultaneously ensuring an appropriate placement of sensors. Lastly, we improve the quality of the results via an approximate gradient descent method that adjusts the sensor thresholds. We demonstrate the effectiveness of our proposed method for several test cases, including a system with multiple switching configurations.

24 POWER TRANSMISSION AND DISTRIBUTION↗
NASA NTRS1984

A finite element method for nonlinear forced vibrations of beams

Techniques for defining a finite element model (FEM) for analysis of nonlinear vibrations in beam structures subjected to harmonic excitation are presented. The resulting model covers longitudinal deformation and inertial effects. The nonlinear oscillations of a beam element under forced excitation are modeled by a harmonic force matrix based on first order approximations of the Jacobian elliptic forcing function. Harmonic force and nonlinear stiffness matrices are derived and the nonlinear forced responses of beams are calculated under various boundary conditions. The results of FEM computations for simply-supported and clamped beams show that midplane stretching caused by large deflections increases the nonlinearity. Axially-restrained beams experience only hardening nonlinearity, while axially-free beams have reduced nonlinearity in deformation and inertia and an increase in linearity due to large deflection.

Mei, C.↗
NASA NTRS1984

The application of satellite data to study the effects of latent heat release on cyclones

Generalized energetics were studied for nonlinear inviscid symmetric instability (SI). It was found that the linear theory fails to predict the stability in certain cases where the basic state is transitional between stability and instability. The initial growth of the SI perturbations can be fairly well approximated by linear theory, but the long time nonlinear evaluations will be bonded energetically if the SI region is finite. However, a further extension of the energetics to conditional symmetric instability (CSI) shows that the nonlinear evolution of circulation will energetically depend much more on the precipitation in a complicated way. By treating the latent heat as a source which is implicitly related to the motion field, the existence, uniqueness and stability of steady viscous (CSI) circulations are studied. Viscous CSI circulations are proved to be unique and asymptotically stable when the heat sources are weak and less sensitive to the motion perturbations. By considering the fact that moist updrafts are narrow and using eddy viscosity of 0(1,000 m squared/s) the stability criterion suggests that some frontal rainbands were probably dominated by the CSI mechanism even in their mature quasi-steady stage.

Clark, J. H. E.↗
DOE OSTI2020

Approximating Trajectory Constraints With Machine Learning – Microgrid Islanding With Frequency Constraints

Here, we introduce deep earning aided constraint encoding to tackle the frequency-constraint microgrid scheduling problem. The nonlinear function between system operating condition and frequency nadir is approximated by using a neural network, which admits an exact mixed-integer formulation (MIP). This formulation is then integrated with the scheduling problem to encode the frequency constraint. With the stronger representation power of the neural network, the resulting commands can ensure adequate frequency response in a realistic setting in addition to islanding success. The proposed method is validated on a modified 33-node system. Successful islanding with a secure response is simulated under the scheduled commands using a detailed three-phase model in Simulink. The advantages of our model are particularly remarkable when the inertia emulation functions from wind turbine generators are considered.

24 POWER TRANSMISSION AND DISTRIBUTION↗
NASA NTRS1985

An Algorithm for Efficient Maximum Likelihood Estimation and Confidence Interval Determination in Nonlinear Estimation Problems

An algorithm for maximum likelihood (ML) estimation is developed with an efficient method for approximating the sensitivities. The algorithm was developed for airplane parameter estimation problems but is well suited for most nonlinear, multivariable, dynamic systems. The ML algorithm relies on a new optimization method referred to as a modified Newton-Raphson with estimated sensitivities (MNRES). MNRES determines sensitivities by using slope information from local surface approximations of each output variable in parameter space. The fitted surface allows sensitivity information to be updated at each iteration with a significant reduction in computational effort. MNRES determines the sensitivities with less computational effort than using either a finite-difference method or integrating the analytically determined sensitivity equations. MNRES eliminates the need to derive sensitivity equations for each new model, thus eliminating algorithm reformulation with each new model and providing flexibility to use model equations in any format that is convenient. A random search technique for determining the confidence limits of ML parameter estimates is applied to nonlinear estimation problems for airplanes. The confidence intervals obtained by the search are compared with Cramer-Rao (CR) bounds at the same confidence level. It is observed that the degree of nonlinearity in the estimation problem is an important factor in the relationship between CR bounds and the error bounds determined by the search technique. The CR bounds were found to be close to the bounds determined by the search when the degree of nonlinearity was small. Beale's measure of nonlinearity is developed in this study for airplane identification problems; it is used to empirically correct confidence levels for the parameter confidence limits. The primary utility of the measure, however, was found to be in predicting the degree of agreement between Cramer-Rao bounds and search estimates.

Murphy, Patrick Charles↗
NASA NTRS2022

Alternative regularizations for Outer-Approximation algorithms for convex MINLP

In this work, we extend the regularization framework from Kronqvist et al. (Math Program 180(1):285–310, 2020) by incorporating several new regularization functions and develop a regularized single-tree search method for solving convex mixed-integer nonlinear programming (MINLP) problems. We propose a set of regularization functions based on distance metrics and Lagrangean approximations, used in the projection problem for finding new integer combinations to be used within the Outer-Approximation (OA) method. The new approach, called Regularized Outer-Approximation (ROA), has been implemented as part of the open-source Mixed-integer nonlinear decomposition toolbox for Pyomo—MindtPy. We compare the OA method with seven regularization function alternatives for ROA. Moreover, we extend the LP/NLP Branch and Bound method proposed by Quesada and Grossmann (Comput Chem Eng 16(10–11):937–947, 1992) to include regularization in an algorithm denoted RLP/NLP. We provide convergence guarantees for both ROA and RLP/NLP. Finally, we perform an extensive computational experiment considering all convex MINLP problems in the benchmark library MINLPLib. The computational results show clear advantages of using regularization combined with the OA method.

Convex Mixed-integer nonlinear programming↗
DOE OSTI2021

Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms

The simulation of large nonlinear dynamical systems, including systems generated by discretization of hyperbolic partial differential equations, can be computationally demanding. Such systems are important in both fluid and kinetic computational plasma physics. This motivates exploring whether a future error-corrected quantum computer could perform these simulations more efficiently than any classical computer. In this work, we describe a method for mapping any finite nonlinear dynamical system to an infinite linear dynamical system (embedding) and detail three specific cases of this method that correspond to previously studied mappings. Then we explore an approach for approximating the resulting infinite linear system with finite linear systems (truncation). Using a number of qubits only logarithmic in the number of variables of the nonlinear system, a quantum computer could simulate truncated systems to approximate output quantities if the nonlinearity is sufficiently weak. Other aspects of the computational efficiency of the three detailed embedding strategies are also discussed.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1988

Nonlinearities in spacecraft structural dynamics

In considering nonlinearities in spacecraft structural dynamics, the following are examined: (1) SCOLE Configuration-Equations of Motion; (2) Modeling Error Sources; (3) Approximate Solutions; (4) Comparison of Model Accuracy; (5) Linear and Nonlinear Damping; (6) Experimental Results; and, (7) Future Work.

Taylor, Larry↗
NASA NTRS1992

Collapse of composite tubes under end moments

Cylindrical tubes of moderate wall thickness such as those proposed for the original space station truss, may fail due to the gradual collapse of the tube cross section as it distorts under load. Sometimes referred to as the Brazier instability, it is a nonlinear phenomenon. This paper presents an extension of an approximate closed form solution of the collapse of isotropic tubes subject to end moments developed by Reissner in 1959 to include specially orthotropic material. The closed form solution was verified by an extensive nonlinear finite element analysis of the collapse of long tubes under applied end moments for radius to thickness ratios and composite layups in the range proposed for recent space station truss framework designs. The finite element analysis validated the assumption of inextensional deformation of the cylindrical cross section and the approximation of the material as specially orthotropic.

Stockwell, Alan E.↗
DOE OSTI2021

Observability Analysis of a Power System Stochastic Dynamic Model Using a Derivative-Free Approach

Serving as a prerequisite to power system dynamic state estimation, the observability analysis of a power system dynamic model has recently attracted the attention of many power engineers. However, because this model is typically nonlinear and large-scale, the analysis of its observability is a challenge to the traditional derivative-based methods. Indeed, the linear-approximation-based approach may provide unreliable results while the nonlinear-technique-based approach inevitably faces extremely complicated derivations. Furthermore, because power systems are intrinsically stochastic, the traditional deterministic approaches may lead to inaccurate observability analyses. In this work, facing these challenges, we propose a novel polynomial-chaos-based derivative-free observability analysis approach that not only is free of any linear approximations, but also accounts for the stochasticity of the dynamic model while bringing a low implementation complexity. Furthermore, this approach enables us to quantify the degree of observability of a stochastic model, what conventional deterministic methods cannot do. The excellent performance of the proposed method has been demonstrated by performing extensive simulations using a synchronous generator model with IEEE-DC1A exciter and the TGOV1 turbine governor.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2022

Quasilinear theory of general electromagnetic fluctuations including discrete particle effects for magnetized plasmas: General analysis

The general quasilinear Fokker–Planck kinetic equation for the gyrophase-averaged plasma particle distribution functions in magnetized plasmas is derived, making no restrictions on the energy of the particles and on the frequency of the electromagnetic fluctuations and avoiding the often made Coulomb approximation of the electromagnetic interactions. Here, the inclusion of discrete particle effects breaks the dichotomy of nonlinear kinetic plasma theory divided into the test particle and the test fluctuation approximation because it provides expression of both the non-collective and collective electromagnetic fluctuation spectra in terms of the plasma particle distribution functions. Within the validity of the quasilinear approach, the resulting full quasilinear transport equation can be regarded as a determining nonlinear equation for the time evolution of the plasma particle distribution functions.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1994

Effect of nonlinear interactions on p-mode frequencies and line widths

We calculate the effect of nonlinear interactions among solar acoustic modes upon the modal frequencies and energy loss rates (or line widths). The frequency shift for a radial p-mode of frequency 3 mHz is found to be about -0.5 microHz. The magnitude of nonlinear frequency shift increases more rapidly with frequency than the inverse mode mass (mode mass is defined as the ratio of energy in the mode to its surface velocity amplitude squared). This frequency shift is primarily due to nonresonant three-mode interactions and is dominated by high l surface gravity waves (f-modes) and p-modes. The line width of a radial p-mode of frequency 3 mHz, due to resonant nonlinear interactions, is about 0.3 microHz. This result is consistent with that of Kumar and Goldreich (1989). We also find, in agreement with these authors, that the most important nonlinear interactions of trapped p-modes involve f-modes and high-frequency p-modes (frequency greater than about 5 mHz) which propagate in the solar photosphere. Thus, using the arguments advanced by Kumar & Goldreich (1989), we conclude that nonlinear couplings cannot saturate the overstable solar p-modes at their small observed amplitudes. Both the nonlinear frequency shifts and line widths, at a fixed frequency, are proportional to the inverse of mode mass which for modes of degree greater than about 100 is approximately l(exp 0.8). Therefore, the frequency of an f-mode of l = 1000, due to nonlinear interactions, is decreased by approximately 0.4%.

Kumar, Pawan↗
DOE OSTI2025

Kernel Manifolds: Nonlinear‐Augmentation Dimensionality Reduction Using Reproducing Kernel Hilbert Spaces

This paper generalizes recent advances on quadratic manifold (QM) dimensionality reduction by developing kernel methods-based nonlinear-augmentation dimensionality reduction. QMs, and more generally feature map-based nonlinear corrections, augment linear dimensionality reduction with a nonlinear correction term in the reconstruction map to overcome approximation accuracy limitations of purely linear approaches. While feature map-based approaches typically learn a least squares optimal polynomial correction term, we generalize this approach by learning an optimal nonlinear correction from a user-defined reproducing kernel Hilbert space. Our approach allows one to impose arbitrary nonlinear structure on the correction term, including polynomial structure, and includes feature map and radial basis function-based corrections as special cases. Furthermore, our method has relatively low training cost and has monotonically decreasing error as the latent space dimension increases. In conclusion, we compare our approach to proper orthogonal decomposition and several recent QM approaches on data from several example problems.

kernel methods↗