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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↗
DOE OSTI2025

Temporally-consistent koopman autoencoders for forecasting dynamical systems

Absence of sufficiently high-quality data often poses a key challenge in data-driven modeling of high-dimensional spatio-temporal dynamical systems. Koopman Autoencoders (KAEs) harness the expressivity of deep neural networks (DNNs), the dimension reduction capabilities of autoencoders, and the spectral properties of the Koopman operator to learn a reduced-order feature space with simpler, linear dynamics. However, the effectiveness of KAEs is hindered by limited and noisy training datasets, leading to poor generalizability. To address this, we introduce the Temporally-Consistent Koopman Autoencoder (tcKAE), designed to generate accurate long-term predictions even with limited and noisy training data. This is achieved through a consistency regularization term that enforces prediction coherence across different time steps, thus enhancing the robustness and generalizability of tcKAE over existing models. We provide analytical justification for this approach based on Koopman spectral theory and empirically demonstrate tcKAE’s superior performance over state-of-the-art KAE models across a variety of test cases, including simple pendulum oscillations, kinetic plasma, and fluid flow data.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Parametric dynamic mode decomposition for reduced order modeling

Dynamic Mode Decomposition (DMD) is a model-order reduction approach, whereby spatial modes of fixed temporal frequencies are extracted from numerical or experimental data sets. The DMD low-rank or reduced operator is typically obtained by singular value decomposition of the temporal data sets. For parameter-dependent models, as found in many multi-query applications such as uncertainty quantification or design optimization, the only parametric DMD technique developed was a stacked approach, with data sets at multiple parameter values were aggregated together, increasing the computational work needed to devise low-rank dynamical reduced-order models. Here in this paper, we present two novel approach to carry out parametric DMD: one based on the interpolation of the reduced-order DMD eigen-pair and the other based on the interpolation of the reduced DMD (Koopman) operator. Numerical results are presented for diffusion-dominated nonlinear dynamical problems, including a multiphysics radiative transfer example. All three parametric DMD approaches are compared.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Data-driven models of nonautonomous systems

Nonautonomous dynamical systems are characterized by time-dependent inputs, which complicates the discovery of predictive models describing the spatiotemporal evolution of the state variables of quantities of interest from their temporal snapshots. When dynamic mode decomposition (DMD) is used to infer a linear model, this difficulty manifests itself in the need to approximate the time-dependent Koopman operators. Our approach is to approximate the original nonautonomous system with a modified system derived via a local parameterization of the time-dependent inputs. The modified system comprises a sequence of local parametric systems, which are subsequently approximated by a parametric surrogate model using the DRIPS (dimension reduction and interpolation in parameter space) framework. The offline step of DRIPS relies on DMD to build a linear surrogate model, endowed with reduced-order bases for the observables mapped from training data. The online step interpolates on suitable manifolds to construct a sequence of iterative parametric surrogate models; the target/test parameter points on these manifolds are specified by a local parameterization of the test time-dependent inputs. Here, we use numerical experimentation to demonstrate the robustness of our method and compare its performance with that of deep neural networks.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Online real-time learning of dynamical systems from noisy streaming data

Abstract Recent advancements in sensing and communication facilitate obtaining high-frequency real-time data from various physical systems like power networks, climate systems, biological networks, etc. However, since the data are recorded by physical sensors, it is natural that the obtained data is corrupted by measurement noise. In this paper, we present a novel algorithm for online real-time learning of dynamical systems from noisy time-series data, which employs the Robust Koopman operator framework to mitigate the effect of measurement noise. The proposed algorithm has three main advantages: (a) it allows for online real-time monitoring of a dynamical system; (b) it obtains a linear representation of the underlying dynamical system, thus enabling the user to use linear systems theory for analysis and control of the system; (c) it is computationally fast and less intensive than the popular extended dynamic mode decomposition (EDMD) algorithm. We illustrate the efficiency of the proposed algorithm by applying it to identify the Van der Pol oscillator, the chaotic attractor of the Henon map, the IEEE 68 bus system, and a ring network of Van der Pol oscillators.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Application of advanced causal analyses to identify processes governing secondary organic aerosols

Abstract Understanding how different physical and chemical atmospheric processes affect the formation of fine particles has been a persistent challenge. Inferring causal relations between the various measured features affecting the formation of secondary organic aerosol (SOA) particles is complicated since correlations between variables do not necessarily imply causality. Here, we apply a state-of-the-art information transfer measure coupled with the Koopman operator framework to infer causal relations between isoprene epoxydiol SOA (IEPOX-SOA) and different chemistry and meteorological variables derived from detailed regional model predictions over the Amazon rainforest. IEPOX-SOA represents one of the most complex SOA formation pathways and is formed by the interactions between natural biogenic isoprene emissions and anthropogenic emissions affecting sulfate, acidity and particle water. Since the regional model captures the known relations of IEPOX-SOA with different chemistry and meteorological features, their simulated time series implicitly include their causal relations. We show that our causal model successfully infers the known major causal relations between total particle phase 2-methyl tetrols (the dominant component of IEPOX-SOA over the Amazon) and input features. We provide the first proof of concept that the application of our causal model better identifies causal relations compared to correlation and random forest analyses performed over the same dataset. Our work has tremendous implications, as our methodology of causal discovery could be used to identify unknown processes and features affecting fine particles and atmospheric chemistry in the Earth’s atmosphere.

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

Deep Koopman Neural Network for Analyzing High-Energy-Density Simulations of Electrical Wire Explosions

Megaampere-scale electrical wire experiments (EWEs) provide a platform for studying magnetohydrodynamic (MHD) instability growth in magneto-inertial fusion (MIF) devices. Even when nonlinear simulations of these experiments can digitally reproduce much of the experimentally observed instability growth, interpreting the results and understanding mode growth and evolution can be non-trivial. As a first step toward providing better interpretation of these simulation features, this work investigates the use of a deep neural network that uses Koopman operator theory to analyze the dynamics of pulsed-power-driven explosions of EWEs. This deep neural network is trained on 1-D resistive MHD simulations of EWEs. This neural network learns to transform the nonlinear data into a lower-dimensional representation where the time dynamics are linear. Layers of this neural network are shown to learn features of the simulations, including the locations of shock waves and different physical regimes of the simulation. Using the learned features, the network can compress a time state of the simulation consisting of 5120 data point into a 36-parameter lower-dimensional latent space embedding. Furthermore, these embeddings are shown to be clustered in the latent space by initial radius and time state.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

pnnl/grid_prediction

Two datadriven predictive approaches, namely, {\em Koopman Operator Theoretic (KOT)-based} model and {\em Graph Neural Network (GNN)-based model}, to enable effective power system state predictions. The KOT-based approaches (Robust DMD, deepDMD) capture the power system evolution as a linear dynamical system on an abstract space. The GNNs model the spatio-temporal correlations using graph convolutional network and are called Spatio-Temporal Graph Convolutional Network (STGCN). These predictive models are trained, tested and compared rigorously based on their predictions of frequencies in the IEEE 68 bus system when subjected to a disturbance. GridSTAGE framework developed at Pacific Northwest National Laboratory is leveraged to generate multiple datasets (in the form of PMU measurements) for training and testing by strategically creating load changes across the spatial locations of the network

Nandanoor, Sai Pushpak↗
DOE OSTI2022

Koopman Model Predictive Control for Eco-Driving of Automated Vehicles

In this paper, we develop a data-driven process for building a model predictive control (MPC) for eco-driving of automated vehicles. The process involves performing system identification in which the non-linear vehicle dynamics model is approximated by the Koopman operator, a linear predictor of higher state-dimension, in a data-driven framework. This approach allows us to formulate the eco-driving problem in a constrained quadratic program that leads to a computationally fast MPC. The MPC is then implemented as a closed-loop control of an electric vehicle in numerical simulations for demonstration.

autonomous vehicle↗
DOE OSTI2020

Computationally Efficient Learning of Large Scale Dynamical Systems: A Koopman Theoretic Approach

In recent years there has been a considerable drive towards data-driven analysis, discovery and control of dynamical systems. To this end, operator theoretic methods, namely, Koopman operator methods have gained a lot of interest. In general, the Koopman operator is obtained as a solution to a least-squares problem, and as such, the Koopman operator can be expressed as a closed-form solution that involves the computation of Moore-Penrose inverse of a matrix. For high dimensional systems and also if the size of the obtained data-set is large, the computation of the Moore-Penrose inverse becomes computationally challenging. In this paper, we provide an algorithm for computing the Koopman operator for high dimensional systems in a time-efficient manner. We further demonstrate the efficacy of the proposed approach on two different systems, namely a network of coupled oscillators (with state-space dimension up to 2500) and IEEE 68 bus system (with state-space dimension 204 and up to 24,000 time-points).

Sinha, Subhrajit↗
DOE OSTI2023

Quantum computing for fusion energy science applications

This is a review of recent research exploring and extending present-day quantum computing capabilities for fusion energy science applications. We begin with a brief tutorial on both ideal and open quantum dynamics, universal quantum computation, and quantum algorithms. Then, we explore the topic of using quantum computers to simulate both linear and nonlinear dynamics in greater detail. Because quantum computers can only efficiently perform linear operations on the quantum state, it is challenging to perform nonlinear operations that are generically required to describe the nonlinear differential equations of interest. In this work, we extend previous results on embedding nonlinear systems within linear systems by explicitly deriving the connection between the Koopman evolution operator, the Perron–Frobenius evolution operator, and the Koopman–von Neumann evolution (KvN) operator. We also explicitly derive the connection between the Koopman and Carleman approaches to embedding. Extension of the KvN framework to the complex-analytic setting relevant to Carleman embedding, and the proof that different choices of complex analytic reproducing kernel Hilbert spaces depend on the choice of Hilbert space metric are covered in the appendixes. Finally, we conclude with a review of recent quantum hardware implementations of algorithms on present-day quantum hardware platforms that may one day be accelerated through Hamiltonian simulation. We discuss the simulation of toy models of wave–particle interactions through the simulation of quantum maps and of wave–wave interactions important in nonlinear plasma dynamics.

Joseph, I. (ORCID:0000000255400840)↗
DOE OSTI2020

Real-Time Modal Analysis of Electric Power Grids - The Need for Dynamic State Estimation: Preprint

We articulate the reason why dynamic state estimation is needed to push the boundaries of the modal analysis of electric power grids in real-time operation. Then, we demonstrate how to unravel linear and nonlinear modes by using the extended dynamic mode decomposition along with estimates of the synchronous generators' rotor angles and rotor speed deviations from nominal speed. The estimated modes are associated with electromechanical oscillations that take place continuously in electric power grids because of imbalances between power generation and demand. The numerical simulations are performed on a synthetic, albeit realistic, 2,000-bus network that was designed to resemble the electric power grid of Texas.

61 RADIATION PROTECTION AND DOSIMETRY↗
DOE OSTI2020

Real-Time Modal Analysis of Electric Power Grids - The Need for Dynamic State Estimation

We articulate the reason why dynamic state estimation is needed to push the boundaries of the modal analysis of electric power grids in real-time operation. Then, we demonstrate how to unravel linear and nonlinear modes by using the extended dynamic mode decomposition along with estimates of the synchronous generators’ rotor angles and rotor speed deviations from nominal speed. The estimated modes are associated with electromechanical oscillations that take place continuously in electric power grids because of imbalances between power generation and demand. The numerical simulations are performed on a synthetic, albeit realistic, 2,000-bus network that was designed to resemble the electric power grid of Texas.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Data-Driven Adaptive Damping Controller for Wind Power Plants with Doubly-Fed Induction Generators

This paper presents an adaptive damping controller for wind power plants in which the turbines are equipped with doubly-fed induction generators. The controller is designed to respond to an input control signal that is triggered according to the system operating conditions. A processing unit continuously estimates the electromechanical modes of oscillation based on real-time streaming data acquired from a phasor measurement unit that is strategically positioned on the grid. The decision to trigger (or not trigger) the control signal is automatic, based on the relative damping of the dominant mode. The modes are estimated using the dynamic mode decomposition algorithm with time-delay embedding. Numerical simulations performed on the two-area system demonstrate that the proposed controller enhances the rotor angle stability for both small-signal and large disturbances, and is adaptive to changing grid conditions.

damping controller↗
DOE OSTI2021

Data-Driven Adaptive Damping Controller for Wind Power Plants with Doubly-Fed Induction Generators: Preprint

This paper presents an adaptive damping controller for wind power plants in which the turbines are equipped with doubly-fed induction generators. The controller is designed to respond to an input control signal that is triggered according to the system operating conditions. A processing unit continuously estimates the electromechanical modes of oscillation based on real-time streaming data acquired from a phasor measurement unit that is strategically positioned on the grid. The decision to trigger (or not trigger) the control signal is automatic, based on the relative damping of the dominant mode. The modes are estimated using the dynamic mode decomposition algorithm with time-delay embedding. Numerical simulations performed on the two-area system demonstrate that the proposed controller enhances the rotor angle stability for both small-signal and large disturbances, and is adaptive to changing grid conditions.

61 RADIATION PROTECTION AND DOSIMETRY↗
DOE OSTI2026

Regional surrogates for predictive control of digital twins

Digital twins of complex systems must involve a model that is fast, generalizable, and usable for real-time control. For example, high-fidelity nonlinear multiphysics simulations can capture laser-material interactions, but are too slow for optimization or model predictive control (MPC). Reduced-order models, used to accelerate such computation, frequently fail to generalize to unseen inputs or control states. We show theoretically that this failure is intrinsic, i.e., that a learned model is non-unique outside the sampled subspace when its low-rank structure arises from limited excitation and clustered eigenvalues, rather than from a user-imposed truncation alone. Motivated by this result, we propose a control-ready regional surrogate-construction framework for both autonomous and nonautonomous dynamics; it employs Koopman lifting to represent nonlinearities, while preserving spatial locality. We illustrate our approach by constructing a control-ready surrogate for the digital twin of a thermal component of additive-manufacturing process. Our surrogate, localized in space through a von Neumann stencil, is learned from noisy high-fidelity simulations that emulate thermal-camera images collected during the manufacturing. It is linear in thermo-physically augmented states so that MPC reduces to a convex quadratic program. The surrogate requires no online correction, generalizes to unseen scan paths and power profiles of the laser, and is more than three orders of magnitude faster than a finite-difference solver. Furthermore, when the MPC sequence computed on the digital twin is applied to this solver, closed-loop temperature regulation is recovered, showing that the surrogate preserves control-relevant input-output behavior.

Data-driven model↗