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At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Data-Driven Operator Theoretic Methods for Global Phase Space Learning

In this work, we propose to apply the recently developed Koopman operator techniques to explore the global phase space of a nonlinear system from time-series data. In particular, we address the problem of identifying various invariant subsets of a phase space from the spectral properties of the associated Koopman operator constructed from time-series data. Moreover, in the case when the system evolution is known locally in various invariant subspaces, then a phase space stitching result is proposed that can be applied to identify a global Koopman operator. A biological system, bistable toggle switch and a second-order nonlinear system example are considered to illustrate the proposed results. The construction of this global Koopman operator is very helpful in experimental works as multiple experiments can't be run at the same time starting from several initial conditions.

Nandanoori, Sai Pushpak↗
DOE OSTI2020

Data Driven Online Learning of Power System Dynamics

With the advancement of sensing and communication in power networks, high-frequency real-time data from a power network can be used as a resource to develop better monitoring capabilities. In this work, a systematic approach based on data-driven operator theoretic methods involving Koopman operator is proposed for the online identification of power system dynamics. In particular, a new algorithm is provided, which unlike any previously existing algorithms, updates the Koopman operator iteratively as new data points are acquired. The proposed algorithm has three advantages: a) allows for real-time monitoring of the power system dynamics b) linear power system dynamics (this linear system is usually in a higher dimensional feature space and is not same as linearization of the underlying nonlinear dynamics) and c) computationally fast and less intensive when compared to the popular Extended Dynamic Mode Decomposition (EDMD) algorithm. The efficiency of the proposed algorithm is illustrated on an IEEE 9 bus system using synthetic data from the nonlinear model and on IEEE 39 bus system using synthetic data from the linearized model.

Sinha, Subhrajit↗
DOE OSTI2022

Data-Driven Operator Theoretic Methods for Phase Space Learning and Analysis

This paper uses data-driven operator theoretic approaches to explore the global phase space of a dynamical system. In this work, we defined conditions for discovering new invariant subspaces in the state space of a dynamical system starting from an invariant subspace based on the spectral properties of the Koopman operator. When the system evolution is known locally in several invariant subspaces in the state space of a dynamical system, a phase space stitching result is derived that yields the global Koopman operator. Additionally, in the case of equivariant systems, a phase space stitching result is developed to identify the global Koopman operator using the symmetry properties between the invariant subspaces of the dynamical system and time-series data from any one of the invariant subspaces. Finally, these results are extended to topologically conjugate dynamical systems; in particular, the relation between the Koopman tuple of topologically conjugate systems is established. The proposed results are demonstrated on several second-order nonlinear dynamical systems including a bistable toggle switch. Our method elucidates a strategy for designing discovery experiments: experiment execution can be done in many steps, and models from different invariant subspaces can be combined to approximate the global Koopman operator.

42 ENGINEERING↗
DOE OSTI2023

Data assimilation in operator algebras

We develop an algebraic framework for sequential data assimilation of partially observed dynamical systems. In this framework, Bayesian data assimilation is embedded in a nonabelian operator algebra, which provides a representation of observables by multiplication operators and probability densities by density operators (quantum states). In the algebraic approach, the forecast step of data assimilation is represented by a quantum operation induced by the Koopman operator of the dynamical system. Moreover, the analysis step is described by a quantum effect, which generalizes the Bayesian observational update rule. Projecting this formulation to finite-dimensional matrix algebras leads to computational schemes that are i) automatically positivity-preserving and ii) amenable to consistent data-driven approximation using kernel methods for machine learning. Moreover, these methods are natural candidates for implementation on quantum computers. Applications to the Lorenz 96 multiscale system and the El Niño Southern Oscillation in a climate model show promising results in terms of forecast skill and uncertainty quantification.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Weighted Composition Operators for Learning Nonlinear Dynamics

Operator theoretic methods in dynamical system have been dominated by the use of Koopman operators and their continuous time counterparts, such as Koopman Generators and Liouville Operators. The advantage gained from their use primarily stems from the ability to extract subspaces and eigenfunctions within a space of observables that are invariant with respect to the Koopman operator over that space. When this occurs, a dynamic mode decomposition of the systems state provides a linear model for the dynamical system. Not all Koopman operators have eigenfunctions that may be exploited in this manner. However, the framework can still be leveraged for approximations using other operators. In this setting, we present a different operator for the study of dynamical systems, the weighted composition operator. These operators are compact for a wide range of dynamics and spaces, and through their interactions with occupation kernels and vector valued kernels, they admit an estimation of the underlying dynamics. Here, this manuscript presents a new algorithm for the data driven study of dynamical systems from data, and also provides two numerical experiments where convergence is achieved as a proof of concept.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Analog and symbolic computation through the Koopman framework

We develop a Koopman operator framework for studying the computational structure of dynamical systems. Specifically, we show that the resolvent of the Koopman operator provides a natural abstraction of halting, yielding a ‘Koopman halting problem’ that is recursively enumerable in general. For symbolic systems, such as those defined on Cantor space, this operator formulation captures reachability between clopen sets, while for equicontinuous systems we prove that the Koopman halting problem is decidable. Our framework demonstrates that absorbing (halting) states in coarse-grained finite automata correspond to Koopman eigenfunctions with eigenvalue one, while cycles in the transition graph impose spectral constraints associated with periodic dynamics. These results provide a unifying perspective on computation in symbolic and analog systems, showing how computational universality is reflected in operator spectra, invariant subspaces, and algebraic structures. Beyond symbolic dynamics, this operator-theoretic lens opens pathways to analyze the computational properties of a broader class of dynamical systems, including polynomial and analog models, and suggests that computational hardness may admit dynamical signatures in terms of Koopman spectral structure.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Data-Driven Approach for Uncertainty Propagation and Reachability Analysis in Dynamical Systems

In this paper, we propose a data-driven approach for uncertainty propagation and reachability analysis in a dynamical system. The proposed approach relies on the linear lifting of a nonlinear system using linear Perron-Frobenius (P-F) and Koopman operators. Here, the uncertainty can be characterized in terms of the moments of a probability density function. We demonstrate how the P-F and Koopman operators are used for propagating the moments. Time-series data is used for the finite-dimensional approximation of the linear operators, thereby enabling data-driven approach for moment propagation. Simulation results are presented to demonstrate the effectiveness of the proposed method.

42 ENGINEERING↗
DOE OSTI2021

Solvability, Operability, and Security for Cyber-Physical Systems: New Computational Methods with Revised Assumptions

In this paper discusses the use and integration of disparate but complementary tools for dealing with solvability, operability, and security challenges in Cyber-Physical Systems (CPSs): the Koopman Operator (KO) for solvability, Disjunctive Programming (DP) for operability, and Multi-Level Optimisation (MLO) for security. These methods can obviate the need for some of the traditional assumptions used in modelling CPSs. This paper demonstrates the methods' capabilities and considers ways to advance each method individually. It concludes by discussing how to integrate the different methods and identify useful synergies generated by doing so.

koopman operator, disjunctive programming, multi-l↗
DOE OSTI2023

Online Output-Based Inertia Estimation of Modern Power Systems

The overall inertia of modern power systems is drastically impacted by the penetration level of converter-based resources (CBRs). The intermittent output of CBRs and their dependence on the weather conditions affirm the need for a realtime inertia estimation approach. This can help the transmission system operators(TSOs) to be continuously aware of the changes in the system inertia and to take suitable control actions. This paper is a step forward to a totally data-driven framework for power system inertia estimation using the available measurements from the phasor measurement units (PMUs). This inertia estimation is done based on the spectral analysis of the so-called Koopman linear operator. Koopman operator considers the relationship between the system's inertia and its dynamical response to slight disturbances in the load. Simulation results show that Koopman operator can approximate the system behavior in a reconstructed linear space hence inertia constant is estimated accurately.

data-driven↗
DOE OSTI2023

Sensor and Actuator Attacks on Hierarchical Control Systems with Domain-Aware Operator Theory

Cyber-Physical Systems (CPSs) provide opportunities for cyber attacks to have physical impacts. Advanced Persistent Threats (APTs) are a subclass of cyber threats that act stealthily to avoid detection and enable long-term attacks. Here, we build on our past work in APT modelling to combine deception-based sensor bias attacks and direct actuator manipulations in attacks against a hierarchical control system. That past work used the Koopman operator to develop a data-driven, domain-aware, optimization-based attacker model. Using an expansion of this model, we compute several different attacks, including multiple simultaneous attacks, against a high-fidelity commercial building emulator and compare the impacts of those attacks to each other. One next step of interest is to construct a defender system, built on the same modelling approach, designed to detect and mitigate such attacks.

koopman operator, Cyber-Physical Security, machine↗
DOE OSTI2022

Nonequilibrium statistical mechanics and optimal prediction of partially-observed complex systems

Abstract Only a subset of degrees of freedom are typically accessible or measurable in real-world systems. As a consequence, the proper setting for empirical modeling is that of partially-observed systems. Notably, data-driven models consistently outperform physics-based models for systems with few observable degrees of freedom; e.g. hydrological systems. Here, we provide an operator-theoretic explanation for this empirical success. To predict a partially-observed system’s future behavior with physics-based models, the missing degrees of freedom must be explicitly accounted for using data assimilation and model parametrization. Data-driven models, in contrast, employ delay-coordinate embeddings and their evolution under the Koopman operator to implicitly model the effects of the missing degrees of freedom. We describe in detail the statistical physics of partial observations underlying data-driven models using novel maximum entropy and maximum caliber measures. The resulting nonequilibrium Wiener projections applied to the Mori–Zwanzig formalism reveal how data-driven models may converge to the true dynamics of the observable degrees of freedom. Additionally, this framework shows how data-driven models infer the effects of unobserved degrees of freedom implicitly, in much the same way that physics models infer the effects explicitly. This provides a unified implicit-explicit modeling framework for predicting partially-observed systems, with hybrid physics-informed machine learning methods combining both implicit and explicit aspects.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

A Convex Data-Driven Approach for Nonlinear Control Synthesis

We consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. We propose a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible. Our method is built on the density-function-based stability certificate that is the dual to the Lyapunov function for dynamic systems. Unlike Lyapunov-based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate. This type of convex problem can be solved efficiently using the machinery of the sum of squares (SOS). For the data-driven part, we exploit the fact that the duality results in the stability theory can be understood through the lens of Perron–Frobenius and Koopman operators. This allows us to use data-driven methods to approximate these operators and combine them with the SOS techniques to establish a convex formulation of control synthesis. The efficacy of the proposed approach is demonstrated through several examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Predicting Critical Transitions in Multiscale Data

Predicting the dynamics of complex nonlinear systems remains a challenging problem both in dynamical systems theory as well as real world science and engineering applications. Data-driven methods utilizing the latest advances in machine learning (ML) provide a promising new paradigm for this task. Our work centered on Reservoir Computing (RC), which has shown itself to be capable of skillfully predicting chaotic dynamics in multiscale systems. In the first part of the work, the focus is on how to improve predictions of critical transitions in a class of slow-fast metastable systems in which the equations are known. An additional goal was to determine whether a relationship exists between RC and Koopman operator theory, to improve the efficiency and broaden the applicability of the approach. In the second part of this work, a variation on the RC model known as Reconstructive Reservoir Computing (RRC) is applied to real-world data to identify anomalies.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Deception-Based Cyber Attacks on Hierarchical Control Systems using Domain-Aware Koopman Learning

Industrial control systems are subject to cyber attacks that produce physical consequences. These attacks can be both hard to detect and protracted. Here, we focus on deception-based sensor bias attacks made against a hierarchical control system where the attacker attempts to be stealthy. We develop a a data-driven, optimization-based attacker model and use the Koopman operator to represent the system dynamics in a domain-aware and computationally efficient manner. Using this model, we compute several different attacks against a high-fidelity commercial building emulator and compare the impacts of those attacks to each other. Finally, we discuss some computational considerations and identify avenues for future research.

koopman operator, Cyber-Physical Security, machine↗
DOE OSTI2023

Deep Koopman learning of nonlinear time-varying systems

Here this paper presents a data-driven approach to approximate the dynamics of a nonlinear time-varying system (NTVS) by a linear time-varying system (LTVS), which results from the Koopman operator and deep neural networks. Analysis of the approximation error between states of the NTVS and the resulting LTVS is presented. Simulations on a representative NTVS show that the proposed method achieves small approximation errors, even when the system changes rapidly. Furthermore, simulations in an example of quadcopters demonstrate the computational efficiency of the proposed approach.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Data-driven Resilience Characterization of Control Dynamical Systems

In this paper, we define and quantify resiliency of a power network and propose data-driven algorithms for computing the same for the power grid. To do this, we use the Koopman operator framework to lift the controlled dynamical system to an abstract (possibly higher) dimensional space, where the evolution is linear. The linear system representation allows us to relate small time local controllability and observability of a general nonlinear control system to the controllability and observability of the lifted linear system. Finally, we define the resiliency of the underlying power grid in terms of the controllability and observability gramians of the lifted linear system. We illustrate the proposed approach to compute the resiliency metrics on time-series data obtained from a microgrid.

koopman operator, resilience, control↗
DOE OSTI2025

Multi-scale, Multi-disciplinary, and Multi-agent Explainable AI with Koopman-Undergirded Learning, Prediction, and Analysis (M3EA KULPA) (Project Closeout Report)

The goal of this project was to develop and use domain-aware machine learning formulations, based on the Koopman Operator (KO), for modelling multi-scale, multi-disciplinary (e.g., multi-physics), and/or multi-agent systems. The project developed these formulations for the following cases: • Systems with dynamics at two separate time scales, • Systems with a bi-level hierarchical control structure, • Systems with bi-level hierarchical control and dynamics at two separate time scales (the lower level controls operating at the faster time scale), and • Systems with n separate but interacting agents/disciplines (with/without control, respectively); the controls for each agent could include bi-level hierarchical control and dynamics at two separate time scales as described above. The project then defined a set of dynamical systems consisting of different nonlinear oscillators that could be used to test these different formulations and then subsequently learned the KO models for those systems. With the KO models, we were able to do the following: • Quantify system stability, including both long-term and transient behavior, • Quantify the effects of feedbacks between the different time scales and agents/disciplines in terms of those feedbacks’ effects on system stability, • Replace a standard Proportional-Integral (PI) control in the hierarchical control structure with a KO-based Linear-Quadratic Regular (LQR), a form of optimal control, • Calculate optimal supervisory control policies a) with and without time scale separated dynamics at the lower level control levels and b) with both PI and KO-based LQR lower level control policies, and • Calculate dynamic Nash equilibria for multi-agent systems where each agent makes its own control decisions.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Data-Centric Approach to Capture Non-Polynomial Nonlinear Dynamics

We propose an analytical construction of observable functions in the extended dynamic mode decomposition (EDMD) algorithm. EDMD is a numerical method for approximating the spectral properties of the Koopman operator. The choice of observable functions is fundamental for applying EDMD to nonlinear problems arising in systems and control. Existing methods either start from a set of dictionary functions and look for the subset that best fits the underlying nonlinear dynamics or rely on machine learning algorithms to “learn” observable functions. Conversely, in this paper, we start from the dynamical system model and lift it through the Lie derivatives, rendering it into a polynomial form. This proposed transformation into a polynomial form is exact and provides an adequate set of observable functions. The strength of the proposed approach is its applicability to a broader class of nonlinear dynamical systems, particularly those with nonpolynomial functions and compositions thereof. Moreover, it retains the physical interpretability of the underlying dynamical system and can be readily integrated into existing numerical libraries. We demonstrate the proposed approach with an application to electric power systems. The modeled system consists of a single generator connected to an infinite bus, where nonlinear terms include sine and cosine functions. The results demonstrate the effectiveness of the proposed procedure in off-attractor nonlinear dynamics for estimation and prediction; the observable functions obtained from the proposed construction outperform methods that use dictionary functions comprising monomials or radial basis functions.

extended dynamic mode decomposition↗