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All sources NASA NTRS DOE OSTI

DOE OSTI2023

Constraining chaos: Enforcing dynamical invariants in the training of reservoir computers

Drawing on ergodic theory, we introduce a novel training method for machine learning based forecasting methods for chaotic dynamical systems. The training enforces dynamical invariants—such as the Lyapunov exponent spectrum and the fractal dimension—in the systems of interest, enabling longer and more stable forecasts when operating with limited data. The technique is demonstrated in detail using reservoir computing, a specific kind of recurrent neural network. Finally, results are given for the Lorenz 1996 chaotic dynamical system and a spectral quasi-geostrophic model of the atmosphere, both typical test cases for numerical weather prediction.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1979

Algebraic methods for the solution of some linear matrix equations

The characterization of polynomials whose zeros lie in certain algebraic domains (and the unification of the ideas of Hermite and Lyapunov) is the basis for developing finite algorithms for the solution of linear matrix equations. Particular attention is given to equations PA + A'P = Q (the Lyapunov equation) and P - A'PA = Q the (discrete Lyapunov equation). The Lyapunov equation appears in several areas of control theory such as stability theory, optimal control (evaluation of quadratic integrals), stochastic control (evaluation of covariance matrices) and in the solution of the algebraic Riccati equation using Newton's method.

Djaferis, T. E.↗

DOE OSTI2026

Lyapunov-Based Iterative Learning of Regions of Attraction for Autonomous Systems

This paper proposes a novel algorithm for estimating the region of attraction of equilibrium points for nonlinear discrete-time autonomous systems. The method iteratively expands an initial estimate of the region of attraction by constructing unions of sublevel sets of learned functions parametrized as neural networks. Unlike conventional techniques that rely on a single global Lyapunov function, the proposed approach provides a collection of local Lyapunov-like functions, enabling richer representations and potentially larger region of attraction estimates. These functions are trained using sampled state-space data, and their Lipschitz continuity ensures that desirable properties extend beyond the training samples. The devised strategy is tested via numerical simulations, demonstrating the effectiveness of the proposed approach.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

Lyapunov-Based Iterative Learning of the Region of Attraction for Autonomous Systems

This presentation introduces a novel algorithm for estimating the region of attraction of equilibrium points for nonlinear discrete-time autonomous systems. The method iteratively expands an initial estimate of the region of attraction by constructing unions of sublevel sets of learned functions parametrized as neural networks. Unlike conventional techniques that rely on a single global Lyapunov function, the proposed approach provides a collection of local Lyapunov-like functions, enabling richer representations and potentially larger region of attraction estimates. These functions are trained using sampled state-space data, and their Lipschitz continuity ensures that desirable properties extend beyond the training samples. The devised strategy is tested via numerical simulations, demonstrating the effectiveness of the proposed approach.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1991

Chaos in the sunspot cycle - Analysis and prediction

The variability of solar activity over long time scales, given semiquantitatively by measurements of sunspot numbers, is examined as a nonlinear dynamical system. First, a discussion of the data set used and the techniques utilized to reduce the noise and capture the long-term dynamics inherent in the data is presented. Subsequently, an attractor is reconstructed from the data set using the method of time delays. The reconstructed attractor is then used to determine both the dimension of the underlying system and also the largest Lyapunov exponent, which together indicate that the sunspot cycle is indeed chaotic and also low dimensional. In addition, recent techniques of exploiting chaotic dynamics to provide accurate, short-term predictions are utilized in order to improve upon current forecasting methods and also to place theoretical limits on predictability extent. The results are compared to chaotic solar-dynamo models as a possible physically motivated source of this chaotic behavior.

Mundt, Michael D.↗

NASA NTRS1992

Detecting and disentangling nonlinear structure from solar flux time series

Interest in solar activity has grown in the past two decades for many reasons. Most importantly for flight dynamics, solar activity changes the atmospheric density, which has important implications for spacecraft trajectory and lifetime prediction. Building upon the previously developed Rayleigh-Benard nonlinear dynamic solar model, which exhibits many dynamic behaviors observed in the Sun, this work introduces new chaotic solar forecasting techniques. Our attempt to use recently developed nonlinear chaotic techniques to model and forecast solar activity has uncovered highly entangled dynamics. Numerical techniques for decoupling additive and multiplicative white noise from deterministic dynamics and examines falloff of the power spectra at high frequencies as a possible means of distinguishing deterministic chaos from noise than spectrally white or colored are presented. The power spectral techniques presented are less cumbersome than current methods for identifying deterministic chaos, which require more computationally intensive calculations, such as those involving Lyapunov exponents and attractor dimension.

Ashrafi, S.↗

NASA NTRS2006

Dynamics and Adaptive Control for Stability Recovery of Damaged Aircraft

This paper presents a recent study of a damaged generic transport model as part of a NASA research project to investigate adaptive control methods for stability recovery of damaged aircraft operating in off-nominal flight conditions under damage and or failures. Aerodynamic modeling of damage effects is performed using an aerodynamic code to assess changes in the stability and control derivatives of a generic transport aircraft. Certain types of damage such as damage to one of the wings or horizontal stabilizers can cause the aircraft to become asymmetric, thus resulting in a coupling between the longitudinal and lateral motions. Flight dynamics for a general asymmetric aircraft is derived to account for changes in the center of gravity that can compromise the stability of the damaged aircraft. An iterative trim analysis for the translational motion is developed to refine the trim procedure by accounting for the effects of the control surface deflection. A hybrid direct-indirect neural network, adaptive flight control is proposed as an adaptive law for stabilizing the rotational motion of the damaged aircraft. The indirect adaptation is designed to estimate the plant dynamics of the damaged aircraft in conjunction with the direct adaptation that computes the control augmentation. Two approaches are presented 1) an adaptive law derived from the Lyapunov stability theory to ensure that the signals are bounded, and 2) a recursive least-square method for parameter identification. A hardware-in-the-loop simulation is conducted and demonstrates the effectiveness of the direct neural network adaptive flight control in the stability recovery of the damaged aircraft. A preliminary simulation of the hybrid adaptive flight control has been performed and initial data have shown the effectiveness of the proposed hybrid approach. Future work will include further investigations and high-fidelity simulations of the proposed hybrid adaptive Bight control approach.

Nguyen, Nhan↗

NASA NTRS2022

Shape Servoing of Deformable Objects using Adaptive Deformation Model Estimation

In this paper, we propose an adaptive shape servoing method to deform a soft object into a desired 3-D shape. The high dimensional representation and the unknown deformation properties of the soft object pose a challenge to actively manipulate its shape. To address this issue, we develop a method to compute the deformation Jacobian matrix in real-time. The Jacobian is estimated using a set of basis functions and its corresponding parameters to capture the dynamics of the system and relate the applied input motion to changes in the soft object's shape. An integral concurrent learning (ICL) based adaptive update law is derived using Lyapunov analysis to estimate the deformation parameters and prove its convergence. A physics-based simulation is used to validate the proposed method and controller by performing manipulation tasks with different desired configurations. The performance is compared with a standard gradient update law to demonstrate the accuracy and robustness of our approach.

Vrithik Raj Guthikonda↗

DOE OSTI2023

A greedy Galerkin method to efficiently select sensors for linear dynamical systems

A key challenge in inverse problems is the selection of sensors to gather the most effective data. In this paper, we consider the problem of inferring the initial condition to a linear dynamical system and develop an efficient control-theoretical approach for greedily selecting sensors. Our method employs a Galerkin projection to reduce the size of the inverse problem, resulting in a computationally efficient algorithm for sensor selection. As a byproduct of our algorithm, we obtain a preconditioner for the inverse problem that enables the rapid recovery of the initial condition. Here, we analyze the theoretical performance of our greedy sensor selection algorithm as well as the performance of the associated preconditioner. Finally, we verify our theoretical results on various inverse problems involving partial differential equations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Safe Physics-Informed Machine Learning for Dynamics and Control

This tutorial paper focuses on safe physics-informed machine learning in the context of dynamics and control, providing a comprehensive overview of how to integrate physical models and safety guarantees. As machine learning techniques enhance the modeling and control of complex dynamical systems, ensuring safety and stability remains a critical challenge, especially in safety-critical applications like autonomous vehicles, robotics, medical decision-making, and energy systems. We explore various approaches for embedding and ensuring safety constraints, including structural priors, Lyapunov and Control Barrier Functions, predictive control, projections, and robust optimization techniques. Additionally, we delve into methods for uncertainty quantification and safety verification, including reachability analysis and neural network verification tools, which help validate that control policies remain within safe operating bounds even in uncertain environments. The paper includes illustrative examples demonstrating the implementation aspects of safe learning frameworks that combine the strengths of data-driven approaches with the rigor of physical principles, offering a path toward the safe control of complex dynamical systems.

Drgona, Jan↗

NASA NTRS2001

Lyapunov-Based Sensor Failure Detection And Recovery For The Reverse Water Gas Shift Process

Livingstone, a model-based AI software system, is planned for use in the autonomous fault diagnosis, reconfiguration, and control of the oxygen-producing reverse water gas shift (RWGS) process test-bed located in the Applied Chemistry Laboratory at KSC. In this report the RWGS process is first briefly described and an overview of Livingstone is given. Next, a Lyapunov-based approach for detecting and recovering from sensor failures, differing significantly from that used by Livingstone, is presented. In this new method, models used are in terms of the defining differential equations of system components, thus differing from the qualitative, static models used by Livingstone. An easily computed scalar inequality constraint, expressed in terms of sensed system variables, is used to determine the existence of sensor failures. In the event of sensor failure, an observer/estimator is used for determining which sensors have failed. The theory underlying the new approach is developed. Finally, a recommendation is made to use the Lyapunov-based approach to complement the capability of Livingstone and to use this combination in the RWGS process.

Haralambous, Michael G.↗

NASA NTRS2002

LYAPUNOV-Based Sensor Failure Detection and Recovery for the Reverse Water Gas Shift Process

Livingstone, a model-based AI software system, is planned for use in the autonomous fault diagnosis, reconfiguration, and control of the oxygen-producing reverse water gas shift (RWGS) process test-bed located in the Applied Chemistry Laboratory at KSC. In this report the RWGS process is first briefly described and an overview of Livingstone is given. Next, a Lyapunov-based approach for detecting and recovering from sensor failures, differing significantly from that used by Livingstone, is presented. In this new method, models used are in t e m of the defining differential equations of system components, thus differing from the qualitative, static models used by Livingstone. An easily computed scalar inequality constraint, expressed in terms of sensed system variables, is used to determine the existence of sensor failures. In the event of sensor failure, an observer/estimator is used for determining which sensors have failed. The theory underlying the new approach is developed. Finally, a recommendation is made to use the Lyapunov-based approach to complement the capability of Livingstone and to use this combination in the RWGS process.

Haralambous, Michael G.↗

NASA NTRS2000

Parametric Modeling and Fault Tolerant Control

Fault tolerant control is considered for a nonlinear aircraft model expressed as a linear parameter-varying system. By proper parameterization of foreseeable faults, the linear parameter-varying system can include fault effects as additional varying parameters. A recently developed technique in fault effect parameter estimation allows us to assume that estimates of the fault effect parameters are available on-line. Reconfigurability is calculated for this model with respect to the loss of control effectiveness to assess the potentiality of the model to tolerate such losses prior to control design. The control design is carried out by applying a polytopic method to the aircraft model. An error bound on fault effect parameter estimation is provided, within which the Lyapunov stability of the closed-loop system is robust. Our simulation results show that as long as the fault parameter estimates are sufficiently accurate, the polytopic controller can provide satisfactory fault-tolerance.

Wu, N. Eva↗

DOE OSTI2021

Numerical solution of singular Lyapunov equations

We consider the numerical solution of large scale singular (continuous-time) Lyapunov equations of the form AX + XA T + BB T = 0, where A is semistable, that is, its spectrum is contained in the left half plane, with the exception of a few semisimple eigenvalues at zero. We also consider the case of a few semisimple eigenvalues on the imaginary axis. We assume that we know these few eigenvalues (zero or imaginary), and that we have or can compute the corresponding invariant subspaces. We use this information to build an appropriate newly proposed subspace on which to project the Lyapunov equations, and then compute a low-rank approximation to the least squares solution. Selected illustrative numerical examples are provided.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

How Hall electric fields intrinsically chaotize and heat ions during collisionless magnetic reconnection

The origin of extreme and anomalous ion heating during magnetic reconnection has been a longstanding mystery. It is shown in this paper that, at the core of the process, the stochastic heating mechanism is responsible for this energization. The in-plane Hall electric fields that intrinsically develop during collisionless reconnection not only accelerate ions, but also thermalize them by inducing chaotic motion. The mechanism explains various observed features of ion heating, e.g., its relation to the guide field strength and the ion mass-to-charge ratio.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1994

Automated path planning of the Payload Inspection and Processing System

The Payload Changeout Room Inspection and Processing System (PIPS) is a highly redundant manipulator intended for performing tasks in the crowded and sensitive environment of the Space Shuttle Orbiter payload bay. Its dexterity will be exploited to maneuver the end effector in a workspace populated with obstacles. A method is described by which the end effector of a highly redundant manipulator is directed toward a target via a Lyapunov stability function. A cost function is constructed which represents the distance from the manipulator links to obstacles. Obstacles are avoided by causing the vector of joint parameters to move orthogonally to the gradient of the workspace cost function. A C language program implements the algorithm to generate a joint history. The resulting motion is graphically displayed using the Interactive Graphical Robot Instruction Program (IGRIP) produced by Deneb Robotics. The graphical simulation has the potential to be a useful tool in path planning for the PIPS in the Shuttle Payload Bay environment.

Byers, Robert M.↗

NASA NTRS2017

An Analysis of the Optimal Control Modification Method Applied to Flutter Suppression

Unlike basic Model Reference Adaptive Control (MRAC)l, Optimal Control Modification (OCM) has been shown to be a promising MRAC modification with robustness and analytical properties not present in other adaptive control methods. This paper presents an analysis of the OCM method, and how the asymptotic property of OCM is useful for analyzing and tuning the controller. We begin with a Lyapunov stability proof of an OCM controller having two adaptive gain terms, then the less conservative and easily analyzed OCM asymptotic property is presented. Two numerical examples are used to show how this property can accurately predict steady state stability and quantitative robustness in the presence of time delay, and relative to linear plant perturbations, and nominal Loop Transfer Recovery (LTR) tuning. The asymptotic property of the OCM controller is then used as an aid in tuning the controller applied to a large scale aeroservoelastic longitudinal aircraft model for flutter suppression. Control with OCM adaptive augmentation is shown to improve performance over that of the nominal non-adaptive controller when significant disparities exist between the controller/observer model and the true plant model.

aeroservoelasti↗

DOE OSTI2021

A Model-Reference Adaptive Direct-Power Control Scheme for Grid-Interactive Inverters

This paper develops an adaptive direct power control scheme for grid-interactive inverters while the control parameters are adaptively adjusted as the grid and filter parameters vary under different conditions. To demonstrate the efficacy and robustness of the adaptive operation, the performance of the adaptive controller is compared with a typical direct power controller by estimating the filter and grid impedances using a model reference parameter estimation technique with small signal injection. The simulation results verify that the developed methodology can track the reference power setpoints under output circuit parameter uncertainties while the typical design leads to instability. To improve the performance of the adaptive technique and eliminate the need for filter parameter estimation, a Lyapunov stability criterion-based model-reference adaptive direct power controller is designed to track the filter impedance. The developed method was tested using MATLAB/Simulink software to demonstrate the effectiveness of the proposed method. The results verify that adaptively tuning the control parameter can still maintain stable operation if the control gains are poorly selected while applying the classical method leads to abnormal operation or instability.

Gursoy, Mehmetcan↗