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Results for “Lyapunov equation”

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 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
NASA NTRS2006

Stochastic Representation of Chaos Using Terminal Attractors

A nonlinear version of the Liouville equation based on terminal attractors is part of a mathematical formalism for describing postinstability motions of dynamical systems characterized by exponential divergences of trajectories leading to chaos (including turbulence as a form of chaos). The formalism can be applied to both conservative systems (e.g., multibody systems in celestial mechanics) and dissipative systems (e.g., viscous fluids). The development of the present formalism was undertaken in an effort to remove positive Lyapunov exponents. The means chosen to accomplish this is coupling of the governing dynamical equations with the corresponding Liouville equation that describes the evolution of the flow of error probability. The underlying idea is to suppress the divergences of different trajectories that correspond to different initial conditions, without affecting a target trajectory, which is one that starts with prescribed initial conditions.

Zak, Michail↗
NASA NTRS2013

Explicit Low-Thrust Guidance for Reference Orbit Targeting

The problem of a low-thrust spacecraft controlled to a reference orbit is addressed in this paper. A simple and explicit low-thrust guidance scheme with constrained thrust magnitude is developed by combining the fundamental equations of motion for constrained systems from analytical dynamics with a Lyapunov-based method. Examples are given for a spacecraft controlled to a reference trajectory in the circular restricted three body problem.

low-thrust guidance↗
NASA NTRS2003

Extension of Liouville Formalism to Postinstability Dynamics

A mathematical formalism has been developed for predicting the postinstability motions of a dynamic system governed by a system of nonlinear equations and subject to initial conditions. Previously, there was no general method for prediction and mathematical modeling of postinstability behaviors (e.g., chaos and turbulence) in such a system. The formalism of nonlinear dynamics does not afford means to discriminate between stable and unstable motions: an additional stability analysis is necessary for such discrimination. However, an additional stability analysis does not suggest any modifications of a mathematical model that would enable the model to describe postinstability motions efficiently. The most important type of instability that necessitates a postinstability description is associated with positive Lyapunov exponents. Such an instability leads to exponential growth of small errors in initial conditions or, equivalently, exponential divergence of neighboring trajectories. The development of the present formalism was undertaken in an effort to remove positive Lyapunov exponents. The means chosen to accomplish this is coupling of the governing dynamical equations with the corresponding Liouville equation that describes the evolution of the flow of error probability. The underlying idea is to suppress the divergences of different trajectories that correspond to different initial conditions, without affecting a target trajectory, which is one that starts with prescribed initial conditions.

Zak, Michail↗
DOE OSTI2021

Lyapunov stability of smart inverters using linearized distflow approximation

Fast-acting smart inverters that utilize preset operating conditions to determine real and reactive power injection/consumption can create voltage instabilities (over-voltage, voltage oscillations and more) in an electrical distribution network if set-points are not properly configured. In this work, linear distribution power flow equations and droop-based Volt–Var and Volt–Watt control curves are used to analytically derive a stability criterion using Lyapunov analysis that includes the network operating condition. The methodology is generally applicable for control curves that can be represented as Lipschitz functions. The derived Lipschitz constants account for smart inverter hardware limitations for reactive power generation. A local policy is derived from the stability criterion that allows inverters to adapt their control curves by monitoring only local voltage, thus avoiding centralized control or information sharing with other inverters. The criterion is independent of the internal time-delays of smart inverters. Simulation results for inverters with and without the proposed stabilization technique demonstrate how smart inverters can mitigate voltage oscillations locally and mitigate real and reactive power flow disturbances at the substation under multiple scenarios. The study concludes with illustrations of how the control policy can dampen oscillations caused by solar intermittency and cyberattacks.

42 ENGINEERING↗
DOE OSTI2024

Divide and conquer: Learning chaotic dynamical systems with multistep penalty neural ordinary differential equations

Forecasting high-dimensional dynamical systems is a fundamental challenge in various fields, such as geosciences and engineering. Neural Ordinary Differential Equations (NODEs), which combine the power of neural networks and numerical solvers, have emerged as a promising algorithm for forecasting complex nonlinear dynamical systems. However, classical techniques used for NODE training are ineffective for learning chaotic dynamical systems. In this work, we propose a novel NODE-training approach that allows for robust learning of chaotic dynamical systems. Here, our method addresses the challenges of non-convexity and exploding gradients associated with underlying chaotic dynamics. Training data trajectories from such systems are split into multiple, non-overlapping time windows. In addition to the deviation from the training data, the optimization loss term further penalizes the discontinuities of the predicted trajectory between the time windows. The window size is selected based on the fastest Lyapunov time scale of the system. Multi-step penalty(MP) method is first demonstrated on Lorenz equation, to illustrate how it improves the loss landscape and thereby accelerates the optimization convergence. MP method can optimize chaotic systems in a manner similar to least-squares shadowing with significantly lower computational costs. Our proposed algorithm, denoted the Multistep Penalty NODE, is applied to chaotic systems such as the Kuramoto-Sivashinsky equation, the two-dimensional Kolmogorov flow, and ERA5 reanalysis data for the atmosphere. It is observed that MP-NODE provide viable performance for such chaotic systems, not only for short-term trajectory predictions but also for invariant statistics that are hallmarks of the chaotic nature of these dynamics.

Chaotic dynamical systems↗
NASA NTRS1990

Laws For Stable Control Of Orientation

Class of globally stable laws for control of orientation of rigid body established. Tradeoffs made among complexity, performance, and information available for mathematical models. New approach involves use of globally nonsingular four-parameter unit-quaternion (Euler-parameter) representation. Attitude dynamics described by nonlinear Euler equation together with nonlinear kinematic equations, which relate representation of attitude to angular velocity of body. Suitable energy-motivated Lyapunov function to analyze stability.

Kreutz, Kenneth K.↗
NASA NTRS1975

Prediction of axial-flow instabilities in a turbojet engine by use of a multistage compressor simulation on the digital computer

A method of estimating the undistorted stall line for an axial-flow compressor by using the digital computer is presented. The method involves linearization of nonlinear dynamic equations about an operating point on a speed line, and then application of the first method of Lyapunov to determine the stability of the nonlinear system from the stability of the linear system. The method is applied to a simulation of the J85 compressor, which utilizes stage stacking and lumped volume techniques for the interstage regions to simulate steady-state and dynamic compressor performance. The stability boundary predicted by the digital simulation compares quite well with the stall line predicted by a dynamic simulation of the J85 compressor programmed on the analog computer. Since previous studies have shown that the analog-predicted stall line agrees well with the stall line of the compressor, the digital method presented is also a good means of estimating the stall line.

Daniele, C. J.↗
NASA NTRS1972

Invariance and stability for bounded uncertain systems.

The positive limit sets of the solutions of a contingent differential equation are shown to possess an invariance property. In this connection the 'invariance principle' in the theory of Lyapunov stability is extended to systems with unknown, bounded, time-varying parameters, and thus to a large and important class of nonautonomous systems. Asymptotic stability criteria are obtained and applied to guaranteed cost control problems.

Peng, T. K. C.↗
DOE OSTI2022

Sensitivity of time-dependent density functional theory to initial conditions

Time-dependent density-functional theory is mathematically formulated through nonlinear coupled time-dependent three-dimensional partial differential equations, and it is natural to expect a strong sensitivity of its solutions to variations of the initial conditions, akin to the butterfly effect ubiquitous in classical dynamics. Since the Schrödinger equation for an interacting many-body system is, however, linear and mathematically the exact equations of the density-functional theory reproduce the corresponding one-body properties, it would follow that the Lyapunov exponents are also vanishing within a density-functional theory framework. Whether for realistic implementations of the time-dependent density-functional theory the question of the absence of the butterfly effect and whether the dynamics provided is indeed a predictable theory was never discussed. At the same time, since the time-dependent density-functional theory is a unique tool allowing us to study the nonequilibrium dynamics of strongly interacting many-fermion systems, the question of predictability of this theoretical framework is of paramount importance. Here our analysis, for a number of quantum superfluid many-body systems (unitary Fermi gas, nuclear fission, and heavy-ion collisions) with a classical equivalent number of degrees of freedom O(10 10 ) and larger, suggests that its maximum Lyapunov exponents are negligible for all practical purposes.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NASA NTRS1972

An approximate method for the synthesis of optimal control of distributed systems.

An approximate method is presented for the synthesis of an optimal control of distributed parameter systems, based on a combination of the ideas of Lyapunov's direct method and those of Bellman's successive approximation in policy space. The method is such that the approximate equation is improved after each iteration in the sense of the performance index being used.

Park, K. E.↗
NASA NTRS1992

Mechanism synthesis and 2-D control designs of an active three cable crane

A Lunar Crane with a suspension system based on a three cable mechanism is investigated to provide a stable end-effector for hoisting, positioning, and assembling large components during construction and servicing of a Lunar Base. The three cable suspension mechanism consists of a structural framework of three cables pointing to a common point that closely coincides with the suspended payload's center of gravity. The vibrational characteristics of this three cable suspension system are investigated by comparing a simple 2-D symmetric suspension model and a swinging pendulum in terms of their analytical natural frequency equations. A study is also made of actively controlling the dynamics of the crane using two different actuator concepts. Also, Lyapunov-based control algorithms are developed to determine two regulator-type control laws performing the system vibrational suppression for both system dynamics. Simulations including initial-valued dynamic responses as well as control performances for two different system dynamics are also presented.

Yang, Li-Farn↗
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 NTRS2003

A Nonlinear, Six-Degree of Freedom Precision Formation Control Algorithm, Based on Restricted Three Body Dynamics

Precision Formation Flying is an enabling technology for a variety of proposed space-based observatories, including the Micro-Arcsecond X-ray Imaging Mission (MAXIM), the associated MAXIM pathfinder mission, and the Stellar Imager. An essential element of the technology is the control algorithm. This paper discusses the development of a nonlinear, six-degree of freedom (6DOF) control algorithm for maintaining the relative position and attitude of a spacecraft within a formation. The translation dynamics are based on the equations of motion for the restricted three body problem. The control law guarantees the tracking error convergences to zero, based on a Lyapunov analysis. The simulation, modelled after the MAXIM Pathfinder mission, maintains the relative position and attitude of a Follower spacecraft with respect to a Leader spacecraft, stationed near the L2 libration point in the Sun-Earth system.

Bauer, Frank↗
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 NTRS1996

Regular and Chaotic Quantum Dynamics of Two-Level Atoms in a Selfconsistent Radiation Field

Dynamics of two-level atoms interacting with their own radiation field in a single-mode high-quality resonator is considered. The dynamical system consists of two second-order differential equations, one for the atomic SU(2) dynamical-group parameter and another for the field strength. With the help of the maximal Lyapunov exponent for this set, we numerically investigate transitions from regularity to deterministic quantum chaos in such a simple model. Increasing the collective coupling constant b is identical with 8(pi)N(sub 0)(d(exp 2))/hw, we observed for initially unexcited atoms a usual sharp transition to chaos at b(sub c) approx. equal to 1. If we take the dimensionless individual Rabi frequency a = Omega/2w as a control parameter, then a sequence of order-to-chaos transitions has been observed starting with the critical value a(sub c) approx. equal to 0.25 at the same initial conditions.

Konkov, L. E.↗
NASA NTRS1988

Globally stable control laws for the attitude maneuver problem - Tracking control and adaptive control

An approach using a globally nonsingular representation is proposed for the attitude control problem of a rigid body. The attitude dynamics are described by the nonlinear Euler equation together with the nonlinear kinematic equations which relate a representation of attitude to the angular velocity of the body. When this approach is combined with an energy-motivated Lyapunov function, a large class of globally stable attitude control laws can be derived. This class includes model-independent tracking control, model-dependent tracking control, and adaptive control, allowing tradeoffs between controller complexity, attainable performance, and available model information.

Wen, John T.↗