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Results for “Stochastic differential equations”

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 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
NASA NTRS2018

A Fast Monte Carlo Method for Model-Based Prognostics Based on Stochastic Calculus

This work proposes a fast Monte Carlo method to solve differential equations utilized in model-based prognostics. The methodology is derived from the theory of stochastic calculus, and the goal of such a method is to speed up the estimation of the probability density functions describing the independent variable evolution over time. In the prognostic scenarios presented in this paper, the stochastic differential equations describe variables directly or indirectly related to the degradation of a monitored system. The method allows the estimation of the probability density functions by solving the deterministic equation and approximating the stochastic integrals using samples of the model noise. By so doing, the prognostic problem is solved without the Monte Carlo simulation based on Euler's forward method, which is typically the most time consuming task of the prediction stage. Three different prognostic scenarios are presented as proof of concept: (i) life prediction of electrolytic capacitors, (ii) remaining time to discharge of Lithium-ion batteries, and (iii) prognostic of cracked structures under fatigue loading. The paper shows how the method produces probability density functions that are statistically indistinguishable from the distributions estimated with Euler's forward Monte Carlo simulation. However, the proposed solution is orders of magnitude faster when computing the time-to-failure distribution of the monitored system. The approach may enable complex real-time prognostics and health management solutions with limited computing power.

Corbetta, M.↗
NASA NTRS2011

Optimal Control via Self-Generated Stochasticity

The problem of global maxima of functionals has been examined. Mathematical roots of local maxima are the same as those for a much simpler problem of finding global maximum of a multi-dimensional function. The second problem is instability even if an optimal trajectory is found, there is no guarantee that it is stable. As a result, a fundamentally new approach is introduced to optimal control based upon two new ideas. The first idea is to represent the functional to be maximized as a limit of a probability density governed by the appropriately selected Liouville equation. Then, the corresponding ordinary differential equations (ODEs) become stochastic, and that sample of the solution that has the largest value will have the highest probability to appear in ODE simulation. The main advantages of the stochastic approach are that it is not sensitive to local maxima, the function to be maximized must be only integrable but not necessarily differentiable, and global equality and inequality constraints do not cause any significant obstacles. The second idea is to remove possible instability of the optimal solution by equipping the control system with a self-stabilizing device. The applications of the proposed methodology will optimize the performance of NASA spacecraft, as well as robot performance.

Zak, Michail↗
NASA NTRS1973

A guidance and navigation system for continuous low thrust vehicles

A midcourse guidance and navigation system for continuous low thrust vehicles is described. A set of orbit elements, known as the equinoctial elements, are selected as the state variables. The uncertainties are modelled statistically by random vector and stochastic processes. The motion of the vehicle and the measurements are described by nonlinear stochastic differential and difference equations respectively. A minimum time nominal trajectory is defined and the equation of motion and the measurement equation are linearized about this nominal trajectory. An exponential cost criterion is constructed and a linear feedback guidance law is derived to control the thrusting direction of the engine. Using this guidance law, the vehicle will fly in a trajectory neighboring the nominal trajectory. The extended Kalman filter is used for state estimation. Finally a short mission using this system is simulated. The results indicate that this system is very efficient for short missions.

Tse, C. J. C.↗
NASA NTRS1973

A guidance and navigation system for continuous low-thrust vehicles

A midcourse guidance and navigation system for continuous low thrust vehicles was developed. The equinoctial elements are the state variables. Uncertainties are modelled statistically by random vector and stochastic processes. The motion of the vehicle and the measurements are described by nonlinear stochastic differential and difference equations respectively. A minimum time trajectory is defined; equations of motion and measurements are linearized about this trajectory. An exponential cost criterion is constructed and a linear feedback quidance law is derived. An extended Kalman filter is used for state estimation. A short mission using this system is simulated. It is indicated that this system is efficient for short missions, but longer missions require accurate trajectory and ground based measurements.

Jack-Chingtse, C.↗
NASA NTRS1974

Offset quadrature communications with decision-feedback carrier synchronization

In order to accommodate a quadrature amplitude-shift-keyed (QASK) signal, Simon and Smith (1974) have modified the decision-feedback loop which tracks a quadrature phase-shift-keyed (QPSK). In the investigation reported approaches are considered to modify the loops in such a way that offset QASK signals can be tracked, giving attention to the special case of an offset QPSK. The development of the stochastic integro-differential equation of operation for a decision-feedback offset QASK loop is discussed along with the probability density function of the phase error process.

Simon, M. K.↗
NASA NTRS1972

Nonlinear filtering for random signals in statistically unknown noise.

Natural and effective formulation of the filtering problem involved in satellite orbit determination, aircraft navigation, and missile tracking. The problem arises because the environment of the sensor keeps changing from time to time, and it is quite impractical and sometimes impossible to collect the statistical data of the noise incurred in the observation. Computable filtering equations are deduced. The idea of invariant imbedding along with stochastic differential calculus is used to derive differential equations for the optimal estimate.

Loo, J. T.↗
NASA NTRS1973

Solution of the stochastic control problem in unbounded domains.

Bellman's dynamic programming equation for the optimal index and control law for stochastic control problems is a parabolic or elliptic partial differential equation frequently defined in an unbounded domain. Existing methods of solution require bounded domain approximations, the application of singular perturbation techniques or Monte Carlo simulation procedures. In this paper, using the fact that Poisson impulse noise tends to a Gaussian process under certain limiting conditions, a method which achieves an arbitrarily good approximate solution to the stochastic control problem is given. The method uses the two iterative techniques of successive approximation and quasi-linearization and is inherently more efficient than existing methods of solution.

Robinson, P.↗
NASA NTRS1975

Stochastic stability and instability of model ecosystems

In this work, we initiate a stability study of multispecies communities in stochastic environment by using Ito's differential equations as community models. By applying the direct method of Liapunov, we obtain sufficient conditions for stability and instability in the mean of the equilibrium populations. The conditions are expressed in terms of the dominant diagonal property of community matrices, which is a suitable mechanism for resolving the central problem of 'complexity vs stability' in model ecosystems. As a by-product of this analysis we exhibit important structural properties of the stochastic density-dependent models, and establish tolerance of community stability to a broad class of nonlinear time-varying perturbations.

Ladde, G. S.↗
NASA NTRS1971

Differential geometric methods in system theory.

Discussion of certain problems in system theory which have been or might be solved using some basic concepts from differential geometry. The problems considered involve differential equations, controllability, optimal control, qualitative behavior, stochastic processes, and bilinear systems. The main goal is to extend the essentials of linear theory to some nonlinear classes of problems.

Brockett, R. W.↗
NASA NTRS1984

Non-Gaussian approach for parametric random vibration of non-linear structures

The dynamic response of a nonlinear, single degree of freedom structural system subjected to a physically white noise parametric excitation is investigated. The Ito stochastic calculus is employed to derive a general differential equation for the moments of the response coordinates. The differential equations of moments of any order are found to be coupled with higher order moments. A non-Gaussian closure scheme is developed to truncate the moment equations up to fourth order. The statistical of the stationary response are computed numerically and compared with analytical solutions predicted by a Gaussian closure scheme and the stochastic averaging method. It is found that the computed results exhibit the jump phenomenon which is typical of the characteristics of deterministic nonlinear systems. In addition, the numerical algorithm leads to multiple solutions all of which give positive mean squares. However, two of these solutions are found to violate the properties of high order moments. One solution preserves the moments properties and demonstrates that the system achieves a stationary response.

Ibrahim, R. A.↗
NASA NTRS1983

Singular Perturbations in Systems and Control

The application of singular perturbation theory to problems in systems and control is addressed. The types of systems considered include linear and nonlinear, discrete and continuous, and stochastic and deterministic. Both ordinary and partial differential equations are covered.

Ardema, M. D.↗
NASA NTRS1969

Optimal stochastic control.

Optimal stochastic control, discussing dynamic mathematical models described by differential equations

Wonham, W. M.↗
NASA NTRS1968

Optimal stochastic control.

Optimal stochastic control of small dynamic systems described by random differential equations, noting selective bibliography on several subjects

Wonham, W. M.↗
NASA NTRS1993

A Discrete Probability Function Method for the Equation of Radiative Transfer

A discrete probability function (DPF) method for the equation of radiative transfer is derived. The DPF is defined as the integral of the probability density function (PDF) over a discrete interval. The derivation allows the evaluation of the PDF of intensities leaving desired radiation paths including turbulence-radiation interactions without the use of computer intensive stochastic methods. The DPF method has a distinct advantage over conventional PDF methods since the creation of a partial differential equation from the equation of transfer is avoided. Further, convergence of all moments of intensity is guaranteed at the basic level of simulation unlike the stochastic method where the number of realizations for convergence of higher order moments increases rapidly. The DPF method is described for a representative path with approximately integral-length scale-sized spatial discretization. The results show good agreement with measurements in a propylene/air flame except for the effects of intermittency resulting from highly correlated realizations. The method can be extended to the treatment of spatial correlations as described in the Appendix. However, information regarding spatial correlations in turbulent flames is needed prior to the execution of this extension.

Sivathanu, Y. R.↗
NASA NTRS1987

Active stability augmentation of large space structures: A stochastic control problem

A problem in SCOLE is that of slewing an offset antenna on a long flexible beam-like truss attached to the space shuttle, with rather stringent pointing accuracy requirements. The relevant methodology aspects in robust feedback-control design for stability augmentation of the beam using on-board sensors is examined. It is framed as a stochastic control problem, boundary control of a distributed parameter system described by partial differential equations. While the framework is mathematical, the emphasis is still on an engineering solution. An abstract mathematical formulation is developed as a nonlinear wave equation in a Hilbert space. That the system is controllable is shown and a feedback control law that is robust in the sense that it does not require quantitative knowledge of system parameters is developed. The stochastic control problem that arises in instrumenting this law using appropriate sensors is treated. Using an engineering first approximation which is valid for small damping, formulas for optimal choice of the control gain are developed.

Balakrishnan, A. V.↗
NASA NTRS2008

Simulation of Stochastic Processes by Coupled ODE-PDE

A document discusses the emergence of randomness in solutions of coupled, fully deterministic ODE-PDE (ordinary differential equations-partial differential equations) due to failure of the Lipschitz condition as a new phenomenon. It is possible to exploit the special properties of ordinary differential equations (represented by an arbitrarily chosen, dynamical system) coupled with the corresponding Liouville equations (used to describe the evolution of initial uncertainties in terms of joint probability distribution) in order to simulate stochastic processes with the proscribed probability distributions. The important advantage of the proposed approach is that the simulation does not require a random-number generator.

Zak, Michail↗