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Time domain expression derived for frequency varying nonperiodic square wave in FDM and FM systems, using orthogonal functions to obtain frequency spectrum
Mathematical representations of a state-plane switching boundary employed in a state-trajectory control law for dc-to-dc converters are derived. Several levels of approximation to the switching boundary equations are presented, together with an evaluation of the effects of nonideal operating characteristics of converter power stage components on the shape and location of the boundary and the behavior of a system controlled by it. Digital computer simulations of dc-to-dc converters operating in conjunction with each of these levels of control are presented and evaluated with respect to changes in transient and steady-state performance.
The depth to which sunlight will penetrate in turbid waters was investigated. The tests were conducted in water with a single scattering albedo range, and over a range of solar elevation angles. Two different techniques were used to determine the depth of light penetration. It showed little change in the depth of sunlight penetration with changing solar elevation angle. A comparison of the penetration depths indicates that the best agreement between the two methods was achieved when the quasisingle scattering relationship was not corrected for solar angle. It is concluded that sunlight penetration is dependent on inherent water properties only.
The semianalytical method is widely used for calculating derivatives of static response with respect to design variables for structures modeled by finite elements. This paper shows that the method can have serious accuracy problems for shape design variables in structures modeled by beam elements. The problem was first discovered in the analysis of a complex car model which is described. Next, it is shown that the same accuracy problem occurs for the simplest beam structures. Finally, the problem is shown to be associated with a high ratio of rigid body rotation to elastic deformation, and a test is developed to diagnose the severity of the problem in a given structure. Several examples are presented to show the validity of this test.
The International Ultraviolet Explorer (IUE) satellite was launched January 26, 1978 into a geosynchronous orbit over South America. From its stationary position, the telescope maintains continuous communication with the control centers at NASA's Goddard Space Flight Center in Greenbelt, Maryland, and at the European Space Agency's (ESA's) Villagranca del Castillo Satellite Tracking Station in Spain. Since its launch in 1978, the satellite has gradually lost four of the original six gyroscopes in the Inertial Reference Assembly (IRA). In August 1985, the fourth of the original six gyros failed and a two-gyro system developed by NASA-GSFC was uplinked to the satellite and is currently in use. A one-gyro system also developed by NASA-GSFC is ready for use in case of another gyro failure. In the event that the sixth gyro should also fail, a zero-gyro system is being developed. The goal of this system is to provide interial target pointing without the use of gyroscopes. The satellite has sun sensors to provide attitude information about two of the three axes. It relies upon the exchange of reaction wheel momenta to determine angular position and rate of the third axis.
Accurate International Space Station (ISS) power prediction requires the quantification of solar array shadowing. Prior papers have discussed the NASA Glenn Research Center (GRC) ISS power system tool SPACE (System Power Analysis for Capability Evaluation) and its integrated shadowing algorithms. On-orbit telemetry has become available that permits the correlation of theoretical shadowing predictions with actual data. This paper documents the comparison of a shadowing metric (total solar array current) as derived from SPACE predictions and on-orbit flight telemetry data for representative significant shadowing cases. Images from flight video recordings and the SPACE computer program graphical output are used to illustrate the comparison. The accuracy of the SPACE shadowing capability is demonstrated for the cases examined.
This article highlights fiscal year 2002 work performed by NASA Glenn Research Center personnel to validate algorithms and data developed in-house to predict shadowing effects on the International Space Station (ISS) solar arrays power generation. The validation effort utilized video footage and on-orbit telemetry for cases spanning a 1-yr period. Validation was required because of the uncertainty of various aspects involved in shadowing analysis. Results show that a good comparison exists between actual and predicted shadowed power system performance for solar array front and backside shadowing.
A numerical solution for the canopy optical depth in an existing microwave-based land surface parameter retrieval model is presented. The optical depth is derived from the microwave polarization difference index and the dielectric constant of the soil. The original procedure used an approximation in the form of a logarithmic decay function to define this relationship, and was derived through a series of lengthy polynomials. These polynomials had to be recalculated when the scattering albedo or antenna incidence angle changes. The new procedure is computationally more efficient and accurate.
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A procedure for deriving analytic partial derivatives of the Lambert problem is presented. Using the universal, cosine based Lambert formulation; first order partial derivatives of the velocities with respect to the positions and times are developed. Taking advantage of inherent symmetries and intermediate variables, the derivatives are expressed in a computationally efficient form. The added cost of computing these partials is found to be approximately 10% to approximately 60% of the Lambert compute cost. The availability of analytic partial derivatives increases optimization speed, efficiency and allows for trajectory optimization formulations that implicitly enforce continuity constraints via embedded Lambert problems.
Analytic partial derivatives for estimating orbital low-thrust parameters via differential correction are developed and compared with two different numerical methods. The formulation is independent of the particular thrust model used and is applicable to all physically possible elliptic orbits. The starting point for the development is the set of variational equations of the elliptic orbital elements in the form due to Lagrange. The first time derivatives of the elements are transformed to derivatives with respect to the space variable, true anomaly, and integrated to first order in closed form in a straightforward general perturbations approach, with one exception: particular attention is given to the mean anomaly as influenced by thrust perturbations in the semimajor axis so that the complete first-order effect is included. The partials of the elements are then taken with respect to any given thrust parameter. Two comparisons are made with numerical methods for computing these thrust partials: numerical quotients and numerical integration of the variational equations for thrust.
The dynamics and control of flexible aerospace structures exercises many of the engineering disciplines. In recent years there has been considerable research in the developing and tailoring of control system design techniques for these structures. This problem involves designing a control system for a multi-input, multi-output (MIMO) system that satisfies various performance criteria, such as vibration suppression, disturbance and noise rejection, attitude control and slewing control. Considerable progress has been made and demonstrated in control system design techniques for these structures. The key to designing control systems for these structures that meet stringent performance requirements is an accurate model. It has become apparent that theoretically and finite-element generated models do not provide the needed accuracy; almost all successful demonstrations of control system design techniques have involved using test results for fine-tuning a model or for extracting a model using system ID techniques. This paper describes past and ongoing efforts at Ohio University and NASA MSFC to design controllers using 'data models.' The basic philosophy of this approach is to start with a stabilizing controller and frequency response data that describes the plant; then, iteratively vary the free parameters of the controller so that performance measures become closer to satisfying design specifications. The frequency response data can be either experimentally derived or analytically derived. One 'design-with-data' algorithm presented in this paper is called the Compensator Improvement Program (CIP). The current CIP designs controllers for MIMO systems so that classical gain, phase, and attenuation margins are achieved. The center-piece of the CIP algorithm is the constraint improvement technique which is used to calculate a parameter change vector that guarantees an improvement in all unsatisfied, feasible performance metrics from iteration to iteration. The paper also presents a recently demonstrated CIP-type algorithm, called the Model and Data Oriented Computer-Aided Design System (MADCADS), developed for achieving H(sub infinity) type design specifications using data models. Control system design for the NASA/MSFC Single Structure Control Facility are demonstrated for both CIP and MADCADS. Advantages of design-with-data algorithms over techniques that require analytical plant models are also presented.
Diffusion-mediated binding of molecules under the influence of discrete spatially confining potentials is a commonly encountered scenario in systems subjected to explicit fields or implicit fields arising from tethering restraints. Here, in this work, we derive analytical expressions for the mean binding time of two random walkers geometrically confined by means of two harmonic potentials in one- and two-dimensional systems, which show excellent agreement with Brownian dynamics simulations. As a demonstration of its utility, we use this theory to maximize the communication speed in existing DNA walkers, obtaining quantitative agreement with previously reported experimental findings. The analytical expressions derived in this paper are broadly applicable to diverse systems, providing ways to characterize communication processes and optimize the rate of signal propagation for sensing and computing applications at the nanoscale.
Conventional Bayesian optimization methods implicitly assume that the data and model being optimized are “perfect.” This assumption leads to inaccurate posterior probability distribution functions (PDFs) when applied to “imperfect” data or models. The new Bayesian optimization framework presented in this report provides a way to parameterize the effect of imperfections usually encountered in a prior PDF of generalized data or a model on the posterior PDF. The effects of imperfections are parameterized by a set of constraints imposed on the posterior expectation values of deviations between the data and the model and on their covariance matrix elements. A particular set of values for these constraints conveys an evaluator’s best estimate of the effect of imperfections on the corresponding posterior expectation values. When a prior PDF of generalized data is assumed to be normal, an expression for a posterior PDF satisfying an arbitrary set of constraints is derived analytically for linear models. An analogous iterative algorithm is given for nonlinear models. The corresponding posterior PDF should be used to estimate any posterior expectation values in the presence of imperfections parameterized by that set of constraints. A posterior PDF of a conventional Bayesian optimization method is recovered analytically when all evaluator-specified constraints are set to zero (i.e., in the absence of any imperfections). The analytical expressions derived in this report for normal PDFs and linear models were verified numerically by a Metropolis–Hastings Monte Carlo method. The methods presented herein could be applied to any kind of data or models, including differential cross-section data or integral benchmark experiments.
Sensitivity analysis is fundamental to the solution of structural optimization problems. Consequently, much research has focused on the efficient computation of static displacement derivatives. As originally developed, these methods relied on analytical representations for the derivatives of the structural stiffness matrix (K) with respect to the design variables (b sub i). To extend these methods for use with complex finite element formulations and facilitate their implementation into structural optimization programs using the general finite element method analysis codes, the semi-analytic method was developed. In this method the matrix the derivative of K/the derivative b sub i is approximated by finite difference. Although it is well known that the accuracy of the semi-analytic method is dependent on the finite difference parameter, recent work has suggested that more fundamental inaccuracies exist in the method when used for shape optimization. Another study has argued qualitatively that these errors are related to nonuniform errors in the stiffness matrix derivatives. The accuracy of the semi-analytic method is investigated. A general framework was developed for the error analysis and then it is shown analytically that the errors in the method are entirely accounted for by errors in delta K/delta b sub i. Furthermore, it is demonstrated that acceptable accuracy in the derivatives can be obtained through careful selection of the finite difference parameter.
One method of producing low-thrust trajectories is the closed-loop Q-Law guidance algorithm. This work wraps Q-Law inside a nonlinear programming problem to search for locally optimal gain combinations and enforce nonlinear constraints on the initial state. Gradient-based trajectory optimization has been shown to benefit greatly when analytical partial derivatives are supplied to the optimizer. This paper presents the Q-Law optimization problem setup and derives the Q-Law thrust vector partial derivatives for the State Transition Matrix required by standard nonlinear optimization software.
We present a new linearization of T-Matrix and Mie computations for light scattering by non-spherical and spherical particles, respectively. In addition to the usual extinction and scattering cross-sections and the scattering matrix outputs, the linearized models will generate analytical derivatives of these optical properties with respect to the real and imaginary parts of the particle refractive index, and (for non-spherical scatterers) with respect to the ''shape'' parameter (the spheroid aspect ratio, cylinder diameter/height ratio, Chebyshev particle deformation factor). These derivatives are based on the essential linearity of Maxwell's theory. Analytical derivatives are also available for polydisperse particle size distribution parameters such as the mode radius. The T-matrix formulation is based on the NASA Goddard Institute for Space Studies FORTRAN 77 code developed in the 1990s. The linearized scattering codes presented here are in FORTRAN 90 and will be made publicly available.