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NASA NTRS1980

Model error estimation for large space systems

In-flight estimation of large structure model errors may have to be carried out in order to detect inevitable deficiencies in large structure controller/estimator models. These error estimates can most efficiently be obtained by the minimization of a quadratic functional of the model errors and on the subsequent analysis of the resulting optimal model error estimates. An integral operator approach to estimation leads to a geometrical interpretation of the model error estimation process. One of the significant insights gained with this interpretation is that the actual but unknown model errors can be decomposed as the sum of two distinct components that are orthogonal in some sense. One of these components is a so-called minimal error vector that retains many of the significant dynamics of the actual errors. The basic ideas in the model error estimation approach are first set forth with a two-dimensional analogy that has most of the essential features of the general estimation problem. The generalized results are then established and their application to a reference large structure model illustrated.

Rodriguez, G.↗

NASA NTRS1981

Landsat-based estimation of California's irrigated land

The procedure developed uses two-phase sampling, stratification and multidate Landsat imagery to produce the estimate. Maximizing the advantages of both spectral data and field patterns available from Landsat, three dates of Landsat imagery are interpreted to provide county-wide estimates of the proportion of land irrigated. Ground data, collected on a subset of the area interpreted on Landsat, are used to calibrate the satellite estimate. The satellite and ground measurements are reduced to proportion data and linked using a regression estimator to produce the estimate of irrigated land. It is estimated that 3.99 million hectares were irrigated with a relative standard error of + or - 1.18% at the 99% confidence level. Acreage tabulations provided by the Department of Water Resources show that the Landsat-based estimate differed by less than 0.4% at the state level.

Wall, S. L.↗

NASA NTRS1982

The use of baseline measurements and geophysical models for the estimation of crustal deformations and the terrestrial reference system

Four possible estimators are investigated for the monitoring of crustal deformations from a combination of repeated baseline length measurements and adopted geophysical models, particularly an absolute motion plate model. The first estimator is an extension of the familiar free adjustment. The next two are Bayesian type estimators, one weak and one strong. Finally, a weighted constraint estimator is presented. The properties of these four estimators are outlined and their physical interpretations discussed. A series of simulations are performed to test the four estimators and to determine whether or not to incorporate a plate model for the monitoring of deformations. The application of these estimations to the maintenance of a new conventional terrestrial reference system is discussed.

Bock, Y.↗

NASA NTRS1983

A function space approach to state and model error estimation for elliptic systems

An approach is advanced for the concurrent estimation of the state and of the model errors of a system described by elliptic equations. The estimates are obtained by a deterministic least-squares approach that seeks to minimize a quadratic functional of the model errors, or equivalently, to find the vector of smallest norm subject to linear constraints in a suitably defined function space. The minimum norm solution can be obtained by solving either a Fredholm integral equation of the second kind for the case with continuously distributed data or a related matrix equation for the problem with discretely located measurements. Solution of either one of these equations is obtained in a batch-processing mode in which all of the data is processed simultaneously or, in certain restricted geometries, in a spatially scanning mode in which the data is processed recursively. After the methods for computation of the optimal estimates are developed, an analysis of the second-order statistics of the estimates and of the corresponding estimation error is conducted. Based on this analysis, explicit expressions for the mean-square estimation error associated with both the state and model error estimates are then developed.

Rodriguez, G.↗

NASA NTRS1984

Sampling system for wheat (Triticum aestivum L) area estimation using digital LANDSAT MSS data and aerial photographs

A procedure to estimate wheat (Triticum aestivum L) area using sampling technique based on aerial photographs and digital LANDSAT MSS data is developed. Aerial photographs covering 720 square km are visually analyzed. To estimate wheat area, a regression approach is applied using different sample sizes and various sampling units. As the size of sampling unit decreased, the percentage of sampled area required to obtain similar estimation performance also decreased. The lowest percentage of the area sampled for wheat estimation with relatively high precision and accuracy through regression estimation is 13.90% using 10 square km as the sampling unit. Wheat area estimation using only aerial photographs is less precise and accurate than those obtained by regression estimation.

Parada, N. D. J.↗

NASA NTRS1983

Estimating crustal deformations from a combination of baseline measurements and geophysical models

Three possible estimation algorithms are presented for the monitoring of crustal deformations from a combination of repeated baseline measurements and prior deformation information. Attention is given to best linear minimum bias estimation, best linear estimation, and Bayesian estimation. The application of the deformation estimators to the maintenance of a conventional terrestrial reference system is considered. It is shown that if no prior deformation model is available, the free adjustment is the preferred algorithm. In the presence of prior information, however, it is better to use a somewhat incorrect model than to ignore it altogether. The best estimation model of the ones tested is the best linear estimation.

Bock, Y.↗

NASA NTRS1985

Testing an Energy Balance Model for Estimating Actual Evapotranspiration Using Remotely Sensed Data

An energy-balance model is used to estimate daily evapotranspiration for 3 days for a barley field and a wheat field near Hannover, Federal Republic of Germany. The model was calibrated using once-daily estimates of surface temperatures, which may be remotely sensed. The evaporation estimates were within the 95% error bounds of independent eddy correlation estimates for the daytime periods for all three days for both sites, but the energy-balance estimates are generally higher; it is unclear which estimate is biassed. Soil moisture in the top 2 cm of soil, which may be remotely sensed, may be used to improve these evaporation estimates under partial ground cover. Sensitivity studies indicate the amount of ground data required is not excessive.

Gurney, R. J.↗

NASA NTRS1985

Kalman-like estimation for static distributed systems Antenna shape from radiation measurements

This paper advances an approach to the determination of shape of static distributed systems. It also illustrates the application of the approach to the problems of surface diagnosis of large parabolic reflectors. The estimation methods developed combine in an optimal sense the information from an elliptic model of the structure and from measurements of the structural deflection and of the far-field pattern changes due to the structural deformation. The estimators have a predictor-corrector structure, quite similar to that of a Kalman filter. The system model is first used to obtain a predicted estimate. A correction term is then added to the prediction to obtain the final state estimate. The relative weighting between prediction and correction terms is determined by an estimator gain. As in a Kalman filter, the estimator gain can be expressed in terms of the state estimation error covariance.

Rodriguez, G.↗

NASA NTRS1986

Determination of lift and drag characteristics of Space Shuttle Orbiter using maximum likelihood estimation technique

This paper presents the technique and results of maximum likelihood estimation used to determine lift and drag characteristics of the Space Shuttle Orbiter. Maximum likelihood estimation uses measurable parameters to estimate nonmeasurable parameters. The nonmeasurable parameters for this case are elements of a nonlinear, dynamic model of the orbiter. The estimated parameters are used to evaluate a cost function that computes the differences between the measured and estimated longitudinal parameters. The case presented is a dynamic analysis. This places less restriction on pitching motion and can provide additional information about the orbiter such as lift and drag characteristics at conditions other than trim, instrument biases, and pitching moment characteristics. In addition, an output of the analysis is an estimate of the values for the individual components of lift and drag that contribute to the total lift and drag. The results show that maximum likelihood estimation is a useful tool for analysis of Space Shuttle Orbiter performance and is also applicable to parameter analysis of other types of aircraft.

Trujillo, B. M.↗

NASA NTRS1988

An investigation of new methods for estimating parameter sensitivities

Parameter sensitivity is defined as the estimation of changes in the modeling functions and the design variables due to small changes in the fixed parameters of the formulation. There are currently several methods for estimating parameter sensitivities requiring either difficult to obtain second order information, or do not return reliable estimates for the derivatives. Additionally, all the methods assume that the set of active constraints does not change in a neighborhood of the estimation point. If the active set does in fact change, than any extrapolations based on these derivatives may be in error. The objective here is to investigate more efficient new methods for estimating parameter sensitivities when the active set changes. The new method is based on the recursive quadratic programming (RQP) method and in conjunction a differencing formula to produce estimates of the sensitivities. This is compared to existing methods and is shown to be very competitive in terms of the number of function evaluations required. In terms of accuracy, the method is shown to be equivalent to a modified version of the Kuhn-Tucker method, where the Hessian of the Lagrangian is estimated using the BFS method employed by the RPQ algorithm. Inital testing on a test set with known sensitivities demonstrates that the method can accurately calculate the parameter sensitivity. To handle changes in the active set, a deflection algorithm is proposed for those cases where the new set of active constraints remains linearly independent. For those cases where dependencies occur, a directional derivative is proposed. A few simple examples are included for the algorithm, but extensive testing has not yet been performed.

Beltracchi, Todd J.↗

NASA NTRS1989

Space shuttle propulsion estimation development verification, volume 1

The results of the Propulsion Estimation Development Verification are summarized. A computer program developed under a previous contract (NAS8-35324) was modified to include improved models for the Solid Rocket Booster (SRB) internal ballistics, the Space Shuttle Main Engine (SSME) power coefficient model, the vehicle dynamics using quaternions, and an improved Kalman filter algorithm based on the U-D factorized algorithm. As additional output, the estimated propulsion performances, for each device are computed with the associated 1-sigma bounds. The outputs of the estimation program are provided in graphical plots. An additional effort was expended to examine the use of the estimation approach to evaluate single engine test data. In addition to the propulsion estimation program PFILTER, a program was developed to produce a best estimate of trajectory (BET). The program LFILTER, also uses the U-D factorized algorithm form of the Kalman filter as in the propulsion estimation program PFILTER. The necessary definitions and equations explaining the Kalman filtering approach for the PFILTER program, the models used for this application for dynamics and measurements, program description, and program operation are presented.

Rogers, Robert M.↗

NASA NTRS1988

Altimetry for non-Gaussian oceans - Height biases and estimation of parameters

The effects of a non-Gaussian ocean on satellite altimetry parameter estimation are discussed. The first part of this paper shows how non-Gaussian ocean parameters affect height estimation for satellites of the Seasat/Geosat/TOPEX type. In the second part, the estimation of the altimeter tracker bias and the non-Gaussian ocean parameters from the altimeter return signal is studied. A new convolution model that facilitates the deconvolution of the ocean surface specular point probability density function is introduced. Next, it is argued that it is not feasible in practice to estimate the electromagnetic bias from noisy altimeter returns. It is then shown that the least squares estimation of the surface parameters in the log-frequency domain has several advantages overestimating parameters in the time domain. The maximum likelihood estimation equations for the estimation of the waveform parameters are derived, and their solution is discussed.

Rodriguez, Ernesto↗

NASA NTRS1990

Kalman filter based range estimation for autonomous navigation using imaging sensors

Rotorcraft operating in high-threat environments fly close to the surface of the earth to utilize surrounding terrain, vegetation, or man-made objects to minimize the risk of being detected by the enemy. Two basic requirements for obstacle avoidance are detection and range estimation of the object from the current rotorcraft position. There are many approaches to the estimation of range using a sequence of images. The approach used in this analysis differes from previous methods in two significant ways: an attempt is not made to estimate the rotorcraft's motion from the images; and the interest lies in recursive algorithms. The rotorcraft parameters are assumed to be computed using an onboard inertial navigation system. Given a sequence of images, using image-object differential equations, a Kalman filter (Sridhar and Phatak, 1988) can be used to estimate both the relative coordinates and the earth coordinates of the objects on the ground. The Kalman filter can also be used in a predictive mode to track features in the images, leading to a significant reduction of search effort in the feature extraction step of the algorithm. The purpose is to summarize early results obtained in extending the Kalman filter for use with actual image sequences. The experience gained from the application of this algorithm to real images is very valuable and is a necessary step before proceeding to the estimation of range during low-altitude curvilinear flight. A simple recursive method is presented to estimate range to objects using a sequence of images. The method produces good range estimates using real images in a laboratory set up and needs to be evaluated further using several different image sequences to test its robustness. The feature generation part of the algorithm requires further refinement on the strategies to limit the number of features (Sridhar and Phatak, 1989). The extension of the work reported here to curvilinear flight may require the use of the extended Kalman filter.

Sridhar, Banavar↗

NASA NTRS1990

Aerodynamic parameters of High-Angle-of attack Research Vehicle (HARV) estimated from flight data

Aerodynamic parameters of the High-Angle-of-Attack Research Aircraft (HARV) were estimated from flight data at different values of the angle of attack between 10 degrees and 50 degrees. The main part of the data was obtained from small amplitude longitudinal and lateral maneuvers. A small number of large amplitude maneuvers was also used in the estimation. The measured data were first checked for their compatibility. It was found that the accuracy of air data was degraded by unexplained bias errors. Then, the data were analyzed by a stepwise regression method for obtaining a structure of aerodynamic model equations and least squares parameter estimates. Because of high data collinearity in several maneuvers, some of the longitudinal and all lateral maneuvers were reanalyzed by using two biased estimation techniques, the principal components regression and mixed estimation. The estimated parameters in the form of stability and control derivatives, and aerodynamic coefficients were plotted against the angle of attack and compared with the wind tunnel measurements. The influential parameters are, in general, estimated with acceptable accuracy and most of them are in agreement with wind tunnel results. The simulated responses of the aircraft showed good prediction capabilities of the resulting model.

Klein, Vladislav↗

NASA NTRS1990

Robust eigensystem assignment for state estimators using second-order models

A novel design of a state estimator is presented using second-order dynamic equations of mechanical systems. The eigenvalues and eigenvectors of the state estimator are assigned by solving the second-order eigenvalue problem of the structural system. Three design method for the state estimator are given in this paper. The first design method uses collocated sensors to measure the desired signals and their derivatives. The second design method uses prefilters to shift signal phases to obtain estimates of the signal derivatives. These two methods are used to build a second-order state estimator model. The third design method is the conventional one which converts a typical second-order dynamic model to a first-order model, and then builds a state estimator based on the first-order model. It is shown that all the three designs for state estimation are similar. A numerical example representing a large space structure is given for illustration of the design methods presented in this paper.

Juang, Jer-Nan↗

NASA NTRS1990

Systems engineering and integration: Cost estimation and benefits analysis

Space Transportation Avionics hardware and software cost has traditionally been estimated in Phase A and B using cost techniques which predict cost as a function of various cost predictive variables such as weight, lines of code, functions to be performed, quantities of test hardware, quantities of flight hardware, design and development heritage, complexity, etc. The output of such analyses has been life cycle costs, economic benefits and related data. The major objectives of Cost Estimation and Benefits analysis are twofold: (1) to play a role in the evaluation of potential new space transportation avionics technologies, and (2) to benefit from emerging technological innovations. Both aspects of cost estimation and technology are discussed here. The role of cost analysis in the evaluation of potential technologies should be one of offering additional quantitative and qualitative information to aid decision-making. The cost analyses process needs to be fully integrated into the design process in such a way that cost trades, optimizations and sensitivities are understood. Current hardware cost models tend to primarily use weights, functional specifications, quantities, design heritage and complexity as metrics to predict cost. Software models mostly use functionality, volume of code, heritage and complexity as cost descriptive variables. Basic research needs to be initiated to develop metrics more responsive to the trades which are required for future launch vehicle avionics systems. These would include cost estimating capabilities that are sensitive to technological innovations such as improved materials and fabrication processes, computer aided design and manufacturing, self checkout and many others. In addition to basic cost estimating improvements, the process must be sensitive to the fact that no cost estimate can be quoted without also quoting a confidence associated with the estimate. In order to achieve this, better cost risk evaluation techniques are needed as well as improved usage of risk data by decision-makers. More and better ways to display and communicate cost and cost risk to management are required.

Dean, ED↗

NASA NTRS1989

Use of robust estimators in parametric classifiers

The parametric approach to density estimation and classifier design is a well studied subject. The parametric approach is desirable because basically it reduces the problem of classifier design to that of estimating a few parameters for each of the pattern classes. The class parameters are usually estimated using maximum-likelihood (ML) estimators. ML estimators are, however, very sensitive to the presence of outliers. Several robust estimators of mean and covariance matrix and their effect on the probability of error in classification are examined. Comments are made about alpha-ranked (alpha-trimmed) estimators.

Safavian, S. Rasoul↗

NASA NTRS1992

Subsonic flight test evaluation of a propulsion system parameter estimation process for the F100 engine

Integrated engine-airframe optimal control technology may significantly improve aircraft performance. This technology requires a reliable and accurate parameter estimator to predict unmeasured variables. To develop this technology base, NASA Dryden Flight Research Facility (Edwards, CA), McDonnell Aircraft Company (St. Louis, MO), and Pratt & Whitney (West Palm Beach, FL) have developed and flight-tested an adaptive performance seeking control system which optimizes the quasi-steady-state performance of the F-15 propulsion system. This paper presents flight and ground test evaluations of the propulsion system parameter estimation process used by the performance seeking control system. The estimator consists of a compact propulsion system model and an extended Kalman filter. The extended Laman filter estimates five engine component deviation parameters from measured inputs. The compact model uses measurements and Kalman-filter estimates as inputs to predict unmeasured propulsion parameters such as net propulsive force and fan stall margin. The ability to track trends and estimate absolute values of propulsion system parameters was demonstrated. For example, thrust stand results show a good correlation, especially in trends, between the performance seeking control estimated and measured thrust.

Orme, John S.↗