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NASA NTRS1984

Reduction of satellite magnetic anomaly data

Analysis of global magnetic anomaly maps derived from satellite data is facilitated by inversion to the equivalent magnetization in a constant thickness magnetic crust or, equivalently, by reduction to the pole. Previous inversions have proven unstable near the geomagnetic equator. The instability results from magnetic moment distributions which are admissible in the inversion solution but which make only small contribution to the computed values of anomaly field. Their admissibility in the solution could result from noisy or incomplete data or from small poorly resolved anomalies. The resulting magnetic moments are unrealistically large and oscillatory. Application of the method of principal components (e.g. eigenvalue decomposition and selective elimination of less significant eigenvectors) is proposed as a way of overcoming the instability and the method is demonstrated by applying it to the region around the Bangui anomaly in Central Africa.

Slud, E. V.↗

NASA NTRS1984

An objective method for forecasting tropical cyclone motion using Nimbus and NOAA-2 infrared measurements

A statistical method for predicting tropical cyclone positions 12, 24, 48 and 72 hours after initial observation is described. Satellite measurements were at the 11.0 micron channel and at equivalent blackbody temperatures which give representations of the structure of the cyclones, as well as the circulation features surrounding them. In all, 197 individual measurements for blackbody temperature were made. Algorithms prepared from a single satellite image included 14 climatological and persistence type variables, and are based on eigenvector preprocessing and a series of independent screening tests. The independent tests showed that the average error made by the algorithms from the single satellite observation were comparable to the 48 hr Joint Typhoon Warning Center (JTWC) forecast and were about 10 percent better for 72 hr forecasts. Forecasts using only climatological persistence variables were found to be 20 percent worse than JTWC for 24 hr forecasts and 10 percent worse for 48 and 72 hr forecasts. When both satellite and nonsatellite variables were included, it was found that the method was comparable to JTWC for the 24 and 48 hr forecasts and about 25 percent better for the 72 hr forecasts.

Hunter, H. E.↗

NASA NTRS1984

Eigenspace design of an active flutter suppression system

An active control system is designed for the suppression of wing flutter in a flight test vehicle. Eigenvalue placement is used to synthesize a full state controller which satisfies performance specifications on control surface activity and which exhibits excellent gain and phase margins. The use of limited state feedback is examined; however, it is found that a simple frequency response matching technique can be used to design a realizable compensator which reproduces the feedback properties of the full state controller. The performance of the control system using this compensator is evaluated at various flight conditions and found to be satisfactory. In addition eigenvector shaping is used to enhance the gust load alleviation capabilities of the flutter control system.

Liebst, B. S.↗

NASA NTRS1984

Design of an optimal output feedback control system with modal insensitivity

This paper deals with the design of an output feedback controller which results in selected modal insensitivity, and at the same time optimizes a quadratic performance index representative of desired system performance for nominal plant parameter values. The approach taken here is to characterize the class of attainable eigenvectors for a given set of eigenvalues (distinct or non-distinct) which lie in a subspace called the 'Modal Insensitivity Subspace'. A constraint is established on the feedback matrix which results in modal insensitivity. Necessary conditions for optimality subject to the constraint on the feedback matrix are given. This forms the basis for a numerical algorithm to compute the optimal feedback gain which analyzed for convergence. To illustrate the procedure, a design is carried out using the lateral dynamics of an L-1011 aircraft.

Raman, K. V.↗

NASA NTRS1986

Modal Analysis and Gain Estimation

Interactive program yields eigenvectors, eigenvalues, and gains for feedback-control systems. Interactive Modal Analysis and Gain Estimation System (IMAGES) provides eigensystem synthesis capability to control-system engineer. IMAGES modular and flexible. Capable of both modal and spectral synthesis of multi-input control systems. IMAGES user-oriented, interactive program that frees engineer to concentrate on eigensystem synthesis. Engineer provided with scratch-pad capability that speeds control-system design.

Jones, R. L.↗

NASA NTRS1984

Sensitivity analysis of discrete structural systems: A survey

Methods for calculating sensitivity derivatives for discrete structural systems are surveyed, primarily covering literature published during the past two decades. Methods are described for calculating derivatives of static displacements and stresses, eigenvalues and eigenvectors, transient structural response, and derivatives of optimum structural designs with respect to problem parameters. The survey is focused on publications addressed to structural analysis, but also includes a number of methods developed in nonstructural fields such as electronics, controls, and physical chemistry which are directly applicable to structural problems. Most notable among the nonstructural-based methods are the adjoint variable technique from control theory, and the Green's function and FAST methods from physical chemistry.

Adelman, H. M.↗

NASA NTRS1984

On the connection between multigrid and cyclic reduction

A technique is shown whereby it is possible to relate a particular multigrid process to cyclic reduction using purely mathematical arguments. This technique suggest methods for solving Poisson's equation in 1-, 2-, or 3-dimensions with Dirichlet or Neumann boundary conditions. In one dimension the method is exact and, in fact, reduces to cyclic reduction. This provides a valuable reference point for understanding multigrid techniques. The particular multigrid process analyzed is referred to here as Approximate Cyclic Reduction (ACR) and is one of a class known as Multigrid Reduction methods in the literature. It involves one approximation with a known error term. It is possible to relate the error term in this approximation with certain eigenvector components of the error. These are sharply reduced in amplitude by classical relaxation techniques. The approximation can thus be made a very good one.

Merriam, M. L.↗

NASA NTRS1985

Development of control laws for a flight test maneuver autopilot for an F-15 aircraft

An autopilot can be used to provide precise control to meet the demanding requirements of flight research maneuvers with high-performance aircraft. The development of control laws within the context of flight test maneuver requirements is discussed. The control laws are developed using eigensystem assignment and command generator tracking. The eigenvalues and eigenvectors are chosen to provide the necessary handling qualities, while the command generator tracking enables the tracking of a specified state during the maneuver. The effectiveness of the control laws is illustrated by their application to an F-15 aircraft to ensure acceptable aircraft performance during a maneuver.

Alag, G. S.↗

NASA NTRS1983

An eigenvalue/eigenvector assignment algorithm using output feedback

An eigenvalue/eigenvector assignment algorithm using stationary output feedback is presented. The algorithm permits assignment of min (n, m + r - 1) eigenvalues and max (m-1, r-1) eigenvectors, where n, m, r refer to the system state, input and output dimensions, respectively. An example is given to illustrate the design procedures.

Mielke, R. R.↗

NASA NTRS1985

Teleconnections in the Southern Hemisphere

Monthly anomalies from climatology of sea level pressure and 500 mb geopotential height are used to study teleconnection patterns in the Southern Hemisphere. The data and method used are described, and correlations between zonally averaged anomalies are given. The zonal wavenumber three patterns and teleconnection patterns are given, and the results of eigenvector analysis with teleconnection patterns at 500 mb are reported. The zonally averaged anomalies display two patterns of variability; an out-of-phase relation between high and low latitudes and a pattern in which anomalies in midlatitudes vary out of phase with anomalies in low and high latitudes. In winter, midlatitudes are dominated by three centers of positive correlations over the oceans while the negative correlations are found at high and low latitudes. The teleconnection pattern in summer at 500 mb shows an out-of-phase relation between heights over the subtropical continents and heights in the midlatitude oceans.

Mo, K. C.↗

NASA NTRS1985

Application of Ritz vectors for dynamic analysis of large structures

The use of an orthogonal set of specially selected Ritz vectors is shown to be very effective in reducing the cost of dynamic analysis by modal superposition. Several mechanical structures are examined, and the Ritz vector approach is compared to the classical eigenvector approach on the basis of cost, accuracy and elapsed analysis (throughput) time. Mathematical proof of the completeness of orthogonal Ritz vectors is provided for the case of a positive definite mass matrix and a symmetric stiffness matrix.

Arnold, R. R.↗

NASA NTRS1985

Optimization of cascade blade mistuning. II - Global optimum and numerical optimization

The values of the mistuning which yield the most stable eigenvectors are analytically determined, using the simplified equations of motion which were developed in Part I of this work. It is shown that random mistunings, if large enough, may lead to the maximal stability, whereas the alternate mistunings cannot. The problem of obtaining maximum stability for minimal mistuning is formulated, based on numerical optimization techniques. Several local minima are obtained using different starting mistuning vectors. The starting vectors which lead to the global minimum are identified. It is analytically shown that all minima appear in multiplicities which are equal to the number of compressor blades. The effect of mistuning on the flutter speed is studied using both an optimum mistuning vector and an alternate mistuning vector. Effects of mistunings in elastic axis locations are shown to have a negligible effect on the eigenvalues. Finally, it is shown that any general two-dimensional bending-torsion system can be reduced to an equivalent uncoupled torsional system.

Nissim, E.↗

NASA NTRS1985

Development of control laws for a flight test maneuver autopilot for an F-15 aircraft

An autopilot can be used to provide precise control to meet the demanding requirements of flight research maneuvers with high-performance aircraft. This paper presents the development of control laws within the context of flight test maneuver requirements. The control laws are developed using eigensystem assignment and command generator tracking. The eigenvalues and eigenvectors are chosen to provide the necessary handling qualities, while the command generator tracking enables the tracking of a specified state during the maneuver. The effectiveness of the control laws is illustrated by their application to an F-15 aircraft to ensure acceptable aircraft performance during a maneuver.

Alag, G. S.↗

NASA NTRS1986

Solution of the symmetric eigenproblem AX=lambda BX by delayed division

Delayed division is an iterative method for solving the linear eigenvalue problem AX = lambda BX for a limited number of small eigenvalues and their corresponding eigenvectors. The distinctive feature of the method is the reduction of the problem to an approximate triangular form by systematically dropping quadratic terms in the eigenvalue lambda. The report describes the pivoting strategy in the reduction and the method for preserving symmetry in submatrices at each reduction step. Along with the approximate triangular reduction, the report extends some techniques used in the method of inverse subspace iteration. Examples are included for problems of varying complexity.

Thurston, G. A.↗

NASA NTRS1986

Eigensystem synthesis for active flutter suppression on an oblique-wing aircraft

The application of the eigensystem synthesis technique to place the closed-loop eigenvalues and shape the closed-loop eigenvectors has not been practical for active flutter suppression, primarily because of the availability of only one control surface (aileron) for flutter suppression. The oblique-wing aircraft, because of its configuration, provides two independent surfaces (left and right ailerons), making the application of eigensystem synthesis practical. This paper presents the application of eigensystem synthesis using output feedback for the design of an active flutter suppression system for an oblique-wing aircraft. The results obtained are compared with those obtained by linear quadratic Gaussian techniques.

Alag, G. S.↗

NASA NTRS1985

Modeling of elastically mounted vertical rotor

The evaluation of the dynamic behavior of a rotating system is possible by means of modal parameters (Eigenvalues and Eigenvectors). A mixed analytical and experimetal approach is used to identify the modal parameters of a specially designed test rig. The modal identification is done both for nonrotating as well as rotating systems. These modal parameters are used to validate a developed Finite Element Model.

Dana, S. S.↗

NASA NTRS1986

Flight control synthesis for flexible aircraft using Eigenspace assignment

The use of eigenspace assignment techniques to synthesize flight control systems for flexible aircraft is explored. Eigenspace assignment techniques are used to achieve a specified desired eigenspace, chosen to yield desirable system impulse residue magnitudes for selected system responses. Two of these are investigated. The first directly determines constant measurement feedback gains that will yield a close-loop system eigenspace close to a desired eigenspace. The second technique selects quadratic weighting matrices in a linear quadratic control synthesis that will asymptotically yield the close-loop achievable eigenspace. Finally, the possibility of using either of these techniques with state estimation is explored. Application of the methods to synthesize integrated flight-control and structural-mode-control laws for a large flexible aircraft is demonstrated and results discussed. Eigenspace selection criteria based on design goals are discussed, and for the study case it would appear that a desirable eigenspace can be obtained. In addition, the importance of state-space selection is noted along with problems with reduced-order measurement feedback. Since the full-state control laws may be implemented with dynamic compensation (state estimation), the use of reduced-order measurement feedback is less desirable. This is especially true since no change in the transient response from the pilot's input results if state estimation is used appropriately. The potential is also noted for high actuator bandwidth requirements if the linear quadratic synthesis approach is utilized. Even with the actuator pole location selected, a problem with unmodeled modes is noted due to high bandwidth. Some suggestions for future research include investigating how to choose an eigenspace that will achieve certain desired dynamics and stability robustness, determining how the choice of measurements effects synthesis results, and exploring how the phase relationships between desired eigenvector elements effects the synthesis results.

Davidson, J. B.↗

NASA NTRS1985

Errors in reduction methods

A mathematical basis is given for comparing the relative merits of various techniques used to reduce the order of large linear and nonlinear dynamics problems during their numerical integration. In such techniques as Guyan-Irons, path derivatives, selected eigenvectors, Ritz vectors, etc., the nth order initial value problem of /y(dot) = f(y) for t greater than 0, y(0) given/ is typically reduced to the mth order (m is much less than n) problem of /z(dot) = g(z) for t greater than 0, z(0) given/ by the transformation y = Pz where P changes from technique to technique. This paper gives an explicit approximate expression for the reduction error e-i in terms of P and the Jacobian of f. It is shown that: (a) reduction techniques are more accurate when the time rate of change of the response y is relatively small; (b) the change in response between two successive stations contributes to the errors at future stations after the change in response is transformed by a filtering matrix H, defined in terms of P; (c) the error committed at a station propagates to future stations by a mixing and scaling matrix G, defined in terms of P, Jacobian and of f, and time increment h. The paper discusses the conditions under which the reduction errors may be minimized and gives guidelines for selecting the reduction basis vector, i.e., the columns of P.

Utku, S.↗