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At least 613 records · Page 34

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NASA NTRS1988

A parallel householder tridiagonalization stratagem using scattered row decomposition

Householder's method for tridiagonalizing a real symmetric matrix, a major step in evaluating eigenvalues of the matrix, is modified into a parallel algorithm for a concurrent machine of message passing type. Each processor of the concurrent machine has its own CPU, communications control and local memory. Messages are passed through connections between processors. Although the basic algorithm is inherently serial, the computations can be spread over all processors by scattering different rows of the matrix into processors, hence the term 'Scattered Row Decomposition'. The steps in the serial and the parallel algorithms are identified. Expressions for efficiency and speedup are given in terms of problem and machine parameters. For a concurrent machine of ring type interconnection, a selected representative problem of large order exhibits efficiency approaching 66 per cent.

Chang, H. Y.↗

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 NTRS1987

Precise computer controlled positioning of robot end effectors using force sensors

A major problem in space applications of robotics and docking of spacecraft is the development of technology for automated precise positioning of mating components with smooth motion and soft contact. To achieve the above objective, a design method was developed for optimally placing the closed-loop poles of a discretized robotic control system at exact prescribed locations inside the unit circle of the complex z-plane. The design method combines the merits of the pole placement and the linear quadratic design approaches. The proposed design procedure is based on the assignment of one real eigenvalue or two complex conjugate (or real) eigenvalues at each design step. The method involves solutions of simple algebraic equations and this is considered to be efficient for on-line or off-line computations. Also, two methods for the linearization of the nonlinear model of a robotic manipulator were presented. Since automatic control of multi-degree freedom robotic manipulators involves high nonlinear equations of systems, a pilot project was proposed involving the control of a one-dimensional system. This simple system can be readily implemented for testing the concepts and algorithms.

Shieh, Leang S.↗

NASA NTRS1987

A Lanczos eigenvalue method on a parallel computer

Eigenvalue analyses of complex structures is a computationally intensive task which can benefit significantly from new and impending parallel computers. This study reports on a parallel computer implementation of the Lanczos method for free vibration analysis. The approach used here subdivides the major Lanczos calculation tasks into subtasks and introduces parallelism down to the subtask levels such as matrix decomposition and forward/backward substitution. The method was implemented on a commercial parallel computer and results were obtained for a long flexible space structure. While parallel computing efficiency for the Lanczos method was good for a moderate number of processors for the test problem, the greatest reduction in time was realized for the decomposition of the stiffness matrix, a calculation which took 70 percent of the time in the sequential program and which took 25 percent of the time on eight processors. For a sample calculation of the twenty lowest frequencies of a 486 degree of freedom problem, the total sequential computing time was reduced by almost a factor of ten using 16 processors.

Bostic, Susan W.↗

NASA NTRS1990

Robust control of systems with real parameter uncertainty and unmodelled dynamics

Two significant contributions have been made during this research period in the research 'Robust Control of Systems with Real Parameter Uncertainty and Unmodelled Dynamics' under NASA Research Grant NAG-1-1102. They are: (1) a fast algorithm for computing the optimal H(sub infinity) norm for the four-block, the two block, or the one-block optimal H(sub infinity) optimization problem; and (2) a construction of an optimal H infinity controller without numerical difficulty. In using GD (Glover and Doyle) or DGKF (Doyle, Glover, Khargonekar, and Francis) approach to solve the standard H infinity norm which required bisection search. In this research period, we developed a very fast iterative algorithm for this computation. Our algorithm was developed based on hyperbolic interpolations which is much faster than any existing algorithm. The lower bound of the parameter, gamma, in the H infinity Riccati equation for solution existence is shown to be the square root of the supremum over all frequencies of the maximum eigenvalue of a given transfer matrix which can be computed easily. The lower band of gamma such that the H infinity Riccati equation has positive semidefinite solution can be also obtained by hyperbolic interpolation search. Another significant result in this research period is the elimination of the numerical difficulties arising in the construction of an optimal H infinity controller by directly applying the Glover and Doyle's state-space formulas. With the fast iterative algorithm for the computation of the optimal H infinity norm and the reliable construction of an optimal H infinity controller, we are ready to apply these tools in the design of robust controllers for the systems with unmodelled uncertainties. These tools will be also very useful when we consider systems with structured uncertainties.

Chang, Bor-Chin↗

NASA NTRS1993

Parallel-vector unsymmetric Eigen-Solver on high performance computers

The popular QR algorithm for solving all eigenvalues of an unsymmetric matrix is reviewed. Among the basic components in the QR algorithm, it was concluded from this study, that the reduction of an unsymmetric matrix to a Hessenberg form (before applying the QR algorithm itself) can be done effectively by exploiting the vector speed and multiple processors offered by modern high-performance computers. Numerical examples of several test cases have indicated that the proposed parallel-vector algorithm for converting a given unsymmetric matrix to a Hessenberg form offers computational advantages over the existing algorithm. The time saving obtained by the proposed methods is increased as the problem size increased.

Nguyen, Duc T.↗

NASA NTRS1987

Piecewise linear approximation for hereditary control problems

Finite dimensional approximations are presented for linear retarded functional differential equations by use of discontinuous piecewise linear functions. The approximation scheme is applied to optimal control problems when a quadratic cost integral has to be minimized subject to the controlled retarded system. It is shown that the approximate optimal feedback operators converge to the true ones both in case the cost integral ranges over a finite time interval as well as in the case it ranges over an infinite time interval. The arguments in the latter case rely on the fact that the piecewise linear approximations to stable systems are stable in a uniform sense. This feature is established using a vector-component stability criterion in the state space R(n) x L(2) and the favorable eigenvalue behavior of the piecewise linear approximations.

Propst, Georg↗

NASA NTRS1978

Inversion of data from diffraction-limited multiwavelength remote sensors. I - Linear case

The remote sensing inverse problem is considered in which the sensor does not necessarily view the same area on the earth at each wavelength, and spatial correlations of the geophysical parameters may be present. Under the conditions of linearity and stationary statistics, the minimum mean-square error solution to the problem of inverting such data is a spatial filter of the Wiener-Kolmogorov class. The resulting remote-sensing system can be characterized by an impulse response matrix in ordinary space or by a transfer matrix in frequency space. A signal-to-noise matrix for the geophysical parameters to be sensed is also defined; this matrix depends on the postulated a priori statistics and on the characteristics of the remote-sensing instrument. The system transfer matrix and the signal-to-noise matrix are simultaneously diagonalizable. The optimum transfer matrix filters out of the estimate vector those eigenparameters for which eigenvalues of the signal-to-noise matrix are less than unity.

Rosenkranz, P. W.↗

NASA NTRS1985

Adaptive Inverse Control for Rotorcraft Vibration Reduction

This thesis extends the Least Mean Square (LMS) algorithm to solve the mult!ple-input, multiple-output problem of alleviating N/Rev (revolutions per minute by number of blades) helicopter fuselage vibration by means of adaptive inverse control. A frequency domain locally linear model is used to represent the transfer matrix relating the higher harmonic pitch control inputs to the harmonic vibration outputs to be controlled. By using the inverse matrix as the controller gain matrix, an adaptive inverse regulator is formed to alleviate the N/Rev vibration. The stability and rate of convergence properties of the extended LMS algorithm are discussed. It is shown that the stability ranges for the elements of the stability gain matrix are directly related to the eigenvalues of the vibration signal information matrix for the learning phase, but not for the control phase. The overall conclusion is that the LMS adaptive inverse control method can form a robust vibration control system, but will require some tuning of the input sensor gains, the stability gain matrix, and the amount of control relaxation to be used. The learning curve of the controller during the learning phase is shown to be quantitatively close to that predicted by averaging the learning curves of the normal modes. For higher order transfer matrices, a rough estimate of the inverse is needed to start the algorithm efficiently. The simulation results indicate that the factor which most influences LMS adaptive inverse control is the product of the control relaxation and the the stability gain matrix. A small stability gain matrix makes the controller less sensitive to relaxation selection, and permits faster and more stable vibration reduction, than by choosing the stability gain matrix large and the control relaxation term small. It is shown that the best selections of the stability gain matrix elements and the amount of control relaxation is basically a compromise between slow, stable convergence and fast convergence with increased possibility of unstable identification. In the simulation studies, the LMS adaptive inverse control algorithm is shown to be capable of adapting the inverse (controller) matrix to track changes in the flight conditions. The algorithm converges quickly for moderate disturbances, while taking longer for larger disturbances. Perfect knowledge of the inverse matrix is not required for good control of the N/Rev vibration. However it is shown that measurement noise will prevent the LMS adaptive inverse control technique from controlling the vibration, unless the signal averaging method presented is incorporated into the algorithm.

Jacklin, Stephen A.↗

NASA NTRS1992

Robust Assignment Of Eigensystems For Flexible Structures

Improved method for placement of eigenvalues and eigenvectors of closed-loop control system by use of either state or output feedback. Applied to reduced-order finite-element mathematical model of NASA's MAST truss beam structure. Model represents deployer/retractor assembly, inertial properties of Space Shuttle, and rigid platforms for allocation of sensors and actuators. Algorithm formulated in real arithmetic for efficient implementation. Choice of open-loop eigenvector matrix and its closest unitary matrix believed suitable for generating well-conditioned eigensystem with small control gains. Implication of this approach is that element of iterative search for "optimal" unitary matrix appears unnecessary in practice for many test problems.

Juang, Jer-Nan↗

NASA NTRS1990

Implicit application of polynomial filters in a k-step Arnoldi method

The Arnoldi process is a well known technique for approximating a few eigenvalues and corresponding eigenvectors of a general square matrix. Numerical difficulties such as loss of orthogonality and assessment of the numerical quality of the approximations as well as a potential for unbounded growth in storage have limited the applicability of the method. These issues are addressed by fixing the number of steps in the Arnoldi process at a prescribed value k and then treating the residual vector as a function of the initial Arnoldi vector. This starting vector is then updated through an iterative scheme that is designed to force convergence of the residual to zero. The iterative scheme is shown to be a truncation of the standard implicitly shifted QR-iteration for dense problems and it avoids the need to explicitly restart the Arnoldi sequence. The main emphasis of this paper is on the derivation and analysis of this scheme. However, there are obvious ways to exploit parallelism through the matrix-vector operations that comprise the majority of the work in the algorithm. Preliminary computational results are given for a few problems on some parallel and vector computers.

Sorensen, D. C.↗

NASA NTRS2003

Probabilistic Aeroelastic Analysis Developed for Turbomachinery Components

Aeroelastic analyses for advanced turbomachines are being developed for use at the NASA Glenn Research Center and industry. However, these analyses at present are used for turbomachinery design with uncertainties accounted for by using safety factors. This approach may lead to overly conservative designs, thereby reducing the potential of designing higher efficiency engines. An integration of the deterministic aeroelastic analysis methods with probabilistic analysis methods offers the potential to design efficient engines with fewer aeroelastic problems and to make a quantum leap toward designing safe reliable engines. In this research, probabilistic analysis is integrated with aeroelastic analysis: (1) to determine the parameters that most affect the aeroelastic characteristics (forced response and stability) of a turbomachine component such as a fan, compressor, or turbine and (2) to give the acceptable standard deviation on the design parameters for an aeroelastically stable system. The approach taken is to combine the aeroelastic analysis of the MISER (MIStuned Engine Response) code with the FPI (fast probability integration) code. The role of MISER is to provide the functional relationships that tie the structural and aerodynamic parameters (the primitive variables) to the forced response amplitudes and stability eigenvalues (the response properties). The role of FPI is to perform probabilistic analyses by utilizing the response properties generated by MISER. The results are a probability density function for the response properties. The probabilistic sensitivities of the response variables to uncertainty in primitive variables are obtained as a byproduct of the FPI technique. The combined analysis of aeroelastic and probabilistic analysis is applied to a 12-bladed cascade vibrating in bending and torsion. Out of the total 11 design parameters, 6 are considered as having probabilistic variation. The six parameters are space-to-chord ratio (SBYC), stagger angle (GAMA), elastic axis (ELAXS), Mach number (MACH), mass ratio (MASSR), and frequency ratio (WHWB). The cascade is considered to be in subsonic flow with Mach 0.7. The results of the probabilistic aeroelastic analysis are the probability density function of predicted aerodynamic damping and frequency for flutter and the response amplitudes for forced response.

Reddy, T. S. R.↗

NASA NTRS1986

Landsat D Thematic Mapper image dimensionality reduction and geometric correction accuracy

To characterize and quantify the performance of the Landsat thematic mapper (TM), techniques for dimensionality reduction by linear transformation have been studied and evaluated and the accuracy of the correction of geometric errors in TM images analyzed. Theoretical evaluations and comparisons for existing methods for the design of linear transformation for dimensionality reduction are presented. These methods include the discrete Karhunen Loeve (KL) expansion, Multiple Discriminant Analysis (MDA), Thematic Mapper (TM)-Tasseled Cap Linear Transformation and Singular Value Decomposition (SVD). A unified approach to these design problems is presented in which each method involves optimizing an objective function with respect to the linear transformation matrix. From these studies, four modified methods are proposed. They are referred to as the Space Variant Linear Transformation, the KL Transform-MDA hybrid method, and the First and Second Version of the Weighted MDA method. The modifications involve the assignment of weights to classes to achieve improvements in the class conditional probability of error for classes with high weights. Experimental evaluations of the existing and proposed methods have been performed using the six reflective bands of the TM data. It is shown that in terms of probability of classification error and the percentage of the cumulative eigenvalues, the six reflective bands of the TM data require only a three dimensional feature space. It is shown experimentally as well that for the proposed methods, the classes with high weights have improvements in class conditional probability of error estimates as expected.

Ford, G. E.↗

NASA NTRSDate not supplied

TPSAS-NF1676L-33433-DND

Reducing aircraft noise is a major objective in the field of computational aeroacoustics. When designing next generation quiet aircraft, it is important to be able to accurately and efficiently predict the acoustic scattering by an aircraft body from a given noise source. Acoustic liners are an effective tool for aircraft noise reduction, and are characterized by a complex valued frequency-dependent impedance. Converted into the time-domain using Fourier transforms, an impedance boundary condition can be used to simulate the acoustic scattering of geometric bodies treated with acoustic liners. This work considers using either an impedance boundary condition or admittance boundary condition to allow acoustic scattering problems to be modeled with geometries consisting of both un-lined and lined surfaces; admittance is defined to be the inverse of impedance. Three different acoustic liner models will be discussed: the Extended Helmholtz Resonator Model, the Three-Parameter Impedance Model, and a Broadband Model. In the Extended Helmholtz Resonator Model and Three-Parameter Impedance Model, the liner impedance is specified at a single frequency; the Broadband Model allows for the investigation of multiple frequencies simultaneously. The impedance and admittance boundary conditions for each model will be derived and coupled with a time-domain boundary integral equation. The scattering solution will be obtained iteratively using spatial and temporal basis functions. Stability will be demonstrated through eigenvalue analysis. Moreover, fast algorithms and high performance computing are considered to both assess and reduce the computational cost of the simulation.

Michelle E Rodio↗

NASA NTRSDate not supplied

Exploring Quantum State Preparation Using Tensor Networks and Sparse Wavefunction Simulations

The variational quantum eigenvalue solver is a powerful hybrid quantum-classical approach that has been suggested as a candidate method to run on near-term quantum hardware for computing ground state electronic energies of molecular systems. However, even for small molecules, the number of variational parameters and qubits required to minimize the electronic energy is beyond the reach of current quantum computers except for small basis sets. We explore a new paradigm for state preparation where we test how much of the optimization can be approximately prepared with classical computers to reduce the number of optimization steps performed using a quantum device. By adapting a recent algorithm for the factorized form of the UCC ansatz, we can study molecular electronic structure problems with up to 64 qubits. In addition, we also test a related approach of using tensor networks to optimize quantum circuits in order to benchmark various lattice models. We present results using these approaches and discuss strategies for incorporating these ideas into variational algorithms involving near-term quantum computers. Our results help demonstrate the strength of the UCC ansatz and address pressing questions about optimal initial parameterizations and circuit construction.

quantum computing↗

NASA NTRS1971

Statistical analysis of the modal properties of large structural systems.

A theory is developed to predict eigenvalue and eigenvector uncertainty in large dynamic models. The uncertainty is based on physical property uncertainty and should not be confused with numerical roundoff, although the method can be extended to include the latter. The theory, when implemented on a computer, is used to analyze the uncertainties in frequencies and mode shapes based on uncertainties in mass, stiffness, modulus of elasticity, etc. The method incorporates a linear statistical model which is quite adequate for handling property uncertainties of 10% or more. The model is not limited to small systems but uses certain statistical assumptions as well as selective matrix manipulations to keep the size of all matrix operations to within the number of degrees of freedom of the system. Examples are given for two longitudinal vibration problems, and the results are supported by a Monte Carlo analysis.

Collins, J. D.↗

NASA NTRS1996

On the Retrieval of Lightning Radio Sources from Time-of-Arrival Data

We examine the problem of retrieving three-dimensional lightning locations from radio frequency Time-Of-Arrival (TOA) measurements. Arbitrary antenna locations are considered. By judiciously differencing measurements that are related to the location of the antennas and their excitation times, the problem is converted from the initial spherical nonlinear form to a system of linear equations. In the linear formalism, the source location and time-of-occurrence is viewed geometrically as an intersection of hyperplanes in the four-dimensional Minkowski space (x,y,z,t). The linear equations are solved to obtain explicit analytic expressions for the location and time variables. Retrieval errors are not interpreted with conventional Geometrical Dilution of Precision (GDOP) arguments as discussed by Holmes and Reedy (1951), but with more recent inversion analyses considered by Twomey (1977). Measurement errors are propagated analytically so that the specific effect of these errors on the solution is clarified. The sensitivity of the solution on the number of antennas used, antenna network geometry, source position, and measurement differencing schemes are discussed in terms of the eigenvalues of the linear system.

Koshak, William J.↗

NASA NTRS1989

The dynamics and control of large flexible space structures - 12, supplement 11

The rapid 2-D slewing and vibrational control of the unsymmetrical flexible SCOLE (Spacecraft Control Laboratory Experiment) with multi-bounded controls is considered. Pontryagin's Maximum Principle is applied to the nonlinear equations of the system to derive the necessary conditions for the optimal control. The resulting two point boundary value problem is then solved by using the quasilinearization technique, and the near minimum time is obtained by sequentially shortening the slewing time until the controls are near the bang-bang type. The tradeoff between the minimum time and the minimum flexible amplitude requirements is discussed. The numerical results show that the responses of the nonlinear system are significantly different from those of the linearized system for rapid slewing. The SCOLE station-keeping closed loop dynamics are re-examined by employing a slightly different method for developing the equations of motion in which higher order terms in the expressions for the mast modal shape functions are now included. A preliminary study on the effect of actuator mass on the closed loop dynamics of large space systems is conducted. A numerical example based on a coupled two-mass two-spring system illustrates the effect of changes caused in the mass and stiffness matrices on the closed loop system eigenvalues. In certain cases the need for redesigning control laws previously synthesized, but not accounting for actuator masses, is indicated.

Bainum, Peter M.↗