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Results for “Eigenvalue algorithm”

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 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
NASA NTRS1985

Instability analysis procedure for 3-level multi-bearing rotor-foundation systems

A procedure for the instability analysis of a three-level multispan rotor systems is described. This procedure is based on a distributed mass elastic representation of the rotor system in several eight-coefficient bearings. Each bearing is supported from an elastic foundation on damped, elastic pedestals. The foundation is represented as a general distributed mass elastic structure on discrete supports, which may have different stiffness and damping properties in the horizontal and vertical directions. This system model is suited to studies of instability threshold conditions for multirotor turbomachines on either massive or flexible foundations. The instability conditions is found by obtaining the eigenvalues of the system determinant, which is obtained by the transfer matrix method from the three-level system model. The stability determinant is solved for the lowest rotational speed at which the system damping becomes zero in the complex eigenvalue, and for the whirl frequency corresponding to the natural frequency of the unstable mode. An efficient algorithm for achieving this is described. Application of this procedure to a rigid rotor in two damped-elastic bearings and flexible supports is described. A second example discusses a flexible rotor with four damped-elastic bearings. The third case compares the stability of a six-bearing 300 Mw turbine generator unit, using two different bearing types. These applications validate the computer program and various aspects of the analysis.

Zhou, S.↗
NASA NTRS1989

Structural robustness with suboptimal responses for linear state space model

A relationship between the closed-loop eigenvalues and the amount of perturbations in the open-loop matrix is addressed in the context of performance robustness. If the allowable perturbation ranges of elements of the open-loop matrix A and the desired tolerance of the closed-loop eigenvalues are given such that max(j) of the absolute value of Delta-lambda(j) (A+BF) should be less than some prescribed value, what is a state feedback controller F which satisfies the closed-loop eigenvalue perturbation-tolerance requirement for a class of given perturbation in A? The paper gives an algorithm to design such a controller. Numerical examples are included for illustration.

Keel, L. H.↗
NASA NTRS2000

Reusable Launch Vehicle Control In Multiple Time Scale Sliding Modes

A reusable launch vehicle control problem during ascent is addressed via multiple-time scaled continuous sliding mode control. The proposed sliding mode controller utilizes a two-loop structure and provides robust, de-coupled tracking of both orientation angle command profiles and angular rate command profiles in the presence of bounded external disturbances and plant uncertainties. Sliding mode control causes the angular rate and orientation angle tracking error dynamics to be constrained to linear, de-coupled, homogeneous, and vector valued differential equations with desired eigenvalues placement. Overall stability of a two-loop control system is addressed. An optimal control allocation algorithm is designed that allocates torque commands into end-effector deflection commands, which are executed by the actuators. The dual-time scale sliding mode controller was designed for the X-33 technology demonstration sub-orbital launch vehicle in the launch mode. Simulation results show that the designed controller provides robust, accurate, de-coupled tracking of the orientation angle command profiles in presence of external disturbances and vehicle inertia uncertainties. This is a significant advancement in performance over that achieved with linear, gain scheduled control systems currently being used for launch vehicles.

Shtessel, Yuri↗
DOE OSTI2025

Block encoding bosons by signal processing

Block Encoding (BE) is a crucial subroutine in many modern quantum algorithms, including those with near-optimal scaling for simulating quantum many-body systems, which often rely on Quantum Signal Processing (QSP). Currently, the primary methods for constructing BEs are the Linear Combination of Unitaries (LCU) and the sparse oracle approach. In this work, we demonstrate that QSP-based techniques, such as Quantum Singular Value Transformation (QSVT) and Quantum Eigenvalue Transformation for Unitary Matrices (QETU), can themselves be efficiently utilized for BE implementation. Specifically, we present several examples of using QSVT and QETU algorithms, along with their combinations, to block encode Hamiltonians for lattice bosons, an essential ingredient in simulations of high-energy physics. We also introduce a straightforward approach to BE based on the exact implementation of Linear Operators Via Exponentiation and LCU (LOVE-LCU). We find that, while using QSVT for BE results in the best asymptotic gate count scaling with the number of qubits per site, LOVE-LCU outperforms all other methods for operators acting on up to qubits, highlighting the importance of concrete circuit constructions over mere comparisons of asymptotic scalings. Using LOVE-LCU to implement the BE, we simulate the time evolution of single-site and two-site systems in the lattice theory using the Generalized QSP algorithm and compare the gate counts to those required for Trotter simulation.

Kane, Christopher F↗
DOE OSTI2024

Numerical eigen-spectrum slicing, accurate orthogonal eigen-basis, and mixed-precision eigenvalue refinement using OpenMP data-dependent tasks and accelerator offload

Performing a variety of numerical computations efficiently and, at the same time, in a portable fashion requires both an overarching design followed by a number of implementation strategies. All of these are exemplified below as we present transitioning the PLASMA numerical library from relying on dependence-driven large tasks to achieving utilization of fine grain tasking and offload to hardware accelerators while keeping its core dependence sets: OpenMP source code pragmas and runtime for most system-level functionality and basic low-level numerical kernels provided directly by hardware vendors or open source projects with vendor contributions. We also present new algorithmic methods and their efficient parallel implementations including fine grained tasking for eigen-spectrum slicing and offload for mixed-precision eigenvalue refinement. We provide performance, scaling, and numerical results showing sizable gains over the available solutions from either the open source and vendor-provided packages.

Luszczek, Piotr↗
NASA NTRS1975

A variable-metric algorithm employing linear and quadratic penalties

A variable-metric algorithm is described that uses both linear and quadratic penalty terms for handling nonlinear constraints. Quadratic penalty coefficients are adjusted in a process which maintains a positive-definite matrix of second partial derivatives of the function without generating the large positive eigenvalues which cause zigzagging and slow convergence. The schemes suggested use inferred second-order properties not only in terms of the variable metric of the Davidson-Fletcher-Powell algorithm (or its relatives) but by estimating of second directional derivatives by fitting cubics to various functions along search directions.

Kelley, H. J.↗
NASA NTRS1977

FLUT - A program for aeroelastic stability analysis

A computer program (FLUT) that can be used to evaluate the aeroelastic stability of aircraft structures in subsonic flow is described. The algorithm synthesizes data from a structural vibration analysis with an unsteady aerodynamics analysis and then performs a complex eigenvalue analysis to assess the system stability. The theoretical basis of the program is discussed with special emphasis placed on some innovative techniques which improve the efficiency of the analysis. User information needed to efficiently and successfully utilize the program is provided. In addition to identifying the required input, the flow of the program execution and some possible sources of difficulty are included. The use of the program is demonstrated with a listing of the input and output for a simple example.

Johnson, E. H.↗
NASA NTRS1984

A solution of Mx(double dot) + Cx(dot) + Kx = 0 applicable to the design of active dampers

A solution is presented for the equations of motion for the damped linear oscillator, Mx(double dot) + Cx(dot) + Kx = 0. The algorithm solves a transformed set of equations in terms of the modal variables of the undamped system and, at the same time, solves the adjoint equation of the transformed problem. The adjoint solution is normalized to give the inverse of the solution matrix of the transformed problem. The normalized inverse is useful in design for direct computation of sensitivity derivatives of damping ratios with respect to damping rates. The algorithm is programmed to reduce storage requirements by a factor of three-fourths compared to standard complex eigenvalue subroutines. A numerical example is included.

Thurston, G. A.↗
NASA NTRS1986

Numerical pole assignment by eigenvalue Jacobian inversion

A numerical procedure for solving the linear pole placement problem is developed which operates by the inversion of an analytically determined eigenvalue Jacobian matrix. Attention is given to convergence characteristics and pathological situations. It is not concluded that the algorithm developed is suitable for computer-aided control system design with particular reference to the scan platform pointing control system for the Galileo spacecraft.

Sevaston, George E.↗
NASA NTRS1989

Quotient-difference type generalizations of the power method and their analysis

The recursion relations that were proposed by W. F. Ford and A. Sidi (Appl. Numer. Math, 4 (1988), pp. 477-489) for implementing vector extrapolation methods are used for devising generalizations of the power method for linear operators. These generalizations are shown to produce approximations to largest eigenvalues of a linear operator under certain conditions. They are similar in form to the quotient-difference algorithm and share similar convergence properties with the latter. These convergence properties also resemble those obtained for the basic LR and QR algorithms. Finally, it is shown that the convergence rate produced by one fo these generalizations is twice as fast for normal operators as it is for nonnormal operators.

Sidi, Avram↗
NASA NTRS1995

Separation analysis, a tool for analyzing multigrid algorithms

The separation of vectors by multigrid (MG) algorithms is applied to the study of convergence and to the prediction of the performance of MG algorithms. The separation operator for a two level cycle algorithm is derived. It is used to analyze the efficiency of the cycle when mixing of eigenvectors occurs. In particular cases the separation analysis reduces to Fourier type analysis. The separation operator of a two level cycle for a Schridubger eigenvalue problem, is derived and analyzed in a Fourier basis. Separation analysis gives information on how to choose performance relaxations and inter-level transfers. Separation analysis is a tool for analyzing and designing algorithms, and for optimizing their performance.

Costiner, Sorin↗
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 NTRS1978

An algorithm for obtaining the minimal realization of a linear time-invariant system and determining if a system is stabilizable-detectable

A definition of centralized fixed modes of a system, which is an adoption of the definition of fixed modes for a decentralized system is made. It is shown that the centralized fixed modes of a system can be easily and efficiently calculated in terms of the eigenvalues of the system, and that the calculation of such fixed modes enables one to determine, in a numerically efficient way, whether a system is controllable, observable, stabilizable, or detectable. An efficient algorithm for reducing a system to minimal realization form is then given. The algorithms have been effectively used on systems of order up to 100.

Davison, E. J.↗
NASA NTRS1994

Using parallel banded linear system solvers in generalized eigenvalue problems

Subspace iteration is a reliable and cost effective method for solving positive definite banded symmetric generalized eigenproblems, especially in the case of large scale problems. This paper discusses an algorithm that makes use of two parallel banded solvers in subspace iteration. A shift is introduced to decompose the banded linear systems into relatively independent subsystems and to accelerate the iterations. With this shift, an eigenproblem is mapped efficiently into the memories of a multiprocessor and a high speedup is obtained for parallel implementations. An optimal shift is a shift that balances total computation and communication costs. Under certain conditions, we show how to estimate an optimal shift analytically using the decay rate for the inverse of a banded matrix, and how to improve this estimate. Computational results on iPSC/2 and iPSC/860 multiprocessors are presented.

DISTRIBUTED MEMORY MULTIPROCES↗
NASA NTRS1987

Polynomial approximation of functions of matrices and its application to the solution of a general system of linear equations

During the process of solving a mathematical model numerically, there is often a need to operate on a vector v by an operator which can be expressed as f(A) while A is NxN matrix (ex: exp(A), sin(A), A sup -1). Except for very simple matrices, it is impractical to construct the matrix f(A) explicitly. Usually an approximation to it is used. In the present research, an algorithm is developed which uses a polynomial approximation to f(A). It is reduced to a problem of approximating f(z) by a polynomial in z while z belongs to the domain D in the complex plane which includes all the eigenvalues of A. This problem of approximation is approached by interpolating the function f(z) in a certain set of points which is known to have some maximal properties. The approximation thus achieved is almost best. Implementing the algorithm to some practical problem is described. Since a solution to a linear system Ax = b is x= A sup -1 b, an iterative solution to it can be regarded as a polynomial approximation to f(A) = A sup -1. Implementing the algorithm in this case is also described.

Tal-Ezer, Hillel↗
DOE OSTI2025

Multi-Objective Optimization of Uranium Target Assembly–3: A Comparison of Genetic and Traditional Methods

Commonly produced as a byproduct of uranium fission, 99 Mo is a key medical isotope that is in high demand in the United States. An international goal is to switch from medical isotope production technologies that require highly enriched uranium to medical isotope production technologies that require only low-enriched uranium. Niowave Inc. is contributing to this goal by developing an accelerator-driven subcritical assembly called the Uranium Target Assembly (UTA). This work compares the performance of Dakota’s Multi-Objective Genetic Algorithm (MOGA) against traditional sensitivity analysis in the neutronic optimization of the UTA-3 system. The design objectives are k-eigenvalue (k eff ) and natural uranium fission power, which are directly correlated with the amount of 99 Mo produced. Dakota:MOGA did not perform as well as human engineering ingenuity in optimization studies with high numbers of input parameters, such as fuel rod type selection and fuel rod placement. However, Dakota:MOGA did outperform traditional sensitivity analysis in optimization studies with fewer than 20 parameters and revealed the degree to which each parameter influences the optimal design space for k eff and natural uranium fission power (to a lesser extent). As the design model became more complex in the final stage of design, the computational resources required to calculate the design objective values in the Monte Carlo N-Particle transport code from selected input parameter combinations limited Dakota:MOGA’s performance, and, unfortunately, human intervention was required to discern the optimal design space. In conclusion, future work will attempt to reduce computational resource constraints by incorporating areduced-order neutronics model into the optimization cycle.

Accelerator-driven systems↗
NASA NTRS1972

An improved general algorithm for arbitrary pole assignment.

A new pole-placement algorithm for multi-input controllable systems is presented. The algorithm can be used to obtain real-valued feedback gains for arbitrarily specified real and complex, distinct or multiple eigenvalues. The multi-input controllable time-invariant system is converted to a single-input system. Pole assignment is achieved by making use of certain properties of the equation TF - AT = D, where A is the open-loop system matrix and F represents a matrix having the desired closed-loop eigenvalues.

Willner, L.↗
NASA NTRS1985

A sequential linear optimization approach for controller design

A linear optimization approach with a simple real arithmetic algorithm is presented for reliable controller design and vibration suppression of flexible structures. Using first order sensitivity of the system eigenvalues with respect to the design parameters in conjunction with a continuation procedure, the method converts a nonlinear optimization problem into a maximization problem with linear inequality constraints. The method of linear programming is then applied to solve the converted linear optimization problem. The general efficiency of the linear programming approach allows the method to handle structural optimization problems with a large number of inequality constraints on the design vector. The method is demonstrated using a truss beam finite element model for the optimal sizing and placement of active/passive-structural members for damping augmentation. Results using both the sequential linear optimization approach and nonlinear optimization are presented and compared. The insensitivity to initial conditions of the linear optimization approach is also demonstrated.

Horta, L. G.↗