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At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
NASA NTRS1982

A general algorithm for solving the algebraic Riccati equation

The generalized eigenvalue problem provides a suitable framework for reliable solutions of many system theoretic, control, and estimation problems. A general algorithm for solving the matrix algebraic Riccati equation (ARE) which utilizes a pencil structure is described here. This algorithm avoids unnecessary inversion of cost or transition matrices, making it a numerically sound way to solve for the gains and/or ARE with singular quadratic costs, for cases satisfying detectability and stabilizability conditions. Examples are solution with discrete dead-beat control, noiseless measurements in Kalman filters and time-delays in discrete-time systems, which cause difficulties in the Hamiltonian standard eigenvalue problem formulation. The ARE algorithm implementatiton and numerical examples are shown.

Walker, R. A.↗
NASA NTRS1978

ORACLS: A system for linear-quadratic-Gaussian control law design

A modern control theory design package (ORACLS) for constructing controllers and optimal filters for systems modeled by linear time-invariant differential or difference equations is described. Numerical linear-algebra procedures are used to implement the linear-quadratic-Gaussian (LQG) methodology of modern control theory. Algorithms are included for computing eigensystems of real matrices, the relative stability of a matrix, factored forms for nonnegative definite matrices, the solutions and least squares approximations to the solutions of certain linear matrix algebraic equations, the controllability properties of a linear time-invariant system, and the steady state covariance matrix of an open-loop stable system forced by white noise. Subroutines are provided for solving both the continuous and discrete optimal linear regulator problems with noise free measurements and the sampled-data optimal linear regulator problem. For measurement noise, duality theory and the optimal regulator algorithms are used to solve the continuous and discrete Kalman-Bucy filter problems. Subroutines are also included which give control laws causing the output of a system to track the output of a prescribed model.

Armstrong, E. S.↗
NASA NTRS1949

Response of a Rotating Propeller to Aerodynamic Excitation

The flexural vibration of a rotating propeller blade with clamped shank is analyzed with the object of presenting, in matrix form, equations for the elastic bending moments in forced vibration resulting from aerodynamic forces applied at a fixed multiple of rotational speed. Matrix equations are also derived which define the critical speeds end mode shapes for any excitation order and the relation between critical speed and blade angle. Reference is given to standard works on the numerical solution of matrix equations of the forms derived. The use of a segmented blade as an approximation to a continuous blade provides a simple means for obtaining the matrix solution from the integral equation of equilibrium, so that, in the numerical application of the method presented, the several matrix arrays of the basic physical characteristics of the propeller blade are of simple form, end their simplicity is preserved until, with the solution in sight, numerical manipulations well-known in matrix algebra yield the desired critical speeds and mode shapes frame which the vibration at any operating condition may be synthesized. A close correspondence between the familiar Stodola method and the matrix method is pointed out, indicating that any features of novelty are characteristic not of the analytical procedure but only of the abbreviation, condensation, and efficient organization of the numerical procedure made possible by the use of classical matrix theory.

Arnoldi, Walter E.↗
NASA NTRS1968

MOP /Matrix Operation Programs system/

MOP /Matrix Operation Programs/ system consists of a set of FORTRAN 4 subroutines which are related through a small common allocation. The system accomplishes all matrix algebra operations plus related input-output and housekeeping details.

Muller, P. M.↗
NASA NTRS1971

Study-simulation of space station dynamics

Matrix algebra translator and executor /MATE/ takes equations describing structural control system environmental interaction problem for flexible spacecraft components and loads them into self programming computer.

Gaitens, M. J.↗
NASA NTRS1972

A linear programming manual

Computer solutions of linear programming problems are outlined. Information covers vector spaces, convex sets, and matrix algebra elements for solving simultaneous linear equations. Dual problems, reduced cost analysis, ranges, and error analysis are illustrated.

Tuey, R. C.↗
NASA NTRS1976

On differential transformations between Cartesian and curvilinear (geodetic) coordinates

Differential transformations are developed between Cartesian and curvilinear orthogonal coordinates. Only matrix algebra is used for the presentation of the basic concepts. After defining the reference systems used the rotation (R), metric (H), and Jacobian (J) matrices of the transformations between cartesian and curvilinear coordinate systems are introduced. A value of R as a function of H and J is presented. Likewise an analytical expression for J(-1) as a function of H(-2) and R is obtained. Emphasis is placed on showing that differential equations are equivalent to conventional similarity transformations. Scaling methods are discussed along with ellipsoidal coordinates. Differential transformations between elipsoidal and geodetic coordinates are established.

Soler, T.↗
NASA NTRS1977

Control logic to track the outputs of a command generator or randomly forced target

A procedure is presented for synthesizing time-invariant control logic to cause the outputs of a linear plant to track the outputs of an unforced (or randomly forced) linear dynamic system. The control logic uses feed-forward of the reference system state variables and feedback of the plant state variables. The feed-forward gains are obtained from the solution of a linear algebraic matrix equation of the Liapunov type. The feedback gains are the usual regulator gains, determined to stabilize (or augment the stability of) the plant, possibly including integral control. The method is applied here to the design of control logic for a second-order servomechanism to follow a linearly increasing (ramp) signal, an unstable third-order system with two controls to track two separate ramp signals, and a sixth-order system with two controls to track a constant signal and an exponentially decreasing signal (aircraft landing-flare or glide-slope-capture with constant velocity).

Trankle, T. L.↗
NASA NTRS1978

Extending reliability: Transformational tailoring of abstract mathematical software

Methods for automatically constructing concrete executable programs from an abstract prototype program by applying transformations based on theorems of matrix algebra and on algebraic properties of programming languages are described. These methods provide a user with highly efficient programs tailored to his environment while maintaining the advantages of high reliability and low cost associated with routines from the best mathematical software libraries. Also, the transformations which produce such programs represent a formal codification of rules for writing linear algebra programs.

Boyle, J. M.↗
NASA NTRS1979

A general and computationally fast formulation for radiative transfer with scattering

A general formulation of monocromatic radiative transfer with scattering has been developed for plane-parallel geometry. The inhomogeneous and nonisothermal medium absorbs, emits, and anisotropically scatters radiation. Surfaces can emit and scatter radiation in any specified manner. The solution procedure uses the fact that phase incoherent scattering is linear in radiative sources. Certain basic scattering functions are then defined and calculated by an adding computer code using matrix algebra. These scattering functions are weighted by the temperature field and summed (superimposed) to obtain the solution for any specific problem. Numerical results for exiting intensities and one-sided heat fluxes from general media bound by one arbitrary surface are presented. These parametric studies demonstrate the effects of scattering particles and surfaces on radiative transfer from inhomogeneous and nonisothermal media. Application of the formulation to radiative equilibrium is also discussed. The conclusion is that all problems in plane-parallel radiative transfer with scattering can be solved by a common and computationally fast algorithm based on this formulation.

Cogley, A. C.↗
NASA NTRS1979

On the overlapping plate method in astrometry

A method is proposed for the simplification of the algorithms of Eichhorn and Russell (1976) for obtaining the solutions of astrometric problems using the overlapping plate method. The method provides the parameter vector (consisting of 'star constants' and plate constants) as well as the covariance matrix of the parameters. A considerable reduction in the amount of matrix algebra required is obtained with the present method.

Jefferys, W. H.↗
NASA NTRS1987

Interactive application of quadratic expansion of chi-square statistic to nonlinear curve fitting

This report contains a detailed theoretical description of an all-purpose, interactive curve-fitting routine that is based on P. R. Bevington's description of the quadratic expansion of the Chi-Square statistic. The method is implemented in the associated interactive, graphics-based computer program. Taylor's expansion of Chi-Square is first introduced, and justifications for retaining only the first term are presented. From the expansion, a set of n simultaneous linear equations is derived, then solved by matrix algebra. A brief description of the code is presented along with a limited number of changes that are required to customize the program of a particular task. To evaluate the performance of the method and the goodness of nonlinear curve fitting, two typical engineering problems are examined and the graphical and tabular output of each is discussed. A complete listing of the entire package is included as an appendix.

Badavi, F. F.↗
NASA NTRS1951

A recurrence matrix solution for the dynamic response of aircraft in gusts

A systematic procedure developed for the calculation of the structural response of aircraft flying through a gust by use of difference equations in the solution of dynamic problems is first illustrated by means of a simple-damped-oscillator example. A detailed analysis is then given which leads to a recurrence matrix equation for the determination of the response of an airplane in a gust. The method takes into account wing bending and twisting deformations, fuselage deflection, vertical and pitching motion of the airplane, and some tail forces. The method is based on aerodynamic strip theory, but compressibility and three-dimensional aerodynamic effects can be taken into account approximately by means of over-all corrections. Either a sharp-edge gust or a gust of arbitrary shape in the spanwise or flight directions may be treated. In order to aid in the application of the method to any specific case, a suggested computational procedure is included. The possibilities of applying the method to a variety of transient aircraft problems, such as landing, are brought out. A brief review of matrix algebra, covering the extent to which it is used in the analysis, is also included. (author)

Houbolt, John C↗
NASA NTRS1977

Exact solution of some linear matrix equations using algebraic methods

A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.

Djaferis, T. E.↗
NASA NTRS1979

Exact solution of some linear matrix equations using algebraic methods

Algebraic methods are used to construct the exact solution P of the linear matrix equation PA + BP = - C, where A, B, and C are matrices with real entries. The emphasis of this equation is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The paper is divided into six sections which include the proof of the basic lemma, the Liapunov equation, and the computer implementation for the rational, integer and modular algorithms. Two numerical examples are given and the entire calculation process is depicted.

Djaferis, T. E.↗