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Modern control theory relies upon the fidelity of the mathematical model of the system. Truncated modes, external disturbances, and parameter errors in linear system models are corrected by augmenting to the original system of equations an 'error system' which is designed to approximate the effects of such model errors. A Chebyshev error system is developed for application to the Large Space Telescope (LST).
Revised and extended versions of a fast, direct (noniterative) numerical Cauchy-Riemann solver are presented for solving finite difference approximations of first order systems of partial differential equations. Although the difference operators treated are linear and elliptic, one significant application of these extended direct Cauchy-Riemann solvers is in the fast, semidirect (iterative) solution of fluid dynamic problems governed by the nonlinear mixed elliptic-hyperbolic equations of transonic flow. Different versions of the algorithms are derived and the corresponding FORTRAN computer programs for a simple example problem are described and listed. The algorithms are demonstrated to be efficient and accurate.
The main body of this paper describes the evolution of the Centaur D-1A Guidance and Software System. Specifically, the performance of the explicit guidance equations, using a linear tangent steering law. Inherent flexibility exists in the equations in that they have multimission capability. They can accommodate both Earth-orbital and Earth-escape missions with either one or two Centaur burns. They can also guide for multi-burn orbital missions. The Centaur performance is indicated in terms of optimality (propellant usage), accuracy, flexibility and computer requirements. In the course of the Centaur Guidance development substantial changes and improvements have been made and more improvements are on the way for the Shuttle/Centaur Guidance. It is the intent of this paper to describe, provide insight into, and identify certain unique aspects of the individual Centaur flight profiles. Mission profile(s) are described narratively with some numerical data given in cases where it may be useful.
In a Benard-type convection problem, the stationary flows of an infinite layer of fluid lying between two rigid horizontal walls and heated uniformly from below are determined. As the temperature difference across the layer increases beyond a certain value, other convective motions appear. These motions areoften cellular in character in that their streamlines are confined to certain well-defined cells having, for example, the shape of rolls or hexagons. A selection principle that explains why hexagonal cells seem to be preferred for certain ranges of the parameters is formulated. An operator-theoretical formulation of one generalized Bernard problem is given. The infinite dimensional problem is reduced to one of solving a finite dimensional system of equations, namely, the selection equations. These equations are solved and a linearized stability analysis of the resultant stationary flows is presented.
The main body of this paper describes the evolution of the Centaur D-1A Guidance and Software System. Specifically, the performance of the explicit guidance equations, using a linear tangent steering law. Inherent flexibility exists in the equations in that they have multimission capability. They can accommodate both earth-orbital and earth-escape missions with either one or two Centaur burns. They can also guide for multi-burn orbital missions. The Centaur performance is indicated in terms of optimality (propellant usage), accuracy, flexibility and computer requirements. In the course of the Centaur Guidance development substantial changes and improvements have been made and more improvements are on the way for the Shuttle/Centaur Guidance. It is the intent of this paper to describe, provide insight into, and identify certain unique aspects of the individual Centaur flight profiles. Mission profile(s) are described narratively with some numerical data given in cases where it may be useful.
In a Benard-type convection problem, the stationary flows of an infinite layer of fluid lying between two rigid horizontal walls and heated uniformly from below are determined. As the temperature difference across the layer increases beyond a certain value, other convective motions appear. These motions are often cellular in character in that their streamlines are confined to certain well-defined cells having, for example, the shape of rolls or hexagons. A selection principle that explains why hexagonal cells seem to be preferred for certain ranges of the parameters is formulated. An operator-theoretical formulation of one generalized Bernard problem is given. The infinite dimensional problem is reduced to one of solving a finite dimensional system of equations, namely, the selection equations. These equations are solved and a linearized stability analysis of the resultant stationary flows is presented.
An algorithm is developed to construct the compliance matrix for a cracked solid in the integral-equation formulation of two-dimensional linear-elastic fracture mechanics. The integral equation is reduced to a system of algebraic equations for unknown values of the dislocation-density function at discrete points on the interval from -1 to 1, using the numerical procedure described by Gerasoulis (1982). Sample numerical results are presented, and it is suggested that the algorithm is especially useful in cases where iterative solutions are required; e.g., models of fiber-reinforced concrete, rocks, or ceramics where microcracking, fiber bridging, and other nonlinear effects are treated as nonlinear springs along the crack surfaces (Ballarini et al., 1984).
Projection methods for computing stationary probability distributions for Markov chain models are presented. A general projection method is a method which seeks an approximation from a subspace of small dimension to the original problem. Thus, the original matrix problem of size N is approximated by one of dimension m, typically much smaller than N. A particularly successful class of methods based on this principle is that of Krylov subspace methods which utilize subspaces of the form span(v,av,...,A(exp m-1)v). These methods are effective in solving linear systems and eigenvalue problems (Lanczos, Arnoldi,...) as well as nonlinear equations. They can be combined with more traditional iterative methods such as successive overrelaxation, symmetric successive overrelaxation, or with incomplete factorization methods to enhance convergence.
The general moment equations for a spin-stabilized vehicle with an inertia-reaction angular rate damper were considered, and it was noted that simplification would result if the damper had a spherical inertia distribution. A control system incorporating such a damper was postulated. The resulting equations were linearized, and conditions for stability were obtained from an analysis of the cubic characteristic equation. Two numerical examples were included.
A new, coordinate-free version of the exact ray-trace equations for optical systems consisting of conic reflecting, refracting and reference surfaces is presented. These equations are differentiated to obtain closed-form optical sensitivity dyadics. For computation, the sensitivities are evaluated in a single global coordinate frame and combined in linearized ray-trace matrix difference equations that propagate the rays and the sensitivities from element to element. One purpose of this analysis is to create optical models that can be directly integrated with models of the instrument structure and control systems for dynamic simulation.
An incompressible three-dimensional laminar boundary-layer flow over a swept wing is used as a model to study both the wall-curvature and streamline-curvature effects on the stationary crossflow instability. The basic state is obtained by solving the full Navier-Stokes (N-S) equations numerically. The linear disturbance equations are cast on a fixed, body-intrinsic, curvilinear coordinate system. Those nonparallel terms which contribute mainly to the streamline-curvature effect are retained in the formulation of the disturbance equations and approximated by their local finite difference values. The resulting eigenvalue problem is solved by a Chebyshev collocation method. The present results indicate that the convex wall curvature has a stabilizing effect, whereas the streamline curvature has a destabilizing effect. A validation of these effects with an N-S solution for the linear disturbance flow is provided.
Bayliss (1975) has studied the question whether in the case of linear differential equations the relationship between the stability of the homogeneous equations and the existence of almost periodic solutions to the inhomogeneous equation is preserved by finite difference approximations. In the current investigation analogous properties are considered for the case in which the damping is quadratic rather than linear. The properties of the considered equation for arbitrary forcing terms are examined and the validity is proved of a theorem concerning the characteristics of the unique solution. By using the Lipschitz continuity of the mapping and the contracting mapping principle, almost periodic solutions can be found for perturbations of the considered equation. Attention is also given to the Lipschitz continuity of the solution operator and the results of numerical tests which have been conducted to test the discussed theory.
An expert system, ETUDES - Expert Time integration control Using Deep and Surface Knowledge System, which addresses the determination of the timestep for time integration of linear structural dynamic equations is described. This timestep may also be applicable for a moderately nonlinear simulation of the same structure. The program also determines whether an explicit or implicit method is most efficient for the particular simulation. A production rule programming system written in OPS5 is used for the implementation of this prototype expert system. Issues relating to the expert system architecture for this application, such as knowledge representation and structure, as well as domain knowledge are discussed. The prototype is evaluated by measuring its performance in various benchmark model problems.
FORTRAN coded out-of-core equation solvers that solve using direct methods symmetric banded systems of simultaneous algebraic equations. Banded, frontal and column (skyline) solvers were studied as well as solvers that can partition the working area and thus could fit into any available core. Comparison timings are presented for several typical two dimensional and three dimensional continuum type grids of elements with and without midside nodes. Extensive conclusions are also given.
A Pointing Covariance Analysis Tool (PCAT) has been developed for evaluating the expected performance of the pointing control system for NASA s Space Interferometry Mission (SIM). The SIM pointing control system is very complex, consisting of multiple feedback and feedforward loops, and operating with multiple latencies and data rates. The SIM pointing problem is particularly challenging due to the effects of thermomechanical drifts in concert with the long camera exposures needed to image dim stars. Other pointing error sources include sensor noises, mechanical vibrations, and errors in the feedforward signals. PCAT models the effects of finite camera exposures and all other error sources using linear system elements. This allows the pointing analysis to be performed using linear covariance analysis. PCAT propagates the error covariance using a Lyapunov equation associated with time-varying discrete and continuous-time system matrices. Unlike Monte Carlo analysis, which could involve thousands of computational runs for a single assessment, the PCAT analysis performs the same assessment in a single run. This capability facilitates the analysis of parametric studies, design trades, and "what-if" scenarios for quickly evaluating and optimizing the control system architecture and design.
The dynamics and stability of a spin stabilized spacecraft with a hinged appendage system are treated analytically and numerically. The hinged system consists of a central hub with masses attached to (assumed) massless booms of fixed length whose orientation relative to the main part can change. The general three dimensional deployment dynamics of such a hinged system is considered without any restriction on the location of the hinge points. The equations of motion for the hinged system, with viscous damping at both hinge points, are linearized about the nominal equilibrium position where the booms are orthogonal to the nominal spin axis for the case of two dimensional and three dimensional motion. Analytic stability criteria are obtained from the necessary condition on the sign of all the coefficients in the system characteristic equation.
A new method for the acceleration of linear and nonlinear time dependent calculations is presented. It is based on the Large Discretization Step (LDS) approximation, defined in this work, which employs an extended system of low accuracy schemes to approximate a high accuracy discrete approximation to a time dependent differential operator. Error bounds on such approximations are derived. These approximations are efficiently implemented in the LDS methods for linear and nonlinear hyperbolic equations, presented here. In these algorithms the high and low accuracy schemes are interpreted as the same discretization of a time dependent operator on fine and coarse grids, respectively. Thus, a system of correction terms and corresponding equations are derived and solved on the coarse grid to yield the fine grid accuracy. These terms are initialized by visiting the fine grid once in many coarse grid time steps. The resulting methods are very general, simple to implement and may be used to accelerate many existing time marching schemes.
The present paper shows that the reflected ion at a supercritical quasi-perpendicular shock wave can excite a purely growing mode propagating parallel to the ambient magnetic field. To discuss the ion heating by such an unstable mode, the self-consistent quasi-linear kinetic equation is solved with the assumption that the present purely growing mode is the dominant unstable mode in the system. In the quasi-linear analysis of the instability, two particular cases are considered: the case of low initial ion beta and that of a high initial ion beta.