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NASA NTRS1988

Parallels between control PDE's and systems of ODE's

System theorists understand that the same mathematical objects which determine controllability for nonlinear control systems of ordinary differential equations (ODEs) also determine hypoellipticity for linear partial differential equations (PDEs). Moreover, almost any study of ODE systems begins with linear systems. It is remarkable that Hormander's paper on hypoellipticity of second order linear p.d.e.'s starts with equations due to Kolmogorov, which are shown to be analogous to the linear PDEs. Eigenvalue placement by state feedback for a controllable linear system can be paralled for a Kolmogorov equation if an appropriate type of feedback is introduced. Results concerning transformations of nonlinear systems to linear systems are similar to results for transforming a linear PDE to a Kolmogorov equation.

Hunt, L. R.↗
NASA NTRS1988

Robust eigenvalue/eigenvector selection in linear state feedback systems

Design procedures are developed to select both eigenvalues and eigenvectors to improve system robustness. A robustness metric based on the determinant of the normalized eigenvector matrix defined. It is shown that this metric significantly speeds up the convergence of robust eigenvector solutions close to the optimal. A procedure to blend multiple inputs to improve achievable system robustness in the lower dimensional pseudo control space has also been derived.

Srinathkumar, S.↗
NASA NTRS1989

Control augmented structural synthesis with dynamic stability constraints

Dynamic stability constraints are included in a computer program that simultaneously synthesizes a structure and its control system. Two measures of stability, the real part of the system complex eigenvalues and the damping ratio, are examined. The procedure for calculating the sensitivities of the two measures of stability to changes in the structure and its control system is explained. The sensitivities are used to formulate an approximate problem that is solved at each design iteration. The effects of structural damping and noncollated controllers on the synthesis process are discussed.

Thomas, H. L.↗
NASA NTRS1988

Stability of semi-discrete approximations for hyperbolic initial-boundary-value problems. II - Asymptotic estimates

The stability of semi-discrete approximations for hyperbolic initial-boundary-value problems is considered. Asymptotic estimates of the eigenvalues of the finite-domain problem are given. These estimates are used to examine questions relating the normal mode analysis of the finite-domain problem and the normal mode quarter-plane analysis of the Gustafsson, Kreiss, and Sundstroem theory.

Warming, Robert F.↗
NASA NTRS1989

Application of spectral collocation techniques to the stability of swirling flows

The linearized stability equations in cylindrical coordinates of a Chebyshev spectral collocation method for the temporal and spatial stability of swirling flows are presently solved with the eigenvalues obtained through the use of the QZ routine. The algorithm thus created is robust and easily adaptable to a range of flow configurations encompassing internal and external flows with minor boundary condition application modifications. Accuracy and efficiency tests of the method are made for the cases of plane Poiseulle, rotating-pipe, and trailing line vortex flows.

Khorrami, Mehdi R.↗
NASA NTRS1989

An analysis of the vertical structure equation for arbitrary thermal profiles

The vertical structure equation is a singular Sturm-Liouville problem whose eigenfunctions describe the vertical dependence of the normal modes of the primitive equations linearized about a given thermal profile. The eigenvalues give the equivalent depths of the modes. The spectrum of the vertical structure equation and the appropriateness of various upper boundary conditions, both for arbitrary thermal profiles were studied. The results depend critically upon whether or not the thermal profile is such that the basic state atmosphere is bounded. In the case of a bounded atmosphere it is shown that the spectrum is always totally discrete, regardless of details of the thermal profile. For the barotropic equivalent depth, which corresponds to the lowest eigen value, upper and lower bounds which depend only on the surface temperature and the atmosphere height were obtained. All eigenfunctions are bounded, but always have unbounded first derivatives. It was proved that the commonly invoked upper boundary condition that vertical velocity must vanish as pressure tends to zero, as well as a number of alternative conditions, is well posed. It was concluded that the vertical structure equation always has a totally discrete spectrum under the assumptions implicit in the primitive equations.

Cohn, Stephen E.↗
NASA NTRS1989

Sequential design of discrete linear quadratic regulators via optimal root-locus techniques

A sequential method employing classical root-locus techniques has been developed in order to determine the quadratic weighting matrices and discrete linear quadratic regulators of multivariable control systems. At each recursive step, an intermediate unity rank state-weighting matrix that contains some invariant eigenvectors of that open-loop matrix is assigned, and an intermediate characteristic equation of the closed-loop system containing the invariant eigenvalues is created.

Shieh, Leang S.↗
NASA NTRS1988

Observability of discretized partial differential equations

It is shown that complete observability of the discrete model used to assimilate data from a linear partial differential equation (PDE) system is necessary and sufficient for asymptotic stability of the data assimilation process. The observability theory for discrete systems is reviewed and applied to obtain simple observability tests for discretized constant-coefficient PDEs. Examples are used to show how numerical dispersion can result in discrete dynamics with multiple eigenvalues, thereby detracting from observability.

Cohn, Stephen E.↗
NASA NTRS1989

Retrieval of aerosol size distribution moments from multiwavelength particulate extinction measurements

Two methods for inferring aerosol size distribution moments from multiwavelength particulate extinction measurements are studied. The methods are an eigenvalue technique that approximates an appropriate moment-weighting function by a linear combination of kernel functions and a conversion ratio approach that uses the ratio of the particulate extinction measurements at two wavelengths to choose a model moment-to-extinction conversion ratio. The techniques are applied to infer the third moment, or volume, of the aerosol size distribution from actual particulate extinction measurements taken as part of the Stratospheric Aerosol and Gas Experiment II during a correlative measurement experiment in Brazil in April 1985.

Livingston, John M.↗
NASA NTRS1988

Dynamic analysis of the joint dominated beam

A method for determining the vibrational modes of the joint dominated beam is presented with reference to space erectable structures. Mathematical models with and without joint imperfection are developed. Joint imperfection is represented by a simple flexibility, k, in the axial direction of the chord members only. An interactive computer program is used to obtain eigenvalue characteristics and mode shapes of the system. Results are presented on frequencies of the given truss beam with and without joint imperfections for three-panel and 10-panel models. Also presented are results of parametric studies of the beam truss models of three and 10 panels with k = 0.05, 0.10, 0.50, and 1.0 x 10 to the -7th.

Abu-Saba, Elias G.↗
NASA NTRS1989

Droplet model for autocorrelation functions in an Ising ferromagnet

The autocorrelation function of Ising spins in an ordered phase is studied via a droplet model. Only noninteracting spherical droplets are considered. The Langevin equation which describes fluctuations in the radius of a single droplet is studied in detail. A general description of the transformation to a Fokker-Planck equations and the ways in which a spectral analysis of that equation can be used to compute the autocorrelation function is given. It is shown that the eigenvalues of the Fokker-Planck operator form (1) a continuous spectrum of relaxation rates starting from zero for d = 2, (2) a continuous spectrum with a finite gap for d = 3, and (3) a discrete spectrum for d greater than 4, where d is the spatial dimensionality. Detailed solutions for various cases are presented.

Tang, Chao↗
NASA NTRS1989

Ideal state reconstructor for deterministic digital control systems

A state reconstructor for deterministic digital systems is presented which is ideal in the following sense: if the plant parameters are known exactly, the output of the state reconstructor will exactly equal the true state of the plant, not just approximate it. Furthermore, this ideal state reconstructor adds no additional states or eigenvalues to the system. Nor does it affect the plant equation for the system in any way; it affects only the measurement equation. While there are countless ways of choosing the ideal state reconstructor parameters, two distinct methods are described here. An example is presented which illustrates the procedures to completely design the ideal state reconstructor using both methods.

Polites, Michael E.↗
NASA NTRS1989

Optimal three finger grasps

Consideration is given to the problem of optimal force distribution among three point fingers holding a planar object. A scheme that reduces the nonlinear optimization problem to an easily solved generalized eigenvalue problem is proposed. This scheme generalizes and simplifies results of Ji and Roth (1988). The generalizations include all possible geometric arrangements and extensions to three dimensions and to the case of variable coefficients of friction. For the two-dimensional case with constant coefficients of friction, it is proved that, except for some special cases, the optimal grasping forces (in the sense of minimizing the dependence on friction) are those for which the angles with the corresponding normals are all equal (in absolute value).

Demmel, J.↗
NASA NTRS1989

Nonaxisymmetric instabilities in thin self-gravitating rings and disks

The stability of geometrically thin self-gravitating rings and disks is studied by computing the eigenvalues and eigenfunctions of linearized normal mode oscillations in these systems. For the vertically averaged incompressible models and models of gaseous and stellar disks with softened gravity, analytic treatment is possible. The existence of oscillatory normal modes is established using a variational principle and infer instability using perturbation theory. Effects due to the Lindblad and corotation resonances are analyzed in detail. The distribution of the ratio of vorticity to surface density is important for determining the stability of the systems.

Papaloizou, John C. B.↗
NASA NTRS1989

The application of TSIM software to act design and analysis on flexible aircraft

The TSIM software is described. This is a package which uses an interactive FORTRAN-like simulation language for the simulation on nonlinear dynamic systems and offers facilities which include: mixed continuous and discrete time systems, time response calculations, numerical optimization, automatic trimming of nonlinear aircraft systems, and linearization of nonlinear equations for eigenvalues, frequency responses and power spectral response evaluation. Details are given of the application of TSIM to the analysis of aeroelastic systems under the RAE Farborough extension FLEX-SIM. The aerodynamic and structural data for the equations of motion of a flexible aircraft are prepared by a preprocessor program for incorporation in TSIM simulations. Within the simulation, the flexible aircraft model may then be selected interactively for different flight conditions and modal reduction techniques applied. The use of FLEX-SIM is demonstrated by an example of the flutter prediction for a simple aeroelastic model. By utilizing the numerical optimization facility of TSIM, it is possible to undertake identification of required parameters in the TSIM model within the simulation.

Kaynes, Ian W.↗
NASA NTRS1989

Numerical simulation of pressure oscillations in a ramjet combustor

Large-eddy simulations of compressible cold flow in a ramjet combustor configuration were performed. The objectives were to investigate the mechanisms for vortex-acoustic interaction in such a device and to develop a simulation method that can be extended to cases with combustion heat release to study combustion instability. From the simulation, it was found that the separated shear layer rolls up into concentrated vortices that merge to form large coherent structures. These vortices interact with the choked nozzle downstream to produce an axial acoustic dipole. The spectrum of the pressure fluctuation at the base of the backward-facing step shows that there are two types of oscillations: an acoustic resonant mode and a vortex-acoustic coupled mode. Based on the flow physics observed in the simulations, a simple one-dimensional model for the vortex-acoustic coupled mode was proposed. The eigenvalue problem based on this model was solved to obtain the frequency of the coupled mode.

Jou, Wen-Huei↗
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 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.↗