Analysis of a control system containing two nonlinearities
Parameter plane method of stability analysis applied to spacecraft attitude control system with two nonlinearities related by nonlinear differential equation
Seltzer, S. M.↗
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Structural Properties and Estimation of Delay Systems
Two areas in the theory of delay systems were studied: structural properties and their applications to feedback control, and optimal linear and nonlinear estimation. The concepts of controllability, stabilizability, observability, and detectability were investigated. The property of pointwise degeneracy of linear time-invariant delay systems is considered. Necessary and sufficient conditions for three dimensional linear systems to be made pointwise degenerate by delay feedback were obtained, while sufficient conditions for this to be possible are given for higher dimensional linear systems. These results were applied to obtain solvability conditions for the minimum time output zeroing control problem by delay feedback. A representation theorem is given for conditional moment functionals of general nonlinear stochastic delay systems, and stochastic differential equations are derived for conditional moment functionals satisfying certain smoothness properties.
Kwong, R. H. S.↗
Description of a computer program and numerical techniques for developing linear perturbation models from nonlinear systems simulations
A numerical technique was developed which generates linear perturbation models from nonlinear aircraft vehicle simulations. The technique is very general and can be applied to simulations of any system that is described by nonlinear differential equations. The computer program used to generate these models is discussed, with emphasis placed on generation of the Jacobian matrices, calculation of the coefficients needed for solving the perturbation model, and generation of the solution of the linear differential equations. An example application of the technique to a nonlinear model of the NASA terminal configured vehicle is included.
Dieudonne, J. E.↗
Effects of internal heat transfer on the structure of self-similar blast waves
An analysis of the problem of self-similar, nonadiabatic blast waves, where both conduction and radiation are allowed to take place, show the problem to be reducible to the integration of a system of six coupled nonlinear ordinary differential equations. Consideration of these equations shows that although radiation tends to produce uniform fields through temperature gradient attenuation, all the energy carried by radiation is deposited on the front and the bounding shock becomes increasingly overdriven. When conduction is taken into account, the distribution of gasdynamic parameters in blast waves in the case of Rosseland diffusion radiation is more uniform than in the case of the Planck emission radiation.
Ghoniem, A. F.↗
Short-term climatic fluctuations forced by thermal anomalies
A two level, global, spectral model using pressure as a vertical coordinate was developed. The system of equations describing the model is nonlinear and quasi-geostrophic (linear balance). Static stability is variable in the model. A moisture budget is calculated in the lower layer only. Convective adjustment is used to avoid supercritical temperature lapse rates. The mechanical forcing of topography is introduced as a vertical velocity at the lower boundary. Solar forcing is specified assuming a daily mean zenith angle. The differential diabatic heating between land and sea is paramterized. On land and sea ice surfaces, a steady state thermal energy equation is solved to calculate the surface temperature. On the oceans, the sea surface temperature is specified as the climatological average for January. The model is used to simulate the January, February and March circulations.
Hanna, A. F.↗
Implicit Total Variation Diminishing (TVD) schemes for steady-state calculations
The application of a new implicit unconditionally stable high resolution total variation diminishing (TVD) scheme to steady state calculations. It is a member of a one parameter family of explicit and implicit second order accurate schemes developed by Harten for the computation of weak solutions of hyperbolic conservation laws. This scheme is guaranteed not to generate spurious oscillations for a nonlinear scalar equation and a constant coefficient system. Numerical experiments show that this scheme not only has a rapid convergence rate, but also generates a highly resolved approximation to the steady state solution. A detailed implementation of the implicit scheme for the one and two dimensional compressible inviscid equations of gas dynamics is presented. Some numerical computations of one and two dimensional fluid flows containing shocks demonstrate the efficiency and accuracy of this new scheme.
Yee, H. C.↗
Implicit total variation diminishing (TVD) schemes for steady-state calculations
The application of a new implicit unconditionally stable high resolution total variation diminishing (TVD) scheme to steady state calculations. It is a member of a one parameter family of explicit and implicit second order accurate schemes developed by Harten for the computation of weak solutions of hyperbolic conservation laws. This scheme is guaranteed not to generate spurious oscillations for a nonlinear scalar equation and a constant coefficient system. Numerical experiments show that this scheme not only has a rapid convergence rate, but also generates a highly resolved approximation to the steady state solution. A detailed implementation of the implicit scheme for the one and two dimensional compressible inviscid equations of gas dynamics is presented. Some numerical computations of one and two dimensional fluid flows containing shocks demonstrate the efficiency and accuracy of this new scheme. Previously announced in STAR as N83-23085
Yee, H. C.↗
Short-term climatic fluctuations forced by thermal anomalies
A two level, global, spectral model using pressure as a vertical coordinate was developed. The system of equations describing the model is nonlinear and quasi-geostrophic (linear balance). Static stability is variable in the model. A moisture budget is calculated in the lower layer only. Convective adjustment is used to avoid supercritical temperature lapse rates. The mechanical forcing of topography is introduced as a vertical velocity at the lower boundary. Solar forcing is specified assuming a daily mean zenith angle. The differential diabatic heating between land and sea is parameterized. On land and sea ice surfaces, a steady state thermal energy equation is solved to calculate the surface temperature. On the oceans, the sea surface temperature is specified as the climatological average for January. The model is used to simulate the January, February and March circulations. Previously announced in STAR as N83-14822
Hanna, A. F.↗
Short-term climatic fluctuations forced by thermal anomalies
A two level, global, spectral model using pressure as a vertical coordinate was developed. The system of equations describing the model is nonlinear and quasi-geostrophic (linear balance). Static stability is variable in the model. A moisture budget is calculated in the lower layer only. Convective adjustment is used to avoid supercritical temperature lapse rates. The mechanical forcing of topography is introduced as a vertical velocity at the lower boundary. Solar forcing is specified assuming a daily mean zenith angle. The differential diabatic heating between land and sea is parameterized. On land and sea ice surfaces, a steady state thermal energy equation is solved to calculate the surface temperature. On the oceans, the sea surface temperature is specified as the climatological average for January. The model is used to simulate the January, February and March circulations. Previously announced in STAR as N83-14822
Hanna, A. F.↗
A global low order spectral model designed for climate sensitivity studies
A two level, global, spectral model using pressure as a vertical coordinate is developed. The system of equations describing the model is nonlinear and quasi-geostrophic. A moisture budget is calculated in the lower layer only with moist convective adjustment between the two layers. The mechanical forcing of topography is introduced as a lower boundary vertical velocity. Solar forcing is specified assuming a daily mean zenith angle. On land and sea ice surfaces a steady state thermal energy equation is solved to calculate the surface temperature. Over the oceans the sea surface temperatures are prescribed from the climatological average of January. The model is integrated to simulate the January climate.
Hanna, A. F.↗
An adaptive pseudospectral method for discontinuous problems
The accuracy of adaptively chosen, mapped polynomial approximations is studied for functions with steep gradients or discontinuities. It is shown that, for steep gradient functions, one can obtain spectral accuracy in the original coordinate system by using polynomial approximations in a transformed coordinate system with substantially fewer collocation points than are necessary using polynomial expansion directly in the original, physical, coordinate system. It is also shown that one can avoid the usual Gibbs oscillation associated with steep gradient solutions of hyperbolic pde's by approximation in suitably chosen coordinate systems. Continuous, high gradient solutions are computed with spectral accuracy (as measured in the physical coordinate system). Discontinuous solutions associated with nonlinear hyperbolic equations can be accurately computed by using an artificial viscosity chosen to smooth out the solution in the mapped, computational domain. Thus, shocks can be effectively resolved on a scale that is subgrid to the resolution available with collocation only in the physical domain. Examples with Fourier and Chebyshev collocation are given.
Augenbaum, Jeffrey M.↗
An adaptive pseudospectral method for discontinuous problems
The accuracy of adaptively chosen, mapped polynomial approximations is studied for functions with steep gradients or discontinuities. It is shown that, for steep gradient functions, one can obtain spectral accuracy in the original coordinate system by using polynomial approximations in a transformed coordinate system with substantially fewer collocation points than are necessary using polynomial expansion directly in the original, physical, coordinate system. It is also shown that one can avoid the usual Gibbs oscillation associated with steep gradient solutions of hyperbolic pde's by approximation in suitably chosen coordinate systems. Continuous, high gradient solutions are computed with spectral accuracy (as measured in the physical coordinate system). Discontinuous solutions associated with nonlinear hyperbolic equations can be accurately computed by using an artificial viscosity chosen to smooth out the solution in the mapped, computational domain. Thus, shocks can be effectively resolved on a scale that is subgrid to the resolution available with collocation only in the physical domain. Examples with Fourier and Chebyshev collocation are given.
Augenbaum, J. M.↗
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.↗
Active stability augmentation of large space structures: A stochastic control problem
A problem in SCOLE is that of slewing an offset antenna on a long flexible beam-like truss attached to the space shuttle, with rather stringent pointing accuracy requirements. The relevant methodology aspects in robust feedback-control design for stability augmentation of the beam using on-board sensors is examined. It is framed as a stochastic control problem, boundary control of a distributed parameter system described by partial differential equations. While the framework is mathematical, the emphasis is still on an engineering solution. An abstract mathematical formulation is developed as a nonlinear wave equation in a Hilbert space. That the system is controllable is shown and a feedback control law that is robust in the sense that it does not require quantitative knowledge of system parameters is developed. The stochastic control problem that arises in instrumenting this law using appropriate sensors is treated. Using an engineering first approximation which is valid for small damping, formulas for optimal choice of the control gain are developed.
Balakrishnan, A. V.↗
The method of Ritz applied to the equation of Hamilton
Without any reference to the theory of differential equations, the initial value problem of the nonlinear, nonconservative double pendulum system is solved by the application of the method of Ritz to the equation of Hamilton. Also shown is an example of the reduction of the traditional eigenvalue problem of linear, homogeneous, differential equations of motion to the solution of a set of nonhomogeneous algebraic equations. No theory of differential equations is used. Solution of the time-space path of the linear oscillator is demonstrated and compared to the exact solution.
Bailey, C. D.↗
Review of probabilistic analysis of dynamic response of systems with random parameters
The various methods that have been studied in the past to allow probabilistic analysis of dynamic response for systems with random parameters are reviewed. Dynamic response may have been obtained deterministically if the variations about the nominal values were small; however, for space structures which require precise pointing, the variations about the nominal values of the structural details and of the environmental conditions are too large to be considered as negligible. These uncertainties are accounted for in terms of probability distributions about their nominal values. The quantities of concern for describing the response of the structure includes displacements, velocities, and the distributions of natural frequencies. The exact statistical characterization of the response would yield joint probability distributions for the response variables. Since the random quantities will appear as coefficients, determining the exact distributions will be difficult at best. Thus, certain approximations will have to be made. A number of techniques that are available are discussed, even in the nonlinear case. The methods that are described were: (1) Liouville's equation; (2) perturbation methods; (3) mean square approximate systems; and (4) nonlinear systems with approximation by linear systems.
Kozin, F.↗
Computational modeling of femtosecond optical solitons from Maxwell's equations
An algorithm is developed that permits the direct time integration of full-vector nonlinear Maxwell's equations. This capability permits the modeling of both linear and nonlinear instantaneous and dispersive effects in the electric polarization in material media. The modeling of the optical carrier is retained. The fundamental innovation is to notice that it is possible to treat the linear and nonlinear convolution integrals, which describe the dispersion, as new dependent variables. A coupled system of nonlinear second-order ordinary differential equations can then be derived for the linear and nonlinear convolution integrals, by differentiating them in the time domain. These equations, together with Maxwell's equations, are solved to determine the electromagnetic fields in nonlinear dispersive media. Results are presented of calculations in one dimension of the propagation and collision of femtosecond electromagnetic solitons that retain the optical carrier, taking into account as the Kerr and Raman interactions.
Goorjian, Peter M.↗
Recursive linearization of multibody dynamics equations of motion
The equations of motion of a multibody system are nonlinear in nature, and thus pose a difficult problem in linear control design. One approach is to have a first-order approximation through the numerical perturbations at a given configuration, and to design a control law based on the linearized model. Here, a linearized model is generated analytically by following the footsteps of the recursive derivation of the equations of motion. The equations of motion are first written in a Newton-Euler form, which is systematic and easy to construct; then, they are transformed into a relative coordinate representation, which is more efficient in computation. A new computational method for linearization is obtained by applying a series of first-order analytical approximations to the recursive kinematic relationships. The method has proved to be computationally more efficient because of its recursive nature. It has also turned out to be more accurate because of the fact that analytical perturbation circumvents numerical differentiation and other associated numerical operations that may accumulate computational error, thus requiring only analytical operations of matrices and vectors. The power of the proposed linearization algorithm is demonstrated, in comparison to a numerical perturbation method, with a two-link manipulator and a seven degrees of freedom robotic manipulator. Its application to control design is also demonstrated.
Lin, Tsung-Chieh↗