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NASA NTRS1993

Robust nonlinear control of vectored thrust aircraft

An interdisciplinary program in robust control for nonlinear systems with applications to a variety of engineering problems is outlined. Major emphasis will be placed on flight control, with both experimental and analytical studies. This program builds on recent new results in control theory for stability, stabilization, robust stability, robust performance, synthesis, and model reduction in a unified framework using Linear Fractional Transformations (LFT's), Linear Matrix Inequalities (LMI's), and the structured singular value micron. Most of these new advances have been accomplished by the Caltech controls group independently or in collaboration with researchers in other institutions. These recent results offer a new and remarkably unified framework for all aspects of robust control, but what is particularly important for this program is that they also have important implications for system identification and control of nonlinear systems. This combines well with Caltech's expertise in nonlinear control theory, both in geometric methods and methods for systems with constraints and saturations.

Doyle, John C.↗
NASA NTRS1993

Explicit robust schemes for implementation of a class of principal value-based constitutive models: Symbolic and numeric implementation

The issue of developing effective and robust schemes to implement a class of the Ogden-type hyperelastic constitutive models is addressed. To this end, special purpose functions (running under MACSYMA) are developed for the symbolic derivation, evaluation, and automatic FORTRAN code generation of explicit expressions for the corresponding stress function and material tangent stiffness tensors. These explicit forms are valid over the entire deformation range, since the singularities resulting from repeated principal-stretch values have been theoretically removed. The required computational algorithms are outlined, and the resulting FORTRAN computer code is presented.

Arnold, S. M.↗
NASA NTRS1993

Wigner expansions for partition functions of nonrelativistic and relativistic oscillator systems

The equilibrium quantum statistics of various anharmonic oscillator systems including relativistic systems is considered within the Wigner phase space formalism. For this purpose the Wigner series expansion for the partition function is generalized to include relativistic corrections. The new series for partition functions and all thermodynamic potentials yield quantum corrections in terms of powers of h(sup 2) and relativistic corrections given by Kelvin functions (modified Hankel functions) K(sub nu)(mc(sup 2)/kT). As applications, the symmetric Toda oscillator, isotonic and singular anharmonic oscillators, and hindered rotators, i.e. oscillators with cosine potential, are addressed.

Zylka, Christian↗
NASA NTRS1992

The challenges and benefits of lunar exploration

Three decades into the Space Age, the United States is experiencing a fundamental shift in space policy with the adoption of a broad national goal to expand human presence and activity beyond Earth orbit and out into the Solar System. These plans mark a turning point in American space exploration, for they entail a shift away from singular forays to a long-term, evolutionary program of exploration and utilization of space. No longer limited to the technical and operational specifics of any one vehicle or any one mission plan, this new approach will involve a fleet of spacecraft and a stable of off-planet research laboratories, industrial facilities, and exploration programs. The challenges inherent in this program are immense, but so too are the benefits. Central to this new space architecture is the concept of using a lunar base for in-situ resource utilization, and for the development of planetary surface exploration systems, applicable to the Moon, Mars, and other planetary bodies in the Solar System. This paper discusses the technical, economic, and political challenges involved in this new approach, and details the latest thinking on the benefits that could come from bold new endeavors on the final frontier.

Cohen, Aaron↗
NASA NTRS1993

Control of nonlinear systems with applications to constrained robots and spacecraft attitude stabilization

This thesis is organized in two parts. In Part 1, control systems described by a class of nonlinear differential and algebraic equations are introduced. A procedure for local stabilization based on a local state realization is developed. An alternative approach to local stabilization is developed based on a classical linearization of the nonlinear differential-algebraic equations. A theoretical framework is established for solving a tracking problem associated with the differential-algebraic system. First, a simple procedure is developed for the design of a feedback control law which ensures, at least locally, that the tracking error in the closed loop system lies within any given bound if the reference inputs are sufficiently slowly varying. Next, by imposing additional assumptions, a procedure is developed for the design of a feedback control law which ensures that the tracking error in the closed loop system approaches zero exponentially for reference inputs which are not necessarily slowly varying. The control design methodologies are used for simultaneous force and position control in constrained robot systems. The differential-algebraic equations are shown to characterize the slow dynamics of a certain nonlinear control system in nonstandard singularly perturbed form. In Part 2, the attitude stabilization (reorientation) of a rigid spacecraft using only two control torques is considered. First, the case of momentum wheel actuators is considered. The complete spacecraft dynamics are not controllable. However, the spacecraft dynamics are small time locally controllable in a reduced sense. The reduced spacecraft dynamics cannot be asymptotically stabilized using continuous feedback, but a discontinuous feedback control strategy is constructed. Next, the case of gas jet actuators is considered. If the uncontrolled principal axis is not an axis of symmetry, the complete spacecraft dynamics are small time locally controllable. However, the spacecraft attitude cannot be asymptotically stabilized using continuous feedback, but a discontinuous stabilizing feedback control strategy is constructed. If the uncontrolled principal axis is an axis of symmetry, the complete spacecraft dynamics cannot be stabilized. However, the spacecraft dynamics are small time locally controllable in a reduced sense. The reduced spacecraft dynamics cannot be asymptotically stabilized using continuous feedback, but again a discontinuous feedback control strategy is constructed.

Krishnan, Hariharan↗
NASA NTRS1991

Review of LFTs, LMIs, and mu

The authors present a tutorial overview of linear fractional transformations (LFTs) and the role of the structured singular value, mu, and linear matrix inequalities (LMIs) in solving LFT problems. The authors first introduce the notation for LFTs and briefly discuss some of their properties. They then describe mu and its connections with LFTs. They focus on two standard notions of robust stability and performance, mu stability and performance and Q stability and performance, and their relationship is discussed. Comparisons with the L1 theory of robust performance with structured uncertainty are considered.

Doyle, John↗
NASA NTRS1992

On the structure of cellular solutions in Rayleigh-Benard-Marangoni flows in small-aspect-ratio containers

Multiple steady flow patterns occur in surface-tension/buoyancy-driven convection in a liquid layer heated from below (Rayleigh-Benard-Marangoni flows). Techniques of numerical bifurcation theory are used to study the multiplicity and stability of two-dimensional steady flow patterns (rolls) in rectangular small-aspect-ratio containers as the aspect ratio is varied. For pure Marangoni flows at moderate Biot and Prandtl number, the transitions occurring when paths of codimension 1 singularities intersect determine to a large extent the multiplicity of stable patterns. These transitions also lead, for example, to Hopf bifurcations and stable periodic flows for a small range in aspect ratio. The influence of the type of lateral walls on the multiplicity of steady states is considered. 'No-slip' lateral walls lead to hysteresis effects and typically restrict the number of stable flow patterns (with respect to 'slippery' sidewalls) through the occurrence of saddle node bifurcations. In this way 'no-slip' sidewalls induce a selection of certain patterns, which typically have the largest Nusselt number, through secondary bifurcation.

Dijkstra, Henk A.↗
NASA NTRS1993

Nearly incompressible fluids. II - Magnetohydrodynamics, turbulence, and waves

The theory of nearly incompressible (NI) fluid dynamics developed previously for hydrodynamics is extended to magnetohydrodynamics (MHD). Based on a singular expansion technique, modified systems of fluid equations are obtained for which the effects of compressibility are admitted only weakly in terms of the different possible incompressible solutions. NI MHD represents the interface between the compressible and incompressible magnetofluid descriptions in the subsonic regime. It is shown that three distinct NI descriptions exist corresponding to each of the three possible plasma beta regimes. The detailed theory of weakly compressible corrections to the various incompressible MHD descriptions is presented, and the implications for the solar wind are discussed.

Zank, G. P.↗
NASA NTRS1992

The eigenvalue spectrum of the Rayleigh equation for a plane shear layer

The eigenvalue spectrum of the Rayleigh equation is examined using three different solution techniques. In particular, a simple second-order finite difference scheme and two spectral methods, the Chebyshev tau and Chebyshev collocation methods, are used to discretize the equation. All of the approximation methods are shown to be capable of predicting the discrete spectrum as well as the continuous spectrum associated with the critical point singularity for the Rayleigh equation. The global eigenvalue methods considered here provide an efficient way of obtaining either an approximation to the complete eigenvalue spectrum or initial guesses for a local shooting procedure for the discrete part of the spectrum.

Liou, William W.-W.↗
NASA NTRS1992

Boundary formulations for three-dimensional continuum structural shape sensitivity analysis

The direct, singular, boundary element analysis formulation is shown to provide a basis for a computationally efficient and accurate shape design sensitivity analysis approach for the structural response of three-dimensional solid objects. The theoretical formulation for surface displacement and traction component sensitivities, and all components of the stress tensor is presented along with a formulation for the recovery of displacement and stress components in the interior of the object under consideration. Discussion of computational issues related to the overall efficiency of these formulations is given, along with numerical results to demonstrate the accuracy and efficiency of this approach.

Kane, J. H.↗
NASA NTRS1992

Dominant modes via model error

Obtaining a reduced model of a stable mechanical system with proportional damping is considered. Such systems can be conveniently represented in modal coordinates. Two popular schemes, the modal cost analysis and the balancing method, offer simple means of identifying dominant modes for retention in the reduced model. The dominance is measured via the modal costs in the case of modal cost analysis and via the singular values of the Gramian-product in the case of balancing. Though these measures do not exactly reflect the more appropriate model error, which is the H2 norm of the output-error between the full and the reduced models, they do lead to simple computations. Normally, the model error is computed after the reduced model is obtained, since it is believed that, in general, the model error cannot be easily computed a priori. The authors point out that the model error can also be calculated a priori, just as easily as the above measures. Hence, the model error itself can be used to determine the dominant modes. Moreover, the simplicity of the computations does not presume any special properties of the system, such as small damping, orthogonal symmetry, etc.

Yousuff, A.↗
NASA NTRS1992

Adaptive filtering and prediction of the Southern Oscillation index

Singular spectrum analysis (SSA), a variant of principal component analysis, is applied to a time series of the Southern Oscillation index (SOI). The analysis filters out variability unrelated to the Southern Oscillation and separates the high-frequency, 2- to 3-year variability, including the quasi-biennial oscillation, from the lower-frequency 4- to 6-year El Nino cycle. The maximum entropy method (MEM) is applied to forecasting the prefiltered SOI. Prediction based on MEM-associated autoregresive models has useful skill for 30-36 months. A 1993-1994 La Nina event is predicted based on data through February 1992.

Keppenne, Christian L.↗
NASA NTRS1993

Determination of solid/porous wall boundary conditions from wind tunnel data for computational fluid dynamics codes

A computational and experimental study has been undertaken to investigate methods of modeling solid and porous wall boundary conditions in CFD codes. The procedure utilizes experimental measurements at the walls to develop a flow field solution based on the method of singularities. This flow field solution is then imposed as a boundary condition in a CFD simulation of the internal flow field. The effectiveness of this method in describing the boundary conditions at the wind tunnel walls using only sparse experimental measurements has been investigated. Verification of the approach using computational studies has been carried out using an incompressible flow solver. The current work demonstrates this technique for low speed flows and compares the result with experimental data obtained from a heavily instrumented variable porosity test section. Position and refinement of experimental measurements required to describe porous wall boundary conditions has also been considered for application to other porous wall wind tunnels.

Beutner, Thomas J.↗
NASA NTRS1993

Finite-surface method for the Maxwell equations in generalized coordinates

A finite-surface technique for the Maxwell equations in generalized nonorthogonal coordinates is developed. It directly applies the integral Faraday's and Ampere's laws to faces of primary and secondary grid cells, respectively. The technique features an accurate treatment of matching conditions at a material interface, grid singularities, and radiation conditions at outer boundaries.

Vinokur, Marcel↗
NASA NTRS1993

A nonsingular model of the open magnetosphere

We present a modified version of the Toffoletto and Hill (1989) open magnetosphere model that incorporates a tail-like interconection field with a discontinuity 10 represent the slow-mode expansion fan that defines the high-latitude tail magnetopause. (The interconnection field is defined as the perturbation on an initially closed magnetosphere model to make it open.) The expansion fan controls the open field line region in the tail, and the intersection of the fan with the tail current sheet is, by design, the x line. The new interconnection field allows greater control of the tail field structure; in particular, it enables us to eliminate the nightside mapping singularity that occurs in previous models when the interplanetary magnetic field is nonsouthward. Also, in contrast to earlier models, the far tail x line extends farther downstream on the flanks than in the center of the tail, consistent with observations.

Toffoletto, F. R.↗
NASA NTRS1993

Vortex/surface interaction

This paper considers the interaction of a vortex generated upstream in a flow field with a downstream aerodynamic surface that possesses a large chord. The flow is assumed to be steady, incompressible, inviscid and irrotational, and the surface to be semiinfinite. The vortex is considered to be a straight vortex filament. To lowest order the problem is modeled using potential theory, where the 3D Laplace's equation for the velocity potential on the surface is solved exactly. The closed-form equation for pressure distribution obtained from this theory is found to have a square root singularity at the leading-edge. It also converges, as x goes to infinity, to the solution of the 2D point-vortex/infinite plane problem. The pressure coefficient presents an anti-symmetric behavior, near the leading-edge and a symmetric behavior as x goes to infinity.

Bodstein, G. C. R.↗
NASA NTRS1992

Testbed for remote telepresence research

The paper describes the design and implementation of a telepresence system, as well as its control hardware and software. The mobile omnidirectional robot has three independent degrees of freedom that permit independent control of translation and rotation, thereby simulating a free-flying robot in a plane. The kinematically redundant robot arm has eight degrees of freedom that assist in obstacle and singularity avoidance. The on-board control computers permit control of the robot from the dual hand controllers via a radio modem system. A head-mounted display system provides the user with a stereo view from a pair of cameras attached to the mobile robotics system. The head tracking camera system moves stereo cameras mounted on a three-DOF platform to coordinate with the operator's head movements. This telepresence system provides a framework for research in remote telepresence, and teleoperations for space.

Adnan, Sarmad↗
NASA NTRS1993

Probability of lensing magnification by cosmologically distributed galaxies

We present the analytical formulae for computing the magnification probability caused by cosmologically distributed galaxies. The galaxies are assumed to be singular, truncated-isothermal spheres without both evolution and clustering in redshift. We find that, for a fixed total mass, extended galaxies produce a broader shape in the magnification probability distribution and hence are less efficient as gravitational lenses than compact galaxies. The high-magnification tail caused by large galaxies is well approximated by an A exp -3 form, while the tail by small galaxies is slightly shallower. The mean magnification as a function of redshift is, however, found to be independent of the size of the lensing galaxies. In terms of the flux conservation, our formulae for the isothermal galaxy model predict a mean magnification to within a few percent with the Dyer-Roeder model of a clumpy universe.

Pei, Yichuan C.↗