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NASA NTRS2014

A Linear Programming Approach to Routing Control in Networks of Constrained Nonlinear Positive Systems with Concave Flow Rates

We consider control design for positive compartmental systems in which each compartment's outflow rate is described by a concave function of the amount of material in the compartment.We address the problem of determining the routing of material between compartments to satisfy time-varying state constraints while ensuring that material reaches its intended destination over a finite time horizon. We give sufficient conditions for the existence of a time-varying state-dependent routing strategy which ensures that the closed-loop system satisfies basic network properties of positivity, conservation and interconnection while ensuring that capacity constraints are satisfied, when possible, or adjusted if a solution cannot be found. These conditions are formulated as a linear programming problem. Instances of this linear programming problem can be solved iteratively to generate a solution to the finite horizon routing problem. Results are given for the application of this control design method to an example problem. Key words: linear programming; control of networks; positive systems; controller constraints and structure.

Positive Systems↗

NASA NTRSDate not supplied

Extending Explicit Guidance Methods to Higher Dimensions, Additional Conditions, and Higher Order Integration

Guidance functions play critical roles in autonomy to steer vehicles and aircraft to the intended target or destination. Explicit guidance (E Guidance) solves the two-point boundary value problem with initial and final conditions for position and velocity. The original formulation of E Guidance involves translational acceleration commands with a direct relationship to time, and it is possible to modify E Guidance for rotational acceleration. Other extensions for E Guidance include higher dimensions, additional conditions, and higher-order integration of the linearly independent E Guidance functions. The most promising extension involves higher-order integration of the E Guidance functions, but it may be physically impractical by initially moving away from the target. This paper provides a brief overview of some methods that extend E Guidance to higher dimensions, utilize additional conditions, or perform higher-order integration, and if they satisfy the two-point boundary value problem.

explicit guidance↗

NASA NTRSDate not supplied

Extending Explicit Guidance Methods to Higher Dimensions, Additional Conditions, and Higher Order Integration

Guidance functions play critical roles in autonomy to steer vehicles and aircraft to the intended target or destination. Explicit guidance (E Guidance) solves the two-point boundary value problem with initial and final conditions for position and velocity. The original formulation of explicit guidance involves translational acceleration commands with a direct relationship to time, and it is possible to modify E Guidance for rotational acceleration. Other extensions for E Guidance include higher dimensions, additional conditions, and higher-order integration of the linearly independent E Guidance functions. The most promising extension involves higher-order integration of the E Guidance functions, but it may be physically impractical by initially moving away from the target. This paper provides a brief overview of some methods that extend E Guidance to higher dimensions, utilize additional conditions, or perform higher-order integration, and if they satisfy the two-point boundary value problem.

explicit guidance↗

NASA NTRS1976

Optimal command generation for tracking a class of discontinuous trajectories

Commands are found to drive a linear system to optimally track a class of prescribed trajectories, each of which contains a point of discontinuity. The paper focuses on the guidance problem of command generation, to be implemented in a feedforward fashion; the necessary additional control or feedback regulation structure is not studied in detail, but assumed to be provided in applications as a conventional error-feedback regulator. Solutions are found for the infinite-time problem which are optimal with respect to a quadratic performance criterion; suboptimal controls which satisfy a continuity condition are also found. The controls have applications to certain problems in aircraft guidance where command trajectories are piecewise continuous. Several examples are worked out in detail, with comparisons with conventional, nonfeedforward solutions to the problem, and a brief discussion of a simpler, suboptimal solution.

Weissenberger, S.↗

NASA NTRS1991

Optimal guidance of aero-assisted transfer vehicles based on matched asymptotic expansions

This paper addresses a number of issues related to the matched asymptotic expansions analysis of skip trajectories, or any class of problems that give rise to inner layers that are not associated directly with satisfying boundary conditions. The procedure for matching inner and outer solutions, and using the composite solution to satisfy boundary conditions is rigorously followed in developing a complete algebraic solution to the problem of inclination change with minimum energy loss. Repeated solution of these algebraic equations along the trajectory, treating each current state as an initial state, constitute a feedback guidance algorithm.

Calise, A. J.↗

NASA NTRS1973

A preliminary design and implementation of the low-thrust simulation and trajectory search program (LOWTRAJ)

The results are presented of one phase of research conducted for the JPL Solar Electric Propulsion (SEP) Navigation Software System development program. It deals only with the problem of designing the flight quality trajectory program, which is a major subset of the entire navigation software system. In this phase of research (breadboard development phase), attempts were made to assess the SEP trajectory software functional requirements, to investigate the program design method satisfying these requirements, to identify the primary anticipated problem areas, and to provide solutions to these problem areas. These efforts culminated in the development of a compact breadboard program. A functional description and the mathematical formulation of the program are presented.

Yen, C. L.↗

NASA NTRS2018

Use of Simple Continuum Solutions in Finite Element Alternating Method for Fracture Problems

The performance of the finite element alternating (FEAM) method for two-dimensional crack problems is studied with respect to a polynomial pressure distribution fitted to the crack face stresses. The FEAM alternates between the analytical solution of crack in an infinite plate subjected to arbitrary polynomial distribution and a finite element solution of an uncracked body to satisfy the required boundary conditions in the crack problem. In this paper, the FEAM is applied to embedded crack and edge crack problems. For embedded crack problems, all of the constant, linear, and quadratic ( N=0,1, or 2, respectively) pressure distributions yield very accurate results with this algorithm with 4 to 5 iterations. The edge crack problems, on the other hand, require much higher order polynomials distributions (N=5 to 6) to yield accurate solutions. For slant edge crack problems, the mode-I stress-intensity factors have better accuracy than the mode-II stress-intensity factors for the same convergence tolerance.

Krishnamurthy, T.↗

NASA NTRS2005

Variations in the Peak Position of the 6.2 micron Interstellar Emission Feature: A Tracer of N in the Interstellar Polycyclic Aromatic Hydrocarbon Population

This paper presents the results of an investigation of the molecular characteristics that underlie the observed peak position and profile of the nominal 6.2 micron interstellar emission band generally attributed to the CC stretching vibrations of polycyclic aromatic hydrocarbons (PAHs). It begins with a summary of recent experimental and theoretical studies ofthe spectroscopic properties of large (>30 carbon atoms) PAH cations as they relate to this aspect of the astrophysical problem. It then continues with an examination of the spectroscopic properties of a number of PAH variants within the context of the interstellar 6.2 micron emission, beginning with a class of compounds known as polycyclic aromatic nitrogen heterocycles (PANHs; PAHs with one or more nitrogen atoms substituted into their carbon skeleton). In this regard, we summarize the results of recent relevant experimental studies involving a limited set of small PANHs and their cations and then report the results of a comprehensive computational study that extends that work to larger PANH cations including many nitrogen-substituted variants of coronene(+) (C24H12(+)), ovalene(+) (C32H14(+)), circumcoronene(+) (C54H18(+)), and circum-circumcoronene(+) (C96H24(+)). Finally, we report the results of more focused computational studies of selected representatives from a number of other classes of PAH variants that share one or more of the key attributes of the PANH species studied. These alternative classes of PAH variants include (1) oxygen- and silicon-substituted PAH cations; (2) PAH-metal ion complexes (metallocenes) involving the cosmically abundant elements magnesium and iron; and (3) large, asymmetric PAH cations. Overall, the studies reported here demonstrate that increasing PAH size alone is insuEcient to account for the position of the shortest wavelength interstellar 6.2 micron emission bands, as had been suggested by earlier studies. On the other hand, this work reveals that substitution of one or more nitrogen atoms within the interior of the carbon skeleton of a PAH cation induces a significant blueshift in the position of the dominant CC stretching feature of these compounds that is suf6cient to account for the position of the interstellar bands. Subsequent studies of the effects of substitution by other heteroatoms (O and Si), metal ion complexation (Fe(+), Mg(+), and Mg(2+)), and molecular symmetry variation-all of which fail to reproduce the blueshift observed in the PANH cations-indicate that N appears to be unique in its ability to accommodate the position of the interstellar 6.2 micron bands while simultaneously satisfying the other constraints of the astrophysical problem. This result implies that the peak position of the interstellar feature near 6.2 micron traces the degree of nitrogen substitution in the population, that most of the PAHs responsible for the interstellar IR emission features incorporate nitrogen within their aromatic networks, and that a lower limit of 1%-2% of the cosmic nitrogen is sequestered within the interstellar PAH population. Finally, in view of the ubiquity and abundance of interstellar PAHs and the permanent dipoles and distinctive electronic structures of these nitrogen-substituted variants, this work impacts a wide range of observational phenomena outside of the infrared region of the spectrum including the forest of unidentified molecular rotational features and the anomalous Galactic foreground emission in the microwave, and the diffuse interstellar bands (DIBs) and other structure in the interstellar extinction curve in the ulhviolet/visible. These astrophysical ramifications are discussed, and the dipole moments and rotational constants are tabulated to facilitate further investigations of the astrophysical role of nitrogen-substituted aromatic compounds.

Hudgins, Douglas M.↗

NASA NTRS1990

Finite element solution of optimal control problems with inequality constraints

A finite-element method based on a weak Hamiltonian form of the necessary conditions is summarized for optimal control problems. Very crude shape functions (so simple that element numerical quadrature is not necessary) can be used to develop an efficient procedure for obtaining candidate solutions (i.e., those which satisfy all the necessary conditions) even for highly nonlinear problems. An extension of the formulation allowing for discontinuities in the states and derivatives of the states is given. A theory that includes control inequality constraints is fully developed. An advanced launch vehicle (ALV) model is presented. The model involves staging and control constraints, thus demonstrating the full power of the weak formulation to date. Numerical results are presented along with total elapsed computer time required to obtain the results. The speed and accuracy in obtaining the results make this method a strong candidate for a real-time guidance algorithm.

Bless, Robert R.↗

NASA NTRS1975

Mars - Satellite origin and angular momentum

The origin of Phobos and Deimos is considered with a view to accounting for the existence of very small satellites with circular orbits in the Martian equatorial plane, and simultaneously for the suspected angular momentum deficiency of the Mars system. All models considered failed to satisfy at least one requirement, and the problem is considered more puzzling than is at first apparent. The Martian angular momentum deficiency, if physically significant, may be unrelated to the present satellites' origin, but might relate to a large ancient satellite, long ago destroyed. Accretion onto Mars of large amounts of asteroidal dust brought in by Poynting-Robertson drag may have some bearing on the angular momentum problem.

Hartmann, W. K.↗

NASA NTRS1981

Buckling and vibration of periodic lattice structures

Lattice booms and platforms composed of flexible members or large diameter rings which may be stiffened by cables in order to support membrane-like antennas or reflector surfaces are the main components of some large space structures. The nature of these structures, repetitive geometry with few different members, makes possible relatively simple solutions for buckling and vibration of a certain class of these structures. Each member is represented by a stiffness matrix derived from the exact solution of the beam column equation. This transcendental matrix gives the current member stiffness at any end load or frequency. Using conventional finite element techniques, equilibrium equations can be written involving displacements and rotations of a typical node and its neighbors. The assumptions of a simple trigonometric mode shape is found to satisfy these equations exactly; thus the entire problem is governed by just one 6 x 6 matrix equation involving the amplitude of the displacement and rotation mode shapes. The boundary conditions implied by this solution are simple supported ends for the column type configurations.

Anderson, M. S.↗

NASA NTRS1985

The stress distribution in pin-loaded orthotropic plates

The performance of mechanically fastened composite joints was studied. Specially, a single-bolt connector was modeled as a pin-loaded, infinite plate. The model that was developed used two dimensional, complex variable, elasticity techniques combined with a boundary collocation procedure to produce solutions for the problem. Through iteration, the boundary conditions were satisfied and the stresses in the plate were calculated. Several graphite epoxy laminates were studied. In addition, parameters such as the pin modulus, coefficient of friction, and pin-plate clearance were varied. Conclusions drawn from this study indicate: (1) the material properties (i.e., laminate configuration) of the plate alter the stress state and, for highly orthotropic materials, the contact stress deviates greatly from the cosinusoidal distribution often assumed; (2) friction plays a major role in the distribution of stresses in the plate; (3) reversing the load direction also greatly effects the stress distribution in the plate; (4) clearance (or interference) fits change the contact angle and thus the location of the peak hoop stress; and (5) a rigid pin appears to be a good assumption for typical material systems.

Klang, E. C.↗

NASA NTRS1985

Burner liner thermal/structural load modeling: TRANCITS program user's manual

Transfer Analysis Code to Interface Thermal/Structural Problems (TRANCITS) is discussed. The TRANCITS code satisfies all the objectives for transferring thermal data between heat transfer and structural models of combustor liners and it can be used as a generic thermal translator between heat transfer and stress models of any component, regardless of the geometry. The TRANCITS can accurately and efficiently convert the temperature distributions predicted by the heat transfer programs to those required by the stress codes. It can be used for both linear and nonlinear structural codes and can produce nodal temperatures, elemental centroid temperatures, or elemental Gauss point temperatures. The thermal output of both the MARC and SINDA heat transfer codes can be interfaced directly with TRANCITS, and it will automatically produce stress model codes formatted for NASTRAN and MARC. Any thermal program and structural program can be interfaced by using the neutral input and output forms supported by TRANCITS.

Maffeo, R.↗

NASA NTRS1986

Boundary force method for analyzing two-dimensional cracked bodies

The Boundary Force Method (BFM) was formulated for the two-dimensional stress analysis of complex crack configurations. In this method, only the boundaries of the region of interest are modeled. The boundaries are divided into a finite number of straight-line segments, and at the center of each segment, concentrated forces and a moment are applied. This set of unknown forces and moments is calculated to satisfy the prescribed boundary conditions of the problem. The elasticity solution for the stress distribution due to concentrated forces and a moment applied at an arbitrary point in a cracked infinite plate are used as the fundamental solution. Thus, the crack need not be modeled as part of the boundary. The formulation of the BFM is described and the accuracy of the method is established by analyzing several crack configurations for which accepted stress-intensity factor solutions are known. The crack configurations investigated include mode I and mixed mode (mode I and II) problems. The results obtained are, in general, within + or - 0.5 percent of accurate numerical solutions. The versatility of the method is demonstrated through the analysis of complex crack configurations for which limited or no solutions are known.

Tan, P. W.↗

NASA NTRS1988

Boundary force method for analyzing two-dimensional cracked plates

Tan, Paul W.↗

NASA NTRS1992

Discrete time learning control in nonlinear systems

In this paper digital learning control methods are developed primarily for use in single-input, single-output nonlinear dynamic systems. Conditions for convergence of the basic form of learning control based on integral control concepts are given, and shown to be satisfied by a large class of nonlinear problems. It is shown that it is not the gross nonlinearities of the differential equations that matter in the convergence, but rather the much smaller nonlinearities that can manifest themselves during the short time interval of one sample time. New algorithms are developed that eliminate restrictions on the size of the learning gain, and on knowledge of the appropriate sign of the learning gain, for convergence to zero error in tracking a feasible desired output trajectory. It is shown that one of the new algorithms can give guaranteed convergence in the presence of actuator saturation constraints, and indicate when the requested trajectory is beyond the actuator capabilities.

Longman, Richard W.↗

NASA NTRS1993

New results on the realizability of Reynolds stress turbulence closures

The realizability of Reynolds stress models in homogeneous turbulence is critically assessed from a theoretical standpoint. It is proven that a well known second-order closure formulated by Shih and Lumley using the strong realizability constraints of Schumann is, in fact, not a realizable model. The problem arises from the failure to properly satisfy the necessary positive second time derivative constraint when a principal Reynolds stress vanishes - a fatal flaw that becomes apparent when the non-analytic terms in their model are made single-valued as required on physical grounds. It is furthermore shown that the centrifugal acceleration generated by rotations of the principal axes of the Reynolds stress tensor can make the second derivative singular at the most extreme limits of realizable turbulence. This previously overlooked effect appears to make it impossible to identically satisfy the strong form of realizability in any version of the present generation of second-order closures. On the other hand, models properly formulated to satisfy the weak form of realizability - wherein states of one or two component turbulence are not accessible in finite time are found to be realizable. However, unlike the simpler and more commonly used second order closures, these models can be ill-behaved near the extreme limits of realizable turbulence due to the way that higher-degree nonlinearities are often unnecessarily introduced to satisfy realizability. Illustrative computations of homogeneous shear flows are presented to demonstrate these points which can have important implications for turbulence modeling.

Speziale, Charles G.↗

NASA NTRS1996

Cascade Optimization Strategy for Aircraft and Air-Breathing Propulsion System Concepts

Design optimization for subsonic and supersonic aircraft and for air-breathing propulsion engine concepts has been accomplished by soft-coupling the Flight Optimization System (FLOPS) and the NASA Engine Performance Program analyzer (NEPP), to the NASA Lewis multidisciplinary optimization tool COMETBOARDS. Aircraft and engine design problems, with their associated constraints and design variables, were cast as nonlinear optimization problems with aircraft weight and engine thrust as the respective merit functions. Because of the diversity of constraint types and the overall distortion of the design space, the most reliable single optimization algorithm available in COMETBOARDS could not produce a satisfactory feasible optimum solution. Some of COMETBOARDS' unique features, which include a cascade strategy, variable and constraint formulations, and scaling devised especially for difficult multidisciplinary applications, successfully optimized the performance of both aircraft and engines. The cascade method has two principal steps: In the first, the solution initiates from a user-specified design and optimizer, in the second, the optimum design obtained in the first step with some random perturbation is used to begin the next specified optimizer. The second step is repeated for a specified sequence of optimizers or until a successful solution of the problem is achieved. A successful solution should satisfy the specified convergence criteria and have several active constraints but no violated constraints. The cascade strategy available in the combined COMETBOARDS, FLOPS, and NEPP design tool converges to the same global optimum solution even when it starts from different design points. This reliable and robust design tool eliminates manual intervention in the design of aircraft and of air-breathing propulsion engines where it eases the cycle analysis procedures. The combined code is also much easier to use, which is an added benefit. This paper describes COMETBOARDS and its cascade strategy and illustrates the capability of the combined design tool through the optimization of a subsonic aircraft and a high-bypass-turbofan wave-rotor-topped engine.

Patnaik, Surya N.↗